CATALOGUE OF SCIENTIFIC PAPERS 1800-1900 SUBJECT INDEX VOLUME I PURE MATHEMATICS CAMBRIDGE UNIVERSITY PRESS WAREHOUSE, C. F. CLAY, MANAGER. aotlbon : FETTEE LANE, E.G. Iagflota: 30, WELLINGTON STREET. F. A. BEOCKHAUS. eta 2ork: G. P. PUTNAM'S SONS. Bombag anto Calcutta: MACMILLAN AND CO., LTD. [All Rights reserved.] ROYAL SOCIETY OF LONDON CATALOGUE OF SCIENTIFIC PAPERS 1800-1900 SUBJECT INDEX VOLUME I PURE MATHEMATICS CAMBRIDGE : AT THE UNIVERSITY PRESS 1908 ARRANGED FOR A COMMITTEE OF THE ROYAL SOCIETY UNDER THE SUPERINTENDENCE OF HERBERT M C LEOD, LL.D., F.R.S. DIRECTOR OF THE CATALOGUE with the assistance of R. J. DALLAS, M.A. KING'S COLLEGE, CAMBRIDGE PREFACE ~VT7"HEN the plan for a Catalogue of Scientific Papers was drawn up in * 1857 by the Royal Society, it was also contemplated that a Subject Index to the papers should be made. During the compilation of the well known twelve volumes of the Catalogue of Authors (1800 1883) much material was collected for the Subject Index by the Committee of the Society which had charge of the work, but nothing was printed. In 1898 it was determined to undertake the continuation for the period 1884 1900, and at the same time to prepare material for a Subject Index for these last seventeen years of the century. It was afterwards decided that one Index should be made to the papers of the whole of the Nineteenth Century ; and to bring this Index into relation with the International Catalogue of Scientific Literature, which deals with publications subsequent to 1900, the Index was to be arranged in accordance with the schedules of the different sciences which form the basis of the International Catalogue. The extent of the work for the period 18841900 proved to be so great that the resources of the Royal Society would have been quite inadequate for its completion. Its continuation has been made possible by liberal donations received from various sources. The Catalogue is especially indebted to Dr Ludwig Mond, F.R.S., a member of the Committee entrusted with its preparation, who has encouraged and forwarded its progress in every way for many years past; in particular, in addition to similar generous donations at earlier stages, Dr Mond recently undertook to provide an extremely liberal subsidy during the three years 1907 9, which has enabled the work to be pressed forward rapidly. The Subject Index will be published as separate Index-volumes for each of the seventeen Sciences of the Schedules of the International Catalogue, viz. Mathematics, Mechanics, Physics, Chemistry, Astronomy, Meteorology, Mineralogy, Geology, Geography, Palaeontology, Biology, Botany, Zoology, Anatomy, Anthropology, Physiology, and Bacteriology. The practical advantage which will accrue from thus classifying the literature of each Science seems to be amply sufficient, in spite of un- avoidable imperfections, to justify the great expense that has been involved. In the past a student interested in a single branch of the subject would probably consult special Treatises and Reports from which he could obtain references to the more important memoirs. But it often happens that such references are transferred simply from one treatise to another, and Preface experience shows that important contributions that have once dropped out of sight may for a long time, or indeed ever afterwards, be overlooked. When the whole literature is set out systematically and exhaustively under appropriate headings as in the present volume, such accidents are guarded against so far as seems to be possible. The present volume dealing with Pure Mathematics is the first instal- ment of the Subject Index : it contains 38748 entries referring to 700 serials. It is believed that these entries cover all the contents of the serials indexed with the exception of about 750 short notes (mostly falling under 6810 and 7210) which were not thought of sufficient permanent interest to be entered. The Index titles for papers published from 1884 to 1900 inclusive have been made by Referees familiar with the subjects, who have consulted the original papers and have made the titles from the contents of the papers and not merely from their headings. It was intended that the Index entries for the papers from 1800 to 1883 should be made from the titles as printed in the Catalogue of Authors, but it was soon found that many of the titles were too indefinite for such minute classification, and reference has had to be made to the original papers in a large number of cases. The latter part of this work was commenced by the late Mr George Griffith of Harrow, whose untimely death was a great loss to the Committee. The entries in the Index are arranged so that reference can be made to the complete titles in the Catalogue of Scientific Papers. Generally the author's name together with the date will indicate the volume in which the title of the paper may be found. These clues are insufficient when the paper is anonymous or occurs in Volume XII or in the additions to Volume VI. They are also at fault for titles marked with an asterisk showing that they belong to previous volumes ; in these cases the number of the volume is given in the Index entry in small Roman numerals within brackets. When an error has been found in an author's name in the Catalogue, it is corrected in the Index by a reference made to the error. The Index contains references to some papers of dates earlier than 1884 which were omitted in previous volumes of the Catalogue : these are indicated by an asterisk placed before the date ; the full titles of these papers will be given in the continuation of the Catalogue of Authors. When an author's personal name does not appear in the heading of an original paper, no attempt has been made to find the name for the Index, but this will be done for the Catalogue of Authors. The Catalogue of Authors for the period 1884 to 1900 is in preparation; the material will not however be quite complete until the Subject Indexes for the various Sciences have been reduced into order. Entries on the same subject are arranged, so far as possible, in order of Preface date irrespective of the authors' names, with the endeavour to present the subject in the historical form. This grouping of the entries, involving modifications of titles prepared by different Referees, or by the same Referee at different times, has been one of the most difficult problems in the preparation of the Index. The abbreviations of the names of the serials used in the Royal Society Catalogue have been further shortened for the Index. As the abbreviations are not uniform in all the volumes, it will be found that the same journal may be indicated by several different abbreviations, but in each case the one selected is that which was used in the volume in which the title of the paper occurs. In the case of serials commencing since 1883, the abbreviations adopted in the International Catalogue have been used as a guide. The list of serials will, it is hoped, be a valuable feature of the Index. It has been drawn up by the Director and contains the names of 701 serials from which the entries in the Index have been taken. Each title is pre- ceded by the abbreviation which represents the serial in the Index ; the date of commencement of the serial is given, and if it is extinct the date of the last volume is added. There are appended symbols representing the names of twenty-three British Libraries in some of which the serials may be found ; where the set is incomplete the symbol is followed by i. The in- formation from which this list has been compiled was obtained, in the first instance, from published catalogues ; subsequently the list was submitted to the custodians of many of the libraries, who kindly marked many serials which had not been found in the catalogues used. The thanks of the Committee for this valuable assistance are due to Mr F. Jenkinson of the Cambridge University Library, Mr E. W. B. Nicholson of the Bodleian Library, the Librarian of the Radcliffe Library, the Librarian of the Patent Office, Dr B. Daydon Jackson and Mr A. W. Kappel of the Linnean Society, Mr W. H. Wesley of the Royal Astronomical Society, Mr C. V. Crook of the Geological Museum, Mr F. W. Clifford of the Chemical Society, and Mr R. W. Chambers of University College, London ; Mr R. Lloyd Praeger obtained the information from the five libraries in Dublin, and Dr Hugh Marshall, F.R.S., from two libraries in Edinburgh and two in Glasgow. Although much care has been expended in making this list as accurate as possible, it is probable that some errors will still be found and the Director will be thankful to any one who will send corrections : portions of the list will be required for the subsequent volumes of the Index. The subjects are arranged under the registration numbers adopted in the International Catalogue; a copy of Schedule A, Mathematics, is pre- fixed to the Index, with indication of the pages on which the titles for the viz Preface different sections occur. It has occasionally been found convenient, in order to save repetition in printing, to group entries under a sub-heading which is not contained in the International Catalogue Schedule. Where this has been done the sub-heading is printed in italics. In some of these cases the words of the sub-heading are understood to exist before the entries following them, and consequently these entries commence with small letters. These minor classifications, being often made mechanically on the basis of the explicit mention of the sub-heading, are not to be taken as exhaustive ; cognate entries may be found elsewhere under the same main heading. The unit of classification is thus the complete numbered heading. The first portion of the present volume of the Index having been sent to press before the whole of the material had been sorted, some slips belonging to earlier sections were discovered during the arrangement of the later ones. These titles have been placed in an Appendix under their appropriate registration numbers. The following referees have assisted at various times in the preparation of the Subject Index in Pure Mathematics : the late Professor J. D. Everett, F.R.S., Mr R. J. Dallas, Miss Alice Everett, Mr R. Hargreaves, Miss W. M. Hudson, Mr H. Knapman, Miss E. Perrin and Mr G. Harold Wilson. The Committee is indebted to them for much valuable help. The arrangement of the contents of this volume for the press has been made by Mr R. J. Dallas, M.A., of King's College, Cambridge. To him and to Miss Brernner and the members of the Catalogue Staff thanks are due for careful and conscientious work. At the request of the other members of the Committee, Mr G. B. Mathews, F.R.S., examined the whole of the proof-sheets, and made sugges- tions for their improvement : he was also consulted from time to time about questions of arrangement, head lines and sub-headings. When the time for going to press approached, the Committee learned with much satisfaction that the Syndics of the Cambridge University Press were willing to undertake the complete risk of printing and publishing, as regards both the Catalogue of Scientific Papers and the Subject Index. It will be the care of the Committee, and it is hoped of the Scientific world generally, to use their best endeavours that this public-spirited action shall not result in financial loss. Finally the thanks of the Committee are due to the officials of the Cambridge Press for their unfailing courtesy in the discharge of a complex task. March 1908. viii SUBJECT INDEX OF PURE MATHEMATICS LIST OF SERIAL PUBLICATIONS WITH THE ABBEEVIATIONS OF THEIR TITLES USED IN THE INDEX, AND LIBRARIES WHERE THE SERIALS CAN BE CONSULTED. The date following the title of a serial indicates the year of its commencement ; if a second date is given it marks the termination of the serial. The letters following the dates indicate libraries where the serials are to be found: if the serial is incomplete, the symbol of the library is followed by i. B.M. British Museum. Glasg.P.S. Camb.P.S. Cambridge Philosophical Library. Camb.U. Cambridge University Library. Glasg.U. Chem.S. Chemical Society. Linn.S. Dub.N.L.I. National Library of Ireland,Dublin. Math.S. Dub.R.C.S. Royal College of Science, Dublin. N.H.M. Dub.R.D.S. Royal Dublin Society. Oxon.B. Dub.R.I.A. Royal Irish Academy, Dublin. Oxon.B. (R.) Dub.T.C. Trinity College, Dublin. Oxon.R. Edinb.R.S. Royal Society of Edinburgh. P.O. Edinb.U. Edinburgh University. R.A.S. Geol.M. Geological Survey Museum, Jer- R.S. myn St. U.C.L. Royal Philosophical Society of Glasgow. Glasgow University. Linnean Society. Mathematical Society. Natural History Museum. Bodleian, Oxford. Deposited in Radcliffe. Radcliffe, Oxford. Patent Office, London. Royal Astronomical Society. Royal Society. University College, London. Abv. Mm. a. Em Abv. 8. Mm. . A. C Ac. Caes. Leop. N. Acta . . Acireale Ac. At A. Cond. Pon. Chauss. .. A. Cons. Arts et IWet . Ac. Nt. C. NT. Acta Acta Mth . (Me"mou-es de la Societe d'Emulation d' Abbeville. Abbeville. Jl833 B.M. ; Camb.U.i. ; Dub.T.C.; Edinb.R.S.i.; Linn.S.i.; ( N.H.M.i.; Oxon.B.i.; R.S.i. . Annales de Chimie, ou Recueil de Memoires concemant la Chimie et les Arts qui en dependent. Paris. 1789 B.M.; Camb.U.; Chem.S.; Dub.R.D.S.i. ; Dub.T.C.i. ; Edinb.R.S.; Edinb.U.; N.H.M.; Oxon.B.i.(R.) ; P.O.; R.S. ; U.C.L. . Nova Acta physico-medica Academies Caes. Leopoldino-Carolinse Naturae Curiosorum. Erlangen, Bonn, Breslau. 1758 Camb.P.S.; Camb.U.; Chem.S.i.; Dub.T.C.; Edinb.R.S.i.; Edinb.U.; Linn.S.i.; Math.S.i.; N.H.M.; Oxon.R.; R.A.S.*.; R.S.; U.C.L.i. See Ac. Nt. C. 1*. Acta n,,,l Caes. Z.eop. Ac. N. Acta. . Atti e Rendiconti dell' Accademia di Scienze, Lettere e Arti dei Zelanti e PP. dello Studio di Acireale. 1890 Camb.P.S.i.; N.H.M.i.; R.S.i. . Annales des Conducteurs des Fonts et Chaussees ; recueil de memoires, etc., concemant le Service de Conducteurs des Fonts et Chaussees. Paris. 1857 P.O. . Annales du Conservatoire des Arts et Metiers. Paris. 1861 B.M.; Camb.U.; Glasg.P.S.i. ; Oxon.B.; P.O.; R.S. See Far. A. Cons. . See Ac. Caes. Loop. N. Acta and Caes. Leop. Ac. N. Acta. . Acta Mathematica. Stockholm. 1882 B.M. ; Camb.P.S. ; Camb.U. ; Dub.T.C. ; Edinb.R.S. ; Edinb.U.; Glasg.U.; Math.S.; Oxon.R.; R.A.S.; R.S.; U.C.L. ix List of Serial Publications A. der Hydros A. diO Aer. J. A. Qen. Civ Aix Mm. Aix Mm. Ac. Alb. X. T Allelod. 8. T. Am. As. P. Am. Eng. & Railroad J, Amiens Ac. Mm Amiens Mm Amiens Mm. Ac Amiens Mm. Ac. Be. . A. Mines ... Am. X. T. Am. J. Be Am. Ph. 8. P. Am. Ph. a. T Annalen der Hydrographie und Maritimen Meteorologie. Heraus- gegeben von der Deutschen Seewarte in Hamburg. Berlin. 1873 B.M.; P.O.i. Annali di Chimica. Milano. 18451900. B.M.; Chem.S.i.; P.O.i. The Aeronautical Journal. London. 1897 B.M.; Camb.U.i.; P.O.; B.S. Annales du G6nie Civil ; Becueil de MSmoires sur les Mathematiques pures et appliqu6es; 1'Astronomie, la Chimie, la Physique, etc. Paris. 186280. B.M.; Camb.U.; Dub.B.C.S.i.; P.O. Annales Hydrographiques. Becueil d'avis, instructions, documents, et memoires relatifs a 1' Hydrographie et a la Navigation. Paris. 1849 B.M.; Ebinb.B.S.i. ; Oxon.B. ; B.A.S.i.; E.S.i. i Memoires de PAcademie des Sciences, Agriculture, Arts et Belles- lettres. Aix. 1819 B.M.; Dub.B.I.A.; N.H.M.i.; Oxon.B.i.; B.S.i. . Transactions of the Albany Institute. Albany. 1830 B.M.; N.H.M.; B.S. .. The Transactions of the Allelodidactic Society. London. 1848. B.S. . . Memoirs of the American Academy of Arts and Sciences. Cambridge and Boston. 1785 B.M.i.; Camb.P.S.; Camb.U.; Dub.B.D.S.i.; Dub.E.I.A.i.; Dub.T.C.i.; Edinb.B.S.; Linn.S. ; N.H.M.; Oxon.B.; P.O.i.; R.A.S.; B.S.; U.C.L.i. See Boat. Am. Ac. Mm. .) Brux. S. As. Bll Bucarest Ac. Rom. A. Bucarest S. Sc. El... Cadiz Period. M. Ci Caen Ac. Mm Cncs. Leop. Ac. N. Acta. Jahresbericht der Schlesischen Gesellschaft fur vaterlandische Cultur. Breslau. 1850 Dub.R.D.S.i.; Dub.R.I.A.i.; N.H.M.; R.S. Bulletin de la Societe Academique de Brest. Brest. 1858 B.M.; Camb.U.i. Giornale di Fisica, Chimica, e Storia Naturale ; Brugnatelli, etc. Pavia. 180827. B.M.; Camb.U.; Dub.T.C.; N.H.M.i.; Oxon.B.; P.O.; R.S. Verhandlungen des Naturforschenden Vereins zu Briinn. Briinn. 1863 Camb.U. i.; Dub.R.I.A.; Linn.S.; N.H.M.; R.S. Bulletins de 1'Academie Royale des Sciences, etc., de Belgique. Bruxelles. 1834 B.M.i. ; Camb.U.; Chem.S.i. ; Dub.R.I.A.; Edinb.R.S. ; Glasg.P.S. i.; Glasg.U.i.; Linn.S.; Math.S.i.; N.H.M.; Oxon.B.; Oxon.R.; P.O.; R.A.S.; R.S. See Brux. Bll. Ac. Centieme Anniversaire de Fondation (17721872) de 1'Academie Royale de Belgique. Bruxelles. 1872. B.M.; Camb.U.; Edinb.R.S.; Glasg.P.S.; Linn.S.; N.H.M.; P.O.; R.A.S.; R.S. } Memoires de 1'Academie Royale des Sciences, des Lettres et des j Beaux-Arts de Belgique. Bruxelles. 1820 B.M.; Camb.P.S.; Camb.U.; Dub.R.I.A.; Dub.T.C.; Edinb.R.S.i.; Edinb.U.; Linn.S.i.; N.H.M.; Oxon.B.(R.); P.O.i.; R.A.S.i.; R.S. See Brux. Mm. Ac. Sc. Annales des Travaux Publics de Belgique. Bruxelles. 1843 B.M.; P.O. Annales des Universitds de Belgique. Bruxelles. 184259. Camb.U.; Oxon.B.; P.O.; R.S.i. See Brux. Ac. Bll. See Brux. Ac. Sc. Mm. ) Memoires Couronne"s et Memoires des Savants Etrangers, publ. par I 1'Acad. Roy. des Sciences, etc. de Belgique. 4to. Bruxelles. 1818 B.M.i.; Camb.P.S.; Camb.U.; Dub.T.C.; Edinb.R.S.t.; Edinb.U.; Glasg.U.i.; Linn.S.i.; N.H.M.; Oxon.B.; P.O.i.; R.A.S.i.; R.S. Memoires Couronn6s et autres Memoires, publ. par 1'Acad. Roy. des Sciences, etc. de Belgique. 8vo. Bruxelles. 1840 B.M.; Camb.P.S.; Camb.U.; Dub.T.C.; Edinb.R.S.; Glasg.U.i.; Linn.S.i.; N.H.M.; Oxon.B.; P.O.; R.A.S.i.; R.S. Annales de 1'Observatoire Royal de Belgique. Bruxelles. 1834 B.M.; Camb.U.; Dub.R.D.S.; Dub.R.I.A.; Edinb.R.S.; Glasg.P.S.i.; Glasg.U.i.; Oxon.B.i.; R.A.S.; R.S.; U.C.L.i. Bulletin de la Societe Beige d'Astronomie. Comptes Rendus des Seances mensuelles de la Societe et Revue des Sciences d'Obser- vation. Astronomie, Meteorologie, Geodesie et Physique du Globe. Bruxelles. 1896 R.A.S. Annales de la Societe Scientifique de Bruxelles. Bruxelles. 1877 B.M.; Dub.N.L.I.i.; Edinb.R.S.; N.H.M. Analele Academiei Romane. Bucuresci. 1880 B.M.; Camb.U.i.; N.H.M.i. Buletinul Societatii de Sciinte Fizice (Fizica, Chimia si Mineralogia) din Bucuresci-Romania. [1892]- [1896]. Buletinul Societatii de Sciinte din Bucuresci-Romania. Bucuresci. (Bulletin de la Societe des Sciences Bucarest-Roumanie.) [1897] Glasg.P.S.; Glasg.U.; N.H.M.; R.S.i.; U.C.L.i. Perif'xlico mensual de Ciencias matematicas y fisicas. Cadiz. 1848. B.M.; R.S. Memoires de 1'Academie des Sciences, Arts et Belles-Lettres de Caen. Caen. 1811 B.M.i.; Camb.U.i.; Dub.T.C.i.; N.H.M.i.; Oxon.B.i.; R.S.i. Nova Acta physico-medica Academise Cses. Leopoldino-Carolinse naturae Curiosorum. Erlangen, Bonn, Breslau. Calif. Ac. F. Camb. and Dubi. Mth. J. Camb. Mm. Anal. 8 Camb. (M.) Mth. M Camb. Mth. J. Camb. Ph. S. P. Camb. Ph. S. T Camb. (U.S.) Mth. M. .. Card. Nt. S. T Carl Rpm Casopis Catania Ac. Oioen. At. . Catania Ac. Ctioen. Bll. Catania At. Ac. Oioen. CE. I.F Chambery Mm. Ac. Sa\ Charleston Md. J. ... Chemnitz B Cherb. Mm. 8. Ac Cherb. Mm. 8. 8c Cherb. 8. 8c. Mm Cherb. 8. 8c. Nt. Mm. Chile A. Un List of Serial Publications 1758 Camb.P.S.; Camb.U.; Chem.S.i.; Dub.T.C.; Edinb.E.S.i.; Edinb.U.; Linn.S.i.; Math.S.i.; N.H.M.; Oxon.R.; R.A.S.i.; E.S. See Ac. Caes. Leop. N. Acta. Proceedings of the California Academy of Natural Sciences. San Francisco. 1854 BJVf.i.; Camb.P.S.i.; Dub.E.I.A.; Edinb.E.S.i.; Glasg.P.S.; Linn.S.i.; N.H.M.; P.O.i.; E.S.i. The Cambridge and Dublin Mathematical Journal; Thomson and Ferrers. Cambridge. 184654. B.M.; Camb.P.S.i.; Camb.U.; Dub.T.C.i.; Edinb.E.S.; Edinb.U.; N.H.M.; Oxon.B.; E.S.; U.C.L. Memoirs of the Cambridge Analytical Society. Cambridge. 1813. B.M.; Camb.U.; Edinb.E.S.; P.O.; E.A.S.; E.S.; U.C.L. The Mathematical Monthly; Eunkle. Cambridge (Massachusetts). 185961. B.M.; Camb.U.; Oxon.B.; P.O.; E.A.S.i.; E.S. ; U.C.L. See Camb. (U.S.) Mth. M. The Cambridge Mathematical Journal. London. 183945. B.M. ; Camb.P.S.; Camb.U.; Dub.T.C.; Edinb.U.; Glasg.U.; Math.S.i.; Oxon.B.i.; E.S.; U.C.L. Proceedings of the Cambridge Philosophical Society. Cambridge. 1866 B.M.i; Camb.P.S.; Camb.U.; Chem.S.i.; Dub.E.D.S. ; Edinb.E.S.; Edinb.U.; Glasg.P.S. i. ; Glasg.U.i.; Linn.S.i.; Math.S.i.; N.H.M.; Oxon.B.i.; Oxon.B.i.; P.O.; B.A.S.i.; E.S.; U.C.L. Transactions of the Cambridge Philosophical Society. Cambridge. 1822 B.M.; Camb.P.S.; Camb.U.; Chem.S.i.; Dub.T.C.i. ; Edinb.E.S.; Edinb.U.; Glasg.P.S.i. ; Glasg.U.; Linn.S.; Math. S.i.; N.H.M.; Oxon.B.; Oxon.E. ; P.O.; E.A.S.; E.S.; U.C.L. See Camb. (M.) Mth. M. Cardiff Naturalists' Society. Eeports and Transactions. Cardiff. 1868 B.M.i.; Camb.U.i.; Dub.E.D.S.; Geol.M.i.; Glasg.P.S.i.; Linn.S.; N.H.M.; Oxon.B.i.; E.S.i. Eepertorium fur physikalische Technik, fur mathematische und astronomische Instrumentenkunde ; Carl. Miinchen. 1865-91. B.M.; Camb.U.i.; Dub.N.L.I.i. ; Oxon.E.; P.O.; E.S. See Exner Rpm. Casopis pro Pestovani Mathematiky a Fysiky. Prag. 1872 B.M. Atti dell' Accademia Gioenia di Scienze Natural! in Catania. Catania. 1825 B.M.; Camb.U.i.; Dub.E.I.A.; Edinb.E.S.i.; Linn.S.i.; N.H.M.; Oxon.B.; E.S. Bulletino mensile della Accademia Gioenia di Scienze Natural! in Catania. Catania. 1888 Dub.E.I.A.; N.H.M.; E.S. See Catania Ac. Ctioen. At. Minutes of Proceedings of the Institution of Civil Engineers, con- taining Abstracts of the Papers and of the Discussions. London. 1837 B.M.; Camb.P.S.; Camb.U.; Dub.E.C.S.; Dub.E.D.S.; Dub.E.I.A.; Edinb.E.S.; Edinb.U.; Glasg.P.S.i.; Glasg.U.; Oxon.B.; Oxon.B.i.; P.O.; E.S.; U.C.L. See I. CE. P. Memoires de la Societe Academique de Savoie. Chambery. 1825 Camb.U.; Dub.E.I.A.; Dub.T.C.; N.H.M.; Oxon.B.; E.S.i. See Sav. Ac. Mm. and Sav. Mm. Ac. Charleston Medical Journal and Eeview; Gaillard, de Saussure, etc. Charleston. 184660. B.M. Bericht der Naturwiss. Gesellsch. zu Chemnitz. Chemnitz. 1859 Edinb.E.S.i.; N.H.M.; E.S.i. Memoires de la Societe Academique de Cherbourg. Cherbourg. 1833 B.M.; Camb.U.i.; Edinb.E.S.i.; N.H.M.i.; Oxon.B.i. {Memoires de la Societe Imperiale des Sciences Naturelles de Cher- bourg. Cherbourg. 1852 B.M.; Camb.P.S.; Camb.U.; Dub.E.I.A.; Edinb.E.S.i.; Linn.S.; N.H.M.; E.A.S.i.; E.S. Anales de la Universidad de Chile. Santiago de Chile. 1843 B.M.i.; Dub.T.C.; N.H.M.i.; Oxon.B.i. V,- Santiago de Chile Un. A. List of Serial Publications Chili S. Sc. Act Christiania F Christ! ania Skr. (Mth. Nt.Kl.) Ciel et Terre Giving C. N Cn. I. P Cn. J Cn. R. 8. P. & T. Coimbral. Con. des Temps Conn. Ac. T. Conn. Mm. Ac Cornwall Pol. 8. Rp. Cornwall Pol. 8. T. Cornwall R. I. J. C. R. . Ore. Ac. Sc. Bll. Crelle J Crelle J. Mth. ... Actes de la Societe Scientifique du Chili (Sociedad cientifica de Chile). Santiago. * 1892 B.M.; Dub.R.D.S.i.; Dub.R.I.A.i.; Edinb.R.S.i.; Linn.S.i.; N.H.M.; R.S.i. Forhandlinger i Videnskabs-Selskabet i Christiania. Christiania. 1859 B.M.; Camb.P.S.; Camb.U. ; Dub.R.I.A.; Edinb.R.S.; Glasg.P.S.; N.H.M.; Oxon.B.; R.S.; U.C.L.i. Skrifter udgivne af Videnskabsselskabet i Christiania. Mathe- matisk-Naturvidenskabelig Klasse. Christiania. 1894 B.M.; Camb.P.S.; Dub.R.I.A.i.; Edinb.R.S.; Glasg.P.S.; N.H.M.; Oxon.B.; R.S.; U.C.L.i. Ciel et Terre. Revue populaire d'Astronomie, de Me"te"orologie et de Physique du Globe. Bruxelles. 1881 B.M.; Edinb.R.S. i.; R.A.S. Der Civilingenieur : Zeitschrift fur das Ingenieunvesen. Freiberg, Leipzig. 1854 B.M.; Camb.U.i.; Dub.R.I.A.i.; P.O. The Chemical News and Journal of Physical Science. London. 1860 Camb.P.S.; Camb.U.i.; Chem.S.; Dub.N.L.I.i.; Dub.R.D.S.i.; Dub.R.C.S.i.; Edinb.U.;Geol.M.; Glasg.P.S.; N.H.M.; Oxon.B.i.; Oxon.R.; P.O.; R.S.; U.C.L.i. Proceedings of the Canadian Institute, Toronto. Toronto. 187990; 1897 B.M.; Camb.P.S.i.; Dub.R.D.S.; Edinb.R.S.i.; Edinb.U.; Glasg.P.S.; Math.S.i.; N.H.M.; Oxon.B.; P.O.; R.A.S.i.; R.S. Transactions of the Canadian Institute. Toronto. 1889 B.M.; Camb.P.S.; Dub.R.D.S.; Edinb.R.S.; Edinb.U.; Glasg.P.S.; Linn.S.; Math.S.i.; N.H.M.; P.O.; R.A.S. ; R.S. The Canadian Journal of Industry, Science, and Art. Toronto. 1853-78. B.M.i.; Dub.R.I.A.; Edinb.R.S.; N.H.M.; P.O.; R.A.S.i.; R.S. Proceedings and Transactions of the Royal Society of Canada. Montreal. 1883 Camb.P.S.; Camb.U.; Chem.S.; Dub.R.D.S.i.; Dub.R.I.A.i.; Dub.T.C.i.; Edinb.R.S.; Geol.M.i.; Glasg.P.S.; Linn.S.; N.H.M.; Oxon.B.; Oxon.R.; P.O.; R.A.S.; R.S.; U.C.L. Institute, jornal scientifico e litterario ; Forjaz. Coimbra. 1853 B.M. Connaissance des Temps, a 1'usage des Astronomes et des Navigateurs. Paris. 1679 B.M.i.; Camb.U.; Dub.T.C.; Oxon.B.; R.A.S.i.; R.S. Transactions of the Connecticut Academy of Arts and Sciences. New Haven. 1866 B.M.; Camb.P.S.; Camb.U.; Dub.R.I.A.; Edinb.R.S.; Glasg.P.S.; Linn.S.; Math.S.i.; N.H.M.; Oxon.R.; P.O.; R.A.S.; R.S. Memoirs of the Connecticut Academy of Arts and Sciences. New Haven. 181016. Linn.S.i.; N.H.M.i.; R.S. ^Reports and Transactions of the Royal Polytechnic Society of | Cornwall. Falmouth. 1833 B.M.; Camb.U.i.; Dub.R.D.S.; Edinb.R.S.i.; Glasg.P.S.i.; Glasg.U.i.; Linn.S.; N.H.M.; Oxon.B.i.; P.O.; R.S.; U.C.L.i. Journal of the Royal Institution of Cornwall. Truro. 1864 B.M.; Camb.U.i.; Dub.R.D.S.; Edinb.R.S.; Glasg.P.S.i.; N.H.M.; Oxon.B.i.; P.O.; R.A.S.i.; R.S.i.; U.C.L.i. Comptes Rendus hebdomadaires des Stances de l'Acade"mie des Sciences. Paris. 1835 B.M. ; Camb.P.S.; Camb.U.; Chem.S.; Dub.R.D.S.; Edinb.R.S.i.; Edinb.U.; Geol.M.i.; Glasg.P.S.i.; Linn.S.; N.H.M.; Oxon.B.; Oxon.R.i.; P.O.; R.A.S.; R.S.; U.C.L. Bulletin International de 1'Academie des Sciences de Cracovie. Cracovie. 1889 B.M.; Camb.U.; Chem.S.; Dub.R.I.A.i.; Edinb.R.S.; Glasg.U.; N.H.M.; Oxon.B.; Oxon.R.i.; R.A.S.i.; R.S.; U.C.L.i. Journal fur die reine und angewandte Mathematik ; Crelle. Berlin. 1826 B.M.; Camb.U.; Dub.N.L.I. ; Dub.T.C.; Edinb.R.S.; Edinb.U.; Glasg.U.; Math.S.i.; Oxon.B.(R.); R.S.; U.C.L. C. B. J Cuyper Rv . Un . . . . Citg.0pt. D. Alpvr. Z Danzig N. Schr. Danzig Schr Delft EC. Pol. A. Des Moines Anal. Devon. As. T D. Gs. Ostas. Mt. Dijon Ac. Mm Dijon Ac. Sc. Mm. Dijon Dim. Ac. ... Dijon Ac. Se Dijon Be. Ac Dingier ... D. Mth. Vr. Jbr. D. Nf. B. . . D. Nf. Tbl. ... D. Nf. Vh D. Nf. Vsm. B Dn. Vcl. Selsk. Skr . List of Serial Publications The Quarterly Journal of the Chemical Society of London. London. 1849 B.M. ; Camb.P.S. ; Camb.U.; Chem.S.; Dub.E.C.S.; DubBDS.; Dub.B.I.A. ; Dub.T.C.; Edinb.E.S.; Geol.M.i.; Glasg.P.S. ; N.H.M.i. ; Oxon.B. ; Oxon.E.i. ; P.O. ; B.S. ; U.C.L. Revue Universelle des Mines, de la Metallurgie, des Travaux Publics, des Sciences, et des Arts, appliques a 1'Industrie; de Cuyper. Paris et Liege. 1857 B.M.; Camb.U.; Dub.B.I.A.i.; Glasg.P.S.i.; Glasg.U.i.; N.H.M.; P.O. See v. Un. Mines. Central-Zeitung fur Optik und Mechanik. Leipzig. 1880 Edinb.U.i.; P.O.i.; B.S.i. Zeitschrift des Deutschen [und des Oesterreichischen] Alpenvereins. Miinchen. 1870 B.M.; Camb.U.; Oxon.B. Neueste Schriften der Naturforschenden Gesellschaft in Danzig. Danzig. 182062. B.M.i.; Dub.E.I.A.i.; N.H.M.; Oxon.B.; B.S. Schriften der Naturforschenden Gesellschaft in Danzig. Danzig. 1863 Camb.P.S.; Camb.U.i.; Dub.B.D.S.i. ; Dub.B.I.A.i.; Edinb.B.S.i.; Linn.S.i.; N.H.M.; Oxon.B.i.; B.S. Annales de 1'Ecole Polytechnique de Delft. Leide. 188597. Camb.P.S.; Camb.U.; Chem.S.i.; Dub.B.D.S.; Dub. B.I. A.; Edinb.B.S.; Math.S.; B.A.S.; B.S. The Analyst : a monthly Journal of Pure and Applied Mathematics. Des Moines, Iowa. 187483. Camb.U.; Edinb.B.S.; B.S. Beports and Transactions of the Devonshire Association for the Advancement of Science, Literature, and Art. Plymouth and London. 1862 Camb.U.;.; Geol.M.; Linn.S.; N.H.M.; Oxon.B.; Oxon. B.i.; P.O.; B.S. Mittheilungen der Deutschen Gesellschaft fur Natur- und Volkerkunde Ostasiens. Yokohama. 1873 B.M.; Edinb.B.S.i. {M4moires de 1'Academie des Sciences, Arts, et Belles-lettres de Dijon. Dijon. 1769 B.M.i.; Camb.U.; Dub.B.D.S.i.; Dub.B.I.A.i.; Dub.T.C.i.; Edinb.B.S. i.; N.H.M.; Oxon.B.i.; B.A.S.; E.S.i. (Seances publiques de 1'Academie des Sciences, Arts, et Belles-lettres \ de Dijon. Dijon. (181029. B.M.i.; N.H.M. Polytechnisches Journal ; Dingier. Stuttgart. 1820 B.M.; Camb.U.; Chem.S.i.; Dub.N.L.I.i.; Dub.B.C.S.i.; Dub.B.D.S.i.; Edinb.B.S.; Glasg.P.S.i. ; Glasg.U.i.; P.O.; B.S.i. Jahresbericht der deutschen Mathematiker-Vereinigung. Berlin, Leipzig. 1890 Camb.U.; Dub.B.I.A.; Edinb.B.S.i.; Math.S.i.; Oxon.B.; B.S. Bericht tiber die Versammlung der Deutschen Naturforscher und Aerzte. 182283. Irregular, see Tageblatt. Camb.U.i.; N.H.M.i.; Oxon. B.i.; B.S.i. See D. Nf. Vsm. B. Tageblatt der... Versammlung Deutscher Naturforscher und Aerzte. 1836 Irregular, see Ber. and Verh. Camb.U. ; N.H.M. ; Oxon. R.f . Verhandlungen der Gesellschaft Deutscher Naturforscher und Aerzte. Leipzig. 1890 Camb.U.; N.H.M.; Oxon.B. See D. Nf. B. DetKongeligeDanske Videnskabernes Selskabs Skrivter. Kiobenhavn. 180118. B.M.; Camb.P.S.i.; Camb.U.; Edinb.B.S.; N.H.M.; Oxon.B.; B.S. See Xiob. Dn. Vd. Selsk. Skr. List of Serial Publications Dorpat Sb Douai Mm. S. Ag. ... Doubs S. Mm. . Dresden Isis Sb Dresden Sb. Isis Dubl. Ph. J. . Dubl. R. S. J. Dubl. S. Sc. P Dubl. S. Sc. T Dunkerque Mm. S. cour. ... Sitzungsberichte der Naturforscher-Gesellschaft zu Dorpat. Dorpat. 1861 Dub.R.I.A.i.; Edinb.R.S. i.; Geol.S.; N.H.M.; R.S.i. Memoires de la Societe d' Agriculture, de Sciences, et d'Arts, seant a Douai. Douai. 1826 B.M.; Camb.U.i.; Oxon.B.i.; R.S.i. Memoires et Comptes Rendus de la Socle" te [Libre] d' Emulation du Doubs. Besancon. 1841 B.M.; N.H.M.i. Sitzungsberichte der Naturwissenschaftlichen Gesellschaft Isis in Dresden. Dresden. 1861 Camb.U.i.; Dub.T.C.; Geol.S.; N.H.M. The Dublin Philosophical Journal and Scientific Review. Dublin. 182526. B.M.; Dub.R.D.S.; Dub.T.C.; Edinb.R.S. i.; N.H.M.; Oxon.B.; Oxon.R.; R.S.i. Journal of the Royal Dublin Society. Dublin. 185675. B.M.;" Camb.P.S.; Camb.U.; Chem.S.i.; Dub.N.L.I.; Dub.R.C.S.; Dub.R.D.S.; Dub.R.I.A. ; Geol.M. ; Glasg.U.i.; Linn.S.; N.H.M.; Oxon.B.; Oxon.R.i.; P.O.; R.A.S.; R.S. The Scientific Proceedings of the Royal Dublin Society. Dublin. 1877 B.M.; Camb.P.S.; Camb.U.; Chem.S.; Dub.N.L.I.; Dub. R.C.S.; Dub.R.D.S.; Dub.R.I.A.; Edinb.R.S.; Geol.M.; Glasg. P.S. ; Linn.S.; Math.S.i. ; N.H.M. ; Oxon.B.; Oxon.R.; P.O.; R.A.S.; R.S.; U.C.L.i. The Scientific Transactions of the Royal Dublin Society. Dublin. 1877 B.M.; Camb.P.S.; Camb.U.; Chem.S.; Dub.N.L.I.; Dub. R.C.S.; Dub.R.D.S.; Dub.R.I.A.; Dub.T.C.; Edinb.R.S.; Geol.M.; Glasg.P.S.; Linn.S.; Math.S.; N.H.M.; Oxon.B.; Oxon.R,; P.O.; R.A.S.; R.S.; U.C.L.i. Memoires de la Societe Dunkerquoise pour 1 'Encouragement des Sciences, des Lettres, et des Arts. Dunkerque. 1853 B.M.; Oxon.B.; R.S.i. Edinb. J. Sc. Edinb. Mth. B. P. Edinb. W. Ph. J. ... . Ph. J Edinb. P. B. S. Edinb. R. S. P.. Edinb. R. S. T Edinb. Sc. S. Arts T.. The Edinburgh Journal of Science, exhibiting a view of the progress of discovery in Natural Philosophy, Chemistry, Mineralogy, Geology, Botany, etc. ; David Brewster. Edinburgh. 18241832. B.M.; Camb.U.; Dub.T.C.; Edinb.R.S.; Edinb.U.; Glasg.P.S.;.; Glasg.U.; N.H.M.; Oxon.B.i.; Oxon.R.; P.O.; R.S. Proceedings of the Edinburgh Mathematical Society. London and Edinburgh. 1883 B.M.; Camb.P.S.; Camb.U.; Edinb.R.S.; Edinb.U.; Glasg.U.; Math.S.; R.S.i. The Edinburgh New Philosophical Journal, exhibiting a view of the progressive Improvements, etc. in the Sciences, etc. ; Robert Jameson. Edinburgh. 182664. B.M.; Camb.P.S.; Camb.U.; Chem.S.i.; Dub.T.C.i.; Edinb.R.S.; Edinb.U.; Glasg.P.S.; Linn.S. i.; N.H.M.; Oxon.B.; Oxon.R.; P.O.; R.A.S.i.; R.S. The Edinburgh Philosophical Journal, exhibiting a view of the Progress of Discovery in Natural Philosophy, etc.; David Brewster and Robert Jameson. Edinburgh. 181926. B.M. ; Camb.P.S.; Camb.U.; Chem.S.; Dub.T.C. Edinb.R.S.; Edinb.U.; Glasg.P.S.; Linn.S.; N.H.M.; Oxon.B.i. Oxon.R.; P.O.; R.A.S.; R.S.; U.C.L.i. ^ Proceedings of the Royal Society of Edinburgh. Edinburgh. (1845 B.M.i.; Camb.P.S.; Camb.U.; Chem.S.i.; Dub.N.L.I. Edinb.R.S.; Edinb.U.; Geol.M.i.; Glasg.P.S.; Linn.S.; Math.S.i. N.H.M.; Oxon.B.i.; Oxon.R.; P.O.i.; R.A.S.; R.S.; U.C.L.i. Transactions of the Royal Society of Edinburgh. Edinburgh. 1788 B.M.; Camb.P.S.; Camb.U.; Chem.S.; Dub.N.L.I.; Dub R.I.A.; Edinb.R.S.; Edinb.U.; Geol.M.i.; Glasg.P.S.; Glasg.U. Linn.S.; Math.S.i.; N.H.M.; Oxon.B.i.; Oxon.R.; P.O.i.; R.A.S. R.S.; U.C.L. See Edinb. T. R. S. Transactions of the Royal Scottish Society of Arts. Edinburgh. 1841 B.M.i.; Camb.U.; Dub.R.D.S.; Edinb.R.S.; Edinb.U.; Glasg.P.S.; Glasg.U.; P.O.; R.S. See Edinb. T. Sc. S. Arts awl 8. Sc. Arts T. List of Serial Publications Edinb. T. R. S Edinb. T. Be. 8. Arts.. Elect. Elektech. Z Eng. 8. T Ens. With. Epinal (VOW.) A. J. Pr. C. . Erfurt Ak. Jb. .. Erfurt N. Acta Erlang. Ps. Bid. 8. Sb. Erlang. Sb. Ps. Bid. 8. Erman Arch. RB. Euro J. d'Ag. ... Eure S. As- Rec. Evk. . Ferussac B1L Sc. With. . Finist. S. Sc. Bll. Firenzc Ac. Georg. At. . Firenze At. Ac. Geory. . Firenze Opusc. Sc Pbrster AL Bauztg Franklin I. J. ... See Edinb. R. S. T. See Edinb. Sc. 8. Arts T. and Sc. S. Arts T. The Educational Times, and Journal of the College of Preceptors. London. 1847 B.M. ; Camb.P.S.i.; Camb.U.i. ; Dub.N.L.L; Math.S.i.; Oxon.B.i.; Oxon.B.i.; B.S.i. The Electrician. London. 1862 B.M.i.; Camb.P.S.i.; Camb.U.i.; Dub.N.L.I.i. ; Dub. E.C.S.i. ; Edinb.E.S.i. ; Edinb.U.i. ; Glasg.P.S.i.; Glasg.U.i. ; Oxon.B.i.; Oxon.E.i.; P.O.; E.S.i.; U.C.L.i. Elektrotechnische Zeitschrift. Berlin, Miinchen. 1880 B.M.; Glasg.U.; P.O. Transactions of the Society of Engineers. London. 1860 B.M.; Camb.U.; Dub.N.L.L; Dub.E.C.S.; Glasg.P.S.i.; Oxon.B.; P.O.; E.S.i. L'Enseignement Math6matique. Eevue Internationale. Paris. 1899 Math.S. Annales de la Societe d ' Emulation du departement des Vosges. Epinal. 1831 B.M.; E.S.i. Journal fur praktische Chemie ; Erdman, etc. deipzig. 1834 B.M.; Camb.U.; Chem.S.; Dub.N.L.I.i.; Dub.E.C.S.i.; Dub.E.D.S.i.; N.H.M.; Oxon.B.; Oxon.B.; E.S.; U.C.L.i. Jahrbucher der koniglichen Akademie gemeinniitziger Wissen- schaften zu Erfurt. Erfurt. I860 B.M.; N.H.M. Nova Acta Academies Electoralis Moguntinse Scientiarum utilium quss Erfurti est. Erfurti. 17991809. B.M.i.; Camb.U.; N.H.M.; Oxon.B.i.; E.S. (Sitzungsberichte der Physikalisch-Medicinischen Societat zu Er- / langen. Erlangen. 1864 B.M.; Camb.P.S.; Dub.E.D.S.; Edinb.E.S.i.; Glasg.U.i.; Linn.S.i.; Math.S.i.; N.H.M.; E.S.i. Archiv fur wissenschaftliche Kunde von Eussland; Erman. Berlin. 184167. B.M.; Camb.U.; N.H.M.; Oxon.B.; E.S.i. Journal d'Agriculture, de Medecine et des Sciences accessoires. Evreux. 1824-29. B.M.; E.S. Eecueil de la Societe d'Agriculture, Sciences, Arts, et Belles-lettres du departement de 1'Eure. Evreux. 183039. B.M.; Camb.U.; Oxon.B.; E.S. Eecueil des Travaux de la Socit Libre d'Agriculture, des Sciences, des Arts et des Belles-Lettres du departement de 1'Eure. Evreux. 1841 B.M.; Camb.U.; E.S. See Eure Bee. 8. Ag. A'Magyar Tud6s Tarsasag' Evkonyvei. Pest. Continued as: A'Magyar Tudomanyos Akademia Evkonyvei. Buda. 183389. B.M.; Edinb.E.S.i.; N.H.M.; Oxon.B.; E.A.S.i.; E.S.; U.C.L.i. Eepertorium der Physik; Exner. Miinchen, Leipzig. 186591. B.M.; Camb.U.i.; Dub.N.L.I.i.; Edinb. U.; Oxon.E.; P.O.; E.S. See Carl Bpm. Bulletin des Sciences Mathematiques, Astronomiques, Physiques, et Chimiques ; Baron de Ferussac. Paris. 182431. B.M.; Edinb.U.i.; Oxon.E.; P.O.; U.C.L. Bulletin de la Societe d'Etudes Scientifiques du Finistere. Morlaix. 1879 N.H.M. \Atti della E. Accademia economico-agraria dei Georgofili. Firenze. (1818 Camb.U.; Dub.T.C.i.; Edinb.E.S.i.; Oxon.B. Collezione d' Opuscoli scientifici, etc. Firenze. 180718. B.M.; N.H.M. Allgemeine Bauzeitung ; Forster. Wien. 1836 B.M.; Camb.U.; P.O. Journal of the Franklin Institute of the State of Pennsylvania. Philadelphia. 1828 B.M.; Camb.U.; Chem.S.i. ; Dub.E.I.A. ; Glasg.P.S.i.; Oxon.B.; P.O.; E.A.S.i.; E.S.; U.C.L.i. List of Serial Publications Fr. Cg. Md. Cliir Congres Medico-Chirurgical de France. Paris. 1863 B.M. Fr. Cg. Sc. Sessions des Congres Scientifiques de France. 183379. B.M.; Camb.U.; N.H.M. Freiburg B Berichte iiber die Verhandlungen der Naturf orschenden Gesellschaf t zu Freiburg i. B. Freiburg i. B. 1855 B.M. ; Camb.U.i. ; Dub.R.I.A. ; Linn.S.i. ; N.H.M. ; Oxon.E. ; B.S. Fr. s. Ag. Mm Memoires d' Agriculture, d'Economie rurale et domestique publics par la Society d' Agriculture. Paris. 1801 B.M.; Edinb.B.S.i.; Oxon.B. Fr. 8. Mn. Bll Bulletin de la Societe Mineralogique de France. Meulan, Paris. 1878 B.M.; Dub.T.C.; Geol.M.; N.H.M.; Oxon.E.; E.S. Fschr. Mth Jahrbuch iiber die Fortschritte der Mathematik. Berlin. 1868 B.M.; Camb.U.; Dub.N.L.L; Dub.E.C.S.; Edinb.U. ; Glasg.P.S.i.; Glasg.U.; Math.S.; Oxon.E.; E.S.; U.C.L. Fschr. Fs Die Fortschritte der Physik. Berlin. 1845 Camb.P.S.; Camb.U.; Chem.S.; Dub.N.L.I.i.; Dub.B.I.A.i.; Edinb.B.S.i.; Edinb.U.; Glasg.U.; Oxon.B.(E.); P.O.; B.A.S.i.; E.S.; U.C.L. Gand. A. Ac G. Arcad Card Apercu Tr Card Mm. Ac. Gard Not. Tr. Ac., Gard Tr. Ac. {' Gen. I. Nt. Mm Gen. Mm. S. Fs. ... Geneva Mm. Ac. Geneva Mm. I. Ligure Geneva TTn. At Gen. a. Ps. Mm Gergonne A. Mth Gg. S. J Gg. S. F. Giaxnb. Vico. Giessen Oberh. Gs. B. Annales Academiae Gandavensis. Gandavi. (Ghent.) 181931. B.M.; Camb.U.; N.H.M.; Oxon.B. Giornale Arcadico di Scienze, etc. Eoma. 1819 B.M.; N.H.M.i.; Oxon.B. Notice [ou Apercu analytique] des Travaux de I'Acade'mie Eoyale du Gard. Nimes. 1807? B.M.; Camb.U.; Oxon.B. See Gard Not. Tr. Ac. and Gard Tr. Ac. Memoires de 1'Academie du Gard. Nimes. 1832 B.M.; Camb.U.; Oxon.B.; E.S.i. See Gard Apercu Tr. Memoires de 1'Institut National Genevois. Geneve. 1854 B.M.; Camb.U.; Dub.E.D.S.; N.H.M.i.; Oxon.B.; E.S. Memoires de la Societe de Physique et d'Histoire Naturelle de Geneve. Geneve. 1821 B.M. ; Camb.P.S.; Camb.U.; Dub.E.I.A.; Dub.T.C.; Edinb.E.S.; Glasg.U.i.; Linn.S.; N.H.M.; Oxon.B.; Oxon.E.i.; B.A.S.i.; E.S.; U.C.L.i. See Gen. S. Ps. Mm. Memorie dell' Accademia Imperiale delle Scienze di Geneva. Geneva. 180914. Camb.U.i.; N.H.M.i.; E.S. Memorie dell' Istituto Ligure. Geneva. 1806. B.M.; Camb.U.; E.S. Atti della E. Universita di Genova. Geneva. 1869 B.M.; Camb.U.; N.H.M. See Gen. Mm. 8. Fs. Annales de Mathematiques, pures et appliquees; Gergonne. Nimes et Paris. 181031. B'.M.; Dub.T.C.; Edinb.U.i.; Glasg.U.; Oxon.B.(E.); E.A.S.i.; E.S.; U.C.L. Journal of the Eoyal Geographical Society of London. London. 183280. B.M.; Camb.P.S.i.; Camb.U.; Dub.E.D.S.; Dub.E.I.A.; Edinb.E.S.; Edinb.U.; Geol.M.; Glasg.P.S.; Glasg.U.; Linn.S.; N.H.M.; Oxon.B.; Oxon.E.; P.O.; E.S. Proceedings of the Eoyal Geographical Society of London. London. 185792. Camb.P.S. i.; Camb.U.; Dub.T.C.; Edinb.E.S.; Geol.M.; Glasg.P.S.i.; Glasg.U.; Linn.S.; N.H.M.; Oxon.B.; Oxon.E.; P.O.i.; E.A.S.i.; E.S. II Giambattista-Vico, giornale scientifico. Napoli. 1857. N.H.M.; Oxon.B.i. See II Giamb. Vico. Berichte der Oberhessischen Gesellschaft fur Natur- und Heilkunde. 1847 B.M.i.; Camb.P.S.; Dub.E.D.S.i.; Dub.E.I.A.i. ; Edinb. E.S.i.; Glasg.P.S.i.; Linn.S.i.; N.H.M.; P.O.i.; B.S.i. xxiii List of Serial Publications Gilbert A. Qill Tech. Rep. Glasg. I. Eng. T. Olasg. P. Ph. B. Glasg. Ph. S. P. Gleanings Be. . . . Ol. Mg OL B. QJ. G. Mt. Gorl. Ab Goteb. Hndl. Gott. Ab Gbtt. Cm. .. Gott. Nr. Oott. Stud Gould As. J '8 Gravcnh. I. Ing. Ts. . . . Grenoble Ac. Delph. Bll. Gr oningcn A. Ac Gruitliuisen N. Analekt. Gruncrt Arch.. Annalen der Physik ; Gilbert. Halle und Leipzig. 17991824. Camb.U.; Chem.S.; Edinb.U.; N.H.M.; Oxon.B.(E.); P.O.; R.S. The Technical Repository; Gill. London. 182227. B.M.; Camb.U.; Edinb.R.S.i.; Oxon.B.; P.O.; R.S.i. Transactions of the Institution of Engineers [and Shipbuiders] in Scotland. Glasgow. 1857 Camb.U.i.; P.O.; U.C.L.i. , Proceedings of the Philosophical Society of Glasgow. Glasgow. 1841 B.M.; Camb.P.S.; Camb.U.; Dub.R.D.S.; Dub.R.I.A.; Edinb.R.S.; Geol.M.i.; Glasg.P.S.; Glasg.U.i.; N.H.M.; Oxon.B.; P.O.i.; R.A.S.; R.S. Gleanings in Science. Calcutta. 182931. B.M.; Edinb.R.S.i.; N.H.M.; U.C.L.t. The Geological Magazine or Monthly Journal of Geology. London. 1864 B.M.; Camb.U.; Dub.N.L.I.; Dub.R.C.S.; Geol.M. ; Glasg.P.S.; Glasg.U.; Linn.S.; N.H.M.; Oxon.R.; P.O.i.; U.C.L. The Quarterly Journal of the Geological Society of London. London. 1845 B.M. ; Camb.P.S.; Camb.U.; Chem.S.i. ; Dub.N.L.I.; Geol.M.; Glasg.U.; Linn.S.; N.H.M.; Oxon.B.; Oxon.R.; P.O.; R.A.S.; R.S.; U.C.L. Giornale di Matematiche ad uso degli Student! delle Universita Italiane; Battaglini. Napoli. 1863 B.M.; Camb.U.; Dub.R.C.S.i.; Dub.R.I.A.i.; Math.S.i.; Oxon.B.; R.S.; U.C.L.i. Abhandlungen der Naturf orschenden Gesellschaf t zu Gorlitz . Gorlitz . 1827 B.M.; Camb.U.i.; Dub.R.D.S.i.; Dub.R.I.A.i.; N.H.M.; R.S. Goteborgs Kongl. Vetenskaps och Vitterhets Samhalles Handlingar. Goteborg. 1850 B.M.; Camb.P.S. i.; Dub.R.D.S.i.; Dub.R.I.A.i.; Edinb. R.S.i.; N.H.M.; R.S.i. Abhandlungen der k. Gesellschaf t der Wissenschaften. Gottingen. 1843 B.M.; Camb.P.S.; Camb.U.; Dub.R.I.A.; Edinb.R.S.i.; Glasg.U.; Linn.S.; N.H.M.; Oxon.B.; Oxon.R.; R.S.; U.C.L.i. Commentationes Societatis Regise Scientiarum GSttingensis. Got- tingae. 17781808. B.M.i.; Camb.U.; Dub.R.I.A.i.; Edinb.R.S.; Glasg. U.i.; N.H.M.; Oxon.B.; R.S.; U.C.L. Commentationes recentiores Societatis, etc. Gottingse. 180837. B.M.; Camb.U.; Edinb.R.S.i.; N.H.M. ; Oxon.B.; Oxon.R.; R.A.S.i.; R.S.; U.C.L. Nachrichten von der k. Gesellschaf t der Wissenschaften und der Georg-Augusts-Universitat zu Gottingen. Gottingen. 1845 B.M.; Camb.P.S.; Camb.U.i.; Dub.R.D.S.i.; Dub.R.I.A.i.; Dub.T.C.i.; Edinb.R.S.; Glasg.U.; Linn.S.; Math.S.i.; N.H.M.; Oxon.B.; Oxon.R.; R.A.S.i.; R.S. Gottinger Studien ; Krische. Gottingen. 184547. N.H.M. ; Oxon.B.i.; R.S.i. The Astronomical Journal ; Gould. Cambridge, Mass. 185161. B.M.; Camb.U.; Glasg.U. i.; Oxon.B.; Oxon.R. i. ; R.A.S.; R.S. .See AS. J. Tijdschrift van het Koninklijk Instituut van Ingenieurs. 's Graven- hage. 1869 B.M. Bulletin de PAcademie Delphinale, ou Soci6te des Sciences et Arts de Grenoble. Grenoble. 1846 B.M.; Camb.U.i.; Oxon.B.; R.S.i. Annales Academies Groninganae. Groningse. 181537. B.M.i.; Camb.U.; Dub.T.C.; Oxon.B.; R.S. Neue Analekten fur Erd- und Himmelskunde. Gruithuisen. Miinchen. 183236. B.M.; R.A.S.; R.S. Archiv der Mathematik und Physik ; Grunert. Greifswald, Leipzig. 1841 B.M.; Camb.U.; Dub.N.L.I.; Dub.R.C.S.; Edinb.U.- Glasg.U.; Math.S.i.; Oxon.B.(R.); R.S.; U.C.L.i. See Arch. Mth. PS. xxiv List of Serial Publications G. Teix. J. Sc. Ha art. Ms. Teyl. Arcb. Haarl. Ntk. Vb. Mtsch. Haidinger Ab Haidinger B Ilain. Mm. S Hain. S. Mm. . . Hall Bij. Halle Ab. Nt. Vr. Halle Nf. Gs. B. Halle Z Halle Z. Nw. Hamb. Mth. Gs. Mt. . . Hammer schmiclt Al. Oestr. Z Hann. Arcbt.-Vr. Z. . . Harm. Z. Arcbt.-Vr. .. Heidi. Nt. Md. Vh Heidi. Vh. Nt. Md Helsingf. Acta. Helsingf. Bd. . Helsingf. Ofv lermbstadt Ms. . . . Jornal de Sciencias Mathematicas e Astronomicas, publicado pelo Dr Francisco Gomes Teixeira. Coimbra. 1878 Math.S.; B.S.i. Archives du Musee Teyler. Haarlem. 1866 B.M.; Camb.P.S.; Camb.U.; Dub.B.I.A.i.; Edinb.K.S.; Glasg.P.S.; N.H.M.; Oxon.B.; E.A.S.; E.S. Natuurkundige Verhandelingen van de [Bataafsche] Hollandsche Maatschappij der Wetenschappen te Haarlem. Haarlem. 1799 B.M.; Camb.U. i.; Dub.R.D.S.; N.H.M.; B.S. Naturwissenschaftliche Abhandlungen ; Haidinger. Wien. 184751. Camb.U.; Chem.S.i.; Edinb.B.S.i.; Linn.S.; N.H.M.; B.A.S.i.; B.S. Berichte iiber die Mittheilungen von Freunden der Naturwissen- schaften in Wien; Haidinger. Wien. 184751. Camb.U.; Chem.S.i.; Edinb.B.S.; Linn.S.; N.H.M.; E.A.S.; B.S. ) Memoires et Publications de la Societe des Sciences, des Arts et des ( Lettres du Hainaut. Mons. 1839 B.M.; Dub.T.C.i.; N.H.M.; Oxon.B.i.; E.S.i. Bijdragen tot de Natuurkundige Wetenschappen ; Hall, etc. Amsterdam. 182632. B.M.; Camb.U.; N.H.M.; B.S. Abhandlungen des Naturwissenschaftlichen Vereins fur Sachsen und Thiiringen in Halle ; Giebel. Berlin. I860 B.M.; Camb.U.; N.H.M.i.; B.S.i. Bericht der Naturforschenden Gesellschaft zu Halle. Halle. 1856 Camb.U.i.; Dub.B.D.S.i.; Dub.B.I.A.i.; Oxon.B.i.; B.S. \ Zeitschrift fur die gesammten Naturwissenschaften ; herausgegeben ( von dem Naturwissenschaftlichen Vereine fur Sachsen und Thiiringen in Halle; Giebel. Berlin. 1853 B.M.; Camb.U.i.; Dub.E.D.S.i.; Dub.R.I.A.i.; Dub.T.C.i.; Edinb.B.S.; Linn.S.; N.H.M.; Oxon.B.; Oxon.B.; B.S. See Z. Ww. Mitteilungen der Mathematischen Gesellschaft in Hamburg. Leipzig. 1889 Math.S. Allgemeine Oesterreichische Zeitschrift fur den Landwirth, Forst- mann und Gartner; Hammerschmidt. Wien. jZeitschr. des Architekten- und Ingenieur- Vereins zu Hannover. | Hannover. 1855 Camb.U.i.; P.O. \Verhandlungen des Naturhistorisch-Medicinischen Vereins zu Heidel- ( berg. Heidelberg. 1857 Camb.U.i.; Chem.S.i.; Dub.B.I.A.; Linn.S. i.; N.H.M.i.; B.S.i. Acta Societatis Scientiarum Fennicse. Helsingfors. 1842 B.M.; Camb.P.S.; Camb.U.i.; Dub.B.I.A.; Edinb.B.S.i.; N.H.M.; Oxon.B.; B.A.S.; B.S. Bidrag till Kannedomen af Finlands Natur och Folk, utgifna af Finska Vetenskaps-Societeten. Helsingfors. 1858 B.M.; Camb.P.S.i.; Camb.U.; Dub.E.I.A.; Edinb.B.S.i.; N.H.M.; Oxon.B.; B.A.S.; B.S. 6'fversigt af Finska Vetenskaps-Societetens Forhandlingar. Hel- singfors. 1853 B.M.; Camb.P.S.i.; Camb.U.i.; Dub.B.D.S.; Dub.B.I.A.; Edinb.E.S.i.; Glasg.U.i.; N.H.M.; Oxon.B.; E.A.S.; E.S. Museum des Neuesten und Wissenswiirdigsten aus dem Gebiete der Naturwissenschaft, der Kiinste, der Fabriken, der Manufakturen, der technischen Gewerbe, der Landwirthschaft, der Produkten- waaren und Handelskunde, und der biirgerlichen Haushaltung, etc.; Hermbstadt. Berlin. 181418. B.M.; B.S. Hertha, Zeitschrift fur Erd-, Volker-, und Staaten-Kunde ; Berghaus und Hoffmann. Stuttgart und Tubingen. 182529. B.M.; Camb.U.; Oxon.B.; E.S.; U.C.L.i. Journal of the Institute of Actuaries. London. 1869 B.M.; Camb.U.; Dub.N.L.I.i.; Oxon.B.; E.A.S.i. I. CE. F. I. gypt. Bll. ............... I. Elect. E. J n Cim II Giamb. Vico n Pout. I. Bin. OTtl. T. IlProgresso Ing Inghirami Opnsc Innsb. Nt. Md. B Intell. Obs. Iowa Ac. Be. F Ir . Ac. Cunningham Mm. Ir. Ac. F. .. Ir. Ac.T List of Serial Publications Minutes of Proceedings of the Institution of Civil Engineers, con- taining Abstracts of the Papers and of the Discussions. London. 1837 B.M.; Camb.P.S.; Camb.U.; Dub.R.C.S.; Dub.R.D.S.; Dub.B.I.A.; Edinb.K.S.; Edinb.U.; Glasg.P.S.i. ; Glasg.U.; Oxon.B.; Oxon.R.i.; P.O.; B.S.; U.C.L. See CE. X. F. Bulletin de 1'Institut Egyptien. Caire. 1859 Camb.P.S.i.; Camb.U.i.; N.H.M.; U.C.L.i. Memoires ou Travaux Originaux pr^sentes et lus a 1'Institut Egyptien publics sous les auspices de S. A. Mohammed Said, vice-roi d'Egypte, sous la direction de M. le Docteur B. Schnepp. Paris & Caire. 1862 Camb.P.S.i.; Camb.U.; N.H.M. Journal of the Institution of Electrical Engineers. London. 1889 B.M.; Camb.U.; Dub.R.D.S.; Edinb.R.S.i.; Edinb.U.; Glasg.U.i.; Oxon.R.i.; P.O.; R.S.; U.C.L. II Cimento, rivista di Scienze, Lettere, ed Arti. Torino. 185255. B.M. II Giambattista-Vico, giornale scientifico. Napoli. 1857. N.H.M.; Oxon.B.i. See Giamb. Vico. II Politecnico; repertorio mensile di studj applicati alia prosperita e coltura sociale. II Politecnico ; repertorio di studj letterar j , scientific! e tecnici . Milano . 183944; 1860 B.M.; P.O. Institution of Mechanical Engineers. Proceedings. Birmingham, London. 1847 B.M.; Camb.P.S.i.; Camb.U.; Dub.R.D.S.; Glasg.U.; P.O.; R.S.; U.C.L. Transactions of the Institution of Mining and Metallurgy, London. London. 1892 B.M.; Camb.U.i.; Geol.S.; N.H.M. ; P.O. II Progresso delle Scienze, Lettere, ed Arti. Napoli. First series undated ; Second series 1832 Camb.U.; Oxon.B. Der Ingenieur ; Zeitschrift fur das gesammte Ingenieurwesen ; Bornemann. Freiberg. 184850. B.M.; P.O. Nuova Collezione di Opuscoli e Notizie di Scienze; Inghirami. Fiesole. 182023. B.M. Berichte des Naturwissenschaftlich-Medizinischen Vereins in Inns- bruck. Innsbruck. 1870 B.M.; Camb.U.; Dub.R.iXS.i.; Dub.R.I.A.i. ; Linn.S.i.; N.H.M.; Oxon.R.; R.S. The Intellectual Observer; a Review of Natural History, Microscopic Research, and Recreative Science. London. 186268. B.M.; Camb.U.; Linn.S.; N.H.M.; Oxon.B.i.; Oxon.R.; P.O.; R.A.S.; R.S.i. Proceedings of the Iowa Academy of Sciences. Des Moines. 1875 B.M.i.; Edinb.R.S.i.; N.H.M.; Oxon.B.i.; P.O.; R.S.i.; U.C.L.i. Royal Irish Academy. " Cunningham Memoirs." Dublin. 1880 B.M.; Camb.P.S.; Camb.U.; Chem.S.i.; Dub.N.L.L; Dub.R.C.S.; Dub.R.D.S.; Dub.R.I.A. ; Dub.T.C.; Edinb.R.S.i.; Edinb.U.; Glasg.P.S.; Linn.S.; N.H.M.i.; Oxon.B.; Oxon.R.; R.A.S.; R.S.i.; U.C.L. Proceedings of the Royal Irish Academy. (Science.) Dublin. 1841 B.M.; Camb.P.S.; Camb.U.; Chem.S.i.; Dub.N.L.L; Dub.R.C.S.; Dub.R.I.A.; Dub.T.C.; Edinb.R.S. ; Edinb.U.; Glasg.P.S.i.; Glasg.U.; Linn.S.; Math.S.i.; N.H.M.; Oxon.B.i.; Oxon.R.; P.O.i. ; R.A.S.; R.S. ; U.C.L.i. Transactions of the Royal Irish Academy. Science. Dublin. 1787 B.M.; Camb.P.S.; Camb.U.; Chem.S.i.; Dub.N.L.L; Dub.R.D.S.; Dub.R.I.A.; Edinb.R.S.; Edinb.U.; Glasg.P.S.; Glasg.U.i.; Linn.S.; Math.S.i.; N.H.M.; Oxon.B.i.; Oxon.R.; P.O.i.; R.A.S.; R.S. ; U.C.L. India Review and Journal of Foreign Science and the Arts. Calcutta. 183738. B.M.; N.H.M.i. List of Serial Publications Isere S. Bll. .. I. & S. I. J. Jap. As. S. T. Jb. Berg- Hw. J. Bt J.dePs J. dePs Jena. Sb Jena. Z J. Gen. Civ. .., J. H. Un. Cir. J. Mcr. Sc. . J. Mines J. Ps. C. . J. Sav J. Tel Kan. Ac. Sc. T. . Kan. TTn. Q Karlsruhe Nt. Vr. Vh. Karsten Arch. . . . Bulletin de la Society de Statistique, des Sciences Naturelles, et des Arts Industriels du departement de 1'Isere. Grenoble. 1838 B.M.i.; N.H.M. ; Oxon.B. ; B.S.i. The Journal of the Iron and Steel Institute. London. 1872 Camb.U. ; Chem.S.i. ; Dub.K.D.S. ; Dub.E.I.A. ; Dub.T.C.; Edinb.U.; Geol.M.i.; Glasg.P.S.i. ; Glasg.U.i.; Oxon.B.; P.O.; B.S.; U.C.L. Transactions of the Asiatic Society of Japan. Yokohama. 1872 B.M. ; Camb.U. ; Edinb.B.S. ; N.H.M. ; Oxon.B. ; Oxon.E.; P.O.*.; B.S. Jahrbuch fur das Berg- und Hiittenwesen im Konigreiche Sachsen. Freiberg. 1873 B.M.; N.H.M.i. ; P.O. The Journal of Botany. London. 1863 B.M.; Camb.U.; Dub.N.L.I.i. ; Dub.K.C.S.; Glasg.U. ; Linn.S. ; N.H.M.; P.O.i. Journal de Physique, de Chimie, et de 1'Histoire Naturelle; de Lametherie etc. Paris. 1794-1823. B.M. ; Camb.U.; Glasg.U. i. ; N.H.M.i.; Oxon.B.; Oxon.B.; E.S.; U.C.L.i. Journal de Physique Theorique et Appliquee; D' Almeida. Paris. 1872 Camb.U.; Dub.B.C.S.; Glasg.U.i.; Oxon.B.; P.O.; E.S. Sitzungsberichte der Jenaischen Gesellschaft fur Medicin und Naturwissenschaft. Jena. 187786. Edinb.B.S.i. ; Linn.S.i.; Oxon.E.; E.S. Jenaische Zeitschrift fur Naturwissenschaft, herausg. von der Medicinisch-Naturwissenschaftlichen Gesellschaft zu Jena. Jena. 1864- B.M.; Camb.P.S. i.; Camb.U.; Chem.S.i.; Dub.N.L.I.i.; Dub.B.D.S.i.; Edinb.E.S. ; Edinb.U.i. ; Glasg.U.i.; Linn.S.; N.H.M.; Oxon.B.; Oxon.E.; E.S.; U.C.L. Journal du Genie Civil des Sciences et des Arts. Paris. 182848. B.M.i.; Camb.U.; P.O. The Johns Hopkins University Circulars. Baltimore. 1879 Camb.P.S. ; Camb.U. ; Dub.N.L.I.i. ; Dub.B.I.A.i. ; Edinb.E.S. i. ; Edinb.U. ; Glasg.P.S. ; Math.S.i. ; N.H.M. ; Oxon.B.; Oxon.E.; E.A.S.i.; E.S.; U.C.L.i. Quarterly Journal of Microscopical Science; Lankester and Busk. London. 1853 B.M.; Camb.P.S.; Camb.U.; Chem.S.i.; Dub.N.L.I. ; Dub.E.C.S. ; Edinb.E.S. ; Edinb.U. ; Glasg.P.S. ; Linn.S. ; N.H.M.; Oxon.B.i. ; Oxon.E.; P.O.; E.S.; U.C.L. Journal des Mines, ou Eecueil de Memoires sur 1' exploitation des Mines, et sur les Sciences et les Arts qui s'y rapportent. Paris. 17941815. B.M.; Camb.U.; Dub.T.C.; Edinb.E.S.; Linn.S.i.; N.H.M.; Oxon.B. (E.) ; E.S.i. The Journal of Physical Chemistry. Ithaca, N.Y. 1896 B.M. ; Camb.P.S. ; Camb.U. ; Chem.S. ; Edinb.E.S. ; Edinb.U.; Glasg.U.; Oxon.E.; P.O.; E.S.i. Journal des Savants. Paris. 1816 B.M.; Camb.U.; Dub.N.L.I.; Dub.T.C.; Edinb.E.S.i.; Glasg.U.i.; Oxon.B.; Oxon.B.; P.O.i.; E.S. Journal Telegraphique publie par le Bureau International des Administrations Telegraphiques. Berne. 1869 P.O. Transactions of the Kansas Academy of Science. Topeka, Kansas. 1872 Camb.P.S.i. ; Dub.E.I.A.; Edinb.E.S.i.; Glasg.P.S.i.; Linn.S.i; N.H.M.; Oxon.B.i.; B.S.i. The Kansas University Quarterly. Laurence, Kansas. 1893 B.M.i. ; Dub.E.D.S.i. ; Dub.E.I.A.i. ; Edinb.E.S.i. ; Glasg. P.S.i.; Math.S.; N.H.M.; E.S. Verhandlungen des Naturwissenschaftlichen Vereins in Karlsruhe. Karlsruhe. 1864 B.M.i.; Dub.E.I.A.; N.H.M. Archiv fiir Mineralogie, Geognosie, Bergbau, und Hvittenkunde ; Karsten. Berlin. 182955. B.M.; Edinb.E.S.i.; Geol.M.; N.H.M.; P.O.; E.S. List of Serial Publications Kastner Arch. Ntl. Kazan Mm . TTn Kazan 8. Nt. (Ps.-Mth.) T, Kazan S. Ps.-Mth. Bll. .. Kazan TTn. Bll Kazan Un. Mm Kharkov Mth. 8. Com.... Kiel Schr. Kiob. Dn. Vd. Sclsk. Af h. Kjob. Dn. Vd. Selsk. Af h. Kiob. Dn. Vd. Selsk. Skr. Kjbb. Dn. Vd. Selsk. Skr. Kjob. Dn. Vd. Selsk. Skr. Kibb.Skr Kiob. Ov Kjob.0v! '.'.".'.'.'.'. Konigsb. Arch. Konigsb. Schr... Krk. Ak. (Mt.-Prz.) Pam. Krk. Ak. (Mt.-Prz.) Rz.... Krk. Ak. (Mt.-Prz.) Hz. * Bp Krk. Roczn. Tow. Nauk. Krk. Roczn. Uniwers. Arcbiv f fir die gesammte Naturlehre ; Kastner. Niimberg. 182435. B.M.; N.H.M.; P.O. Scientific Memoirs published by the Imperial University of Kazan. Kazan. 1834 B.M.i.; Camb.U.; Dub.E.I.A. ; Edinb.E.S.i.; Glasg.P.S.i.; Linn.S.i.; B.S.i. See Kazan Un. Mm. Proceedings of the Physico-Mathematical Section of the Naturalists' Society of the Imperial University of Kazan. Kazan. 188390. U.S. Bulletin de la Societe Physico-Mathematique de Kasan. Kasan. 1891 Dub.B.I.A.i.; Edinb.B.S.i. ; B.S.i. Bulletin of the Imperial University of Kazan. Kazan. 1865 B.M.i. ; Camb.U.i. ; Edinb.B.S.i.; Linn.S.i. See Kazan Mm. Un. Communications and Proceedings of the Mathematical Society of the Imperial University of Kharkov. Kharkov. 1879? E.S.i. Schriften der Universitat zu Kiel. Kiel. 185580. B.M.; Camb.U.; Dub.R.I.A.; Edinb.B.S.i.; N.H.M.i.; Oxon.B.; E.S. I Det Kongelige Danske Videnskabernes Selskabs naturvidenskabelige \ og mathematiske Afhandlinger. Kiobenhavn. ( 182446. B.M. ; Dub.T.C. ; Edinb.E.S. ; Linn.S.i. ; N.H.M. ; E.S. ( Det Kongelige Danske Videnskabernes Selskab Skrivter. Kiobenhavn. {180118. B.M.; Camb.P.S. i.; Camb.U.; Edinb.E.S.; N.H.M. ; I Oxon.B.; B.S. See Dn. Vd. Selsk. Skr. /Det Kongelige Danske Videnskabernes Selskabs Skrifter. Natur- videnskabelig og Mathematisk Af deling. Kj^benhavn. 1 1849 B.M. ; Camb.U. i. ; Edinb.E.S. ; Linn.S. ; N.H.M. ; E.A.S. ; | E.S.; U.C.L.t. /Oversigt over det Kongelige Danske Videnskabernes Selskabs For- handlinger. Kjdbenhavn. Jl806 Camb.P.S.; Camb.U.i.; Chem.S.i. ; Dub.E.D.S.i. ; Dub. E.I.A.i. ; Dub.T.C.i. ; Edinb.E.S.i. ; Glasg.U.i. ; Linn.S.i. ; ( N.H.M.i.; Oxon.E.; P.O.i.; E.A.S.i.; E.S. Konigsberger Archiv fiir Naturwissenschaften und Mathematik ; Bessell, etc. Konigsberg. 1812. B.M. ; Camb.U.; Linn.S. Schriften der koniglichen Physikalisch-Oekonomischen Gesellschaft zu Konigsberg. Konigsberg. I860 B.M.; Camb.P.S.; Dub.E.I.A. ; Edinb.E.S.i.; Linn.S.; N.H.M.; P.O.i.; E.S. Kosmos. Czasopismo polskiego Towarzystwa przyrodnik6w imienia Kopernika. [Cosmos. The Journal of the Polish Society of Naturalists founded in honour of Copernicus.] Lwow. 1876 B.M.; N.H.M. Pamitnik Akademii Umiejtnosci w Krakowie. Wydzia-l- Mate- matyczno-Przyrodniczy. [Memoirs of the Academy of Science in Cracow. Section of Mathematics and Natural Science.] Krakow. 1874 B.M.; Edinb.E.S.i.; N.H.M. Eozprawy Wydziatu Matematyczno - Przyrodniczego Akademii Umieje,tnosci. [Proceedings of the Section of Mathematics and Natural Science of the Academy of Science.] Krak6w. 1874 B.M.; Edinb.E.S.i.; N.H.M. Eocznik Towarzystwa Naukowego z Uniwersytetem Krakowskim Polaczonego. Krakowie. [Annals of the Scientific Society of the Polish University of Krakow. Krakow.] 181772. B.M. Lane. Hist. 8. T i Proceedings and Papers of the Lancashire and Cheshire Historic Lane. T. Hist. 8 { Society. Liverpool . 184954. Continued as: Transactions, etc., 1855 B.M. ; Camb.U.i.; Dub.E.I.A.i.; Edinb.E.S.i.; Glasg.P.S.i.; Oxon.B. i.; E.S. List of Serial Publications Laus. Bll. S. Vd. Laus. 8. Vd. Bll. . Leic. S. T. Leijd. A. Ac Leip. Ab. Jablon. Gs Leip. Ab. rath. Ps. Leip. As. 6s. Vjschr. Leip. B Leip. Jablon. Preisschr. Leip. Mth. Ps. Ab Leip. Mtlx. Ps. B Le mans S. R. Tr. . . . Les Mondes Liege A. Ac.. Liege Mm. S. Sc. Liege S. Sc. Mm. Lille Mm Lille Mm. S LilleS. Mm Lille Tr. .. Lille Tr. Mm Lindenau Z ( Bulletin des Seances de la Societe Vaudoise des Sciences Naturelles. | Lausanne. 1842 Dub.E.D.S.i. ; Dub.E.I.A.i. ; Edinb.E.S.i. ; Linn.S.; N.H.M.; Oxon.B. i.; E.S. The Transactions of the Leicester Literary and Philosophical Society. 1835 Camb.U.; Dub.K.D.S.; Geol.S.; Glasg.P.S.; Linn.S.i.; N.H.M.i. ; Oxon.B.; P.O.; U.C.L. Annales Academies Lugduno-Batavse. Leijden. 181575. B.M.; Camb.U.; Dub.T.C.i.; N.H.M.; Oxon.B.; B.S.i.; U.C.L.i. Abhandlungen bei Begriindung der k. Sachsischen Gesellschaft der Wissenschaften am Tage der zweihundertjahrigen Geburtsfeier Leibnizens; herausg. v. d. Jablonowski'schen Gesellschaft zu Leipzig. Leipzig. 1846. Camb.U.; Dub.E.I.A.; Edinb.E.S.; N.H.M.; E.A.S.; E.S. Abhandlungen der Mathematisch-Physischen Classe der Koniglich Sachsischen Gesellschaft der Wissenschaften. Leipzig. 1852 B.M.; Camb.P.S.; Camb.U.; Dub.E.I.A.; Edinb.E.S.; Glasg.U. ; Math.S.i. ; N.H.M.; Oxon.B.; P.O.; E.A.S.; E.S. ; U.C.L. i. See Leip. Mth. Fs. Ab. Vierteljahrsschrift der Astronomischen Gesellschaft. Leipzig. 1866 B.M.; Camb.P.S.i.; Camb.U.; Dub.E.I.A.; Edinb.E.S.; Oxon.B.; E.A.S.; E.S. Berichte liber die Verhandlungen (Math.-Phys. Classe) der Konig- lich Sachsischen Gesellschaft der Wissenschaften zu Leipzig. Leipzig. 1846 B.M.; Camb.P.S.; Camb.U.; Dub.E.I.A.; Edinb.E.S.; Glasg.U.; Math.S.i.; N.H.M.; Oxon.B.; Oxon.E.; E.A.S.; E.S.; U.C.L.i. See Leip. Mth. Ps. B. Preisschriften gekront und herausgegeben von der Fiirstlich Jab- lonowski'schen Gesellschaft zu Leipzig. Leipzig. 1847 B.M. ; Camb.U. ; Dub.E.I.A. ; Edinb.E.S.i. ; N.H.M. ; Oxon.B.; E.A.S.i.; B.S.i.; U.C.L.i. See Leip. Ab. Mth. Fs. See Leip. B. Analyse des Travaux de la Societe [Eoyale] des Arts du Mans. Le Mans. 1820. N.H.M.; E.S. Les Mondes, Eevue hebdomadaire des Sciences et de leurs Appli- cations aux Arts, et a 1'Industrie; l'Abb6 F. Moigno. Paris. 1863 B.M.; Camb.U.i.; Dub.N.L.I.i.; Glasg.P.S.i.; Oxon.E.; P.O.; B.S.i. L'Institut; Journal des Academies et Societes Scientifiques de la France et de 1'Etranger. Paris. 1833 B.M.i.; Camb.U.; Dub.T.C.; Edinb.E.S.i.; Glasg.P.S.i.; N.H.M.; Oxon.B. (E.); P.O.i.; E.S.i. Annales Academise Leodiensis. Liege. 181727. B.M.; Camb.U.; Dub.T.C.; N.H.M.; Oxon.B.; E.S. Memoires de la Societe [Eoyale] des Sciences, de 1' Agriculture, et des Arts a Liege. Liege. 1843 B.M.; Camb.U.; Dub.T.C.; Edinb.E.S.i.; Glasg.P.S.i. ; Glasg.U.i.; Linn.S.i.; N.H.M.; Oxon.B.; P.O.; E.S. Memoires de la Societe [Eoyale] des Sciences, etc. a Lille. Lille. 1827 B.M.; Camb.U.; Dub.T.C.; N.H.M. ; Oxon.B.; Oxon.E.; E.S.i. Eecueil des Travaux de la Socie'te' d' Amateurs des Sciences, de 1' Agriculture, et des Arts a Lille. Lille. 181927. B.M.; Camb.U.; Dub.T.C.; N.H.M.; Oxon.B.; Oxon.E.; E.S. Travaux et Memoires de 1'Universite de Lille. Lille. 1889 Camb.P.S.; Camb.U.; Dub.E.D.S.; Dub.E.I.A.; N.H.M.; E.S. Zeitschrift f iir Astronomic und verwandte Wissenschaften ; Lindenau. Tubingen. 181618. B.M.; Camb.U.; E.A.S.; E.S. xxix c 2 Liouv. J. Liouv. J. Mth. . Lisb. A. Lisb. J. Sc. Mth. Lisb. Mm. Lisb. Mm. Ac. Sc. [ Lisb. Mm. Sc. ] L. Mth. S. F. . . V.-St Lotos Xiouvain A. Ac Lpldina Lpool Lt. Ph. S. P L. Ps. S. P Lucca At. Ac Lum. Elect. Lund. Acta XTn Lund Phys. Sallsk. Ts. Lund . Tin. Acta Lux. I. R. Pb. ... Lux. S. Sc. Mm Lux. S. Sc. Nt. Lyon Ac. Mm. (Be.) . . . . Lyon Ac. Sc. Mm List of Serial Publications Journal de Mathe'matiques pures et appliquees, fonde par Joseph Liouville. Paris. 1836 B.M. ; Camb.U. ; Dub.E.I.A. ; Edinb.R.S.i. ; Edinb.U.; Glasg.P.S.i.; Glasg.U.; Math.S.i.; Oxon.B.(B.); E.S.; U.C.L. Annaes das Sciencias e Lettras, publicados debaixo dos auspicios da Academia Eeal das Sciencias: Sciencias mathematicas, physicas, historico-naturaes, e medicas. Lisboa. 1857 B.M.; Dub.E.I.A.; Edinb.E.S.i.; N.H.M.; E.A.S.i.; B.S.i. See A. Sc. Historia e Memorias da Academia Real das Sciencias de Lisboa. Lisboa. 1797 B.M.; Camb.U.; Edinb.B.S.; N.H.M.; Oxon.B.; E.A.S.; E.S.i. See Lisb. Mm., Lisb. Mm. Ac. Sc. and Lisb. Mm. Sc. Jornal de Sciencias mathematicas, physicas e naturaes. Publicado sob os auspicios da Academia E. das Sciencias de Lisboa. Lisboa. 1868 B.M.; Camb.U.; Dub.E.D.S.i. ; Edinb.E.S.i.; Linn.S. ; Math.S.i.; N.H.M.; Oxon.B.; E.A.S.; E.S.; U.C.L.i. See Lisb. Ac. Sc. Mm. Proceedings of the London Mathematical Society. London. 1865 B.M. ; Camb.P.S. ; Camb.U.; Dub.E.I.A.; Dub.T.C. ; Edinb.E.S.; Edinb.U.; Glasg.U.; Math.S.; Oxon.B.i.; Oxon.B.; E.S.; U.C.L. Die landwirthschaftlichen Versuchs-Stationen. Organ fur wissen- schaftliche Forschungen auf dem Gebiete der Landwirthsehaft. Dresden, Chemnitz. 1859 B.M.i.; Camb.U.; Chem.S.i.; Oxon.B.; P.O.i.; E.S.i. Lotos. Zeitschrift fur Naturwissenschaften. Prag. 185195. B.M.; Camb.U.; Dub.E.I.A.i.; N.H.M. Annales Academise Lovaniensis. Bruxelles, Louvain. 182127. B.M.; Camb.U.; Dub.T.C.; Oxon.B.; E.S. Leopoldina : amtliches Organ der Kaiseiiichen Leopoldino-Caro- linischen Deutschen Akademie der Naturforscher. Dresden. 1859 B.M.; Camb.P.S.; Camb.U.i.; Edinb.E.S.i.; Linn.S.; N.H.M.; E.A.S.i.; E.S. Proceedings of the Literary and Philosophical Society of Liverpool. London, Liverpool. 1844 B.M.; Camb.U.i.; Chem.S.i.; Dub.E.I.A.; Edinb.E.S.i.; Glasg.P.S.; Linn.S.; N.H.M.; Oxon.B.i.; P.O.i.; E.A.S.i.; E.S.; U.C.L.i. Proceedings of the Physical Society of London. London. 1874 B.M.; Camb.P.S.; Camb.U.; Chem.S.; Dub.E.C.S.; Dub. E.D.S.; Dub.E.I.A.; Dub.T.C.; Edinb.E.S.i.; Glasg.P.S.i.; Glasg.U.i.; Math.S.; Oxon.B.; Oxon.E.; P.O.; E.A.S.; E.S.; U.C.L.i. Atti della E. Accademia Lucchese di Scienze, Lettere, ed Arti. Lucca. 1821 B.M.; Camb.U.; Dub.T.C.i.; Oxon.B.i. La Lumiere Electrique. Journal universel d'Electricite". Paris. 187994. B.M.; Glasg.U.i.; P.O. Acta Universitatis Lundensis. Lunds Universitets Ars-skrift. Afdel- ningen for Mathematik och Naturvetenskap. Lund. 1864 B.M.; Camb.P.S.; Camb.U.; Dub.E.I.A.; Edinb.E.S.i.; Linn.S.i.; N.H.M.; Oxon.B.; E.S. See Lund. Tin. Acta. Physiografiska Sallskapets Tidskrift. Lund. 183738. Camb.U.; N.H.M.; E.S. See Lund. Acta TTn. Publications de 1'Institut Eoyal Grand-Ducal de Luxembourg. Section des Sciences Naturelles et Math&natiques : ci-devant "Societe des Sciences Naturelles." Luxembourg. 1870 Dub.B.D.S.; Dub.E.I.A.; Edinb.E.S.i.; N.H.M.; E.S.i. [Societe des Sciences Naturelles du Grand-Duche de Luxembourg. j Luxembourg. 1185369. Dub.E.I.A.; E.S. (Memoires de 1' Academic des Sciences, Belles-Lettres et Arts de ( Lyon. Classe des Sciences. Lyon, Paris. List of Serial Publications Lyon A. a. Ag. Lyon Mm. Ac Lyon Mm. Ac. Sc. Lyon S. A. Ag. . . . Lyon 3. Ag. A. ... Lyon TTn. A Maclurian Lyceum Ct. Macon Ac. A Macon S. C. R Madras E ng. Rp Madras J. Madrid Ac. Ci. Mm Madrid Mm Madrid Rv. .. Mag. Ale. Ets Mag. Ak. Ets. (Mth. Term.) Magdeb. Wt. Vr. Jbr. u. Ab Mag. Tud. Alt. Etlt. (Mth.) Mag. Tud. Ak. Ets Majocchi A. Fis. C Manch. Lt. Ph. S. Mm.... Mancb. Lt. Ph. S. Mm. & P. .. 1845 B.M.; Camb.U.; Edinb.B.S.i.; Linn.S.i.; N.H.M.; Oxon.B.; B.S.i. See Lyon Mm. Ac. and Lyon Mm. Ac. Sc. Annales des Sciences physiques et naturelles, d' Agriculture, et d'Industrie, publiees par la Societe d' Agriculture, etc. Annales de la Societe d' Agriculture, Histoire Naturelle et Arts Utiles de Lyon. Lyon. 1838 B.M.; Camb.U.; Dub.B.I.A.; Linn.S.; N.H.M.; Oxon.B.; P.O.; B.S. See Lyon S. A. Ag. and Lyon 8. Ag. A. See Lyon Ac. Mm. (Sc.) and Lyon Ac. Sc. Mm. See Lyon A. S. Ag. Annales de 1'Universite de Lyon. Paris, Lyon. 1891 B.M.; Edinb.B.S.; N.H.M.i.; B.S.i. Contributions of the Maclurian Lyceum to the Arts and Sciences. Philadelphia. 182729. B.M.; Linn.S.i.; N.H.M. Annales de 1'Acade'mie de Macon, Societe des Arts, Sciences, Belles- Lettres et d' Agriculture. Macon. 1851 B.M. E.S.i. Compte Bendu des Travaux de la Societe (d' Agriculture,) des Sciences, Arts et Belles-Lettres, de Macon. Macon. 180752. B.M.i.; E.S.i. Beports, etc. on various professional subjects connected with the duties of the Corps of Engineers of the Madras Presidency; Capt. J. T. Smith, F.B.S. Madras. 183946. P.O.; B.S. The Madras Journal of Literature and Science. Madras. 1833 B.M.i.; Camb.U.; Dub.N.L.I.i.; Linn.S.i.; N.H.M.; Oxon. B.i.; P.O.; B.A.S.i.; B.S.i.; U.C.L.i. IMemorias de la Beal Academia de Ciencias. Madrid. 1850 B.M.; Camb.U.i. ; Dub.B.I.A.; Dub.T.C.; Edinb.E.S.; Linn.S.i.; N.H.M.; Oxon.B.; B.A.S.i.; B.S.i.; U.C.L.i. Bevista de los Progresos de las Ciencias exactas, fisicas, y naturales. Madrid. 185086. B.M.; Dub.E.D.S.i.; Edinb.B.S.i.; N.H.M.; Oxon.B.i.; E.A.S.i.; B.S.i. Magyar akademiai Ertesito. [Beport of the Hungarian Academy.] Pest. 184059. B.M. Magyar akademiai Ertesito. A mathematikai es term6szettudo- manyi osztalyok kozlonye. [Beport of the Hungarian Academy. Communications of the Mathematical and Natural Science sections.] Pest. 186065. B.M.; Camb.P.S.i.; B.S. Jahresbericht und Abhandlungen des Naturwissenschaftlichen Vereins zu Magdeburg. Magdeburg. 1869 B.M.; B.S.i. Ertekez6sek a Mathematikai Osztaly korebol. Kiadja a Magyar Tudomanyos Akademia. [Memoirs on Mathematical subjects. Published by the Hungarian Academy of Science.] Pest. 1867-94. B.M.; Edinb.B.S.i.; B.S. A Magyar Tudomanyos Akademia Ertesitoje. [Beport of the Hungarian Academy of Science.] Pest. 1867 B.M. Annali di Fisica, Chimica, e Matematiche, col Bulletino dell' Industria meccanica e ehimica ; Majocchi. Milano. 184150. B.M.; B.S. Memoirs of the Literary and Philosophical Society of Manchester. London and Manchester. 17851887. B.M.; Camb.P.S.; Camb.U.; Chem.S.i.; Dub.B.D.S.; Dub.B.I.A.; Glasg.P.S.i.; Linn.S.; Math.S.i.; N.H.M.; Oxon.B.; Oxon.B.; P.O.; B.A.S.i.; B.S.; U.C.L.i. See Mancb. Mm. Ph. a., Manch. Ph. S. Mm. and Manch. S. Mm. Memoirs and Proceedings of the Manchester Literary and Philo- sophical Society. Manchester. xxxi Manch. Lt. Ph. 8. P. Blanch. Mm. Ph. s. Blanch. Ph. 8. Mm. Blanch. Ph. 8. P. ..'. Blanch. 8. Mm Blanch. 8. P Mart. Schr Mars. Fac. Sc. A. Mathematician .. Mathesis Mbl . Nt. . Medley I. Eng Mengelwerk Wisk. M.-et-L. Mm. 8. Ac. Met. 8. QJ Metz Ac. Mm Metz Mm. Ac Mex. Arch. Com. Sc Blex. Beg. Trim., Mex. 8. "Alzate" Mm. Big. Ntvd. Midi. Ntlist. BUI. At. X. Lomb. List of Serial Publications 1888 BM; Camb.P.S. ; Camb.U. ; Chem.S. ; Dub.E.I.A.i.; Edinb.E.S.; Glasg.P.S.; Linn.S.; Math.S.; N.H.M.; Oxon.B.; OxonE.; P.O.; E.A.S.; E.S.; U.C.L.i. Proceedings of the Literary and Philosophical Society of Manchester. Manchester. 185787. B.M.; Camb.P.S.; Camb.U.; Chem.S.; Dub.E.D.S.; DubE.I.A.; Dub.T.C.; Edinb.E.S.; Glasg.P.S.; Linn.S.; Math. S.i.; N.H.M.; Oxon.B.; P.O.; E.A.S.; E.S.; U.C.L.i. See Blanch. Ph. 8. P. I See Blanch. Lt. Ph. 8. BIm. See Blanch. Lt. Ph. 8. P. See Blanch. Lt. Ph. 8. Mm. See Blanch. Lt. Ph. 8. P. Schriften der Gesellschaft zur Beforderung der gesammten Natur- wissenschaften zu Marburg. Marburg. 1823 B.M.i.; Camb.U.; N.H.M.; Oxon.E.; E.S. Annales de la Faculte des Sciences de Marseille. Marseille, Paris. 1891 B.M.; Camb.P.S.; Dub.E.I.A.; Edinb.E.S.; Glasg.P.S.; Linn.S.; Math.S.i.; N.H.M.; E.A.S.; E.S. The Mathematician ; Davies, etc. London. 184550. B.M.; Camb.U.; Dub.T.C.; Oxon.B.; E.S. Mathesis. Eecueil Mathematique.... Gand, Paris. 1881 B.M.; Camb.U. Maanblad voor Natuurwetenschappen, uitgegeven door de Sectie voor Natuurwetenschappen van het Genootschap ter Bevordering van Natuur-, Genees- en Heelkunde. Amsterdam. 1871 N.H.M. Professional Papers on Indian Engineering; Major J. G. Medley. Eoorkee. 186468. P.O.i.; E.S. Mengelwerk van uitgeleezene en andere Wis- en Natuurkundige Ver- handelingen ; door het Genootschap der Mathematische Weten- schappen, etc. Amsterdam. 17961816. E.S. See Amst. Mengelwerk. The Messenger of Mathematics. Cambridge, London. 1862 B.M.; Camb.P.S.; Camb.U.; Dub.N.L.I.i.; Dub.E.C.S.i.; Dub.E.D.S. i.; Edinb.E.S.i. ; Edinb.U.; Glasg.U. ; Math.S.i.; Oxon.B.; Oxon.E.; E.S.; U.C.L. Memoires de la Societe Academique de Maine et Loire. Angers. 1857 B.M.; Camb.U.; N.H.M.; E.S.i. . Quarterly Journal of the Eoyal Meteorological Society. London. 1873 Camb.U.; Dub.E.I.A.i.; Dub.T.C.i.; Edinb.E.S.; Linn. S.i.; Oxon.E.; P.O.; E.A.S.; E.S. . Meteorologische Zeitschrift. Berlin. 1884 Camb.U.; Edinb.E.S.; P.O.; E.S. , \Memoires de I'Acad&nie (Eoyale, Imperiale) de Metz. Metz. J1821 B.M.; Camb.U.; Dub.T.C.; N.H.M.; Oxon.B.; E.S.i. . Archives de la Commission Scientifique du Mexique, publiees sous les auspices du Ministere de 1'Instruction Publique. Paris. 186569. Oxon.B.; E.S. . Eegistro trimestre, 6 Coleccion de Memorias de Historia, Literatura, Ciencias, etc., por una Sociedad de Literates. Mexico. 183233. Memorias de la Sociedad Cientifica "Antonio Alzate." Mexico. 1887 B.M.i.; Camb.P.S.; Dub.E.I.A.; Edinb.E.S.; Linn.S.i.; Math.S.i.; N.H.M.i.; E.A.S.; E.S.; U.C.L.i. . Magazin for Naturvidenskaberne ; Lundh, etc. Christiania. 182336. B.M.; N.H.M.i.; E.S. . Monatshefte fur Mathematik und Physik. Wien. 1890- B.M.; Camb.U.; Edinb.U.; Math.S.i.; N.H.M. .. The Midland Naturalist. London, Birmingham. 187893. Camb.U.; Geol.M.; Linn.S.; P.O. . Atti dell' I. E. Istituto Lombardo di Scienze, Lettere, ed Arti. Milano. 185864. B.M.; Camb.U.; Edinb.E.S.; N.H.M.; Oxon.B.; E.S. xxxii List of Serial Publications Mil. Effem Mil. Effem. As. . Mil. Ephem. As.. Mil. G. I. Lomb. Mil. G. a. Inc. Mil. I. Lomb. G Mil. I. Lomb. Mm.. Mil. I. Lomb. Bd. MiL Mm. I. tomb Mil. Mm. I. Lomb. Ven. Mitan Arb. Kurland. Gs. Mi tan Quatember Mitan Send. Kurland. Gs. Mntp. Ac. Mm. Mntp. Ac. PV. . . . Mntp, Ac. Sc. Mm. Mntp. Mm. Ac j. Mntp. Mm. Ac. Sect. Sc. } Mntp. Mm. Ac. Sect. Md. Mntp. Rec. Bll. Mod. Ac. Sc. Mm. Mod. Mm. Ac. Sc. Mod. Mm. 8. It. Mod. S. It. Mm. (Effemeridi Astronomiche di Milano. Con Appendice di Osservazioni \ e Memorie Astronomiche. Milano. (1806 Camb.U.; Oxon.B.; B.A.S.t. Ephemerides Astronomicae, ad Meridianum Mediolanensem supputatffl ab Angelo de Cesaris. Accedit Appendix cum Observationibus et Opusculis. Mediolani. (Milan.) 1774_1805. Oxon.B. Giornale dell' I. B. Istituto Lombardo di Scienze, Lettere, ed Arti, e Biblioteca Italiana ; compilata da varj dotti nazionali e stranieri. Milano. 184156. B.M.; N.H.M.; Oxon.B.; B.S. See Mil. I. Lomb. G. Giornale della Societad' Incorragiamento delle Scienze, etc. stabilita in Milano. Milano. 180865. B.M.; Camb.U. See Mil, a. I. Lomb. Memorie dell' I. B. Istituto Lombardo di Scienze, etc. Milano. 1843 B.M.; Camb.P.S.; Camb.U.; Dub.B.I.A.; Dub.T.C.; Edinb.B.S.t.; Math.S.t.; N.H.M.; Oxon.B.; B.A.S.t.; B.S.; U.C.L.t. See Mil. Mm. I. Lomb. Beale Istituto Lombardo di Scienze e Lettere. Bendiconti. Milano. 1864 B.M.; Camb.P.S.; Camb.U.; Dub.B.I.A.; Edinb.B.S.t.; Math.S.t.; N.H.M.; Oxon.B.t.; B.A.S.t.; B.S.; U.C.L.t. See Mil. I. Lomb. Mm. Memorie dell' I. B. Istituto del regno Lombardo- Veneto. Milano. 181938. B.M.; Camb.U.; N.H.M.; Oxon.B. i.; B.S. Arbeiten der Kurlandischen Gesellschaft fur Literatur und Kunst. Mitau. 184751. B.M.; Camb.U. Die Quatember; Zeitschrift fur naturwissenschaftl., geschichtl., philolog. und gemischte Gegenstande; von Trautvetter. Mitau. 182930. B.M. Sendungen der Kurlandischen Gesellschaft fur Literatur und Kunst. Mitau. 1840-47. B.M. The Mineralogical Magazine and Journal of the Mineralogical Society of Great Britain and Ireland. Truro, London. 1876 B.M.; Camb.U.; Chem.S.t.; Dub.N.L.I.; Geol.M.; N.H.M.; Oxon.B.(B-); P.O.; B.S. Academic des Sciences et Lettres de Montpellier. Memoires de la Section des Sciences. Montpellier. 1847- B.M.; Camb.U.; Dub.E.D.S.; Dub.B.I.A.; Edinb.E.S.; Linn.S.t.; N.H.M.; Oxon.B.; B.A.S.; E.S.; U.C.L.t. See Mntp. Ac. Sc. Mm., Mntp. Mm. Ac. and Mntp. Mm. Ac. Sect. Sc. Extraits des Proces-Verbaux des Seances de 1' Academic des Sciences et Lettres, etablie a Montpellier. Montpellier. 184754. N.H.M.t. See Mntp. Ac. Mm. Memoires de 1' Academic des Sciences et Lettres : Section de la MMecine. Montpellier. 1849 B.M.; Camb.U.; Dub.E.D.S.; Dub.B.I.A.; Edinb.B.S.t.; Oxon.B.; B.S. Eecueil des Bulletins publies par la Socie"te Libre des Sciences, etc., de Montpellier. Montpellier. 180314. B.M.; Camb.U.; Oxon.B.t. \ Memorie della Begia Accademia di Scienze, Lettere ed Arti di / Modena. Modena. 1833 B.M.; Dub.B.I.A.t.; Dub.T.C.t.; Edinb.E.S.; Math.S.i.; N.H.M.; Oxon.B.t.; U.C.L.t. ^Memorie di Matematica e di Fisica della Societa Italiana delle | Scienze. Modena. 1782 B.M.t.; Camb.P.S.; Camb.U.t.; Dub.B.I.A.; Edinb.B.S.t.; Linn.S.t.; Oxon.B.t.; B.A.S.t.; B.S.; U.C.L.t. See Bm. 8. It. Mm., Verona Mm. 8. It. and Verona 8. It. Mm. Mod. S. Nt. At. Moigno An. Cosmos... Moigno Cosmos Mondes(les) Mon. Sc Mosc. Bll. 8. Nt. Mosc. Obs. A. Mosc. S. Nt. Bll. Mosc. S. Sc. Bll. Mosc. Un . Mm Mosc. TTn. Mm. (Ps.-Mth Mth. Oz Mth. Misc Mth. Nt. B. Ung. Mth. Term. Ets Mth. Termt. ts Mth. Ts Mulhousc Bll. Mulhouse Bll. S. In. Mulhouse 8. In. Bll. Munch. Ab Munch. Ak. Ab Munch. Sb. List of Serial Publications Atti della Societa dei Naturalisti di Modena. Modena. 1883- Camb.U.; Dub.R.I.A.i.; N.H.M. .. Annuaire du Cosmos ; Moigno. Paris. 185970. B.M. ( Cosmos. Revue Encyclopedique Hebdomadaire des Progress des I Sciences: Moigno. Paris. ' 1852 B.M.; Camb.U.; Dub.T.C.; Edinb.R.S.i.; N.H.M.i.; Oxon.B.; P.O.i ; R.A.S.i.; R.S. Le Moniteur Scientifique du Chimiste et du Manufacturier ; Quesne- ville. Paris. 1857 B.M.; Chem.S.i.; Dub.R.C.S.i.; Oxon.B.; P.O.; R.A.S.i. Bulletin de la Societe Imperiale des Naturalistes. Moscou. 1829 B.M. ; Camb.P.S.; Camb.U.; Chem.S.i.; Dub.R.D.S.; Dub.R.I.A.; Edinb.R.S.i.; Glasg.U.i.;Linn.S.;N.H.M.;Oxon.B.i.; Oxon.R.; P.O.i.; R.A.S.i.; R.S. See Mosc. 8. Nt. Bll. Annales de 1'Observatoire de Moscou; Bredichin. Moscou. 1874 B.M.i.; Camb.U.; R.A.S.; R.S. .. See Mosc. Bll. 8. Nt. .. Bulletin of the Imperial Society of Lovers of Natural Science, Anthropology and Ethnography, in connection with the Imperial University of Moscow. [In Russian.] Moscow. 1865 B.M.i.; Edinb.R.S.i.; N.H.M.i. .. Scientific Memoirs of the Imperial University of Moscow. [In Russian.] Moscow. B.M.i.; Chem.S.i.; N.H.M.i. .) Scientific Memoirs of the Imperial University of Moscow. Section Phys.-Math. Moscow. 188096. Chem.S. .. Mathematische Annalen ; Clebsch. Leipzig. 1869 B.M.; Camb.P.S.; Camb.U.; Dub.N.L.I.i.; Dub.R.C.S.i.; Dub.R.D.S.i.; Dub.T.C.i.; Edinb.U.; Glasg.U.; Math.S.; Oxon.R.; R.S.; U.C.L. .. The Mathematical Gazette. London. 1894 B.M.; Camb.U.; Math.S.; U.C.L.i. .. The Mathematical Miscellany; Gill. New York. 183639. U.C.L. .. Mathematische und naturwissenschaftliche Berichte aus Ungarn. Berlin. 1882 Camb.P.S.; Chem.S.; Edinb.R.S.; R.A.S.i.; R.S. (Mathematikai e"s termeszettudomanyi Ertesito. Kiadja a Magyar Tudomanyos Akademia. [Mathematical and Natural Science Report, published by the Hungarian Academy of Science.] Budapest. 1883 B.M.i.; Edinb.R.S.; N.H.M.; R.S. .. Mathematisk Tidsskrift. Kjpbenhavn. 1859 B.M.; Camb.U.; Oxon.B.; R.S.i. .. (Bulletin de la Societe Industrielle de Mulhouse. Mulhouse. .. J1828 B.M.i.; Camb.U.i.; Chem.S.i.; Dub.R.C.S.i.; Dub.T.C.i.; .. ( Glasg.P.S.i.; Glasg.U.i.; Oxon.B.i.; P.O. .. i Abhandlungen der mathematisch-physikalischen Classe der Konigl. .. { Bayerischen Akademie der Wissengchaften. Miinchen. 1832 B.M. ; Camb.P.S.; Camb.U.; Chem.S.i.; Dub.T.C.; Edinb.R.S.; Edinb.U.; Glasg.U.; Linn.S.; Oxon.B.; Oxon.R.; P.O.; R.A.S.i.; R.S. .. Sitzungsberichte der Mathematisch-Physikalischen Classe der K. B. Akademie der Wissenschaften zu Miinchen. Miinchen. 1871 B.M. ; Camb.P.S.; Camb.U.; Chem.S.; Dub.T.C.; Edinb.R.S.; Glasg.U.i.; Linn.S.; Oxon.B.; Oxon.R.; P.O.; R.A.S.; R.S.; U.C.L.i. . . Denkschrif ten der Konigl. Bayerischen Akademie der Wissenschaften zu Miinchen. Miinchen u. Salzbach. 180824. B.M.; Camb.P.S.; Camb.U.; N.H.M.; Oxon.R.; P.O.; R.S. .. Sitzungsberichte der Konigl. Bayerischen Akademie der Wissen- schaften zu Miinchen. Miinchen. 186070. B.M.; Camb.P.S.; Camb.U.; Chem.S.; Dub.R.D.S.i.; Dub.R.I.A.i. ; Dub.T.C.i. ; Edinb.R.S. ; Linn.S. ; N.H.M. ; Oxon.B.; P.O.i.; R.A.S.; R.S. List of Serial Publications Nap. Ac. Font. At.. Nap. Ac. Sc. Mm. . Nap. At. Ac. .. Nap. At. Ac. Sc. Nap. At. I. Inc. N. A. Mtn. Nouvelles Annales de Mathematiques. Paris. 1842 B.M.; Camb.U.; Dub.T.C.; Edinb.U.; Glasg.U.; Math.S.i.; Oxon.B.(R.); R.S.; U.C.L.i. Nancy Mm. S. Sc. Memoires de la Societe [Royale] des Sciences, Lettres, et Arts de Nancy. Nancy. 1833 B.M.; Camb.U. i.; N.H.M.i.; Oxon.B.; R.S.i. Nancy a. Sc. Bll Bulletin de la Societe des Sciences de Nancy. Nancy, Paris. 1873 B.M.; N.H.M.; R.S. N. Antol. Sc Nuova Antologia di Scienze, Lettere ed Arti. Firenze e Roma. 1866 B.M.; Dub.N.L.I.i.; N.H.M. Nap. Ac. At Atti della Reale Accademia delle Scienze e Belle Lettere; Sezione della Societa R. Barbonica. Napoli. 181951. B.M. ; Camb.U.; Dub.R.D.S.; N.H.M.; Oxon.B.; R.A.S.i.; R.S. Atti della R. Accademia delle Scienze Fisiche e Matematiche. Napoli. 186382; 1888 B.M. ; Camb.U.; Dub.R.I.A.; Edinb.R.S.; Glasg.U.i.; Linn.S.i.; Math.S.i.; N.H.M.; Oxon.B.i.; Oxon.R.; R.A.S.i.; R.S. See Nap. At. Ac. and Nap. At. Ac. Sc. Atti dell' Accademia Pontaniana di Napoli. Napoli. 1832 B.M.; Camb.U.; Dub.R.D.S.i.; N.H.M.; R.S.i.; U.C.L.i. Memorie della R. Accademia delle Scienze, etc. Napoli. 185257. B.M.; Camb.U.; Dub.R.D.S.; Edinb.R.S.; Linn.S. ; N.H.M.; Oxon.B.; Oxon.R.; R.A.S.; R.S. See Nap. Mm. Ac. Sc. I See Nap. Ac. At. Atti del Real Istituto d' Incorraggiamento alle Scienze Natural! di Napoli. Napoli. 1811 B.M.; Camb.U.; Edinb.R.S.i.; N.H.M.; Oxon.B.; P.O.; R.S.i. See Nap. I. Inc. At. Nap. At. S. Font Atti della Societa Pontaniana di Napoli. Napoli. 181047. B.M.; Camb.U.; N.H.M.; R.S. See Nap. S. Font. At. Nap. I. Inc. At See Nap. At. I. Inc. Nap. Mm. Ac. Sc See Nap. Ac. Sc. Mm. Nap. Ms Museo di Letteratura e Filosofia ; Gatti . Napoli . 184262. B.M.; Oxon.B. Nap. Rd Rendiconto dell' Accademia delle Scienze Fisiche e Matematiche. Napoli. 1862 Camb.U.; Dub.R.I.A.; Edinb.R.S.; Glasg.U.i.; Linn.S.i.; Math.S.; N.H.M.; Oxon.R.i.; P.O.; R.A.S.; R.S.; U.C.L.i. Nap. S. Font. At See Nap. At. S. Pont. N. Arch. Wisk Nieuw Archief voor Wiskunde . Amsterdam . 1875 Camb.P.S.i.; Edinb.R.S.i.; Math.S. N. A. Sc. Nt Nuovi Annali delle Scienze natural!; Alessandrini, Bertoloni, Gherardi, e Ranzani. Bologna. 183854. Camb.U.; N.H.M.; Oxon.B.i.; R.S. See Bologna N. A. N. Cim II Nuovo Cimento, Giornale di Fisica, Chimica e Storia Naturale. Pisa. 1855 B.M.; Camb.P.S.; Camb.U.; Chem.S.i.; Edinb.R.S.i.; N.H.M.; Oxon.R.i.; P.O.i.; R.S. Nebr. Un. Stud University Studies. Published by the University of Nebraska. Lincoln, Nebraska. 1888 B.M.; Camb.P.S.; Camb.U.; Dub.R.I.A.; Edinb.R.S.; Oxon.B.; R.S. N. Eng. I. Mn. E. T Transactions of the North of England Institute of Mining Engineers. Newcastle-upon-Tyne. 1852 B.M.; Camb.U.; Edinb.R.S.i.; Oxon.B.i.; P.O.i.; R.S. Neucn. Bll (Bulletin de la Societe des Sciences Naturelles de Neuchatel. Neuch. S. Sc. Bll \ Neuchatel. 1844 B.M.i. ; Camb.U.; Dub.R.I.A.; Edinb.R.S.i.; N.H.M.; Oxon.B.i.; R.A.S.i.; R.S.i. NH. Rv The Natural History Review and Quarterly Journal of Science. London and Dublin. List of Serial Publications 185460. B.M.; Camb.U.; Dub.K.D.S.; Dub.T.C.; Glasg.P.S.; Linn.S.; N.H.M.; Oxon.R.; P.O.; U.C.L.i. The Natural History Review; a Quarterly Journal of Biological Science. London. Nicholson J. Niort Mm. de r Athenee N. Jb. Mn N. Mg. Ntvd Vd. Skr. .. N. KB. 8. Nt. Mm. (Mth. W. 8. W. Ph. 8. T H. 8. W. B. 8. J N. 8. W. R. 8. T. . W. Ts. Mth Niirnb. Nt. Gs. Ab Nv. Archt. T N.-Vorp. Mt. IT. Y. Am. Mth. 8. Bll. W. V. Am. Mth. 8. T. IT. T. Mth. 8. Bll. . 1861-65. B.M.; Camb.P.S.; Camb.U.; Dub.R.D.S. Edinb.R.S.; Glasg.P.S.; Glasg.U.; Linn.S.; N.H.M. Oxon.R.; P.O.; R.S. Journal of Natural Philosophy, Chemistry, and the Arts Dub.T.C. ; Oxon.B.i.; Nicholson. London. 17971813. B.M. ; Camb.U.; Chem.S.i. ; Dub.R.I.A.i. ; Dub. T.C.i.; Edinb.R.S.; Edinb.U.; Glasg.P.S.i.; Glasg.U.; N.H.M.; Oxon.B.; Oxon.R.; P.O.; R.A.S.i.; R.S.; U.C.L. Memoires de 1'Athenee de Niort, Societ4 Libre des Sciences et des Arts du departement des Deux-Sevres. Niort. 1808(9). B.M.; Oxon.B.; R.S.i. Neues Jahrbuch fur Mineralogie, Geologic und Palaontologie. Stuttgart. 1833 B.M.; Camb.U.; Chem.S.i.; Dub.N.L.I.i.; Dub.R.D.S.i.; Geol.M.; N.H.M. ; Oxon.R.; R.S. Nyt Magazin for Naturvidenskaberne. Christiania. 1838 Camb.U.i.; Edinb.R.S. i.; Linn.S.i.; N.H.M.; R.S. Det Kongelige Norske Videnskabersselskabs Skrifter i det 19de Aarhundrede. Kjabenhavn og Trondhjem. 1817 B.M.; Camb.U.t.; Dub.T.C.i.; Edinb.R.S.i.; N.H.M.; R.S. See Thrond. Skr. Memoirs of the Mathematical Section of the New Russian Society of Naturalists. [In Russian.] Odessa. 1878- B.M.i.; Dub.R.I.A.; Math.S.i.; R.S.i. Transactions of the Philosophical Society of New South Wales. Sydney. 1866. Camb.P.S.; Camb.U.; Chem.S. ; Edinb.R.S.i.; Glasg.U.; Linn.S.; Oxon.B.; R.A.S.; R.S. Journal and Proceedings of the Royal Society of New South Wales. Sydney. 1876 B.M. ; Camb.P.S. ; Camb.U. ; Chem.S.i. ; Dub.T.C. ; Edinb.R.S.i.; Geol.M.i.; Glasg.P.S. i.; Glasg.U.i.; Linn.S.i.; N.H.M.; Oxon.B.; Oxon.R.i.; P.O.i.; R.A.S.; R.S. Transactions of the Royal Society of New South Wales. Sydney. 186775. B.M.; Camb.P.S.; Camb.U.; Chem.S.i.; Dub.R.D.S.; Dub.R.I.A. ; Dub.T.C. ; Edinb.R.S.i. ; Glasg.U.i. ; N.H.M. ; R.A.S.; R.S. Nature: a weekly illustrated Journal of Science. London. 1870 B.M.; Camb.P.S.; Camb.U.; Chem.S.; Dub.N.L.L; Dub. R.C.S. ; Dub.R.D.S. ; Dub.R.I.A. ; Dub.T.C. ; Edinb.R.S. ; Edinb.U.; Geol.M.; Glasg.P.S.; Glasg.U.i.; Linn.S.; Oxon.B.; Oxon.R.; P.O.; R.A.S.; R.S.; U.C.L. Nyt Tidsskrift for Mathematik. Kjabenhavn. 1890 B.M.; Math.S.i. Abhandlungen der Naturhistorischen Gesellschaft zu Niirnberg. Niirnberg. 1852 B.M.i.; Camb.U.; Dub.R.I.A.; N.H.M.; R.S.i. Transactions of the Institution of Naval Architects. London. 1860 B.M. ; Camb.U.; Dub.R.I.A.; Edinb.U.; P.O.; R.S. ; U.C.L.i. Mittheilungen aus dem Naturwissenschaftlichen Vereine von Neu- Vorpommern und Rugen. Berlin. 1869 B.M.; Camb.U.; Dub.R.D.S.; Dub.R.I.A.; N.H.M. Bulletin of the American Mathematical Society. New York. 1895 B.M. ; Camb.P.S.; Camb.U.; Dub.T.C.; Edinb.R.S.; Edinb.U.; Glasg.P.S.; Glasg.U.; Math.S.; Oxon.B.; Oxon.R.; R.S.i. Transactions of the American Mathematical Society. Lancaster, Pa. and New York. 1900 Camb.P.S.; Camb.U.; Dub.R.I.A.; Dub.T.C.; Edinb.R.S.; Edinb.U.; Glasg.U.; Math.S.; Oxon.B.; Oxon.R.; R.S. Bulletin of the New York Mathematical Society. New York. 189294. B.M.; Camb.P.S.; Camb.U.; Edinb.R.S.; Glasg.P.S.; Math.S.; Oxon.B.; Oxon.R.; R.A.S. xxxvi List of Serial Publications N. Z.I.T Obs Oestr. Z. Brgw Oken Isis OPUBC. Mt. Fis Orv.-Termt. Ets. (Termt. Szak) Fadova Ac. At. e Mm. . . . Fadova Mm. Ac Fadova N. Sag Fadova Rv. Period. . . Palermo Ac. At Palermo At Palermo Cir. Mt. Rd. Palermo G. Sc. Nt. Falomba Rac. Far. A. Cons. . . Far. Ac. Sc. Mm. Far. A. EC. Norm. Far. An. Met Far. An. S. Met Far. A. Obs Transactions and Proceedings of the New Zealand Institute. Wellington. 1868 B.M.; Camb.P.S.i. ; Camb.U. ; Dub.K.D.S.; Dub.K.I.A. ; Edinb.E.S.; Edinb.U.; Geol.M.i.; Glasg.P.S.i.; Linn.S.; N.H.M.; Oxon.B.; Oxon.E.; P.O.; B.S.; U.C.L.i. The Observatory. A monthly Eeview of Astronomy. London. 1878 Camb.P.S.; Camb.U.; Dub.T.C.i.; Edinb.E.S.; Oxon.B.; P.O.; E.A.S. Oesterreichische Zeitschrift fiir Berg- und Hiittenwesen ; von Otto Freiherrn von Hingenau. Wien. 1853 B.M.; P.O. Isis, oder Encyclopadische Zeitung; Oken. Jena. 181748. B.M.i. ; Camb.U. ; Edinb.U. ; Linn.S.i. ; N.H.M. ; Oxon.B. (E.); B.S. Opuscoli matematici e fisici di diversi Autori. Milano. 183234. E.S. Orvos-Termeszettudomanyi Ertesit6 a Kolozsvari Orvos-Termeszettu- domanyi Tarsulat es az Erdelyi Muzeum-Egylet Termeszettu- domanyi Szakosztalyanak [Medical and Natural History Proceedings of the Sections of the Klausenburg Medical and Natural History Society and of the Natural History Section of the Museum Association of Transylvania.] Kolozsvar [Klausenburg]. 1879 N.H.M.; B.S.i. Atti e Memorie della E. Accademia di Scienze, Lettere ed Arti in Padova. Nuova serie. Padova. 1885 Edinb.E.S.; N.H.M. Memorie dell' Accademia di Scienze, Lettere, ed Arti di Padova. Padova. 1809. B.M.; Camb.U.; N.H.M.; Oxon.B.; E.S. Nuovi Saggi dell' Accademia di Scienze, Lettere, ed Arti di Padova. Padova. 181783. B.M.i.; Camb.U.i.; Dub.E.I.A.i.; Dub.T.C.i.; Edinb. E.S.; N.H.M.; Oxon.B.i.; E.S.i. Eevista Periodica dei Lavori della I. E. Accademia di Scienze, Lettere, ed Arti di Padova. Padova. 1851 B.M.; Edinb.B.S.i.; N.H.M.; E.S.i. ( Atti dell' Accademia di Scienze, Lettere ed Arti di Palermo. Palermo. -11845 B.M.; Camb.U.i.; Dub.E.I.A.; Dub.T.C.; N.H.M.; Oxon. ( B.i.; E.A.S.i.; E.S. Eendiconti del Circolo Matematico di Palermo. Palermo. 1887 B.M. ; Camb.P.S.; Camb.U.; Dub.E.I.A.; Dub.T.C.; Math.S.; E.S. Giornale di Scienze natural! ed economiche, pubblicato per cura del Consiglio di Perfezionamento annesso al E. Istituto Tecnico di Palermo. Palermo. 1865 B.M.; Camb.U.; Dub.E.D.S.i.; E.S. Eaccolta di Lettere, etc. intorno alia Fisica ed alle Mathematiche ; Palomba. Eoma. 184548. B.M.i. Annales du Conservatoire des Arts et Metiers. Paris. 1861 B.M.; Camb.U.; Glasg.P.S.i.; Oxon.B.; P.O.; E.S. See A. Cons. Arts et Met. M^moires de 1'Academie des Sciences de 1'Institut de France. Paris. 1816 B.M.; Camb.U.; Dub.N.L.I.i.; Dub.E.D.S.i.; Dub.T.C.i.; Edinb.E.S. i. ; Edinb.U.; N.H.M.; Oxon.B.; Oxon.E.; P.O.i.; E.A.S.i.; E.S.; U.C.L. See Far. Mm. Ac. Sc. Annales scientifiques de 1'Ecole Normale Superieure. Paris. 1864 B.M.; Camb.P.S.i.; Camb.U.; Dub.N.L.I.i.; Dub.E.C.S.i.; Dub.E.D.S.i.; Edinb.E.S.i. ; Edinb.U.i. ; Glasg.U.i. ; Oxon.B.; E.S. See Far. EC. Norm. A. ^Annuaire de la Societe Meteorologique de France. Paris. {1849 B.M.; Camb.U.i.; Dub.T.C.i.; Edinb.E.S.i.; E.S.i. Annales de 1'Observatoire de Paris; memoires publiees par U. J. Le Verrier. Paris. xxxvii List of Serial Publications Far. A. Pon. Chauss. Par. BU. 8. Encour. Par. Bll. 8. Gg . Par. EC. Norm. A. Par. EC. PoL Cor. Par. EC. PoL J. Par. Og. 8. Bll Par. Ing. Civ. Mm. Par. J. EC. Pol. .. Par. Mm. Ac. Be. Par. Mm. de I'l. .. Par. Mm. Ing. Civ.. Par. Mm. Sav. Etr. Par. Mm. 8. Sav. Par. Obs. A. Par. Ob. A. (Mm.) Par. Poids et TO.es. PV. Par. 8. Amat . Tr. Par. 8. Ap. 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Bll. Far. 8. Fnlm. PV Far. T. Nauk Sc. Fam. . . Par. Tr. 8. Amat Perpignan. Bll. 8. Ag. Fyr. Orient Peterm. Mt. ... Phil. Trans Ph. Mg Phm. J Pisa A. Scuola Norm. Pisa A. Tin. Tosc Pisa A. Un. Tosc. Sc. Pisa N. O Pisa 8. Tosc. At. (PV.) .. Pistoja At. Ac Fogg. A -PoHgrafo List of Serial Publications Seances des Ecoles Normales. Paris. 180001. R.S.; U.C.L. See Par. Bll. 8. Gg. and Par. Gg. 8. Bll. Bulletin de la Societe Mathematique de France. Paris. 1873 B.M. ; Camb.P.S. ; Camb.U. ; Edinb.E.S. ; Math.S. ; Oxon.E.; E.A.S.; B.S. See Far. Bll. 8. Fhlxn. Memoires publics par la Soci^te" Philomathique a 1'occasion du Centenaire de sa Fondation. Paris. 1888. B.M.; Edinb.E.S.; N.H.M.; E.A.S.; E.S. Nouveau Bulletin des Sciences de la Societe Philomathique de Paris. Paris. 18071813; 182526; 183233. B.M.i.; Camb.U.; Dub.T.C. ; N.H.M.; P.O.i.; E.S.; U.C.L. Extraits des Proces-Verbaux des Seances de la Societe Philctaathique. Paris. 183663. N.H.M.; E.S. 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B.M.; Camb.U.; N.H.M.; Oxon.B.; E.S. Annalen der Physik und Chemie ; Poggendorff. Leipzig. 1824 B.M. ; Camb.P.S.i.; Camb.U.; Chem.S.; Dub.E.I.A.; Dub.T.C.; Edinb.E.S.; Edinb.U.; Glasg.P.S.i.; Glasg.U. i.; N.H.M.; Oxon.B.(E-); P.O.; E.S.; U.C.L.i. See A. Ps. C. II Poligrafo: Giornale di Scienze, Lettere, ed Arti; Orti. Verona. 183045. B.M.; Oxon.B. xxxix Pop. As. Frace Mt.-Flr PragAb Prag Cesk6 Ak. Fr. Jos. Prag Ceske Ak. Fr. Jos. Prag Fr. Jos. Ac. sc. mi. (JlftTi. Nt.) Frag Jb. Bohm. Ms Prag Sb Presse Sc Ps. Kv. .. Ps. Z. List of Serial Publications Poly tech nische Mittheilungen, unter Mitwirkung von Professoren hoherer technischer Lehranstalten. Tubingen. 184446. B.M.; B.S. Popular Astronomy. Northfield, Minnesota. 1894 B.M.; Glasg.U.; K.A.S. Prace Matematyczno-Fizyczne. Warsaw. 1888 Camb.P.S.; Camb.U.; Chem.S.i.; Dub.B.I.A.i.; Math.S.; E.S.i. Abhandlungen der k. Bohmischen Gesellschaft der Wissenschaften. Prag. 180492. B.M.i.; Camb.P.S.; Camb.U.i.; Dub.E.I.A.i.; Edinb. 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Sitzungsberichte der k. Bohmischen Gesellschaft der Wissenschaften in Prag. Prag. 1859 Camb.P.S.; Camb.U.i.; Dub.E.D.S.; Dub.E.I.A.; Edinb. E.S.i.; Linn.S.i.; N.H.M.; E.S.; U.C.L.i. Presse Scientifique des Deux Mondes. Paris. 186066. B.M.; E.S.i. The Physical Be view. New York, London, Berlin. 1894 B.M.; Camb.P.S.; Camb.U.; Dub.E.C.S.; Edinb.E.S.; Edinb.U.i.; Oxon.E.; P.O.; B.S. Physikalische Zeitschrift. Leipzig. 1899 Camb.P.S.; Edinb. U.; Oxon.E.; E.S. Queb. T Queb. T. Lt. 8 Quetelet Cor. BXth. . QJ. QJ. Be. { Transactions of the Literary and Historical Society of Quebec. Quebec. 1829 B.M.i.; Dub.E.I.A.; Edinb.B.S.i.; Glasg.P.S.i.; Linn.S.i.; [ N.H.M.; B.S.i. Correspondance Mathematique et Physique; publi^e par MM. Gamier et Quetelet. Gand et Bruxelles. 182539. B.M.; Camb.U.; E.A.S.i.; E.S.; U.C.L. The Quarterly Journal of Pure and Applied Mathematics. London. 1855 B.M.; Camb.P.S.; Camb.U.; Dub.N.L.L; Dub.T.C. ; Edinb.B.S.i.; Edinb.U. ; Glasg.P.S.i.; Glasg.U.; Math.S.i. ; Oxon.B.; Oxon.B.; P.O.; B.A.S.i.; E.S.; U.C.L. The Journal of Science and the Arts ; edited at the Eoyal Institution of Great Britain. London. Continued as : Quarterly Journal of Science, Literature and Arts. London. 181630. B.M.; Camb.U.; Chem.S.; Dub.T.C.; Edinb.E.S.; Glasg.U.; Oxon.B.; Oxon.E.; E.S.; U.C.L. Railroad & Bng. J The Bailroad and Engineering Journal. New York. 188792. B.M.; P.O. Ranuzzi An. Og Annuario geografico Italiano ; Banuz z i . Bologna . 184445. B.M.; Camb.U. xl List of Serial Publications Bee. Mth. (Moscou) ... R, E. Pp Rheinl. Westphal. Sb. Riga Cor.-Bl. B. I. J Bio Arch. Falestr B. I. P. .. Bm. At Bm. At. IV. Line. Im. At. B. Ac. Bm. Cor. Sc Bm. Mm. Osserv. Bm. N. Line. At. . . . Bm. IT. Line, Mm. Bm. B. Ac. Line. At. ... Bm. B. Ac. Line. Mm. Bm. B. Ac. Line. Bd. Bm. B. Ac. Line. T Bm. S. Cor Bm. S. It. Mm. . . Rochester (W. Y.) Ac. Sc. P.... Bot. N. Vh. Kecueil mathematique. Publie par la Societe Mathematique de Moscou. [In Eussian.] Moscou. 1866 R.S. Papers on subjects connected with the duties of the Corps of Royal Engineers. London. 1844 Camb.U.; Geol.M.i.; Glasg.U.i.; P.O.i. Sitzungsbericht des Naturhistorischen Vereins der Preussischen Rheinlande und Westphalens. Bonn. 1844 Dub.R.D.S.i. ; Dub.R.I.A.i. ; Edinb.R.S.; Linn.S.i.; N.H.M.; Oxon.R.; R.S.i. Correspondenzblatt des Naturforscher- Vereins zu Riga. Riga. 1846 B.M.; Dub.R.I.A.i.; N.H.M.; R.S.i. Journal of the Royal Institution of Great Britain. London. 180203; 183031. Camb.U.i.; Chem.S.i.; Dub.R.D.S.; Edinb. R.S.i.; Glasg.P.S.i.; Linn.S.i. ; N.H.M.i. ; Oxon.R.; P.O.i.; R.A.S.i.; R.S.; U.C.L.i. Archives da Palestra Scientifica do Rio de Janeiro. Rio de Janeiro. 1858. N.H.M.; R.S. Notice of the Proceedings at the meetings of the members of the Royal Institution, with Abstracts of the Discourses delivered at the Evening Meetings. London. 1851 B.M. ; Camb.U. ; Chem.S. ; Dub.R.I.A. ; Dub.T.C. ; Edinb.R.S.; Geol.M.; Glasg.P.S.; Glasg.U.; Linn.S.; N.H.M.; Oxon.R.; P.O.; R.A.S.; R.S.; U.C.L. (Atti dell' Accademia Pontificia dei Nuovi Lincei. Roma. - 1 1847 B.M. ; Dub.R.I.A. ; Edinb.R.S. ; N.H.M. ; Oxon.B.i. ; ( R.A.S.i.; R.S. See Bm. N. Line. At. Atti della Reale Accademia dei Lincei. Roma. 187083. B.M.; Camb.P.S.; Camb.U.i.; Chem.S.i.; Dub.R.D.S.; Dub.R.I.A.; Linn.S.; Math.S.; N.H.M.; Oxon.B. ; Oxon.R.i.; R.A.S.i.; R.S.; U.C.L.i. See Bm. B. Ac. Line. At. Corrispondenza Scientifica in Roma per le avanzamento delle Scienze, etc. Roma. 1848 See Bm. S. Cor. Memorie dell' Osservatorio dell' Universita Gregoriana del Collegio Romano, diretto dai PP. della Compagnia di Gesu. Roma. 185063. B.M.; R.A.S.; R.S. See Bm. At. ".,,,/ Bm. At. N. Line. Memorie della Pontificia Accademia dei Nuovi Lincei. Roma. 1887 Dnb.R.D.S.; Dub.R.I.A.; Edinb.R.S.; N.H.M.; R.S. See Bm. -*t. B. Ac. Atti della R. Accademia dei Lincei. Memorie della Classe di Scienze fisiche, matematiche e naturali. Roma. 1877 B.M.i.; Camb.P.S.; Camb.U.; Chem.S.; Dub.R.I.A.i. ; Edinb.R.S.; Glasg.P.S.i.; Glasg.U.i.; Linn.S.; Math.S.; N.H.M.; Oxon.B.; Oxon.R.; P.O.i.; R.A.S.; R.S.; U.C.L. Atti della R. Accademia dei Lincei. Rendiconti. Roma. 1885 B.M.; Camb.P.S.; Camb.U.; Chem.S.; Dub.T.C.; Edinb. R.S.; Glasg.U.i.; Linn.S.; Math.S.; N.H.M.; Oxon.B.; Oxon.R.; R.A.S.; R.S.; U.C.L. Atti della R. Accademia dei Lincei. Transunti. Roma. 187784. B.M. ; Camb.P.S.; Camb.U.; Chem.S.; Dub.R.I.A.; Edinb.R.S.; Glasg.P.S.i.; Linn.S.; Math.S.; N.H.M.; Oxon.B.- Oxon.R.; R.A.S.; R.S.; U.C.L. See Bm. Cor. Sc. Memorie di Matematica e di Fisica, della Societa Italiana delle Scienze. Napoli e Roma. 1782 B.M.i.; Camb.P.S.; Camb.U.i.; Dub.R.I.A.; Edinb.R.S.i.- Linn.S.i.; Oxon.B.i.; R.A.S.i.; R.S.; U.C.L.i. See Mod. Mm. S. It., Verona Mm. S. It. and Verona 8. It. Mm. Proceedings of the Rochester Academy of Sciences. 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D Denkschrif ten der Bussischen Geographischen Gesellschaf t zu St Petersburg. Weimar. 1849. B.M.; Camb.U.; B.S. R S P Proceedings of the Boyal Society of London. London. 1832 B.M.; Camb.P.S.; Camb.U.; Chem.S.; Dub.N.L.I.i. ; Dub.B.C.S.; Dub.B.D.S.; Dub.B.I.A.; Dub.T.C.; Edinb.B.S.; Edinb.U.; Geol.M.; Glasg.P.S.; Glasg.U.i.; Linn.S.i.; Math.S.i.; N.H.M.; Oxon.B.i.; Oxon.B.; P.O.; B.A.S.; B.S.; U.C.L. Rs. Ps.-C. S. J Journal of the Bussian Physico-Chemical Society of the Imperial University of St Petersburg. [In Bussian.] St Petersburg. 1869 Camb.P.S.i.; Chem.S.; Edinb.B.S.i.; N.H.M. R. 8. Yearbook Yearbook of the Boyal Society of London. (Biography. 1900.) Rv. Artl. Bevue d'Artillerie. Paris, Nancy. 1872 B.M.; P.O. Rv. Brazil. Bevista Brazileira, Jornal de Sciencias, Lettras, e Artes; Oliveira. Bio de Janeiro. 185761. B.M.; N.H.M.; B.S.i. Rv. Cours. Se Bevue des Cours Scientifiques de la France et de 1'Etranger; Eug. Yung et Em. Alglave. Paris. 1863-70. B.M.; Edinb.B.S.i.; Edinb.U.; N.H.M. ; Oxon.E. ; P.O.; B.S. Rv. Ligure Bivista Ligure, giornale di Lettere, Scienze, etc. Geneva. 1843. B.M.; N.H.M.; Oxon.B. Rv. Mar IBevue maritime et coloniale. Paris. Rv. Mar. et Col J1861 B.M.; Oxon.B.; Oxon.B.; P.O. Rv. Mt. Bivista di Matematica. Torino. 189195. Camb.U.; Oxon.B.; E.S. Rv. Mtb Bevue de Mathematiques. (Bivista di Matematica.) Turin. 1896 Camb.U.; Oxon.B.; Oxon.B.; B.S. Rv. Be La Revue Scientifique de la France et de 1'Etranger. Paris. 1871- B.M.; Camb.U.; Edinb.B.S.; Edinb.U.; N.H.M.; Oxon.B,; P.O.; B.A.S.i.; B.S. Rv. Sc.-lnd Bivista Scientifico-Industriale delle principali scoperte ed invenzioni fatte nelle scienze e nelle Industrie. Firenze. 1869- P.O. Rv. Un. Mines Bevue Universelle des Mines , de la Metallurgie , etc . Lie"ge & Paris . 1857 B.M.; Camb.U.; Dub.B.I.A.i.; Glasg.P.S.i.; Glasg.U.i.; N.H.M.; P.O. See Cuyper Rv. Tin. The Transactions of the South African Philosophical Society. Cape Town. 1878 B.M.; Camb.P.S.; Camb.U.i.; Chem.S.; Edinb.B.S.; Edinb.U.; Glasg.P.S.; Linn.S.i.; N.H.M.; Oxon.B.; Oxon.R.; P.O.; B.A.S.; B.S. s. Afr. QJ. The South African Quarterly Journal; edited at the African Insti- tution. Cape Town. 183035. B.M.i.; Edinb.B.S.i.; N.H.M. Santiago de Chile Un. A. Anales de la Universidad de Chile. Santiago de Chile. 1843 B.M.i.; Dub.T.C.; N.H.M.i.; Oxon.B.i. See Chile A. Un. SartheS. Bll Bulletin de la Societe d' Agriculture, etc., de la Sarthe. Le Mans. 1833 B.S.i. xlii List of Serial Publications S. Aust. R. a. T Transactions and Proceedings and Eeport of the Royal Society of South Australia. Adelaide. 1877 Camb.P.S.i.; Camb.U.i.; Chem.S.i.; Dub.E.I.A.i.; Edinb. E.S.; Linn.S.i.; N.H.M.; P.O.; E.A.S.; B.S.i. Sav. Ac. Mm ( Memoires de la Societe Academique de Savoie. Chambery. Sav. Mm. Ac /1825 Camb.U.; Dub.E.I.A.; Dub.T.C.; N.H.M.; Oxon.B.; E.S.i. See Chambery Ittm. Ac. Sav. Sc. Abs Science Abstracts. Physics and Electrical Engineering. London. 1898 Camb.P.S.; Camb.U.; Chem.S.; Edinb.E.S.i.; Edinb.U.; Glasg.P.S.; Oxon.E.; P.O.; E.A.S.i.; E.S.; U.C.L. Sch. Gs. KT. D Neue Denkschriften der allgemeinen Schweizerischen Gesellschaft fiir die gesammten Naturwissenschaften. Neuchatel, Zurich, etc. 1837 B.M.; Camb.P.S.; Camb.U.; Dub.E.D.S.i.; Dub.B.I.A.i.; Edinb.E.S.; Linn.S.i.; N.H.M.; Oxon.B.; E.S. See Zur. N. D. Sch. Gs. Sch. Gs. Vh Verhandlungen der Schweizerischen Gesellschaft fiir die gesammten Naturwissenschaften. Aarau, etc. 1823 B.M.i.; Edinb.E.S.i.; Linn.S.i.; N.H.M.; E.S. See Sch. Nf. Gs. Vb. Schlomilch Z Zeitschrif t fiir Mathematik und Physik ; Schlomilch. Leipzig. 1856 B.M.; Camb.U.; Dub.N.L.I.i.; Dub.E.D.S.i.; Dub.B.I.A.i.; Dub.T.C.i.; Edinb.U.; Glasg.U.i.; Math.S.i.; Oxon.B. (E.); E.S.; U.C.L.t. See Z. Mth. Ps. Sch. Nf. Gs. Vh See Sch. Gs. Vh. Schumacher As. Ab Astronomische Abhandlungen ; Schumacher. Altona. 182325. B.M.; Camb.U.; Dub.E.D.S.; Edinb.E.S.; E.A.S.; E.S. Schweigger J Journal fiir Chemie und Physik ; Schweigger. Niirnberg. 181129. Edinb.E.S.; N.H.M.; Oxon.E.; P.O. Continued as: Neues Jahrbuch der Chemie und Physik ; Schweigger-Seidel. Berlin. 183133. B.M.; Chem.S.; Edinb.E.S.; N.H.M.; Oxon.E.; P.O.; E.S. Science. Cambridge, Mass. , and New York. 1883 B.M. ; Dub.N.L.I.i.; Dub.E.C.S.i. ; Edinb.E.S.; Edinb. U.i.; Glasg.P.S.i.; N.H.M.; Oxon.E.i.; P.O.; E.A.S.i. Transactions of the Eoyal Scottish Society of Arts. Edinburgh. 1841 B.M.t.; Camb.U.; Dub.E.D.S.; Edinb.E.S.; Edinb.U.; Glasg.P.S.; Glasg.U.; P.O.; E.S. See Edinb. Sc. S. Arts T. and Edinb. T. Sc. S. Arts. Sid. Mess The Sidereal Messenger. Northfield, Minn. 188391. B.M.; E.A.S. Siena At. Ac Atti deL' Accademia delle Scienze di Siena detta de' Fisio-critici. Siena. 1761 B.M.; Camb.U.i.; Dub.E.I.A.i.; Dub.T.C.i.; N.H.M.t.; Oxon.B.; E.S.i. The American Journal of Science and Arts ; Silliman. New Haven. 1818 B.M.; Camb.P.S.i.; Camb.U.; Chem.S.i.; Dub.N.L.I.i.; Dub.E.C.S.i.; Dub.T.C.i.; Edinb.E.S.; Edinb.U.; Geol.M.; Glasg.P.S.; Glasg.U.i.; N.H.M.; Oxon.B.; Oxon.E.; P.O.; E.A.S.i.; E.S. See Am. J. Sc. Skandia. Tidskrift for Vetenskap och Konst; utgifven af Svenska Litteratur-Foreningen. Upsala. 183337. B.M. IForhandlingar vid det af Skandinaviska Naturforskare och Lakare hallna Mote Gotheborg, etc. 1839 Oxon.B.i.; E.S.i. Forhandlingerne ved de Skandinaviske Naturforskeres...Mfde.... Gotheborg, etc. 1839 B.M.; N.H.M.; E.S.i. Smiths. Ct Smithsonian Contributions to Knowledge. Washington. 1848 B.M.; Camb.P.S.; Camb.U.; Chem.S.; Dub.T.C.; Edinb. E.S. ; Edinb.U.; Geol.M. i. ; Glasg.P.S.; Glasg.U.i.; Linn.S. ; N.H.M.; Oxon.B.; Oxon.B.i.; P.O.i.; E.A.S.; E.S.; U.C.L.i. Smiths. Misc. Col Smithsonian Miscellaneous Collections. Washington. 1862 B.M.; Camb.P.S.; Camb.U.; Chem.S.; Dub.T.C.; Edinb. E.S. ; Edinb.U.; Geol.M.; Glasg.P.S.; Glasg.U.; Linn.S.; N.H.M.; Oxon.B.; Oxon.E.i.; P.O.i.; E.A.S. ; E.S.; U.C.L.i. List of Serial Publications Spet. It. Steierm. Mt. smiths. Rp Annual Report of the Board of Regents of the Smithsonian Insti- tution. Washington. 1846 B.M.i.; Camb.P.S.; Camb.U.; Dub.T.C.; Edinb.R.S.i.; Geol.M.; Glasg.P.S.i.; Glasg.U.; Linn.S.i.; Math.S.i.; N.H.M.i.; Oxon.B.; Oxon.R.i.; P.O.i.; R.A.S.i.; R.S.i.; U.C.L.i. Memorie della Societa degli Spettroscopisti Italian!, raccolte e pubblicate per cura del Prof. P. Tacchini. Palermo. 1872 B.M.i.; Camb.U.; Edinb.R.S.i.; P.O.; R.A.S.; R.S. Mittheilungen des Naturwissenschaftlichen Vereins fur Steiermark. Graz. 1863 B.M.; Camb.U.i.; Dub.R.I.A.; Edinb.R.S.i.; Linn.S.i.; N.H.M.; R.S.; U.C.L.i. ... Bericht iiber die Thatigkeit der St. Gallischen Naturwissenschaft- lichen Gesellschaft. St. Gallen. I860 N.H.M.; R.S.i. ... The Transactions of the Academy of Science of St. Louis. St. Louis. 1856 B.M.; Dub.R.I.A.; Edinb.R.S.; Glasg.P.S.; Linn.S.i.; N.H.M.; Oxon.B.; P.O.i.; R.S. (Kongliga Svenska Vetenskaps-Akademiens Handlingar. Stockholm. "-1 1739 B.M.; Camb.P.S.; Camb.U.; Chem.S.i.; Dub.R.I.A.; ( Edinb.R.S.i.; Glasg.P.S.i.; Linn.S.i.; N.H.M.; R.A.S.; R.S. .. Bihang till Kongl. Svenska Vetenskaps-Akademiens Handlingar. Stockholm. 1872 B.M. ; Camb.P.S.; Camb.U.; Chem.S.; Dub.R.I.A.; .. Edinb.R.S.; Glasg.P.S.; Linn.S.i.; N.H.M.; R.A.S.; R.S. .. Ofversigt af Kongl. Vetenskaps-Akademiens Forhandlingar. Stock- holm. 1844 B.M. ; Camb.P.S.; Camb.U.; Chem.S.i.; Dub.R.I.A.; Edinb.R.S.i.; Glasg.P.S.i.; Glasg.U.i.; Linn.S.i.; N.H.M.; Oxon.R.; R.A.S.; R.S.; U.C.L.i. .. Lefnadsteckningar ofver Kongl. Svenska Vetenskaps Akademien... ledamSter. Stockholm. 186973. Chem.S.i.; Dub.R.I.A.; Edinb.R.S.; Glasg.P.S.; Linn.S.i.; R.A.S.; R.S. . . Memoires de 1'Academie Imperiale des Sciences de St. Petersbourg. St. Petersbourg. 1803 B.M.; Camb.P.S.; Camb.U.; Chem.S.i.; Dub.R.D.S.; Dub.R.I.A.; Dub.T.C.; Edinb.R.S.; Edinb.U. ; Glasg.U.i.; N.H.M.; Oxon.B.; Oxon.R.; P.O.i.; R.S.; U.C.L.i. See St. Pet. Ac. Sc. Mm., St. Pet. Mm. and St. Pet. Mm. Ac. Sc. .. Bulletin Scientifique public par 1'Academie Imperiale des Sciences de St. Petersbourg. St. Petersbourg. 183642. Bulletin de la Classe Physico-mathematique de 1'Academie Imperiale des Sciences de St. Petersbourg. St. Petersbourg and Leipzig. 1843-59. Bulletinde 1'Academie des Sciences de St.Petersbourg. St.Petersbourg. 1860 B.M.i.; Camb.P.S.; Camb.U.; Chem.S.i.; Dub.R.I.A.; Edinb.R.S.; Glasg.P.S.i.; Glasg.U.; Linn.S.i.; N.H.M.; Oxon.B.; Oxon.R.i.; P.O.i.; R.A.S.i.; R.S. See St. Pet. Bll. Ac. Sc. and St. Pet. Bull. Sc. .. See St. Pet. Ac. Mm. .) Memoirs of the Imperial Academy of Science. [In Russian.] St. Petersburg. Not the same as St. Pet. Ac. Mm. 186295. B.M.; Dub.R.I.A. . . Nova Acta Academies Scientiarum Imperialis Petropolitame. Petropoli. 1783-1802. B.M.; Camb.U.; Edinb.R.S.; Linn.S.i.; N.H.M.; Oxon.B.; Oxon.R.; P.O.; R.A.S.i.; R.S.; U.C.L. .. Recueil des Actes des Seances Publiques de 1'Academie Imperiale des Sciences de St. Petersbourg. St. Petersbourg. 182748. B.M.; Camb.U.; Chem.S.i.; Dub.R.I.A.i.; Edinb.R.S.; N.H.M.; Oxon.B.; Oxon.R.i.; R.A.S.; R.S. " | See St. Pet. Ac. Sc. Bll. j See St. Pet. Ac. Mm. . . Memoires presentes a 1'Academie Imperiale des Sciences de St. Peters- bourg par divers Savans. St. Petersbourg. xliv St. Gal. B St. Louis AC. T. Stockh. Ac. XZndl Stockh. Ak. Kndl Stockh. Ak. ZZndl. Bh. Stockh. Ofv Stockfc. Vt. Ak. Lefn. St. Pet. Ac. Mm. St. Pet. Ac. Sc. Bll St. Pet. Ac. Sc. Mm St. Pet. Ac. Sc. Mm. ( St. Pet. Ac. Sc. N. Acta. St. Pet. Ac. Sc. Rcc. .. St. Pit. Bll. Ac. Sc St. Pet. Bll. Sc St. Pet. Mm St. Pet. Mm. Ac. Sc St. Pet. Mm. Sav. E tr. . List of Serial Publications Strasb. Mm. S. Sc. Strasb. S. Sc. N. Mm. Svea Sym. Met. IVIg. .. Tasm. B. S. P. Taylor Sc. Mm Tel. E. J. Terr. Mag Texas Ac. Sc. T Thomson A. Ph Thrond. Skr TillochPh. Mg Tok. Coll. Sc. J. Tok. Un. Mm. . Tor. Ac. Mm. Tor. Ac. Sc. At. ... Tor. Ac. Sc. Mm. . . . 183159. B.M.; Camb.U.; Edinb.K.S.; Glasg.U.; Linn.S. ; N.H.M.; E.A.S.; B.S.; U.C.L.i. Memoires de la Societe des Sciences Naturelles de Strasbourg. Strasbourg. 183070. B.M.; Dub.B.I.A.i.; Dub.T.C.i.; N.H.M. Nouveaux Memoires de la Societe des Sciences, Agriculture, et Arts du De'partement du Bas-Rhin. Strasbourg. 183242; 185968. Camb.U.i.; N.H.M.; Oxon.B.; B.S.i. Svea. Tijdskrift for Vetenskap och Konst. Upsala. 181831. Symons's Monthly Meteorological Magazine. London. 1866 Camb.U.; P.O.; B.S. Monthly Notices of Papers and Proceedings of the Eoyal Society of Tasmania. Hobart. 1863 B.M.i.; Camb.P.S. i.; Dub.E.D.S.; Edinb.B.S.i.; Linn.S.i.; N.H.M.; E.A.S.; E.S. Scientific Memoirs, selected from the Transactions of Foreign Academies and Learned Societies and from Foreign Journals; Taylor. London. 183752. B.M.; Camb.U.; Chem.S.i.; Edinb.E.S.; Linn.S.i.; N.H.M. ; Oxon.B.(B.); P.O.; B.A.S.i.; E.S.; U.C.L. Journal of the Society of Telegraph Engineers. London. 1872 B.M.; Camb.P.S. ; Camb.U. i. ; Dub.T.C.i.; Oxon.B.; Oxon.E.; P.O.; E.S.; U.C.L. Terrestrial Magnetism [and Atmospheric Electricity]. An Inter- national Quarterly Journal. Chicago, Cincinnati, Baltimore. 1896 Camb.U.i.; E.S. Transactions of the Texas Academy of Science. Austin. 1892 Camb.P.S.; Edinb.E.S.; Glasg.P.S.; Math.S.i.; N.H.M.; E.S. Annals of Philosophy, or Magazine of Chemistry, Mineralogy, Mechanics, and the Arts; Thomson. London. 181326. B.M.; Camb.P.S.; Camb.U.; Chem.S.i.; Edinb.E.S.i.; Linn.S.; N.H.M.; Oxon.B.; Oxon.E.; P.O.; E.A.S.; E.S.; U.C.L.i. Det Kongelige Norske Videnskabersselskabs Skrifter i det 19de Aarhundrede. Kjtfbenhavn og Throndhjem. 1817 B.M.; Camb.U.i.; Dub.T.C.i.; Edinb.E.S.i.; N.H.M.; E.S. See Norske Vd. Skr. The Philosophical Magazine, comprehending the various branches of Science, the Liberal and Fine Arts, Geology, Agriculture, Manufactures, and Commerce. London. 17981826. B.M.; Camb.P.S.; Camb.U.; Chem.S.i.; Edinb. E.S.i.; Edinb.U.; Glasg.P.S.; Glasg.U.i.; Linn.S.i.; N.H.M.; Oxon.B.; Oxon.E.; P.O.; E.A.S.; E.S.; U.C.L. The Journal of the College of Science, Imperial University, Japan. Tokio, Japan. 1887 B.M.; Camb.P.S.; Camb.U.; Chem.S.; Dub.T.C.; Edinb. E.S. ; Edinb.U.i.; Glasg.P.S.i. ; Linn.S.i.; Math.S.; N.H.M.; Oxon.B.; Oxon.E.; P.O.; E.S.; U.C.L. Memoirs of the Science Department, University of Tokio, Japan. Tokio, Japan. 187985. Camb.U.; Chem.S.i.; Dub.E.D.S.; Dub.E.I.A.; Edinb. E.S.i.; Glasg.P.S.; Linn.S.i.; N.H.M.; Oxon.B.; Oxon.E.; B.A.S.i.; E.S.; U.C.L.i. Memorie della E. Accademia delle Scienze di Torino. Torino. 1818 B.M.i.; Camb.P.S.; Camb.U.; Dub.E.I.A.; Edinb.E.S.; Glasg.U.i.; Linn.S.; N.H.M.; Oxon.B.; P.O.; E.A.S.; E.S.; U.C.L.i. See Tor. Ac. Sc. Mm., Tor. Mm. Ac., Turin Ac. Mm. and Turin Mm. Ac. Atti della E. Accademia delle Scienze di Torino. Torino. 1865 B.M.; Camb.P.S.; Camb.U.; Chem.S.i.; Dub.E.I.A.; Edinb.E.S.; Glasg.U.i.; Linn.S.; N.H.M.; Oxon.B.; P.O.i. ; E.A.S.; E.S.; U.C.L.i. See Tor. At. Ac. Sc. See Tor. Ac. Mm. xlv d2 List of Serial Publications Tor. At. Ac. Be Tor. Lav. Be. Fis. Mt. Tor. Mm. Ac Tortolinl A Toul. Ac. Sc. Bll. . . . T oul. Ac. Sc. Mm. . . . Toul. Fac. Sc..A. ... Tool. Mm. Ac TouL Mm. Ac. Sc. . . Toul. Obs. A. .. Toul. 8. H. Nt. Bll. Ts. Mth Ts. Mt. Fys Turin Ac. Mm. Turin Mm. Ac. . . . TTn. Serv. I. J Ups. Arsk Ups. N. Acta S. Sc. Ups. 8. Sc. N. Acta Tips. Vet. S. Arsk. . . V. 8. Coast Sv.Rp... Utr. A. Ac Utr. Aant. Frv. On. Vars. S. Wt. Tr. (C. E. Ps. C.) Vars. 8. Kt. Tr. (Mm.) See Tor. Ac. Sc. At. Notizia storica del lavori fatti dalla Classe di Scienze Fisiche e Matematiche della E. Accademia delle Scienze negli anni 1864 65. Torino. 1869. Linn.S.; E.A.S.; E.S. See Tor. Ac. Mm. Annali di Scienze, Matematiche, e Fisiche ; Tortolini. Eoma. 185057. B.M.; Camb.U. i.; Dub.E.D.S.; Dub.T.C.; Edinb.U.; Glasg.U.i.; Math.S.i.; Oxon.B.(B.); E.S.; U.C.L. See A. Mt. Bulletin de 1'Academie des Sciences, Inscriptions et Belles-lettres de Toulouse. Toulouse. 189899. Dub.E.LA.; Edinb.E.S.; N.H.M.; E.S. Memoires de 1'Academie des Sciences, Inscriptions et Belles-Lettres de Toulouse. Toulouse. 1782 B.M.; Camb.U.; Dub.E.LA.; Edinb.E.S.i. ; N.H.M.; Oxon.B.i.; E.A.S.i.; E.S. See Toul. Mm. Ac. and TouL Mm. Ac. Sc. Annales de la Faculte des Sciences de Toulouse pour les Sciences Math^matiques et les Sciences Physiques. Paris. 1887 Camb.P.S.; Camb.U.; Edinb.E.S.; Math.S.; Oxon.E.; E.S. | See Toul. Ac. Sc. Mm. Annales de 1'Observatoire Astronomique, Magnetique et Me'te'orolo- gique de Toulouse. Paris. 1880 B.M.; Edinb.E.S.; Math.S.i.; E.A.S.; E.S. Bulletin de la Societe d'Histoire Naturelle de Toulouse. Toulouse. 1867 N.H.M. Tidsskrift for Mathematik. Kjefbenhavn. 185989. B.M.; Camb.U.; Math.S.i.; Oxon.B. Tidskrift for Matematik och Fysik, tillegnad den Svenska Elementar- Undervisningen. Upsala. 186874. B.M.; E.S.i. (Memoires de l'Acad6mie Eoyale des Sciences de Turin. Turin. <1784 1816. B.M.; Dub.B.I.A.i.; Edinb.E.S.; Linn.S.; Oxon.B.; ( P.O.; E.A.S.; E.S.; U.C.L. See Tor. Ac. Journal of the Eoyal United Service Institution. London. 1858 B.M. ; Camb.U. ; Dub.N.L.I.i. ; Edinb.U. ; Oxon.B.i. ; P.O.; E.S. Upsala Universitetets Arsskrift. Upsala. 1861 B.M.; Camb.U.; Dub.E.D.S.i.; Dub.E.I.A.i.; Edinb.E.S.; Linn.S.i.; Math.S.i.; N.H.M.; Oxon.B.; E.A.S.i.; E.S. | Nova Acta Eegise Societatis Scientiarum Upsaliensis. Upsalise. 1773 B.M.; Camb.U.; Dub.E.I.A.i.; Edinb.E.S.i.; Glasg.U.i.; Linn.S.; Math.S.i.; N.H.M.; Oxon.B.i.; Oxon.E.; E.A.S.i.; ( E.S.i.; U.C.L.i. Arsskrift utgifven af Kongl. Vetenskaps- Societe ten i Upsala. Upsala. 186061. B.M.; Camb.U.; Edinb.E.S.; N.H.M.; E.S. Eeports of the Superintendent of the Coast Survey, showing the Progress of the Survey from year to year. Washington. 1851 Camb.U.; Dub.E.I.A.i.; Edinb.E.S.; N.H.M.; E.A.S.; E.S.; U.C.L.i. AnnalesAcademiffiBheno-Trajectinse. TrajectiadEhenum. (Utrecht.) 181537. B.M.; Camb.U.; N.H.M.; Oxon.B.; E.S.i. Aanteekeningen van het Verhandelde in de Sectie-Vergaderingen van het Provinciaal Utrechtsch Genootschap van Kunsten en Wetenschappen. Utrecht. 1846 Dub.E.D.S.; Edinb.E.S.; E.S. Travaux de la Societe des Naturalistes de Varsovie. Comptes Eendus de la Section de physique et de chimie. Varsovie. [In Eussian.j 1889 Math.S.; N.H.M. Travaux de la Societe des Naturalistes de Varsovie. Memoires. Varsovie. [In Eussian.] 189196. Math.S.; N.H.M. xlvi List of Serial Publications Ven. Aten Ven. Aten. At Ven. Aten. Esercit. Ven. At. X Ven. I. At Ven. Esercit. Aten. Ven. I. Mm Ven. Mm. I Verona Mm. S. It. . Verona S. It. Mm. . Viet. R. S. P. .. Viet. R. S. T. Viet. T. Ph. I. Viet. T. R. S. V. Nost. Eng. Voigt Mg.. Wash. Nat. Ac. Wash. Ph. 8. Bll. W. Eng. J Westf. Vr. Jbr. Wet. Gs. Jbr. .. Wiad. Mt WienAk. D Atti delle Adunanze dell' I. B. Istituto Veneto di Scienze, Lettere, ed Arti. Venezia. 1841 B.M.; Dub.E.D.S.i.; Dub.B.I.A.i.;Edinb.B.S.i.; Linn.S.i.; E.S.i. L'Ateneo Veneto: Eivistamensiledi Scienze, Lettere ed Arti. Venezia. 1881? Dub.B.D.S.i.; E.S.i. Atti dell' Ateneo Veneto. Venezia. 1864 Dub.E.D.S.; E.S.i. Esercitazioni Scientifiche e Letterarie dell' Ateneo di Venezia. Venezia. 183760. B.M.i.; Dub.T.C.i.; Oxon. B.i.; E.S.i. See Ven. Esercit. Aten. f Atti del Eeale Istituto Veneto di Scienze, Lettere ed Arti. Venezia. "(1841 B.M.; Edinb.E.S.i.; Linn.S.i.; Math.S.i.; N.H.M.; E.S.i. See Ven. Aten. Esercit. IMemorie del Eeale Istituto Veneto di Scienze, Lettere, ed Arti. I Venezia. 1843 B.M. ; Camb.U. ; Dub.E.I.A.i. ; Linn.S.i.; N.H.M. ; Oxon.B.i.; E.S. /Memorie di Matematica e di Fisica della Societa Italiana delle 1 Scienze. Modena e Verona. 1 1782 B.M.i.; Camb.P.S.; Camb.U.i.; Dub.E.I.A.; Edinb.E.S.i.; ( Linn.S.i.; Oxon.B.i.; E.A.S.i.; E.S.; U.C.L.i. See Mod. Mm. S. It, and Rm. S. It. Mm. Proceedings of the Eoyal Society of Victoria. Melbourne. 1889 B.M. ; Camb.P.S. ; Camb.U. ; Dub.E.I.A. ; Dub.T.C. Edinb.E.S.; Edinb.U.; Glasg.P.S.; N.H.M.; Oxon.B.; Oxon.E. P.O.; E.S.; U.C.L.i. Transactions and Proceedings of the Eoyal Society of Victoria. Melbourne. 1861 B.M.; Camb.P.S.; Camb.U.; Dub.E.D.S.; Dub.E.I.A.; Edinb.E.S.; Edinb.U.; Glasg.P.S.i.; Glasg.U.i.; Linn.S.i.; N.H.M.; Oxon.B.; Oxon.B.i.; P.O.; E.A.S.; E.S. See Viet. T. R. 8. Transactions of the Philosophical Institute (afterwards Eoyal Society) of Victoria. Melbourne. 185560. B.M.; Camb.U.i.; Dub.E.I.A.; Edinb.E.S.; Glasg. P.S.i.; Linn.S.; N.H.M.; Oxon.B.; P.O.; E.A.S.; E.S. See Viet. R. S. T. Van Nostrand's Engineering Magazine. New York. 186985. B.M.; P.O.; E.S.i. . Magazin fur den neuesten Zustand der Naturkunden, mit Eiicksicht auf die dazu gehorigen Hiilfswissenschaften ; Voigt. Jena und Weimar. 17971806. B.M.; Camb.U.; N.H.M.; E.S. . Memoirs of the National Academy of Sciences. Washington. 1866 B.M.i.; Camb.P.S.; Camb.U.i.; Dub.E.I.A.; Edinb.E.S.; Math.S.i.; N.H.M.; Oxon.B.i.; Oxon.E.; P.O.; E.S.; U.C.L.i. Bulletin of the Philosophical Society of Washington. Washington. 1874 B.M.; Camb.P.S.; Edinb.E.S.; Linn.S.; N.H.M.; Oxon.B.; P.O.; E.A.S.; E.S. . The West of England Journal of Science and Literature. Bristol. 183536. B.M.; Camb.U.; Edinb.E.S.; N.H.M.; Oxon.B.; P.O. . Jahres-Bericht des Westf alischen Provinzialvereins fur Wissenschaf t und Kunst. Miinster. 1873 N.H.M. . Jahresbericht der Wetterauischen Gesellschaft fur die gesammte Naturkunde. Hanau. 1843 Dub.E.I.A.; E.S.i. . Wiadomosci Matematyczne. Warsaw. 1897 Camb.P.S.; Math.S. . Denkschriften der Kaiserlichen Akademie der Wissenschaf ten. Mathematisch-Naturwissenschaftliche Classe. Wien. 1850 B.M.; Camb.P.S.i.; Camb.U.; Chem.S.i.; Dub.E.I.A.; Edinb.E.S.; Edinb.U.; Linn.S.; N.H.M.; Oxon.B.(E); P.O.i.; E.A.S.; E.S.; U.C.L.i. See Wien D. xlvii List of Serial Publications Wien Ak. Bb. Wien Aim. Wien Berg-Hm. Jb. Wien D Wien Jb. Pol. I. .. "Wien Jbr. Ober-Realsch. Inn. Btadt Wien Met. Z Wien Mt. Gg. Gs Wien BB ) Wien Sb J Wise. Ac. T Wttrtb. Jh. Wurzb. Ps. Did. Sb. Wurzb. Ps. Md. Vh. Wiirzb. Vh. ... Sitzungsberichte der Mathematisch-Naturwissenschaftlichen Classe der Kaiserlichen Akademie der Wissenschaften. Wien. 1848 B.M.; Camb.P.S.i.; Camb.U.; Chem.S.i.; Dub.R.I.A.; Dub.T.C.; Edinb.R.S.i.; Glasg.U.i.; Linn.S.; N.H.M.; Oxon.B.; Oxon.B.; P.O.i.; R.A.S.i.; B.S.; U.C.L.i. See Wien BB. and Wien Sb. Almanach der Kaiserlichen Akademie der Wissenschaften. Wien. 1851 B.M.; Camb.P.S.i.; Camb.U.; Dub.R.I.A.i.; Edinb.B.S.i.; Oxon.B.; P.O.i.; R.A.S.i.; R.S.i.; U.C.L.i. Anzeiger der Kaiserlichen Akademie der Wissenschaften: Math.- Naturwiss. Classe. Wien. 1864 Camb.U.; Linn.S.; N.H.M.; Oxon B.; R.S.i. Berg- und Huttenmannisches Jahrbuch der k. k. Schemnitzer Bergakademie und der k. k. Montan-Lehranstalten zu Leoben und Pfibram ; Gustav Faller. Wien. 1851 B.M.i.; P.O.i. See Wien Ak. D. Jahrbuch des k. k. Polytechnischen Instituts in Wien; Prechtl. Wien. 181939. B.M.; Camb.U.; Oxon.B.; P.O. Jahresbericht der offentlichen Ober-Realschule in der innern Stadt. Wien. 185963. Zeitschrif t der Oesterreichischen Gesellschaf t fur Meteorologie . Wien . 186685. Dub.R.D.S.; Edinb.R.S.; P.O.; R.S. Mittheilungen der k. k. Geographischen Gesellschaft. Wien. 1857 B.M.; Dub.R.I.A.i.; Dub.T.C.i.; N.H.M.; OxonB.; R.S. See Wien Ak. Bb. Transactions of the Wisconsin Academy of Sciences, Arts and Letters. Madison. 1872 B.M. ; Camb.P.S.; Dub.R.I.A.; Edinb.R.S.; N.H.M.; Oxon.R.i.; P.O.i.; R.S. Jahreshefte des Vereins fur vaterlandische Naturkunde in Wiirttem- berg. Stuttgart. 1845 B.M.; Camb.U.; Dub.R.D.S. i.; Dub.T.C.i.; Linn.S.i.; N.H.M.; R.S. Sitzungsberichte der Physikalisch-Medicinischen Gesellschaft zu Wiirzburg. Wiirzburg. 185962. 1881 Camb.P.S.; Camb.U.; Chem.S.i.; Dub.R.I.A.; Linn.S.i.; Oxon.R.i.; R.S. i Verhandlungen der Physikalisch-Medicinischen Gesellschaft. Wiirz- \ burg. 185060. 1868 B.M.i. ; Camb.P.S. ; Camb.U. ; Chem.S.i. ; Dub.R.I.A.; Linn.S.; N.H.M.; Oxon.R.; R.S.; U.C.L.i. ZachCor. Zach M. Cor Z. Bauw :. Z. Xnstk Z. Mth. Ps. .. Z. Ww. Correspondance Astronomique, Geographique, Hydrographique, et Statistique ; von Zach. Genes. 181826. B.M.; R.A.S.; R.S. Monatliche Correspondenz zur Beforderung der Erd- und Himmels- Kunde; von Zach. Gotha. 180013. Oxon.B.; R.A.S.; R.S.; U.C.L. Zeitschrif t fiir Bauwesen; herausg. unter Mitwirkung der konigl. technischen Bau-Deputation und des Architecten- Vereins zu Berlin. Berlin. 1851 B.M.; Camb.U.i.; P.O. Zeitschrift fiir Instrumentenkunde. Organ fiir Mittheilungen aus dem gesammten Gebiete der wissenschaftlichen Technik. Berlin. 1881 B.M.; Camb.U.; Chem.S.; Edinb.U.; Oxon.R.; P.O.; R.A.S.; R.S. Zeitschrift fiir Mathematik und Physik; Schlomilch. Leipzig. 1856 B.M.; Camb.U.; Dub.R.D.S. i.; Dub.R.I.A.i.; Dub.T.C.i.; Edinb.U.; Glasg.U.i.; Math.S.i.; Oxon.B.(R.); R.S.; U.C.L.i. See Schlomilch Z. Zeitschrift fiir die gesammten Naturwissenschaften; herausgegeben von dem Naturwissenschaftlichen Vereine fiir Sachsen und Thiiringen in Halle; Giebel. Berlin. xlviii Z. Fsychol Zur. Mschr. . Ziir.N. D. Scb. Gs.. Zwick. Vr. m. Jbr. List of Serial Publications 1853 B.M. ; Camb.U.f. ; Dub.N.L.I.t. ; Dub.B.D.S.t. ; Dub. B.I.A.i.; Dub.T.C.i.; Edinb.B.S.; Linn.S.; N.H.M.; Oxon.B.; Oxon.E.; B.S. See Halle Z. and Halle Z. Nw. Zeitschrift fiir Psychologie und Physiologic der Sinnesorgane. Hamburg, Leipzig. 1890 B.M.; Camb.U.; Edinb.U.; Oxon.B.; Oxon.E.; E.S.; U.C.L. Monatsschrift des Wissenschaftlichen Vereins in Zurich ; Hitzig, etc. Zurich. 185659. B.M.; Camb.U.; N.H.M.; Oxon.B.; E.S. Mittheilungen der Naturforschenden Gesellschaft in Zurich. Zurich. 184756. Chem.S.i.; Dub.E.I.A.; Edinb.E.S.i.; Linn.S.; N.H.M.; E.A.S.; E.S. Neue Denkschriften der allgemeinen Schweizerischen Gesellschaft fiir die gesammten Naturwissenschaften Neuchatel. Zurich, etc. 1837 B.M.; Camb.P.S.; Camb.U.; Dub.E.D.S.i; Dub.E.I.A.i.; Edinb.E.S.; Linn.S.i.; N.H.M.; Oxon.B.; E.S. See Scb. OB. N. 3>. Vierteljahrsschrift der Naturforschenden Gesellschaft in Zurich. Zurich. 1856 B.M.; Camb.P.S.; Camb.U.t.; Chem.S.i.; Dub.E.I.A.; Edinb.E.S.; Linn.S.i.; Math.S.t.; N.H.M.; E.A.S.; E.S. Jahresbericht des Vereins fiir Naturkunde zu Zwickau. Zwickau. 1874 N.H.M.; E.S.i. xlix PURE MATHEMATICS SCHEDULE OF CLASSIFICATION ADAPTED FROM SCHEDULE A OF THE INTERNATIONAL CATALOGUE OF SCIENTIFIC LITERATURE. 0000 Philosophy 0010 History. Biography ... History Arabian Mathematics Greek Mathematics Indian Mathematics Biography 0020 Periodicals. Reports of Insti- tutions, Societies, Congresses, etc 0030 General Treatises, Text Books, Dictionaries, Tables, Collected Works 0032 Bibliographies ... 0035 Tables of Mathematical Func- tions 0040 Addresses, Lectures, etc., of a general character . 0050 Pedagogy 0060 Institutions 0070 Nomenclature ... 0080 Instruments, including Calcu lating Machines. Models 34 Instruments 34-38 Calculating Machines 38-39 Models 39-40 0090 Aids to Calculation, Graphical Processes ... ... ... 40 ARITHMETIC AND ALGEBRA. Foundations of Arithmetic. 0400 General 42 0410 Rational numbers ; arithmeti- cal operations ... ... 43 Scales of Notation 45-46 0420 Existence of irrational and transcendental numbers ; in- finite processes adapted to rational numbers ... ... 46 0430 Aggregates 49 PAGE 1 3 . 3-17 4 . 9-10 11 17-25 ti- 0800 0810 0820 0830 0840 Universal Algebra. General ... Calculus of Operations General theory of complex numbers. (See also 2870.) Quaternions Ausdehnungslehre ; vector an- alysis PAGE 50 51 53 56 59 es, 25 i ed 25 ... 26 1C- 0850 0860 0870 1200 Matrices Other special sorts of complex numbers Algebra of Logic Theory of Groups. General . . . 62 63 64 65 ... 26 'a 29 ... 30 ... 32 33 1210 1230 Discrete groups of finite and of infinite order (including groups of permutations). (See also 2450, 4440.) Continuous groups of finite and of infinite order 66 72 ALGEBRA AND THEORY OF NUMBERS. 1590 General 75 Elements of Algebra. General ... ... ... ... 75 Rational polynomials ; divisi- 1600 1610 1615 1620 bility ; reducibility 77 Algebraic inequalities 79 Permutations, combinations, partitions, distributions. Binomial and multinomial coefficients 80 Binomial coefficients 80-81 Chess 81-82 Combinations ... 82-83 Magic squares ... 84-85 Partition of numbers 85-86 Schedule of Classification Partitions Permutations PAGE 86-87 87-88 Fiqurate numbers ... 89 Special series 91-92 Probabilities (including com- bination of observations). Theory of errors ... 92 Least Squares Method 97-98 Probabilities ... 100-103 Problems ... 101-102 Theory of statistics. Actu- arial mathematics ... 104 Mortality ... 114-116 Population ... 117-118 Calculus of differences; inter- polation 121 Interpolation ... 1 23-1 25 Linear Substitutions. General 126 Determinants ... ... 127 Discriminants and resultants 136 Elimination ... 137-139 Characteristic properties of linear substitutions ; types of linear substitutions. (See also 1210.) General theory of quantics Binary forms Ternary forms Special developments asso- ciated with forms in more than three variables 1625 Finite Summation. Recurring 1630 1635 1640 2000 2010 2020 2030 2040 2050 2060 2070 2400 2410 2420 2430 140 142 146 153 lf> 5 2440 Theory of Equations. General Elements of the theory of algebraicequations; exis- tence of roots, symmetric functions; rational frac- tions ; partial fractions Existence of roots Symmetric functions 162-163 Reality, multiplicity, separa- tion, of roots 163 Equations of the second, third and fourth orders : other particular equations Equations, 2nd order ... Equations, 2nd, 3rd and 4th orders 170 Equations, 3rd order 170-171 Irreducible case ... 171 Equations, 3rd and 4th orders 171-172 Equations, 4th order 172-173 Trinomial Equations 173-174 Numerical solution of equa- tions (including graphical methods) 174 156 158 1 59 169 169 PAGE Numerical solution of equations in general 174 Approximate determina- tion of roots 174-175 Graphical methods 177-178 2450 General resolution of equa- tions ; theory of Galois. (See also 1210.) 178 Equations, 5th order 180-181 Equations, 6th order ... 181 2460 Simultaneous equations ; in- cluding linear equations 184 Linear equations ... 185-186 2470 Transcendental equations ... 187 Theory of Numbers. 2800 General 188 2810 Divisibility ; linear con- gruences ... ... 193 Divisibility ... 193-194 2815 Continued fractions and in- determinate equations 197 Continued fractions 197-201 Indeterminate equations 201-204 1st order ... 202-203 2nd order, 3rd order, 4th order, 5th order ... 203 2820 Quadratic residues 204 2830 Quadratic binary forms ... 205 2840 Quadratic forms of three or more variables ; bilinear forms. (See also 2060, 2070.) 209 Decomposition of numbers into sums of squares 209-2 1 Bilinear forms ... 21 3-2 1 4 2850 Congruences other than linear ; cubic and higher residues 214 2860 Forms of higher degree which cannot be considered as products of linear factors. (See also 2060, 2070.) ... 218 Fermafs theorem on z n +y n =z n ... 218-219 2870 Forms of higher degree which can be considered as pro- ducts of linear factors ; algebraic numbers ; ideals. (See also 0820.) 220 2880 Application of trigonometri- cal functions to arithmetic ; cyclotomy 222 Binomial equations ... 222 2890 Application of other trans- cendental functions to arithmetic 224 2900 Distribution of prime num- bers 225 2910 Special numbers and numeri- cal functions Bernoulli's numbers 228 228-229 Schedule of Classification 2920 Irrationality and transcen- dence of particular num- bers, such as e and TT. (See also 6810.) ... ... 232 e 232-233 TT... 233 3190 General ANALYSIS. 233 Foundations of Analysis. 3200 General 234 3210 Theory of functions of real variables 235 3220 Infinite series ; infinite pro- ducts and other infinite processes. (See also 5610, 5620.) 238 Infinite series ... 238-252 Binomial theorem 238-239 Convergence ... 239-241 Expansion ... 242-244 Power series ... 245-246 Summation ... 248-250 Trigonometric series 250-251 Infinite products ... 252-253 Other infinite processes ... 253 3230 Principles and elements of the differential calculus 253 3240 Taylor's series. Maxima and minima ; other analytical applications of the differen- tial calculus 259 Lagrange's series ... 261 Other series 261 Maxima and Minima 262-263 Maxima 264 Minima 264 Taylor's series ... 265-266 3250 Principles and elements of the integral calculus. Ap- proximate integration. Mechanical quadrature 267 Integral calculus 268 Integration ... 270-272 3260 Definite integrals (simple) ... 274 Evaluation ... 277-278 Transformation ... 280 3270 Multiple integrals 283 Double integrals 284-285 3280 Calculus of variations ... 286 Theory of Functions of Complex Variables. 3600 General 289 3610 Uniform functions of one variable 292 3620 Multiform functions of one variable ; Rieinann sur- faces. (See also 4060, 6420, 8075.) 296 Biemann surfaces 296-297 3630 Expansions in series of func- tions, other than powers of the variable ... ... 297 3640 Functions of several variables 298 Algebraic Functions and their Integrals. 4000 General 299 4010 Algebraic functions of one variable 301 4020 Algebraic functions of several variables 302 4030 Logarithmic, circular, ex- ponential functions ... 303 Circular functions 303-305 Expansion ... 303-304 Logarithmic and exponen- tial functions 305-311 Logarithmic tables ... 308 4040 General properties of elliptic functions and single theta functions ; addition-theo- rem. (See also 8050, 8060.) 311 Addition 311 Addition-theorem 311-312 Doubly periodic func- tions 312-313 Elliptic functions... 313-319 Elliptic integrals 319-321 Theta functions ... 323-324 4050 Multiplication,division,trans- formation of elliptic func- tions; modular functions. (See also 4440.) 325 Division 325 Modular equations ... 326 Modular functions . . . 326 Multiplication ... ... 327 Transformation ... 327-329 4060 Abelian integrals. (See also 8050, 8060.) 329 A belian functions 329-331 Abelian integrals . . . 331-333 Abel's theorem ... 333-334 4070 Periodic functions of several variables ; general theta functions 335 Other Special Functions. 4400 General 338 4410 Eulerian functions 339 Eulerian integrals ... 340 Gamma functions 340-342 4420 Legendre's functions; Bessel's functions ; hypergeo- metric functions ... 342 Bessel's functions 343-344 Cylindrical functions ... 344 Hypergeometric functions 344-345 Hyper geometric series 345-346 Legendre's functions 346-347 Legendre's polynomials 347 Spherical harmonics 347-348 liii Schedule of Classification 4430 Polymorphic functions. Other functions which may be de- fined by definite integrals. (See also 4860.) 349 4440 Automorphic functions. (See also 1210, 4050.) 350 Fuchsian functions 350-351 4450 Other functions which may be defined by linear differential equations. Lame's functions ... 351 Lame's functions 352 4460 Functions which may be de- fined by functional equa- tions. (See also 6030.) ... 353 4470 Integral functions 353 Differential Equations. 4800 General. (See also 8080 Con- nexes.) 354 4810 Existence theorems for ordinary and partial differential equations ... 356 Integrability 356 4820 Methods of solution and re- duction of ordinary differen - tial equations ... ... 357 Equations 358 Integrals 359 Integration ... 359-362 Solution 362-363 4830 Methods of solution and re- duction of partial differen- tial equations of first order, including the differential equations of theoretical dynamics ... ... ... 364 Equations ... ... 364 Integration ... 365-367 Solution ... 367-368 Theory 368 4840 Methods of solution and re- duction of partial differen- tial equations of the second and higher orders . . . 368 Equations ... 369-371 Integrals 371 Integration ... 371-373 Solution, ... ... ... 374 4850 General theory of ordinary linear equations. (See also 4450.) 375 Equations ... 377-380 Integrals ... 381-382 Integration . . . 383-386 Solution 386-387 4860 Integration of ordinary linear equations by definite inte- grals. (See also 4430.) ... 388 4870 General theory of ordinary equations, not linear, of the first order 389 Riccatts equation 390-391 Solutions ... 391-392 4880 General theory of ordinary equations, not linear, of order higher than first ... 392 Differential Forms and Differential Invariants. 5200 General 393 5210 Linear differential forms; Pfaffians 394 5220 Differential forms of the second and higher orders. (See also 8450.) 396 5230 Transformation of differential forms, including tangen- tial (or contact) trans- formations ... ... 396 Transformation ... 397-398 5240 Differential invariants. (See also 1230.) 398 Analytical Methods connected with Physical Problems. 5600 General 400 5610 Harmonic Analysis ; Fourier's series. (See also 3220.) 400 Fourier's series ... 400-401 5620 Harmonic Analysis ; series other than Fourier's. Spherical and ellipsoidal harmonics. (See also 3220.) 403 5630 Generalities on the differen- tial equations of mathe- matical physics ... ... 404 5640 Integration of the differential equations of mathematical physics by series ... ... 406 5650 Integration of the differential equations of mathematical physics by definite integrals 407 5660 Dirichlet's problem and ana- logous problems affected by boundary conditions ... 407 Difference Equations and Functional Equations. 6000 General 409 6020 Solution of equations of finite differences 409 6030 Solution of functional equa- tions. (See also 4460.) ... 411 GEOMETRY. 6390 General 412 Foundations. 6400 General 412 6410 Principles of geometry ; non- Euclidean geometries ; hyperspace ... ... 413 liv Schedule of Classification PAGE 4- Dimensional Space 413-414 n- Dimensional Space 414415 Fundamental Principles 415-416 Non-Euclidean geometry 417-418 Theory of Parallels 419-420 6420 Topology of space and hyper- space. (See also 3620, 8075.) 421 6430 Methods of analytical geo- metry. (See also 0840.) 423 Coordinates ... 425-427 Elementary Geometry. 6800 General 429 6810 Planimetry ; straight lines, and circles ... ... 431 Circles. (See also 2880.) 431-435 Circle touching 3 given circles. (Apollonian problem.} 432 Quadrature of the Circle 434 Ratio of Diameter to Circumference. (See also 2920.) ... 434-435 Polygons. (See also 7210.) 437-438 Regular polygons 438-439 Porisms. (See also 4050 and 7210.) ... 439-440 Pythagorean theorem . . . 440 Quadrilaterals. (See also 7210.) Straight lines Triangles (including Cir- cles related thereto). (For Conic Sections related to Triangles see 7210.) ... 442^49 Circles of the Triangle 443-444 Malfattfs problem ... 444 Nine-point or Feuerbach Circle 444 Remarkable Points of the Triangle ... 446-447 6820 Stereometry ; straight lines, planes, and spheres ; poly- hedra 449 Mensuration of volumes 450-451 Polyhedra. (See also 8075.) 45 2-456 Regular Polyhedra ... 454 Tetrahedra ... 455-456 Spheres 457-460 Contacts 458-459 6830 Trigonometry, plane and spherical ... ... ... 461 Division of Angles and Circular Arcs 462-463 Trisection ... 462-463 Formulae of Plane Trigo- nometry ... 463-465 Plane Triangles 466-467 Spherical Trigonometry 468-471 441 44^ PAGE Formulas of Spherical Trigonometry 468-469 Spherical Triangles 470-471 6840 Descriptive geometry ; ' per- spective ......... 472 Anharmonic or Cross- Ratio ...... 472-473 Axonometry . . . 473-474 Involution ... ... 475 Perspective ... 475-476 Practical Solid Geometry 477 -478 Protective Geometry 478-480 Stereographic Projec- tions, and other Pro- jections of the Sphere. (See also 8840 and Geography 83.) ... 480 Trans versals . . . 480-48 1 Geometry of Conies and Quadrics. 7200 481 7210 482 483 484 485 7230 General ......... Equations relating to Conies and Quadrics Metrical and projective pro- perties of conies ... ... Conies determined by given Points or Tan- gents, and Methods of Determination . . . Contacts ......... Curvature of Conies. (See also 8430.) ... 485-486 Ellipses ...... 486-487 Equations and Coordi- nates ...... 487-488 Foci ...... 488-489 Hyperbolas ...... 489 Normals ... 490-491 Parabola ... 491-492 Pascal's hexagram, Bri- anckon's hexagon, and connected figures. (See also 807 5.)' ... 492-493 Polars and Polar Re- ciprocals ... 493-494 Polygons related to Conies. (See also 6810.) ......... 494 Poncelet's Polygons, Por- isms. (See also 4050 and 6810.) ... 494-495 Quadrilaterals, Quad- rangles, and related Conies. (See also 6810.) ......... 496 Tangents ... 496-497 Triangles, and related Conies. (See also 6810.) ...... 497-499 Systems of conies. (See also 8070 and 8090.) ...... 499 Iv Schedule of Classification PAGE Conies with Common Focus or Foci ... 500 7240 Metrical and projective pro- perties of quadric surfaces 502 Axes of Quadric Surf aces 502-503 Ellipsoid 504 Equations of Quadrics. (See also 2040 and 7200.) 504-505 > Foci and Focal Properties of Quadrics ... 505-506 Generation and Genera- tors of Quadrics ... 506 Hyperboloid and Hyper- bolic Paraboloid 506-507 Inscribed, Circumscribed, and Conjugate Figures. (See also 7210 and 8075.) 507-508 Intersections of Quadrics. (See also 7660, 8440.) 508-509 Normals to Qaadrics ... 510 Plane Sections of Quad- rics 510-511 Quadrics determined by Points, Lines and Tangent Planes, and Methods of Determina- tion 512 7260 Systems of quadric surfaces. (See also 8070, 8090.) ... 513 Confocal Quadrics ... 513 Algebraic Curves and Surfaces of degree higher than the second. (See also 8070.) 7600 General 514 Curves ... ... ... 515 7610 Metrical and projective pro- perties of algebraic plane curves of degreehigherthan the second. (See also 8030, 8075,8430.) 517 Curves, 2nd and 3rd degrees ... ... 519 Curves, 3rd degree 519-522 Curves, 3rd and 4th, 3rd and 5th degrees ... 522 I Curves, 4th degree 522-524 Curvet, 5th, 5th and 6th, 6th, 14th, mth degrees 524 Curves, nth degree 524-525 \ Curves, nth and pth de- rees 525 plane. algebraic ^curves. (See also 8030.) 530 Cartesian ovals ... ... 530 Curves, 3rd degree 531-532 Curves, 3rd and 4th de- grees 532 Curves, 4th degree 532-533 Curves, 5th, 6th degrees 533 PAGE Lemniscates ... 533-534 7640 Algebraic surfaces of degree higher than the second. (See also 8040.) 536 Contacts, and Tangent Lines and Planes. (See also 8450.) ... 536-537 Surfaces, 2nd and 3rd, 2nd, 3rd and 4th, de- grees ... ... ... 539 Surfaces, 3rd degree 539-540 Configurations. (See also 8075.) 539 Surfaces, 3rd and 4th, 4th, 4th and 5th, 5th, (m + n)th, nth degrees 540 7650 Special algebraic surfaces ... 541 Anallagmatic surfaces. (See also 8020.) ... 541 Hummer's surface. (See also 8060 and 8080.) 542-543 Steiner's surface ... 544 Surfaces, 3rd degree, 3rd and 4th degrees . . . 545 Surfaces, 4th degree 545-546 Surfaces, 5th, 6th, 7th, 8th, nth, (n + 2)th de- grees 546 Tore or Anchor-ring 546-547 Wave surface ... ... 547 7660 Skew algebraic curves. (See also 7240, 7660, 8030, 8050. ) 548 Curves, 3rd degree 548-549 Curves, 3rd and 4th de- grees 549 Curves, 4th degree 549-550 Curves, 5th and higher degrees ... ... 550 Transformations and General Methods for Algebraic Configurations. (References to papers dealing with enumera- tive geometry may be found in sections 8000 to 8100.) 8000 General 552 8010 Collineation ; duality ... 552 8020 Other algebraic transforma- tions. (See also 6810, 7650.) 556 8030 Groups of points on an algebraic curve ; genus of curves ; principle of corre- spondence. (See also 7610, 7630, 7660.) 560 Groups of points 562-563 Involutions ... 563-564 Principle of correspond- ence 564-565 8040 Groups of curves and points on an algebraic surface ; genus of surfaces. (See also 7640.) 566 Ivi Schedule of Classification 8050 Application of transcendental functions to algebraic curves. (See also 4040, 4060, 7660.) 568 8060 Application of transcendental functions to algebraic sur- faces. (See also 4040, 4060, 7650.) 569 8070 Enumerative geometry. (See also 7230, 7260, 7600-7660.) 570 8075 Special configurations of points, lines, planes or other elements. Space partitioning. (See also 3620, 6820, 7210, 7610, 7640.) 573 Special configurations of points, lines, planes or other elements... 573-575 Space partitioning 575-576 8080 Line geometry. Connexes, complexes, congruences ; higher elements of space. (See also 7650.) 576 Line geometry ... 576-577 Connexes ... ... ... 577 Complexes 577-580 Congruences ... 580-582 Higher elements of space 582 8090 Systems (linear, and not linear) of curves and surfaces ... ... ... 582 Systems of curves. (See also 7230.) ... 583-585 Systems of surfaces 585-586 8100 Algebraic configurations in hyperspace ... ... 586 ^-dimensional space 589 Infinitesimal Geometry ; applications of Differential and Integral Cal- culus to Geometry. 8400 General 589 8410 Principles of infinitesimal geometry ... ... ... 590 8420 Kinematic geometry. (See also Mechanics 0420.) ... 591 Curve of pursuit ... 591 Cycloids 592 Displacement ... 592-593 Z 'cycloids ... ... 593 ^oocycloids ... ... 594 Motion 595-596 Roulettes 597 8430 Curvature of plane curves ; other applications of the differential calculus to plane curves. (See also 7210.) 598 Curves. (See also 7610, 7630.) 600-602 PAGE Envelopes ... .,. 602 Evolutes 602-603 Pedals 604-605 Radii of curvature 605-606 Trajectories ... ... 606 8440 Curvature of skew curves ; other applications of the differential calculus to skew curves. (See also 7240, 8420. ) 607 Curvature 607 Curves 607-608 8450 Curvature of surfaces ; cur- vilinear coordinates, and other applications of the differential calculus to surfaces. (See also 5220, 8810, 8840, 8850.) ... 610 Curvature of surfaces 610-616 Indicatrix ... 611-612 Lines of curvature 612-613 Measure of curvature 613-614 Radii of curvature ... 614 Surface of centres ... 615 Surfaces 615-616 Theorems 616 Curvilinear coordinates 616-618 Other applications of the differential calculus to surfaces ... 618-621 Surfaces 619-621 8455 Differential geometry of con- gruences and other applica- tions of the differential cal- culus to elements of space 621 8460 Eectification and quadrature of curves ; areas and vol- umes of surfaces. Other applications of the integral calculus to geometry ... 622 Areas 622-624 Quadrature . . . 626-62 7 Rectification ... 627-629 Volumes ... 629-630 8470 Special transcendental curves. (For cycloids, epicycloids and trochoids see 8420.) 631 Spirals 632 8480 Special transcendental sur- faces 632 8490 Hypergeometric configura- tions and higher ele- ments of hyperspace ... 633 Differential Geometry ; applications of Differential Equations to Geometry. 8800 General 634 8810 Determination of curves on surfaces ... ... ... 634 Curves on surfaces 635-636 Geodesies 636-638 Ivii Schedule of Classification 8830 Geodesies on quadrics. (See also Geography 70.) Loxodromes Minimal surfaces ... Surfaces determined by rela- tions of curvature and by other differential proper- ties Surfaces of constant cur- vature Conformal and other repre- sentations of surfaces on others. (See also 3600, 6840 and Geography 70- 95.) PAGE 638 639 640 642 643 645 8850 PAGE Conformal representation 645-646 646-647 647-648 648 649-650 650-651 Projection Representation ... Deformation of surfaces Deformation Surfaces ... Orthogonal and isothermic surfaces. (See also 5630.) 651 Isothermic surfaces ... 651 Orthogonal surfaces 651-653 8870 Hypergeometric configura- tions and higher ele- ments of hyperspace ... 653 APPENDIX 0000, 0010, 0032, 0035, 0050 ... 654 0080, 0090, 0400, 0410 655 0420, 0430, 0830, 0840, 0850, 0870, 1200, 1210 656 1230, 1600, 1620, 1625, 1630 ... 657 1635, 2000, 2010, 2020, 2030, 2040, 2050, 2060 658 2070, 2400, 2420, 2430, 2440, 2450, 2460, 2800, 2810, 2815, 2830 659 2840, 2850, 2880, 2890, 2900, 2920, 3200, 3220, 3230, 3240, 3250 660 3260, 3270, 3280, 3600, 4010, 4040, 4050, 4060 661 4420, 4440, 4800, 4830, 4840, 4850, 4870, 5200, 5210, 5230 ... 662 5240, 5600, 5610, 5620, 5630, 5660, 6000, 6400, 6410, 6420, 6430, 6800, 6810 663 6820, 6830, 6840 664 7210, 7230, 7240, 7610, 7630, 7640 665 7660, 8010, 8020, 8075, 8080, 8090, 8420, 8430, 8460, 8820 ... 666 Iviii INDEX TO SCIENTIFIC PAPERS FOR THE NINETEENTH CENTURY. PURE MATHEMATICS. 0000 Philosophy. A posteriori truth. Bing, F. (xn) Ts. Mth. 3 (1879) 1-. . (Bing). Lorenz, L. V. (xn) Ts. Mth. 3 (1879) 57-. . Bing, F. (xn) Ts. Mth. 3 (1879) 66-. . Lorenz, L. V. (xn) Ts. Mth. 3 (1879) 118-. . Bing, F. (xn) Ts. Mth. 3 (1879) 122-. Aesthetic value of line forms. Wieher, C. Karlsruhe Nt. Vr. Vh. 11 (1896) (Ab.) 47-. Algebra of relations, principles. Schroder, F. [1900] N. Y. Am. Mth. S. Bll. 7 (1901) 176-. Analysis, pure, metaphysics. Piola, G. Bb. It. 97 (1840) 315-. Aristotle, difference between natural philosophy and mathematics according to. Dutordoir, . Brux. S. Sc. A. 24 (1900) (Pt. 1), 52-. Arithmetization of math'ematics. Pierpont, J. N. Y. Am. Mth. S. Bll. 5 (1899) 394-. Axioms. Zehender, W. von. Z. Psychol. 19 (1899) 41-. "Begriffsschrift," notation in algebra of logic. Frege, G. [1879-82] Jena. Sb. (1879) 29-; (1883) 1-. of Peano and Frege. Frege, G. Leip. Mth. Ps. B. 48 (1896) 361-. Choice of forms, importance in mathematics. Clariana y Bicart, L. Fschr. Mth. (1891) 50-. Comte's views on philosophy of mathematics. Vasiljev, A. V. Kazan S. Ps.-Mth. Bll. 8 (1899) (Prot.) 11- ; Ens. Mth. 2 (1900) 157-. Conception of direction. Dixon, E. T. B. A. Ep. (1891) 572. space. Hoppe, . D. Nf . Tbl. (*1875) 142-. , an extension of conception of length. Graberg, F. Ziir. Vjschr. 31 (1886) 339-. , further development. Graberg, F. Mr. Vjschr. 32 (1887) 191-. Converse of a theorem, truth of; Hauber's criterion, applications and generalisations. Gunther, S. Arch. Mth. Ps. 56 (1874) 26-. VOL. I. Correlation of the different branches of ele- mentary mathematics. Hayward, R. B. Nt. 33 (1886) 543-. Definition of real number. Burali-Forti, C. [1900] N. Y. Am. Mth. S. Bll., 7 (1901) 158, 170-. Definitions. Laurent, H. Ens. Mth. 2 (1900) 194-. . Peano, G. [1900] N. Y. Am. Mth. S. Bll. 7 (1901) 158, 168-. , theory. Gergonne, J. D. Gergonne A. Mth. 9 (1818-19) 1-. , Whewell's views. Hennessy, J. P. B. A. Ep. (1858) (Pt. 2) 3-. Demonstration, indirect. De Morgan, A. Ph. Mg. 4 (1852) 435-. Demonstrations, reduction to simplest form. Barre de Saint -Venant, . Par. S. Phlm. PV. (1849) 45-. Equality, notion of. Burali-Forti, G. Ens. Mth. 1 (1899) 246-. Equation of the beautiful. Lagout, E. A. Gen. Civ. 2 (1863) 141-. Ex absurdo proof. Brunei, . Bordeaux S. Sc. Mm. 4 (1894) xxxii-. Failure of attempt to deduce inductive principles from the theory of probabilities. Bryant, S. Ph. Mg. 17 (1884) 510-. Foundations of the theory of analytical func- tions. Timcenko,!. N.Es. S. Nt. Mm. (Mth.} 12 (1892) 256 pp. ; 16 (1899) 216 pp. ; 19 (1899) xv + 183 pp.; Fschr. Mth. (1899) 48-. Freedom of the will and physical determinism. Hoppe, E. Arch. Mth. Ps. 11 (1892) 339-. Function of algebra, chief, formulation and simplification of formulae. Ermakow, W. P. Fschr. Mth. (1896)47. Fundamental operations of arithmetic, the five. Kupffer, K. Dorpat Sb. 10 (1895) 359-. , order of precedence. Capelli, A, Nap. Ed. 39 (1900) 138-. principles of human knowledge. Laurent, H . Ens. Mth. 1 (1899) 381-. General philosophy, aid rendered by mathe- matics. Cauchy,A.L. C. E. 21 (1845) 134-. Genius in mathematics. Studnicka, F. J. Casopis 2 (*1873) 57-; 8 (*1879), 85-; Fschr. Mth. (*1873) 54 ; (*1879) 56. 0000 Philosophy Geometric calculus, principles of. Macfarlane, A. [1900] N. Y. Am. Mth. S. Bll. 7 (1901) 158, 178. methods considered from the philosophical and historical points of view. Galdeano, Z. G. de. Fschr. Mth. (1892) 500. Geometry applied to practical life. Pearson, K. Nt. 43 (1891) 273-. as a purely logical system. Fieri, M. [1900] N. Y. Am. Mth. S. Bll. 7 (1901) 158, 171-, 234. Hegel and metaphysics of fluxional calculus. Smith, W. B. Edinb. K. S. T. 25 (1869) 491-. Induction, Aristotle's account. Whewell, W. [1850]. Camb. Ph. S. T. 9 (1856) 63-. and deduction. Jones, E. E. G. Nt. 46 (1892) 293-. . Russell, F. G. Nt. 46 (1892) 586-. . Jones, E. E. G. Nt. 46 (1892) 587-. . Dixon, E. T. [1892] Nt. 47 (1892- 93) 10-. . Jones, E. E. C. Nt. 47 (1892-93) 78-. , mathematical. Colecchi, 0. II Progresso, 17 (1837) 55-. , . Wundt, W. Ph. Stud. 1 (*1883) 90-. , , applications. Valeriani, V. G. Mt. 15 (1877) 34-. in mathematics. Hadamard, J. [1900] N. Y. Am. Mth. S. Bll. 7 (1901) 158, 181. Inference, theory of mathematical. Stokes, G. J. Am. As. P. (1899) 71. Infinite and infinitesimal quantities. Doolittle, . Smiths. Misc. Col. 33 (1888) Art. 1, 133-. (Wash. Ph. S. Bll. 6 (1884).) numbers, various opinions. Cantor, G. Stockh. Ak. Hndl. Bh. 11 (1887) No. 19, 10pp. , , Cantor's article. Enestrom, G. Stockh. 6'fv. (1885) No. 10, 69-. real integers, and their relation to each other. Cantor, . D. Nf. [B.] (*1883) 57. Infinitesimal calculus, metaphysic of. Sa- poretti, A. Bologna Ac. Sc. Mm. 5 (1895-96) 309-. Infinity, conception of in mathematics and natural science. Cranz, C. Ph. Stud. 11 (1895) 1-. , mathematical and metaphysical. Frances- chinis, F. M. Ven. Aten. Esercit. 1 (1827) 317-. Introduction to course of mathematics. Ger- gonne, J. D. Gergonne A. Mth. 21 (1830-31) 305-. Intuition and logic in mathematics. Poincare", H. N. Y. Am. Mth. S. Bll. 7 (1901) 71-. Iteration as the fundamental mathematical operation. Schapira, H. D. Nf. Vh. (1893) (Th. 2, Halfte 1), 8-; D. Mth. Vr. Jbr. 3 (1894) 88-. Language, scientific basis of an artificial. Mac- farlane, A. N. Y. Mth. S. Bll. 1 (1892) 189-. Law of continuity. An apparent violation. Boget, P. M. Ph. Mg. 3 (1828) 118-, 203-. Logic and intuition in mathematics and in teaching. Poincart, H. Ens. Mth. 1 (1899) 157-. Philosophy 0000 Logic, mathematical. Peano, G. As. Fr. C. B. (1894) (Pt. 2) 222-. Logical equalities with three terms. Poretsky, P. [1900] N. Y. Am. Mth. S. Bll. 7 (1901) 158, 172-. Mathematical form, theory. Kempe, A. B. E. S. P. 38 (1885) 393- ; Phil. Trans. 177 (1887) 1- ; E. S. P. 42 (1887) 193-. problems, and the development of mathe- matics. Hilbert, D. Ens. Mth. 2 (1900) 349-. reasoning, nature of. Poincare, H. [1894] Kazan S. Ps.-Mth. Bll. 8 (1899) (Prot.) 74-. science, relation of parts to fundamental idea. Price, B. Ashmol. S. T. 2 (1849) 23 pp. sciences, modern classifications. Dickstein, S. Wiad. Mt. 2 (1898) 181- ; Fschr. Mth. (1898) 42. studies and deductive inference. Pirez, E. Mex. S. "Alzate" Mm. 8 (1894) 315-. theory of composition of sensations. De La Rive, L. Gen. S. Ps. Mm. 30 (1890) No. 4, 99 pp. thought. Newcomb, S. [1893] N. Y. Mth. S. Bll. 3 (1894) 95-. Mathematics applied to natural history. Bu- quoy, G. von. Oken Isis (1825) 1266-. , basis of. Lipps, G. F. Ph. Stud. 9 (1894) 151-, 358-; 10 (1894) 169- ; 11 (1895) 254-; 14 (1898) 157-. , certainty in. Burja, . Berl. Mm. Ac. (1802) 3-. , experimental groundwork. Hoppe, B. D. Nf . Tbl. (*1876) (Beil.) 60-. , first notions. Liotard, . Gard Aperpu Tr. (1822) 171-. , fundamental notions. Cugnin, E. Ev. Sc. 6 (1896) 193-, 264-. , principles. Alexander, S. Silliman J. 7 (1849) 178-, 329-. , Her hart's view of place in education. Greenstreet, W. J. Mth. Gz. No. 2 (1894) 9-. and " natural history" ) r d, MS. memoir on. Gauss, C. F. Gott. Nr. (1893) 617-. "Introduction Arithmetique " (de Gerase) Livre II, ch. 9 and 20. Martin, H. Torto- lini A. 8 (1857) 429-. Invariant theory, history of development and account of present condition. Meyer, F. D. Mth. Vr. Jbr. 1 (1892) 79-. "Invention nouvelle en 1'Algebre" (reproduc- tion of 16th century work by Albert Girard) . Bierens de Haan, D. N. Arch. Wisk. 11 (1884) 83-. Irrational square-roots, calculation of, and the invention of continued fractions. Wertheim, G. Z. Mth. Ps. 42 (1897) (Suppl.) 147-. Kaiser, F. Arch. Mth. Ps. 49 (1869) 81-. , Constantijn, and Descartes. Korteweg, D. J. Arch. Neerl. 22 (1888) 422-. Hypergeometric series, Gauss' works on, errors in. Simon, H. Z. Mth. Ps. 32 (1887) (H.- It.-Ab.) 99-. , history. Schulze, K. Hamb. Mth. Gs. Mt. 1 (1889) 110-. Imaginaries, geometric representation. Hoiiel, J. Bordeaux Mm. S. Sc. 5 (1867) i-. Imaginary elements in geometry, historical monograph on. Eamorino, A. G. Mt. 35 (1897) 242- ; 36 (1898) 317-. numbers, history. StudniZka, F. J. Casopis 13 (1884) 49-, 254- ; Fschr. Mth. (1884) 37-. quantities in analysis, development. Kiess- ling, K. J. Hamb. Mth. Gs. Mt. 1 (1889) 24-. , history of theory. Beman, W. W. Am. As. P. (1897) 33-. 11 0010 History Irrationality of e and it, on early proofs of. Pringsheim,A. Munch. Ak. Sb. 28 (1899) 325-. Irrationals, quadratic, of ancients and their methods of development. Giinther, S. Z. Mth. Ps. 27 (1882) (Suppl.) 1-. Italian contributions to mathematics. Riccardi, P. Bologna Ac. Sc. Mm. 6 (1896-97) 755- ; 7 (1897) 371-. mathematical works 1800-16. Anon. (vi. 138.) Bb. It. 1 (1816) 225-, 358-; 2, 60-. sources for historico-mathematical know- ledge. Favaro, A. Bb. Mth. (1889) 113-. Jacobi's correspondence with Legendre. Jacobi, C. G. J. Bll. Sc. Mth. As. 8 (1875) 287- ; 9 (1875) 38-, 51-, 126-. posthumous works. Lejeune-Dirichlet, G. Crelle J. 42 (1851) 91-. theorem, extension, priority claim. Meyer, U. H. Grunert Arch. 22 (1854) 474-. theta formulae, date and origin. Kronecker, L. Berl. Ak. Sb. (1891) 653-. Japan, history of mathematics in. Westphal, A. (xn) [1876] D. Gs. Ostas. Mt. 1 (1873-76) (Heft 9) 54-. Japanese arithmetic, use of geometry in. Berson, . Toul. Ac. Sc. Mm. 3 (1891) 268-. mathematics. Fujisawa, . [1900] N. Y. Am. Mth. S. Bll. 7 (1901) 69. Jaquemet, Pere, two letters. Marre de Marin, E.A. Bll. Sc. Mth. As. 4 (1880) (Pt. 1) 200-. Jewish mathematics. Steinschneider, M. Bb. Mth. (1893) 65-, 105- ; (1894) 37-, 79-, 99-; (1895) 19-, 43-, 97-; (1896) 32-, 77-, 109- ; (1897) 13-, 35-, 73-, 103- ; (1898) 5-, 33-, 79-; (1899) 1-, 37-, 97-. (150150). Steinschneider, M. Z. Mth. Ps. 44 (1899) (Suppl.) 471-. Johann von Gemunden. Curtze, M. Bb. Mth. (1896) 4. Johannes de Lineriis, nationality. Curtze, M. Bb. Mth. (1895) 105-. Johannes de Muris, "Das Quadripartitum," and practical arithmetic in the 14th century. Nagl, A. Z. Mth. Ps. 34 (1889) (Suppl.) 135-. Saxonia's commentary on Introductio Alcabitii. Steinschneider, M. Bb. Mth. (1891) 114-. Jordanus Nemorarius, certain algebra problems. Wertheim, G. Bb. Mth. 1 (1900) 417-. , "Tractatus de numeris datis." Treut- lein, P. Z. Mth. Ps. 24 (1879) (Suppl.) 125-. , . Curtze, M. [1880-91] Lpldina. 18 (1882) 26-; Z. Mth. Ps. 36 (1891) (H.-lt. Ab.) 1-, 41-, 81-, 121-. 1 . Daublebsky von Sterneck, R. Mh. Mth. Ps. 7 (1896) 165-. Josephus Sapiens or Hispanus, Gerbert on. Weissenborn, H. Bb. Mth. (1893) 21-. Hispanus, Gerbert on. Curtze, M. Bb. Mth. (1894) 13-. (reply to Curtze). Suter, H. Bb. Mth. (1894) 84. Kepler's logarithms, and letters of Kepler. Frinch, . Grunert Arch. 24 (1855) 286-. Kochanski and Leibnitz, correspondence be- tween. Dickstein, S. Fschr. Mth. (1897) 9-; Krk. Ak. (Mt.-Prz.) Rz. 13 (1898) I-/ History 0010 Lagrange and Euler's correspondence. Cantor, M. Z. Mth. Ps. 23 (1878) (H.-lt. Ab.) 1-. , letters. Genocchi, A. Tor. Ac. Sc. At. 9 (1874) 746-. , letters to Euler, published by Boncompagni. Genocchi, A. [1877] Tor. Ac. Sc. At. 12 (1876) 350-. , , four. Lagrange, J. L. (x) Tor. Ac. Sc. At. 9 (1873-74) 756-. , works. Bertrand, J. J. Sav. (1869) 257-. Lancashire geometers. Wilkinson, T. T. Manch. Ph. S. Mm. 11 (1854) 123-. and ancient geometrical analysis. Wilkinson, T. T. Lane. T. Hist. S. 8 (1856) 75-. Least squares method, historical note. Abbe, C. Am. J. Sc. 1 (1871) 411-. , history. Merriman, M. Des Moines Anal. 4 (1877) 33-, 140-. , priority claims. Gerling, C. L. Gott. Nr. (1861) 273-. Legendre, Adrien Marie. M., F. Bb. Un. 52 (1833) 45-. Legendre 's correspondence with Jacobi. Le- gendre, A. M. Bll. Sc. Mth. As. 8 (1875) 287-; 9 (1875) 38-, 51-, 126-. "Theory of numbers," error in. Dujardin, . C. B. 119 (1894) 843-, 934. Leibnitz, correspondence. Jacobi, C. G. J. Berl. B. (1850) 426-. on determinants. GerJiardt, K. I. Berl. Ak. Sb. (1891) 407-. MSS. published by Gerhardt. Weissenborn, H. Schlomilch Z. 1 (1856) 240-. and Pascal. Gerhardt, C. I. Berl. Ak. Sb. (1891) 1053-. , theorem; and a letter of Lagrange. Genocchi, A. Tor. At. Ac. Sc. 4 (1868-69) 263-. Leibnitz's discovery of algorithm of higher analysis, 200th anniversary of. Gerhardt, C. I. Berl. Ac. Mb. (1875) 588-. Leo, Byzantine mathematician of 9th cen- tury. Heiberg, J. L. Bb. Mth. (1887) 33-. Leonardo da Vinci, geometrical constructions by. Cantor, M. Hamb. Mth. Gs. Mt. 2 (1890) (Festschr., Tl. 1) 8-. Vinci's geometrical construction for ellipse. Rulf, W. Mh. Mth. Ps. 9 (1898) 30-. Leonardo Pisano, life and works. Boncompagni, B. Bm. At. 5 (1851-52) 5-, 208-. , three of his writings published by Bon- compagni. Genocchi, A. Tortolini A. 6 (1855) 161-, 218-, 273-, 345-. , . Terquem, O. Tortolini A. 7 (1856) 106-. , treatise on square numbers. Woepcke, F. Liouv. J. Mth. 20 (1885) 54-. , work on cubic equations. Woepcke, F. Liouv. J. Mth. 19 (1854) 401-. Pisano's "Liber Quadratorum," problems. Genocchi, A. Tortolini A. 6 (1855) 186-, 251-. writings, and their relation to Arabian works. Woepcke, F. Km. At. 10 (1856-57) 236-; 12 (1858-59)230-, 399-; 14 (1860-61) 211-, 241-, 301-, 343-. 12 0010 History Letter "x," origin of denoting the unknown by. Wertheim, G. Z. Mth. Ps. 44 (1899) (H.-lt. Ab.) 48. Liber karastonis sive de statera. Steinschneider, M. (vm) [1862] A. Mt. 5 (1863) 54-. Liber trium fratrum. Kinkelin, H. Basel Vh. 3 (1863) 511-. (9th century work on geometry). Curtze, M. Ac. Nt. C. N. Acta 49 (1887) 105-. , proposition of an approximate cal- culation in. Vaux, C. de. Bb. Mth. (1898) 1-. Lie, Marius Sophus, communications from, 1869-71. Sylow, L. Christiania Skr. (Mth.- Nt. Kl.) (1899) No. 9, 15 pp. , , scientific work. Zorawski, K. Wiad. Mt. 3 (1899) 85-. Limits of error of 7- etc. deduced from the b known limits of error of a, b, respectively. Mehmke, R. Fschr. Mth. (1888) 1289-. Linear differential equations. Radelfinger, F. G. [1899] Wash. Ph. S. Bll. 14 (1906) 21-. Lobacevskij as algebraist. Vasiliev, A. N. Y. Mth. S. Bll. 3 (1894) 231-. , centenary of birth, celebration at Imperial University, Kasan. Weyr, E. Casopis 25 (1896) 1-; Fschr. Mth. (1896) 15. Logarithm of sum or difference, and solution of quadratic equation by circular functions, Cavalieri's methods, 1639. Govi, G. Em. R. Ac. Line. At. 3 (1876) (Pte. 2) 173-. Logarithmic tables, early history. Glaisher, J. W. L. Ph. Mg. 44 (1872) 291-, 500-. , , and their calculators. Glaisher, J. W. L. Ph. Mg. 45 (1873) 376-. , the first. Hejzlar, F. Casopis 3 (* 1874) 49-. , progress to accuracy. Glaisher, J. W. L. As. S. M. Not. 33 (1873) 330-. Logarithms, Gaussian, origin. Bouniakowsky, V. [1862] St Pet. Ac. Sc. Bll. 5 (1863) 471-. , history, critique. Matzka, W. Grunert Arch. 34 (1860) 341-. , and literature. Gieswald, . Grunert Arch. 26 (1856) 316-. , , notes. Cajori, F. Z. Mth. Ps. 44 (1899) (Suppl.) 31-. , notation of. Draenert, . D. Nf. Tbl. (*1876) (Suppl.) 173-. Logic, precursors in mathematical. Peano, G. Bv. Mt. 4 (1894) 120. , . Vacca, G. Ev. Mth. 6 (1896-99) 121-. , . Gergonne, J. D. Ev. Mth. 6 (1896-99) 183-. Longitude determination at sea, history. Gel- cich, E. Z. Mth. Ps. 44 (1899) (Suppl.) 105-. Malfatti's problem, history. Baker, M. [1877] Smiths. Misc. Col. 20 (1881) Art. 2, 113-. (Wash. Ph. S. Bll. 2 (1880).) MS. No. 14836 of the K. Hof- u. Staatsbiblio- thek, Miinchen. Curtze, M. Z. Mth. Ps. 40 (1895) (Suppl.) 75-. MSB., some old. Wittstein, A. Z. Mth. Ps. 40 (1895) (H.-lt. Ab.) 121-. History 0010 Marci , Joannes Marcus (Jan Marck) . Studnidka, F. J. Bohm. Gs. Ws. Jbr. (1890) xxxii pp. Martinus Eex and his geometry. Dickstein, S. Fschr. Mth. (1895) 10. Mascheroni's geometry of the compass. Cay- ley, A. Mess. Mth. 14 (1885) 179-. Mathematical analysis, development of theories in, during 19th century. Picard, E. [1899] Wiad. Mt. 4 (1900) 173-. anecdotes. Ferroni, P. Verona S. It. Mm. 7 (1794) 319-. historical research, purpose and results of recent. Gilnther,. D.Nf.Tbl. (*1875) 130-. history, present problems. Loria, G. [1892] Bb. Mth. (1893) 39-. instruments of Count G. Suardi of Brescia. Gelcich, E. Z. Mth. Ps. 30 (1885) (H.-lt. Ab.) 1-. knowledge previous to science. Bobynin, V. V. Bb. Mth. (1889) 104-. papyrus in Greek language. Cantor, M. Z. Mth. Ps. 38 (1893) (H.-lt. Ab.) 81-. and physical disputations at Leipzig, 1512- 26. Suter, H. Bb. Mth. (1889) 17-. science, history (Giinther) . Favaro, A . Yen. I. At. 3 (1877) 913-. science in Eussia, development. Bobynin, W. W. [V. V.-]. Fschr. Mth. (1890) 3. sciences, progress of. Zegers, J. Chile A. Un. (1852) 35-. , among European nations. Bobynin, V. V. Bb. Mth. (1892) 110-. , since 1789. Delambre, J. B. J. Par. Mm. de PI. (1808) (H.) 171-. theory of life assurance (historical). Samot, D. N. Arch. Wisk. 5 (*1879) 35-. Mathematics, development from the most remote times to 15th century. Vashchenko-Zakhar- chenko, M. [1882] Bordeaux S. Sc. Mm. 5 (1883) 259-. , historiography of . Cantor, . [1900] N. Y. Am. Mth. S. Bll. 7 (1901) 57-. , history. Piani, D. (vm) Bologna Opusc. Sc. N. Col. (1824) 191-. , . Enestrom, G. (xii) Ts. Mth. 3 (1879) 113-, 161-. , , in antiquity, series of articles. Tannery, P. Bll. Sc. Mth. As. 8 (1884); Bll. Sc. Mth. 9 (1885); 10 (1886); 11 (1887). , , applied to the solution and statement of some questions. Bobynin, W. W. [V. V.]. Fschr. Mth. (1892) 77. , by Boncompagni. Chasles, M. C. E. 34 (1852) 889-. , , contribution to. Wappler, E. Z. Mth. Ps. 34 (1889) (Suppl.) 147- ; 45 (1900) (H.-lt. Ab.) 7-. , for high-schools. Enestrom, G. (xn) Ts. Mth. 4 (1880) 62-. , , in Middle Ages, contributions to. Heiberg, J. L. Z. Mth. Ps. 35 (1890) (H.-lt. Ab.) 41-, 81-. , , a question of. Eubini, R. G. Mt. 17 (1879) 149-. , , recent researches. Gilnther, S. D. Nf. [B.] (1877) 83-. , , . Enestrom, G. Bb. Mth. (1887) 3 13 0010 History Mathematics, &c., history, proposed method of writing. Luca, F. de. II Giamb. Vico 4 (1857) 451-. , present state of some branches. Smith, H. J. S. [1876] L. Mth. S. P. 8 (1876-77) 6. , use of history in teaching. Heppel, G. Nt. 48 (1893) 16-. Matthew Stewart's theorem. Mackay, J". S. Edinb. Mth. S. P. 10 (1892) 90-. Maurolico, Francesco, on Archimedes and Apollonius. Flauti, V. Nap. Ed. 5 (1856) 112-. , , arithmetic, historical notes. Fontana, M. Bologna Mm. I. It. 2 (1808) 275-. Meaning, traditional, of word "geometry." Zeuthen, H. G. Kjrfb. Ov. (1893) 330-. Mediaeval translations of Euclid's elements. Curtze, M. Bb. Mth. (1896) 1-. Middle ages, progressions in. Curtze, M. Bb. Mth. (1895) 113-. Modern organization of mathematics. Gal- deano, Z. G. de. Fschr. Mth. (1899) 68. Mobius and Weiske, recollections of. Baltzer, E. Leip. Mth. Ps. B. 37 (1885) 1-. Monge, Gaspard, life and works. Mercadier, . Par. Mm. de PI. 24 (1854) (H.) 1-; Bv. Sc. 1 (1894) 779-. Mortality tables before Halley, history. Enes- trom, G. Stockh. Ofv. (1896) 157- ; Fschr. Mth. (1896) 187. Moscow Math. Soc., historical sketch. Mlod- zeevskij, B. K. [1894] Bee. Mth. (Moscou) 18 (1896) xxx-. Multiplication, abridged. Curtze, M. Z. Mth. Ps. 40 (1895) (H.-lt. Ab.) 1-. Napier, John, biography and works. Gravelaar, N. L. W. A. Amst. Ak. Vh. (Sect. 1) 6 (1899) No. 6, 159 pp. , , inventor of logarithms. Biot, J. B. J. Sav. (1835) 354-; Con. des Temps (1838) , , review of memoirs on. Biot, J. B. J. Sav. (1835) 257-. Nasr ad-Din and Begiomontanus. Braunmiihl, A. von. Ac. Nt. C. N. Acta 71 (1898) 31-. National mathematics, memorials. Iznoskov, I. Kazan S. Nt. (Ps.-Mth.) P. 3 (1885) 88-. Negative numbers, introduction. Fouche, M. N. A. Mth. 12 (1893) 164-. Newton, (Sir) Isaac, Brewster's Life of. Biot, J. B. J. Sav. (1832) 193-, 263-. , , Memoirs of. Biot, J. B. J. Sav. (1855) 589-, 662-. and his correspondence with Cotes. Biot, J. B. J. Sav. (1852) 133-, 217-, 269-. Pascal, forged correspondence between. Chasles, M. C. B. 65-72 (1867-71) passim. , . Brewster, (Sir) D. C. B. 65 (1867) 261-, 537-, 653-, 717-, 769-, 770-, 825-, 925-. . . Chevreul, M. E. C B 69 (1869) 305-. , portraits of. Crompton, S. [1866-67] Manch. Lt. Ph. S. P. 6 (1867) 1-; 7 (1868) 3. Newton's classification of cubic curves Ball W. W. E. L. Mth. S. P. 22 (1891) 104-. ' History 0010 Newton's influence on modern geometry. Graham, E. II. Nt. 42 (1890) 139-. Principia, a section of, in relation to modern geometry. Taylor, C. Camb. Ph. S. P. 3 (1880) 359-. , singular passage in. Tait, P. G. Edinb. B. S. P. 13 (1886) 72-. Nine-point circle, history. Mackay, J. S. Edinb. Mth. S. P. 11 (1893) 19-. Nipsus, rule, and new theorem. Poinsot, L. C. B. 28 (1849) 581-. Non-Euclidean geometry, brief historical sketch. Prosper, V. E. Fschr. Mth. (1893-94) 78. , progress in. Halsted, G. B. Am. As. P. (1899) 53-. Notation, systems used by various races. Hum- boldt, F. H. A . von. Crelle J. 4 (1829) 205-. Numerals, earliest history. Cantor, M. (vi. Adds.) D. Nf. Vsm. B. 34 (1858) 135-. , transition from Eoman to Arabic. Herford, B. Manch. Lt. Ph. S. P. 12 (1873) 91-. Numeration of ancient Mexicans. Simeon, . Mex. Arch. Com. Sc. 3 (1867) 523. , origin. Savary, A. Par. Tr. S. Amat. 2 (1808) 71-. Oppermann and Steen. Zeuthen, H. G. N. Ts. Mth. 10 (A) (1899) 33-; Fschr. Mth. (1899) 80. Optic and catoptric anamorphoses, history. Euoss, H. Z. Mth. Ps. 39 (1894) (H.-lt. Ab.) Oresme, Nicole, works. Curtze, M. Bll. Sc. Mth. As. 6 (1874) 57-. Oriental and occidental mathematics, connec- tion. Giinther, S. Lpldina. 13 (1877) 38-. origin of ir = 3. Mahler, E. Z. Mth. Ps. 27 (1882) (H.-lt. Ab.) 207-. Painvin, L., works. Bll. Sc. Mth. As. 9 (1875) 188-. Paoli, Pietro. Mod. Ac. Sc. Mm. 1 (1898) 105-. Pappus and Kepler, relation between. Giinther, S. Bb. Mth. (1888) 81-. Papyri recently discovered, dealing with mathe- matics. Heiberg, J. L. Kjtfb. Ov. (1900) Pascal denies authorship of certain papers pre- sented to Academy. Breton (de Champ), . C. B. 68 (1869) 710-, 862-, 969-. and Lalouvere. Tannery, P. Bordeaux S. Sc. Mm. 5 (1890) 55-; 4 (1894) 251-. Pascal's "Essais pour les coniques." Mac- donald, W. J. Edinb. Mth. S. P. 2 (1884) 19-. letters, supposed. Lindeldf, L. L. [1867] Helsingf. Ofv. 10 (1868) 17-. MS. notes. Faugere, . C. B. 65 (1867) 202-, 340-, 455-, 643-, 702; 67 (1868) 497. . Benard, . C. E. 65 (1867) 203-. Persian work on mathematics and astronomy. Tytler, J. As. S. J. 4 (1837) 254-. Petrus de Dacia and his writings. Enestrom, G. Stockh. Ofv. (1885) No. 3, 15-, No. 8, 65-; (1886) 57-; Fschr. Mth. (1885) 4-. Pitiscus: " Thesaurus mathematicus. " Burck- hardt, F. [1868] Basel Vh. 5 (1873) 159-. Pitiscus' " Trigonometria " (A.D. 1595), account of. Gravelaar, N. L. W. A. N. Arch. Wisk. 3 (1898) 253-; Fschr. Mth. (1898) 32-. 14 0010 History Plane curves, history of investigation on. Loria, G. Wiad. Mt. 2 (1898) 203-. , Boberval's method. Geer, P. van. N. Arch. Wisk. 11 (1884) 28-; Fschr. Ps. (1884) (Ab. 1) 217-. Platonic number. Giinther, S. Lpldina. 18 (1882) 149-. Pliny, remarks on passage in. Giinther, S. Arch. Mth. Ps. 55 (1873) 147-. Pliicker, works. Bertrand, J. J. Sav. (1867) 269-. Polish mathematical literature, 1873-92. Dik- stein, S. Kosmos (Lw.) 20 (1895) 352-. Polyhedra and numerals, history. Lindemann, F. Munch. Ak. Sb. 26 (1897) 625-. Popular account of new fields of mathematics. Miller, G. A. Science 11 (1900) 528-. Porism, old. Davies, T. S. Mathematician 3 (1850) 75-, 140-, 225-, 311- ; (Suppl.) 42-. Poristic triangles and polygons, historical notes. Mackay, J. S. Edinb. Mth. S. P. 5 (1887) 62-. "Practica Geometries" (12th century tract). Curtze, M. Mh. Mth. Ps. 8 (1897) 193- ; 9 (1898) 266-. Problem by V. Viviani. Geoghegan, E. Nt. 38 (1888) 78. Problems on numbers from Fermat, some. Pepin, T. Bm. N. Line. Mm. 8 (1892) 85-. , some inverted (an old arithmetic book). C., R. A. Educ. Times 48 (1895) 29-. Projectile curve, discovery of parabolic form. WoMwill, E. Z. Mth. Ps. 44 (1899) (Suppl.) 577-. Properties of figure composed of a triangle and the squares on its sides. Mackay, J. S. Edinb. Mth. S. P. 6 (1888) 2-. Prosthaphasresis method in trigonometry, his- tory. Braunmuhl, A. von. Z. Mth. Ps. 44 (1899) (Suppl.) 15-. Quadratic irrationalities, approximate calcula- tion, history. Hunrath, K. Z. Mth. Ps. 33 (1888) (H.-lt. Ab.) 1-. residues, historical account of law of reci- procity. Baumgart, 0. Z. Mth. Ps. 30 (1885) (H.-lt. Ab.) 169-, 241-. , study. Pepin, T. Bm. N. Line. Mm. 16 (1899) 229-. Quadratum geometricum, anonymous work on. Curtze, M. Z. Mth. Ps. 40 (1895) (H.-lt. Ab.) 161-. Quadratures before integral calculus, especially Format's. Zeuthen, H. G. Kjtfb. Ov. (1895) 37. Quaternions, Gauss' claim to invention of. Tail, P. G. [1899] Edinb. E. S. P. 23 (1902) 17-. , Klein's view of. Knott, C. G. [1899] Edinb. B. S. P. 23 (1902) 24-. Quintic equation, history of theory. Pierpont, J. Mh. Mth. Ps. 6 (1895) 15-. Babbis, irrationalities. Mahler, E. Z. Mth. Ps. 29 (1884) (H.-lt. Ab.) 41-. Baoul de Laon, arithmetical treatise. Nagl, A. Z. Mth. Ps. 34 (1889) (Suppl.) 85-. Bates of interest in ancient and modern times. Hodge, W. B. Assur. Mg. 6 (*1857) 301- ; 7 (*1858) 311- ; 8 (*1860) 68-; 9 (*1861) 61-. History 0010 Beckoning of chances (reproduction of 16th century work by Benedict Spinoza). Bier ens de Haan, D. N. Arch. Wisk. 11 (1884) 75-. Bheticus, "Canon Doctrines Triangulorum " and Vieta's "Canon Mathematicus." Hun- rath, K. Z. Mth. Ps. 44 (1899) (Suppl.) 211-. Bhind papyrus, interpretation. Favaro, A. Mod. Ac. Sc. Mm. 19 (1879) 89-. 2 on reducing to fractions with numerator 1. Bobynin, V. V. Bb. Mth. (1890) 109-. , a table in the. Mansion, . Brux. S. Sc. A. 12 (1888) (Ft. 1) 44-. Bhythmomachy. Wappler, E. Z. Mth. Ps. 37 (1892) (H.-lt. Ab.) 1-. Bichelot's posthumous papers, extract. Konigs- berger, L. Bpm. Mth. 1 (1877) 191-, 340-. Biemann and his significance for modern mathematics. Klein, F. D. Nf. Vh. (1894) (Th. 1) 57-. Bobertus (? Johannes) Anglicus and his quad- rant. Steinschneider, M. Bb. Mth. (1896) 102-. . Tannery, P. Bb. Mth. (1897) 3-. , " Tractatus Quadrantis," Germ, transl., 1477. Curtze, M. Z. Mth. Ps. 44 (1899) (Suppl.) 41-; 45 (1900) (H.-lt. Ab.) 41-. Buffini, Paolo, and the theory of groups. Burkhardt, H. Z. Mth. Ps. 37 (1892) (Suppl.) 119-. Bussia, mathematical teaching, history. Boby- nin, V. V. Ens. Mth. 1 (1899) 77-. , sketch of development of physico-mathe- matical sciences in. Bobynin, V. V. [1886] Fschr. Mth. (1887) 2. Saccheri, a precursor of Legendre and Lobacev- skij. Beltrami, E. Bm. B. Ac. Line. Bd. 5 (1889) (Son. 1) 441-. , Lobacevskij. Vasiljev, A. V. Kazan S. Ps.-Mth. Bll. 3 (1893) (Prot.) 53-. , researches on postulate of parallels. Mansion, P. Brux. S. Sc. A. 14 (1890) (Pt. 2) 46-. Scheubel, Johannes, German algebraist of 16th century. StaigmiJller, H. Z. Mth. Ps. 44 (1899) (Suppl.) 429-. Schlafli (Ludwig) and Steiner (Jakob), corre- spondence between. Graf, J. H. Bern Mt. (1896) 61-. " Seqt," so-called, of Egyptian mathematicians. Cantor, M. Wien Ak. Sb. 90 (1885) (Ab. 2) 475-. Signs of arithmetical operations, origin of some. Le Paige, C. Brux. S. Sc. A. 16 (1892) (Pt. 2) 70-. , + and - , history. De Morgan, A. [1864] Camb. Ph. S. T. 11 (Pt. 1) (1866) 203-. , , . Clifton, E. B. [1865] Camb. Ph. S. T. 11 (Pt. 1) (1866) 213-. , invention, and use by Leonardo da Vinci. De Morgan, A. (vi. Adds.) Ph. Mg. 20 (1842) 135-, 230-. + and -, use to denote "plus" and "minus." Enestrom,G. Stockh. Ofv. (1894) 243-; Fschr. Mth. (189394) 59-. 15 0010 History Simson line, historical note on so-called. Muir, T. Edinb. Mth. S. P. 3 (1885) 104-. " Sinus," origin of word. Ruska, J. Z. Mth. Ps. 40 (1895) (H.-lt. Ab.) 126-. Sluse, Be'ne' Francois de, geometrician of 17th century. Le Paige, C. Ciel et Terre 7 (188687) 536-, 570-. , , letter. Marre, A. Lisb. J. Sc. Mth. 10 (1885) 81-. Snellius, problem of. Verdam, G. J. Arch. Mth. Ps. 2 (1842) 210-. Solution of cubic, claims of Cardan and Tar- taglia. Zeuthen, H. G. Kjtfb. Ov. (1893) 303-. Sovero, Bartolomeo, mathematician of 17th century. Favaro, . Bv. Sc. 33 (1884) 592. Spain, history of mathematics, during 16th century. Enestrom, G. Bb. Mth. (1894) 33-. , mathematics in. Galdeano, Z. G. de. Ens. Mth. 1 (1899) 6-. Spanish writings of 16th and 17th centuries, certain. Vicuna, G. Bb. Mth. (1890) 33-. Spherical trigonometry formulae, history. Tod- hunter, I. Ph. Mg. 45 (1873) 98-. , historical sketch. Poreckij, P. S. Kazan S. Nt. (Ps.-Mth.) P. 5 (1887) 152-. Square root of quadrinomial surd. Bier ens de Haan,D. N. Arch. Wisk. 3 (*1877) 208-. Steiner (Jakob) and Schlafli (Ludwig), corre- spondence between. Graf, J. H. Bern Mt. (1896) 61-. Stereographic projection, history. Gunther, . D. Nf. Tbl. (1887) 227-. Stevin, Spinoza and Girard, certain works of. Enestrom, G. Stockh. Ofv. (1884) No. 9, 191- ; Fschr. Mth. (1884) 16-. Styrian mathematicians before Kepler. Pein- lich, R. Arch. Mth. Ps. 54 (1872) 470-. Summation of series of cubes (passages from MSS.). Woepcke, F. (vm) A. Mt. 5 (1863) 147- ; Liouv. J. Mth. 9 (1864) 337-; 10 (1865) 83-. Sweden, history of arithmetic. Hultman, F. W. Ts. Mt. Fys. 1 (1868) 1-, 53-, 149-, 245-; 2 (1869) 57-, 105- ; 3 (1870) 7-, 49-, 241- ; 4 (1871) 5-, 97-, 209-. , mathematical sciences before 1679. Dahlin, E. M. Ups. Arsk. (1875) iv + 146pp. , mathematics up to 1600. Enestrom, G. Stockh. Ofv. (1889) 489-; Fschr. Mth. (1889) 3-. , mathematics, Peder Mansson, three small treatises. Enestr&m, G. Bb. Mth. (1888) Swedenborg, Emanuel, as a mathematician. Enestrom, G. [1889] Stockh. Ak. Hndl. Bh. 15 (Afd. 1) (1890) No. 12, 29 pp. ; Fschr. Mth. (1890) 20. Swedenborg's mathematical works. Enestrom, G. Stockh. Ofv. (1889) 529-. Switzerland, history of mathematics. Wolf, R. Bern Mt. (1845) 121-, 137- ; (1846) 161-, 209- ; (1847) 101-, 129-, 161- ; 1848 46-, 217-, 269-. History 0010 Switzerland, history of mathematics and physics. Wolf, R. Bern Mt. (1847) 68-. Symbols, origin. Davis, H. S. Pop. As. 5 (1898) 402-. Symmedian point, early history. Mackay, J. S. Edinb. Mth. S. P. 11 (1893) 92-. "Ta Yen," so-called rule, in Europe. Curtze, M. Z. Mth. Ps. 41 (1896) (H.-lt. Ab.) 81-. Talmud, mathematics. Mahler, E. Z. Mth. Ps. 31 (1886) (H.-lt. Ab.) 121-. as source for mathematical history. Mahler, . D. Nf. Tbl. (*1882) 125-. ' Taurinus, Franz Adolph. Contribution to the previous history of non-Euclidian geometry. Stackel, P. Z. Mth. Ps. 44 (1899) (Suppl.) 399. Terminology of the oldest mathematical writings in the German language. Mailer, F. Z. Mth. Ps. 44 (1899) (Suppl.) 301-. Tests for divisibility of numbers (historical). Fontes, . Toul. Ac. Sc. Mm. 5 (1893) 459-. "Thabit Ben Korra," Steinschneider, remarks. Curtze, M. (vn) [1873] Z. Mth. Ps. 19 (1874) 95-. Theory of numbers in XV. century. Curtze, M. Bb. Mth. (1895) 37-. , Format's contention with Wallis. Wertheim, G. Z. Mth. Ps. 44 (1899) (Suppl.) 555-. , origin and development. Studnicka, F. J. Casopis 4 (* 1874) 1-. Torricelli and first rectification of a curve. Loria, G. Em. E. Ac. Line. Ed. 6 (1897) (Sem. 2) 318-. . Bordiga, G. Em. E. Ac. Line. Ed. 7 (1898) (Sem. 1) 28-. Towarzystwo Nauk Scisi'ych (Society of the exact sciences) in Paris, 1870-82, origin and development. Folkierski, W. Prace Mt.- Fiz. 6 (1895) 151- ; Fschr. Mth. (1895) 43. " Tractatus de Abaco," end of 12th or beginning of 13th century. Curtze, M. Z. Mth. Ps. 43 (1898) (H.-lt. Ab.) 122-. Trapezium, Brahmegupta's. Zeuthen, H. G. (xn) Ts. Mth. 6 (1876) 168-, 181- ; 6 (1882) 97-; 1 (1883) 145-. in Euclid, Hero, and Brahmegupta. Weiss- enborn, H. Z. Mth. Ps. 24 (1879) (Suppl.) 167-, 185-. Trigonometrical canon , earliest . De Morgan, A . As. S. M. Not. 6 (1843-45) 221-. Trigonometry in antiquity. Ideler, C. L. Zach M. Cor. 26 (1812) 3-. . Zeuthen, H. G. Bb. Mth. 1 (1900) 20-. Christian middle ages. Curtze, M. Bb. Mth. 1 (1900) 321-. , history. Braunmuhl, A. von. Ac. Nt. C. N. Acta 71 (1898) 1-. Trisection of angle, classification of solutions. Baker, M. Smiths. Misc. Col. 33 (1888) Art. 4, 96-. (Wash. Ph. S. Bll. 10 (1888).) Trochoids, history. Hoza,F. Casopis 1 (* 1872) Twilight, problem of the shortest. Zelbr, K. Z. Mth. Ps. 41 (1896) (H.-lt. Ab.) 121-, 153-. 1C 0010 Biography Tycho Brahe's table of sines. Studnicka, F. J. Prag Sb. (1899) (Mth.-Nt.) No. 39, 4 pp. Unverzagt's linear coordinates. Eudio, F. Z. Mth. Ps. 44 (1899) (Suppl.) 383-. Vandermonde, Charles Auguste. Lacepede, B. G. E. (le comte de). Par. Mm. de PI. 1 (1798) (H.) xix-. , Christian names. Simon, H. Z. Mth. Ps. 41 (1896) (H.-lt. Ab.) 83-. Viete, Frangois, discoveries. (Quadrature of circle, &c.) Bitter, F. As. Fr. C. B. (1879) 143-. , , inventor of modern algebra. Eitter, F. As. Fr. C. E. (1892) (Pt. 2) 17-. , , new algebra. Eitter, F. As. Fr. C. E. (1892) (Pt. 2) 177-. _, _, trigonometry. Eitter, F. As. Fr. C. E. (1892) (Pt. 2) 208-. Wallis's investigation of expression for IT. Cayley, A. QJ. Mth. 23 (1889) 165-. "Wargentin, P. W., and the so-called Halley method in mathematical statistics. Enestrom, G. Z. Mth. Ps. 44 (1899) (Suppl.) 81-. Weierstrass, his significance for pure mathe- matics. Vasiljev, A. V. Kazan S. Nt. (Ps.- Mth.) P. 4 (1886) 21-. , mathematical work. Poincare, H. Acta Mth. 22 (1899) 3-. Wronski, mathematical discoveries. Dickstein, S. Bb. Mth. (1892) 48-, 85-; (1893) 9-; (1894) 49-, 85-; (1896) 5-. Ziegler's commentary on " Saphea " of Zarkali. Enestrom, G. Bb. Mth. (1896) 53-. BIOGEAPHY. Abel, Niels Henrik. Mon. Sc. 12 (1870) 660- ; Bordeaux S. Sc. Mm. 1 (1884) 368 pp. Abria, Jeremie Joseph Benoit. Bordeaux S. Sc. Mm. 3 (1893) 301-. Adams, John Couch. Am. J. Sc. 43 (1892) 248 ; As. J. 11 (1892) 112 ; Educ. Times 45 (1892) 70-; Eng. S. T. (1892) 251- ; Lpldina. 28 (1892) 56 ; L. Ps. S. P. 11 (1892) (Ann. Meet.) 11-; Manch. Lt. Ph. S. Mm. & P. 5 (1892) 186- ; Nt. 45 (1892) 301- ; Obs. 15 (1892) 173- ; Am. Ac. P. 27 (1893) 444-; As. S. M. Not. 53 (1893) 184-, [348] ; Edinb. E. S. P. 20 (1895) i-. Airy, (Sir) George Biddell. Am. J. Sc. 43 (1892) 248 ; A. Ps. C. (Berl. Ps. Gs. Vh. 11) 45 (1892) 601- ; As. J. 11 (1892) 96, 168 ; As. Nr. 129 (1892) 33-; As. S. M. Not. 52 (1892) 212- ; C. E. 114 (1892) 91- ; 115 (1892) 1117- ; Eng. S. T. (1892) 244-; I. CE. P. 108 (1892) 391- ; Lpldina. 28 (1892) 54 ; Manch. Lt. Ph. S. Mm. & P. 5 (1892) 185- ; N. S. W. E. S. J. 26 (1892) 2- ; Nt. 45 (1892) 232- ; Obs. 15 (1892) 74- ; E. S. P. 51 (1892) i- ; Science 19 (1892) 64-; Am. Ac. P. 27 (1893) 446-; Edinb. E. S. P. 19 (1893) i- ; Kazan S. Ps.- Mth. Bll. 2 (1893) (Prot.) 24-; Met. S. QJ. 19 (1893) 97. Albeggiani, Giuseppe. Palermo Cir. Mt. Ed. 7 (1893) 39-. Alberti, Leon Battista. Bb. Mth. (1895) 9-. Biography 0010 Alvord, (Gen.) Benjamin. Smiths. Misc. Col. 33 (1888) Art. 2, 127-. (Wash. Ph. S. Bll. 7 (1885).) Ameseder, Adolf. Mh. Mth. Ps. 2 (1891) 479. Ampere, Andre Marie. Smiths. Ep. (1872) 111-; Amiens Ac. Mm. 7 (1881) 261- ; A. Tel. 16 (1889) 84-. Audouard, Etienne Prosper. Brest S. Ac. Bll. 20 (1895) 475-. August, Friedrich Wilhelm Oscar. Lpldina. 36 (1900) 46-. Avery, Eobert Stanton. Wash. Ph. S. Bll. 12 (1895) 435-. Azzarelli, Mattia. Em. N. Line. At. 51 (1897) 49-. Babbage, Charles. As. S. M. Not. 32 (1872) 101- ; Devon. As. T. 5 (1872) 30-; Am. Ac. P. 8 (1873) 455- ; Smiths. Ep. (1873) 162-. Babinet, Jacques. Ev. Cours. Sc. 3 (1872) 409- ; Kv. Sc.-Ind. 4 (1872) 294-. Bach, . Nancy S. Sc. Bll. (1885) xxxi-. Badon-Ghijben, Jacob. Amst. Jb. Ak. (1873) 20-. Baehr, George Frederik Willem. Amst. Ak. Vs. 7 (1899) 131-. Battaglini, Giuseppe. Ven. I. At. (1893-94) 1419- ; G. Mt. 32 (1894) 205- ; Nap. Ed. 33 (1894) 49-; Palermo Cir. Mt. Ed. 8 (1894) 180- ; Em. E. Ac. Line. Mm. 1 (1894) 558-, 665 ; Ev. Mt. 4 (1894) 91-; Tor. Ac. Sc. At. 29 (1894) 678-. Bauernfeind, Carl Maximilian von. Lpldina. 30 (1894) 161 ; 31 (1895) 62-, 78-, 94-; Forster Al. Bauztg. 60 (1895) 31- ; Munch. Ak. Sb. 25 (1896) 161-. Bellavitis, Giusto. Bll. Sc. Mth. As. 4 (1880) 343-; Em. E. Ac. Line. T. 5 (1881) 15-; Z. Mth. Ps. 26 (1881) (H.-lt. Ab.) 153- ; Ven. I. At. 8 (1882) 395- ; Nap. Ac. Pont. At. 15 1883 (Pt. 1) 5-; Em. S. It. Mm. 6 (1887) Ixxiii-. Beltrami, Eugenia. Nt. 61 (1899-1900) 568- ; Ven. I. At. (1899-1900) (Pt. 1) 72-; A. Mt. 4 (1900) 151- ; Bologna Ed. 4 (1900) 91- ; C. E. 130 (1900) 677- ; 131 (1900) 1037- ; Ens. Mth. 2 (1900) 173- ; G. Mt. 38 (1900) 355- ; Mil. I. Lomb. Ed. 33 (1900) 241- ; Nap. Ed. 39 (1900) 74-; Palermo Cir. Mt. Ed. 14 (1900) 275-; Em. E. Ac. Line. Ed. 9 (1900) (Sem. 1) 139- ; Tor. Ac. Sc. At. 35 (1900) 355- or 541- ; Wiad. Mt. 4 (1900) 266-; Kazan S. Ps.-Mth. Bll. 10 (1901) (Prot.) 32-; Munch. Ak. Sb. 30 (1901) 345-. Berg, Franciscus JoJumnes van den. 's Gravenh. I. Ing. Ts. (1891-92) (Verg.) 133; N. Arch. Wisk. 1 (1895) 1- ; Amst. Ak. Jb. (1897) 97-. Bertrand, Joseph Louis Francois. Nt. 61 (1899- 1900) 614- ; C. E. 130 (1900) 961- ; 131 (1900) 1033- ; G. Mt. 38 (1900) 171- ; J. Sav. (1900) 257-, 312- ; Lpldina. 36 (1900) 129- ; Science 11 (1900) 637 ; St Pet. Ac. Sc. Bll. 12 (1900) xxxiv ; Wiad. Mt. 4 (1900) 267-. Betti, Enrico. Nap. Ed. 31 (1892) 143-; N. Cim. 32 (1892) 5- ; Palermo Cir. Mt. Ed. 6 (1892) 245-; Ev. Mt. 2 (1892) 151- ; A. Mt. 20 (1892-93) 256 ; Ven. I. At. (1892-93) 609-. Biervliet, Albert Marie Joseph van. Brux. S. Sc. A. 15 (1891) (Pt. 1) 86. 0010 Biography Bjorling, Emanuel Gabriel. Stockh. Vt. Ak. Lefn. 2 (1878-85) 89-. Bobek, Karl. Mh. Mth. Ps. 11 (1900) 97-. Boltshauser, Giovanni Adamo. Sav. Ac. Mm. 6 (1897) Ixxxiii-. Bolyai de Bolya, Johann. Bordeaux Mm. S. Sc. 5 (1867) 203- ; Z. Mth. Ps. 42 (1897) (Suppl.) 133-. Bolyai de Bolya, (Farkas) Wolfgang. Bordeaux Mm. S. Sc. 5 (1867) 192- ; Mag. Tud. Ak. Etk. (Mth.) 11 (1885) No. 9, 36 pp. Bolzani, Josef For tunatus. Kazan Un. Bll. 12 (1876) 208-. Boncompagni, (il principe) Baldassarre. Em. N. Line. At. 47 (1894) 161- ; Z. Mth. Ps. 39 (1894) (H.-lt. Ab.) 201-. Bonnaire, Antoine Francois Donat. Caen Ac. Mm. (1891) (Ft. 2) 95, 95-. Bonnet, Pierre Ossian. C. E. 115 (1892) 1115-; 117 (1893) 1014- ; Lpldina. 28 (1892) Boole,' George. E. S. P. 15 (1867) vi-; Leic. S. T. (1835-79) 272-. Booth, (Rev.) James. Educ. Times 31 (1878) 126-. Boschi, Pietro. Bologna Ed. (1887-88) 10-. Bouquet, Jean Claude. Bll. Sc. Mth. 9 (1885) 301- ; C. E. 101 (1885) 585-; 102 (1886) 1267- ; N. Antol. Sc. 83 (1885) 404. Bour, Jacques Edmond Emile. Par. S. Phlm. Bll. 3 (1866) 119- ; N. A. Mth. 6 (1867) 145- ; Eec. Mth. (Moscou) 2 (1867) (Pt. 2) 44-. Brassai, Samuel. Mag. Tud. Ak. Ets. 10 (1899) 276-. Brassinne, Philippe Emile. Toul. Ac. Sc. Mm. 7 (Sem. 2) (1885) 25-. Bravais, Auguste. Par. Ac. Sc. Mm. 35 (1866) xxiii-; Smiths. Ep. (1869) 145-. Breton, Philippe. Isere S. Bll. 27 (1892) 391-. Bretschneider, Carl Anton. Z. Mth. Ps. 24 (1879) (H.-lt. Ab.) 73-. Brianchon, Charles Julien. Ev. Sc. 1 (1894) 592-. Brioschi, Francesco. A. Mt. 26 (1897) 343-; C. E. 125 (1897) 1139- ; Em. E. Ac. Line. Ed. 6 (1897) (Sem. 2) 353- ; Nt. 57 (1897-98) 279 ; Crelle J. Mth. 119 (1898) 259 ; G. Mt. 36 (1898) 51- ; L. Mth. S. P. 29 (1898) 721- ; Manch. Lt. Ph. S. Mm. & P. 42 (1898) xxxix ; Mth. A. 50 (1898) 477- ; N. Antol. Sc. 157 (1898) 170- ; Nap. Ed. 37 (1898) 3- ; Ev. Sc.-Ind. 30 (1898) 28 ; St. P<*t. Ac. Sc. Bll. 8 (1898) xxvii-; Wiad. Mt. 2 (1898) 66-; Wien. Aim. 48 (1898) 332- ; Mil. I. Lomb. Ed. 32 (1899) 108- ; Miinch. Ak. Sb. 28 (1899) 449-; Mathesis 20 (1900) 112- ; Palermo Cir. Mt. Ed. 14 (1900) 262-. Brisse, Charles Michel. 3. de Ps. 7 (1898) [637] ; Nt. 59 (1898-99) 80 ; N. Y. Am. Mth. S. Bll. 5 (1899) 211-. Broch, Ole Jacob. Christiania F. (1889) (Ov.) 12-; Nt. 39 (1889) 375 ; Ev. Sc. 43 (1889) 218. Bubendey, Georg Heinrich (sen.). Lpldina. 25 (1889) 215. Buchheim, Arthur. Nt. 38 (1888) 515-. Buka, Felix. Lpldina. 33 (1897) 49 ; D. Mth. Vr. Jbr. 6 (1899) (Heft 1) 23-. Biography 0010 Bumakovskij , Viktor Jakovlevic. Lpldina. 25 (1889) 219 ; Fschr. Mth. (1890) 30 ; Kharkov Mth. S. Com. 2 (1891) 149-. Buys-Ballot, Christoph Heinrich Diederich. Brux. Ac. Bll. 19 (1890) 180- ; Lpldina. 26 (1890) 58 ; Manch. Lt. Ph. S. Mm. & P. 3 (1890) 167- ; Nt. 41 (1890) 371 ; Met. S. QJ. 17 (1891) 61-; Met. Z. 8 (1891) 1-; Sym. Met. Mg. 25 (1891) 8 ; Amst. Ak. Jb. (1899) 59-. Casey, John. L. Mth. S. P. 22 (1891) 477-; Lpldina. 27 (1891) 155 ; Mathesis 11 (1891) 13 ; E. S. P. 49 (1891) xxiv-. Casorati, Felice. Bb. Mth. (1891) 1- ; Lpldina. 27 (1891) 52 ; Palermo Cir. Mt. Ed. 5 (1891) 236- ; Mil. I. Lomb. Ed. 25 (1892) 1206-. Catalan, Eugene Charles. Nt. 49 (1893-94) 437 ; Brux. Ac. Bll. 27 (1894) 326-, 330- ; Lpldina. 30 (1894) 112 ; Mathesis 14 (1894) 33 ; Kazan S. Ps.-Mth. Bll. 4 (1895) (Prot.) 84. , , letters to some mathematicians. Liege S. Sc. Mm. 17 (1892) No. 5, 22 pp. ; 18 (1895) No. 5, 36 pp. , , mathematical work. Liege S. Sc. Mm. 12 (1885) No. 1, 38 pp. Cauchy, (le baron) Augustin Louis. J. Sav. (1869) 205- ; Ev. Sc. 9 (1898) 97-. Cayley, Arthur. J. H. Un. Cir. [14 (1894-95)] 51 ; Nt. 51 (1894-95) 323 ; Bll. Sc. Mth. 19 (1895) 189- ; C. E. 120 (1895) 233- ; Crelle J. Mth. 115 (1895) 349- ; L. Mth. S. P. 26 (1895) 546- ; Manch. Lt. Ph. S. Mm. & P. 9 (1895) 235- ; Mth. A. 46 (1895) 462- ; N. Y. Am. Mth. S. Bll. 1 (1895) 133- ; Obs. 18 (1895) 112- ; Em. E. Ac. Line. Ed. 4, 1895 (Sem. 1) 177- ; E. S. P. 58 (1895) i-; Ev. Sc. 3 (1895) 188; As. S. M. Not. 56 (1896) 191- ; . Kazan S. Ps.-Mth. Bll. 5 (1896) (Prot.) 29-. Cebysev (Tchebichef), Pafnutij Lvovic. C. E. 119 (1894) 1048- ; Kharkov Mth. S. Com. 4 (1895) 263- ; Lpldina. 31 (1895) 55 ; Nt. 52 1895 345 ; Science 1 (1895) 129- ; St Pet. Ac. Sc. Bll. 2 (1895) x-, 189- ; Fschr. Mth. (1899) 15- ; Mathesis 20 (1900) 67-. Cellerier, Charles. Gen. S. Ps. Mm. 31 (1890- 93) xviii-. Challis, James. As. S. M. Not. 43 (1883) 160- ; Obs. 6 (1883) 23. Charles, Jacques Alexandre Cesar. Par. Mm. Ac. Sc. 8 (1829) (H.) 73-. Chasles, Michel. Bologna Ed. (1881) 37-; E. S. P. 32 (1881) i- ; Kharkov Mth. S. Com. (1881) 23-; Kosmos (Lw.) 6 (1881) 555-; Ev. Sc. 50 (1892) 801-. Chelini, Domenico. Bll. Sc. Mth. As. 3 (1879) 228-; Em. N. Line. At. 32 (1879) 152- ; Em. E. Ac. Line. T. 3 (1879) 54-; Em. S. It. Mm. 6 (1887) Iviii-. Chio, Felice. Ev. Sc.-Ind. 3 (1872) 61. Clebsch, Rudolf Friedrich Alfred. Gott. Nr. (1872) 550- ; Mil. I. Lomb. Ed. 5 (1872) 1041- ; Ts. Mth. 2 (1872) 190- ; G. Mt. 11 (1873) 44- ; 12 (1874) 29- ; Mth. A. 6 (1873) 197- ; 7 (1874) 1-; A. Mt. 6 (1875) 153-. Close, (Maj.-Gen.) Frederick. L. Mth. S. P. 31 (1900) 488-. Coaldey, George W. Am. J. Sc. 46 (1893) 484 ; Lpldina. 29 (1893) 204 ; Nt. 48 (1893) 398. 18 0010 Biography Cockle, (Sir) James. As. S. M. Not. 55 (1895) 192 ; L. Mth. S. P. 26 (1895) 551- ; Manch. Lt. Ph. S. Mm. & P. 9 (1895) 215- ; R. S. P. 59 (1896) xxx-. Coffin, John Hunting ton Crane. Am. Ac. P. 25 (1890) 312- ; Am. J. Sc. 39 (1890) 246. Correnti, Cesare. Firenze Ac. Georg. At. 11 (1888) xxxix-. Cotterill, Thomas. L. Mth. S. P. 12 (1880-81) 217-. Craig, Thomas. J. H. Un. Cir. [19 (1899- 1900)] 67. Dalgas, Gustavo. Firenze Ac. Georg. At. 11 (1888) xxxv-. Dauge, Felix. Mathesis 19 (1899) 177-. David, Claude Alexandre. Lille S. Mm. 1 (1865) 452-. David, Jean Marie. Toul. Ac. Sc. Mm. 2 (1890) 528-. Davidov, Avgust Julievic. Rec. Mth. (Moscou) 15 [1891] 1-; Fschr. Mth. (1890) 29. Davies, Griffith. Assur. Mg. 5 (1855) 337-. Decrue, David. Sch. Nf. Gs. Vh. (1892) 208-. De la Gournerie, Jules Antoine Rene Maillard. Par. Ac. Sc. Mm. 44 (1888) cxxxvii-; A. Cons. Arts et Met. 8 (1896) 23-. Delsaulx, (le rev. pere) Joseph. Brux. S. Sc. A. 15 (1891) (Ft. 1) 86-. De Morgan, Augustus. Ts. Mth. 1 (1871) 94- ; As. . M. Not. 32 (1872) 112-. Despeyrous, Theodore. Toul. Ac. Sc. Mm. 7 (Sem. 2) (1885) 100-. Dienger, Josef. Crelle J. Mth. 115 (1895) 350. Dodgson, (Rev.) Charles Lutwidge (Pseudonym: Lewis Carroll). Nt. 57 (1897-98) 279-, 303 ; N. Y. Am. Mth. S. Bll. 4 (1898) 241. Dodson, James. I. Act. J. 14 (1869) 341-. Drechsler, Hermann Adolf. Dresden Isis Sb. (1888) 43 ; Lpldina. 24 (1888) 171 ; 26 (1890) 193-. Drobisch, Moritz Wilhelm. Leip. Mth. Ps. B. 48 (1896) 697-. Du Bois-Reymond, Paul. Crelle J. Mth. 104 (1889) 352- ; Lpldina. 25 (1889) 112 ; Mth. A. 35 (1890) 457- ; Munch. Ak. Sb. 20 (1891) 415. Duhamel, Jean Marie Constant. Ts. Mth. 2 (1872) 143-. Dupin, (le baron) Pierre Charles Francois. A. Cons. Arts et Met. 10 (1873) 233-; Ts. Mth. 3 (1873) 64; Par. Ac. Sc. Mm. 44 (1888) i-. Durege, Heinrich. Lpldina. 29 (1893) 110; Wien Aim. 43 (1893) 264-. Eisenstein, Gotthold. Z. Mth. Ps. 40 (1895) (Suppl.) 143-. Elliott, Ezekiel Brown. Science 11 (1888) 261 ; Wash. Ph. S. Bll. 11 (1892) 470-. Ellis (Sharpe), Alexander John. L. Mth. S. P. 21 (1891) 457- ; Nt. 43 (1891) 20 ; R. S. P. 49 (1891) i-. Euler, Leonhard. Basel Vh. 7 (1885) (Anhang) 39- ; Barcel. Ac. Mm. 1 (1892-1900) 241-. Fambri, Paulo. N. Antol. Sc. 153 (1897) 131-. Fedorenko, Ivan. As. Nr. 120 (1889) 319- ; Lpldina. 25 (1889) 52. Ferroni, Pietro. Firenze Ac. Georg. At. 7 (1830) 33-. Biography 0010 Fink, Karl. Lpldina. 24 (1888) 58 ; D. Mth. Vr. Jbr. 7 (1899) (Heft 1) 33-. Finlaison, John. Assur. Mg. 10 (1863) 147-. Fleck, (I'abbe) Joseph Martin. Metz Ac. Mm. 79 (1900) 193-. Fowler, (Sir) Robert Nicholas. L. Mth. S. P. 22 (1891) 476-. Frost, (Rev.) Percival. L. Mth. S. P. 29 (1898) 726-; Nt. 58 (1898) 131; N. Y. Am. Mth. S. Bll. 4 (1898) 557 ; Science 7 (1898) 858; R. S. P. 64 (1899) vii-. Gabba, Alberto. Brescia At. Cm. (1876) 100-. Gabelli, Pasquale. Yen. Aten. 1 (1886) 72-, 209-. Gadolin, Axel Wilhelm. Nt. 47 (1892-93) 232 ; Helsingf. Acta 19 (1893) No. 18, 24 pp. Galois, Evariste. Par. EC. Norm. A. 13 (1896) 197-. Gascheau, Gabriel. Toul. Ac. Sc. Mm. 6 (Sem. 2) (1884) 17-. Gasco, Luis Gonzaga. Bb. Mth. 1 (1900) 225- ; D. Mth. Vr. Jbr. 8 (1900) (Heft 1) 26-. Gaskin, (Rev.) Thomas. As. S. M. Not. 48 (1888) 161- ; R. S. P. 46 (1890) i-. Gasparis, Annibale de. Acireale Ac. At. 3 (1891) viii; As. Nr. 129 (1892) 247-; Nap. Ac. Pont. At. 22 (1892) (Append.) 37-; 28 (1898) No. 1, 7 pp. ; Nap. I. Inc. At. 5 (1892) 9; Nap. Rd. 31 (1892) 65-; Nt. 45 (1892) 612; Obs. 15 (1892) 231- ; As. S. M. Not. 53 (1893) 225-. Gauss, Johann Friedrich Karl. R. S. P. 7 (1856) 589-; Casopis 6 (1877) 145- ; Nt. 15 (1877) 533-; 28 (1883) 272-; Science 9 (1899) 697-. Genocchi, Angelus Joannes Bartholomaeus Aloysius. Nap. Rd. 28 (1889) 79; Tor. Ac. Sc. Mm. 39 (1889) 463-. Gergonne, Joseph Diez. Mntp. Mm. Ac. Sect. Md. 3 (1862) 191-. Gerhardt, Karl Immanuel. Lpldina. 35 (1899) 136; Bb. Mth. 1 (1900) 205- ; D. Mth. Vr. Jbr. 8 (1900) (Heft 1) 28-. Germain, (Mile.) Sophie. Ausl. 5 (1832) 623-. Gibbes, Lewis R. Am. J. Sc. 49 (1895) 80. Gierster, Josef. D. Mth. Vr. Jbr. 2 (1893) 44-. Gilbert, Louis Philippe. Brux. S. Sc. A. 16 (1892) (Pt. 1) 102- ; Mathesis 12 (1892) 57; Par. S. Phlm. Bll. 4 (1892) 138-. Gill, Charles. Assur. Mg. 6 (1857) 216-. Giorgini, Gaetano. G. Mt. 31 (1893) 23-. Gogu, Constantin. Bucarest S. Sc. Bll. (1897) 103-, 109-. Gompertz, Benjamin. As. S. M. Not. 26 (1866) 104- ; Assur. Mg. 13 (1867) 1-. Graindorge, Joseph. Mathesis 16 (1896) 48. Grassmann, Hermann Gunther. Z. Mth. Ps. 23 (1878) (H.-lt. Ab.) 69-; Mth. A. 14 (1879) Graves, (Rt. Rev.) Charles. Nt. 60 (1899) 277. Graves, John Thomas. R. S. P. 19 (1871) xxvii . Gray, Peter. I. Act. J. 26 (1887) 301- ; As. S. M. Not. 48 (1888) 163-. Gretschel, Heinrich Friedrich. Lpldina. 28 (1892) 57; D. Mth. Vr. Jbr. 2 (1893) 42-. Grofe, Gustav von. Dorpat Sb. 11 (1896) 186-. 19 B2 0010 Biography Gromeka, Ippollt Stepanovic. Kazan S. Nt. (P*. Mth.) P. 8 (1890) 163 (bis)-, 168-, 181- ; Fschr. Mth. (1890) 30. Gronau, Johann Friedrich Wilhelm. Danzig Schr. 7 (Heft 1) (1888) i-. Grunert, Johann August. Arch. Mth. Ps. 55 (1873) 1-; Wien Aim. (1873) 145-. Gunther, Paul. D. Mth. Vr. Jbr. 1 (1892) 10 ; Z. Mth. Ps. 37 (1892) (H.-lt. Ab.) 46-. Guthrie, Francis. J. Bt. 37 (1899) 528; Nt. 61 (1899-1900) 84. Gylde"n, Johann August Hugo. C. B. 123 (1896) 771- ; Lpldina. 32 (1896) 189; Obs. 19 (1896) 446; Nt. 55 (1896-97) 38, 158- ; Acta Mth. 20 (1897) 397-; As. Nr. 142 (1897) 49-; As. S. M. Not. 57 (1897) 222-; Bll. As. 14 (1897) 289-; Wiad. Mt. 1 (1897) 31-. Hachette, Jean Nicolas Pierre. Fr. S. Ag. Mm. (1834) 143-. Hallauer, Octave Rene. Mulhouse S. In. Bll. 54 (1884) 139- ; Science 4 (1884) 306-. Halphen, Georges Henri. C. E. 109 (1889) 994-; 110 (1890) 489-; Liouv. J. Mth. 5 (1889) 345-; Palermo Cir. Mt. Ed. 3 (1889) 210- ; Em. E. Ac. Line. Ed. 5 (1889) (Sem. 1) 815- ; J. H. Un. Cir. [9 (1889-90)] 26; Bll. Sc. Mth. 14 (1890) 62-; Par. EC. Pol. J. 60 (1890) 137-. Hamilton, (Sir) William Rowan. Am. J. Sc. 42 (1866) 293-; As. S. M. Not. 26 (1866) 109- ; Ir. Ac. P. 9 (1867) 307-. Hankel, Hermann. Mth. A. 7 (1874) 583-. Hargreave, Charles James. E. S. P. 16 (1868) xvii-. Harnack, Carl Gustav Axel. Dresden Isis Sb. (1888) (Ab.) 3-; Lpldina. 24 (1888) 110- ; Mth. A. 32 (1888) 161- ; Z. Mth. Ps. 33 (1888) (H.-lt. Ab.) 121-. Haughton, (Rev.) Samuel. Gl. Mg. 4 (1897) 573-; I. ME. P. (1897) 514-; Nt. 57 (1897- 98) 55-; Gl. S. QJ. 54 (1898) Ixvi-; E. S. P. 62 (1898) xxix-. Haydon, Frank Scott. L. Mth. S. P. 18 (1886- 87) 392. Heilermann, Hermann. Z. Mth. Ps. 45 (1900) (H.-lt. Ab.) 57. iellwig, Karl ~ Hellwig, Karl Franz. Erfurt Ak. Jb. 24 (1898) 292. Heppel, George Hastings. Assur. Mg. 10 (1863) 83. Hermite, Charles. Manch. Lt. Ph. S. Mm & P. 45 (1901) xxxviii-. Herr, Josef Philipp. Lpldina. 20 (1884) 221- ; As. Nr. Ill (1885) 79-. Herschel, (Sir) John F. W. Brux. Ac. Bll. 31 (1871) 478-; Smiths. Ep. (1871) 109- ; Am. Ph. S. P. 12 (1872) 217- ; As. S. M. Not. 32 (1872) 122- ; Edinb. E. S. P. 7 (1872) 543-; E. S. P. 20 (1872) xvii-; Am. Ac. P. 8 (1873) 461- ; Wien Aim. (1873) 147-. Hesse, Ludwig Otto. Ts. Mth. 4 (1874) 190- ; Crelle J. Mth. 79 (1875) 345-. Hilgard, Julius Erasmus. Am. Ac. P 26 (1891) 370- ; Am. J. Sc. 41 (1891) 526; Lpldina. 27 (1891) 108; Nt. 44 (1891) 87; Eailroad & Eng. J. 65 (1891) 283-; Science 17 (1891) 272; Wash. Ph. S. Bll. 12 (1895) 462. Biography 0010 Hill, Thomas. Am. Ac. P. 27 (1893) 426-. Hirst, Thomas Archer. Lpldina. 28 (1892) 59 ; Nt. 45 (1892) 399-; As. S. M. Not. 53 (1893 218- ; E. S. P. 52 (1893) xii-; L. Ps. S. P. 12 (1894) (Ann. Meet. 1893) 9. Hochheim, Karl Adolf. Lpldina. 34 (1898) 130. Holmgren, Hjalmar Josef. Acta Mth. 7 (1885) 392. Romans, Sheppard. I. Act. J. 34 (1899) 144-. Hopkinson, John. Elektech. Z. 19 (1898) 617 ; I. ME. P. (1898) 534-; L. Mth. S. P. 29 (1898) 727-; Nt. 58 (1898) 419- ; I. CE. P. 135 (1899) 338-; L. Ps. S. P. 16 (1899) (Ann. Meet.) 9 ; E. S. P. 64 (1899) xvii-. Hoppe, Ernst Reinhold Eduard. Lpldina. 36 (1900) 132. Homer, Johann Caspar. Ziir. Vischr. 41 (1896) (Festschr. Th. 1) 79-. Horner, Johann Jakob. Ziir. Vjschr. 41 (1896) (Festschr. Th. 1) 227-. Hoiiel, Guillaume Jules. Lpldina. 22 (1886) 167- ; Bordeaux S. Sc. Mm. 4 (1888) 1-. Hunyady, Jend. Mag. Tud. Ak. Ets. 2 (1891) Imseneckij, Vasllij Grigorjevic. Kazan S. Ps.- Mth. Bll. 2 (1893) (Prot.) 15-; 3 (1893) 37-; Eec. Mth. (Moscou) 18 1896) 347-; Fschr. Mth. (1896) 19. Isbister, Alexander Kennedy. Educ. Times 36 (1883) 189- ; Peterm. Mt. 30 (1884) 101-. Jacobi, Carl Gustav Jacob. Grunert Arch. 22 (1853) 158-. Janni, Vincenzo. Nap. Ac. Pont. At. 22 (1892) (App.) 83-. Jeffery, Henry Martyn. Cornwall Pol. S. Ep. (1891) 16-; L. Mth. S. P. 22 (1891) 479-; E. S. P. 50 (1892) vii-; Cornwall E. I. J. 11 (1895) 208-. Jellett, (Rev.) John Hewitt. Nt. 37 (1888) 396. Jellicoe, diaries. I. Act. J. 24 (1884) 17-. Kelland, Philip. E. S. P. 29 (1879) vii-; Edinb. E. S. P. 10 (1880) 321-. Kendall, E. Otis. Nt. 59 (1898-99) 300; N. Y. Am. Mth. S. Bll. 5 (1899) 272. Kieseritzky, Johann Georg Gustav. Lpldina. 32 (1896) 180-. Kirchhoff, Gustav Robert. A. di C. 6 (1887) 380; Am. J. Sc. 34 (1887) 496; Berl. B. 20 (1887) 2771- ; Elektech. Z. 8 (1887) 457-; Lpldina. 23 (1887) 216; Nt. 36 (1887) 606- ; Obs. 10 (1887) 396; Tor. Ac. Sc. At. 23 (1887-88) 2-; Am. Ac. P. 23 (1888) 370- ; As. Nr. 118 (1888) 47-; A. Tel. 15 (1888) 96; L. Ps. S. P. 9 (1888) (Ann. Meet.) 12-; Phm. J. 18 (1888) 376; Wien Aim. 38 (1888) 193-; Gott. Ab. 35 (1889) (Mth.) 10 pp.; Smiths. Ep. (1889) 527-; E. S. P. 46 (1890) vi . Kirkman, (Rev.) Thomas Penyngton. Manch. Lt. Ph. S. Mm. & P. 9 (1895) 238-. Klein, Benno. D. Mth. Vr. Jhr. 1 (1892) 9. Koch, JoJiann Rudolf. Bern. Mt. (1891) xxii-. Kornuch-Trockij, Jakov Petrovic. As. Nr. 145 (1898) 127- ; Kazan S. Ps.-Mth. Bll. 7 (1898) (Prot.) 115-. Kossak, Ernst. Lpldina. 28 (1892) 56. 20 0010 Biography Kotelmikov, Petr Ivanovic. Kazan S. Nt. (Ps.- Mth.) P. 5 (1887) 225-; Fschr. Mth. (1887) 20. Kovalevskij, Sofja Vasiljevna. Bll. Se. Mth. 15 (1891) 212- ; Christiania F. (1891) (Ov.) 8-; Crelle J. Mth. 108 (1891) 88; Lpldina. 27 (1891) 59-; Nt. 43 (1891) 375-; Palermo Cir. Mt. Ed. 5 (1891) 121- ; Ev. Mt. 1 (1891) 21-; A. Mt. 19 (1891-92) 201- ; As. & Asps. 11 (1892) 281- ; Acta Mth. 16 (1892-93) '385-; Eec. Mth. (Moscou) 16 (1893) 1-. Kraus, Ludwig. Casopis 15 (1886) 49-; Fschr. Mth. (1886) 22. Kronecker, Leopold. Lpldina. 27 (1891) 207; 28 (1892) 94-; A. Ps. C. (Berl. Ps. Gs. Vh. 11) 45 (1892) 595-; C. E. 114 (1892) 19-; Mathesis 12 (1892) 18; N. Y. Mth. S. Bll. 1 (1892) 173- ; Berl. Ak. Ab. (1893) 22 pp. ; D. Mth. Vr. Jbr. 2 (1893) 5-; Mth. A. 43 (1893) 1-; Munch. Ak. Sb. 22 (1893) 202. Kummell, diaries Hugo. Wash. Ph. S. Bll. 13 (1900) 404-. Kummer, Ernst Eduard. C. E. 116 (1893) 1163- ; Lpldina. 29 (1893) 110; D. Mth. Vr. Jbr. 3 (1894) 13-; Mathesis 14 (1894) 40-; Munch. Ak. Sb. 24 (1895) 140-. Kunze, Karl Ludwig Albrecht. Lpldina. 27 (1891) 78-, 94-. Laborie, Edmond. Angers S. Sc. Bll. (1891) 236. Lagrange, Joseph Louis (comte) de. Par. Mm. del'I. (1812) (H.) xxvii-; Ts. Mth. 1 (1877) 129-. Laguerre, Edmond Nicolas. C. E. 103 (1886) 424-; 104 (1887) 1643- ; Par. EC. Pol. J. 56 (1886) 213- ; N. A. Mth. 6 (1887) 105- ; 7 (1888) 352. Lalanne, Leon Louis Chretien. C. E. 114 (1892) 569-; 115 (1892) 1118-. Lamarle, Anatole Henri Ernest. Brux. Ac. Bll. 39 (1875) 360-. Lame, Gabriel. A. Mines 1 (1872) 271- ; Ev. Sc. 14 (1878) 720-. Langren, Michel-Florent van. Ciel et Terre 12 (1891-92) 241-, 297-. Laplace, Pierre Simon (le marquis) de. Hall Bij. 2 (1826) 284-; Par. Mm. Ac. Sc. 10 (1831) (H.) 81-. Lavagna, Giovanni Maria. Ev. Sc.-Ind. 2 (1871) 40-. Le Besgue, Victor Amedee. N. A. Mth. 16 (1877) 116-. Lefavour, Edward Brown. Wash. Ph. S. Bll. 11 (1892) 488-. Legendre, Adrien Marie. Smiths. Ep. (1867) 137. Lejeune Dirichlet, Gustave Pierre. Tortolini A. 3 (1860) 221-, 283-. Letnikov, Aleksej Vasiljevic. Eec. Mth. (Mos- cou) 14 (1890) i-; Fschr. Mth. (1889) 21. Liagre, Jean Baptiste Joseph. Ciel et Terre 11 (1890-91) 541- ; Brux. Ac. Bll. 21 (1891) 77-. Lie, Marius Sophus. Tor. Ac. Sc. At. 34 (1898) 235- or 363-; Nt. 59 (1898-99) 445-; Arch. Mth. Ntvd. 21 (1899) No. 1, xxiii pp.; C. E. 128 (1899) 525-; 129 (1899) 1062; D. Nf. Vh. (1899) (Th. 2, Halfte 1) 4-; Ens. Mth. 1 Biography 0010 899) 156; Leip. Mth. Ps. B. 51 (1899) xi-; Mth. S. P. 30 (1899) 334-; Lpldina. 35 (1899) 60; N. Ts. Mth. 10 (B) (1899) 48; N. Y. Am. Mth. S. Bll. 5 (1899) 367-; Em. E. Ac. Line. Ed. 8 (1899) (Sem. 1) 281-, 360- ; Science 9 (1899) 447-; St. Pet. Ac. Sc. Bll. 10 (1899) xxxiii-; Bb. Mth. 1 (1900) 166- ; D. Mth. Vr. Jbr. 8 (1900) (Heft 1) 30-; Kazan S. Ps.-Mth. Bll. 9 (1900) (App.) 32 pp.; Manch. Lt. Ph. S. Mm. & P. 43 (1900) xxxiii-; Mathesis 20 (1900) 228-; Mth. A. 53 (1900) 1-; Munch. Ak. Sb. 30 (1901) 339-. Ligin, W. N. Fschr. Mth. (1900) 29. Ligowski, Wilhelm. D. Mth. Vr. Jbr. 4 (1897) 46. Liouville, Joseph. Edinb. E. S. P. 14 (1888) 83 (bis)-. Lobacevskij , Nikolaj Ivanovic. Kazan Un. Mm. (1857) (Bk. 4) 153- ; Bll. Sc. Mth. As. 1 (1870) 66-, 324-, 384-; Mathesis 13 (1893) 117-, 193; Ev. Sc. 52 (1893) 636-; Kazan S. Ps.-Mth. Bll. 3 (1893) (Prot.) 27-; 4 (1895) (Prot.) 4-; Eec. Mth. (Moscou) 17 (1895) 474-; Z. Mth. Ps. 40 (1895) (Suppl.) 205- ; Bordeaux S. Sc. Mm. 1 (1896) 307-. Lobatto, Rehuel. Amst. Jb. Ak. (1866) 25-; Arch. Mth. Ps. 49 (1869) 332-. Lommel, Eugen Cornelius Joseph von. Elek- tech. Z. 20 (1899) 458; Lpldina. 35 (1899) 160- ; D. Mth. Vr. Jbr. 8 (1900) (Heft 1) 47-. Loomis, Elias. Lpldina. 25 (1889) 214; Met. Z. 6 (1889) 466-; Nt. 40 (1889) 401; Obs. 12 (1889) 358; Eailroad & Eng. J. 63 (1889) 435; Am. Ac. P. 25 (1890) 324-; Am. J. Sc. 39 (1890) 427-; Met. S. QJ. 16 (1890) 101; Sid. Mess. 9 (1890) 241- ; Smiths. Ep. (1890) 741- ; Sym. Met. Mg. 24 (1890) 122. Lovering, Joseph. Am. J. Sc. 43 (1892) 167; Lpldina. 28 (1892) 102; Am. Ac. P. 27 (1893) 372-. Lubbock, (Sir) John William. As. S. M. Not. 26 (1866) 118- ; E. S. P. 15 (1867) xxxii-. Luca, Ferdinando de. II Polit. 18 (1870) 612-. Lucas, Edouard. Mathesis 11 (1891) 217. Luther, Eduard. Lpldina. 23 (1887) 216; 24 (1888) 52; Obs. 10 (1887) 432-; As. Nr. 118 (1888) 31-; As. S. M. Not. 48 (1888) 169- ; Kimigsb. Schr. 28 (1888) (Ab.) 105-. Lutter, Ndndor (Ferdinand). Lpldina. 28 (1892) 53; Mag. Tud. Ak. Ets. 3 (1892) 531-. Luvini, Giovanni. Acireale Ac. At. 3 (1891) vii. M'Cann, H. W. Beng. As. S. P. (1884) 113-. Machovec, Franz. Mh. Mth. Ps. 3 (1892) 403. Maddem, Lorenzo. Acireale Ac. At. 2 (1891) xv. Mainardi, Gaspare. Em. E. Ac. Line. T. 3 (1879) 126-. Majer, Jozef. Wiad. Mt. 3 (1899) 302-. Makeham, William Matthew. I. Act. J. 30 (1893) 1-. Maksimovic, Vladimir Pavlovic. Kazan S. Nt. (Ps.-Mth.) P. 8 (1890) 53- ; Fschr. Mth. (1890) 21 0010 Biography Malmsten, Carl Jolian. Acta Mth. 7 (1885) 394; Goteb. Hndl. 26 (1891) 67-. Marsilly, (le gen.) Louis Joseph Auguste de Commines de. Bone Ac. Hip. Bll. 24 (1891) (C. E. 1890) 1-. Masdea, Giuliano. Nap. I. Inc. At. 1 (1899) 20. Mathieu, Emile Leonard. Lpldina. 26 (1890) 215; Nancy S. Sc. Bll. (1891) 1-; N. Y. Mth. S. Bll. 1 (1892) 156-, 236. Matthes, Corel Johannes. Amst. Ak. Jb. (1882) 1-. Maus, Henri. Brux. Ac. Bll. 26 (1893) 173-. Maxwell, James Clerk. Edinb. K. S. P. 10 (1880) 331- ; Kosmos (Lw.) 6 (1881) 129- ; E. S. P. 33 (1882) i-. Mayr, Alois. Lpldina. 26 (1890) 217. Mehler, Gustav Ferdinand. Lpldina. 31 (1895) 170; Mth. A. 48 (1897) 603-. Meissel, Ernst Daniel Friedrich. As. Nr. 137 (1895) 239-; Lpldina. 31 (1895) 102. Melinikov, Michail Ivanovic. Kazan S. Nt. (Ps.-Mth.) P. 4 (1886) 49-. Merrifield, Charles Watkins. L. Ps. S. P. 5 (1884) (Ann. Meet.) 13-; Nt. 29 (1884) 270; Nv. Archt. T. 25 (1884) 309- ; K. S. P. 36 (1884) i-. Meyer, Arnold. Ziir. Vjschr. 42 (1897) 65-; D. Mth. Vr. Jbr. 5 (1901) (Heft 1) 18-. Meyer, Friedrich. D. Mth. Vr. Jbr. 8 (1900) (Heft 1) 59-. Miller, Thomas. Edinb. E. S. P. 19 (1893) liv-. Minding, Ernst Ferdinand Adolf. Z. Mth. Ps. 45 (1900) (H.-lt. Ab.) 113-. Minich, Serafino Eafaele. Yen. I. At. 1 (1882- 83) 1095-. Minnigerode, B. Lpldina. 32 (1896) 143-. Mister, Jean Nicolas. Mathesis 18 (1898) 241. Mitchell, Oscar Howard. J. H. Un. Cir. 8 (1888-89) 110. Mobius, August Ferdinand. Arch. Mth. Ps. 49 (1869) (Lt. B.) clxxxxv. Mollinger, Otto. Lpldina. 23 (1887) 54. Moigno, (I'abbe) Francois Napoleon Marie. A. Tel. 11 (1884) 289; Les Mondes 8 (1884) 443-; Nt. 30 (1884) 291-. Moseley, (Canon) Henry. Nv. Archt. T. 13 (1872) 328-. Mossotti, Ottaviano Fabrizio. As. S. M. Not. 24 (1864) 87-. Moth, Franz Xaver. Wien Aim. (1879) 172-. Nagel, Christian Heinrich. Mathesis 7 (1887) Nash, Alfred Moses. L. Mth. S. P. 26 (1895) 557-. ' Neumann, Franz Ernst. C. B. 120 (1895) 1189- ; D. Nf. Vh. (1895) (Th. 1) 27-, (Th. 2, Halfte 1) 5-; Elektech. Z. 16 (1895) 353; Gott. Nr. (1895) 248-; Nt. 52 1895 176; Lpldina. 32 (1896) 51-, 63-; Wien Aim. 46 (1896) 271- ; D. Mth. Vr. Jbr. 4 (1897) 54-; R. S. P. 60 (1897) viii-. Newton, Hubert Anson. As. Nr. 141 (1896) 407- ; Asps. J. 4 (1896) 237; Nt. 54 (1896 394; Am. J. Sc. 3 (1897) 359-; As. S. M. Not. 57 (1897) 228-; N. Y. Am. Mth. S. Bll. 3 (1897) 169- ; Pop. As. 4 (1897) 236-; E. S. P. 63 (1898) i-. Biography 0010 Nobert, Friedrich Adolph. N.-Vorp. Mt. 15 (1884) 38-. Novarese, Enrico. Ev. Mt. 2 (1892) 35. Ofterdinger, Felix Ludwia. Bb. Mth. (1896) 50-; Lpldina. 32 (1896) 103. Oliver, James Edward. Nt. 51 (1894-95) 587; Science 1 (1895) 544-; Am. Ac. P. 31 (1896) 367-. Olivier, Theodore. A. Cons. Arts et Met. 8 (1896) 21-. Omerique, Antonio Hugo de. Fschr. Mth. (1895) 15. Oppermann, Ludvig Henrik Ferdinand. Ts. Mth. 1 (1883) 137-. Orlov, Aleksandr Petrovic. Kazan S. Nt. (Ps.- Mth.) P. 7 (1889) 396-. Paciuolo, Lucas. Z. Mth. Ps. 34 (1889) (H.-lt. Ab.) 81-, 121-. Padova, Ernesto. Em. E. Ac. Line. Ed. 5 (1896) (Sem. 1) 284-. Padula, Fortunato. Em. E. Ac. Line. T. 6 (1882) 30-. Palmieri, Luigi. Lpldina. 32 (1896) 147; Nap. Ac. Pont. At. 26 (1896) (Necrol. No. 5) 15 pp. ; Nap. I. Inc. At. 9 (1896) 12-; Nap. Ed. 35 (1896) 236-. Pankiewicz, Jan. Wiad. Mt. 3 (1899) 302. Paolis, Eiccardo de. Palermo Cir. Mt. Ed. 6 (1892) 208-. Parkinson, (Eev.) Stephen. E. S. P. 45 (1889) i-; As. S. M. Not. 50 (1890) 167-. Peck, William Grey. Eailroad & Eng. J. 66 (1892) 146. Peirce, Benjamin. Pop. As. 3 (1896) 49-. Penot, Jean Fleury Achille. Mulhouse S. In. Bll. 56 (1886) 505-. Perigal, Henry. Aer. J. 2 (1898) 67; L. Mth. S. P. 29 (1898) 732-; Nt. 58 (1898) 131; Obs. 21 (1898) 282-; Science 7 (1898) 858; As. S. M. Not. 59 (1899) 226-; Met. S. QJ. 25 (1899) 223-. Pestalozzi, Ham Jakob. Zur. Vjschr. 41 (1896) (Festschr. Th. 1) 123-. Pettko, Jdnos. Mag. Tud. Ak. Ets. 3 (1892) 429-. Petzval, Josef. Mh. Mth. Ps. 2 (1891) 480; Wien Aim. 42 (1892) 182-. Piani, Domenico. Bologna Ac. Sc. Mm. 1 (1871) 3-. Plana, Giovanni Antonio Amedeo. As. S. M. Not. 24 (1864) 89-; Edinb. E. S. P. 5 (1866) 293; Bll. Sc. Mth. As. 5 (1873) 65-; Par. Ac. Sc. Mm. 38 (1873) cvii-; Em. S. It. Mm. 6 (1887) Ixxxvi-. Plarr, Gustav. Nt. 45 (1892) 419. Plch, Kornel. Casopis 19 (1890) 51-. Pliicker, Julius. E. S. P. 17 (1869) Ixxxi-; Bll. Sc. Mth. As. 3 (1872) 59-; 5 (1873) 313- ; Gott. Ab. 16 (1872) 40 pp.; G. Mt. 11 (1873) 153-. Poinsot, Louis. Par. Ac. Sc. Mm. 45 (1899) Ixxiii-. Poncelet, Jean Victor. A. Cons. Arts et Met. 7 (1895) 1-. Ponte Horta, Francisco da. G. Teix. J. Sc. 14 (1900) 3-. Popov, Aleksandr Thedorovich. Kazan S. Nt. (Ps.-Mth.) P. 1 (1883) (No. 9) 13-. 22 0010 Biography Prediger, Carl. D. Mth. Vr. Jbr. 4 (1897) 51-. Price, (Rev.) Bartholomew. Nt. 59 (1898-99) 229-; As. S. M. Not. 59 (1899) 228-; L. Mth. S. P. 30 (1899) 332-; L. Ps. S. P. 16 (1899) (Ann. Meet.) 10-; Science 9 (1899) 188; E. S. Yearbook (1900) 185-. Prowe, Leopold Friedrich. Z. Mth. Ps. 33 (1888) (H.-lt. Ab.) 89-. Puchewicz, Wtadysiaiv. Wiad. Mt. 3 (1899) 302. Puiseux, Victor Alexandre. Bll. Sc. Mth. As. 8 (1884) 234-; Par. Ac. Sc. Mm. 44 (1888) Ixvii-. Quet, Antoine. Les Mondes 9 (1884) 582-. Quetelet, Lambert Adolphe Jacques. Amst. Ak. Wet. P. (1873-74) (No. 9) 1-; Arch. Sc. Ps. Nt. 49 (1874) 350- ; Bll. Sc. Nord 6 (1874) 62-; Brux. Ac. Bll. 37 (1874) 245-; 38 (1874) 816- ; Bv. Sc.-Ind. 6 (1874) 38-; As. S. M. Not. 35 (1875) 176- ; Edinb. E. S. P. 8 (1875) 474-; Mag. Tud. Ak. Ets. 8 (No. 16) (1875) 241- ; E. S. P. 23 (1875) xi-. Quick, (Eev.) Eobert Herbert. Educ. Times 44 (1891) 185-. Eaynard, Arthur Cowper. L. Mth. S. P. 26 (1895) 554-. Eealis, Savin. N. A. Mth. 5 (1886) 200-. Eenjifo, Ismael. Santiago de Chile Un. A. 100 (1898) 527-. Eesal, Henri Ame. A. Mines 10 (1896) 625-; C. E. 123 (1896) 435-; Liouv. J. Mth. 2 (1896) 455-; Lpldina. 32 (1896) 147- ; Manch. Lt. Ph. S. Mm. & P. 41 (1897) liii. Eeuschle, Carl Gustav. Leip. As. Gs. Vjschr. 10 (1875) 178- ; Z. Mth. Ps. 21 (1876) (H.-lt. Ab} 1-. Ehodes, Edivard Hawksley. L. Mth. S. P. 26 (1895) 558. Eibaucour, Albert. Mathesis 13 (1893) 270-. Eibeiro de Sousa Pinto, Rodrigo. G. Teix. J. Sc. 12 (1895) 2-. Eiccardi, Pietro. Mod. Ac. Sc. Mm. 2 (1900) xxii-. Eiemann, Georg Friedrich Bernhard. Gott. Nr. (1867) 305- ; Eec. Mth. (Moscou) 3 (1868) (Pt. 2) 153- ; K. S. P. 16 (1868) Ixix-. Bitter, Ernst. Lpldina. 31 (1895) 218; Nt. 52 (1895) 600 ; D. Mth. Vr. Jbr. 4 (1897) 52-. Eoberts, Samuel Oliver. L. Mth. S. P. 31 (1900) 282-; Mth. Gz. 1 (1900) 278. Eoche, Edouard Albert. C. E. 96 (1883) 1171- ; Lille S. Mm. 14 (1885) 17-. Eogers, William Augustus. Am. J. Sc. 5 (1898) 322; As. Nr. 146 (1898) 343-; Asps. J. 7 (1898) 390- ; Ps. Ev. 6 (1898) 315- ; Science 7 (1898) 447-. Eosenberger, Johann Carl Ferdinand. Elektech. Z. 20 (1899) 690; Lpldina. 35 (1899) 182; Bb. Mth. 1 (1900) 217-. Eossi Ee, Vincenzo de. Em. N. Line. At. 42 (1889) 83-. Eubini, Raffaele. Nap. Ed. 29 (1890) 134- ; Nap. Ac. Pont. At. 21 (1891) 275-. Sagastume, Demetrio. Arg. S. Ci. A. 47 (1899) 97-. Saint-Venant, Adhemar Jean Claude Barre de. A. Pon. 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Berl. Mb. (1860) 714-. , 19th roots of unity, of primes up to 1000. Eeuschle [C. G.]. Berl. Mb. (1860) 719-. , 23rd roots of unity, of primes up to 1000. Eeuschle \C. G.]. Berl. Mb. (1860) 725-. , 29th roots of unity, of primes up to 1000. Eeuschle [C. G.]. Berl. Mb. (1860) 729-. Compound interest. Weninger, V. (xn) Mag. Ak. Ets. (Mth. Term.) 6 (1865) 207-. Congruence M= a a b? (Mod. N), specimen table. Cayley, A. QJ. Mth. 9 (1868) 95-. Connecting curves (on railroads). Eibette, . (vra) A. Cond. Pon. Chauss. 5 (1861) 108-. Cremona's transformation between two planes. Eoberts, S. [1872] L. Mth. S. P. 4 (1871- 73) 121-. Tables of Functions 0035 Cube roots of primes to 31 places. Drach, S. M. [1877] Mess. Mth. 7 (1878) 86-. Definite integrals. Bierens de Haan, D. Amst. Vh. 5 (1858) 1-. , (Bierens de Haan's). Bellavitis, G. Yen. At. (1858-59) 423-. , supplement. Bierens de Haan, D. [1861] Amst. Vh. Ak. 10 (1864) 92 pp. A m O"-MI(m). Cayley, A. [1879] Camb. Ph. S. T. 13 (1883) 1-. 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Norm. 1 (1800) 16-, 268-, 381- ; (pte. 2) 3-, 130- ; 2 (1800) 116-, 302- ; 3 (1800) 24-; 4 (1800) 32-, 223-; 5 (1800) 201- ; 6 (1800) 32-. . Lagrange, J. L. de. Par. Se. EC. Norm. 1 (1800) (pte. 2) 3-, 34-; 3 (1800) 227-, 276-, 463-; 4 (1800) 401- ; 10 (1801) 5-. (elementary), 1795. Lagrange, J. L. de. Par. EC. Pol. J. 2 (1812) (7 e et 8 e call.) 173-. Mathematical problems. Hilbert, D. Gott. Nr. (1900) 253-. theories, etc. (Presidential address, 1883.) Cayley, A. B. A. Bp. (1883) 3-. tripos, the. (Presidential address.) Glaisher, J. W. L. L. Mth. S. P. 18 (1886-87) 4-. Mathematics. Kempe, A. B. [1894] L. Mth. S. P. 26 (1895) 5-. , influence on physical science. Bastarrica, J. Chile A. Un. (1852) 41-. and natural science, mutual relations. Wild, J. St Gal. B. (1885-86) 101-. Phys. Sect. Address, 1865. Spottiswoode, W. B. A. Kp. 35 (1865) (Sect.) 1-. , , 1869. Sylvester, J. J. B. A. Bp. 39 (1869) (Sect.) 1-. . , , 1870. Maxwett, J. C. B. A. Ep. 40 (1870) (Sect.) 1-. , , 1871. Tait, P. 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Pedagogy 0050 Universities and technical schools, address. Klein, E. D. Mth. Vr. Jbr. 7 (1889) (Heft 1) 39-. 0050 Pedagogy. Actuarial competence, requisites. Sutton, W. I. Act. J. 16 (*1872) 434-. Actuaries, importance of mathematical studies. Sprague, T. B. I. Act. J. 18 (*1876) 403-. , mathematical education. Kiepert, L. D. Nf. Vh. (1894) (Th. 2, Hdlfte 1) 16-. , . Fehr, H. Ens. Mth. 1 447-. Actuary, education. Porter, H. W. Mg. 4 (*1854) 108-. Algebra, the first chapter. Anon. Educ. Times 39 (1886) 326-. in schools. Heppel, G. Mth. Gz. No. 4 (1895) 25-. , teaching, in 1676. (Henrion's "Euclid.") Mansion, . Brux. S. Sc. A. 19 (1895) (Ft. 1) 101-. American colleges, mathematics. Eddy, H. T. Am. As. P. (1884) 37-. Analytical geometry, combination of planimetry and stereometry in teaching. Eipert, L. As. Fr. C. E. (1900) (Pt. 2) 1036-. Application of mathematical and mechanical principles to technical problems. Heun, K. D. Mth. Vr. Jbr. 5 (1901) (Heft 1) 91-. Arithmetic and algebra, elementary teaching. Eecknagel, G. D. Nf. Vh. (1899) (Th. 2, Hdlfte 1) 279-. , the first stage. Quick, (Eev.) E. H. Educ. Times 41 (1888) 227-. , the five fundamental operations. Kupffer, K. Dorpat Sb. 10 (1895) 359-. , teaching. Edinb. Mth. S. Comm. Edinb. Mth. S. P. 6 (1888) 89-. , , method. Elowson, G. Ts. Mt. Fys. 2 (1869) 40-. , , . Moesta, C. W. Chile A. Un. (1852) 322-. Arithmetical problems in secondary schools, treatment. Burali-Forti, C. Ev. Mt. 1 (1891) 31-. Blind, teaching of the congenitally. Penjon, . Gergonne A. Mth. 3 (1812-13) 308-. Combination of solid with plane geometry in teaching. Candida, G. Ens. Mth. 1 (1899) 204-. Descriptive geometry, first lesson. Budelot, G. Ens. Mth. 1 (1899) 286-. , lacuna in the teaching. Tilly, (Maj.) de. Brux. S. Sc. A. 9 (1885) (Pt. 2) 95-. Educational testimony concerning the calculus. Meech, L. W. Des Moines Anal. 1 (1874) 133-. Elliptic functions, teaching of theory. Mansion, . Brux. S. Sc. A. 24 (1900) (Pt. 1) 54-. Equations, choice of examples in teaching. Breton \de Champ}, P. N. A. Mth. 2 (1843) 539-. Errors in mathematical text books. Beyer, . Grunert Arch. 3 (1843) 113-. Euclid and teaching of geometry. Magnus, P. Educ. Times 33 ( 1880) 81-. 30 0050 Pedagogy Examination questions, composition. Laisant, C. A. Ens. Mth. 1 (1899) 120-. . , solutions. Sprague, T. B. Assur. Mg. 10 (*1863) 45-; 13 (*1867) 253-; I. Act. J. 14 (* 1869) 147-. , _ ( a correction). Adler, M. N. I. Act. J. 14 (*1869) 242. Examinations, judging in mathematical. Bordoni, A. [1841] Mil. Mm. I. Lomb. 1 (1843) 189-. , problems suitable. Wittstein, T. Grunert Arch. 7 (1846) 27-. Exposition of pure mathematics. Delarue, D. M. (xn) [1869] Eec. Mth. (Moscou) 4 (1869-70) (Ft. 2) 81-. Formulae, special application of mnemonics. Hoffmann, L. Forster Al. Bauztg. 10 (1845) 362-. Geometrical drawing, teaching. Henrici, 0. B. A. Ep. (1895) 608. teaching, reform. Fiedler, W. Ziir. Vjschr. 22 (1877) 82-. , . Wittek, H. D. Nf. Vh. (1894) (Th. 2, Hdlfte 1) 317-. Geometry and analysis, higher, Naples Uni- versity, 1870-71. Bernardinis, G. de, <& Fuortes, T. G. Mt. 9 (1871) 233-. , elementary, new trend in. Ciamberlini, C. Ev. Sc.-Ind. 30 (1898) 99-, 113-, 153-. , , teaching. Develey, . Bb. Un. 8 (1818) 257-. , , . Barclay, A. J. G. Edinb. Mth. S. P. 2 (1884) 24-. , , . Diekmann, . D. Nf. Tbl. (1888) 342-. , experiment in teaching. Fraser, A. Y. Edinb. Mth. S. P. 5 (1887) 94-. , first teaching. Laisant, C. A. Ens. Mth. 1 (1899) 339-. , higher, teaching. Chasles, M. Liouv. J. Mth. 12 (1847) 1-. , natural. Mault, A. Tasm. E. S. P. (1886) 103-. , practical, development. (Graphics.) Anon. Elect. 28 (1892) 298-. , , . ( .) Thompson, S. P. Elect. 28 (1892) 332-. , solid, teaching in higher schools. Hildebrandt, C. D. Nf. Vh. (1897) (Th. 2, Halfte 1) 34-. , , at Strassburg. Krazer, A. D. Mth. Vr. Jbr. 8 (1900) (Heft 1) 119-. in schools. Heppel, G. Edue. Times 50 (1897) 181-. , teaching. Dallas, E. J. Nt. 59 (1898- 99) 416. , . Eeinhardt, N. V. Kazan S. Ps.-Mth. Bll. 8 (1899) (Prot.) 19-. versus Euclid. Minchin, G. M. Nt. 59 (1898-99) 369-. Graduated system of teaching, especially in geometry. Bazala, J. D. Nf. Vh. (1894) (Th. 2, Hdlfte 1) 313-. Higher mathematics at Italian Universities, teaching. Pascal, E. Ev. Mt. 3 (1893) 170-. Hyperbolic functions in teaching. Barbarin, P. Ens. Mth. 2 (1900) 443-. 31 Pedagogy 0050 Imaginaries, treatment in higher school teaching. Pietzker, F. [1898] D. Mth. Vr. Jbr. 7 (1899) (Heft 1) 154-. Indices and surds, teaching. Genese, E. W Mth. Gz. 1 (1900) 339-. Infinitesimal calculus, course. Obrecht, A. Santiago de Chile Un. A. 87 (1894) 533-, 657-, 715- ; 88 (1894) 39-; 90 (1895) 45-; 91 (1895) 677-; 93 (1896) 105-, 241- ; 103 (1899) 699-, 887-; 104 (1899) 39-, 201-, 359-, 747-, 1061- ; 106 (1900) 79-, 375-, 545-, 949-, 1097-. Infinity in elementary mathematics. Lamarle, E. [1851] Brux. Ac. Sc. Mm. 27 (1853) 31 pp. Inverted problems (an old arithmetic book). C., H. A. Educ. Times 48 (1895) 29-. Languages and mathematics in state schools of modern Germany. Baumann, H. Educ. Times 37 (1884) 98-. Leibnitz, calculus of, and its teaching. Arnoux, E. As. Fr. C. E. (1900) (Pt. 2) 1089-. Mathematical history, 8 years of teaching. Favaro, A. Bb. Mth. (1887) 49-. , lectures in Moscow University, pro- gramme of course. Bobynin, V. V. Fschr. Mth. (1890) 87; Bb. Mth. (1891) 79-. at Munich. Steinschneider, M. Bb. Mth. (1897) 113-. , university course. Enestrom, G. Bb. Mth. (1890) 1-. instruction in France. Pierpont, J. N. Y. Am. Mth. S. Bll. 6 (1900) 225-. problems, and the development of mathe- matics. Hilbert, D. Ens. Mth. 2 (1900) 349-. teaching. Hudson, W. H. H. Educ. Times 46 (1893) 135-. . Biraben, F. Arg. S. Ci. A. 48 (1899) 106-, 156-. . Duclaux, E. Ev. Sc. 11 (1899) 353-. . Garciade Galdeano, Z. As. Fr. C. E. (1900) (Pt. 2) 1052-. . Perry, J. Nt. 62 (1900) 317-. . Heaviside, O. Nt. 62 (1900) 548-. , changes. Piola, G. (vm) [1840] Bb. It. 2 (1841) 159-. , curiosities in early English. Heppel, G. Educ. Times 38 (1885) 145-. , elementary. Laisant, . Ev. Sc. 11 (1899) 358-. , general principles. Galdeano, Z. G. de. Ens. Mth. 1 (1899) 195-. at German Universities, especially Gottingen. Fano, G. Ev. Mt. 4 (1894) 170-. in Germany, history. Giinther, S. Ens. Mth. 2 (1900) 237-. in higher grade schools (gymnasiums and lyceums). Catania, S. Ev. Mt. 5 (1895) 33-. Holland. Cardinaal, J. Ens. Mth. 2 (1900) 317-. , methods. Grunert, J. A. Arch. Mth. Ps. 42 (1864) 348-. in preparation for the " Ecole Poly- technique." Laurent, H. Ens. Mth. 1 38-. 0050 Pedagogy Mathematical teaching, reform. Mair, D. Nt. 62 (1900) 389. , . Woollen, H. Nt. 62 (1900) 436. , . Beard, W. F. Nt. 62 (1900) 466. , . Stromeyer, C. E. Nt. 62 (1900) 523. in Eussia. Bobynin, V. V. Ens. Mth. 1 (1899) 420-. , history. Bobynin, V. V. Ens. Mth. 1 (1899) 77-. schools. Lie, S. Christiania F. (1884) No. 16, 8pp.; (1885) No. 11, 10 pp. . Tafelmacher, A. Santiago de Chile Un. A. 85 (1893) 35-. , simplification of elementary. Heppel, G. Educ. Times 44 (1891) 463-. in United States. Safford, T. H. N. Y. Mth. S. Bll. 3 (1894) 4-. universities in 1894. Galdeano, Z. G. de. Fschr. Mth. (1893-94) 118. , problems and methods. Klein, F. D. Mth. Vr. Jbr. 7 (1899) (Heft 1) 126-. , use of history. Heppel, G. Nt. 48 (1893) 16-. , practical applications. Bettazzi, E. Ens. Mth. 2 (1900) 14-. Mathematics for astronomy and navigation. Wilson, T. Mth. Gz. No. 4 (1895) 27-. at certain German Universities. Dillner, G. Ups. Arsk. (1876) 37 pp. , elementary, economy of thought. Meyer, W. F. D. Mth. Vr. Jbr. 7 (1899) (Heft 1) 147-. in girls' schools. Bryant, (Mrs.) . Educ. Times 31 (*1878) 145-. , how to study. Degen, C. F. Kiob. Dn. Yd. Selsk. Skr. 2 (1801-2) 75-. , relations between university and high school in domain of. Schotten, H. D. Nf. Vh. (1898) (Th. 2, Halfte 1) 20-. in Eussia, history of study. Bobynin, V. Bb. Mth. (1888) 103-. schools. Heppel, G. Educ. Times 39 (1886) 385-. as a scientific and pedagogic aid. Bugaev, N. V. (xn) [1869] Eec. Mth. (Moscou) 3 (1868) (Pt. 2) 183-. in Spain. Galdeano, Z. G. de. Ens. Mth. 1 (1899) 6-. in 1893-94. Galdeano, Z. G. de. Fschr. Mth. (1893-94) 117. teaching of physics, importance. Calzecchi, T. Ev. Sc.-Ind. 30 (1898) 65-. Mensuration, note. Lefevre, A. Texas Ac. Sc. T. 2 (No, 2) (1899) 39-. Multiplication, method of performing. Berdelle, C. As. Fr. C. E. (1884) (Pt. 2) 584-. Notion of the infinite in teaching elementary geometry. Eipert, L. Ens. Mth. 2 (1900) 127-. . (Correction of error in Eipert's paper, p. 129.) Niewenqlowski, B. Ens. Mth. 2 (1900) 219. a limit in teaching. Clasen, . Braunschw. Vr. Nt. Jbr. (11) (1899) 159-. on number and limit in teaching. Prinqs- heim, A. D. Mth. Vr. Jbr. 6 (1899) (Heft 1) 73-. Institutions 0060 Problem solving as a discipline. Hall, A. Wash. Ph. S. Bll. 11 (1892) 598-. Pure mathematics for engineering students. Hathaway, A. S. [1900] N. Y. Am. Mth. S. Bll. 7 (1901) 266-. Quadratic equation, theory. Macfarlane, A. Ens. Mth. 2 (1900) 363-. equations, solution. Mathews, G. B. Nt. 57 (1897-98) 463-. Seducing results of measurement. Rndel, K. D. Nf. Vh. (1893) (Th. 2, Halfte 1) 217-. Eole of mathematics in general education. Budelot, G. Ens. Mth. 1 (1899) 222. Eules of multiplication, their nature and teaching. Baron, B. Ens. Mth. 1 (1899) 317-. Scientific method in teaching. Binet, A. Ens. Mth. 1 (1899) 29-. Smith's-Prize papers. Cayley, A. Mess. Mth. 3 (1874) 165- ; 4 (1875) 6- ; 6 (1877) 173-. Statutes for testing teachers, and Mathematical course. Hauck, G. D. Nf. Vh. (1899) (Th. 2, Halfte 1) 17-. . (Prussian.) Weber, H. D. Nf. Vh. (1899) (Th. 2, Halfte 1) 9-. . ( .) Scliotten, H. D. Nf . Vh. (1899) (Th. 2, Halfte 1) 26-. Statutory tests for teachers in Bavaria. Eudel, K. D. Nf. Vh. (1899) (Th. 2, Halfte 1) 29-. Synthetic and projective geometry, need of, in analytical geometry. Davidescu, D. A. Bucarest S. Sc. Bll. (1900) 19-. Systematisation of mathematics. Ellis, A. J. E. S. P. 10 (1859-60) 85-. Technical High Schools in America. Engels, H. Giving. 40 (1894) 13-. instruction, especially its geometry. Paganini, C. Mil. G. I. Lomb. 1 (1841) 291-. teaching in Chili. Nogues, A. F. [1893- 94] Santiago de Chile Un. A. 85 (1893) 277-, 345-, 515-. Training of teachers of science in higher schools, in view of modern requirements. Schwalbe, B. D. Nf. Vh. (1896) (Th. 2, Halfte 1) 15-. Trigonometry, teaching. Aragon,A. Mex. S. "Alzate" Mm. 5 (1891) 69-. , of elements. Fehr, H. Ens. Mth. 1 (1899) 45-. 'Unitary' method in teaching proportion. Beta. Educ. Times 32 (*1879) 259-. Universities and technical schools, address. Klein, F. D. Mth. Vr. Jbr. 7 (1899) (Heft 1) 39-. University lectures on infinitesimal calculus. Pringsheim, A. D. Mth. Vr. Jbr. 7 (1899) (Heft 1) 138-. Vector theory, teaching. Fontene, G. Ens. Mth. 1 (1899) 50-. 0060 Institutions. Franklin Institute during 75 years. Thurston, E. H. Franklin I. J. 149 (1900) 81-. 32 0070 Nomenclature 0070 Nomenclature. "Al-djebr," exact meaning of word. Vaux, C. de. Bb. Mth. (1897) 1-. Algorithm, meaning of term. Hunrath, K. Bb. Mth. (1887) 70. Antiparallels. Langley, E. M. Nt. 40 (1889) 460-. "Autotomic" curves. Basset, A. B. Nt. 62 (1900) 572. . Orchard, H. L. [1900] Nt. 63 (1900- 01)7. . Thorn, A. S. [1900] Nt. 63 (1900- 01) 7. . Basset, A. B. [1900] Nt. 63 (1900- 01) 82. Chinese arithmetical notation. Edkins, . Nt. 37 (1888) 65. "Conjugate angles," new geometrical term. Worthington, A. M. Nt. 47 (1892-93) 8. Cuneiform numerical signs, transmission. Bobynin, V. Bb. Mth. (1893) 18-. Curves without double points. Richmond, H. [1900] Nt. 63 (1900-01) 58. Decimal division of circle (i.e. of right angle). Hatzfeld, A. Bv. Sc. 48 (1891) 655-. Determinant notation. Nan-son, E. J. Ph. Mg. 44 (1897) 396-. Differential notation and teaching. Poincare, H. Ens. Mth. 1 (1899) 106-. Duodecimal system. Heaviside, 0. Elect. 37 (1896) 489. Functions, growth of notation for. Enestrom, G. Bb. Mth. (1891) 89-. General and logico-mathematical notation. Oliver, J. E. A. Mth. 4 (1888) 186-. Geometrical square, nomenclature. Weissen- born, H. Bb. Mth. (1888) 37. Geometrography or systematic notation for geometric constructions. Lemoine, E. G. Teix. J. Sc. 12 (1895) 114-. Gnomon theorem (Euc. I. 43). Allman, G. J. Bb. Mth. (1887) 22. Kardaga. Steinschneider,M. Bb. Mth. (1891) 114. Language, mathematical and scientific. Anon. (vi 543). Gergonne A. Mth. 12 (1821-22) 322-. Logarithms, notation. Draenert, . D. Nf. Tbl. (*1876) (Suppl.) 173-. Nomenclature. Dantas Pereira, J. M. Lisb. Mm. Ac. Sc. 10 (1830) (pte. 2) 197-. , geometrical. Fontene, G. Ens. Mth. 2 (1900) 134- . of metric system. Rautenfeld-Lindenruh, H. von. Eiga Cor.-Bl. 38 (1895) 85. Notation. Lubbock, J. W. [1829] As. S. Mm. 4 (1831) 471-. . Sylvester, J. J. Ph. Mg. 7 (1854) 50-. . Houzeau, J. C. Brux. Ac. Bll. 39 (1875) 487-. , algebraic. Jarrett, T. [1827] Camb. Ph. S. T. 3 (1830) 65-. , . Cottins, J. V. Science 20 (1892) 359. Nomenclature 0070 Notation, arithmetical. McColl, H. Educ. Times 33 (*1880) 111. , calculus. Couto, M. V. do, & Simoes Margiochi, F. Lisb. Mm. Ac. Sc. 3 (pte. 2) (1814) 48-. of decimal fractions. Gravelaar, N.L.W.A. N. Arch. Wisk. 4 (1899) 54-; Fschr. Mth. (1899) 37. , numerical. Prevost, P. Bb. Un. 3 (1816) 306- . , questions concerning. Eipert, L. Ens. Mth. 1 (1899) 139. useful in analysis. Ocagne, M. d\ Par. S. Mth. Bll. 15 (1887) 156-. TT to denote 3-14159..., first use. Enestrom, G. Bb. Mth. (1889) 28. , . Ball, W. W. E. Bb. Mth. (1894) 106. Printing mathematical symbols. Mitchell, C. T., and others. Elect. 29 (1892) 652,, et seq. . Thompson, S. P. Nt. 46 (1892) 513. mathematics. Cassie, W. [1892] Nt. 47 (1892-93) 8-. . Jackson, M. J. Nt. 47 (1892-93) 227. Quantitative comparisons, a common error in speech. Johnson, G. H. Science 22 (1893) 352. Eoulettes. Willson, F. N. Am. As. P. (1887) 59-. Signs (for functions, operations, &G.). Hoppe, R. Arch. Mth. Ps. 61 (1877) 323-. Spoken numeration, from international point of view. Berdelle, C. Ens. Mth. 1 (1899) 269-. Symbols. Hall, A. [1889] A. Mth. 5 (1889- 90) 19. of non-existence. Doolittle, M. H. Wash. Ph. S. Bll. 11 (1892) 603. Terminology. Laisant, C. A. Ens. Mth. 1 (1899) 22-. and conceptions. Gyory, S. (xn) Mag. Tud. Ak. Ets. 3 (No. 12) (1869) 131-. , mathematical. Roberts, S. [1882] L. Mth. S. P. 14 (1882-83) 5-. , suggestions. Muirhead, R. F. Edinb. Mth. S. P. 13 (1895) 143. Trigonometry, development of notation. Braun- milhl, A. von. Bb. Mth. 1 (1900) 64-. Umbral notation. Harley, (Rev.) R. B. A. Up. (1887) 600-. Uniform notation for mathematical and physi- cal quantities. Winkler, E., t& Keck, . Giving. 30 (1884) 285-. . Keck, . Hann. Archt.- Vr. Z. 30 (1884) 285-. Use of + and to denote ' ' plus ' ' and ' ' minus. ' ' Enestrom, G. Stockh. Ofv. (1894) 243-; Fschr. Mth. (1893-94) 59-; Bb. Mth. (1899> 105-. terms "square" and "cube" in Algebra. Taylor, W. B. Wash. Ph. S. Bll. 11 (1892V 590-. Variation, variable, &c., remarks on the words. Strauch, G. W. Grunert Arch. 7 (1846) 221-. 33 0080 Instruments 0080 Instruments, including Calculating Machines. Models. INSTEUMENTS. Algebraical machines. Torres, L. As. Fr. C. R. (1895) (Pt. 2) 90-. Angle, correction for tilt of horizontal circle. Matzka, W. Grunert Arch. 13 (1849) 113-. , instrument for dividing into equal parts. Gouin, E. Les Mondes 12 (1866) 470-. measurer. Heineken, N. J. Intell. Obs. 8 (1886) 197-. for schools. Fischer, F. W. [1876] Arch. Mth. Ps. 61 (1877) 99-. , mechanical division. Perrin, E. [1875] Par. S. Mth. Bll. 4 (1876) 85-. _, . Brocard, H. [1876] Par. S. Mth. Bll. 5 (1877) 43-. Antigraph, a reversing drawing instrument. Marcus, S. Wien SB. 17 (1855) 282-. Arcs, instrument for drawing. Alderson, Jas. Gill Tech. Hep. 10 (1829) 128-. Area, moment of inertia, and centre of gravity of a plane area, apparatus for giving. Col- lignon, E. (ix) A. Pon. Chauss. 3 (1872) 223-. Arithmoplanimeter. Lalanne, L. Par. A. Pon. Chauss. 20 (1840) 3-. Axonometric drawing, protractors for, prepara- tion. Anon, (vi 316) Giving. 7 (1861) 83-. Besson, Jacques, (t!569), (Mathematical in- struments). Beck, T. Giving. 36 (1890) 191-. Calculating machine. Cunq's "machine a debits," for calculating quantity of water flowing through an orifice. Sornin, J. Toul. Mm. Ac. 6 (1856) 346-. Calculimeter (sort of slide rule). Fenton, J. J. Viet. B. S. P. 2 (1890) 84-. Chinese abacus or souan-pan. Rodet, L. Par. S. Mth. Bll. 8 (1880) 158-. Circles, instrument for dividing on paper. Balfour, J. M. (vi Adds.) [1859] Edinb. T. Sc. S. Arts 5 (1861) 149-. with remote centres, construction. Fischer, E. Dingier 294 (1894) 133-. Circular arcs, instrument for use of mechani- cians for comparison of. Sipos, P. (vi Adds.) . Berl. Sm. D. Ab. 2 (1796) 201-. Compass, bicentric. Maze (I'abbe) C. Les Mondes 8 (1884) 703-. for drawing spirals. Hartung, A. Dingier 266 (1887) 12-. , elliptic, new. Jost, K. Wien Ak. Sb. 95 (1887) (Ab. 2) 251-. , hyperbolic. Eeichelt, K. F. Cztg. Opt. 17 (1896) 208. Compasses, copying , a simple form of panto- graph. Sedlaczek, J. Wien SB. 16 (1855) 180-. , elliptic. Burchard, L. Carl Bpm. 13 (1877) 528-. , improvement. Mollinger, 0. Sch. Gs. Vh. (1850) 52-. , proportional. Narrien, J. Thomson A Ph. 8 (1816) 338-. Instruments 0080 Compasses, proportional. Butkevlch, F. S. (xn) [1873] Mosc. S. Sc. Bll. 39 (No. 2) (1880) 128-. _, _. Palacios, D. Mex. S. " Alzate " Mm. 7 (1893) 193-. , trisecting, Laisant. Brocard, H. Par. S. Mth. Bll. 3 (1875) 47-. Conic section compass. Meyn, R. Giving. 8 (1862) 247-. compasses. Frizell, J. P. Camb. (M.) Mth. M. 1 (1859) 262-. , Drzewiecki's for direct construction of conies with mathematical pen. Kuhn, M. Carl Bpm. 10 (1874) 420-. sections, instrumentfor drawing. Rebicek, G. Carl Bpm. 11 (1875] 64-. Curve description; kinograph. Engelmeier, P. K. [1886-89] Mosc. S. Sc. Bll. 65 (No. 1) (1890) 89-. for drawing curves (elastic lamina) ' ' tira- curve-opisometer." Rossi Re, V. de. Bm. N. Line. At. 33 (1880) 232-. tracer. Pirani, E. Arch. Mth. Ps. 1 (1884) 113-. and root finder. Bashforth, F. B. A. Bp. (1845) (pt. 2) 3-. tracing apparatus. Cayley, A. L. Mth. S. P. 4 (1871-73) 345-. , designing of apparatus for. TJiallmayer, V. Carl Bpm. 14 (1878) 713-. machine. Dechevrens, M. C. B. 130 (1900) 1616- . , mechanical. Jessop, C. M. QJ. Mth. 22 (1887) 151-. Curves described by Jopling's apparatus, nature. Tredgold, T. Tilloch Ph. Mg. 62 (1823) 211-. , mechanical description. Russell, W. H. L. [1869] B. S. P. 18 (1870) 72-. Curvimeter (for finding length of curves). Crane, W. R. Kan. Un. Q. 4 (1896) 121-. Cycloids, description. Kracht,H. Dingier 289 (1893) 108-. Descriptive geometry, apparatus for illustrating. Hoza, F. Arch. Mth. Ps. 53 (1871) 188-. , teaching. Weissand, . N. A. Mth. 14 (1855) 47-. Diagonal scale principle applied to angular measurement in circular slide-rule. Campbell, J. R. L. Mth. S. P. 6 (1875) 73-. Diagrammometer, cypher. Kozlov, V. S. Bs. Ps.-C. S. J. 23 (Ps.) (1891) 180- ; C. E. Ill (1890) 166-. Difference engine, new. Grant, G. B. Am. J. Sc. 2 (1871) 113-. Division of angles and arcs of circles. Murphy, I. J. Belfast NH. S. Ep. & P. (1892-93) 47-. straight line into equal parts, machine for. Soave, F. Verona Mm. S. It. 8 (1799) 56-. toothed wheels, machine for. Glasser, A. von. Dingier 252 (1884) 56-. Drawing instruments, especially ellipsographs in theory and practice. Fischer, E. Dingier 255 (1885) 188-, 217-, 261-. Egg-oval, perfect, machine for drawing ; and advantageous method of producing curvilineal figures. Hay, D. R. Edinb. N. Ph. J. 40 (1846) 331-. 34 0080 Instruments Ellipses, paradox in tracing. Lametz, . Metz Ac. Mm. 79 (1900) 69-. Ellipsograph. Leninn, M. St Pet. Ac. Sc. Bll. 5 (1839) 377-. . Thallmayer, V. Dingier 226 (1877) 237-; 227 (1878) 337-. . Sidersky, D. Arch. Mth. Ps. 65 (1880) 420- . . Hendricks, J. E. Des Moines Anal. 10 (1883) 161-. . Grant, G. B. Franklin I. J. 122 (1886) 301. . Schromm, F. Z. Instk. 12 (1892) 139-. . Delone, N. Mosc. S. Sc. Bll. 83 (No. 2) (1894) 11-; Fschr. Ps. (1895) (Ab. 1) 351. . Riefler, C. Z. Instk. 16 (1896) 115-. of Proclus. Blake, E. M. [1898] Am. J. Mth. 22 (1900) 146-. and protractor. Burnliam, S. W. Pop. As. 4 (1897) 181-, 336. Ellipsographs. Zistl, M. Cztg. Opt. 5 (1884) 245-, 258-, 265-. . Hartmann, W. Z. Instk. 11 (1891) 285-. Ellipsometer. Henry, F. (x) A. Pon. Chauss. 3 (1872) 459-. . Jannettaz, E. Fr. S. Mn. Bll. 15 (1892) 237- ; 16 (1893) 205-. Elliptographs, and a mechanical rocker for detecting oscillations. Alexander, T., & Thomson, A. W. Ir. Ac. T. 29 (1887-92) 673-. Epi- and hypo-cycloids, apparatus for tracing. Thallmayer, V. Dingier 228 (1878) 312-. Equations, apparatus for graphic solution. Lill, E. Les Mondes 16 (1868) 325-. of form \ n + aA 2 + 6X + c = 0, Mehmke's ap- paratus for solution. Mlodzeevskij, B. K. Mosc. S. Sc. Bll. 92 (No. 1) (1896) 36-; Fschr. Mth. (1896) 72. with imaginary roots, apparatus for graphic solution. Lill, E. N. A. Mth. 7 (1868) 363-. , machine for solution. Girard, P. van. Wien Jb. Pol. I. 14 (1829) 62-. , . Bashforth, F. B. A. Ep. (1845) (pt. 2) 3-. , . Boys, C. V. L. Ps. S. P. 7 (1886) 355-; Ph. Mg. 21 (1886) 241-. , . Cunynglwme, H. L. Ps. S. P. 7 (1886) 360- ; Ph. Mg. 21 (1886) 260-. , . Ocagne, M. d'. J. de Ps. 5 (1896) 310-. , . Meslin, G. C. B. 130 (1900) 888-. , of linear. Veltmann, W. Z. Instk. 4 (1884) 338-. , simultaneous linear. Thomson, (Sir) W. (Lord Kelvin). [1878] B. S. P. 28 (1879) 111-. , transcendental. Herschel (Sir) J.F. W. [1832] Camb.Ph. S. T. 4 (1833) 425-. , mechanical solution. Kempe, A. B. [1872] Mess. Mth. 2 (1873) 51-. , . Guarducci, F. Bm. K. Ac. Line. Mm. 7 (1891) 219-. , 3rd order, mechanical solution by a quadri- lateral linkage. Greenhill,A.G. Mess. Mth. 5 (1876) 162-. Instruments 0080 Equations, 3rd order, solution by slide-rule. Bour, E. [J. E. E.] C. E. 44 (1857) 22-. , 4th and 5th orders, solution by moving mechanisms. Ameseder, A. Wien Ak. Sb. 93 (1886) (Ab. 2) 380- . to 7th order, machine for solution. Lalanne, L. C. E. 11 (1840) 859-. , (Lalanne). Cauchy, A. L. G. E. 11 (1840) 959-. Generation of orthogonal projections of plane curves, mechanical. Delaunay, N. Z. Mth. Ps. 40 (1895) 242-. Genesis of quantity demonstrated by means of two mechanical instruments. Araldi, (Capi- tano) A. (vn) [1828] Mod. Mm. Ac. Sc. 1 (1858) (pte. 4) 90-. Geometrical drawing, apparatus for. Zmurko, W. [L.] (xn) Kosmos (Lw.) 5 (1880) 44-. Geometrography applied to different solutions of a problem. Lemoine, E. Par. S. Mth. Bll. 20 (1892) 132-. Harmonic analyser. Henrici, 0. [1894] L. Ps. S. P. 13 (1895) 77-; Ph. Mg. 38 (1894) 110-. . LeConte, J. N. Ps. Ev. 7 (1898) 27-. . Michelson, A. A. <& Stratton, S. W. Am. J. Sc. 5 (1898) 1-. for graphic determination of coefficients in Fourier series. Bervy, N. Mosc. Obs. A. 3 (Pt. 2) (1896) 109-. , Henrici and. Coradi's. Grabowski, L. Wiad. Mt. 4 (1900) 236-. , reading amplitude and epoch directly. Sharp, A . [1894] L. Ps. S. P. 13 (1895) 89- ; Ph. Mg. 38 (1894) 121-. , remarks on Henrici's. Perry, J. L. Ps. S. P. 13 (1895) 97- ; Ph. Mg. 38 (1894) 125-. , simple form. Yule, G. U. L. Ps. S. P. 13 (1895) 403- ; Ph. Mg. 39 (1895) 367-. Helical epicycloids, instrument for describing. Burnaby, S. B. Amt. Mch. S. J. 3 (1879) 115-, 160-, 194-, 228-. Herpolhode of Poinsot, and an apparatus of Darboux and Koenigs. Gilbert, P. Brux. S. Sc. A. 14 (1890) (Pt. 2) 25-. Hydraulic integration of differential equations. Petrovitch, M. Am. J. Mth. 20 (1898) 293-. Hyperbolas, apparatus for drawing. Thall- mayer, V. Dingier 228 (1878) 106-. , method of drawing. Burch, G. J. Ph. Mg. 41 (1896) 72-. , , Burch's. Wadsworth, F. L. 0. Ph. Mg. 41 (1896) 372-. Hyperbolograph. Cunynghame, H. [1886] L. Ps. S. P. 8 (1887) 67-; Ph. Mg. 22 (1886) 138-. Instruments for ovals and for drawing in per- spective. B., R. Nicholson J. 9 (1804) 122-. Integral of product of 2 given functions, instru- ment for calculating. Thomson, (Sir) W. (Lord Kelvin). E. S. P. 24 (1876) 266-. Integraph, new. Barker, A. H. Sc. Abs. 1 (1898) 381. Integrating differential equations of first order, Petrovich's apparatus. Price, W. A. Ph. Mg. 49 (1900) 487-. Integration, mechanical. Abdank-Abakanowicz, B. C. E. 94 (1882) 783-. 35 02 0080 Instruments Integration, mechanical. Nehls, C. Giving. 20 (1874) 71- ; 21 (1875) 131-, 199-, 261-. , , applied to three bodies' problem. Strom- gren, E. Stockh. Ofv. (1900) 443-. , , of general linear differential equation of any order with variable coefficients. Thomson, (Sir) W. (Lord Kelvin). E. S. P. 24 (1876) 271-. , , linear differential equations of 2nd order with variable coefficients. Thomson, (Sir) W. (Lord Kelvin). E. S. P. 24 (1876) 269-. , , the product of two functions. Suther- land, W. Ph. Mg. 20 (1885) 175- ; 21 (1886) 141-. Integrator, Amsler's. Nehls, C. Hann. Z. Archt. Vr. 18 (1872) 534-. . Abakanowicz,B. (xn) Krk. Ak. (Mt.-Prz.) Bz. & Sp. 7 (1880) 75-. , cubical, description. McElroy, G. W. A. Mth. 3 (1887) 105-. , disk-globe-and-cylinder. Thomson, (Prof.) Jas. E. S. P. 24 (1876) 262-. , instrument for graphic integration. Abdank- Abakanowicz, B. C. E. 92 (1881) 402-, 515-. , mechanical, Amsler-Laffon's. Merrifield, C. W. Nv. Archt. T. 21 (1880) 252-. , , for areas, etc. Deprez, M. C. E. 73 (1871) 785-; Les Mondes 27 (1872) 10-. , -, ((Xdx+Ydy). Cayley, A. B. A. Ep. (1877) (Sect.) 18-. , , work done by a force. Fuchs, K. Mth. Termt. Ets. 13 (1895) 239-; Mth. Nt. B. Ung. 13 (1897) 144-. , new form. Marinovitch, B. Lum. Elect. 17 (1885] 529-. , model. Napoli, D. & Abdank-Abakano- wicz, . C. E. 101 (1885) 592-. , Zmurko's. Skibinski,P.K. Kosmos(Lw.) 9 (1884) 185- ; Wierf Ak. D. 53 (1887) (Ab. 2) 35-. Integrators. Hummel, . Dingier 275 (1890) 17-. , mechanical. Abdank-Abakanowicz, B. Lum. Elect. 18 (1885) 49-, 110-, 161-, 249-, 306-, 393-, 535-, 589-; 24 (1887) 3-, 54-, 161-. , two mechanical. Fraser, A. Y. Edinb. Mth. S. P. 4 (1886) 29-. Integrometer, new. Abdank-Abakanowicz, B. C. E. 95 (1882) 1047-. Involution and evolution of numbers, instru- ment for. Roget, P. M. [1814] Phil. Trans. (1815) 8-. Least squares, instrument for. Bouniakowsky , V. Pogg. A. 107 (1859) 463-. Linear differential equation of 2nd order, general mechanism for integrating. Tait, P. G. [1876] Edinb. E. S. P. 9 (1878) 118-. Link-work apparatus for describing curves. Janse, L. N. Arch. Wisk. 9 (*1882) 61-. Linkages for tracing conic sections. Stronq, W. M. A. Mth. 8 (1893-94) 181-. x m . Freeland, F. T. [18791 Am. J Mth 3 (1880) 316-. Logical machine, Jevons' . Sonnenschein, E. A (xi) [1883] Birm. Ph. S. P. 4 (1885) 72-. 36 Instruments 0080 Logical machine, new. Marquand, A. Am. Ac. P. 21 (1886) 303-. Mathematical instruments, models etc., ex- hibition in Munich. Dyck, W. D. Mth. Vr. Jbr. 3 (1894) 39-. Mechanisms, new: projector, ellipsograph, . . . Delaunay, N. Bll. Sc. Mth. 19 (1895) 240-. Mensurator and coelometer. Adams, M. L. Mth. S. P. 5 (1874) 141-. Military surveying circle. Delanges,P. Bologna Mm. I. It. 1 (pte. 2) (1806) 63-. Mirror for drawing in perspective. Schone- mann, P. Z. Nw. 10 (1874) 102-. Multiplication by a mechanical device. Ber- delle, C. As. Fr. C. E. (1887) (Pt. 2) 210-. Nautical instruments, and calculating appara- tus. Gelcich, E. Cztg. Opt. 5 (1884) 242-, 253-, 268-, 277-; 6 (1885) 9 ; 7 (1886) 265-. Nicolas's gyrograph for tracing arcs of large circles. Gobin, A. Lyon A. S. Ag. 8 (1864) 520-. Nurnberg, German Museum, mathematical collection. Giinther, S. Lpldina. 14 (1878) 93-, 108-. , , instruments. Giinther, S. D. Nf. Vh. (1893) (Th. 2, Halfte 1) 32. Observations, mechanical calculation of results. Favaro, A. Yen. I. At. 2 (1876) 559-. Pantograph ; proof of chief theorem, etc. Haussner, A. Arch. Mth. Ps. 65 1880) 334-. , scenographic, for projecting geometric figures. Magistrini, G. B. Bologna Opusc. Sc. 1 (1817) 289-. Pantographs. Ott, A. Cztg. Opt. 17 (1896) 161-. , proof of properties (l r + 2 r + 3 r x r ) . Li- gowski, W. [1879] Arch. Mth. Ps. 65 (1880) 329-. Parabola, apparatus for drawing. Thallmayer, V. Dingier 227 (1878) 430-. Parabolograph. Delone, N. B. Es. Ps.-C. S. J. 29 (Ps.) (1897) 76. Parallel lines, instrument for drawing. Nichol- son, W. Nicholson J. 2 (1799) 429-. , . Suverkrop, J. P. [1883] Sc. S. Arts T. 11 (1887) 73-. rule, new. B., R. Nicholson J. 9 (1804) 212-. , . Boswell, J. W. Nicholson J. la (1806) 196- " Perspecteur-calquer," instrument for drawing in perspective. Breton, . Isere S. Bll. 27 (1892) 659-. Perspective drawer. Brauer, E. Z. Mth. Ps. 43 (1898) 163-. drawing instrument. Brunei-Bar ennes, de. Dingier 34 (1829) 325-. , Verly's instrument, report. Rodolphe, . Metz Mm. Ac. 20 (183^39) 258-. , scale for dividing vanishing lines. Cumber- land, G. Nicholson J. 16 (1807) 1-. Plagiograph aliter skew pantigraph. Sylvester, J. J. Nt. 12 (1875) 168. .history. Sylvester, J. J. Nt. 12 (1875) 214-. Plane surface, formation, law of progress of accuracy. Airy, (Sir) G. B. Ph. Mg. 42 (1871) 107-. ' 0080 Instruments Planigraph, projecting tortuous curve on plane. Delone, N. B. [1895] Vars. S. Nt. Tr. (1895-96) (C. R., Ps. C.) No. 2, 3 pp. Planimeter. Amsler, J. Ziir. Vjschr. 1 (1856) 41-, 101-. , Amsler's. Decker, G. Dingier 140 (1856) 27- ; 141 (1856) 29-. _, __. Weisbach, J. Giving. 4 (1858) 1-. , . Wittstein, T. Hann. Z. Archt. Vr. 7 (1861) 289-. , . Petsch, . A. Pon. Chauss. 16 (1868) 124-. , . Combes, C. P. M. A. Mines 19 (1871) 278-. , . Bramwell, F. J. B. A. Rp. 42 (1872) 401-. , . Purvis, F. P. Ph. Mg. 48 (1874) 11-. , . Rateau, A. As. Fr. C. E. (1896) (Pt. 2) 130-. , , proof of principle. Steggall, . Edinb. Mth. S. P. 7 (1889) 25-. , , theory. Zhblkovskil,A. K. (xn) [1865] Kazan Un. Mm. (1864, Ps.-Mth. No. I), 219-. , , . Aldis, W. S. N. Eng. I. Mn. E. T. 28 (1879) 211-. , , . Thire, A. A. Mines 11 (1887) 121-. , Bouniakovsky's and modification. Decker, G. Dingier 140 (1856) 27-. , construction. Schulten, N. G. of. (vm) Helsingf. Ofv. 5 (1863) 245-. , Coradi's. ? Coradi, . V. Nost. Eng. Mg. 32 (1885) 465-. , hatchet. Hill, F. W. [1894] L. Ps. S. P. 13 (1895) 229-; Ph. Mg. 38 (1894) 265-. , history. Favaro, A. Forster Al. Bauztg. 38 (1873) 68-, 93-. , . Wolf, R. Ziir. Vjschr. 37 (1892) 111-; 38 (1893) 3-. - of Hohmann and Coradi. Fischer, G. Cztg. Opt. 5 (1884) 74-. , Lippincott. Greenhill, A. G. [1899] Sc. Abs. 3 (1900) 234. , Maass-, description. Gunther, F. Cztg. Opt. 7 (1886) 247-. , new. Eckert, , <& Hamann, . Cztg. Opt. 18 (1897) 125. , , description. Maevskij, J. A. [1889] Vars. S. Nt. Tr. (1889-90) (C. R., Ps. C.) No. 2, 7-. , form. Maxwell, J. C. Edinb. T. Sc. S. Arts 4 (1856) 420-. , (Beverley). Balfour, J. M. [1865] Edinb. Sc. S. Arts T. 7 (1868) 198-. , (1852). Baranoffsky, E. Helsingf. Acta 4 (1856) 3-. , (1855). Bouniakoicsky, V. [1855] St Pet. Ac. Sc. fill. 14 (1856) 152-. polar. Amsler, J. Dingier 140 (1856) 321- ; 141 (1856) 326-. , , Amsler's. Decker, G. Dingier 141 (1856) 330-. , , . Sehinz, E. Bern Mt. (1857) 152-. , , . NehU, C. Giving. 20 (1874) 71-. , , , results. Hartogh Heys van Zou- teven, H. Arch. Neerl. 5 (1870) 440-. Instruments 0080 Planimeter, polar, determination of constant. Schell, A. Wien SB. 56 (1867) (Ab. 2) 325-. , , general theory. Schell, A. Wien SB. 58 (1868) (Ab. 2) 189- ; Carl Epm. 11 (1875) , ', note. Resal, H. C. B. 77 (1873) 509-. , Eichard. ? Anon. Sc. Abs. 1 (1898) 635. , rolling, theory. Fenner, P. Cztg. Opt. 7 (1886) 133-. , solutions of electric problems. Atkinson, L. B. Elect. 22 (1889) 702- ; 23 (1889) 29-. , Trunk's. Hiibner, A. Arch. Mth. Ps. 44 (1865) 337-. , Wetli's. Stampfer, S. Wien SB. (1850) (Ab. 1) 134-. Planimeters. Moberg, A . (vm) [1859] Helsingf. Ofv. 5 (1863) 53-. . Beck, A. Eiga Cor.-Bl. 40 (1898) 99-. , free. Bouniakowsky , V. St Pet. Ac. Sc. Bll. 2 (I860) 567-. and similar instruments. Kohlmorgen, 0. Westf. Vr. Jbr. (1890) 126-. of Wetli, Decher, and Amsler. Miller, W. H. Ph. Mg. 16 (1858) 230-. Problems of 3rd degree, instrument for solution. Simon, C. Les Mondes 4 (1864) 547-. Proportion -table, equivalent to sliding -rule 13' 4" long. Everett, J. D. Ph. Mg. 32 (1866) 350-. Protractor, semicircular, improved form. Stewart, J. N. Z. I. T. 3 (1870) 316-. for trisecting angles. Alexander, J. H. Silliman J. 7 (1849) 243-. Quadrature, instrument for. Boileau, P. C. E. 15 (1842) 594-. Eefraction through prism; apparatus to illus- trate deviation of ray. Marcucci, S. Ev. Sc.-Ind. 27 (1895) 101-. Scale, accuracy of, and limit of perception. Neovius, E. Helsingf. Ofv. 23 (1881) 71-. " Scoliograph, unicursal," for drawing algebraic curves. Boguslawsky, A. [1881] Mosc. S. Nt. Bll. 57 (1882) (Pt. 2) 72-. Screw, use in mathematical instruments. Carl, P. Carl Epm. 2 (1867) 37-. Sectograph (for dividing lines and angles, &c.) . Jones, T. Tilloch Ph. Mg. 42 (1813) 401-. Sine and cosine quadrant. Schreiber, C. Gztg. Opt. 7 (1886) 38-. Spherical integrator, modification of a form of. Smith, F. J. Ph. Mg. 22 (1886) 147-. triangles, instrument for determining, by mechanical action. Penrose, F. C. As. S. M. Not. 36 (1876) 281-. Spherometer. Mayer, A. M. Am. J. Sc. 32 (1886) 61-. , new. Kruss, H. Cztg. Opt. 12 (1891) 63-. Spirals, Johnson's volutor for drawing. Booth, J. B. A. Ep. (1860) (Pt. 2) 60-. Square with parallel lines. Chaussenot, (I'Aine). Par. Bll. S. Encour. 43 (1844) 4-. Stanhope logical and arithmetical machine. Harley, (Rev.) R. N. S. W. E. S. J. 24 (1890) 121-. 37 0080 Calculating Machines Stereoscopic drawing apparatus. Jensen, J. [1870] Arch. f. Ap. 4 (1871) 233-. Straight line apparatus, Hart's. Darboux, G. Bll. Sc. Mth. As. 3 (1879) 144-. Stupakoff's comparometer. Basford, G. M. Am. Eng. & Eailroad J. 73 (1899) 59, 169. Tachymeter, Cairo's. Puissant, L. C. E. 3 (1836) 140, 200, 245-, 384. Tractonograph (instrument for describing curves of "tractory" nature). Kleritj, L. Dingier 305 (1897) 234-, 260- . of Kleritj, and rod-planimeter. Korselt, A. Z. Mth. Ps. 43 (1898) 312-. Trigonometer, instrument for solving spherical triangles. Braun, K. (xn) Mth. Term. Ets. 1 (1883) 313- ; Mth. Nt. B. Ung. 1 (1882-83) 283-. , Savy's. Le Grand, G. Aube Mm. S. Ag. 5 (1854) 235-. Trisection of angle, instrument for. Burt, T. S. Beng. As. S. J. 1 (1832) 499-. , . Tate, T. Ph. Mg. 19 (1860) 261- . , . Cassani, P. Yen. Aten. At. 1 (1878) 163-. , . Longcliamps, G. de. As. Fr. C. B. (1893) (Pt. 2) 190-. .mechanical. Russell, A. H. Nt. 43 (1891) 547-. angles, test. Ronisch, A. Cztg. Opt. 16 (1895) 253-. Vernier for line or curve with unequal divisions. Artur, J. F. Par. Bll. S. Encour. 50 (1851) 676-. , new form. Ludloic, H. H. V. Nost. Eng. Mg. 27 (1882) 196. or Nonius, priority. Breusing, A. [1879] As. Nr. 96 (1880) 129-. Verniers. Matzka, W. Grunert Arch. 34 (1860) 334-. CALCULATING MACHINES. Addition machine. Goss, J. Tilloch Ph. Mg. 41 (1813) 166-. Arithmetical machine, Cebysev's. Bool, V. G. von. Mosc. S. Sc. Bll. 91 (M. 1) (1894) 12-. with continuous motion. Tchebitchef, P. L. Bv. Sc. 4 (1882) 402-. , supplementary explanation. Cebysev, P. L. Mosc. S. Sc. Bll. 91 (No. 1) (1894) 22. machines, history. Mehmke, R. Prace Mt.- Fiz. 6 (1895) 177-. Arithmometer, Thomas's. Ettingshausen, A. von. Wien SB. 24 (1857) 16-. , . Reuleaux, F. Giving. 8 (1862) 182- Hirn, G. A. Les Mondes 3 (1863) 433- 198- 25-. Junge, A. Z. Mth. Ps. 9 (1864) Zillmer, . I. Act. J. 15 (*1870) , . Cavallero, A. Cuyper Bv. Un. (1880) 309-. Calculating Machines 0080 Arithmometer, Thomas's, actuarial use. Gray, P. I. Act. J. 17 (*1873) 249-; 18 (*1875) 20-, 123-. , , adaptation of assurance formulae to. Hannyngton, J. Assur. Mg. 12 (*1866) 184. , , use. Hannyngton, (Maj.-Gen.). I. Act. J. 16 (*1872) 244-. , use. Him, G. A. A. G6n. Civ. 2 (1863) (pte. 2) 113-, 153-. , . Hancock, W. J. I. Act. J. 16 (*1872) 265-. , in constructing tables. Garment, Z). I. Act. J. 22 (*1881) 368-; 23 (*1882) 232. Calculating apparatus, arithmetical puzzles. Lucas, E. Bv. Sc. 45 (1890) 1-. based on Napier's rods. Bridge, J. Ph. Mg. 9 (1880) 191-. circle. Weiss, P. C. B. 131 (1900) 1289-. instruments and machines. (2) Logarithmic scales. (3) Graphic processes. Ocagne,M.d\ A. Cons. Arts et Met. 5 (1893) 231-, 323-; 6 (1894) 42-. machine. Babbage, C. Edinb. Ph. J. 8 (1823) 122-. . Olivier, T. Par. Bll. S. Encour. 45 (1846) 153-. . Nystrom, J. W. Franklin I. J. 21 (1851) 262-. . Selling, E. D. Nf. Tbl. (1887) 234-; Wiirzb. Ps. Md. Sb. (1888) 148- ; Cztg. Opt. 9 (1888) 3-. . Levdnen, S. Helsingf. Ofv. 38 (1896) 26-. . Siedentopf, H. N.-Vorp. Mt. 30 (1899) xvii-. or arithmometer. Hoyan, . (vi Adds.) Par. Bll. S. Encour. 21 (1822) 355-. , . Thomas, . Moigno Cosmos 4 (1854) 72-, 186-. . , Babbage's. Baily, F. Tilloch Ph. Mg. 63 (1824) 355-. , . Menabrea, L. F. Taylor Sc. Mm. 3 (1843) 666-. , . Brit. Ass. Comm. B. A. Bp. (1878) 92-. , . Menabrea, (le gen.) L. F. C. B. 99 (1884) 179-. , , mechanical arrangements. Babbage, (Maj.-Gen.) H. P. B. A. Bp. (1888) 616-. for definite integrals. Moseley, H. B. A. Bp. (1841) (Pt. 2) 35-. , duplex. Kiittner, W. Dingier 300 (1896) 199- . , n(x) for (x) repeated n times, where n is fractional]. Appell, P. Par. S. Phlm. Bll. 3 (1879) 166. 52 0810 Calculus of Operations Operation, laws. Ellis, A. J. E. S. P. 10 (1859-60) 85-. Operations equivalent to their adjuncts. Bor- tolotti, E. Em. E. Ac. Line. Ed. 7 (1898) (Sem. 2) 74-. in general. Bourlet, C. C. E. 124 (1897) 348-. , remarkable sequence. Brunei, G. Bordeaux S. Sc. P.-V. (1896-97) 86-. , theorems. Grolous, J. Par. S. Phlm. Bll. 12 (1875) 81-. Periodicity, generalisation of notion. Brunei, . Bordeaux S. Sc. Mm. 4 (1888) xxvi-. Quotient- and product-calculation. Guldberg, A. S. Ts. Mth. 2 (1884) 84-, 161- ; Fschr. Mth. (1884) 342. Eeduction of Cayley 's operation to polar operations. Capelli, A. Mth. A. 29 (1887) 331-. Signs + and , nature in symbolical algebra. Gregory, D. F. Camb. Mth. J. 3 (1843) 153-. Symbolic calculus, principles. Cesdro, E. (xn) Mathesis 3 (1883) 10-. equation, Tr m ir n u + qpu = 0, where symbols ""m' ""n) P combine according to the laws "K-mP = pTT m +l ' ' Jr m' n 'n = lr n' lr m, + a ( n ~ m ) P' Boole, G. Camb. and Dubl. Mth. J. 2 (1847) 7-. equations, certain. Piiicherle, S. Bologna Ac. Sc. Mm. 5 (1895-96) 663-. , employment. Cauchy, A. L. C. E. 17 (1843) 449-. , interchange of symbols. Graves, C. Ir. Ac. P. 6 (1853-54) 144-. , resolution of non-linear. Cauchy, A. L. C. E. 22 (1846) 235-. , theorems and formulae deduced from. Cauchy, A. L. C. E. 17 (1843) 377-. expansions. Russell, W. H. L. E. S. P. 14 (1865) 329-. methods, proofs of (1) formula for nth differential coefficient, (2) Euler's theorems on quantics, (3) Staudt's theorem on Ber- noulli's numbers. BuchJieim, A. Mess. Mth. 11 (1882) 143-. relations &c. Johnson, W. W. Mess. Mth. 11 (1882) 191-. theorem involving repeated differentiations. Glaisher, J. W. L. [1879] Camb. Ph. S. P. 3 (1880) 269-. theorems, Crofton. Glaisher, J. W. L. QJ. Mth. 16 (1879) 257-. variability of several dimensions. Formenti, C. Mil. I. Lomb. Ed. 16 (1883) 288-, 351-. versus operative expansion. Hathaway, A. S. Des Moines Anal. 5 (1878) 38-. Symbolical expression, properties and applica- tion to attraction of ellipsoid. Cayley, A. Camb. Mth. J. 3 (1843) 62-. Symbols of operation. Crofton, M. W. Am. J. Mth. 4 (1881) 269-. Complex Numbers 0820 Symbols, relation of certain. Cockle, (Sir) J. [1879] QJ. Mth. 17 (1881) 20-, 208-. Ternary quadric operators on products of forms of any orders. Walker, J. J. L. Mth. S. P. 20 (1889) 110-. Transmutations. Bourlet, C. Par. S. Mth. Bll. 25 (1897) 132-. Uniform substitutions and Babbage's problem. Iaggi,E. N. A. Mth. 19 (1900) 483-. , convergence. Lemeray, E. M. N. A. Mth. 16 (1897) 306- ; 17 (1898) 75-. regarded as function of n. Krug, A. Wien. Ak. D. 57 (1890) 151-. 0820 General theory of complex numbers. (See also 2870.) d'Alembert's equation (1 + W^T) m = (1 - h x/^T)" 1 . Cossali, P. [1783] Mod. Mm. S. It. 9 (1802) 231-. Algebraic couples. Hamilton, (Sir) W. R. [1835] Ir. Ac. T. 17 (1837) 293-. . Cayley, A. Ph. Mg. 27 (1845) 38-. . Hargreave, C. J. B. A. Ep. (1857) 184-. equations, theory. Gegenbauer, L. Mh. Mth. Ps. 6 (1895) 12-. integers. Landsberg, G. D. Mth. Vr. Jbr. 4 (1897) 111-. numbers, new foundation for theory. Hensel, K. D. Mth. Vr. Jbr. 6 (1899) (Heft 1) 83-. and transcendent equations, complex num- bers derived from. Hug, J. C. Zur. Vjschr. 7 (1862) 270-. triplets. Graves, C. Ir. Ac. P. 3 (1847) 51-, 57-, 80-, 105-. Calculation with p + qj-l. Ohm, M. D. Nf. Vsm. B. (1845) 73-. Chessboard, mathematical theory. Problem of n queens on board of n 2 spaces. Giinther, S. Arch. Mth. Ps. 56 (1874) 281-. Complex and bicomplex numbers, theory. Benthem, A. Arch. N6erl. 12 (1877) 113-. coordinates in plane geometry. Berlin, M. Lund. Acta Un. 9 (1872) (Mth.) No. 4, 45 pp. elements in analytical geometry, significance. Stolz, 0. Mth. A. 4 (1871) 416-. factors. Cauchy, A. L. C. E. 24 (1847) 347-. forms, real representations. Segre, C. Mth. A. 40 (1892) 413- . geometry, contribution. Sforza, G. Q. Mt. 30 (1892) 159-. integers. Lanavicensis (Sylvester, J. J.). QJ. Mth. 4 (1861) 94-, 124-. magnitudes, addition-logarithms for. Mehmke, R. Z. Mth. Ps. 40 (1895) 15-. number, cosine, significance in plane geo- metry. Thiele, T. N. (xn) Ts. Mth. & (1876) 177-. 53 0820 General theory of complex numbers j. Complex numbers. Genocchi, A. Liou Mth. 19 (1854) 281-. . Pad, P. G. Mt. 11 (1873) 244-. . Zolotarev, E. I. [1877] St. Pet. Ac. Sc. Bll. 24 (1878) 310-. , analysis. Denzler, W. Ziir. Mt. 3 (1853-55) 560-. , with application to equation ar x + j/ A =2\ Kummer, E. E. Berl. Ab. (1857) (Mth.) , application to n dimensions. Mollame, V. G. Mt. 31 (1893) 271-. comprised in formula a + b *j c. Pepin, T. Liouv. J. Mth. 1 (1875) 317-. , elementary theory. Hoiiel, J. Bor- deaux Mm. S. Sc. 5 (1867) 1- ; 6 (1868) 1- ; 8 (1870) 97-; 1 (1876) 1-. formed from roots of unity. Kummer, E. E. Liouv. J. Mth. 12 (1847) 185- ; 16 (1851) 377-; Berl. Mb. (1861) 1051- ; (1863) 21-. . Kronecker, L. Berl. Mb. (1863) 340-. . Fuchs, L. [1864] Crelle J. 65 (1866) 74-. , prime factors. Kummer, E. E. Crelle J. 35 (1847) 327-. and 2nd factor of class number. Kummer, E. E. Berl. Mb. (1870) 855-. , simplest representation. Kummer, E. E. Berl. Mb. (1870) 409-. w n = l, ideal prime factors. Kummer, E. E. Berl. Ab. (1856) (Mth.) 1-. Xth roots of unity. Kummer, E. E. Crelle J. 40 (1850) 93-, 117-. 31st roots of unity. Kummer, E. E. Berl. Mb. (1870) 755-. , Gaussian representation of. Breuer, A. Arch. Mth. Ps. 12 (1894) 337-. geometric representation of a new kind. Kotelinikov, A. P. Kazan S. Ps.-Mth. Bll. 6 (1896) (Prot.) 26. , significance (Schlomilch). Unfer- dinger, F. Arch. Mth. Ps. 52 (1871) 252-. , introduction. Baltzer, R. [1882] Crelle J. Mth. 94 (1883) 87-. , mechanical interpretation. Zbikovskij, A. K. Kazan S. Nt. (Ps.-Mth.) P. 6 (1888) 6. , multiplication, geometric significance. Orelli, J. Ziir. Vjschr. 20 (1875) 443-. , new method for graphical representation. Brill, J. Mess. Mth. 17 (1888) 80-. and their functions, theory. Hertz, K., & Dicksttin, S. Par. T. Nauk Sc. Pam. 7 (*1875) Art. 3, 60 pp. , theorem of Dedekind, proof. Study, E. Gott. Nr. (1898) 1-. , theory. Lejeune-Dirichlet, G. Berl. Ab. (1841) 141-. , . Kummer, E. E. Berl. B. (1846) 87-; Crelle J. 35 (1847) 319-. , . Wantzel, L. C. B. 24 (1847) 430-. , . Bachmann, P. Crelle J. 67 (1867) , . Dedekind, E. C. B. 90 (1880) 1205-. 0820 Complex numbers, theory. Zolotarev, E. J. Liouv. J. Mth. 6 (1880) 51-, 129-. units. Kronecker, L. [1845] Crelle J. Mth. 93 (1882) 1-. . Lejeune-Dirichlet, G. Berl. B. (1846) 103-. . Kronecker, L. Crelle J. 53 (1857) 176- ; C. B. 96 (1883) 93-, 148-, 216-. . Molk, J. Bll. Sc. Mth. As. 7 (1883) (Pt. 1) 133-. , addition to memoir on. Kronecker, L. C. B. 99 (1884) 765-. , expressions formed from. Kummer, E. E. Crelle J. 50 (1855) 212-. Conjugate functions and imaginary logarithms. Hamilton, (Sir) W. R. B. A. Ep. (1834) 519-. Cube root of >Ja+ bi in form x + yi, simple solu- tion. Simony, O. Arch. Mth. Ps. 55 (1873) 72-. roots, imaginary, number of (Lockhart). Homer, W. G. Thomson A. Ph. 8 (1816) 279-, 388-. of unity, theorem of algebra, geometry deduced from. Appell, P. C. B. 84 (1877) 540-. Directive numbers, theorems, and proof of Cauchy's theorem on closed contours. Tran- son, A. N. A. Mth. 7 (1868) 97-. Discriminants of finite ' ' fields. ' ' Dedekind, R. Gott. Ab. 29 (1882) Art. 2, 56 pp. Every equation in one unknown can be satisfied by a complex value of this un- known, geometric proof. Wittstein, T. Grunert Arch. 6 (1845) 225-. Format's theorem, generalization for Galois number fields. Moore, E. H. N. Y. Am. Mth. S. Bll. 2 (1896) 189-. G. C. M. of 2 complex numbers, number of divisions. Dupre, A. Liouv. J. Mth. 13 (1848) 333-. Geometrical representation of \/-l, considera- tion of objections. Warren, J". Phil. Trans. (1829) 241-. Goniometry, application of principles given by Warren. Terrot, C. H. [1847] Edinb. B. S. T. 16 (1849) 345-. Imaginaries. Buee, . [1805] Phil. Trans. (1806) 23-. . Fran^ais, J. F. Gergonne A. Mth. 4 (1813-14) 222-, 364-. . Volpicelli, P. G. Arcad. 52 (1831) 221-. . Bellavitis, G. Poligrafo 5 (1831) 15-; 12 (1832) 211-. . Arenstein, J. Haidinger B. 3 (1847) 292-. . Bellavitis, G. Yen. At. 6 (1847) 459-; Yen. Mm. I. 4 (1852) 243-. . Arenstein, J. Haidinger Ab. 2 (1848) (Ab. 2) 43-. . Cockle, Jas. Ph. Mg. 36 (1850) 281-. . Grunert, J. A. Grunert Arch. 20 (1853) 121-. . Sabatie, J. N. A. Mth. 2 (1863) 206-. . Cheyrezy, U. N. A. Mth. 3 (1864) 445-. . Sexe, S. A. N. Mg. Ntvd. 16 (1869) 1-; 17 (1870) 233-; Arch. Mth. Ntvd. 7 (1882) 115-. 54 0820 General theory of complex numbers 0820 Imaginaries, d'Alembert's theorem. Du Bour- guet, . Gergonne A. Mth. 4 (1813-14) 20-. , . Gergonne, J. D. Gergonne A. Mth. 4 (1813-14) 23-. , . Barsotti, G. Lucca At. Ac. 11 (1842) 393-. in algebra. Janssen van Eaaij, W. H. L. Haarl. Ms. Teyl. Arch. 4 (1896) 53-. , application to mechanics. Durege, . Ziir. Vjschr. 7 (1862) 293-. , summation of trigonometric series, and calculation of integrals. Michaelis, J. P. Lux. S. Sc. Mm. 7 (1864) 172-. , theory of curves. Durege, H. Arch. Mth. Ps. 42 (1864) 1-. , bisignal univalent. Kirkman, T. P. Ph. Mg. 37 (1850) 292-. , Bolyai's theory. Stdckel, P. Mth. Termt. fits. 17 (1899) 259-; Mth. Nt. B. Ung. 16 (1899) 263-. , calculation with, and geometrical theorems. Zech, P. Grunert Arch. 16 (1851) 354-. , calculus. Carmichael, E. Ph. Mg. 6 (1853) 273-. , construction for. Cayley, A. Mess. Mth. 10 (1881) 1-. , defective notation. Mathematicus.[Pseudon.~\ (vi Adds.) Nicholson J. 29 (1811) 254-; 31 (1812) 193-. , doctrine. Walton, W. Camb. and Dubl. Mth. J. 8 (1853) 101-. , in coordinate geometry. Walton, W. Camb. and Dubl. Mth. J. 7 (1852) 234-. , employment in geometry. Laguerre, . N. A. Mth. 9 (1870) 163-, 241-. , of space. Laguerre, . N. A. Mth. 11 (1872) 14-, 108-, 241-. , Fresnel's interpretation. Lippich, F. Prag Sb. (1864) (Pt. 2) 73-. , geometric application. Schou, (Lieut.) T. V. (xn) Ts. Mth. 1 (1865) 129-. , construction. Drobisch, M. W. Leip. B. 2 (1848) 171-. , meaning. Zehfuss, G. Grunert Arch. 32 (1859) 234-. , representation. Argand, . Gergonne A. Mth. 4 (1813-14) 133- ; 5 (1814-15) 197-. , . Durege, . Ziir. Vjschr. 3 (1858) 358. , . Lie, S. Crelle J. 70 (1869) 346- ; Christiania F. 22 (1870) 16-, 107-. , , applications to cubic equations, etc. Wittstein, T. Grunert Arch. 7 (1846) 402-. , significance of form a + 1) V - 1. Mutter, G. W. Grunert Arch. 1 (1841) 397-. (Grones). Toblini, G. Poligrafo 14 (1833) 37-. , history of theory. Betnan, W. W, Am. As. P. (1897) 33-. , how to avoid. Sexe, S. A. Arch. Mth. Ntvd. 4 (1879) 145-. , , and elementary theory. Sexe, S. A. Arch. Mth. Ntvd. 9 (1884) 205-. , interpretation in mathematical physics. Chancaurtais, E. B. de. C. E. 68 (1869) 127-. Imaginaries, new theory. Cauchy, A. L. C. E. 24 (1847) 1120-. , , various applications. Cauchy, A. L. C. E. 25 (1847) 129-. , octaves. Cayley, A. Ph. Mg. 30 (1847) 257. , paradox resulting from employment. Mathematicus . [Pseudon.] (viAdds.) Nichol- son J. 30 (1812) 209-. , real sum of two, Lagrange's equation, proof. Plana, G. Tor. Mm. Ac. 16 (1857) 97-. , reality. Matzka, W. Bohm. Gs. Ab. 6 (1851) 179-. , relation of V - 1 to geometrical perpen- dicularity. Paterson, J. Am. As. P. (1851) 1-. , representation, etc. Ballauff, L. Grunert Arch. 5 (1844) 259-. , , method. Duport, . Par. EC. Norm. A. 9 (1880) 301-. , , . Amstein, H. Laus. S. Vd. Bll. 18 (1882) 1-. , , new. Duport, . C. E. 88 (1879) 1071-. , simplification. Nordmark, Z. Ups. N. Acta S. Sc. 8 (1821) 136-. , 8-square. Cayley, A. [1882] Am. J. Mth. 4 (1881) 293-. , . Sylvester, J. J. (xn) J. H. Un. Cir.Jl] (1882) 203. , (V). Tait, P. G. Edinb. E. S. P. 7 (1872) 607-. Point and line calculus, with special reference to parallels. Graefe, E. Arch. Mth. Ps. 15 (1897) 34-, vii-. Potential of closed circuit, quaternion investi- gation. Tait, P. G. QJ. Mth. 4 (1861) 143-. Problem of Fermat, to find point, sum of whose distances from 3 given points is least, solu- tion. Tait, P. G. [1867] Edinb. E. S. P. 6 (1869) 165-. Quaternion analysis, geometrical proofs of theorems by. Hamilton, (Sir) W. R. Ir. Ac. P. 5 (1853) 407-. , new geometrical interpretation. Brill, J. Camb. Ph. S. P. 6 (1889) 156-. associative principle. Hathaway, A. 8. [1895] N. Y. Am. Mth. S. Bll. 2 (1896) 43-. , definition. Christie, A. S. Wash. Ph. S. Bll. 11 (1892) 579-. differences, an equation in. Tait, P. G. Edinb. E. S. P. 12 (1884) 561-. differentiation. M'Aulay, A. [1890] Edinb. E. S. P. 18 (1892) 98-. equations, solution of a large class. Sylvester, J. J. C. E. 98 (1884) 651-. forms of propositions in fluid motion. Butcher, J. G. [1876] L. Mth. S. P. 8 (1877) 174-. formulae. Allardice, R. E. Edinb. Mth. S. P. 7 (1889) 8-. for quantification of curves, surfaces and barycentres. Stringham, W. I. Am. J. Mth. 2 (1879) 205-. 57 0830 Quaternions Quaternion formulae for reduction of multiple integrals. Carvallo, E. Par. S. Mth. Bll. 18 (1890) 80-. , function, development. Graves, C. Ir. Ac. P. 6 (1853-54) 385-. group, the. Miller, G. A. 37 (1898) 312-. innovations. Heaviside, O. Nt. 49 (1893- 94) 246. integrals, some. Tait, P. G. [1870] Edinb. E. S. P. 7 (1872) 318-, 784-. integration, a. Tait, P. G. [1871] Edinb. B. S. P. 7 (1872) 434-. , theorems in. McAulay, A. Mess. Mth. 14 (1885) 26-. invariants of linear vector functions, and quaternion determinants. Joly, G. J. Ir. Ac. P. 4 (1896-98) 1-. and its depreciators. Knott, G. G. Edinb. Mth. S. P. 11 (1893) 62-. method, intrinsic nature of. Tait, P. G. [1894] Edinb. B. S. P. 20 (1895) 276-. note on a geometrical problem. Tait, P. G. [1889] Edinb. E. S. P. 16 (1890) 315-. notes. Tait, P. G. Edinb. E. S. P. 15 (1889) 379-. , from number to. Fleuri, G. N. S. W. E. S. J. 28 (1894) 65-. " Path," Tait's, note. Martin, H. [1867] Edinb. E. S. P. 6 (1869) 121-. proof of reciprocal curves in space. Hamilton, (Sir) W. B. B. A. Ep. (1862) (pt. 2) 4. Samuel Eoberts' theorem of 4 co- intersecting spheres. Walker, J. J. L. Mth. S. P. 12 (1880-81) 147-. - triple generation of three-bar motion. Hart, H. [1882] Mess. Mth. 12 (1883) 32-. transformation, a. Tait, P. G. Edinb. E. S. P. 5 (1863) 115-. transformations. Tait, P. G. Edinb. E. S. P. 7 (1872) 501-. . Gwyther, R. F. Manch. Lt. Ph. S. P. 19 (1880) 118-. treatment of linear complex. Buchheim, A. Mess. Mth. 13 (1884) 120-. variables, functions. StudniCka, F. J. Prag Sb. (1894) (Mth.-Nt.) No. 26, 8 pp. Quaternions. Hamilton, (Sir) W. B. [1843 Ir. Ac. T. 21 (1848) 199-. . Cayley, A. Ph. Mg. 26 (1845) 141-, 208-. . Tait, P. G. Mess. Mth. 1 140-, 203-. . Kelland, P. . Ladd, C. 172-. . Stringham, W. I. 310-. . Wood, De V. Des Moines Anal. 7 (1880) 11-, 33-, 65-, 121-. . Christie, A. S. Des Moines Anal. 7 (1880) 185-. . Ball, B. S. Nt. 48 (1893) 391. . Studnitlca, F. J. Prag Sb. (1893) (Mth.- Nt.) No. 47, 10 pp. [1843] 141-, (1862) 78-, Mth. 2 (1863) 136-. Des Moines Anal. 4 (1877) Am. Ac. P. 13 (1878) Quaternions 0830 Quaternions. Studnifka, F. J. Casopis 23 (1894) 145-. 209- ; Fschr. Mth. (1893-94) 135. and algebra of vectors. Gibbs, J. W. Nt. 43 (1891) 511-. Am. Ph. S. P. ----- . Tait, P. G. Nt. 43 (1891) 608. ----- . Gibbs, J. W. Nt. 47 (1892- 93) 463-. ----- . Gibbs, J. W. Nt. 48 (1893) 364-. , application, new. Everett, J. D. B. A. Ep. (1874) (Sect.) 11. , applications to exponential and other functions. Studnicka, F. J. Prag Sb. (1894) (Mth.-Nt.) No. 7, 18 pp. , geometry of straight lines and complexes. Valentiner, H. Kitfb. Ov. (1899) 655-. , -- mechanics. Sebuev, G. N. Kazan S. Ps.-Mth. Bll. 2 (1893) 111-; Fschr. Ps. (1893) (Ab. 1) 299. , and physics. Molenbroek, P. Amst. Ak. Vh. (Sect. 1) 2 (1894) No. 3, 38 pp. ; Fschr. Mth. (1893-94) 1315-. , metrical geometry. Versluys, J. N. Arch. Wisk. 2 (*1876) 135-. , -- physics. Tait, P. G. QJ. Mth. 6 (1864) 279-; Ph. Mg. 29 (1890) 84-. , --- . McAulay, A. [1892] Nt. 47 (1892-93) 151. , projective geometry. Chapman, C. H. Am. J. Mth. 14 (1892) 115-. , study of quadrics. Papelier, . Par. S. Mth. Bll. 17 (1889) 182-. , -- theory of linear complex and linear congruence. Buchheim, A. Mess. Mth. 12 (1883) 129-. , rotation. Cayley, A. Ph. Mg. 33 (1848) 196-. and "Ausdehnungslehre." Gibbs, J. W. Nt. 44 (1891) 79-. --- . Tait, P. G. Nt. 44 (1891) 105-. , -- , application to different kinds of uniform space. Cox, H. [1882] Camb. Ph. S. T. 13 (1883) 69-. , calculus, theorem, general. Graves, C. Ir. Ac. P. (1850) 347-. , Col. Hime's outlines of. Macfarlane, A. Ph. Mg. 39 (1895) 135-. , continued fractions in W. B. Ph. Mg. 3 (1852) 371- ; 4 303; 5 (1853) 117-, 236-, 321-. , -- , and quadratic equations. Hamilton, (Sir) W. B. Ir. Ac. P. 5 (1853) 219-, 299-. , development of the A -process in. S)iaw, J. B. Am. J. Mth. 19 (1897) 193-. , elements. Bartlett, W. P. G. Camb. (M.) Mth. M. 2 (1860) 29-, 97-, 128-, 195-. , . Peddie, W. Edinb. Mth. S. P. 11 (1893) 130-. , establishment of fundamental properties. McAulay, A. Mess. Mth. 18 (1889) 131-. , extensions. Hamilton, (Sir) W. B. Ph. Mg. 7 (1854) 492-; 8 (1854) 125-, 261- ; 9 (1855) 46-, 280-. , and fourfold space. HatJiaway, A. S. [1897-1900] N. Y. Am. Mth. S. Bll. 4 (1898) 54-; 7 (1901) 243, 245-. 58 Hamilton, (Sir) (1852) 0830 Quaternions Quaternions, functions resembling. Cockle, Jas. Ph. Mg. 33 (1848) 435-. , Gauss' claim to invention. Tait, P. G. [1899] Edinb. B. S. P. 23 (1902) 17-. , general case of. Sylvester, J. J. C. B. 99 (1884) 117-. , geometrical interpretation. Donkin, W. F. Ph. Mg. 36 (1850) 489-. _, . Spottiswoode, W. Ph. Mg. 37 (1850) 108-. , geometry of. Paterson, J. Camb. and Dubl. Mth. J. 9 (1854) 241-. , theory. Hamilton, (Sir) W. E. Ir. Ac. P. 3 (1847) 31-. , involution and evolution. Sylvester, J. J. Ph. Mg. 16 (1883) 394-. , Klein's view. Knott, C. C. [1899] Edinb. B. S. P. 23 (1902) 24-. , and method of equipollences. Bellavitis, G. Yen. At. (1857-58) 334-; Mod. Mm. S. It. 1 (1862) 126-. , multinomial theorems. Wilkinson, (Eev.) M. M. U. [1890] Edinb. E. S. P. 17 (1891) 149-. , multiplication, new proof of associative property. Versluys, J. [1871] Arch. Neerl. 7 (1872) 177-. , nabla (V). Kimura, S. A. Mth. 10 (1895-96) 127-. , nonions, sedenions, &c. Sylvester, J. J. (xn) [1883] J. H. Un. Cir. [3] (1884) 7-. , notes. Boole, G. (vi Adds.) Ph. Mg. 33 (1848) 278-. , . Anthony, E. Mess. Mth. 10 (1881) 66-. , . Kimura, S. Nt. 52 (1895) 366. , number-theory. Hurwitz, A. Gott. Nr. (1896) 313-. as a practical instrument of physical re- search. McAulay, A. Ph. Mg. 33 (1892) , principles elementarily explained. Padel- letti, D. G. Mt. 20 (1882) 1-. , , elementary, establishment on analytical basis. Plarr, G. [1874-77] Edinb. E. S. T. 27 (1876) 175- ; 28 (1879) 37-. , proof of Frobenius' theorem on their relation to complex numbers. Grissemann, F. X. Mh. Mth. Ps. 11 (1900) 132-. , property. Mounier, G. J. D. N. Arch. Wisk. 8 (*1882) 81-. , reduced form. Studnicka, F, J. Prag Sb. (1875) 183-. , relative forms. Peirce, C. S. (xn) J. H. Un. Cir. [1] (1882) 179. , and spherical geometry. MSbius, A. F. Leip. B. 11 (1859) 138-. , theory. Versluys, J. N. Arch. Wisk. 1 (*1875) 41-, 97-; 2 (*1876) 135-. , . Sarran, E. A. Tel. 8 (1881) 52-, 113-. , . Stephanos, C. Mth. A. 22 (1883) 589-. , . Weyr, E. C. B. 98 (1884) 906-, 1320-. , . Holder, 0. Gott. Nr. (1889) 34-. , . Study, E. N.-Vorp. Mt. 31 (1900) 1-. Ausdehnungslehre 0840 Quaternions and vector analysis, principles. Peddie, W. Edinb. Mth. S. P. 11 (1893) 85-. vectors. Knott, C. G. Nt. 48 516-. Quotient of vectors in space, product of matrices. Wettum, T. B. van. N. Arch. Wisk. 1 (1895) 68-; Fschr. Mth. (1893- 94) 226. Quotients of space-directed lines. Elliott, E. B. Smiths. Misc. Col. 33 (1888) Art. 4, 106- ; (Wash. Ph. S. Bll. 10 (1888).) Surfaces, cubic, class treated by quaternions. Chace, A. B. Am. J. Mth. 2 (1879) 315-. , quadric, treated by quaternions. Lowell, A. L. Am. Ac. P. 13 (1878) 222-. , theorems. Hamilton, (Sir) W. E. Ir. Ac. P. 4 (1850) 306-. Symbolic cubic, particular case. Plarr, G. [1881] Edinb. B. S. P. 11 (1882) 342-. equation satisfied by symbol of operation on a quaternion, existence. Hamilton, (Sir) W. E. Ph. Mg. 24 (1862) 127-. integration of differential equations in quaternions. Pocklington, H. C. [1899] Camb. Ph. S. P. 10 (1900) 59-. Tait's elementary treatise, note on paragraphs 439, 440. Plarr, G. C. E. 86 (1878) 1454-. Trigonometrical coefficients. Dierke, G. A. Educ. Times 41 (1888) 171. Trigonometry, plane, by quaternions. Wood, De V. Des Moines Anal. 9 (1882) 19-. Use of V - 1 and alternate numbers i, j, k in study of infinitesimal deformations. Perna, A. Palermo Cir. Mt. Ed. 12 (1898) 322-. Vector expressions for curves. Joly, C. J. Ir. Ac. P. 4 (1896-98) 374-. Vectorial equation of the circumference. Eaimondi, E. G. Mt. 25 (1887) 219-. function of 1st degree p. Alberti, V. G. Mt. 37 (1899) 375-. Vectors and quaternions, theory. Beez, . Z. Mth. Ps. 41 (1896) 35-, 65-. Wave-surface, electromagnetic, quaternion treatment. Tait, P. G. [1894] Edinb. B. S. P. 21 (1897) 165-. , Fresnel's quaternion investigations. Tait, P. G. QJ. Mth. 3 (1860) 190-. 0840 Ausdehnungslehre; vector analysis. Analytical geometry, use of imaginaries. Stille, W. A. Des Moines Anal. 1 [1874] 181-. Area of any closed circuit in space. Hayward, E. B. L. Mth. S. P. 4 (1871-73) 289-. Associated poles, and their representation by products. Grassmann, H. Gott. Nr. (1872) 567-. Ausdehnungen, multiple, simplest theorems. Hoppe, E. Arch. Mth. Ps. 64 (1879) 189-. "Ausdehnungslehre." Grassmann, H. Grunert Arch. 6 (1845) 337-. , geometric representation of imaginaries from standpoint of. Schlegel, S. F. V. Z. Mth. Ps. 23 (1878) 141-. 59 0840 Ausdehnungslehre ; vector analysis "Ausdehnungslehre," Grassmann's. Schlegel, V. Z. Mth. Ps. 41 (1896) (H.-lt. Ab.) 1-, , , applications. Clifford, W. K. Am. J. Mth. 1 (1878) 350-. , , to mechanics. Schlegel, V. Giving. 40 (1894) 59-. , , properties of circles. Cox, H. QJ. Mth. 25 (1891) 1-. , , relation to modern geometrical methods. Schlegel, S. F. V. Z. Mth. Ps. 24 (1879) 83-. , mechanics according to principles of. Grassmann, H. G. Mth. A. 12 (1877) 222-. , and modern algebra. Grassmann, H. G. Mth. A. 7 (1874) 538-. , place in the associative algebra of quaternion type. Joly, C. J. Ir. Ac. P. 6 (1900-02) 13-. Barycentre of curvature of plane curves. Steiner, J. G. Arcad. 101 (1844) 257-; 102 (1845), 1-, 121-. Barycentric calculus, application to problems in analytical geometry. Minding, E. F. A. Crelle J. 5 (1829) 397-. based on composition of straight lines. Mdbius, A. F. Crelle J. 28 (1844) 1-. Complex function, construction, theorem of Picard. Stolz, O. [1880] Innsb. Nt. Md. B. 11 (1881) 27-. functions, geometric representation. Wittstein, T. Grunert Arch. 7 (1846) 411-. variables, extension to 3 dimension space. Hall, T. P. Iowa Ac. Sc. P. 6 (1899) 202-. Composition of simultaneous and successive vectors. MacFarlane, A. Science 4 (1896) 346. Crystallography, parameters of, and vectors of higher order. Voigt,W. Gott. Nr. (1900) 355. Curves with branches in several planes, existence. (Geometrical interpretation of J - 1 and fractional powers). Gregory, D. F. Camb. Mth. J. 1 (1839) 259-. Directed magnitudes. Baiter, A. L. Rochester (N. Y.) Ac. Sc. P. 3 (1900) 162-. Directions and rotations, arithmetic. Berdelle, C. As. Fr. C. R. (1887) (Ft. 2) 197-. Directive algebra, application. Neuberg, J. (xn) Mathesis 1 (1881) 123-. , to geometry. Transon, A. N. A. Mth. 7 (1868) 145-, 193-, 241-. arithmetic. - Berdelle, C. As. Fr. C. R. (1888) (Pt. 2) 109-. Distance, theory. Clifford, W. K. B. A. Rp. 39 (1869) (Sect.) 9. Diihring's vector conceptions. Wessely, K. Arch. Mth. Ps. 9 (1890) 393-. Electromagnetism, application of vector func- tions. Petrini, H. Ups. S. Sc. N. Acta 15 (1895) No. 14, 60 pp. Equations in linear and vector functions, note on solution. Tait, P. G. Edinb. R. S. P. 21 (1897) 497-. Equipollences, calculus, origin. Laquiere, E. M. As. Fr. C. R. (1881) 76-. 0840 Equipollences, graphic solution of geometrical problems. Bellavitis, G. [1842] Yen. Mm. I. 1 (1843) 225-. , method. Bellavitis, G. A. Sc. Lomb. Yen. 7 (1837) 243-; 8 (1838) 17-, 85-; Mod. Mm. S. It. 25 (1855) 225-. , origin. Bellavitis, G. Ven. I. Mm. 19 (1876) 449-. , solution by. Bellavitis, G. G. Teix. J. Sc. 1 (1878) 145- ; 2 (1880) 3-. , proof of theorems by. Bellavitis, G. Ven. At. 6 (1847) 53-. , solution of a question. Monteiro, A. S. G. Teix. J. Sc. 2 (1880) 97-. External multiplication (i.e. use of alternate numbers) in algebra. Fehr, H. N. A. Mth. 14 (1895) 74-. Flat-space coordinates. Young, W. H. L. Mth. S. P. 30 (1899) 54-. Forces, composition, deduced from general principles. De Morgan, A. [1859] Camb. Ph. S. T. 10 (1864) 290-. Function of complex argument applied to lemniscate geometry, relationship, and kine- matics. Holzmiiller, F. G. Z. Mth. Ps. 21 (1876) 325-. Functions of vectors, which are themselves vec- tors. Burkhardt, H. Gott. Nr. (1893) 155-. Geometric calculus, essay. Peano, G. Tor. Ac. Sc. At. 31 (1895) 552- or 952-. , principles. Macfarlane, A. [1900] N. Y. Am. Mth. S. Bll. 7 (1901) 158, 178. multiplication of surfaces. Chessin, A. S. A. Mth. 9 (1894-95) 70-. sums and differences. Barre de Saint- Venant, . C. R. 21 (1845) 620-. Geometry of arbitrary magnitudes. Bugaev, N. V. [1888] Rec. Mth. (Moscou) 14 (1890) 394-; Fschr. Mth. (1891) 725-. Grassmann's geometrical theories. Ziicet, A. A. Mth. 2 (1885-86) 1-, 25-. method. Carvallo, E. N. A. Mth. 11 (1892) 8-; 12 (1893) 454-. applied to prove theorems in deter- minants. Milller, E. Z. Mth. Ps. 44 (1899) 28-. in projective geometry. Burali-Forti, C. Palermo Cir. Mt. Rd. 10 (1896) 177-, 254-; 11 (1897) 64-. methods, applications. Mehmke, E. Z. Mth. Ps. 37 (1892) 305-. , [centroid of quadrilateral and pentagon]. Caspary, F. N. A. Mth. 17 (1898) 389-. , to elementary mechanics. Genese, R. W. As. Fr. C. R. (1894) (Pt. 2) 211-. , quadric curves and surfaces. Miiller, E. Crelle J. Mth. 115 (1895) 234-. , vectors in a plane. Caspary, F. N. A. Mth. 18 (1899) 248-. numbers in intrinsic geometry. Cesaro, E. Rm. R. Ac. Line. Rd. 3 (1894) (Sem. 1) 367-, 621. Hertz's and Poynting's equations. Donati, L. Bologna Ac. Sc. Mm. 7 (1897) 633-. Images in the representation of functions, theory. Harris, R. A. A. Mth. 4 (1888) 65-, 128. 60 0840 Ausdehnungslehre ; vector analysis 0840 Imaginary binary straight lines. Thieme, F. E. Arch. Mth. Ps. 59 (1876) 426-. geometry (higher). Bey el, C. Zur. Vjschr. 31 (1886) 20-. of Lobatschewsky. Battaglini, G. Nap. Ed. 6 (1867) 157-. , applications of formulae. Suvorov, T. M. (xn) [1881] Kazan S. Nt. (Ps.- Mth.) P. 1 (1883) [No. 5] 4-. or lateral curves, main features of theory. Thieme, F. E. Arch. Mth. Ps. 58 (1876) 185-. straight lines. Thieme, F. E. Arch. Mth. Ps. 58 (1876) 218-. values of a variable, geometric interpreta- tion. Moitchot, . C. B. 61 (1865) 113-. Line geometry on principles of ' ' Ausdehnungs- lehre." Miiller, E. Mh. Mth. Ps. 2 (1891) 267-. Lineal multiplication of two factors. Collins, J. V. Am. As. P. (1899) 69-. Linear vector equation of central pedal of conic. Hart, H. [1882] Mess. Mth. 12 (1883) 32-. equations, solution in special conditions. Elfrinkhof, L. van. N. Arch. Wisk. 19 (1892) 132- ; Fschr. Mth. 24 (1895) 84. and vector function. Tait, P. G. Edinb. E. S. P. 21 (1897) 160-, 310- ; 22 (1900) 547-. vector functions, scalar invariants of two. Joly, C. J. [1895] Ir. Ac. T. 30 (1892-96) 709-. , theory. Joly, C. J. [1894] Ir. Ac. Tr. 30 (1892-96) 587-. Lineo-linear vector functions. Hathaway, A. S. Am. As. P. (1892) 59-. Loci of the equations p = (f> w e and p = (p v \f/ v e. Hyde, E. W. Z. Mth. Ps. 42 (1897) 122-. Magnitudes possessing extension, measurement. Study, E. Wien Ak. Sb. 91 (1885) (Ab. 2) 100-. Mixed products. Mueller, E. Mth. A. 48 (1897) 589-. Moments of inertia (Grassmann's method). Mehmke, R. Mth. A. 23 (1884) 143-. Non-congruent quantities, geometrical division. Hyde, E. W. A. Mth. 4 (1888) 9-. Operations on imaginaries considered geometri- cally. Henkle, W. D. Des Moines Anal. 1 (1874) 5-. Parallelogram of forces, analytical proof. Imschenetsky, W. G. Fschr. Ps. (1889) (Ab. 1) 290-. Plane vectors, linear transformation. Peano, G. Tor. Ac. Sc. At. 31 (1895) 113- or 157-. Point pairs, and circles in space, treated by Grassmann's method. Miiller, E. Mh. Mth. Ps. 7 (1896) 77-. Polar affections of geometrical magnitudes in piano. (Direction of lines, etc.) " Lanivicensis," (Sylvester, J. J.). [18641 Mess. Mth. 3 (1866) 11-. Polars, theory, and algebraic configurations, application of Ausdehnungslehre. Grass- mann, H. G. [1877] Crelle J. Mth. 84 (1878) 273-. Quadruple algebra ; and equations of 5th order. Cockle, Jas. Ph. Mg. 35 (1849) 434-. Quaternions and " Ausdehnungslehre." Gibb, J. W. Nt. 44 (1891) 79-. . Tait, P. G. Nt. 44 (1891) 105-. complex variable theory, calculus including. Lipschitz, R. 0. S. C. B. 91 (1880) 619-, 660-. , Hamilton's, place in Ausdehnungslehre. Grassmann, H. G. Mth. A. 12 (1877) 375-. and vector analysis. Gibbs, J. W. Nt. 48 (1893) 364-. , principles. Peddie, W. Edinb. Mth. S. P. 11 (1893) 85-. vectors. Knott, C. G. Nt. 48 (1893) 516-. Quotients of space directed lines. Elliott, E. B. Smiths. Misc. Col. 33 (1888) Art. 4, 106- ; (Wash. Ph. S. Bll. 10 (1888).) Beduction of vectors and metric properties. Andrade, J. C. B. 125 (1897) 394-. Bigid body, motion of (exercise in Grassmann's methods). Liiroth, J. Z. Mth. Ps. 43 (1898) 243-. Botations. Burali Forti, C. Bll. Sc. Mth. 23 (1899)82-. Screw as a unit in the Grassmannian system of the sixth order. Hyde, E. W. A. Mth. 8 (1893-94) 38-. Self -con jugate linear and vector functions, property. Tait, P. G. [1871] Edinb. B. S. P. 7 (1872) 498-. Spherical geometry according to Grassmann. Miiller, E. Mh. Mth. Ps. 3 (1892) 365-; 4 (1893) 1-, 93; 9 (1898) 269-. Symbolic forms. O'Brien, M. [1851] Phil. Trans. (1852) 161-. Symbolical geometry and mechanics. O'Brien, M. [1847] Camb. Ph. S. T. 8 (1849) 497-. Use of v 1 and alternate numbers i, j, k in study of infinitesimal deformations. Perna, A. Palermo Cir. Mt. Bd. 12 (1898) 322-. Vector analysis. Chapman, C. H. J. H. Un. Cir. [10 (1890-91)] 46-. . Tait, P. G. Nt. 47 (1892-93) 225-. . Donati, L. Bologna Ac. Sc. Mm. 7 (1897) 11-. . Farkas, G. Mth. Termt. Ets. 16 (1898) 321- ; Mth. Nt. B. Ung. 16 (1899) 111-; Orv.-Termt. Ets. (Termt. Szak) (1899) 91-; (1900) 1-, (Rv.)l-. . Burkliardt, H. D. Mth. Vr. Jbr. 5 (1901) (Heft 1) 43-. , direct and logarithmic, &c. Malcor, (Capt.) . Bv. Mar. et Col. 76 (1883) 735-; 77 (1883) 44-; 78 (1883) 416-. , a notation. Henrici, 0. B. A. Bp. (1897) 560-. , reality of roots of symbolical cubic. Tait, P. G. [1867] Edinb. B. S. P. 6 (1869) 92-. conditions, integrability. Tait, P. G. [1877] Edinb. E. S. P. 9 (1878) 527-. , derivatives of a. Daniels, F. Z. Mth. Ps. 45 (1900) 203-. differentiation. MacFarlane, A. [1900] Wash. Ph. S. Bll. 14 (1906) 73-. fields and fields of force. Broca, A. C. B. 130 (1900) 109-. 61 0850 Matrices Vector fields, properties. Donati, L. Bologna Ac. Sc. Mm. 7 (1897) 427-. , theory, as preliminary to study of elec- tricity and magnetism. Ferraris, G. Tor. Ac. Sc. Mm. 47 (1897) 259-. functions. Burkhardt, H. Mth. A. 43 (1893) 197-. quantities, certain. Bjerknes, V. Christiania Skr. (Mth.-Nt. Kl.) (1898) No. 4, 27pp. ratios considered as trigonometric functions of angles. Stringham, W. I. (xn) [1879] J. H. Un. Cir. [1] (1882) 35. Vectorial coordinates. Sylvester, J. J. Ph. Mg. 31 (1866) 287-, 380-. geometry on complexes of second order having a centre of figure. Genty, . Liouv. J. Mth. 8 (1882) 299-. Vectors, application to central motion. Cas- tellano, F. Tor. Ac. Sc. At. 29 (1894) 80-. , calculation with Ptolemy's theorem gene- ralised. Riecke,. Grunert Arch. 32 (1859) 470-. of ellipse and related curves, sums and products. Schlomilch, 0. X. Z. Mth. Ps. 26 (1881) 59-. , kinematic applications of theory. Cas- tellano, F. Ev. Mt. 2 (1892) 19-. and linear transformations, with physical applications. Carvallo, E. Mh. Mth. Ps. 2 (1891) 177-, 225-, 311-. , , note on Carvallo's paper. Peano, G. Mh. Mth. Ps. 5 (1894) 136. quaternions. Knott, C. G. Nt. 48 (1893) 148-. . Lodge, A. Nt. 48 (1893) 198-. . Macfarlane, A. Nt. 48 (1893) 540-. , relation among four. Tail, P. G. [1889] Edinb. E. S. P. 16 (1890) 88-. , scalar relations connecting six. Wilkinson, (Rev.) M. M. U. [1889] Edinb. E. S. P. 16 (1890) 773-. , theory. Zeiliger, D. N. N. Es. S. Nt. Mm. (Mth.) 11 (1890) 1-, 149- ; 13 [?1891] 1-; Fschr. Ps. (1890) (Ab. 1) 239. , , analysis. Peano, G. Tor. Ac. Sc. At. 33 (1897) 313- or 513-. , recent innovations. Knott, C. G. [1892] Edinb. E. S. P. 19 (1893) 212-. , , teaching. Fontene, G. Ens. Mth 1 (1899) 50-. versus quaternions. Heaviside, 0. Nt. 47 (1892-93) 533-. Macfarlane, A. Nt. 48 (1893) 75. which can be compounded, theory. Berruti, G. Tor. Ac. Sc. At. 29 (1894) 115, 531-. 0850 Matrices. Algebraic resolvents. Rados, G. Mth. Termt Birational transformations, prime factors Kantor, S. Mh. Mth. Ps. 10 (1899) 54-. Matrices 0850 Characteristics of systems of three dimensions. Suvorov, T. M. (xn) Kazan Un. Mm. 7 (1871) 3-. Complex numbers. Poincare, H. C. E. 99 (1884) 740-. Differential equation to a curve of any order. Sylvester, J. J. Nt. 34 (1886) 365-. Elementary universal geometry, new terms. Sylvester, J. J. Nt. 32 (1885) 576-. Equations in matrices. Sylvester, J. J. J. H. Un. Cir. [3] (1884) 122. multiple quantity. Sylvester, J. J. Nt. 31 (1885) 35-. Exchangeable matrices. Frobenius, G. Berl. Ak. Sb. (1896) 601-. Hamilton's quadratic equation, and the general unilateral equation in matrices. Sylvester, J. J. Ph. Mg. 18 (1884) 454-. a general method for unilateral equations in matrices of any order. Sylvester, J. J. C. E. 99 (1884) 621-. quaternions and Cayley's matrices. Wettum, T. B. van. N. Arch. Wisk. 17 (1890) 206- ; Fschr. Mth. (1890) 699-. Harriot's law on algebraic equations, extension to matrices. Sylvester, J. J. C. E. 98 (1884) 1026-. Infinite and imaginary, use in service of finite and real. Sylvester, J. J. Nt. 32 (1885) 103-, 271. ' Interpolation-formula in matrices, extension of Sylvester's. Buchheim, A. Ph. Mg. 22 (1886) 173. Invariants, arithmetical, of a rectangular matrix. Smith, H. J. S. L. Mth. S. P 4 (1871-73) 236-. Inverse orthogonal matrices, etc. Sylvester, J. J. Ph. Mg. 34 (1867) 461-. Involutant of two binary matrices. Cay ley, A. Mess. Mth. 20 (1891) 136-. Involutants and allied species of invariants to matrix systems. Sylvester, J. J. (xn) [1883-84] J. H. Un. Cir. [3] (1884) 9-, Involution of two matrices of second order. Sylvester, J. J. B. A. Ep. (1883) 430-. Linear equation in matrices of any order, general resolution. Sylvester, J. J. C. E. 99 (1884) 409-, 432-, 481. trinomial equation in matrices of any order. Sylvester, J. J. C. E. 99 (1884) 527-, 576. Matrical equation px = xq. Sylvester, J J C. E. 99 (1884) 67-, 115-. qQ-Qq' = 0. Cayley, A. Mess. Mth. 14 (1885) 176-. 00 = 00. Taber, H. Am. Ac. P. 26 (1891) 64-. Matrices. Tyler, H. W. Erlang. Ps. Md. S Sb. 23 (1891) 33-. , application of theory to linear differential equations. Brill, J. Camb. Ph. S. P 8 (1895) 201-. and associated systems of complex quan- tities. Weyr,E. Prag Sb. (1887) (Mth.-Nt.) 616-. , classification. Weyr, E. C. E. 100 (1885) 966. G2 0850 Matrices Matrices, fundamental proposition. Weyr, E. Prag Sb. (1884) (Mth.-Nt.) 148- ; Fschr. Mth. (1885) 107. , multiplication of, identical relation. Kretowski, W. Krk. Ak. (Mt.-Prz.) Bz. 6 (1893) 151- ; Fschr. Mth. (1893-94) 217. , note. Brill, J. [1895-97] L. Mth. S. P. 27 (1896) 35-; 28 (1897) 368-. , product of two. Sardi, C. G. Mt. 5 (1867) , ' . Janni, V. G. Mt. 11 (1873) 357-. , . Bonolis, A. G. Mt. 34 (1896) 375-. and quaternions. Elfrinkhof, L. van. N. Arch. Wisk. 1 (1895) 88-; Fschr. Mth. (1893-94) 226-. satisfying a certain equation. Brunei, . Bordeaux S. Sc. Mm. 4 (1888) xxii. , spherical geometry theorem proved by. Chrystal, G. Edinb. Mth. S. P. 2 (1884) 45-. of substitution. Brunei, . Bordeaux S. Sc. Mm. 4 (1888) viii. , theorem by Cayley, proof. Forsyth, A. R. Mess. Mth. 13 (1884) 139-. , theory. Cayley, A. Phil. Trans. 148 (1858) 17-; 156 (1866) 25-. , . Buchheim, A. L. Mth. S. P. 16 (1884-85) 63-. , . Weyr, E. C. B. 100 (1885) 787-; Prag Sb. (1887) (Mth.-Nt.) 358-. , . Taber, H. Am. J. Mth. 12 (1890) 337-. , , identities in. Taber, H. Am. J. Mth. 13 (1891) 159-. Matrix, a. Ravut, L. N. A. Mth. 17 (1898) us-. (a, b) , . . ax + b . ', and function _, , Mess. Mth. (1880) 104-. of third order, square root of. Cayley, A. Edinb. B. S. P. 7 (1872) 675-. which represents a vector. Chapman, C. H. Am. J. Mth. 13 (1891) 363-. Monothetic equations. Sylvester, J. J. C. B. 99 (1884) 13-. Non-degenerate matrices, Sylvester's theorem. Taber, H. Am. Ac. P. 27 (1893) 46-. Notation for matrices, etc. Cayley, A. Crelle J. 50 (1855) 277-. Quaternion equation qQ-Qq' = Q. Cayley, A. Mess. Mth. 14 (1885) 108-. matrix. Wettum, T. B. van. N. Arch. Wisk. 18 (1891) 168- ; Fschr. Mth. (1891) . Elfrinkhof, L. van. N. Arch. Wisk. 19 (1892) 143- ; Fschr. Mth. (1892) 141-. symbols, application to matrix. Taber, H. [1890] L. Mth. S. P. 22 (1891) 67-. Boots of matrices. Metzler, W. H. Am. J. Mth. 14 (1892) 326-. unitary matrices. Sylvester, J. J. C. B. 94 (1882) 396-. zeroidal matrices. Brunei, G. C. B. 106 (1888) 467-. Split matrix, properties. Sylvester, J. J. (xn) J. H. Un. Cir. [1] (1882) 210-. Other Complex Numbers 0860 Trinomial unilateral quadratic equation in matrices of second order. Sylvester, J. J. QJ. Mth. 20 (1885) 305-. Vectors and linear transformations, with physi- cal applications. Carvallo, E. Mh. Mth. Ps. 2 (1891) 177-, 225-, 311-. , , note on Carvallo's paper. Peano, G. Mh. Mth. Ps. 5 (1894) 136. 0860 Other special sorts of complex numbers. Associative imaginaries. Cayley, A. (xn) J. H. Un. Cir. [1] (1882) 211-. Biquaternions. Clifford, W. K. L. Mth. S. P. 4 (1873) 381-. , application to plane geometry. Com- bebiac, . Par. S. Mth. Bll. 26 (1898) 259-. , coplanar, or doubly complex quantities, theory. Seydler, A. J. Prag Sb. (1881) 80-. , geometrical interpretation of results. Ha- milton, (Sir) W. R. Ir. Ac. P. 5 T1853) 388-. , memoir. Buchheim, A. Am. J. Mth. 7 (1885) 293-. Classes of numbers derived from real numbers by dexter and sinister operations. Burali- Forti, C. Tor. Ac. Sc. At. 29 (1894) 382-. Complex functions, generalisation of founda- tions. Scheffers, G. Leip. Mth. Ps. B. 45, (1893) 828-; 46 (1894) 120-. numbers extended. Vahlen, T. Konigsb. Schr. 38 (1897) [72]-. formed from n principal units. Weier- strass, K. Gott. Nr. (1884) 395-. . Schwarz, H. A. Gott. Nr. (1884) 516-. . Dedekind, R. Gott. Nr. (1885) 141-. . Holder, 0. Gott. Nr. (1886) 241-. . Weyr, E. Bll. Sc. Mth. 11 (1887) 205-. . Scheffers, G. Leip. Mth. Ps. B. 41 (1889) 290-. . Schur, F. Mth. A. 33 (1889) 49-. . Gegenbauer, L. Mh. Mth. Ps. 2 (1891) 429-. . Hilbert, D. Gott. Nr. (1896) 179-. , general. Dedekind, R. Gott. Nr. (1887) 1-. and groups of transformations. Study, E. Leip. Mth. Ps. B. 41 (1889) 177. of n dimensions. Petersen, J. Gott. Nr. (1887) 489-. , real systems. Carton, E. C. B. 124 (1897) 1296-. , systems. Study, E. Gott. Nr. (1889) 237-. , . Carton, E. C. B. 124 (1897) 1217-. , of higher. Molten, T. Mth. A. 41 (1893) 83-; 42 (1893) 308-. 63 0870 Algebra of Logic Complex numbers, systems, reducibility. Schef- fers, G. Mth. A. 41 (1893) 601-. , , reduction to typical forms. Schef- fers, G. Mth. A. 39 (1891) 293-. , , theory. Hausdorff, F. Leip. Mth. Ps. B. 52 (1900) 43-. Elliptic and hyperbolic numbers. Levi-Civita, T. Ven. I. At. (1892-93) 1765-. Nonions analogous to quaternions, quantities forming a group. Sylvester, J. J. C. E. 97 (1883) 1336- ; 98 (1884) 273-, 471-. , system. Peirce, C. S. (xn) J. H. Un. Cir. [2] (1883) 86-. , ( , represented by 9 matrices). Sylvester, J. J. (xn) J. H. Un. Cir. [1] (1882) 241- ; [2] (1883) 46, 86. Non-quaternion number-systems containing no skew units. Starkweather, G. P. Am. J. Mth. 21 (1899) 369-. Octonions. McAulay, A. B. S. P. 59 (1896) 169-. Pluquaternions, and homoid products. Kirk- man, T. P. Ph. Mg. 33 (1848) 447-, 494-. Seven fundamental keys, system. Fontene, G. Par: S. Mth. Bll. 27 (1899) 171-. Symbolical and double numbers. Philippoff, M, Z. Mth. Ps. 37 (1892) 298-. Systems of numbers, calculation. Scheffers, G. Leip. Mth. Ps. B. 41 (1889) 400-. one-vectors in space of n dimensions. Young, W. H. L. Mth. S. P. 29 (1898) 478-. Trinomial complex numbers. Schwering, K. Acta Mth. 10 (1887) 57-. Triquaternions. Combebiac, . Par. S. Mth. Bll. 27 (1899) 180-. Types of algebraical magnitudes arising by composition of two other types. Hensel, K. Crelle J. Mth. 105 (1889) 329-. Units of an w-fold space, applications. Chap- man. C. H. Am. J. Mth. 10 (1888) 225-. 0870 Algebra of Logic. Abacus, logical. Jevons, W. S. Manch. Lt. Ph. S. P. 5 (1866) 161-. Algebra of binary relatives. Schroeder, E. Mth. A. 46 (1895) 144-.' logic. Peirce, C. S. Am. J. Mth. 3 (1880) 15-. , a contribution to the philosophy of notation. Peirce, C. S. Am. J Mth 7 (1885) 180-. , formulae. Peano, (?. Ev. Mt. 1 (1891) 24-, 182-. , principles. Macfarlane, A. B. A. Ep. (1879) 262-. , . Peano, G. Ev. Mt. 1 (1891) , remark. Korselt, A. Mth. A. 44 (1894} 156-. relations, principles. Schroder, F [1900] N. Y. Am. Mth. S. Bll. 7 (1901) 176-. Boole's analysis of logic. Harley, E. B. A. Ep. 36 (1866) (Sect.) 3-. laws of thought. Young, G. P. Cn J 10 (1865) 161-. 0870 Algebra of Logic Boole's laws of thought. Ellis, E. L. B. A. Ep. 40 (1870) (Sect.) 12-. . Harley, E. B. A. Ep. 40 (1870) (Sect.) 14-. Calculus of equivalent statements . Mac Coll,H. [1896-97] L. Mth. S. P. 28 (1897) 156-, 555- ; 29 (1898) 98-. and integration limits. McColl, H. L. Mth. S. P. 9 (1877-78) 9-, 177- ; 10 (1878-79) 16-; 11 (1879-80) 113-. logic. Boole, G. Camb. and Dubl. Mth. J. 3 (1848) 183-. . Cayley, A. QJ. Mth. 11 (1871) 282-. , Boole's, improvement. Peirce, C. S. [1867] Am. Ac. P. 7 (1868) 250-. , cycle of operations of. Schroder, E. Mth. A. 12 (1877) 481-. , functional theorems. Nagy, A. Ev. Mt. 2 (1892) 177-. Deductive logic, principles. Vailati, G. Ev. Mt. 1 (1891) 127-. Dependence, property of, as a relation. Vailati, G. Ev. Mt. 2 (1892) 161-. Dialectics, rational. Gergonne, J. D. Gergonne A. Mth. 7 (1816-17) 189-. Elimination problem in identical calculus. Schroder, E. D. Nf. Tbl. (1885) 353-. Expressions for hypothetical and disjunctive propositions. Graves, C. Ir. Ac. P. 4 (1850) 147-. Formal properties of algebraic operations. Burali-Forti, C. Ev. Mth. 6 (1896-99) 141-. Implicational and equational logic. McColl, H. [1880] Ph. Mg. 11 (1881) 40-. Inverse or inductive logical problem. Jevons, W. S. [1871] Manch. Lt. Ph. S. P. 11 (1872) 65-. Jevons-Clifford problem. Nagy, A. Mh. Mth. Ps. 5 (1894) 331-. Law of roots in logic. Poretsky, P. Ev. Mth. 6 (1896-99) 5-. Logic, algebraic analogues. Ellis, A. J. E. S. P. 21 (1873) 497-. , application to mathematics. Debrun, B. Niort. Mm. de l'Athene"e 1 (1808) 1-. , . Matzka, W. Grunert Arch. 6 (1845) 353-. , formal. Ellis, A. J. E. S. P. 20 (1872) 307-. of mathematics. Peirce, C. S. [1867] Am. Ac. P. 7 (1868) 402-. number. Peirce, C. S. Am. J. Mth. 4 (1881) 85-. relatives, notation. Peirce, C. S. [1870] Am. Ac. Mm. 9 (pt. 2) (1873) 317-. , symbolic. MacColl, H. [1900] N. Y. Am. Mth. S. Bll. 7 (1901) 158, 166-. , treatment. Murphy, J. J. Manch. Lt. Ph. S. Mm. & P. 4 (1891) 201-. Logical calculus. Poreckij, P. S. Kazan S. Nt. (Ps.-Mth.) P. 7 (1889) 260-. . Nagy, A. G. Mt. 28 (1890) 1-. equalities, laws. Poreckij, P. S. Kazan S. Ps.-Mth. Bll. 8 (1899) 33-, 129-, 183- ; 10 (1901) 50-, 132-, 191- ; 11 (1901) 17-. equations. Johnson, W. E. [1900] N. Y. Am. Mth. S. Bll. 7 (1901) 158, 177-. 64 0870 Algebra of Logic Logical equations, methods of solution, and inverse method of mathematical logic. Poryetskii, P. S. (xn) [1882] Kazan S. Nt. (Ps.-Mth.) P. 2 (1884) 161-. form of geometrical theorems. Macfarlane, A. A. Mth. 3 (1887) 154-. functions, theorems concerning. Nagy, A. Ev. Mth. 6 (1896-99) 21-. notation, a new system. Murphy, J. J. Manch. Lt. Ph. S. Mm. & P. 2 (1889) 22-. propositions, on the transformations. Murphy, J. J. Manch. Lt. Ph. S. Mm. 8 (1884) 132-. quantities, graphical representation. Nagy, A. Em. E. Ac. Line. Ed. 6 (1890) (Sem. 2) 50-, 373-, 393. spectrum. Macfarlane, A. Ph. Mg. 19 (1885) 286-. , method, application to Boole's pro- blem. Macfarlane, A. Am. As. P. (1890) 57-. Magnitudes, theory. Burali-Forti, C. Ev. Mt. 3 (1893) 76-. Mathematical logic, first principles. Poryetskii, P. S. (xn) [1880] Kazan S. Nt. (Ps.-Mth.) P. 1 (1883) [Ufa 2)] 2-. , some applications in arithmetic. B unite - kij, E. Fschr. Mth. (1897) 63. , studies. Peano, G. Tor. Ac. Sc. At. 32 (1896) 361- or 565-. Mechanical performance of logical inference. Jevons, W. S. [1869] Phil. Trans. 160 (1870) 497-. Notations, various, for expressing propositions of logic. Venn, J. [1880] Camb. Ph. S. P. 4 (1883) 36-. Numerically definite propositions. Boole, G. [1868] Camb. Ph. S. T. 11 (pt. 2) (1869) 396-. reasoning, system. Jevons, W. S. [1870] Manch. Lt. Ph. S. Mm. 4 (1871) 330-. Eelation between the logical theory of classes and the geometrical theory of points. Kempe, A. B. [1890] L. Mth. S. P. 21 (1891) 147-. Eelations between logical magnitudes. Nagy, A . Mh. Mth. Ps. 4 (1893) 147-. members of a system. Amicis, E. de. Ev. Mt. 2 (1892) 113-. Straight lines, fundamental geometry of. Vaikiti, G. Ev. Mt. 2 (1892) 71-. Syllogism, on the. Harris, J. R. J. H. Un. Cir. [3] (1884) 130-. Syllogisms of transposed quantity. De Mor- gan, A. Camb. Ph. S. T. 10 (1864) 355*-. Symbolical language, growth and use. McColl, H. [1881] Manch. Lt. Ph. S. Mm. 7 (1882) 225-. Symbols of logic, and theory of the syllogism. De Morgan, A. [1846-50] Camb. Ph. S. T. 8 (1849) 379- ; 9 (1856) 79-. Types of compound statement involving 4 classes. Clifford, W. K. Manch. Lt. Ph. S. P. 16 (1877) 88-. VOL. I. Theory of Groups. General 1200 Theory of Groups. 1200 General Adjoint operations. Pincherle, S. Bologna Ed. 2 (1898) 130-. Anharmonic curves. Bioche, C. Par. S. Phlm. Bll. 5 (1893) (C. R. No. 10) 2. Constancy of factor-groups, Holder's theorem. Brown, G. L. N. Y. Am. Mth. S. Bll. 1 (1895) 232-. Distributive operations and homography. Pin- cherle, S. Mil. I. Lomb. Ed. 29 (1896) 397-. Finite and infinite groups. Bettazzi, R. Tor. Ac. Sc. At. 31 (1895) 362- or 506-. Functional substitution equations. Bottcher, E. Wiad. Mt. 4 (1900) 233-. Group-characteristics. Frobenius, G. Berl. Ak. Sb. (1896) 985-. , composition. Frobenius, G. Berl. Ak. Sb. (1899) 330-. and subgroups, relations. Frobenius, G. Berl. Ak. Sb. (1898) 501-. Group and number. Bettazzi, R. Tor. Ac. Sc. At. 31 (1895) 304- or 446-. theory and its application. Zorawski, K. Krk. Ak. (Mt.-Prz.) Ez. 6 (1893) 289-; Crc. Ac. Sc. Bll. (1893) 145-. Groups, complete theory. Kirkman, T. P. [1863] Manch. Ph. S. P. 3 (1862-63) 133-, 161- ; 4 (1865) 171-. depending on n =l. Cayley, A. Ph. Mg. 7 (1854) 40-, 408 ; 18 (1859) 34-. of functions. Pincherle, S. Bologna Ac. Sc. Mm. 6 (1884) 205-. , functions given by. Kirkman, T. P. Manch. Lt. Ph. S. Mm. & P. 4 (1891) 315- ; 5 (1892) 23-. and generators, systems. Fulco, P. Catania Ac. Gioen. At. 12 (1899) Mm. 12, 14 pp. of interchangeable elements. Frobenius, G. , <& Stickelberger, L. [1878] Crelle J. Mth. 86 (1879) 217-. involution, transformations. Montesano, D. Yen. I. At. (1887-88) 1425-. and many -valued functions. Kirkman, T. P. [1861] Manch. Ph. S. Mm. 1 (1862) 274-. of order 8 !/2. Schottenfels, I. M. N. Y. Am. Mth. S. Bll. 6 (1900) 440-. related to regular bodies in n -dimensional space. Schoute, P. H. B. A. Ep. (1894) 562-. , systems of. Dickson, L. E. N. Y. Am. Mth. S. Bll. 3 (1897) 265-. , theory. Cayley, A. [1877-89] L. Mth. S. P. 9 (1877-78) 126- ; Am. J. Mth. 11 (1889) 139-. , and graphical representation. Cayley, A. Am. J. Mth. 1 (1878) 174- . Imprimitive groups. Kuhn, H. W. N. Y Am Mth. S. Bll. 7 (1901) 113, 116-. and soluble groups. Soderberg, J. T. [1899] Ups. S. Sc. N. Acta 20 (1904) No. 1, 26 pp. Infinite order, groups of. Miller, G. A. [1900] N. Y. Am. Mth. S. Bll. 7 (1901) 121-. 65 1200 Theory of Groups. General Invariants of groups of linear substitutions. Molien, T. Berl. Ak. Sb. (1897) 1152-. Inversion groups on quadric surfaces. Kasner, E. N. Y. Am. Mth. S. T. 1 (1900) 430-. Linear differential equations, arithmetical de- velopments. Fricke, R. D. Mth. Vr. Jbr. 2 (1893) 59-. , form of integrals. Johanson, A. M. Stockh. Ofv. (1889) 583-. , theory. Volterra, V. Em. S. It. Mm. 6 (1887) No. 8, 104 pp. groups of functions. Pincherle, S. Bologna Ac. Sc. Mm. 6 (1884) 101-. , report on recent progress. Dickson, L. E. [1899] N. Y. Am. Mth. S. Bll. 6 (1900) 13-. Maximum groups. Kirkman, T. P. Manch. Ph. S. P. 3 (1862-63) 59-. Non-isomorphic simple groups of the same order. Schottenfels, I. M. [1898] A. Mth. 1 (1900) 147-. -modular groups. Kirkman, T. P. (m) [1862] Manch. Ph. S. Mm. 2 (1865) 204-. Popular account of new fields of mathematics. Miller, G. A. Science 11 (1900) 528-. Prime factors of the group-determinant. Fro- benius, G. Berl. Ak. Sb. (1896) 1343-. Eational and linear substitutions, the group of. Cordone, G. Tor. Ac. Sc. At. 31 (1895) 472- or 804- . Eegions of discontinuity for groups of real linear substitutions of a complex variable. Fricke, R. Gott. Nr. (1895) 360-. Eepresentation of finite groups by linear sub- stitutions. Frobenius, G. Berl. Ak. Sb. (1897) 995-; (1899) 482-. Euled surfaces admitting homographic auto- morphic transformation. Bioche, C. Par. S. Phlm. Bll. 4 (1892) 130-. Singular transformations of groups. Taber, H. N. Y. Am. Mth. S. Bll. 6 (1900) 199-. . Newson, H. B. N. Y. Am. Mth. S. Bll. 6 (1900) 431-. Sub-groups of doubly infinite series of simple groups. Wiman, A. Stockh. Ak. Hndl. Bh. 25 (1900) (Afd. 1) No. 2, 47 pp. general projective group. Taber, H. N. Y. Am. Mth. S. Bll. 2 (1896) 221-, 346. Substitution groups, fundamental theorem. Miller, G. A. [1895] N. Y. Am. Mth. S. Bll. 2 (1896) 75-. Substitutions. Zochios, . C. E. 120 (1895) 766-. Symmetrical group, characteristics. Frobenius, G. Berl. Ak. Sb. (1900) 516-. Test of a simple group. Miller, G. A. N. Y. Am. Mth. S. Bll. 1 (1895) 124-. Uniform substitutions. Koenigs, G. Bll. Sc. Mth. As. 7 (1883) 340-. . Podetti, F. G. Mt. 35 (1897) 264-. Vanishing groups. Cockle, Jas. Camb. and Dubl. Mth. J. 6 (1851) 174- ; 7 (1852) 114- ; 8 (1853) 50-. Discrete Groups 1210 1210 Discrete groups of finite and of infinite order (including groups of permutations). See also 2450, 4440. Abelian general linear group, new definition. Dickson, L. E. N. Y. Am. Mth. S. T. 1 (1900) 91-. groups. Heffter, L. Crelle J. Mth. 119 (1898) 261-. . Searpu, U. G. Mt. 38 (1900) 225-. and linear modular systems. Fund, 0. Hamb. Mth. Gs. Mt. 3 (1900) 371-. , and their application to number theory. Zsigmondy, K. Mh. Mth. Ps. 7 (1896) 185-. Abstract group isomorphic with the symmetric group on k letters. Dickson, L. E. L. Mth. S. P. 31 (1900) 351-. groups of degree 32. Bagnera, G. A. Mt. 2 (1899) 263-. order *1 and J*!. Moore, E. H. [1896] L. Mth. S. P. 28 (1897) 357-. simple group of order 2 7 . 3 6 . 5 . 7, a certain. Dickson, L. E. N. Y. Am. Mth. S. T. 1 (1900) 353-, 509. Alternating group, uniform character. Beke, E. Mth. A. 49 (1897) 581-. groups. Kbnig, G. (xii) Mth. Tennt. Ets. 1 (1883) 213- ; Mth. Nt. B. Ung. 1 (1882-83) 191-. Analytical representation of a heptagram. Rogers, L. J. [1890] L. Mth. S. P. 22 (1891) 37-. Arithmetical group, laws of reciprocity and sub- groups. Stouff, X. Par. 6c. Norm. A. 10 (1893) 295-. Automorphic functions of two variables. Pi- card, E. Acta Mth. 1 (1882) 297- ; C. B. 97 (1883) 1045-. modular groups. Fricke, R. Gott. Nr. (1896) 91-. Binary forms with linear automorphic trans- formations. Klein, C. F. [1875] Mth. A. 9 (1876) 183-. Cayley's color-diagrams, representations of finite groups. Maschke, H. Am. J. Mth. 18 (1896) 156-. Characteristics of types 3 & 4 and their sub- stitution groups. Pascal, E. A. Mt. 20 (1892-93) 163- ; Bm. E. Ac. Line. Ed. 2 (1893) (Sem. 1) 120-. Commutator groups. Miller, G. A. N. Y. Am. Mth. S. Bll. 4 (1898) 135-. Commutators of given group. Miller, G. A. [1899] N. Y. Am. Mth. S. Bll. 6 (1900) 105-. Composite groups. Bumside, W. Mess. Mth. 24 (1895) 82-. , formation. Holder, O. Mth. A. 46 (1895) 321-. , theory. Netto, E. Crelle J. Mth. 78 (1874) 81-. Configuration of 140 straight lines in space. Maschke, H. Mth. A. 36 (1890) 190-. 66 1210 Discrete Groups of Finite and of Infinite Order 1210 Congruence-group contained in the group of linear fractional substitutions. Moore, E. H. Am. As. P. (1892) 62. groups of prime rank. Gierster, J. Mth. A. 22 (1883) 111-. , theorem. Brioschi, F. Mil. I. Lomb. Ed. 12 (1879) 483-. Congruences, class of groups denned by. Burn- side, W. L. Mth. S. P. 25 (1894) 113- ; 26 (1895) 58-. Conjugate system of linear substitutions, for- mation. Baraniecki, M. A. Par. T. Nauk Sc. Pam. 10 (*1878) Art. 3, 8 pp. Connexion in series and theory of substitutions. Hoyer, P. Mth. A. 42 (1893) 58- ; 49 (1897) 39-. Crystalline structure and regular point groups. Rohn, K. Leip. Mth. Ps. B. 51 (1899) (Mth.) 445-. Cubic surfaces with automorphic collineations. Bobek, K. Mh. Mth. Ps. 10 (1899) 122-, 307-. , polar hexahedra. Cremona, L. D. Nf. B. (*1877) 93-. Cyclic subgroups of simple ternary linear fractional group in a Galois field. Dickson, L. E. Am. J. Mth. 22 (1900) 231-. Direct product groups. Miller, G. A. N. Y. Am. Mth. S. T. 1 (1900) 66-. Discontinuity in certain homographies. Fricke, E. Mth. A. 47 (1896) 557-. Discontinuous groups. Poincare, H. C. E. 94 (1882) 840-. in biquadratic field. Fricke, R. Gott. Nr. (1892) 268-. and quadratic forms. Picard, E. C. E. 95 (1882) 763-. of real linear substitutions (Poincare's). Kepinski, S. Krk. Ak. (Mt.-Prz.) Ez. 6 (1893) 37- ; Fschr. Mth. (1893-94) 207-. Doubly transitive groups of degree n and order n(n-l). Burnside, W. Mess. Mth. 25 (1896) 147-. 2 TO and order 2 (2 m - 1) . Burn- side, W. Mess. Mth. 25 (1896) 187-. substitutions of three degrees. Maillet, E. Par. S. Mth. Bll. 25 (1897) 189-. Exchangeable and decomposable groups. Maillet, E. Par. S. Mth. Bll. 28 (1900) Finite group, definition. Bettazzi, R. Tor. Ac. Sc. At. 32 (1896) 240 or 352-. groups. Frobenius, G. Berl. Ak. Sb. (1895) 163-. . Maillet, E. QJ. Mth. 29 (1898) 250-. . Bauer, M. Mth. Tennt. fits. 17 (1899) 611- ; N. A. Mth. 19 (1900) 508-. , class of. Burnside, W. Camb. Ph. S. T. 18 (1900) 269-. contained in group of quadratic homo- geneous transformations. Autonne, L. C. E. 97 (1883) 567-. linear group, determination. Jordan, C. Nap. Ac. At. 8 (1879) No. 11 41 pp. Finite groups contained in linear groups, class. Jordan, C. Par. S. Mth. Bll. 5 (1877) 175-. of correlations. Kantor, S. Crelle J. Mth. 116 (1896) 171-. whose degree is the fifth power of a prime. Bagnera, G. A. Mt. 1 (1898) 137-. of linear substitutions, fundamental pro- position in theory. Burkhardt, H. Mth. A. 41 (1893) 309-. , theorem on. Klein, F. D. Nf. Vh. (1896) (Th. 2, Hdlfte 1) 5 ; D. Mth. Vr. Jbr. 5 (1901) (Heft 1) 57. transformations of a variable. Gordan, P. Mth. A. 12 (1877) 23-. . Cayley, A. [1879] Mth. A. 16 (1880) 260-, 439-. . Fano, G. Mh. Mth. Ps. 7 (1896) 297-. , report on. Miller, G. A. [1898] N. Y. Am. Mth. S. Bll. 5 (1899) 227-. -, sets of operations in relation to. White- head, A. N. E. S. P. 64 (1899) 319-. , theory. Valentiner, H. Kjtfb. Dn. Vd. Selsk. Skr. 5 (1889-91) 63- (with French resume). , . Burnside, W. [1893-95] L. Mth. S. P. 25 (1894) 9-; 26 (1895) 191-, 325-, 559. , . Bauer, M. N. A. Mth. 19 (1900) 59-. Functions, cyclic. Dziobek, O. Arch. Mth. Ps. 69 (1883) 265-. , linear rational, taken according to prime modulus, and substitutions to which they lead. Serret, J. A. C. E. 48 (1859) 112-, 178-, 237-. , , of several variables, values. Capelli, A. G. Mt. 14 (1876) 141-. , number of values, by permutations of letters. Bertrand, J. Par. 6c. Pol. J. 30 8 cah. (1845) 123-. , , (Bertrand). Serret, J. A. Liouv. J. Mth. 14 (1849) 135-. , , . Despeyrons, C. Liouv. J. Mth. 10 (1865) 55-. , polynomial, admitting of reduction in number of variables. Sylvester, J. J. Ph. Mg. 5 (1853) 119-. proper to represent substitution of prime number of letters. Rogers, L. J. Mess. Mth. 21 (1892) 44-. reproduced by substitution. Cauchy, A. L. C. E. 19 (1844) 1436-. of several quantities, nature, formation, and substitutions rendering invariable. Mathieu, E. Liouv. J. Mth. 6 (1861) 241- ; N. A. Mth. 1 (1862) 227-. 5 or 6 letters, especially those doubly "transitive. Cauchy, A. L. C. E. 21 (1845) 1401- ; 22 (1846) 1-. 4, 5, or 6 letters. Serret, J. A. Liouv. J. Mth. 15 (1850) 45-. 6 letters, unsymmetrical, historical origin. Sylvester, J. J. Ph. Mg. 21 (1861) 369-. 7 letters. Hermite, C. C. E. 57 (1863) 750-. . Brioschi, F. C. E. 95 (1882) 665-, 814-, 1254-. 67 1210 Discrete Groups of Finite and of Infinite Order 1210 Galoisian groups. Perott, J. Par. S. Mth. Bll. 21 (1893) 61-. Generators of a symmetric group. Capelli, A . Q. Mt. 35 (1897) 354-. Geometrical operations, certain. Fleuri, G. N. S. W. E. S. J. 26 (1892) 60-. Group belonging to irreducible equations of prime degree. Sylow, L. [1867] Christiania F. 10 (1868) 105-. of discrete operations, definition. Dyck, W. [1881-82] Mth. A. 20 (1882) 1- ; 22 (1883) 70-. modular equations. Gierster, J. Leip. Mth. Ps. B. 37 (1885) 291- ; Mth. A. 26 (1886) 309-. theory. Guccia, G. B. Palermo Cir. Mt. Ed. 1 (1887) 388-. and algebraic equations. Foglini, G. Em. N. Line. Mm. 3 (1888) 3-; 7 (1891) 175-. . Bolza, 0. Am. J. Mth. 13 (1891) 59-. . Soderberg, J. T. Ups. S. Sc. N. Acta 15 (1895) No, 2, 38 pp. , geometrical treatment. Paolis, E. de. Em. S. It. Mm. 7 (1890) No. 6, 164 pp. , investigations. Frobenius, G. Crelle J. Mth. 101 (1887) 273-. and regular solids. Schoute, P. H. N. Arch. Wisk. 20 (1893) 100-. of 360 collineations in plane. Fricke, E. D. Nf. Vh. (1896) (Th. 2, Halfte 1) 5. Groups, Burnside's theory. Miller, G. A. N. Y. Am. Mth. S. Bll. 5 (1899) 249-. which can be represented as multiply transi- tive substitution groups. Miller, G. A. Mess. Mth. 28 (1899) 104-. of class n - u, degree n, at least u - 1 times transitive. Maillet, E. Par. 6c. Pol. J. 5 (1900) 47-. degree p and order p(p + l)ir, p being prime, ir a divisor of p-1. Sylow, L. Christiania Skr. (Mth. Nt. Kl.) (1897) No. 9, 19pp. , especially substitutions permutable with given one. Capelli, A. G. Mt. 14 (1876) 66-. formed of a finite number of linear substitu- tions. Jordan, C. C. E. 83 (1876) 1035- ; 84 (1877) 1446-. of linear substitutions. Bianchi, L. Mth. A. 42 (1893) 30-. with coefficients belonging to imaginary quadratic corpora. Bianchi, L. Mth. A. 40 (1892) 332-. complex coefficients, some classes of. Bianchi, L. Mth. A. 43 (1893) 101-. integral complex coefficients with applications to theory of numbers. Bianchi, L. Em. E. Ac. Line. Ed. 6 (1890) (Sent. 1) 331- ; Mth. A. 38 (1891) 313-. , theory. Loewy, A. Mth. A. 53 (1900) 225-. , list of systems. Dickson, L. E. NY Am Mth. S. Bll. 5 (1899) 470-. of n dimensions. Frattini, G. Em. E. Ac. Line. T. 8 (1884) 260-. Groups of numbers in algebraic corpora. Weber, H. Mth. A. 48 (1897) 433-; 49 (1897) 83-; 50 (1898) 1-. whose order is a number without square factors. Holder, O. Gott. Nr. (1895) 211-. power of prime, determination. Young, J. W. A. Am. J. Mth. 15 (1893) 124-. , theory. Bauer, M. Mth. Termt. fits. 18 (1900) 133-. of order p m q i . Jordan, C. Liouv. J. Mth. 4 (1898) 21-. p^q 2 , p and q being primes. Le Va- vasseur, E. P. C. E. 128 (1899) 1152-. 16p, p being an odd prime. Le Va- vasseur, E. P. C. E. 129 (1899) 26-. P 3 q. Western, A. E. L. Mth. S. P. 30 (1899) 209-. whose orders are products of three prime factors. Cole, E. N., & Glover, J. W. Am. J. Mth. 15 (1893) 191-. of orders p 3 , pq z , pqr, p*. Holder, 0. Mth. A. 43 (1893) 301-. which r arbitrary points in space of n di- mensions can represent. Schlegel, V. Arch. Mth. Ps. 10 (1891) 283-. several times transitive, elementary proper- ties. Cesaro, E. G. Mt. 22 (1884) 47-. and substitutions, new basis for theory. Hoyer, P. Mth. A. 51 (1899) 445-. , theorem. Cayley, A. Mth. A. 13 (1878) 561-. , theorems. Kirkman, T. P. Manch. Ph. S. P. 2 (1860-62) 73-. , theory. Burnside, W. Mess. Mth. 23 (1894) 50-; 24 (1895) 191-. Holoedric transformation into itself of a given group. Moore, E. H. [1895] N. Y. Am. Mth. S. Bll. 1 (1895) 61-; 2 (1896) 33-. and transitive isomorphs of symmetric or alternant groups. Maillet, E. Liouv. J. Mth. 1 (1895) 5-. Holomorph of the cyclical group, and some subgroups. Miller, G. A. QJ. Mth. 31 (1900) 382-. Homogeneous substitution groups, theory. Molien, T. Dorpat Sb. 11 (1896) 259-. Hyperabelian groups, special class. Bourget, H. Toul. Fac. Sc. A. 12 (1898) D, 90 pp. Hyperelliptic curves of class 3, allowing auto- morphic transformation. Wiman, A. [1895] Stockh. Ak. Hndl. Bh. 21 (Afd. 1) (1896) No. 1, 23 pp. Hypergeometric functions with the branch points (2, 3, 7), (2, 4, 7), arithmetical theory. Fricke, E. Mth. A. 41 (1893) 443-. Hypoabelian group, first, generalised. Dick- son, L. E. QJ. Mth. 30 (1899) 1-. groups. Dickson, L. E. N. Y. Am. Mth. S. Bll. 4 (1898) 495-. Icosahedral group, construction. Bagnera, G. Palermo Cir. Mt. Ed. 11 (1897) 87-. substitutions, the sixty. Cayley, A. QJ. Mth. 27 (1895) 236-. Imprimitive and soluble groups. Soderberg, J. T. Ups. S. Sc. N. Acta 20 (1904) No. 1, 26pp. 1210 Discrete Groups of Finite and of Infinite Order 1210 Imprimitive substitution groups of degree ten. Miller, G. A. N. Y. Am. Mth. S. Bll. 1 (1895) 67-. Induced linear substitutions, groups. Rados, G. Mth. Termt. Ets. 17 (1899) 44-; Mth. Nt. B. Ung. 17 (1901) 227-. Intransitive groups, construction. Bolza, 0. Am. J. Mth. 11 (1889) 195-. linear substitution-groups. Maschke, H. Mth. A. 52 (1899) 363-. substitution groups of ten letters. Miller, G. A. QJ. Mth. 27 (1895) 99-. Invariance of factors of a substitution-group. Pierpont, J. Am. J. Mth. 18 (1896) 153-. Invariant expression for Poincare's substitu- tions. Brun, F. de. Stockh. Ofv. (1890) 265-; Fschr. Mth. (1890) 410. Invariants attached to Klein's group 168. Boulanger, A. C. E. 130 (1900) 107-. of a quadrangle by protective transformation. Lovett, E. 0. A. Mth. 12 (1898-99) 79-. relative to the Hessian group. Boulanger, . C. E. 122 (1896) 178-. of substitutions of a Kleinian group. Bagnera, G. Em. E. Ac. Line. Ed. 7 (1898) (Sem. 1) 340-. of ternary monomial groups. Skinner, E. B. [1900] N. Y. Am. Mth. S. Bll. 7 (1901) 244, 256. Isomorphism, double, in group theory. Frattini, G. Em. E. Ac. Line. Ed. 2 (1893) (Sem. 1) 253-. of a group with itself. Burnside, W. L. Mth. S. P. 27 (1896) 354-. of groups of substitutions. Capelli, A . G. Mt. 16 (1878) 32-. of simple linear groups. Dickson, L. E. N. Y. Am. Mth. S. Bll. 6 (1900) 323-. Isomorphisms, group of. Miller, G. A. N. Y. Am. Mth. S. Bll. 6 (1900) 337-. , groups of. Miller, G. A. C. E. 130 (1900) 316-. , with same groups of. Miller, G. A. N. Y. Am. Mth. S. T. 1 (1900) 395-. of (Gp)*, p a prime greater than 3, study of. Le Vavasseur, R. P. Par. EC. Norm. A. 16 (1899) 377-. a Hamiltonian group to itself, the simple. Miller, G. A. N. Y. Am. Mth. S. Bll. 5 (1899) 292-. substitution group to itself, the simple. Miller, G. A. Ph. Mg. 45 (1898) 234-. Kirkman's problem, application of theory of groups to. Burnside, W. Mess. Mth. 23 (1894) 137-. Kleinian groups. Poincarg, H. C. E. 93 (1881) 44-; Acta Mth. 3 (1883) 49-. with two variables. Pezzo, P. del. Nap. Ed. 32 (1893) 123-. Klein's group of ( + !)! 7i-ary collineations. Moore, E. H. Am. J. Mth. 22 (1900) 336-. Kummer surface, birational transformations of. Hutchinson, J. I. [1900] N. Y. Am. Mth. S. Bll. 7 (1901) 211-. Lagrange and theory of groups. Pierpont, J. N. Y. Am. Mth. S. Bll. 1 (1895) 196-. Limit of transitivity of non-alternate groups. Jordan, C. (x) Par. S. Mth. Bll. 1 (1873) 40-; Liouv. J. Mth. 1 (1895) 35-. . Miller, G. A. N. Y. Am. Mth. S. Bll. 4 (1898) 140-. substitution-groups. Bochert, A. Mth. A. 29 (1887) 27-; 33 (1889) 572-; 40 (1892) 176-. Linear differential equations with algebraic integrals. Jordan, C. [1877] Crelle J. Mth. 84 (1878) 89-. group of Borchardt's moduli. Maschke, H. Gott. Nr. (1887) 421- ; Mth. A. 30 (1887) 496. groups of finite order with three variables. Picard, E. Par. S. Mth. Bll. 15 (1887) 152-. including the Abelian group. Dickson, L. E. QJ. Mth. 31 (1900) 60-. isomorphic with the simple group of order 25920. Dickson, L. E. C. E. 128 (1899) 873-. as quotient groups. Dickson, L. E. N. Y. Am. Mth. S. T. 1 (1900) 30-. homogeneous group. Dickson, L. E. N. Y. Am. Mth. S. Bll. 5 (1899) 120-. groups. Taber, H. N. Y. Am. Mth. S. Bll. 2 (1896) 336-; 3 (1897) 121. defined by an invariant. Dickson, L. E. Mth. A. 52 (1899) 560-. in a Galois field, etc. Dickson, L. E. Am. J. Mth. 21 (1899) 193-. substitutions on mq variables, etc. Dickson, L. E. L. Mth. S. P. 30 (1899) 200-. substitution groups, coefficients. Maschke, H. Mth. A. 50 (1898) 492-. and quadratic forms of Dirichlet and Hermite. Bianchi, L. Em. E. Ac. Line. Ed. 7 (1891) (Sem. 2) 3-. substitutions applied to theory of groups. Laurent, H. N. A. Mth. 16 (1897) 149-. Magic squares, application to group theory. Maillet, E. Toul. Ac. Sc. Mm. 6 (1894) 258-. Metacyclic groups and neighbouring configura- tions. Heffter, L. Mth. A. 50 (1898) 261-. , and triple systems. Heffter, L. D. Mth. Vr. Jbr. 5 (1901) (Heft 1) 67-. Multiplication forms with application to theory of automorphic forms. Ritter, E. Mth. A. 44 (1894) 261-. Non-isomorphic simple groups. Dickson, L. E. N. Y. Am. Mth. S. Bll. 6 (1900) 268, 276-. Normal divisors of prime index of a finite group. Bagnera, G. Em. E. Ac. Line. Ed. 7 (1898) (San. 1) 63-. Number of values a function can take by permutation of its letters. Cauchy, A. L. Par. EC. Pol. J. 10 (1815) (17 e cah.) 1-; C. E. 21 (1845) 593-, 668-, 727-, 779-, 1356-. . Spitzer, S. [1848] Haidinger Ab. 3 (1850) (Ab. 2 43-. . Serret, J. A. Liouv. J. Mth. 15 (1850) 1-. . Mathieu, E. [1858-59] Liouv. J. Mth. 5 (1860) 9-. 69 1210 Discrete Groups of Finite and of Infinite Order Number of values a function can take by permutation of its letters. Jordan, C. Par. F,c. Pol. J. 22 (1861) (cah. 38) 113-. . Gosiewski, W. Par. T. Nauk S"c. Pam. 2 (*1872) 1-; (x) Bll. Sc. Mth. As. 6 (1874) 154-. . Bochert, A. Mth. A. 33 (1889) 584-; 40 (1892) 157-. . Carton, E. Par. S. Mth. Bll. 22 (1894) 230-. . Bochert, A. Mth. A. 49 (1897) 113-. of rational functions of given number of letters, etc. Spina, C. Em. At. N. Line. 21 (1868) 182-. variables in irreducible group. Molien, T. Dorpat Sb. 11 (1896) 277-. Octahedron, forms connected with the. Pittarelli, G. Em. E. Ac. Line. Ed. 4 (1888) (San. 1) 509-. Operation groups. Cesaro, E. Em. E. Ac. Line. Ed. 2 (1886) (Sem. 2) 35-. . Le Vavasseur, R. P. C. E. 122 (1896) 180-, 516-, 711-. . Miller, G. A. C. E. 128 (1899) 227-, 628. , generation. Frattini, G. Em. E. Ac. Line. Ed. 1 (1885) 281-, 455-; 2 (1886) (Sem. 1) 16-. of order Sp, p being any prime number. Miller, G. A. Ph. Mg. 42 (1896) 195-. p a , p being any prime number. Miller, G. A. Mess. Mth. 27 (1898) 119-. f order less than 64, and order 2p 3 , p being prime. Miller, G. A. QJ. Mth. 30 (1899) 243-. , theorems on. Miller, G. A. [1896] N. Y. Am. Mth. S. Bll. 3 (1897) 111-. symbols involving Galois' imaginaries. Le Vavasseur, E. P. Toul. Ac. Sc. Mm. 9 (1897) 247-. Operations, a group of. Brunei, G. Bordeaux S. Sc. PV. (1896-97) 30-. and substitutions, groups of. Maillet, E. As. Fr. C. E. (1897) (Pt. 2) 190-. Orthogonal group in a Galois field. Dickson, L. E. N. Y. Am. Mth. S. Bll. 4 (1898) 196-. , groups derived from. Dickson, L. E. [1898-99] N. Y. Am. Mth. S. Bll. 4 (1898) 382-, 560 ; Calif. Ac. P. (Mth. Ps.) 1 (1903) 29-, 47-. Perfect groups.- Miller, G. A. Am. J. Mth. 20 (1898) 277-. Permutable substitutions. Baraniecki, M. A. Par. T. Nauk Sc. Pam. 7 (*1875) Art. 7, 35pp. Permutants, note. Bilenki, H. N. A. Mth. 19 (1900) 213-. Permutation-groups of 8 things. Wiman, A. Gott. Nr. (1897) 55-. Permutations, relation between number of alternate and quasi-alternate. Andre, D. Par. S. Phlm. Bll. 8 (1896) 5-. , transformation and classification. Sprague, T. B. Edinb. Mth. S. P. 9 (1891) 59-. 1210 Position of inflection points in a plane cubic. Disteli, M. Ziir. Vjschr. 35 (1890) 145-. Possible types of plane quartics, multiply homological with themselves. Ciani, E. Palermo Cir. Mt. Ed. 13 (1899) 347-. Primitive groups. Jordan, C. Liouv. J. Mth. 16 (1871) 283-; (x) Par. S. Mth. Bll. 1 (1873) 175-. . Rudio, F. Crelle J. Mth. 102 (1888) 1-. of class n - 1 and degree n. Maillet, E. Par. S. Mth. Bll. 25 (1897) 16-. n-2 and degree n. Maillet, E. As. Fr. C. E. (1898) (Pt. 2) 157-. , classification. Jordan, C. C. E. 73 (1871) 853-. containing a given substitution, limit of degree. Jordan, C. C. E. 78 (1874) 1217- ; Crelle J. Mth. 79 (1875) 248-. of degree less than 17, enumeration. Miller, G. A. C. E. 124 (1897) 1505- ; 125 (1897) 673. for first 17 degrees, enumeration. Jordan, C. C. E. 75 (1872) 1754-. , holohedrically isomorphic with groups several times transitive. Maillet, E. C. E. 124 (1897) 351-; Liouv. J. Mth. 3 (1897) 277-. , mode of formation. Maillet, E. QJ. Mth. 27 (1895) 119-. substitution groups of degree 10, 15, and 16. Miller, G. A. L. Mth. S. P. 28 (1897) 533-; Am. J. Mth. 20 (1898) 229-; QJ. Mth. 31 (1900) 228-. Primitivity of groups. Dyck, W. C. E. 96 (1883) 1024-. and transitivity of discrete group. Dyck, W. [1881-82] Mth. A. 20 (1882) 1-; 22 (1883) 70-. Quartic of Klein, theorem concerning. Ciani, E. Mil. I. Lomb. Ed. 33 (1900) 565-. Quaternary group of 51840 linear substitutions. Morrice, G. G. [1889-92] L. Mth. S. P. 21 (1891) 58-; 23 (1892) 213-. substitutions with ternary Hessian for sub-group. Maschke, H. Gott. Nr. (1888) 78-; Mth. A. 33 (1889) 317-. quadratic forms, and polyhedral groups. Bianchi, L. A. Mt. 21 (1893) 237-; 23 (1895) 1-; Em. E. Ac. Line. Ed. 3 (1894) (Sem. 1) 3-. Quotient G/H in the theory of groups, note on so-called. Cayley, A, Am. J. Mth. 15 (1893) 387-. Eational functions, theory. Beke, E. Mth. A. 47 (1896) 441-. Eeduction of the Galois group of an equation by adjunction of arbitrary irrationalities. Bucca, F. Palermo Cir. Mt. Ed. 14 (1900) 122-. Eegular substitution groups of order less than 48. Miller, G. A. QJ. Mth. 28 (1896) 232-. Eepresentation, analytical, of substitutions. Polignac, C. de. Par. S. Mth. Bll. 9 (1881) 59-. , , for a prime number of elements. Grandi, A. G. Mt. 19 (1881) 238-. , , , theorem generalised. Grandi, A. Mil. I. Lomb. Ed. 16 (1883) 101-. 70 Discrete Groups of Finite and of Infinite Order 1210 Kesolvable groups. Frobenius, G. Berl. Ak. Sb. (1893) 337-; (1895) 1027-. Simple group of order 504. Burnside, W. Mth. A. 52 (1899) 174-. between 1092 and 2001, proof of non-existence. Ling, G. H., & Miller, G. A. Am. J. Mth. 22 (1900) 13-. 360 operations. Fricke, E. Gott. Nr. (1896) 199-. plane collineations. Gerbaldi, F. Palermo Cir. Mt. Rd. 12 (1898) 23-; 13 (1899) 161- ; 14 (1900) 66-. homographies. Wiman, A. Mth. A. 47 (1896) 531-. 504 operations. Fricke, R. Mth. A. 52 (1899) 321-. groups, doubly infinite system of. Moore, E. H. [1893] N. Y. Mth. S. Bll. 3 (1894) 73-. as far as order 660. Cole, E. N. Am. J. Mth. 15 (1893) 303-. of order less than 200. Holder, 0. Mth. A. 40 (1892) 55-. from order 201 to order 500. Cole, E. N. Am. J. Mth. 14 (1892) 378-. , several classes. Miller, G. A. [1899] L. Mth. S. P. 31 (1900) 148-. , some. Miller, G. A. Par. S. Mth. Bll. 28 (1900) 266-. , sub-groups of doubly infinite series. Wiman, A. Stockh. Ak. Hndl. Bh. 25 (1900) (Afd. 1) No. 2, 47 pp. , triply infinite system. Dickson, L. E. QJ. Mth. 29 (1898) 169- ; 30 (1899) 383-. linear groups of order 25920, etc. Dickson, L. E. [1899] L. Mth. S. P. 31 (1900) 30-. transitive groups, note on. Miller, G. A. [1899] N. Y. Am. Mth. S. Bll. 6 (1900) 103-. 600-cell hyper-polyhedron and its self-corre- sponding displacements. Oss, S. L. van. [1899] Amst. Ak. Vh. (Sect. 1) 7 (1901) No. 1, 18 pp. ; Nt. 61 (1899-1900) 304. Space, division by linear transformation groups. Dyck, W. Leip. Mth. Ps. B. 35 (1883) 61-. Special sub-groups of class one. Fricke, R. Mth. A. 30 (1887) 345-. Sub-groups of the Betti-Mathieu group. Dickson, L. E. Am. J. Mth. 22 (1900) 49-. llth class in the modular group. Rougier, J. Mars. Fac. Sc. A. 6 (1897) Fasc. 1, 112 pp. Substitution groups. Miller, G. A. C. R. 122 (1896) 370- ; 123 (1896) 91-, 204. . Maillet, E. Par. S. Mth. Bll. 24 (1896) 85-. , composition. Capelli, A. Rm. R. Ac. Line. Mm. 19 (1884) 262-. whose degree does not exceed eight. Miller, G. A. Am. J. Mth. 21 (1899) 287-. , general solution of problem of. Hoyer, P. Mth. A. 52 (1899) 550-. of given order. Maillet, E. Toul. Fac. Sc. A. 9 (1895) D, 22 pp. ; 10 (1896) A, 20 pp. with integral coefficients. Viterbi, A. G. Mt. 36 (1898) 346-. 1210 Substitution groups whose order = degree. Le Vavasseur, R. P. C. R. 120 (1895) 822-, 899-, 1206- ; 121 (1895) 238-. ---- is 4. Miller, G. A. Ph. Mg. 41 (1896) 431-. ----- the product of two primes. Miller, G. A. N. Y. Am. Mth. S. Bll. 2 (1896) 332-. producible from given number of elements, list of all. Miller, G. A. N. Y. Am. Mth. S. Bll. 2 (1896) 138-. -- , theorems on. Sylow, L. Mth. A. 5 (1872) 584-. -- , theory of. Netto, E. Crelle J. Mth. 103 (1888) 321-. -- , , two laws in. Hoyer, B. Mth. A. 46 (1895) 539-. -- for 2, 3, 4, 5, 6, 7 and 8 letters. Cayley, A. QJ. Mth. 25 (1891) 71-, 137-. , 3, 4, 5, 6 and 7 letters. Askicith, E. H. QJ. Mth. 24 (1890) 111-. -- of 6, 7 and 8 letters. Cole, F. N. N. Y. Mth. S. Bll. 2 (1893) 184- ; 3 (1894) 169. -- 8 letters. Askwith, E. H. QJ. Mth. 24 (1890) 263-. ----- . Miller, G. A. N. Y. Mth. S. Bll. 3 (1894) 168-. ---- and 9 letters. Miller, G. A. N. Y. Mth. S. Bll. 3 (1894) 242-. --- 9 letters. Askwith, (Rev.) E. H. QJ. Mth. 26 (1893) 79-. ----- , list. Cole, F. N. QJ. Mth. 26 (1893) 372-. Substitutions of form n being prime. Brioschi, F. Gott. Nr. (1869) 491-. n letters which leave k letters un- altered, number of. Durfee, W. P. J. H. Un. Cir. [4] (1885) 23. , particularly for n = 3, 4, 5, 6, in relation to groups of Pascal's hexagram, geometrical interpretation. Veronese, G. A. Mt. 11 (1882-83) 93-. on p n letters where p is prime. Dickson, L. E. A. Mth. 11 (1896-97) 65-, 161-. represented by transpositions. Caldarera, G. Catania Ac. Gioen. At. 9 (1896) Mm. 7, 16pp. , theorem in. Cayley, A. C.R. 67 (1868) 784-. , . Frattini, G. [1884] Rm. R. Ac. Line. Mm. 18 (1883) 487-. , . Giwlice, F. Palermo Cir. Mt. Rd. 1 (1887) 222-. , theorems, general. Jordan, C. C. R. 66 (1868) 836-. , theory. Betti, E. Tortolini A. 6 (1855) 5-. , . Bachmann, P. Liouv. J. Mth. 10 (1865) 209-. , . Janni, G. G. Mt. 9 (1871) 280- ; 10 (1872) 193- ; 11 (1873) 1-, 71-, 257-. _, _. Jordan, C. G. Mt. 10 (1872) 116; Liouv. J. Mth. 17 (1872) 351- ; C. R. 76 (1873) 952-. , , and applications. Netto, E. Arch. Mth. Ps. 62 (1878) 225-. 71 1210 Discrete Groups Substitutions, theory, fundamental theorem, new proof. Netto,E. Mth. A. 13 (1878) 249-. , (Netto). Miller, G. A. A. Mth. 1 (1900) 71-. , , primitive groups of, etc. Jordan, C. C. E. 79 (1874) 1149-. Surfaces admitting an infinite discontinuous group of transformations. Painleve, P. C.B. 126 (1898) 512-. Sylow's theorem. Frobenius, G. Crelle J. Mth. 100 (1887) 179-. , application to symmetric group. Eadcig, A. A. Kharkov Mth. S. Com. 5 [1896?] 1-. , extension. Miller, G. A. N. Y. Am. Mth. S. Bll. 4 (1898) 323-. , generalisation. Frobenius, G. Berl. Ak. Sb. (1895) 981-. theorems on groups, illustration. Cayley, A. Mess. Mth. 23 (1894) 59-. Symmetric and alternant homographic groups. Maschke, H. Mth. A. 51 (1899) 253-. . Wiman, A. Mth. A. 52 (1899) 243-. permutation-groups of n things. Wiman, A. Gott. Nr. (1897) 191-. group, note. Burnside, W. [1896] L. Mth. S. P. 28 (1897) 119-. System of linear forms, theorem on. Frattini, G. Bm. B. Ac. Line. Bd. 2 (1886) (Sem. 1) 132-. Systems of groups, fundamental theorem relative to two. Cauchy, A. L. C. B. 22 (1846) 630-. Tactic disclosing existence of 4-valued function. Sylvester, J. J. Ph. Mg. 21 (1861) 515-. Ternary substitution-groups of finite order, which leave a triangle unchanged. Maschke, H. Am. J. Mth. 17 (1895) 168-. Theta-fuchsian functions. Ovidio, E. d\ Tor. Ac. Sc. At. 29 (1894) 741-. Transitive functions, reduction to intransitive. Cauchy, A. L. C. E. 21 (1845) 1199-. group, G120 of six letters, invariant functions. Dziobek, 0. Arch. Mth. Ps. 68 (1882) 225-. Transitive groups. Maillet, E. A. Mt. 25 (1897) 219-. of degree n and class n- 1. Maillet, E. Par. S. Mth. Bll. 26 (1898) 249-. whose degree is the square of a prime. Sylmv, L. Acta Mth. 11 (1887-88) 201-. isomorphic to alternating group of six. Miller, G. A. Am. Ph. S. P. 36 (1897) 208-. , proofs and theorems. Netto, E. Crelle J. Mth. 83 (1877) 43-. substitution groups. Bochert, A. Mth. A. 40 (1892) 176- ; 49 (1897) 133-. . Miller, G. A. Mess. Mth. 28 (1899) 84-. of degree 12. Miller, G. A. N. Y. Am. Mth. S. Bll. 1 (1895) 255-. , list. Miller, G. A. QJ. Mth. 28 (1896) 193-. 13 and 14. Miller, G. A. QJ. Mth. 29 (1898) 224-. 17. Miller, G. A. QJ. Mth. 31 (1900) 49-. isomorphic to a given group. Miller, G.A. G. Mt. 38 (1900) 63-. Continuous Groups 1230 Transitive substitution groups of nine letters. Cole, F. N. N. Y. Mth. S. Bll. 2 (1893) 250-. 10 and of 11 letters, list. Cole, F. N. QJ. Mth. 27 (1895) 39-. 24 letters, etc. Wilkinson, A.C.L. QJ. Mth. 30 (1899) 157-. whose order is the cube of any prime: the non-regular. Miller, G. A. A. Mth. 10 (1895-96) 156-. product of three primes: the non-regular. Miller, G.A. Zttr. Vjschr. 42 (1897) 68-. of order 8p, p being any prime. Miller, G. A. Ph. Mg. 43 (1897) 117-. whose orders are products of three primes. Miller, G. A. N. Y. Am. Mth. S. Bll. 3 (1897) 213-. of same order and degree. Frattini, G. Em. E. Ac. Line. At. 14 (1883) 143-. Triple systems. Moore, E. H. Mth. A. 43 (1893) 271- ; Palermo Cir Mt. Ed. 9 (1895) 86. . Heffter, L. Mth. A. 49 (1897) 101-. . Moore, E. H. N. Y. Am. Mth. S. Bll. 4 (1898) 11-. , theory. Netto, E. Mth. A. 42 (1893) 143. , . Vries, J. de. Palermo Cir. Mt. Ed. 8 (1894) 222-. , transitive. Moore, E. H. Mth. A. 50 (1898) 225-. tangent planes of cubic surface, proof of Cayley's theorem. Kohn, G. Mh. Mth. Ps. 2 (1891) 343-. Two groups in five dimension space. Engel, F. D. Mth. Vr. Jbr. 8 (1900) (Heft 1) 196-. Uniform group and linear transformation. Kneser, A. Mth. A. 28 (1887) 125-. Universal invariant for finite groups of linear substitutions. Moore, E. H. Mth. A. 50 (1898) 213-. 1230 Continuous groups of finite and of infinite order. Algebraic groups, certain. Cartan, E. C. B. 120 (1895) 544-. of straight lines and planes. Carda, K. Mh. Mth. Ps. 11 (1900) 31-. surfaces admitting continuous transitive group of transformations. Fano, G. Palermo Cir. Mt. Bd. 10 (1896) 1-. with continuous group of birational trans- formations into them selves. Castelnuovo, G., <& Enriques, F. C. E. 121 (1895) 242-. . Painleve, P. C. E. 121 (1895) 318-. infinite automorphic transformations. Enriques, F. Yen. I. At. (1892-93) 1590- ; (1893-94) 638-. . Fano, G. Em. E. Ac. Line. Ed. 4 (1895) (Sem. 1) 149- ; Palermo Cir. Mt. Bd. 11 (1897) 240-. Canonical form, reduction of finite and con- tinuous group of transformations to. Cartan, E. Am. J. Mth. 18 (1896) 1-. 72 1230 Continuous Groups of Finite and of Infinite Order 1230 Canonical form, reduction of group of trans- formations to. Lie, S. Leip. Mth. Ps. B. 41 (1889) 277-. Complex numbers and groups of transforma- tions. Study, E. Leip. Mth. Ps. B. 41 (1889) 177. , systems of, and transformation groups. Study, E. Mh. Mth. Ps. 1 (1890) 283-. Conformal representation; Poincare's theory; substitution groups. Brun, F. de. Stockh. Ofv. (1890) 135- ; Fschr. Mth. (1890) 410. Contact transformations, finite groups contained in the group of linear. Autonne, L. C. E. 102 (1886) 313-. , irreducible groups. Lie, S. Leip. Mth. Ps. B. 41 (1889) 320-. , theory. Lie, M. S. Arch. Mth. Ntvd. 1 (1876) 194-. Continuity of infinitesimal transformation groups. Slocum, S. E. Am. Ac. P. 36 (1901) 83-. Continuous binary linearoid groups, and corre- rnding differential equations. Wilczynski, J. Am. J. Mth. 22 (1900) 191-. and discontinuous groups, correspondence between. Bettazzi, R. A. Mt. 16 (1888-89) 49-. group of Darboux rotations. Souslow, G. D. Nf. Vh. (1895) (Th. 2, Hfilfte 1) 12-. defined by any given group of finite order. Burnside, W. L. Mth. S. P. 29 (1898) 207-, 546-. of Jc lonquieres' transformations in a plane. Enriques, F. Em. E. Ac. Line. Ed. 2 (1893) (Sem. 1) 532-. in three dimensions. Bianchi, L. Em. S. It. Mm. 11 (1898) 267-. of transformations with 28 parameters in theory of deformation of surfaces. Staeckel, P. C. E, 121 (1895) 396-. groups. Lie, S. Christiania F. (1884) No. 8, 4 pp., No. 9, 4 pp. ; (1885) No. 15, 4 pp. . Poincare, H. C. E. 128 (1899) 1065- ; Camb. Ph. S. T. 18 (1900) 220-. , application to singular points of differen- tial equations. Marotte, F. C. E. 123 (1896) 867-. , theory of functions. Lie, S. C. E. 114 (1892) 334-. , . Painleve, P. C. E. 118 (1894) 845-. and differential equations with fixed critical points. Painleve, P. C. E. 130 (1900) 1171-. , Lie's, application. Autonne, L. C. E. 112 (1891) 570-. , theory. Lovett, E. 0. [1897] N. Y. Am. Mth. S. Bll. 4 (1898) 59-. , . Slocum, S. E. Am. Ac. P. 35 (1900) 237-, 481-. , . Eettger, E. W. Am. J. Mth. 22 (1900) 60-. , , application. Drach, . C. E. 116 (1893) 1041-. , problems in theory. Bouton, C. L. A. Mth. 1 (1900) 93-. Continuous groups of projective transforma- tions. Newton, H. B. [1895-96] Kan. Un. Q. 4 (1896) 71-, 169-, 243-; 5 (1896) 81. quadratic transformations of the plane. Bohlmann, G. Gott. Nr. (1896) in space of n dimensions, determination. Medolaghi, P. Em. E. Ac. Line. Ed. 8 (1899) (Sem. 1) 291-. , theorem in. Koch, H. von. Stockh. Ofv. (1894) 311-. , theory. Campbell, J. E. N. Y. Am. Mth. S. Bll. 4 (1898) 407-. imprimitive groups of point transformations. Fano, G. Em. E. Ac. Line. Ed. 7 (1898) (Sem. 1) 302- . Cremona cubic group, finite groups. Autonne, L. C. E. 101 (1885) 53-; 105 (1887) 267-. group with any number of variables, com- plete groups contained in a. Del Re, A. Em. E. Ac. Line. Ed. 6 (1890) (Sem. 2) 271-. , finite groups in. I. Quadratic groups. II. Cubic groups. Autonne, L. Liouv. J. Mth. 1 (1885) 431- ; 2 (1886) 49-. quadratic group, finite groups in. Autonne, L. C. E. 98 (1884) 565-; Liouv. J. Mth. 4 (1888) 177-, 407-. , irreducible groups of finite order in. Autonne, L. C. E. 103 (1886) 1176-. groups. Autonne, L. C. E. 104 (1887) 1422-. transformations, continuous groups of. Noether, M. D. Nf. Vh. (1896) (Th. 2, Halfte 1) 11-. , . Fano, G. Mh. Mth. Ps. 9 in plane, continuous groups of. Enriques, F. Em. E. Ac. Line. Ed. 2 (1893) (Sem. 1) 468-. , , and certain projective transformation groups. Fano, G. Palermo Cir. Mt. Ed. 10 (1896) 16-. of space, continuous primitive groups of. Fano, G. Tor. Ac. So. At. 33 (1897) 284- or 480-. Differential equations admitting finite groups. Lie, S. Mth. A. 25 (1885) 71-. a group of transformations. Lie, S. Arch. Mth. Ntvd. 9 (1884) 431-. infinitesimal transformations. Paczmvski, J. Prace Mt.-Fiz. 7 (1896) 178-. , algebraic and group theory. Loewy, A. Mh. Mth. Ps. 8 (1897) 225-. which possess fundamental integrals. Lie, S. Leip. Mth. Ps. B. 45 (1893) 341-. invariants of equation of 2nd order. Tresse, A. Leip. Jablon. Preisschr. 32 (1896) 87 pp. Dynamical problems, whose solutions admit of infinitesimal transformation. Stackel, P. Leip. Mth. Ps. B. 45 (1893) 331-. Dynamics, differential equations of, application of Lie's transformation guoups. Stackel, P. Leip. Mth. Ps. B. 49 (1897) 411-. 73 1230 Continuous Groups of Finite and of Infinite Order 1230 Finite continuous group of transformations, investigation of the finite equations of a. Vessiot, E. Toul. Fac. Sc. A. 10 (1896) C, 26pp. groups. Autonne, L. Liouv. J. Mth. 3 (1887) 63-. , complete systems and differential invariants of. Lindelof, E. Helsingf. Acta 20 (1895) No. 1, 62 pp. , determination of equations. Vessiot, E. C. R. 120 (1895) 77-. of Lie's transformations, decom- position. Maillet, E. C. E. 130 (1900) 1536^. the plane. Williams, F. B. Am. Ac. P. 35 (1900) 95-. , structure. Cartan, . C. E. 116 (1893) 962-. of transformations. Engel, F. Leip. Mth. Ps. B. 38 (1886) 83-. primitive groups of Lie's transformations. Maillet, E. C. E. 130 (1900) 1602-. transformations of a projective group, gene- rated by the infinitesimal transformations of the group. Engel, F. Leip. Mth. Ps. B. 44 (1892) 279-; 45 (1893) 659-. General projective group, case of degeneration. Engel, F. C. E. 118 (1894) 397-. Geometry of functional space. Pincherle, S. Bologna Ed. 1 (1897) 85-. Greatest continuous groups of certain systems of differential equations, finite character. Engel, F. Leip. Mth. Ps. B. 46 (1894) 297-. Group and invariant, the conceptions. Lie, S. Leip. Mth. Ps. B. 39 (1887) 83-. of parameters, canonical form. Schur, F. Leip. Mth. Ps. B. 41 (1889) 229-. theory applied to hydrodynamics. Wilczyn- ski, E. J. N. Y. Am. Mth. S. T. 1 (1900) 339-, 509. for equations of the 2nd degree with n variables. Werner, H. Mth. A. 35 (1890) 113-. , geometrical treatment. Paolis, R. de. Em. S. It. Mm. 7 (1890) No. 6, 164 pp. , minor contributions. Engel, F. Leip. Mth. Ps. B. 39 (1887) 89-; 43 (1891) 47-, 308-, 585-; 44 (1892) 43-, 292-; 45 (1893) 360- ; 46(1894)25-. , recent investigations. Lie, S. Leip. Mth. Ps. B. 44 (1892) 297-. Groups and algebraic surfaces. Picard, E. Palermo Cir. Mt. Ed. 9 (1895) 244-. appearing in the theory of analytical func- tions. Medolaghi, P. C. E. 126 (1898) 1188-. of circular transformations. Newson, H. B. [1897] N. Y. Am. Mth. S. Bll. 4 (1898) isomorphous to the group of all the trans- formations of one variable. Medolaghi, P. Palermo Cir. Mt. Ed. 12 (1898) 188-. Hyperfuchsian groups. Poincare, H. C. E. 98 (1884) 503-. Infinite continuous groups. Lie, M. S. Chris- tiania F. (1883) No. 12, 56 pp. ; Leip. Mth. Ps. Ab. 21 (1895) 43-. Infinite continuous groups, theory. Medolaghi, P. A. 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Mm. 13 (1886) 33-, 399. , connected with (a + b) n - a* - b n . Sadun, E. Ev. Mt. 4 (1894) 189- ; 5 (1895) 19-. developed from- = - + ^^.-. Tait,P.G. p q q p [1877] Edinb. E. S. P. 9 (1878) 416-. Identity, algebraic, on differences of 4 letters. Glaisher, J. W. L. [1878] Mess. Mth. 8 (1879) 45-. , , reciprocals of cyclic differences. Saint-Germain, de. Caen Ac. Mm. (1892) (Pt. 1) 15-. of polynomials, principle. Roy, A. Mathe- matician 3 (1850) 1-. -- Waring's, an. Virieu, J. de. N. A. Mth. 1 (1862) 45-. , Waring's analytical proof. Serret, P. N. A. Mth. 7 (1848) 199-. . Ifa + 6 + c=0, (b-c c-a a-b\ - + r- + - I a b c J b Glaisher, J. W. L. [1880] Mess. Mth. 10 (1881) 54-. Irreducible factors of x n -l. Kronecker, L. Liouv. J. Mth. 19 (1854) 177-. 78 1610 Rational polynomials Multiplication and division, algebraic. Salim- beni, L. Verona Mm. S. It. 7 (1794) 482-. Polynomial, complete, number of terms. Ger- gonne, J. I). Gergonne A. Mth. 13 (1822-23) 282-. , condition for factor of form x n - a n . Zeipel, E. V. von. Ups. N. Acta S. Sc. 4 (1863) 24pp. , . Bjarling, C. F. E. Arch. Mth. Ps. 55 (1873) 429-. , determination by integrals of its partial differentials. Mossbrugger, L. GrunertArch. 4 (1844) 210-. , development, case. Weihrauch, K. Z. Mth. Ps. 26 (1881) 127-. , without binomial theorem. Brassinne, E. Toul. Mm. Ac. 1 (1851) 184-. 4th degree, values of x for which it is a square. Calzolari, L. G. Mt. 7 (1869) 317-. , linear factorisation. Kiipper, K. Casopis 14 (1885) 28-; Fschr. Mth. (1886) 60. , nth power. Studnicka, F. J. Prag Sb. (1879) 375-; Casopis 9 (*1880) 49-. (x 2 - 1)", property. Polignac, C. de. [1874] Par. S. Mth. Bll. 3 (1875) 19-. Polynomials, class. Appell, P. Par. EC. Norm. A. 9 (1880) 119-. , factorisable in linear factors. Moutard, . C. E. 119 (1894) 42-. , division and extraction of roots. Gergonne, J. D. Gergonne A. Mth. 4 (1813-14) 297-. , equal factors of . Ostrogradsky, M. A. C.E. 42 (1856) 930-. , geometrical application of theorem. Laisant, C. A. As. Fr. C. E. (1898) (Pt. 2) 73-. , powers. Brianchon, C. J. Par. EC. Pol. J. 15 (1837) (25 cah.) 158-. , . Casinelli, A. (s. L.) Bologna N. Cm. 4 (1840) 453-. , . Lewis, W. J. Franklin I. J. 3 (1842) 44-. , , divisibility. Crelle, A. L. Berl. Ab. (1839) 1-, 299-. , product of two, (Gauss). Terquem, O. N. A. Mth. 3 (1844) 47-. , properties. De Forest, E. L. Des Moines Anal. 7 (1880) 39-, 73-, 105-. , , method of demonstrating. De Forest, E. L. Des Moines Anal. 10 (1883) 97-. , rational factors. Gergonne, J. D. Gergonne A. Mth. 12 (1821-22) 309-. , , geometric construction. Winckler, A. Wien Sb. 53 (1866) (Ab. 2) 326-. , roots. Sebuev, G. N. Kazan S. Nt. (Ps.- Mth.} P. 7 (1889) 394-. of 3rd degree in one variable. Nordlund, K. P. Ts. Mt. Fys. 4 (1871) 185-. Power of polynomial, expression by symmetric functions. Piarron de MondSsir, E. S. As. Fr. C. E. (1877) 154-. Powers of (a^ + a<& 2 + ) n , theorem. Tralles, J. G. Berl. Ab. (1816-17) 56-. Principle of greatest and least exponents. Ap- plications in theory of algebraic functions. Bugaev, N. V. [1889] Eec. Mth. (Moscou) 14 (1890) 553-. Algebraic inequalities 1615 Quotient and product functions, certain. Guld- berg, A. S. Christiania F. (1884) No. 4, 18pp. remainder in (x - l)P/x m . Catalan, E. C. Liege S. Sc. Mm. 15 (1888) No. 1, 142-. quadrics to sums of squares. Laurent, H. N. A. Mth. 19 (1880) 12-; 20 (1881) 38-. Series. ~2A n (n)x n . Grunert, J. A. Crelle J. 25 (1843) 240- . Square root extraction, of 6-nomials, 5-nomials, etc. Eeade, J. Thomson A. Ph. 11 (1818) 31-. Sum of a series connected with transvectants. Gordan, P. Palermo Cir. Mt. Ed. 12 (1898) 326-. Sums of like powers of (linear) expressions. Bierens de Haan, D. N. Arch. Wisk. 5 (*1879) 208-. powers of first n numbers, geometrical illustrations. Bochow, K. Magdeb. Nt. Vr. Jbr. u. Ab. (1896-98) 165-. Ternary forms, conditions for linear factors. Brill, A. [1896-98] Mth. A. 50 (1898) 157- ; D. Mth. Vr. Jbr. 5 (1901) (Heft 1) 52-. Theorems of Bezout and Euler. Poussart, A. Ens. Mth. 2 (1900) 136-. Transformation of expression (la - cy)* + (ex - az) 2 + (ay - bx)*. Smith, Arch. Camb. Mth. J. 1 (1839) 155-. (cy - bz) 2 + (az - ex) 2 + (bx - ay) 2 . N., M. N. Camb. Mth. J. 2 (1841) 9-. E. G. Grunert Arch. 27 (1856) 482-. x n + y n expressed in terms of x + y and xy. Valeriani, V. G. Mt. 12 (1874) 208-. 1615 Algebraic inequalities. Inequalities, applications. Brashmann, N. D. (xn) Mosc. Un. Mm. 10 (1835) 381-. , calculus. Fourier, J. B. J. Par. S. Phlm. N. Bll. (1826) 99-. , . Gergonne, J. D. Gergonne A. Mth. 17 (1826-27) 134-. , extension of theorem in. Rogers, L. J. Mess. Mth. 17 (1888) 145-. , generalisation of formula of Tschebyscheff (Cebysev). Jensen, . Bll. Sc. Mth. 12 (1888) 134-. , means, and imaginaries. Grunert, J. A. Grunert Arch. 1 (1841) 268-. , note on some, and their application to in- tegral calculus. La Vallee-Poussin, C. de. Brux. S. Sc. A. 16 (1892) (Pt. 1) 8-. , notes. Face, F. C. Mess. Mth. 18 (1889) 90-. , number of conditions expressed by. Sludskil, T. A. (xn) [1873] Eec. Mth. (Moscou) 7 (1874-75) (Pt. 1) 419-. , problem. (x m + y m > or < z m when x y + y 2 = z' 2 ). Grunert, J. A. Grunert Arch. 20 (1853) 356-. 79 1620 Permutations, combinations, etc. Binomial coefficients 1620 Inequalities, theorem. Glaisher, J. W. L. Mess. Mth. 8 (1879) 140-. of Tschebyscheff's (Cebysev), proof. Markov, A. A. (xn) Kharkov Mth. S. Com. (1883) 105- ; Fschr. Mth. (1885) 168, 171. {(n-l)Sa 2 >2Sa&}. Grunert, J. A. Grunert Arch. 26 (1856) 105-. ma m ~ l >^ -j-->mb m - 1 , proof and applica- tions. Fort, 0. Schlomilch Z. 7 (1862) 46-. Inequality concerning sums of products of numbers. Schiappa Monteiro, A. [1900] Lisb. J. Sc. Mth. 6 (1902) 12-. , conditions, calculus. Navier, C. L. M. H. Par. S. Phlm. Bll. (1825) 66-, 81-. , of, Fourier. Cournot, A. A. (\i Adds.) Fe"russac Bll. Sc. Mth. 6 (1826) 1-. _ a2nf2 " 1 > !Ltl . Walton, W. Mess. Mth. a (a 2n - 1) n 8 (1879) 130-. ^1^ lies between^ a^and?^. Walker, 3fl-yl q q y G. F. [1882] Mess. Mth. 12 (1883) 37-. mx m ~ l (x - 1) ^ x m - 1 > m (x - 1) . Chrystal, G. Edinb. Mth. S. P. 6 (1888) 29-. , a m b" + b m c n + c m a n > a n b m + b n c m + c n a m . Segar, H. W. Mess. Mth. 22 (1893) 47-. Mean value. Holder, O. GStt. Nr. (1889) 38-. 1620 Permutations, combinations, partitions, distributions. Bi- nomial and multinomial co- efficients. Algorithms of arithmetical functions. Cesaro, E. G. Mt. 23 (1885) 175-. Arithmetical cube. Laisant, C. A. As. Fr. C. B. (1891) (Ft. 2) 8-. tetrahedron. Laisant, C. A. Par. S. Mth. Bll. 19 (1891) 18-. triangle, history. Fontes, . As. Fr. C. B. (1893) (Ft. 2) 236-. , properties. Laisant, C. A. As. Fr. C. E. (1891) (Ft. 2) 1-; (1893) (Ft. 2) 206-. Arrangement of (1) cards, (2) counters : two problems. Crelle, A. L. Crelle J. 44 (1852) 15 counters in order, generalisation. Malmsten, C.^J. Goteb. Hndl. 17 (1882) 77-. game (Josephspiel). Curtze, M. Bb. Mth. (1894) 116 ; (1895) 34-. (Josephus's problem), generalisation. Tait, P. G. [1898] Edinb. B. S. P. 22 (1900) 165-. and probabilities, question. Tait, P. G. Edinb. B. S. P. 8 (1873) 37-. in rows. Brunei, G. Bordeaux S. Sc. PV. (1894-95) 28-. Arrangements, compound, question in. Sylvester, J. J. B. S. P. 12 (1862-63) 561-. , generalisation of theorem of Cauchy. Sylvester, J. J. Ph. Mg. 22 (1861) 378-. Arrangements, problem. Bouniakmcsky, V. [1857] St Pet. Ac. Sc. Bll. 16 (1858) 67-. , . Cayley, A. [1877] Edinb. B. S. P. 9 (1878) 338-. , . Tait, P. G. Edinb. B. S. P. 10 (1880) 400-. , Tait's problem. Muir, T. [1877] Edinb. B. S. P. 9 (1878) 382-. , . Cayley, A. [1877] Edinb. B. S. P. 9 (1878) 388-. , . Muir, T. [1881] Edinb. B. S. P. 11 (1882) 187-. of m things of one sort and n things of another sort, under conditions of priority. Whitworth, W. A. [1878] Mess Mth. 8 (1879) 105-. Binomial' 'calculus." Walsh, J. TillochPh. Mg. 63 (1824) 443-. BINOMIAL COEFFICIENTS. Euler, L. [1778] St Pet. Ac. Sc. N. Acta 15 (1806) 33-. Stockier, F. de B. G. Lisb. Mm. Ac. Sc. 2 (1799) 480-. Pessuti, G. Mod. S. It. Mm. 11 (1804) 446-. Mazio, N. G. Arcad. 22 (1824) 27-. Arndt, F. Grunert Arch. 3 (1843) 256-. Horta, F. Lisb. A. 2 (1858) 98-. Le Tauniac, A. N. A. Mth. 20 (1861) 147-. Lucas, E. N. A. Mth. 9 (1870) 308-. Schlegel, S. F. V. Z. Mth. Ps. 23 (1878) 263-. analytic faculties. Trudi, N. [1873] Nap. Ac. At. 6 (1875) No. 3, 16 pp. Euler's theorem. Wolfers, J. P. Arch. Mth. Ps. 45 (1866) 411-. formulae. Tardy, P. G. Mt. 3 (1865) 1-. . Delannoy, H. As. Fr. C. B. (1890) (Ft. 2) 35-. a function of, and a series. Siacci, F. G. Mt. 10 (1872) 349-. and their interpolation. Euler, L. St Pet. Ac. Sc. Mm. 9 (1824) 57-. note. Busche, E. [1899] Hamb. Mth. Gs. Mt. 3 (1900) 379-. properties. Schlomilch, 0. Grunert Arch. 1 (1841) 431- ; 2 (1842) 434-. . Tirelli, F. G. Mt. 14 (1876) 318-. , and application to combinations. Baur, C. W. Z. Mth. Ps. 32 (1887) 218-. derived from complex numbers. Studnicka, F. J. Prag Sb. (1877) 92-. property. Zdrahal, A. Casopis 11 (*1882) 47-; Fschr. Mth. (*1882) 190. . Ekama, H. N. Arch. Wisk. 19 (1892) 105- ; Fschr. Mth. (1892) 156. in relation to gamma function and figurate numbers. Matthiessen, . D. Nf. Tbl. (*1868) 128- . sum of series involving. Bonolis, A. G. Mt. 11 (1873) 233-. n (n-l)(n-2) , . ,, 1 - + V gg ~ etc> derlved from expansion for cos nix. Schlomilch, 0. Grunert Arch. 7 (1846) 46-. 80 1620 Binomial Coefficients sum of squares. Szily, K. Mth. Termt. Ets. 11 (1893) 330- ; Mth. Nt. B. Ung. 12 (1895) 84-. by Taylor's series. Crelle, A. L. Berl. Ab. (1843) 1-. theorem. Graves, C. [1865] Ir. Ac. P. 9 (1867) 297-. . Hermes, J. Arch. Mth. Ps. 8 (1890) 269-. theorems. Catalan, E. C. Liege S. Sc. Mm. 15 (1888) No. 1, 41-, 133-, 250-. transformation of integral powers into. Trudi, N. [1873] Nap. Ac. At. 6 (1875) A T o. 4, 13 pp. Binomial theorem, alternative proof. Schuster, A. Metz Ac. Mm. 72 (1893) 163-. , case. Catalan, E. Brux. Ac. Bll. 16 (1888) 194. , new form. Mendizdbal Tamborrel, J. de. Mex. S. "Alzate" Mm. 4 (1890) 117- ; Fschr. Mth. (1890) 256. before Newton. Marre, A. N. A. Mth. 5 (1846) 488-. for positive integral index. Bouvier, L. C. Gergonne A. Mth. 14 (1823-24) 205-. . Garcet, . N. A. Mth. 19 (1860) 32-. as case of a general theorem. Grunert, J. A. Grunert Arch. 1 (1841) 67-. , proofs. Woodcock, T. B. A. Ep. (1889) 501-. , property. Andre, H. d\ N. A. Mth. 5 (1846) 121-. Binomials. Hello, M. P. de. Lisb. Ac. Sc. Mm. (Cor.) 4 (Pte. 1) (1815) 41-. Card dealing problem, Gergonne's. Dickson, L. E. N. Y. Am. Mth. S. Bll. 1 (1895) 184-. game, question. Le Cointe, J. L. A. (vni) A. Mt. 5 (1863) 108-. games of patience. Badoureau, A. [1881] Bv. Sc. 2 (1882) 475-. , problem. Vachette, A. N. A. Mth. 5 (1846) 394-. , (i.e. playing cards) and remark on theory of substitutions. Petersen, J. P. C. (xn) Ts. Mth. 1 (1871) 74-. trick. Gergonne, J. D. Gergonne A. Mth. 4 (1813-14) 276-. puzzle and a law of numbers. Steen, A. (xn) Ts. Mth. 1 (1865) 49-; 2 (1866) 1-; 3 (1867) 1-; 6 (1870) 1-, 181- ; 3 (1873) 156-. . Hertzsprung, S. (xn) Ts. Mth. 3 (1867) 97-. . Lorenz, L. V. (xn) Ts. Mth. 1 (1871) 53-. Cards. Theory of whist. Badoureau, . Ev. Sc. 36 (1885) 587-. Chess. Bishop's move. Cherriman, J. B. (xn) Cn. E. S. P. & T. 1 (1883) (Sect. 3) 19. Chessboard geometry, and indeterminate equa- tion ax + by c. Laquiere, E. M. Par. S. Mth. Bll. 7 (1879) 85-. , theorem on polygons having vertices at VOL. I. Chess 1620 centres of squares. Laisant, C. A. Par. S. Mth. Bll. 6 (1878) 156-. Chessboard geometry, theorem on triangle having vertices at centres of squares. Lucas, E. [1877] Par. S. Mth. Bll. 6 (1878) 9-. problem. Lavernede, T. Gard Mm. Ac. (1838-39) 151-. . "Entre chiens et loup." Lucas, E. Ev. Sc. 5 (1883) 555-. problems. Lucas, E. Ev. Sc. 17 (1879) 154- ; 18 (1880) 948-; 19 (1880) 36-, 375-; 1 (1881) 408-, 783- ; 2 (1882) 365-. . Perott, J. Par. S. Mth. Bll. 11 (1883) 173-. End games. Cunningham, (Lt.-Col.) A. B. A. Ep. (1894) 564. Knight's move. Brewster, (Sir) D. Edinb. Ph. J. 9 (1823) 236-. . Bouvard, A. Dijon Ac. Mm. (1830) 23-. . Roget, P. M. (vi Adds.) Ph. Mg. 16 (1840) 305-. . Moon, R. Camb. Mth. J. 3 (1843) 233-. . Minding, E. F. A. [1847] St Pet. Ac. Sc. Bll. 6 (1848) 209-. . Volpicelli, P. C. E. 31 (1850) 314-. . Clausen, T. Grunert Arch. 21 (1853) 91-. . Anon, (vi 883) N. A. Mth. 13 (1854) 181-. . Polignac, C. de. C. E. 52 (1861) 840-. . Geynet, A. C. E. 60 (1865) 484. . Urusov, (Prince) S. S. (xu) [1865] Eec. Mth. (Moscou) 2 (1867) (Pt. 1) 189-. _ _. Regnier, A. Les Mondes 32 (1873) 507-. . F rost t A. H. [1876] QJ. Mth. 14 (1877) 123-. . Flye Sainte-Marie, C. Par. S. Mth. Bll. 5 (1877) 144-. . Polignac, C. (Prince) de. [1880] Par. S. Mth. Bll. 9 (1881) 17-. . Lucas, E. Ev. Sc. 6 (1883) 370-. . Gilbert, G. K. [1885] Smiths. Misc. Col. 33 (1888) Art. 2, 88. (Wash. Ph. S. Bll. 7 (1885).) . Hofmann, F. N. A. Mth. 5 (1886) 224-. . Jordan, C. Palermo Cir. Mt. Ed. 2 (1888) 59-. . Brunei, . Bordeaux S. Sc. Mm. 4 (1894) liii. . Wolf, R. Ziir. Vjschr. 39 (1894) 147-. , generalisation of problem. Fitting, F. Z. Mth. Ps. 45 (1900) 137-. illustrating an electrical problem. Brunei, . Bordeaux S. Sc. Mm. 2 (1891) xxxiv-. , number of reentrant moves in half- boards. Laquiere, E. M. [1880] Par. S. Mth. Bll. 9 (1881) 11-. , regular solutions of Euler's problem. Laquiere, E. M. Par. S. Mth. Bll. 8 (1880) 82-, 132-. 81 F 1620 Chess Knight's move, solution by geometry of posi- tion. Volpicelli, P. Em. At. E. Ac. 25 (1872) 87-, 364-; 26 (1873) 49-, 241-. , (Volpicelli) . Tarry, H. Les Mondes 28 (1872) 60-. moves over squares of 5, 6, 7, 8, and extension to higher squares. Frost, A. H. QJ. Mth. 14 (1877) 354-. Mathematical analysis applied to chess. (Brief resume.) Janisch, . St Pet. Ac. Sc. Bll. 6 (1863) 473-. Problem, solution by aid of numerical functions. Bugaev, N. V. (xn) [1878] Eec. Mth. (Moscou) 9 (1878-81) (Pt. 1) 355-. 8 queens problem. Glaisher, J. W. L. Ph. Mg. 48 (1874) 457-. . Eohn, K. Dresden Isis Sb. (1889) (Ab.) 89-. . Sprague, T. B. Edinb. Mth. S. P. 8 (1890) 30- ; 17 (1899) 43-. , generalisation. Gunther, S. Arch. Mth. Ps. 56 (1874) 281-. n queens problem. Parmentier, T. As. Fr. C. E. (1883) 197-. Sylvester's anallagmatic chess-board pattern. Lucas, E. As. Fr. C. E. (1877) 213-. Class of numbers arising in certain expansions. Ocagne, M. d'. Am. J. Mth. 9 (1887) 353-. Classes, division into. Harding, P. J. Mth. Gz. No. 6 (1895) 59. Coefficients in expansion of (x + y) n . Laisant, . As. Fr. C. E. (1888) (Pt. 2) 72-. ._ f ?fz*Y*. Mathieu, J. J. A. N. A. Mth. 2 (1863) 509-. COMBINATIONS. Kirkman, T. P. (n Adds.) Ph. Mg. 5 (1853) Buniakovskij, V. Ja. St. Pet. Ac. Sc. Mm. (Es.) 21 (*1872) 261-. Mantel, W. N. Arch. Wisk. 11 (1884) 190. Thiele, T. N. Kjtfb. Ov. (1895) 415-, xv-. application to explanation of Arbogast's method. De Morgan, A. Camb. and Dubl. Mth. J. 6 (1851) 35-. of a certain number of successive integers having a sum below a given limit, number of. Marsano, G. B. G. Mt. 19 (1881) 156- ; 20 (1882) 249-. complete. Levret, . N. A. Mth. 1 (1842) 87-. . Melon, A.G. N. A. Mth. 8 (1869) 168-. . Niewenglmvski, G. H. N. A. Mth. 13 (1874) 285-. compound. Cayley, A. Manch. Lt. Ph. S. P. 16 (1877) 113-. Crocchi's theorem (on sums of combinations of powers and homogeneous products). Sylvester, J. J. (xn) [1882] J. H. Un. Cir. [2] (1883) 2. ( ) proof. Franklin, F. (xn) [1882] J. H. Un. Cir. [2] (1883) 24. Combinations 1620 of 28 dominoes. Eeiss, M. [1859] A. Mt. 5 (1871-73) 63-. of elements dispersed in a plane. Mantel, . As. Fr. C. E. (1883) 171-. and factorial numbers, theorems. Catalan, E. Brux. Ac. Bll. 25 (1893) 238-, 430-. formula. Catalan, E. C. Liouv. J. Mth. 7 (1842) 511-. , new. Studnicka, F. J. Prag Sb. (1879) 295-. formulae. Grand, , & EocJiat, . Gergonne A. Mth. 3 (1812-13) 213-. for expanding power of binomial. Gergonne, J. D. Gergonne A. Mth. 17 (1826-27) 356-. a game of. Brianchon, C. J. Par. EC. Pol. Cor. 3 (1814-16) 387-. number of given class having a given sum. Boschi, P. Bologna Ac. Sc. Mm. 5 (1883) 805-. probabilities and life contingencies. Hardy, P. Assur. Mg. 2 (*1852) 151-, 259-. problem. Euler, L. [1779] St Pet. Ac. Sc. Mm. 3 (1811) 57-. . Durrande, J. B. Gergonne A. Mth. 7 (1816-17) 334-. . Catalan, E. C. Liouv. J. Mth. 3 (1838) 111- ; 5 (1840) 264 ; 6 (1841) 74. . Chevillard, A. J. N. A. Mth. 3 (1844) 537-. . Kirkman, T. P. Camb. and Dubl. Mth. J. 2 (1847) 191- ; 5 (1850) 255-. . Anstice, E. E. Camb. and Dubl. Mth. J. 7 (1852) 279- ; 8 (1853) 149-. . Prouhet, E. N. A. Mth. 12 (1853) 85-. . Kelland, P. [1855] Edinb. E. S. T. 21 (1857) 359-. . Polignac, (Prince) C. de. [1868] L. Mth. S. P. 2 (1869) 69-. . Robertson, A. Edinb. Mth. S. P. 4 (1886) 78-. . Hertzsprung, S. Ts. Mth. 5 (1887) 13- ; Fschr. Mth. (1887) 198-. . Brunei, . Bordeaux S. Sc. Mm. 5 (1895) xiv-. problems. Encontre, D. Mntp. Eec. Bll. 2 (1805) 68-. . Anon, (vi 538) Gergonne A. Mth. 11 (1820-21) 165-. . Weiss, A. Crelle J. 34 (1847) 255-; 38 (1849) 107-. . Sinram,T. Arch. Mth. Ps. 63 (1879) 445-. . Holtze,A. Arch. Mth. Ps. 11 (1892) 284-. proof for number of combinations of n things taken r at a time. Scott, W. H. Mess. Mth. 5 (1876) 18-. of theorem. Forstemann, W. A. Crelle J. 13 (1835) 237-. . Crone, C. (xn) Ts. Mth. 1 (1877) 153-. questions in. De Morgan, A. Assur. Mg. 5 (*1855) 93-. regular, and applications. Ai\dre, D. Par. EC. Norm. A. 5 (1876) 155-. with repetition. Martynowski, J. Liege Mm. S. Sc, 16 (1861) 53-. . Catalan, E. C. Li^ge Mm. S. Sc. 2 (1867) 1-. 82 1620 Permutations, Combinations, Partitions, Distributions, etc. 1620 Combinations, continued. with repetition. Soufflet, . Les Mondes 17 (1868) 110- ; 24 (1871) 337-. . Motta Pegado, L.'P. da. Lisb. J. Sc. Mth. 8 (1880) 1-. . Gelin, . Mathesis 6 (1886) 175-. in sets having odd and even numbers of in- versions. Metzler, W. H. Am, J. Mth. 22 (1900) 55-. simple. Andre, D. N. A. Mth. 10 (1871) 221-. special class. Bouniakowsky, V. [1842] St Pet. Ac. Sc. Mm. 5 (1844) 297-i kind. Buniakovskij, V. J. [1871] St Pet. Ac. Sc. Mm. (Es.) 20 (*1872) (Suppl.) No. 2, 71 pp. theorem. Whitworth, W. A. [1878] Mess. Mth. 8 (1879) 82-. . Christie, E. W. D. L. Mth. S. P. 20 (1889) 119-. of H. F. Baker. Gegenbauer, L. Wien Ak. Sb. 102 (1893) (Ab. 2 a) 951-. theorems. Hessel, J. F. C. Grunert Arch. 7 (1846) 295-. . Kirkman, T. P. (n Adds.) Ph. Mg. 4 (1852) 209- ; (m) Camb. and Dubl. Mth. J. 8 (1853) 38-. . Andre, D. N. A. Mth. 12 (1873) 84-. , new. Oettinger, L. Grunert Arch. 15 (1850) 241- ; Crelle J. 53 (1857) 322-. theory. Baur, C. W. Schlomilch Z. 2 (1857) 267-. . Bie, L. H. N. Ts. Mth. 6 (A) (1895) 81-. , and application to theory of numbers. Stern, M. A. Crelle J. 21 (1840) 91-, 177-. and variations, number and sums. Wasmund, C. Grunert Arch. 21 (1853) 228-. Combinatory aggregation, analysis. Sylvester, J. J. Ph. Mg. 24 (1844) 285-. analysis. Jacobi, C. G. J. Crelle J. 22 (1841) 372-. , foundations of new theory. MacMalion, (Mai.) P. A. Phil. Trans. (A) 194 (1900 361-. , game of mousetrap. Cayley, A. QJ. Mth. 15 (1878) 8-. , . Steen, A. QJ. Mth. 15 (1878) 230-. , generalisation of elementary. Traversa, N. G. Mt. 35 (1897) 174-. ) inversion of fundamental idea of Hindenburg's. Roth, F. Arch. Mth. Ps. 70 (1884) 427-; 2 (1885) 82-. , Mairan's problem. Cournot, A. A. Ferussac Bll. Sc. Mth. 11 (1829) 93-. , new method in. MacMahon, (Maj.) P. A. Camb. Ph. S. T. 16 (1898) 262-. , problem. Spottiswoode, W. Ph. Mg. 3 (1852) 349-. , _. Atl dre, D. Par. S. Mth. Bll. 5 (1877) 150-. , . Ocagne, M. d\ N. A. Mth. 19 (1880) 44-. , , solution. Eamus, C. Crelle J. 11 (1834) 353-. Combinatory analysis, summation formulae of Steiner, proof. Gudermann, C. [1829] Crelle J. 5 (1830) 402-. , theorems. Scherk, H. F. Crelle J. 11 (1834) 226- ; 20 (1840) 360-. formula, applications. Gerbaldi, F. G Mt. 18 (1880) 308-. numbers, graphic representation. Laisant, C. A. As. Fr. C. E. (1893) (Pt. 2) 298-. problem. Vachette, A. N. A. Mth. 7 (1848) 421-. . Clausen, T. Grunert Arch. 21 (1853) 93-. . Cantor, M. Schlomilch Z. 2 (1857) 103-. , Steiner. Eeiss, M. [1856] Crelle J. 56 (1859) 326-. , . Brunei, G. Bordeaux S. Sc. PV. (1896-97) 37-. problems. Thibault, . N. A. Mth. 4 (1845) 627-. , class. Hertzsprung, S. Ts. Mth. 4 (1886) 154- ; Fschr. Mth. (1887) 198. , four. Schroder, E. Z. Mth. Ps. 15 (1870) 361-. products, formulae and coefficients for cal- culation. Wasmund, C. Grunert Arch. 34 (1860) 440-. symbol of Euler's, use in analysis. Ter- quem, 0. Liouv. J. Mth. 4 (1839) 177-. systems, order and class. Prouhet, E. Liouv. J. Mth. 1 (1856) 321-. theorem. Stern, M. A. Crelle J. Mth. 95 (1883) 102-. of Stern, proof. Terquem, O. Liouv. J. Mth. 3 (1838) 556-. Compositions of numbers, theory. MacMahon, P. A. [1892] Phil. Trans. (A) 184 (1894) 835. Connected elements, notation for and distribu- tion. Polignac, C. de. As. Fr. C. E. (1884) (Pt. 2) 37-. Contact and isolation, a problem in permuta- tions. Fortey, H. L. Mth. S. P. 15 (*1883- 84) 98-. Cycle, problem of completing a. Waelsch, E. Wien Ak. Sb. 90 (1885) (Ab. 2) 160-. Derangements. Eodrigues, O. Liouv. J. Mth. 4 (1839) 236-. . Henry, C. N. A. Mth. 20 (1881) 5-. . Musso, G. Ev. Mt. 4 (1894) 109-. Dirichlet's function [coefficient of \/& in (x + -Jb) n ], Daublebsky von Sterneck, E. Mh. Mth. Ps. 7 (1896) 342-. Distribution of points on a line. Comstock, M. L. Des Moines Anal. 1 (1874) 156-. Diversities, number of. Lipschitz, . C. E. 102 (1886) 602-. Double of odd number, partition into two odd numbers. Maria, P. C. G. Mt. 4 (1866) 1-, 65-, 193-. 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Mth. 4 (1839) 317-. , when n is large. Serret, J. A. C. B. 50 (1860) 662-. (faculties). Oettinger, L. Crelle J. 33 (1846) 1-, 117-, 226-, 329-; 35 (1847) 13-; 38 (1849) 162-, 216- ; 44 (1852) 26-, 147-. and faculties, treatment by limits. Ohm, H. Crelle J. 39 (1850) 23-. , W. G. Homer's method. Homer, J. [1872] QJ. Mth. 12(1873)258-. , new method. Nicholson, P. Tilloch Ph. Mg. 53 (1819) 412-. , theorem. Andre, D. (ix) Par. S. Mth. Bll. 1 (1873) 84-. , numerical values. Kotel'nikov, P. I. (xn) Kazan Un. Mm. ((1848) Bk. 4) 9-. , ratios of reciprocal. Cauchy, A. L. C. B. 17 (1843) 921-, 1159-. , theorem. Gomes Teixeira, E. Arch. Mth. Ps. 2 (1885) 265-. Faculties, geometric, and transformation of logs, into definite integrals. Cauchy, A. L. C. B. 17 (1843) 779-. and powers ; analogy. Ampere, A.M. Ger- gonne A. Mth. 15 (1824-25) 369-. , quotients of two. Schlomilch, 0. Schlo- milch Z. 3 (1858) 322-. , remarks on some. Guibert, M. P. A. N. A. Mth. 19 (1860) 213-. , theorem. Schlomilch, 0. Grunert Arch. 7 (1846) 331-. , theory. Bessel, F. W. Konigsb. Arch. 1 (1812) 241-.' , . Kramp, C. Gergonne A. Mth. 3 (1812- 13) 1-, 114-, 325-. , . Miiller, A. Crelle J. 11 (1834) 361-. Faculty coefficients. Studnicka, F. J. Prag Ceske Ak. Fr. Jos. Bz. (Trida 2) 9 (1900) Art. 17, 10pp.; Fschr. Mth. (1900) 156-. , determination. Schlomilch, 0. Grunert Arch. (1848) 445-. , relations. Schlomilch, O. Grunert Arch. 9 (1847) 333-. Fractional coefficients. Barsotti, G. Lucca At. Ac. 12 (1843) 411-. Game of draughts, problem. Lamarle, E. [1852] Brux. Ac. Sc. Mm. 27 (1853) 47 pp. Gordian knot (bar and rings), theory. Dassen, C. C. Arg. S. Ci. A. 44 (1897) 337-. jumping frog, arrangement of counters on board. Berg, F. J. van den. N. Arch. Wisk. 12 (1886) 38-; Fschr. Mth. (1885) 121. loto. Du Hays, . Liouv. J. Mth. 7 (1842) 192-. played with 15 pieces on square board of 16 holes. Bellavitis, G. Yen. I. At. 6 (1880) 901-. of solitaire. Reiss, M. Crelle J. 54 (1857) 344-. . Hermary, . As. Fr. C. B. (1879) 284-. , as problem of analysis and situation. Suremain-de-Missery, A. Dijon Ac. Mm. (1841-42) 58-. Grouping in games. Andre, D. Par. S. Phlm. Bll. 1 (1899) 139- ; 2 (1900) 45-, 77-. of instruments for simultaneous telegraphy. Zetzsche, E. J. Tel. 15 (1891) 201-, 229-, 255-. problems, analytical treatment. Hoyer, P. Mth. A. 50 (1898) 499-. Homogeneous products of n things of r dimen- sions, proof. Walton, W. [1869] QJ. Mth. 10 (1870) 219-. Hyper-magic arithmetical spaces. Laisant, C. A. Par. S. Mth. Bll. 22 (1894) 28-. squares, general limitations affecting. Roberts, S. [1892] L. Mth. S. P. 24 (1893) 37-. La tour d'Hanoi (problem in permutations). Allardice, R. E., & Eraser, A. Y. Edinb. Mth. S. P. 2 (1884) 50-. Latin squares. Cayley, A. Mess. Mth. 19 (1890) 135-. , Euler's. Maillet, E. As. Fr. C. B. (1894) (Pt. 2) 244-. Magic cubes in n dimensions, way of represent- ing in plane. Schlegel, . Par. S. Mth. Bll. 20 (1892) 97-. cyclovolute. Nulty, E. [1834-44] Am. Ph. S. T. 5 (1837) 205- ; 10 (1853) 17-. parallelepipeds. Harmuth, T. [1881] Arch. Mth. Ps. 67 (1882) 238-. rectangles with odd numbers of sides. Harmuth, T. Arch. Mth. Ps. 66 (1881) 413-. Magic squares. Moon, R. Camb. Mth. J. 4 (1845) 209-. Clausen, T. Grunert Arch. 21 (1853) 97-. Gyory, S. (xn) Mag. Ak. Ets. (1856) 77-. Thompson, W. H. QJ. Mth. 10 (1870) 186-. Drach, S. M. Mess. Mth. 2 (1873) 169-, 187. Cayley, A. Mess. Mth. 6 (1877) 168. Frisby, E. Wash. Ph. S. Bll. 3 (1880) 143-. Marchand, D. Les Mondes 2 (1882) 106-, 247-. Rocquigny, F. [G.] de. Les Mondes 2 (1882) 163. 84 1620 Magic Squares Arnoux, G. As. Fr. C. E. (1885) (Pt. 1) 94. Schoute, P. H. As. Fr. C. E. (1885) (Pt. 2) 152-. Coccoz, . As. Fr. C. E. (1886) (Pt. 2) 130-. Frolow, M. As. Fr. C. E. (1886) (Pt. 2) 170-. Clauss, F. Arch. Mth. Ps. 6 (1888) 424-. Berg, F. J. van den. N. Arch. Wisk. 16 (1889) 1- ; Fschr. Mth. (1889) 203. Parmentier, (le gen.) . As. Fr. C. E. (1890) (Pt. 2) 88-. Coccoz, . As. Fr. C. E. (1892) (Pt. 2) 136-, [1213]; (1894) (Pt. 2) 163-, 1219; (1895) (Pt. 2) 102-. Ball, W. W.R. Mess. Mth. 23 (1894) 65-. Iznoskov, I. Kazan S. Ps.-Mth. Bll. 5 (1896) (Prot.) 48-. algebra. Homer, J. QJ. Mth. 11 (1871) 57-, 123-, 213-. application of group theory. Maillet, E. Toul. Ac. Sc. Mm. 6 (1894) 258-. with base eight. Coccoz, . As. Fr. C. E. (1893) (Pt. 2) 171-, 1079. case. Holditch, H. QJ. Mth. 6 (1863) 181-. of knight's march. Beverley,W. (viAdds.) Ph. Mg. 33 (1848) 101-. n? numbers. Monteiro, A. S. G. Teix. J. Sc. 3 (1881) 81-. and cubes. Frost, A. H. QJ. Mth. 7 (1866) 92-. . Barnard, F. A. P. Wash. Nat. Ac. Mm. 4 (Pt. 1) (1888) 209-. . Mayor, P. Laus. S. Yd. Bll. 27 (1892) 243-. cubes, etc., theory. Moon, E. Camb. and Dubl. Mth. J. 1 (1846) 160-. doubly magic. Laquiere, E. M. As. Fr. C. E. (1880) 243-. etc. Harmuth, T. Arch. Mth. Ps. 66 (1881) 286-. Euler's. Boije af Gennas, C. 0. [1898] Stockh. Ak. Hndl. Bh. 24 (Afd. 1) (1899) No. 2, 15 pp. and Gauss. Gunther, S. Z. Mth. Ps. 21 (1876) (Suppl.) 61-. where highest number is prime, law. Tychsen, C. (XH) Ts. Mth. 1 (1865) 55-. most perfect forms. McClintock, E. Am. J. Mth. 19 (1897) 99-. and other such arrangements. Smyth, B. B. Kan. Ac. Sc. T. 14 (1896) 46-. proof of theorem. G anther, S. Arch. Mth. Ps. 57 (1875) 285-. Mixtures, problem. Coupy, E. N. A. Mth. 6 (1847) 14-. li'ads formed with n 2 elements. Brunei, G. Bordeaux S. Sc. PV. (1894-95) 56-. v cycles of 2c + l elements. Brunei, G. Bor- deaux S. Sc. PV. (1898-99) 71-. Nasik cubes. Frost, A. H. QJ. Mth. 8 (1867) 74. -, general properties. Frost, A. H. [1877] QJ. Mth. 15 (1878) 93-, 366-. squares of any order, construction. Frost, A. H. L. Mth. S. P. 27 (1896) 487-. Permutations, etc. 1620 Nasik squares, general properties. Frost, A. H. [1877] QJ. Mth. 15 (1878) 34-. Network representing benzene. Brunei, G. Bordeaux S. Sc. Mm. 4 (1894) xviii-. Notation [a a a 2 . . . a p ] < m) to represent expansion of mth power of polynomial ^ + 0,2 + ... + a p . Ocagne, M. d'. N. A. Mth. 2 (1883) 220-. Number of complete arrangements satisfying given conditions. Andre, D. C. E. 87 (1878) 838-; Par. S. Mth. Bll. 7 (1879) 43. distributions with repetition of given class and weight. Caldarera, G. G. Mt. 31 (1893) 161-. partitions of a rational integer into sums. Hermes, J. Mth. A. 45 (1894) 371- ; 47 (1896) 281-. permutations of n elements presenting sequences. Andre, D. C. E. 97 (1883) 1356-. substitutions falling into given number of cycles. Schroder, E. Arch. Mth. Ps. 68 (1882) 353-. ways of getting sum not exceeding n, from terms of series 1,2, 3.... Plainer, G. Mil. I. Lomb. Ed. 21 (1888) 690-, 702-. Numbers, 4m + 2 as sum of 12 squares. Liouville, J. Liouv. J. Mth. 5 (1860) 143-. Partition analysis applied to properties of any system of consecutive integers. MacMahon, (Maj.) P. A. [1899] Camb. Ph. S. T. 18 (1900) 12-. Partition of numbers. Warburton, H. [1847] Camb. Ph. S. T. 8 (1849) 471-. Battaglini, G. Nap. Ac. Sc. Mm. 2 (1855-57) 353-. Cayley, A. [1855-57] Phil. Trans. (1856) 127- ; (1858) 47-; Ph. Mg. 13 (1857) 245-. Brioschi, F. Tortolini A. 8 (1857) 5-. Sylvester, J. J. QJ. Mth. 1 (1857) 81-, 141-, 198- ; Tortolini A. 8 (1857) 12-. Volpicelli, P. Tortolini A. 8 (1857) 22-. Bellavitis, G. Tortolini A. 2 (1859) 137-. Lanavicensis, (Pseud.) [J. J. Sylvester'] Ph. Mg. 18 (1859) 283-. Trudi, N. (xi) Nap. Ac. At. 2 (1865) No. 23, 50pp. Catalan, E. C. N. A. Mth. 8 (1869) 407-. Buniakovskij, V. Ja. [1874] St Pet. Ac. Sc. Mm. (Rs.) 25 (*1875) (Suppl.) No. 1, 29pp. Glaisher, J. W. L. C. E. 80 (1875) 255-. Laguerre, E. Par. S. Mth. Bll. 5 (1877) 76-. Faa de Bruno, F. C. E. 86 (1878) 1189-, 1259. David, . C. E. 90 (1880) 1344- ; 91 (1880) 621-. Catalan, E. C. Liege S. Sc. Mm. 12 (1885) No. 2, 19-, 56-, 264-; 13 (1886) 314-. Pomey, J. B. N. A. Mth. 4 (1885) 408-. Polignac, C. de. C. E. 104 (1887) 1688-. Glosel, K. Mb.. Mth. Ps. 7 (1896) 133-, 290. Mathews, G. B. L. Mth. S. P. 28 (1897) 486-. 85 1620 Partition of Numbers addition to theory. Csorba, G. Mth. Termt. Ets. 17 (1899) 189-. application of theorem of Eisenstein. Catalan, . As. Fr. C. E. (1886) (Pt. 1) 86. certain special. MacMahon, (Capt.) P. A. QJ. Mth. 21 (1886) 367-. formulae. Pepin, . Liouv. J. Mth. 4 (1888) 83-. graphic proof of theorem of Euler's. Sylvester, J. J. C. E. 96 (1883) 1110-. whose graphs possess symmetry. MacMahon, (Maj.) P. A. [1898] Camb. Ph. S. T. 17 (1899) 149-. number of ways of breaking up a number with one law of subordinate partition. Volpicelli, P. Em. At. 10 (1856-57) 43-, 122-. outlines of seven lectures. Sylvester, J. J. L. Mth. S. P. 28 (1897) 33-. points of analysis depending on. Trudi, N. [1877] Nap. Ac. At. 8 (1879) No. 1, 88 pp. problem. Chrystal, G. Edinb. Mth. S. P. 2 (1884) 49-. and other problems, application of Arbogast's method of derivations. David, . Liouv. J. Mth. 8 (1882) 61-. proof of theorem. Volpicelli, P. C. E. 40 (1855) 1150-. . Zuchristian, J. Mh. Mth. Ps. 4 (1893) 185-. question. Boschi, P. Bologna Ac. Sc. Mm. 1 (1880) 555-. theory. MacMahon, P. A. [1896] Phil. Trans. (A) 187 (1897) 619- ; 192 (1899) 351-. two theorems. Volpicelli, P. N. A. Mth. 14 (1855) 314-. verification formulae. Glaisher, J. W. L. E. S. P. 24 (1876) 250-. Partition of pentagonal numbers, graphic proof of Euler's theorem. Sylvester, J. J. C. E. 96 (1883) 743-. polygon by diagonals, series for. Binet, J. P. M. Liouv. J. Mth. 8 (1843) 394-. , problem. Ocagne, M. d'. Par. S. Mth. Bll. 28 (1900) 157-. -series, note. Cayley, A. Am. J. Mth. 6 (1884) 63-. , solution of problem, and types of abacus. MacMahon, (Maj.) P. A. Par. S. Mth. Bll. 26 (1898) 57-. of sum of money, number of ways. Schubert, H. C. H. Hamb. Mth. Gs. Mt. 1 (1889) 269. table for binary quantics, specimen. Cayley, A. Am. J. Mth. 4 (1881) 248-. tables, non-unitary. Cayley, A. Am. J. Mth. 7 (1885) 57-. PAETITIONS. Cayley, A. QJ. Mth. 1 (1857) 183-. Franklin, F. Am. J. Mth. 2 (1879) 187- ; (xn) J. H. Un. Cir. [2] (1883) 72. Ely, G. S. J. H. Un. Cir. [3] (1884) 76-. Partitions 1620 Barcroft, D. J. H. Un. Cir. [5] (1885-86) 64. algebraic expression. Herschel, (Sir) J. F. W. Phil. Trans. (1850) 399-. theorems connected with theory. Forsyth, A. R. [1895] L. Mth. S. P. 27 (1896) 18-. of complex numbers (theorem contained in theorem of Jacobi's). Sylvester, J. J. C. B. 96 (1883) 1276-. compound, general theory. Sylvester, J. J. Ph. Mg. 16 (1858) 371-. , method of graphs applied to. Ely, G. S. Am. J. Mth. 6 (1884) 382-. constructive theory. Sylvester, J. J. Am. J. Mth. 5 (1882) 251- ; 6 (1884) 334-. , Sylvester's. Jenkins, M. Am. J. Mth. 7 (1885) 74-. and derivations. Glaisher, J. W. L. B. A. Ep. (1874) (Sect.) 11-. double. Cayley, A. Ph. Mg. 20 (1860) 337-. formulae of verification. Glaisher, J. W. L. B. A. Ep. (1875) (Sect.) 11-. fundamental theorem. Sylvester, J. J. (xn) [1882] J. H. Un. Cir. [2] (1883) 22. graphical method. Sylvester, J. J. (xii) J. H. Un. Cir. [2] (1883) 70-. isomerism, graphical representation. Brunei, G. Bordeaux S. Sc. PV. (1898-99) 108-. , problem (to find number of compounds that can be formed by n m-valent atoms and (ra - 2) n + 2 univalent atoms) . Franklin, F. Am. J. Mth. 1 (1878) 365-. miscellaneous theorems. Glaisher, J. W. L. Mess. Mth. 5 (1876) 164-. of non-pentagonal number. Jacobi, C. G. J. Crelle J. 32 (1846) 164-. note. Glaisher, J. W. L. Ph. Mg. 49 (1875) 307-. . Mitchell, 0. H. (xn) J. H. Un. Cir. [1] (1882) 210. . Ely, G. S. (xn) J. H. Un. Cir. [1] (1882) 211. of a polygon. Cayley, A. L. Mth. S. P. 22 (1891) 237-. problem. Cayley, A. Mess. Mth. 7 (1878) 187-. . Brunei, . Par. S. Mth. Bll. 12 (1884) 78-. , doubly recurring. Betti, E. Tortolini A. 8 (1857) 48-. of Fibonacci, on [u n + l=u n + u n -l]. Cesaro, E. G. Teix. J. Sc. 6 (1885) 17-. , Fibonacci, peculiarities. Fontes, . As. Fr. C. B. (1894) (Pt. 2) 217-. , general term found as with recurring series. Andre, D. Par. EC. Norm. A. 7 (1878) 375-. , geometrical, fractional number of terms. Choquet, C. N. A. Mth. 1 (1842) 74-. of quotients, Euclid's summation. Zeuthen, H. G. (xn) Ts. Mth. 6 (1876) 168-, 181- ; 6 (1882) 97- ; 1 (1883) 145-. , self-repeating. Warburton, H. Camb. Ph. S. T. 9 (1856) (Pt. 4) 471-. , whose series of differences are recurring. Raabe, J. L. Baumgartner Z. 6 (1829) 307-. , summation. (Dienger, Grunert Arch. 8 213-214.) Arndt, F. Grunert Arch. 8 (1846) 383-. , of certain. Tait, P. G. Edinb. Mth. S. P. 3 (1885) 107-. , class. Ocagne, M. d'. C. B. 115 (1892) 790-, 904. , , successive. Dantas Pereira, J. M. Lisb. Mm. Ac. So. 2 (1799) 168-. Sum of a combinatory series, properties. Unferdinger, F. Grunert Arch. 26 (1856) 227-. cubes = a square. Catalan, E. C. [1866] . Rm. At. N. Line. 20 (1867) 1-, 77-. of secondary numbers, theorem. Barbier, E. Bll. Sc. Mth. As. 7 (1883) 42-. first n products of p consecutive numbers. Laisant, A. N. A. Mth. 6 (1867) 366-. products of n quantities taken m together. Trowbridge, D. Des Moines Anal. 1 (1874) 154-. r different terms of series. Hammond, J. L. Mth. S. P. 7 (1875-76) 119-. reciprocals. Bouvier, L. C. Gergonne A. Mth. 15 (1824-25) 39-. . Blissard, J. QJ. Mth. 6 (1863) 242-. . Worontzof, M. M. QJ. Mth. 8 (1867) 185-, 310-. of first 1000 integers. Adams, J. C. B. A. Bp. (1877) (Sect.) 14-. Summation of series, general term of which is na n , a being a given number. Bertrand, J. N. A. Mth. 8 (1849) 354-. Summation of terms of recurring series, forma- tion of integers by. Maillet, E. As. Fr. C. B. (1896) (Pt. 2) 78-. Summations, successive, l m + 2 m + 3 m + . . . x m . Lucas, E. [1877] Mess. Mth. 7 (1878) Sums of 4 algebraic series. Glaisher, J. W. L. [1877] Mess. Mth. 7 (1878) 119-. powers of reciprocals of integers. Schlo- milch, O. X. Leip. Mth. Ps. B. 29 (1877) 106-. products of an arithmetical progres- sion, taken 2, 3, 4 ... together, calculation. Lampe, E. Fschr. Mth. (1893-94) 409. natural numbers. Mollame, V. N. A. Mth. 5 (1886) 364-. numbers, theorem. Lerch, . G. Teix. J. Sc. 8 (1887) 161-. series l n + 2 + . . . x n , l n - 2 n + . . . x n . Glaisher, J. W. L. QJ. Mth. 30 (1899) 166-. Special series. Collins, E. St Pet. Ac. Sc. Mm. 8 (1817-18) 242-. Schlomilch, O. Grunert Arch. 10 (1847) 342-. Thacker, A. Camb. and Dubl. Mth. J. 5 (1850) 243-. Mourgues, , N. A. Mth. 10 (1851) 78-. Catalan, E. C. N. A. Mth. 15 (1856) 230-. Siacci, F. (vm) A. Mt. 4 (1861) 46-. Lucas, E. N. A. Mth. 9 (1870) 49-. Amigues, E. N. A. Mth. 10 (1871) 79-. Dostor, G. Arch. Mth. Ps. 54 (1872) 70-; 57 (1875) 222-. Lampe, E. [1877] Crelle J. Mth. 84 (1878) 270-. Lucas, E. N. A. Mth. 16 (1877) 18-. Pelndr, M. Casopis 6 (*1877) 279-; Fschr. Mth. (*1877) HO-. Dostor, G. Arch. Mth. Ps. 63 (1879) 435-; 64 (1879) 310-. Marchand, D. N. A. Mth. 20 (1881) 140-. Schier, 0. Wien Ak. Sb. 85 (1882) (Ab. 2) 503-. Boije af Gennas, C. 0. Stockh. Ak. Hndl. Bh. 10 (1885) No. 4, 6 pp. Slavik, J. Casopis 16 (1887) 121- ; Fschr. Mth. (1887) 240. Locchi, E. Mh. Mth. Ps. 4 (1893) 85-. Nielsen, N. N. Ts. Mth. 4 (B) (1893) 1-; Fschr. Mth. (1893-94) 410-. Barbette, E. Mathesis 14 (1894) 105-. Jung, V. Casopis 28 (1899) 124- ; Fschr. Mth. (1899) 247. Tr 3 . r=l Genocchi, A. A. Mt. 7 (1865) 151-. Cantor, M. Z. Mth. Ps. 11 (1866) 248-; 12 (1867) 170-. 91 1630 Finite Summation Unferdinger, F. Arch. Mth. Ps. 48 (1868) Amigues, E. N. A. Mth. 10 (1871) 117-. Glaisher, J. W. L. Mess. Mth. 4 (1875) 168-. Eocquigny, F. [G.] de. Les Mondes 51 (1880) 153-. geometrical summation. Glaisher, J. W. L. (vn) [1873] Mess. Mth. 3 (1874) 5. problem of Boncompagni, solution. Matthies- sen, L. Z. Mth. Ps. 13 (1868) 348-. Bretschneider, C. A. Grunert Arch. 1 (1841) 415-. Borchardt, C. W. Berl. Mb. (1857) 301-. Brioschi, F. [1857] Tortolini A. 1 (1858) 43-. Dostor, G. Arch. Mth. Ps. 64 (1879) 353-. Dostor, G. Arch. Mth. Ps. 64 (1879) 356-. Puiseux, V. Liouv. J. Mth. 11 (1846)477-. Gilbert, P. N. A. Mth. 8 (1869) 434-. formulae. Lionnet, E. N. A. Mth. 1 (1842) 175-. . Rispal, A. N. A. Mth. 6 (1847) 273-. "2" (a + n - 1) fc-i, "i" [a + n - 1]<* M*-i. n _Pepin (lepere). N^A. Mth. 15 (1856) 27-. 2 (x + ih)n, *~2 (-l) i (* + tft)'. Bjorling, JE. G. Stockh!lk. Hndl. 2 (1857-58) 18 pp. 2 f-j , Jacobian formula involving. Stern, M. A. Crelle J. Mth. 84 (1878) 216-. r=2+l S r 2 or r 3 . Dostor, G. Arch. Mth. Ps. 64 (1879) 361-. F(r). Dostor, G. r= 459-, 513-. 2 (2r 2 ) and 2 (2r- I 2 "*). Glaisher, J. W. L. Mess. Mth. 20 (1891) 172-. 2 (r 2 ). Glaisher, J. W. L. Mess. Mth. 20 (1891) 120- r 28 (1899) 29-. N. A. Mth. 18 (1879) 1630 Probabilities (including combination of observations). Theory of errors. Accuracy of recollection, measurement. Cat- tell, J. McK. Science 2 (1895) 761-. Approximate formulae relating to large num- bers. La Place, P. S. (Marquis) de. Par. Mm. de PI. (1809) 353-, 559-. Probabilities 1630 Approximations, numerical. Guyou, . N. A. Mth. 8 (1889) 165-. Arithmetical events, study. Cesaro, E. Brux. Mm. Cour. 4. 47 (1886) No. 9, 14 pp. Association ratios (of phenomena in observa- tions). Doolittle, M. H. Smiths. Misc. Col. 33 (1888) Art. 4, 83-, 94-. (Wash. Ph. S. Bll. 10 (1888).) Averages, theory. Local probability. Doo- little, M. H. Wash. Ph. S. Bll. 11 (1892) 596-. Bernoulli's inverse theorem. Meyer, A. Brux. Ac. Bll. 23 (1856) 148-. theorem. Mansion, . Brux. S. Sc. A. 16 (1892) (Pt. 1) 85-. . Goedseels, . Brux. S. Sc. A. 19 (1895) (Pt. 1) 4-. . Mansion, . Brux. S. Sc. A. 22 (1898) (Pt. 1) 3-. . Eayleigh, (Lord). Ph. Mg. 47 (1899) . ' Eggenberger, J. Z. Mth. Ps. 45 (1900) 43-. and approximate probability. Nekrasov, P. A. [1898] Eec. Mth. (Moscou) 20 (1899) 485-, 535-; Fschr. Mth. (1899) 214, 215. binomial coefficients. Nasimov, P. S. Fschr. Mth. (1888) 211. , gamma-functions and Laplace's integral. Eggenberger, J. Bern. Mt. (1893) 110-, 236. , inversion. Monro, C. J. (x) L. Mth. S. P. 5 (1873-74) 74-, 145-. , . Nekrasov, P. A. Mosc. S. Sc. Bll. 65 (No. 1) (1890) 45-; Fschr. Mth. (1890) 235. , , simple proof. Gosiewski, W. Krk. Ak. (Mt.-Prz.) Ez. 13 (1886) 153- ; Fschr. Mth. (1885) 176-. , proof by Euler's summational formula. Lagrange, C. Brux. Ac. Bll. 31 (1896) 439-. , repeated application. Andrade, J. C. R. 116 (1893) 1281-, 1482. Best reading of an instrument for measuring a given quantity. Baum, F. G. Ps. Rv. 8 (1899) 181-. Betting on races. Du Chatenet, M. N. A. Mth. 5 (1886) 327-, 380-, 408-. Bravais's formulae for regression, etc., with skew correlation. Yule, G. U. R. S. P. 60 (1897) 477-. Broken diamond, problem. Cesaro, E. G. Mt. 24 (1886) 124-. Buffon's problem of the needle. Barbier, E. Liouv. J. 5 (1860) 273-. . Sylvester, J. J. Acta Mth. 14 (1890-91) 185-. Causation process, results. Paterson, J. Am. As. P. (1856) 29-. Chance. Whitwarth, W. A. [1871] Mess. Mth. 1 (1872) 163-. and choice, problems supplementary to. Whitworth, W. A. Mess. Mth. 4 (1868) 152-, 193-. , continuity. Davis, E. W. [1897] Nebr. Un. Stud. 2 (1894-1902) 131-. , doctrines. Saint, W. Nicholson J. 21 (1808) 204-. 92 1630 Probabilities. Theory of Errors 1630 Chance, doctrines. Opsimath. [Pseud.] Nichol- son J. 21 (1808) 210-. , , doubts respecting. Opsimath. [Pseud.] (vi Adds.) Nicholson J. 21 (1808) 66-. Chances, determination of e x as a limit in calculation of. Andrade, . Par. EC. Pol. J. 64 (1894) 225-. , doctrine. H., B. (vi Adds.) Nicholson J. 21 (1808) 121-. , , calculation in. Young, Dr T. Nichol- son J. 18 (1807) 116-. , paradox in. Gallon, F. Nt. 49 (1893-94) 365-. , . Shorter, L. E. Nt. 49 (1893-94) 413. , problem. Day, H. G. QJ. Mth. 9 (1868) 354-. , , experimental solution. Mendenhall, T. C. Am. As. P. (1879) 190-. , problems. Wilson, J. M. Ph. Mg. 31 (1866) 170-. , theory in Boole's laws of thought. Wilbra- ham, H. Ph. Mg. 7 (1854) 465-. , , reply to Wilbraham. Boole, G. Ph. Mg. 8 (1854) 87-. Chessboard, chance of each piece cheeking king. Taylor, H. M. Ph. Mg. 1 (1876) 221-. , use for solution of problems in probability. Delannoy, H. As. Fr. C. E. (1889) (Pt. 2) 43-; (1895) (Pt. 2) i 70-. Combination of deviations. Ocagne, M. d'. N. A. Mth. 14 (1895) 133-. measures of one magnitude. Bertrand, J. C. E. 106 (1888) 701-. observations. Frisiani, P. [1859] Mil. I. Lomb. Mm. 11 (1870) 21 pp. . Lipschitz, R. C. E. Ill (1890) 163-. . Pizzetti, P. Geneva Un. At. [11] (1892) 113-. . Levanen, S. Helsingf. Ofv. 37 (1895) 19-; Fschr. Mth. (1895) 246. . Equations, linear, resulting from observations of equal weight, solution. San- tini, G. [1869] Yen. Mm. I. 14 (1868) 477-. , theory. Gauss, C. F. Gott. Cm. 5 (1819-22) 33-, 63-; 6 (1823-27) 57-; 7 (1828- 31) 89-. , . Donkin, W. F. Ashmol. S. T. (1844) 71 pp.; Liouv. J. Mth. 15 (1850) 297-. , . Kleiber, I. A. Kazan S. Nt. (Ps.-Mth.) P. 6 (1888) 148- ; Fschr. Mth. (1888) 220-. , , Gauss's. Nasimov, P. S. Fschr. Mth. (1889) 218-. _. Kriiger, L. Gott. Nr. (1897) 146-. , , generalised. Newcomb, S. Am. J. Mth. 8 (1886) 343-. , weights of British noblemen during last three generations. Galton, F. Nt. 29 (1884) 266-. Combinations and factorial numbers, theorems. Catalan, E. Brux. Ac. Bll. 25 (1893) 238-, 430-. Combining probabilities of same event, possi- bility. Terrot, C. H. [1856] Edinb. E. S. T. 21 (1857) 369-. Comparison of probabilities that are functions of one variable. Blaschke, E. Wien Ak. D. 54 (1888) (Ab. 2) 105-. Compensation, arithmetic mean in calculating. Dorna, A. Tor. At. Ac. Sc. 4 (1868-69) 757-. of errors. Lorenz, L. V. (xn) Ts. Mth. 2 (1872) 1-, 125-. . Zachariae, G. (xn) Ts. Mth. 2 (1872) 97-, 182-. . Bruns, H. Leip. Mth. Ps. Ab. 13 (1887) 515-. , effect of weights. Paucker, M. G. von. Mitau Arb. Kurland. Gs. 9 (1851) 183-. , Laplace's method. Dienger, J. [1874] Wien Ak. D. 34 (1875) (Ab. 2) 21-. , a law of duality in. Reina, V. Tor. Ac. Sc. At. 29 (1894) 433-. proportional errors. Jung, G. Mil. I. Lomb. Ed. 13 (1880) 238-. Conscription, drawing of lots in, problem arising from. Rasch, J. W. N. Arch. Wisk. 1 (1895) 206- ; Fschr. Mth. (1893-94) 342-. Constant aberrations in shooting, precession of shot. Cranz, C. D. Nf. Vh. (1897) (Th. 2, Halfte 1) 6. Continued fractions, application to calculation of probabilities. Markov, A. A. Kazan S. Ps.-Mth. Bll. 9 (1900) 29-; Fschr. Mth. (1899) 215. , probability of appearance of given number. Wiman, A. Stockh. Ofv. (1900) 829-. , numbers of given magnitude. Broden, T. Stockh. Ofv. (1900) 239-. Correlated averages. Edgeworth, F. Y. Ph. Mg. 34 (1892) 190-. , new method of treating. Edgeicorth, F. Y. Ph. Mg. 35 (1893) 63-. Correlation of averages for four variables, example. Bryant, (Mrs) S. Ph. Mg. 36 (1893) 372-. between organs, note on calculation. Edge- worth, F. Y. Ph. Mg. 36 (1893) 350-. Criterion that given deviations for correlated variables may have arisen from random sampling. Pearson, K. Ph. Mg. 50 (1900) 157-. Crofton, integral of. Petrini, H. Ts. Mth. 6 (1889) 39-; Fschr. Mth. (1888) 305-. Curve of errors, quadrature. Gram, J. P. (xn) Ts. Mth. 1 (1883) 65-; Fschr. Mth. (1884) 242. Definite integral, evaluation by theory of probabilities. Schoute, P. H. As. Fr. C. E. (1880) 258-. , form for series in theory of probabilities. Carpmael, C. [1885] Cn. E. S. P. & T. 3 (1886) (Sect. 3) 101-. integrals and application to probabilities. La Place, P. S. (Marquis) de. Par. Mm. de l'I. (1810) 279-. Determination of ie by needle throwing. Levanen, S. Helsingf. Ofv. 34 (1892) 36-; Fschr. Mth. (1892) 203. 93 1630 Probabilities. Theory of Errors 1630 Determination of some unknowns in geodesy. Berardinis, G. de. G. Mt. 25 (1887) 313-. unknowns by a large number of approxi- mate equations of 1st degree. Cauchy,A. L. C. E. 36 (1853) 1114-. Development of Lp + (l--.p)] m approximates to probability curve as m increases. De Forest, E. L. Des Moines Anal. 6 (1879) 65-. Digits, relative frequency in natural numbers. Newcomb, S. Am. J. Mth. 4 (1881) 39-. Distribution of frequency of barometric heights. Pearson, K., & Lee, A. [1897] Phil. Trans. (A) 190 (1898) 423-. points on a line. Bartlett, W. P. G. Camb. (M.) Mth. M. 1 (1859) 84-. signs of residual errors. Seeliger, H. [1879] As. Nr. 96 (1880) 49-. Division, eventualities in. Cesaro, E. A. Mt. 13 (1885) 269-. of a straight line, probability problem. Berg, F. J. van den. N. Arch. Wisk. 18 (1891) 42-; Fschr. Mth. (1891) 233-. Duration of play. Delannoy, . Par. S. Mth. Bll. 16 (1888) 124-. , problem. Rouche, E. C. R. 106 (1888) 47-. , . MacMahon (Maj.) P. A. L. Mth. S. P. 20 (1889) 195-. , , demonstration. Bertrand, . C. E. 106 (1888) 49-. , some questions of. Rouche, E. C. E. 106 (1888) 253-. Earthquakes, lunar and solar periodicities. Schuster, A. Nt. 56 (1897) 321. v Ellipse of error, theory. Czuber, (or Cuber) E. [1880] Wien Ak. Sb. 82 (1881) (Ab. 2) 698-. or hyperbola, questions of probability. Cesaro, E. G. Mt. 22 (1884) 44-. Equation, 2nd order, probability that roots are real. Bouniakou-sky , V. [1835] St Pet. Ac. Sc. Mm. 3 (1838) 341-. Equations, 1st order, in number greater than number of unknowns; geometrical applica- tions. Terquem,0. N. A. Mth. 5 (1846) 551-. , normal, solution by indirect elimination. Schott, C. A. U. S. Coast Sv. Ep. (1855) 255-. Error function and symmetric functions. Thiele, T. N. Kjb. Ov. (1899) 135- ; Fschr. Mth. (1899) 216-. , law of. Edgeworth, F. Y. Ph. Mg. 16 (1883) 300-. , . Venn, J. Nt. 36 (1887) 411-. , . Edgeworth, F. Y. Nt. 36 (1887) 482-. , . Backhouse, T. W. Nt. 36 (1887) 531. , . Schols, C. M. Amst. Ak. Vs. [1] (1893) 194- ; Fschr. Ps. (1893) (Ab. 1) 16-. , , in case of correlated variations. Burbury, S. H. B. A. Ep. (1895) 621-. , , compound. Edgeicorth, F. Y. Ph. Mg. 41 (1896) 207-. , , and correlated averages. Edgeworth, F. Y. Ph. Mg. 34 (1892) 429-, 518-. , , when differences of observations are large. Oudemam, J. A. C. Amst. Ak. Vs. [2] (1894) 12-. Error, law of, and elimination of chance. Edgeworth, F. Y. Ph. Mg. 21 (1886) 308-. , , empirical proof. Edgeworth, F. Y. Ph. Mg. 24 (1887) 330-. , facility of. Watson, H. W. Birm. Ph. S. P. 7 (1889-91) 289-. , for geometric mean. M'Alister, D. B. A. Ep. (1879) 267. , of, in sum of n independent quantities each accurate to nearest degree. Carpmael, C. (xn) [1882] On. E. S. P. & T. 1 (1883) (Sect. 3) 9-. , frequency. Tait, P. G. [1865] Edinb. E. S. T. 24 (1867) 139-. , . Galton,F. Ph. Mg. 49 (1875)33-. in position of a point in space. De Forest, E. L. Conn. Ac. T. 6 (*1882-85) 123-. , theory. De Forest, E. L. Des Moines Anal. 9 (1882) 33-, 65-. target shooting. De Forest, E. L. Conn. Ac. T. 7 (1885-88) 1-. Errors, accidental, Gauss' law, proof. Gosi- ewski, W. Prace Mt.-Fiz. 2 (1890) 223-; Fschr. Mth. (1890) 235. __ ( _, law. Pellerin, . As. Fr. C. E. (1898) (Pt. 2) 156-. , , points in theory. Faye, H. C. E. 106 (1888) 783-. , actual and probable, of interpolated values derived from tables by means of first differ- ences. Woodward, M. S. Des Moines Anal. 9 (1882) 143-, 169-. , determination of modulus. Edgeworth, F. Y. Ph. Mg. 21 (1886) 500-. in 2 dimensions, law of facility. De Forest, E. L. Des Moines Anal. 8 (1881) 3-, 41-, 73-. , distribution of residual, after combination. Seeliger, H. Munch. Ak. Sb. 29 (1900) 3-. , exercises in calculation. Edgeworth, F. Y. Ph. Mg. 36 (1893) 98-. , law. Le Maire, A. Brux. S. As. Bll. 1 (1896) 124-, 207-. of observation. Damien, B. C. (xn) Bll. Sc. Nord 15 (1883) 121-. , extension of law of probability. Adan, (Capit.) E. Brux. Ac. Bll. 23 (1867) 148-. , formula expressing probability. Pizzetti, P. Em. E. Ac. Line. Ed. 5 (1889) (Sem. 1) 191 -. , formulae for calculation. Maggi, G. A. Mil. I. Lomb. Ed. 15 (1882) 351-. , law. Kummell, G. H. Des Moines Anal. 3 (1876) 133-, 165- ; 6 (1879) 80-. , . Bertrand, J. C. E. 106 (1888) 153-. , . Tisserand, E. C. E. 106 (1888) 231-. , . Pizzetti, P. Em. E. Ac. Line. Ed. 1 (1892) (Sem. 1) 380-. , of facility, and method of least squares. Glaisher, J. W. L. As. S. Mm. 39 (1872) 75-. , , proof. Crofton, M. W. [1869] Phil. Trans. 160 (1870) 175-. , mean error of 2nd order. Paucker, M. G. von. Mitau Arb. Kurland. Gs. 8 (1850) 104-. 94 1630 Probabilities. Theory of Errors 1630 Errors of observations, probability of sums of powers of, etc. Helmert, F. R. Z. Mth. Ps. 21 (1876) 192-. , signs of residuals. De Forest, E. L. Des Moines Anal. 5 (1878) 65-. , , grouping. De Forest, E. L. Des Moines Anal. 5 (1878) 1-. , theory. De Morgan, A. [1861] Camb. Ph. S. T. 10 (1864) 409-. , . Lorenzcmi, G. Padova Ac. At. e Mm. 5 (1889) 203-. , . Czuber, E. Mh. Mth. Ps. 1 (1890) 457-. , . Veltmann, W. As. Nr. 131 (1893) 1-. , , theorems. Winckler, A. Wien Sb. 53 (1866) (^6. 2) 6-. , physical basis for theory. Burton, C. V. L. Ps. S. P. 10 (1890) 276-; Ph. Mg. 28 (1889) 480-. in plane and space. Schols, C. M. [1874- 86] Amst. Ak. Vh. 15 (1875) 67 pp. ; Delft EC. Pol. A. 2 (1886) 123-. , demonstration of law. Schols, C. M. Delft EC. Pol. A. 3 (1887) 195-. , relative and absolute. Gaucherel, E. N. A. Mth. 14 (1855) 145 -. } _ t theory. Jaufroid, B. N. A. Mth. 15 (1856) 154-. from single causes, composition. Kummell, C. H. As. Nr. 103 (1882) 177-. , theory. Estienne, . C. K. 130 (1900) 66-. , , application to levelling. Ocagne, M. d'. C. E. 120 (1895) 717-. , , correction formulae for series of observa- tions. De Forest, E. L. Des Moines Anal. 4 (1877) 79-, 107-. , , corrections, repeated, limit. De Forest, E. L. Des Moines Anal. 5 (1878) 129-. , , , , of terms of series. De Forest, E. L. Des Moines Anal. 5 (1878) 65-. , , , unsymmetrical, of terms of series, and their limits. De Forest, E. L. Des Moines Anal. 6 (1879) 140-, 161- ; 7 (1880) 1-. , , criticism of formula of Gauss. Czuber, E, Mh. Mth. Ps. 2 (1891) 459-. , , elementary. De Forest, E. L. Des Moines Anal. 8 (1881) 137-. , , foundations. Miiller, F. Mh. Mth. Ps. 2 (1891) 61-. , , function occurring in. Moser,C. Sch. Nf. Gs. Vh. (1898) 45-. , , and method of least squares. Goedseels, E. [1896] Brux. S. As. Bll. 1 (1896) 253- ; 2 (1896-97) 6-. , , removal of "side coefficients." Kummell, . Smiths. Misc. Col. 33 (1888) Art. 3, 41. (Wash. Ph. S. Bll. 8 (1885).) , , singular law in. Bertrand, J. C. E. 105 (1887) 779-, , , test by target shooting. Kummell, C. H. Smiths. Misc. Col. 33 (1888) Art. 1, 138-. (Wash. Ph. S. Bll. 6 (1884).) Evolution, mathematical theory. Pearson, K. [1893] Phil. Trans. (A) 185 (1895) 71-. Evolution, mathematical theory. (Skew varia- tion in homogeneous material.) Pearson, K. [1895] Phil. Trans. (A) 186 (1896) 343-. , . (Eegression, heredity, and pan- mixia.) Pearson, K. [1895] Phil. Trans. (A) 187 (1897) 253-. , . (Eeproductive selection.) Pearson, K. E. S. P. 59 (1896) 301-. , . (Telegony in man, etc.) Pearson, K. [1896] E. S. P. 60 (1897) 273-. , . (Spurious correlation.) Pearson, K. E. S. P. 60 (1897) 489-. , . ( .) Galton, F. E. S. P. 60 (1897) 498-. , . (Eelative variation and correlation in civilised and uncivilised races.) Lee, (Miss) A., & Pearson, K. E. S. P. 61 (1897) 343-. , . Pearson, K., d- Filon, L. N. G. [1897] Phil. Trans. (A) 191 (1898) 229-. , . (Speeds of American trotting horses as hereditary data.) Galton, F. [1897] E. S. P. 62 (1898) 310-. , . (Law of ancestral heredity.) Pearson, K. E. S. P. 62 (1898) 386-. , . (Inheritance of cephalic index.) Fawcett, (Miss) C. D., & Pearson, K. E. S. P. 62 (1898) 413-. , . (Eeconstruction of stature of pre- historic races.) Pearson, K. [1898] Phil. Trans. (A) 192 (1899) 169-. , . (Genetic selection.) Pearson, K. [1898] Phil. Trans. (A) 192 (1899) 257-. , . (Inheritance of fertility in man.) Pearson, K., & Lee, A. [1898] Phil. Trans. (A) 192 (1899) 279-. , . (Inheritance of fecundity in thorough- bred racehorses.) Pearson, K., & Bramley- Moore, L. [1898] Phil. Trans. (A) 192 (1899) 290-. , . (Inheritance of characters not quantitatively measurable.) Pearson, K., & Lee, A. [1899] Phil. Trans. (A) 195(1901)79-. , . (Magnitude of certain coefficients of correlation.) Pearson, K. [1899] E. S. P. 66 (1900) 23-. , . (Law of reversion.) Pearson, K. E. S. P. 66 (1900) 140-. , . (Correlation of characters not quantitatively measurable.) Pearson, K. [1900] Phil. Trans. (A) 195 (1901) 1-. , . ( .) Pearson, K. E. S. P. 66 (1900) 241-. , . (Fertility and homogamy.) Pearson, K. E. S. P. 66 (1900) 316-. , . Evolution in man. (Correlation of skull.) Lee, A., d- Pearson, K. [1900] Phil. Trans. (A) 196 (1901) 225-. , . (Homotyposis.) Pearson, K. [1900] Phil. Trans. (A) 197 (1901) 285-. Factorials, theorems deduced from probabilities. Godfray, H. Mathematician 3 (1850) 89-. Factors for minimum probable error of a mean. Caitchy, A.L. C. E. 37 (1853) 326-. Finite expansion, with least error. Buchwaldt, F. Ts. Mth. 2 (1884) 33-, 192; Fschr. Mth. 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Bll. 6 (1884).) 1630 Observation results, numerical, adjustment. Sprague, T. B. B. A. Bp. (1883) 446-. Observations, a posteriori evaluation of trust- worthy character of a series of. Bertrand, J. C. B. 106 (1888) 887-. , combination of results. Powalky, C. JR. As. S. M. Not. 34 (1874) 476-. , compensation and mean error. Paucker, M. G. von. Mitau Arb. Kurland. Gs. 6 (1849) 41-; 9 (1851) 183-. , corrections of personal error. Fechner, G. T. Leip. B. 13 (1861) 57-. , determination of accuracy. Gauss, C. F. Lindenau Z. 1 (1816) 185-. , . Hauber, G. F. Baumgartner Z. 7 (1830) 286-. , direct, calculation of result of system of. Pizzetti, P. Liege S. Sc. Mm. 15 (1888) No. 3, 32 pp. , doubtful. Edgeworth, F. Y. Ph. Mg. 23 (1887) 364-. , , criterion for rejection. Peirce, B. Gould As. J. 2 (1852) 161-. , , Peirce 's criterion. Gould, B. A. Gould As. J. 4 (1856) 81-. , , . Airy, G. B. Gould As. J. 4 (1856) 137-. , , (Airy's criticism). Winlock, J. Gould As. J. 4 (1856) 145-. , , . Peirce, B. Am. Ac. P. 13 (1878) 348-. , , rejection. Stone, E. J. As. S. M. Not. 28 (1868) 165-. , , . Glaisher, J. W. L. As. S. M. Not. 33 (1873) 391-. , , . Stone, E. J. [1873] As. S. M. Not. 34 (1874) 9-. , , . Doolittle, . Smiths. Misc. Col. 33 (1888) Art. 1, 152-. (Wash. Ph. S. Bll. 6 (1884).) , equations of condition. Grunert, J. A. Grunert Arch. 21 (1853) 453-. , functions of, accuracy. Gerling, C. L. Grunert Arch. 38 (1862) 379-. , incomplete, introduction into probability- calculations. Kiittner, W. Z. Mth. Ps. 29 (1884) 193-. , limits of accuracy. Paucker, M. G. von. Mitau Send. Kurland. Gs. 1 (1840) 73-. , to measure of precision. Adan, (Capi- taine) E. Brux. Ac. Bll. 22 (1866) 480-. , methods of detecting unknown inequalities in. Stokes, G. G. B. S. P. 29 (1879) 122-. , probable existence of a cause of error in a series. Liagre, J. Brux. Ac. Bll. 22 (1855) (pte. 2) 15-. , mean. Svanberg,J. Stockh. Ak. Hndl. (1821) 388-. , reduction. Edgeworth, F. Y. Ph. Mg. 17 (1884) 135-. relating to several quantities, new method of reducing. Edgeworth, F. Y. Ph. Mg. 25 (1888) 184-. , (Edgeworth's). Turner, H. H. Ph. Mg. 24 (1887) 466-. represented by diagrams. Earnshaw, S. Ph. Mg. 27 (1845) 127-. I G2 1630 Probabilities, etc. Observations, results, probable mean. Anon. (vi 542) Gergonne A. Mth. 12 (1821-22) 181. , , properties. Quetelet, L. A. J. Brux. Ac. Bll. 19 (1852) (pte. 2) 303-. , series of, translation into continuous for- mulae. Merrifield, C. W. [1880] L. Mth. S. P. 12 (1880-81) 4-. and statistics. Edgeworth, F. Y. [1885] Camb. Ph. S. T. 14 (1889) 138- ; Camb. Ph. S. P. 6 (1889) 101-. , theory, and method of least squares. Thiele, T. N. Bll. Sc. Mth. 14 (1890) 73-. , true average. Smith, E. H. Nt. 37 (1888) 464. , values making greatest error a minimum. Cauchy, A. L. [1814] Par. EC. Pol. J. 13 (1831) (20 e cah.) 175-. Observed points, new method of representing, suitable for explorers. Ldska, V. Prag Sb. (1895) (Mth.-Nt.) No. 41, 6 pp.; Fschr. Mth. (1895) 248. Personal equation. Gonnessiat, F. Lyon Un. A. 3 (Fasc. 2) (1892) 167 pp. Petersburg problem. Orchard, W. Assur. Mg. 2 (*1852) 185. . Czuber, (or Cuber) E. Arch. Mth. Ps. 67 (1882) 1-. . Catalan, E. C. Lie~ge S. Sc. Mm. 15 (1888) No. 1, 248-. . Wolf, E. Zur. Vjschr. 34 (1889) 264-. . Lupton, S. Nt. 41 (1890) 165-. . Seydler, . C. E. 110 (1890) 326-. , paradox analogous to. Bertrand, J. C. E. 105 (1887) 831-. Physical laws deduced from empirical curves. Pucci, E. N. Cim. 27 (1890) 220- ; 28 (1890) 111-. Plane, calculation of mean situation from more than three points. Weisbach, J. Civing. 16 (1870) 397-. Plausible value of a variable magnitude. Es- tienne, . C. E. 130 (1900) 393-. Points problem. Mansion, P. Brux. Mm. Cour. 8. 21 (1870) (No. 1) 13 pp. . Badoureau, A. Par. EC. Pol. J. 27 ( = cah. 44) (1874) 123-. . Catalan, E. C. Liege S. Sc. Mm. 12 (1885) No. 2, 65-. Position of a point determined by many measurements. Hammer, E. Z. Mth. Ps. 43 (1898) 105-. , errors. Dahlander, G. B. Stockh. Ofv. 14 (1857) 405- ; 15(1858)49-; 18(1861) 63-. in a plane, composition of law of errors for. Ocagne, d'. Par. S. Mth. Bll. 23 (1895) 65-. , probability of error in. Br avals, A . Par. Mm. Sav. Etr. 9 (1846) 255-. Pothenot's problem, theory of errors in. Andrce, . As. Nr. 47 (1358) 193-. Precision of a system of measurements. Bertrand, J. L. F. C. E. 106 (1888) , determination. Bertrand, J. C. E. 106 (1888) 440-. Probabilities 1630 Presumable error in evaluation of the angles of a triangle. Bertrand, J. C. E. 106 (1888) 967-. Principle of least action and probability theory, connexion between. Gosiewski, W. Prace Mt.-Fiz. 1 (1888) 97-; Fschr. Ps. (Ab. 1) 253-. PROBABILITIES. Breevilt, C. [1788] Mengelwerk Wisk. Vh. 1 (1796) 81-. Oettinger, L. Munch. Ab. 2 (1831-36) 199-, 243-. Poisson, S. D. C. E. 2 (1836) 395-. Mondesir, E. Liouv. J. Mth. 2 (1837) 3-. Bienayme, J. Par. S. Phlm. PV. (1840) 23-. Oettinger, L. Grunert Arch. 1 (1841) 113-. Hamilton, (Sir) W. B. B. A. Ep. (1843) (pt. 2) 3-. Oettinger, L. Crelle J. 26 (1843) 217-, 311- ; 30 (1846) 232-, 296-; 34 (1847) 153- ; 36 (1848) 221-, 296-; 42 (1852) 213-. Bordoni, A. Mil. G. I. Lomb. 4 (1852) 31-. Wolf, E. Bern Mt. (1852) 133-. Cayley, A. Ph. Mg. 6 (1853) 259. Dedekind,R. Ziir. Vjschr. 4 (1859) 346-; 5 (1860) 66-. Cayley, A. Ph. Mg. 23 (1862) 352-, 470-. Matthiessen, L. Arch. Mth. Ps. 47 (1867) 457-. Meyer, A. Liege S. Sc. Mm. 4 (1874) (No. 2) 446 pp. Eocquigny, F. [G.] de. Les Mondes 51 (1880) 652-. a priori. Edgeworth, F. Y. Ph. Mg. 18 (1884) 204-. , of errors of observation. Pizzetti, P. G. Mt. 27 (1889) 77-. accidental errors. Seeliger, H. As. Nr. 97 (1880) 289-. . Gosiewski, W. Prace Mt.-Fiz. 1 (1888) 1-; Fschr. Mth. (1888)227. , (Gosiewski). Baraniecki, M. A. Prace Mt.-Fiz. 2 (1890) 220- ; Fschr. Mth. (1890) 235-. . Eussjan, C. N. Es. S. Nt. Mm. (Mth.) 9 (1889) 134- ; Fschr. Mth. (1889) 219-. (Eussjan). Sonin, N. J. [1889] Vars. S. Nt. Tr. (1889-90) (C. E., Ps. C.) No. 4, 2-. , application to educational statistics. Maximowitsch, W. P. Fschr. Mth. (1888) 229. adjustment and computation. Lazarus, W. I. Act. J. 20 (*1378) 410-. analogy, and least squares. Donkin, W. F. QJ. Mth. 1 (1857) 152-. analysis. Binet, J. P. M. C. E. 19 (1844) 373-. application. Brian, L. Abv. Mm. S. Em. (1838-40) 251-. . Ghibaldan, I. Bucarest S. Sc. Bll. 6 (1897) 479- ; 7 (1898) 225- ; 8 (1899) 18-. to comparison of causes favouring a re- curring phenomenon. Steen, A. Kjob. Ov. (1866) 8-. 100 1630 Probabilities application to dynamical systems. Zermelo, E. Ps. Z. 1 (1900) 317-. geodesy. La Place, P. S. (Marquis) de. Con. des Temps (1820) 422-. , levelling operations, &c. La Place, P. S. (Marquis) de. A. C. 12 (1819) 337-. geodetic measurements. Hansen, P. A. As. Nr. 9 (1831) 189-, 237-. legal subjects. Jerwood, J. Devon. As. T. 2 (pt. 2) (1868) 578-. natural philosophy. La Place, P. S. (Marquis) de. Con. des Temps (1818) 361-. petrography. Cesaro, G. Brux. Ac. Mm. 53 (1895-98) (No. 1) 55 pp. , practical. Littrow, J. J. von. Baum- gartner Z. 9 (1831) 433-. to science of humanity. Quetelet, L. A. J. Brux. Ac. Bll. 36 (1873) 19-. variations in the transparency of air. Seidel, L. Miinch. Sb. 2 (1863) 320-. average deviations. Hardy, G. F. I. Act. J. 27 (1889) 214-. and averages, theory. Nekrasov, P. A. Bee. Mth. (Moscou) 21 (1901) 579-. correction of a formula. Laquiere, E. M. Par. S. Mth. Bll. 8 (1880) 74-. criticism of Laplace. Ruffini, P. Pisa N. G. 1 (1822) 201-. definitions and principles of calculus of. Gosieicski, W. Wiad. Mt. 4 (1900) 137- ; Fschr. Mth. (1900) 230. demonstration of Gauss, rigour of a. Bertrand, J. C. B. 106 (1888) 563-. drawing lots. Jouffret, E. Bv. Mar. 38 (1873) 5-, 523-, 957-. errors of calculations performed in duplicate. Glaisher, J. W. L. (vn) [1873] Mess. Mth. 3 (1874) 106- . events which seldom happen, their possible frequency being unlimited. Seidel, P. L. von. Miinch. Ak. Sb. 6 (1876) 44-. experiments. Baker, A. Cn. I. P. 5 (1888) 194-. gaming, with application to definite integrals. Mayer, D. E. Par. EC. Pol. J. 3 (1897) 153-. general theorem, elementary proof. Tchebiclieff, P. Crelle J. 33 (1846] 259-. generalisation of a problem in. Azzarelli, M. Em. N. Line. At. 45 (1892) 72-. integration of recurring series. Ocagne, M. d'. Par. S. Mth. Bll. 15 (1887) 143-. judgments. Bertrand, J. Par. S. Phlm. Mm. Cent. (1888) 69-. in courts of appeal. Guibert, A. C. B. 7 (1838) 650-. , especially in criminal cases. Poisson, S. D. C. B. 1 (1835) 473-. and evidence. Bienayme, J. Par. S. Phlm. PV. (1838) 93-. judicial errors. Ostrogradsky, M. A. St. Pet. Ac. Sc. Mm. 3 (1838) (Bll. Sc.) xix-. juries' verdicts. Coste, L. M. P. Liouv. J. Mth. 7 (1842) 169-. , proportion of condemnations. Poisson, S. D. C. B. 5 (1837) 355-, 459-. UNIVERSITY OF CALIFORNIA^ JBANTA JBAftBARA COLLEGE LIBRAE^ Probabilities: Problems 1630 legal, decisions of majorities. Guibert, A. Liouv. J. Mth. 3 (1838) 25-. losses in war. Bouniakowsky, V. [1846] St. Pet. Ac. Sc. Mm. 6 (Pt. 1) (1850) 233-. . Lafrogne, (le It.) H. Bv. Mar. et Col. 142 (1899) 275-. principle, new. Bienayme, J. Par. S. Phlm. PV. (1840) 37-. , . Catalan, E. C. Brux. Ac. Bll. 44 (1877) 463-; 8 (1884) 72-. Probabilities : Problems. Plana, G. Turin Mm. Ac. (1811-12) 355-. Rangoni, L. (xi) Mod. S. It. Mm. 18 (Mt.) (1820) 518-. Nieuport, C. F. F. de. Brux. Ac. Sc. Mm. 3 (1826) 143-. Lubbock, J. W. [1828] Camb. Ph. S. T. 3 (1830) 141-. Bernard, . Par. EC. Pol. J. 15 (cah. 24) (1835) 264-. Catalan, E. C. Liouv. J. Mth. 2 (1837) 469-; 6 (1841) 75-. Levy, A. N. A. Mth. 1 (1842) 179-. Ostrogradsky, M. A. [1846] St Pet. Ac. Sc. Bll. 6 (1848) 321-. Terquem, 0. N. A. Mth. 5 (1846) 218-, 260. Dedekind, R. Crelle J. 50 (1855) 268-. Lazarus, W. I. Act. J. 15 (* 1870) 245-. Sutton, W. I. Act. J. 15 (*1870) 452-; 16 (*1872) 151-. Lemoine, E. (x) Par. S. Mth. Bll. 1 (1873) 39-. Halphen, G. H. (x) Par. S. Mth. Bll. 1 (1873) 221-. Jordan, C. (x) Par. S. Mth. Bll. 1 (1873) 256-, 281-. Bienayme, J. Par. S. Mth. Bll. 2 (1874) 153-. Levdnen, S. Helsingf. Ofv. 17 (1875) 112-. Hoppe, R. Arch. Mth. Ps. 61 (1877) 410-. Bianco, 0. Z. G. Mt. 16 (1878) 169-. Woolhouse, W. S. B. I. Act. J. 21 (*1879) 204-. Lemoine, E. As. Fr. C. B. (1880) 183-. Cesaro, E. (xn) Mathesis 2 (1882) 177-, 232; 3 (1883) 233-. Gosiewski, W. (xn) [1882] Krk. Ak. (Mt.- Prz.) Bz. & Sp. 10 (1883) 137-, liii-. Lemoine, E. N. A. Mth. 3 (1884) 118- ; As. Fr. C. B. (1885) (Pt. 2) 50-. . Mg. 22 Weill, . Par. S. Mth. Bll. 14 (1886) 158-. Edgeworth, F. Y. Ph. 22 (1886) 371-. Cesaro, E. Palermo Cir. Mt. Ed. 1 (1887) 299-. Mayer, E. C. B. 108 (1889) 391-. Pdnek, A. Casopis 20 (1891) 105- ; Fschr. Mth. (1891) 232. Makeham, W. M. I. Act. J. 29 (1892) 475-. Lemoine, E. As. Fr. C. B. (1895) (Pt. 1) 182. Lindelof, L. Helsingf. Ofv. 42 (1900) 79-. Hermite, C. Helsingf. Ofv. 42 (1900) 88-. analytical. Schlomilch, 0. Grunert Arch. 9 (1847) 372-. Bernoulli, etc. Malfatti, G. F. Verona S. It. Mm. 1 (1782) 768-. 101 1630 Probabilities: Problems Bernoulli and Lagrange. Abbati, P. (viAdds.) Mod. Mm. S. It. 19 (1821) 385-. Bertrand, generalisation. Barbier, E. C. E. 105 (1887) 407. , solution. Andr6, D. C. E. 105 (1887) 436-. cards. Cantor, M. Schlomilch Z. 2 (1857) 410-. class. Sylvester, J. J. B. A. Ep. 35 (1865) (Sect.) 8-. collimator for adjustment of azimuth circle. Airy, G. B. B. A. Ep. (1850) (pt. 2) 1-. competitive examinations. Edgeworth, F. Y. Ph. Mg. 30 (1890) 171-. conditions limiting solution. Boole, (?. Ph. Mg. 8 (1854) 91-. election. Korteweg, D. J. N. Arch. Wisk. 1 (*1875) 157- ; 2 (*1876) 40-. evidence. Doolittle, M. H. Wash. Ph. S. Bll. 11 (1892) 602. game of pool. Arcais, F. d'. Padova Ac. At. e Mm. 16 (1900) 219-. graphic solution. Favaro, A. Yen. I. At. 3 (1877) 1025-. Laurent. Pdnek, A. Casopis 20 (1891) 94-; Fschr. Mth. (1891) 231. note. Laquiere, E. M. Par. S. Mth. Bll. 8 (1880) 79-. and relative values of pieces in game of chess. Lemoine, E. As. Fr. C. E. (1880) 179-. solution. Plana, G. Mod. S. It. Mm. 18 (1818) 31-. . Brashmann, N. D. (xn) Mosc. Un. Mm. 10 (1835) 523-. . Poisson, S. D. Liouv. J. Mth. 2 (1837) 373-. . Hansen, P. A. As. Nr. 16 (1839) 9-, 27-. . Meyer, A. Brux. Ac. Bll. 15 (1848) (pte. 2) 508. . Roger, E. Liouv. J. Mth. 17 (1852) 202-. . Blazejarski, E. N. A. Mth. 1 (1862) 131-. . Liagre, J. B. J. Brux. Ac. Bll. 17 (1864) 365-. . Terrier, (Prof.) . Neuch. Bll. 9 (1873) . Hendricks, J. E. Des Moines Anal 3 (1876) 152-. . Spitzer, S. D. Nf. B. (*1877) 97-. . Piuma, C. M. G. Mt. 17 (1879) 360-. . Spitzer, S. Arch. Mth. Ps. 64 (1879) 74-. . Bertrand, J. C. E. 105 (1887) 369, 437-. , application of. Poreckij, P. S. Kazan S. Nt. (Ps-Mth.) P. 5 (1887) 83- ; Fschr. Mth. (1887) 200. , geometrical. Lemoine, E. [18821 Par S. Mth. Bll. 11 (1883) 13-. solved by Poisson, indeterminateness. Bertrand J. C. E. 106 (1888) 636-. and theorems. Catalan, E. Brux. Ac. Mm. 46 (1886) (No. 2) 16 pp. two. Andrade, . C. E. 130 (1900) 395-. Probabilities, etc. 1630 repeated trials. Bertrand, J. C. E. 94 (1882) 185- ; 105(1887) 1201-. subjective, mathematical and objective. Vassilief, A. [1900] N. Y. Am. Mth. S. Bll. 7 (1901) 158, 183. the surprise ratio. Bostwick, A. E. Science 3 (1896) 66-. theorem. Luchterhandt, A . E. Grunert Arch. 2 (1842) 65-. . Dienger, J. Prag Ab. 5 (1872) 44 pp. . De Forest, E. L. Des Moines Anal. 7 (1880) 169-. . Bertrand, J. C. E. 114 (1892) 701-. of Bienayme, elementary proof. Bertrand, J. C. E. 81 (1875) 458, 491-. , demonstration. Wythoff, W. A, N. Ts. Mth. 4 (B) (1893) 77-. , new. Simmons, (Rev.) T. C. L. Mth. S. P. 26 (1895) 290-. , , application. Bienayme, J. C. E. 81 (1875) 417-. theorems. Franklin, F. (xn) [1881] J. H. Un.^Cir. [1] (1882) 128. . Cebysev, P. St Pet. Ac. Sc. Mm. (Rs.) 55 (1887) (Suppl.) No. 6, 16 pp. ; Acta Mth. 14 (1890-91) 305-. "Theorie Analytique" (Laplace). Glaisher, J. W. L. [1872] Mess. Mth. 2 (1873) 62-. -- (-). Laurent, H. N. A. Mth. 12 (1873) 176-. -- ( ). Tychsen, C. Kjob. Ov. (1876) 12-. theory. Encontre, D. Mntp. Eec. Bll. 1 (1803) 9-. . Bellavitis, G. Poligrafo 4 (1836) 3-. . De Morgan, A. [1837] Camb. Ph. S. T. 6 (1838) 423-. . Ellis, E. L. Camb. Mth. J. 4 (1845) 127-. . Boole, G. Camb. and Dubl. Mth. J. 6 (1851) 286. . Donkin, W. F. Ph. Mg. 1 (1851) 353-, 458-; 2 (1851) 55-. . Boole, G. Ph. Mg. 2 (1851) 96- ; 7 (1854) 29-; 9 (1855) 165-. . Bellavitis, G. Yen. At. (1856-57) 299-. . Newcomb, S. Am. Ac. P. 4 (1857-60) 433-; Camb. (M.) Mth. M. 1 (1859) 136-, 233-, 331- ; 2 (1860) 134-, 272-; 3 (1861) 119-, 343-. . Boole, G. Phil. Trans. (1862) 225-. . Plrogov, N. N. Es. Ps.-C. S. J. 19 (Ps.) (1887) 19-. . Lindsay, T. Pop. As. 4 (1897) 240-. , and application to gunnery. Sabudski, N. Fschr. Mth. (1898) 197-. and applications. Czuber, E. D. Mth. Vr. Jbr. 7 (1899) (Heft 2) viii + 279 pp. evidence. De Morgan, A. [1846-50] Camb. Ph. S. T. 8 (1849) 379-; 9 (1856) 79-. Ellis, E. L. [1842] Camb. Boole, G. Ph. Mg. 8 question treated by d'Alembert. Maupin, G. Par. S. Mth. Bll. 23 (1895) 185-. Delannoy, . Par. S. Mth. Bll. 23 (1895) 262-. , foundations. Ph. S. T. 8 (1 , general me (1854) 431-. , geometrical, interpretation of 0. Simmons, (Rev.) T. C. As. Fr. C. E. (1893) (Pt. 1) 175. , , problems. Vivanti, G. Ev. Mt. 1 (1891) 69-. 102 1630 Probabilities, Theory of Errors 1630 theory and practice. Wolf, R. Bern Mt. (1849) 97-, 183- ; (1850) 85-, 209- ; (1851) 17- ; (1853) 25-. , principle. Young, J. R. B. A. Ep. (1844) (ft. 2) 1-. , proposition. Liapounoff, A. St Pet. Ac. Sc. Bll. 13 (1900) 359-. . Protection by mimicry. Blakiston, T., & 'Alexander, T. Nt. 29 (1884) 405-. , two multiple integrals in. Czuber, E. Mh. Mth. Ps. 2 (1891) 119-. Probability of being hit by a projectile. Keller, 0. A. Mines 19 (1891) 622-. and certainty, nature and limits. Miller, G. [1793] Ir. Ac. T. 5, 199-. of composite events. Simmons, (Rev.) T. C. As. Fr. C. E. (1896) (Pt. 2) 1-. . Delannoy, H. Par. S. Mth. Bll. 26 (1898) 64-. curve, asymmetrical. De Forest, E. L. Des Moines Anal. 9 (1882) 135-, 161- ; 10 (1883) 1-, 67-. , . Edgeworth, F. Y. Ph. Mg. 41 (1896) 90-. of error of observation. Schlomilch, 0. Z. Mth. Ps. 17 (1872) 87-. in result of a certain computation. Franklin, F. (xn) [1881] J. H. Un. Cir. [1] (1882) 165. writing a series of numbers. Nipher, F. E. Am. J. Sc. 12 (1876) 79-. errors arising from uncertainty of last Z" re. Glaisher, J. W. . (vn) [1873] s. Mth. 3 (1874) 59-. , elementary proofs of laws. Laquiere, E. M. Par. S. Mth. Bll. 9 (1881) 69-. , law. Petitcol, (Capt.) . N. A. Mth. 4 (1885) 441- ; Ev. Mar. et Col. 91 (1886) 70-. .,.--, Ocagne, M. d\ Par. S. Phlm. Bll. 6 (1894) (C. .R. No. 9) 2-. , . Gosieicski, W. Prace Mt.-Fiz. 9 (1898) 117- ; Fschr. Mth. (1898) 186-. of observations. Cauchy, A. L. C. E. 37 (1853) 264-. position. Reina, V. Em. E. Ac. Line. Ed. 6 (1897) (Sent. 1) 107-. in sum or mean of observations. Jullien, (le pere) M. Tortolini A. 1 (1858) 76-, 149-, 227-. , exact and popular estimates, comparison. Terrot, C. H. Edinb. E. S. P. 2 (1851) 228-. of finding isolated shoals. Littlehales, G. W. A. Mth. 9 (1894-95) 163-. , formulae. Jordan, C. C. E. 65 (1867) 993-. of hitting a target. Merriman, M. [1897] N. Y. Am. Mth. S. Bll. 4 (1898) 88, 91-. , local. Crofton, M. W. [1867] L. Mth. S. P. 2 (1869) 54-. , , applied to straight lines drawn at random in a plane, and to proofs of theorems in integral calculus. Crofton, M. W. Phil. Trans. 158 (1868) 181-. , , proof of theorem. Heutnann, C. N. Ts. Mth. 4 (B) (1893) 25-; Fschr. Mth. (1893-94) 343. Probability, local, question in. Berg, F. J. van den. N. Arch. Wisk. 9 (*1882) 32-; 10 (1884) 202. , , theorem of Crofton. Cavallin, C. B. S. Ts. Mth. 2 (1884) 1-; Fschr. Mth. (1884) 181. that marriage of man over forty will be fruitful. Sprague, T. B. [1879] Edinb. E. S. P. 10 (1880) 202-. of means. Poisson, S. D. Con. des Temps (1827) 273- ; (1832) 3-. . Lobatscheivsky, N. Crelle J. 24 (1842) 164-. . Lobatto, R. Arch. Wisk. Gn. 2 (1860-63) 96-. , Laplace's rule. Bienayme, J. Par. Mm. Sav. Etr. 5 (1838) 513- ; Par. S. Phlm. PV. (1839) 42-; (1840) 19-. , Poisson generalised. Hauber, C. F. Baumgartner Z. 7 (1830) 406-. that n-gon can be formed from n parts of straight line. Berg, F. J. van den. N. Arch. Wisk. 18 (1891) 63-; Fschr. Mth. (1891) 234. , physical basis. Edgeworth, F. Y. Ph. Mg. 16 (1883) 433-. of system of errors, law of. Gosiewski, W. Prace Mt.-Fiz. 3 (1892) 33-; Fschr. Mth. (1892) 206- . that two numbers are relatively prime. Cesdro,E.- (xn) J. H. Un. Cir. [2] (1883) 85. Probable error, comparison of two values. LUroth, J. As. Nr. 87 (1876) 209-. of constants. Hagen, G. H. L. Berl. Ak. Sb. (1883) 1169-. , determination. Lilroth, J. As. Nr. 73 (1869) 187-. , . Jordan, W. As. Nr. 74 (1869) 209-. , (Jordan). Andres, C. G. G. As. Nr. 74 (1869) 283-. , from differences of m equally exact observations. Andrts, C. C. G. As. Nr. 79 (1872) 257-. , experimental determination of constant. Adcock, R. J. Des Moines Anal. 7 (1880) 189. , Peter's formula. Helmert, F. R. As. Nr. 88 (1876) 113-. , . Johnson, W. W. N. Y. Mth. S. Bll. 2 (1893) 57-. of a series of observations. Guyou, E. C. E. 106 (1888) 1282-. , calculation. Mees, R. A . Z. Mth. Ps. 20 (1875) 145- ; 21 (1876) 126-. from sum of variations. Fechner, G. T. A. Ps. C. Jubelbd. (1874) 66-. errors, limit, in observations. Cesaris, G. A. de. Mil. Effem. As. (1811) 102-. inference. Terrot, C. A. Edinb. E. S. P. 2 (1851) 288-. intersection of non-concurrent lines, geo- metrical determination of point of. Ocagne, M. d\ Par. EC. Pol. J. 63 (1893) 1-. value of least errors in a series of observa- tions. Bertrand, J. C. E. 106 (1888) 786-. 103 1630 Probabilities, etc. Probable value of a series of observations, mechanical representation. Newcomb, S. As. S. M. Not. 33 (1873) 573. values of constants in u=a+by. Anon. 's Gravenh. I. Ing. Ts. (1884-85) (Verg.) 68-. Random scattering of points on a surface. Kleiber, J. Ph. Mg. 24 (1887) 439-. Beckoning of chances (reproduction of 16th century work by Benedict Spinoza). Bier ens de Haan, D. N. Arch. Wisk. 11 (1884) 75-. Eeduction of observation equations in more than one observed quantity. Kummell, C. H. Des Moines Anal. 6 (1879) 97-. results of experiments. Lupton, S. Ph. Mg. 31 (1891) 418-. , criticism. Lupton's method. Pickering, S. U. Ph. Mg. 32 (1891) 90-. , objections raised to Pickering's method. Hayes, E. H. Ph. Mg. 32 (1891) 99-. Eesolution of equations given by method of least squares in a particular case. Port, A. C. E. 105 (1887) 491-. Eestricted number of observations, interpreta- tion. Vallier, E. C. K. 128 (1899) 654-. , . Hatt, . G. K. 128 (1899) 893-. Eesultant error, law. Schols, C. M. Delft EC. Pol. A. 3 (1887) 140-. Boots of the equation e** -^ (R-&) - 0. Markoff, A. St Pet. Ac. Sc. Bll. 9 (1898) 435-. Scoring at rifle matches. Mallock, A. Nt. 60 (1899) 366-. . Burbury, S. H. Nt. 60 (1899) 412. . Mallock, A. Nt. 60 (1899) 412. Speculation, theory. Bachelier, L. Par. EC. Norm. A. 17 (1900) 21-. Stationary motion in an infinite system of molecules. Burbury, S. H. L. Mth. S. P. 29 (1898) 225-. Stock-exchange, probabilities connected with. Dormoy, E. As. Fr. C. E. (1886) (Pt. 1) 79-; (1887) (Pt. 2) 214-. Success of prediction, measure of. Peirce, C. S. Science 4 (1884) 453-. Sum of least variations, estimation and utility. Fechner, G. T. [1874] Leip. Mth. Ps. Ab. 11 (1878) 1-. Surveying, possible error in determination of a point. Soulages, E. Arg. S. Ci. A. 44 (1897) 306-. with usual instruments, judging error in. Weisbach, J. Baumgartner Z. 4 (1837) 24-. Tables of numbers, error due to omitted decimal places. Frischauf, J. Z. Mth. Ps. 16 (1871) 178-. Target, firing at. Bertrand, J. C. B. 107 (1888) 205-. questions, probability applied to. Bertrand, J. C. B. 106 (1888) 232-, 387-, 521-. , . Jung, G. C. E. 106 (1888) 1001-. Testimonies or judgments, combination. Boole, G. Edinb. E. S. T. 21 (1857) 597-. Actuarial Mathematics 1635 Testimony and arguments. Abbott, (Rev.) T. K. Ph. Mg. 28 (1864) 12-. for facts contrary to analogy. Young, M. [1798] Ir. Ac. T. 7 (1800) 79-. , formulae for value. Bonicelli,P. (viAdds.) Majocchi A. Fis. C. 12 (1843) 167-. in legal cases, measure of force. Tozer, J. [1843] Camb. Ph. S. T. 8 (1849) 143-. Todhunter's History of Theory of Probability, passages. Bianco, O. Z. [1877] G. Mt. 16 (1878) 26-. Tradition and probability. Macfarlane, A. I. Act. J. 22 (*1881) 356-. Transit observation, frequency of occurrence of each digit in lOths of seconds in chrono- graph observations. Eastman, J. R. [1873] Wash. Ph. S. Bll. 1 (1874) 85-. Triangulation, interpolation of a point by method of least squares. Jordan, W. Z. Mth. Ps. 16 (1871) 164-. Trigonometric measures, mean error of results. Borsch, (Dr) [A.] Arch. Mth. Ps. 46 (1866) 40-. problem, surveying. Gauss, C. F. As. Nr. 1 (1823) 81-. surveys, combination of errors in connexion with. Kriiger, L. Gott. Nr. (1900) 1-. Variability of organs, and Gauss's law of devia- tion. Ludwig, F. Z. Mth. Ps. 43 (1898) 230-. Verification of predictions. Doolittle, M. H. Smiths. Misc. Col. 33 (1888) Art. 2, 122-. (Wash. Ph. S. Bll. 7 (1885).) Weighing of errors, theorems. Paucker, M. G. von. Mitau Arb. Kurland. Gs. 9 (1851) 141-. Weight of observations where mean of squares of errors is composed of several parts. Hel- mert, F. R. As. Nr. 89 (1877) 225-, 241-. and precision of an observation. Bertrand, J. C. E. 105 (1887) 1099-. Weighted causes, and Bessel's method. Paucker, M. G. Mitau Arb. Kurland. Gs. 9 (1851) 170-. 1635 Theory of statistics. Actuarial mathematics. Accident assurance, and connected questions. Tiselius, H. Stockh. Ofv. (1895) 561-. Accumulations of capital in United Kingdom, in 1875-85. Gi/en, R. Nt. 41 (1890) 211-. Actuarial science, application of mathematics. Stuckey, J. J. Aust. As. Ep. (1893) 280-. Adjustment of mortality tables. Lazarus, W. I. Act. J. 16 (* 1872) 424-. . Woolhouse, W. S. B. I. Act. J. 21 (* 1879) 37-, 56-. . Higham, J. A. I. Act. J. 23 (*1882) 335-; 24 (1884) 44-. , Corneille and Landre's method. Danielewicz, B. Wiad. Mt. 1 (1897) 92-. , Gompertz's method. Gray, P. Assur. Mg. 7 ('1858)121-. , graphic method. Sprague, T. B. I. Act. J. 26 (1887) 77-. 104 Theory of Statistics. Actuarial Mathematics 1635 Adjustment of mortality tables, graphic me- thod. Woolhouse, W. S. B. I. Act. J. 26 (1887) 420-. , method. M. Assur. Mg. 6 (*1857) 357-. , . Woolhouse, W. S. B. I. Act. J. 29 (1892) 236-. , new method. Woolhouse, W. S. B. I. Act. J. 15 (*1870) 389-. , and weight of observations. McCay, C. F. I. Act. J. 22 (*1881) 24-. tables, Galloway's method. Watkins, C. Assur. Mg. 6 (*1857) 178, 360. , Woolhouse's formula. Higham, C. D. I. Act. J. 26 (1887) 50-. Age of parents as influencing sex and number of children. Brown, S. Assur. Mg. 3 (* 1853) 17-. Ages at which life assurance is effected. Chandler, S. C. (jun.). I. Act. J. 17 (*1873) 56-. Algebra of commercial exchanges. Cossali, P. [1812] Mod. Mm. S. It. 16 (1813) 124-. Allowance during sickness, tables. Higham, J. A. Assur. Mg. 7 (*1858) 112-. American life, law of. Nichols, W. S. I. Act. J. 19 (* 1876) 28-. Amount assured, and sum reserved, connection between. Jellicoe, C. Assur. Mg. 5 (* 1855) 100-. of 1 for fraction of a year. Farren, E. J. Assur. Mg. 3 (*1853) 335-. . Chisholm, D. Assur. Mg. 3 (* 1853) 336-. . Filipowski, H. Assur. Mg. 3 (*1853) 338-; 4 (1854) 243-, 253. and present value of regularly increasing annuity, formulae. Shaw, C. G. Assur. Mg. 5 (*1855) 152. or decreasing annuities, formulae. Galsworthy, E. H. Assur. Mg. 5 (*1855) 53-. Annual premium for life assurance. Gray, P. Assur. Mg. 2 (* 1852) 95-. term assurance on two joint lives. Sprague, T. B. Assur. Mg. 8 (*1860) 59-. and single payments, table connecting. Smith, H. A. Assur. Mg. 11 (*1864) 176-. premiums for endowment assurance. Smith, H.A. I. Act. J. 22 (1881) 143-. . McDonald, J. I. Act. J. 32 (1896) 447-. Annuities. Horner, W, G. Thomson A. Ph. 8 (1816) 279-, 388-. . Huet, . N. A. Mth. 5 (1846) 346-. . Nievengloski, G. H. N. A. Mth. 6 (1847) 8-. . -Dupain, J. C. N. A. Mth. 2 (1863) 464-. . Guldberg, A. S. (xn) Sk. Nt. Mot. F. (1880) 194-. . Bryan, G. H. Mth. Gz. 1 (1900) 8-. . Loney, . Mth. Gz. 1 (1900) 38-. , approximate calculation. Lobatto, R. [1861] Amst. Vh. Ak. 10 (1864) 32 pp. , . Bmmiakowsky, V. [1877] St Pet. Ac. Sc. Bll. 24 (1878) 437-. 1635 Annuities, assurance, etc., approximate calcu- lation. Stamkart, F. J. Amst. Vs. Ak. 1 (1866) (Ntk.) 95-. and assurances, improved theory. Wool- house, W. S. B. I. Act. J. 15 (1870) 95-; 18 (*1875) 151-. , (Woolhouse's). Button, W. I. Act. J. 15 (*1870) 307-. , Milne's treatise, correction of error. Hoskins, H. I. Act. J. 17 (*1873) 192-. 3. We. Mg. 36 (1850) 8-. on successive lives. Weddle, T. Ph. on assured lives, value. Berridge, G. W. I. Act. J. 19 (*1876) 351-. , calculation. Lubbock, J. W. [1828] Camb. Ph. S. T. 3 (1830) 141-. __, _. Wrede, F. J. [1869] Stockh. Ofv. 26 (1870) 5-. , . Fagnart, E. Mathesis 16 (1896) 64-. under certain conditions, values. Hardy, P. Assur. Mg. [1 (* 1851)] 1 bis-. , certain determining amounts of continuous. Makeham, W. M. I. Act. J. 15 (*1870) 432-. Annuities-certain, rate of interest. Hardy, G. F. I. Act. J. 23 (*1882) 266-. , theory. Makeham, W. M. I. Act. J. 14 (1869) 189-. , value. Orchard, W. Assur. Mg. [1 (*1851)] 100 big-. , formula. Hansen, C. N. Ts. Mth. 11 (A) (1900) 65-. , formulae in D. Jones's work. Macfadyen, J. E. Assur. Mg. 13 (* 1867) 200. , . McCuaig, C. Assur. Mg. 13 (*1867) 251-. involving three lives, approximations. Hardy, G. F. I. Act. J. 23 (*1882) 274-. and life-assurance. Tychsen, C. (xn) Ts. Mth. 5 (1875) 49-, 97-. assurances and premiums, calculation of worth. Hultman, F. W. Ts. Mt. Fys. 2 (1869) 63-, 113-, 209-. expectations. Landre, C. L. N. Arch. Wisk. 12 (1886) 174- ; Fschr. Mth. (1885) 194-. on lives. Willich, C. M. Assur. Mg. 7 (*1858) 273-. on three joint lives, approximate values. Hardy, P. Assur. Mg. 3 (*1853) 330-. , four, five and six joint lives, tables. Shaw, C. G. Assur. Mg. 5 (*1855) 180-. on several lives. Quiquet, A. C. B. Ill (1890) 337-. for long terms, approximations. Brown, S. Assur. Mg. [1(* 1851)] 20-. , mean annual extinction in chief Portuguese offices. Silva, D. A. da. Lisb. J. Sc. Mth. 1 (1868) 175-. payable half yearly or quarterly. Sprague, T. B. Assur. Mg. 13 (*1867) 188-, 201-, 305-. m times a year. Hansen, C. N. Ts. Mth. 11 (B) (1900) 58-. , question. Lindelof, L. L. Acta Mth. 3 (1883) 97-. and rate of interest, theorems for deter- mining. Benwell, J. B. Tilloch Ph. Mg. 49 (1817) 179-. 105 1635 Theory of Statistics. Actuarial Mathematics 1635 Annuities, tables, comparison. Lubbock, J. W. [1829] Camb. Ph. S. T. 3 (1830) 321-. , of present values. Bowditch, J. I. Am. Ac. Mm. 1 (1333) 73-. , theorems for determining. Benwell, J. B. [1816-17] Thomson A. Ph. 8 (1816) 119- ; Tilloch Ph. Mg. 48 (1816) 449-; 50 (1817) 164- ; 51 (1818) 6-. , theory. Landre, C. L. N. Arch. Wisk. 15 (1888) 1-; Fschr. Mth. (1888) 235-. Annuity, analytic formula and table. Benwell, J. B. Tilloch Ph. Mg. 57 (1821) 161-. and assurance formulas, reduction. Wool- house, W. S. B. I. Act. J. 17 (*1873) 171-. _ tables for three lives. Braid, W. Assur. Mg. 6 (*1857) 109, 115-. certain, approximation to rate of interest. McKenzie, D. J. McG. I. Act. J. 23 (*1882) 405-. and life annuity, analogy between. King, G. I. Act. J. 20 (*1878) 435-. , . Ryan, G. H. I. Act. J. 25 (1886) 443-. computation on Makeham's hypothesis. McClintock, E. I. Act. J. 18 (*1875) 242-. forborn, determination. Hardy, P. Assur. Mg. 7 (*1858) 1-. , Lubbock's formula for approximation. Sprague, T. B. I. Act. J. 18 (*1875) 305-. , market value. Wilding, R. I. Act. J. 22 (*1881) 61-. problem, and Arbogast's development. De Morgan, A. Assur. Mg. 12 (* 1866) 206-. , old and new solutions. Gray, P. Assur. Mg. 13 (* 1867) 60. question. C., J. I. Act. J. 14 (*1869) at one rate deduced from that at another. Meikle, J. Assur. Mg. 3 ("1853) 325-. , rate of interest. Meikle, J. Assur. Mg. 5 (*1855) 152-. , . De Morgan, A. Assur. Mg. 8 (*1860) 61-. , . McLauchlan, J. J. I. Act. J. 18 (*1875) 290-. tables, method of checking. De Morgan, A. Assur. Mg. 2 (*1852) 390-. on three lives, application of Gompertz's law. Woolhouse, W. S. B. Assur. Mg. 10 (*1863) 121-. , note on Woolhouse's paper. De Morgan, A. Assur. Mg. 10 (*1863) 237-. -, rule for value. De Morgan, A. (vi Adds.) ' Ph. Mg. 15 (1839) 337-. value with final payment. Cherriman, J. B. I. Act. J. 21 (* 1879) 299-. Apportionable annuities. Sprague, T. B. Assur. Mg. 13 (*1867) 358-; 14 (*1869) 36-; 15 (*1870) 244. Apportionment, problem in. Sprague, T. B. I. Act. J. 19 (*1876) 372-. Arbitrary rates of life premium. S., H. A. Assur. Mg. 6 (*1857) 297-. Association of electors by lot. Bertrand, J. C. B. 106 (1888) 17-. Assurance on adventure so as to cover interest. Mercator. [Pseudon.] (vi Adds.) Bost. Mm. Am. Ac. 1 (1785) 183-. Assurance, analytical theory. Fourier, J. B. J. A. C. 10 (1819) 177-. and annuities, formulas. Enestrom, G. Stockh. Ofv. (1894) 561- ; Fschr. Mth. (1893-94) 364-. in another's favour. Dienger, J. Arch. Mth. Ps. 42 (1864) 333-. of capital and income. Unferdinger, F. Grunert Arch. 26 (1856) 408-. in stock. Danielewicz, B. Wiad. Mt. 2 (1898) 32-. companies, amalgamation. Jellicoe, C. Assur. Mg. 7 (*1858) 254-. , . Sprague, T. B. Assur. Mg. 7 (*1858) 355-. , equitable transference of business between. Sprague, T. B. Assur. Mg. 7 (*1868) 301-. company, testing solvency. Younger, S. Assur. Mg. 7 (*1858) 183-, 295-. , . S.,H. A. Assur. Mg. 7 (*1858) 294, 353-. , comparison of actual with expected ex- periences. Low, G. M. I. Act. J. 18 (*1875) 195-. on continent, progress. Brown, S. Assur. Mg. [1 (*1851)] 293-; 2 (* 1852) 16-. , demonstration of certain formulas. Adler, M. N. Assur. Mg. 12 (1866) 52-. , a fallacy concerning. Gray, P. I. Act. J. 14 (*1869) 63-. of invalid lives. Younger, S. Assur. Mg. 11 (*1864) 49-. , Younger's plan. Brabrook, E. W. Assur. Mg. 10 (*1863) 349-. , . Smith, H. A. Assur. Mg. 10 (*1863) 352-; 11 (* 1864) 180. , . Gray, P. Assur. Mg. 10 (*1863) 354-. against invalidity. Lazarus, W. I. Act. J. 15 (*1870) 143-. and old age, German method. Young, T. E. I. Act. J. 29 (1892) 269-. issue. Tucker, R. Assur. Mg. 5 (*1855) 350-. , premiums. Sprague, T. B. I. Act. J. 28 (1890) 350-. , rate of premium. Day, A. Assur. Mg. 8 (*1860) 127-. , life against life, value. Bailey, A. H. Assur. Mg. 9 (M861) 299. , , and for n years longer. Tucker, R. Assur. Mg. 4 (*1854) 250-. , , t years longer. CJiatham, J. I. Act. J. 25 (1886) 439-. loan (kind of life assurance). Vesz, J. A. (xn) Mag. Tud. Ak. Etk. (Mth.) [1] (No. 3) (1868) 20 pp. payable at instant of death. Younger, S. Assur. Mg. 7 (*1858) 238. pensions, theory. Enestrom, G. Stockh. Ofv. (1895) 197-, 243-; Fschr. Mth. (1895) 249-, 250-. premiums on ill health. Danielewicz, B. Wiad. Mt. 1 (1897) 170-. of recently selected lives, premiums. Sprague, T.B. I. Act. J. 20 (* 1878) 95-. 106 Theory of Statistics. Actuarial Mathematics 1635 Assurance against sickness. Moser, C. Arch. Sc. Ps. Nt. 6 (1898) 364-. , some new forms. Fix, . Ev. Sc. 3 (1895) 171- , state-loans, &c. Oettinger, L. Grunert Arch. 36 (1861) 189-, 453-; 37 (1861) 125-, 365-; 38 (1862) 263-. of survivor, formula for premium. Lobatto, R. [1865] Arch. Neerl. 1 (1866) 46-. to survivor, new method of calculating. Reboul, E. Assur. Mg. 9 (*1861) 1-. of unsound lives, practice concerning. Humphreys, G. I. Act. J. 18 (*1875) 178-. for widows, theory. John, G. Jb. Berg- Hw. (1888) 42-. Auxiliary life contingency tables, history. Hendriks, F. Assur. Mg. [1 (*1851)] 1-, 12 Ms-. Average duration of life. Merrifield, C. W. Assur. Mg. 6 (*1857) 175-. , calculation for Prussia. Dieterici, C. F. W. Berl. Ab. (1858) (Ph.) 433-. life office, determination of an. King, G. I. Act. J. 20 (*1878) 300-. mortality rates, groupings according to age. Gray, W. T. I. Act. J. 25 (1886) 369-. premium, theory. Harvey, C. J. I. Act. J. 24 (1884) 296-. prices, official ("mercuriali"). Pareto, E. (x) Em. E. Ac. Line. At. 26 (1873) 599-; 1 (1875) 3-. Averages, and application to rate of exchange. Drobisch, M. W. Leip. B. 23 (1871) 25-. , general. Morrison, R. Assur. Mg. 12 (*1866) 350- ; 13 (* 1867) 39-, 161-. and life contingencies, use of integral cal- culus. Younger, S. Assur. Mg. 7 (*1858) 71-. , percentage of. Preece, W. H. (xi) Tel. E. J. 2 (1873) 293-. Bank-notes, mathematical theory of. Walras, L. [1879] Laus. S. Vd. Bll. 16 (1880) 553-. Banking, mathematical theory. Edgeworth, F. Y. B. A. Ep. (1886) 777-. Barrett's method and questions of nomen- clature. De Morgan, A. Assur. Mg. 10 (*1863) 301-. Benefits, calculation of certain. Sprague, T. B. I. Act. J. 24 (1884) 327-. , complex, calculation. King, G. I. Act. J. 26 (1887) 276-. paid several times a year and complete annuities. King, G. I. Act. J. 24 (1884) 418-. , present or future, formula. Macfadyen, J. R. I. Act. J. 17 (*1873) 381-. Births and deaths at Brussels, laws. Quetelet, L. A. J. Brux. Ac. Sc. Mm. 3 (1826) 495-. in Hanau. Moller, (Dr) von. (vi Adds.) Wet. Gs. Jbr. (1861-63) 59-. , Stuttgart 1847. Cless, G. Wiirtb. Jh. 4 (1849) 120-. , deaths and marriages in Prussia, 1820- 1834. Hoffmann, J. G. Berl. Ab. (1841) (Th. 3) 1-. of different sexes, effect of legitimacy on ratio. Prevost, P. Bb. Un. 42 (1829) 139-. 1635 Births, proportion of girls and boys. Poisson, S. D. Par. Mm. Ac. Sc. 9 (1830) 239-. , statistics (Versailles). Berigny, A. Par. An. M6t. (1852) 281-. in various states of Europe and particularly Prussia. Dieterici, C. F. W. Berl. Ab. (1855) (Ph.) 321-. Bond bought at premium, rate of interest. Todhunter, R. I. Act. J. 33 (1898) 356-. Bonus distribution, system. Rothery, H. J. I. Act. J. 30 (1893) 131-. , theory. Andras, H. W. I. Act. J. 32 (1896) 320-. Bonuses as affecting annuities. Macfadyen, J. R. I. Act. J. 14 (*1869) 364-. Building debts, discharge by capital or rent. Wittstein, T. Hann. Z. Archt.-Vr. 7 (1861) 107-. Calculation in economics. Durini, G. N. [1834] Nap. At. I. Inc. 6 (1840) 1-. Cape of Good Hope Mutual Life Assurance Company, report. Sprague, T. B. I. Act. J. 20 (*1878) 358-. Carlisle and other tables, relations. Henry, J. Assur. Mg. 11 (* 1864) 89-, t . (Criticism of Henry's paper.) Farr, W. Assur. Mg. 11 (*1864) 109-. table of mortality, Milne's method in con- struction. Sutton, W. I. Act. J. 24 (1884) 110-, 186-. Census. Lubbock, (Sir) J. W. (vi Adds.) Ph. Mg. 7 (1830) 416-. from 1801-1881, account of. Burridge, A. F. I. Act. J. 25 (1886) 83-. , principles of statistics applied. Hough, F. B. Am. As. P. 17 (1868) 145-. theorem, applications. Listing, J. B. Gott. Nr. (1867) 430-. Centenarians in France (census of 1886). Levasseur, E. C. E. 107 (1888) 71-. Chances, doctrine, and subdivision of risks. Young, Dr T. QJ. Sc. 22 (1827) 84-. Changes of population of Great Britain of various classes, 1811-1821. Harvey, G. QJ. Sc. 16 (1823) 203-. Chisholm's policy tables, method of using. McKenzie, D. J. McG. I. Act. J. 28 (1890) 247-. tables, advantages. B., W. F. Assur. Mg. 8 (*1860) 110-. Civil life, mathematics relating to. Michelsen, J. A. C. (vi Adds.) Berl. Sm. D. Ab. 3 (1799) (Mth.) 36-. Claim acceleration reserve. Higham, C. D. I. Act. J. 24 (1884) 76; 26 (1887) 478. Classification of policies to obtain a table of mortality. Brown, S. Assur. Mg. 4 (* 1854) 282-. ' Coinage ; introduction of decimal system. Farey, J. Tilloch Ph. Mg. 49 (1817) 15-. and money of account, subsidiary principle applied. Elliott, E. B. Am. As. P. 24 (1875) 87-. , ternary denominational system, prerogative. Sylvester, J. J. (xii) [1881] J. H. Un. Cir. [1] (1882) 132. Coins, nomenclature, &c. Costaz, . [1831] Par. Mm. Ac. Sc. 12 (1833) 599-. 107 1635 Theory of Statistics. Actuarial Mathematics Combined marriage and mortality table. Sprague, T. B. I. Act. J. 21 (*1879) 406-. Commutation, or D and N, tables. Chisholm, J. I. Act. J. 14 (1869) 20Q-. formulae, certain. Smith, H. A. Assur. Mg. 8 (* 1860) 116- ; I. Act. J. 19 ( 1876) 143. system, use of difference columns. Laundy, S.L. Assur. Mg. 8 ( 1860) 168-. tables of D. Chisholm. Meikle, J. Assur. Mg. 7 (* 1858) 297-. Complete annuities, formulas. Sheppard, H. N. I. Act. J. 32 (1896) 458-. annuity, formulae for purchase. Sorley, J. I. Act. J. 20 (* 1878) 454-. , market value. Baden, A. I. Act. J. 17 (*1873) 447-. , . Carr, T. I. Act. J. 18 (* 1875) 224-. payable by instalments, approximate value. Hoskins, H. I. Act. J. 19 (1876) 143-. m equal instalments each year, value. Carr, T. I. Act. J. 18 (*1875) 247-. Composition of decremental forces, theory. Makeham, W. M. I. Act. J. 18 (*1875) 317-. for leave to reside abroad. Campbell, E. Assur. Mg. 9 (*1861) 167-. Compound survivorship annuity. Hardy, G. F. I. Act. J. 34 (1899) 93-. problem, solution. Makeham, W. M. Assur. Mg. 10 (*1863) 241-. Constancy of number of marriages and moral statistics. Quetelet, L. A. J. Brux. Ac. Bll. 5 (1858) 89-. Constructive total loss, doctrine. Hopkins, M. Assur. Mg. [1 ( 1851)] 63-. Consumption in relation to life assurance. Lyon, T. G. I. Act. J. 30 (1893) 120- ; 31 (1895) 155. Contingencies, distinct, problems involving. MakeMm, W.'M. Assur. Mg. 13 (*1867) 109-. Contingent assurances, calculation. Colenso, F. E. I. Act. J. 32 (1896) 185-. with certain limitations. Otter, W. C. Assur. Mg. 7 (*1858) 239-. , premiums. Sunderland, A. W. I. Act. J. 26 (1887) 58-. , problem. Lidstone, G. J. I. Act. J. 35 (1901) 480-. parts of terminable annuity. Gray, P. Assur. Mg. 11 (*1864) 172-. reversionary interests. Sang, E. Assur. Mg. [1 (*1851)] 18W-, 151-, 287-. . Jellicoe, C. Assur. Mg. 2 (*1852) . Tucker, JR. Assur. Mg. 5 (*1855) 162, 239. reversions. Paterson, W. B. I. Act. J. 30 (1893) 536-. Copyhold, enfranchisement, tables. Smyth, E, I. Act. J. 21 (*1879) 381-. fine enfranchisement. Mathews, G. S. I. Act. J. 28 (1890) 1-. 1635 Copyhold, fines payable on renewal, formulae. Ryley, E. Assur. Mg. 4 (* 1854) 367-. Corn-law sliding scale, expression for discon- tinuous law. Smith, Arch. Camb. Mth. J. 3 (1841) 152. Criminals, decay of morality in relation to number of youthful. Hoffmann, J. G. Berl. B. (1838) 1-. Cumulative vote. Langley, E. M. Mth. Gz. 1 (1900) 160. Curves representing statistics. Edgeworth, F. Y. B. A. Ep. (1899) 825. D, N columns of a certain company. Morgan, W. Assur. Mg. 12 (*1866) 235-. and N formula for term assurance. Smith, H.A. I. Act. J. 19 (*1876) 143. Dacca census. Walters, H. As. Kesearches 17 (1832) 534-. Date of death and payment, period between. Farren, E. J. Assur. Mg. 3 (*1853) 234-. Death strain in a life office, computation. Ryan, G. H. I. Act. J. 30 (1893) 196-. , true measure. Higham, C. D. I. Act. J. 20 ( 1878) 153-. Deaths, excess over births in coloured popula- tion of Boston. Boudin, J. C. M. F. J. C. B. 61 (1865) 746-. Debt, gradual discharge. Drobisch, M. W. Leip. B. 2 (1848) 1-. Decimal systems of money etc. Grubb, T. [1855] Dubl. E. S. J. 1 (1856-57) 21-. Decrement of human life, formula. Young, Dr T. Phil. Trans. (1826) (pt. 3) 281-. Deductions from sums assured, calculation. Hardy, G. F. I. Act. J. 32 (1896) 153-. Deferred annuities, tables. Stephenson, J. W. Assur. Mg. 12 (*1866) 178- ; 13 (1867) assurances and premiums. Rothery, H. J. I. Act. J. 28 (1890) 483-. and other life annuities, approximation. Ivory, H. Assur. Mg. 4 (*1854) 29 1-. De Moivre's formula. Gray, P. Assur. Mg. 12 (*1866) 232-. , method of obtaining. Charlon, . I. Act. J. 15 (*1870) 141-. Deterioration, estimating increase of rate to meet. Smith, H. A. Assur. Mg. 10 (*1863) 120. Determination of present value of mine on the Eand. Hellmann, F. I. Mn. Metall. T. 6 (1898) 229-. Development and health, estimation from re- ports of recruiting officers. Bischoff, T. L. W. Munch. Sb. (1867) (I.) 282-. Dice for statistical experiments. Galton, F. Nt. 42 (1890) 13-. Differential tariffs. Courau, J. Arg. S. Ci. A. 50 (1900) 86-. Direct taxation, true measure of liability. Jellicoe, C. Assur. Mg. 3 (*1853) 1-. Discounted bonus premiums, rationale. Moir, H. I. Act. J. 36 (1902) 4-. Discounting of bills, mathematical theory. Necaev, N. V. Kazan S. Nt. (Ps.-Mth.) P. 3 (1885) 95-. , . Preobralenskij, V. V. Kazan S. Nt. (Ps.-Mth.) P. 3 (1885) 97-. 108 1635 Theory of Statistics. Actuarial Mathematics 1635 Diseased lives, valuation reserve necessary. Sorley, J. I. Act. J. 20 (*1378) 342-. Dispositions of code of criminal instruction. Gergonne, J. D. Gergonne A. Mth. 9 (1818- 19) 306-. Distribution by months of conceptions and births. Villerme,L. A. Sc. Nt. 24 (1831) 130-. , pure theory of. Berry, A. B. A. Ep. (1890) 923-. Doctrine of successive lives. Gray, P. Assur. Mg. 2 (*1852) 1-, 271-. Doctrines of political economy, mathematical exposition. Whewell, W. [1829-31] Camb. Ph. S. T. 3 (1830) 191- ; 4 (1833) 155-. population. Farr, W. B. A. Ep. (1877) (Sect.) 174-. Drummond's rule for calculating relative im- portance of boroughs, attempt to prove. M'Intyre, (Dr). (vi Adds.) Ph. Mg. 11 (1832) 360-. Drunkenness in towns of England and Wales. Poynting, J. H. Manch. Lt. Ph. S. P. 16 (1377) 211-. Economic science and statistics section of British Association, address. Duff, M. E. G. B. A. Ep. 37 (1867) (Sect.) 132-. , . Brown, S, B. A. Ep. 38 (1868) (Sect.) 144-. , . Northcote, (Sir) S. B. A. Ep. 39 (1869) (Sect.) 173-. , . Jevons, W. S. B. A. Ep. 40 (1870) (Sect.) 178-. , . Neaves, (Lord). B. A. Ep. 41 (1871) (Sect.) 191-. , . Fawcett, (Prof.) H. B. A. Ep. 42 (1872) (Sect.) 217-. , . Campbell, (Sir) G. B. A. Ep. (1876) (Sect.) 186-. utility of means of transport, especially railways. Perozzo, L. Em. E. Ac. Line. Mm. 1 (1894) 476-. Election methods. Nanson, E. J. [1882] Viet. E. S. T. 19 (1883) 197-. problem, to secure proportional representa- tion of parties. Phragmen, E. Stockh. Ofv. (1894) 133-. Elections, counting votes at general. Cauchy, A. L. C. E. 26 (1848) 469-. , multiple, theory. Phragmen, E. Stockh. Ofv. (1396) 181-. and representative system. Gergonne, J. D. Gergonne A. Mth. 6 (1315-16) 1-; 10 (1819- 20) 281-. Electoral districts, problem. Thiele, T. N. N. Ts. Mth. 1 (B) (1890) 56-; Fschr. Mth. (1890) 232-. , system; and the minority. Lorenz, L. N. Ts. Mth. 1 (B) (1890) 40-; Fschr. Mth. (1890) 232-. Endowment assurance, demonstration of for- mula. Sprague, T. B. Assur. Mg. 8 (*1360) 111-. , easy method of calculating reserves. Weninger, V. (xn) Mag. Ak. Ets. (Mth. Term.) 6 (1365) 241-. , formula. M., A. Assur. Mg. [1 (*1851)] Endowment assurance policies. Sunderland, A.W. I. Act. J. 28 (1890) 257-. payable at ages 50, 55, 60, 65, tables. Garment, I). I. Act. J. 23 (*1882) 224-. , periodical valuation. Thomson, H. A I. Act. J. 34 (1899) 1-. , tables. Besant, A. D. I. Act. J. 31 (1895) 366-. assurances, grouping for valuation. Alten- burger, J. I. Act. J. 34 (1899) 150- ; 35 (1901) 332-. on two joint lives, approximate calcula- tion. Lidstone, G. J. I. Act. J. 33 (1898) 354-. , valuation in groups. Lidstone, G. J. I. Act. J. 34 (1899) 61-, 510-. policies some time in force, valuation. Samot, D. J. A. I. Act. J. 20 (*1878) 344-. value, deducing annuity and assurance value. Laundy,S.L. Assur. Mg. 11 (* 1864) Entailed estates, problem concerning. Sprague, T. B. I. Act. J. 19 (*1876) 38-. Equal partition in inheritance, effect of age difference. Mayette, J. Macon Ac. A. 6 (1388) 360-. Equated time of payment. Levine, A. I. Act. J. 33 (1898) 539-. Equation of payments. Walton, W. Camb. Mth. J. 2 (1841) 188-. . Schulze, L. E. Grunert Arch. 36 (1861) 177-. . (Malcolm's rule.) Benwell, J. B. Tilloch Ph. Mg. 49 (1817) 269-. Equitable Society's experience. Young, Dr T. Ph. Mg. 4 (1828) 339-. Estate duty. Lame, G. N. A. Mth. 7 (1848) 360-. , mathematical law of progression. Lame, G. C. E. 27 (1848) 125-. , . Dienger, J. Grunert Arch. 12 (1849) 401-. Exchange, mathematical theory. Walras, L. [1873] Laus. S. Vd. Bll. 12 (1874) 317-. value of money, mathematical theory. Vilfredo-Pareto, . As. Fr. C. E. (1894) (Pt. 2) 1180-. Expectation and estate. Westergaard, H. (xn) Ts. Mth. 6 (1876) 11-. Expected deaths and expected claims, estima- tion. Ryan, G. H. I. Act. J. 26 (1887) 249-. Expenses of life assurance companies, measure. Macfadyen, J. E. I. Act. J. 19 (*1876) 153-, 445-. , . Sprague, T. B. I. Act. J. 19 (*1876) 167-, 305-, 447-. "Exposed to risk," new formula for deducing. King, G. I. Act. J. 27 (1889) 218-. Extra mortality. White, F. W., & Whittall, W. J. H. I. Act. J. 24 (1884) 385-. premium. Macfadyen, J. E. I. Act. J. 17 (*1873) 77-. , conditional payments. Younger, S. Assur. Mg. 10 (*1863) 268-. for foreign climates. Meikle, J. I. Act. J. 19 (*1376) 268-. 109 1635 Theory of Statistics. Actuarial Mathematics 1635 Extra premiums on deteriorated lives, dispens- ing with. Hakeham, W. M. I. Act. J. 17 (*1873) 153-. risk, data for fixing premiums. Jellieoe, C. Assur. Mg. 7 (*1868) 131-. risks, adjustment. Makeham, W. M, I. Act. J. 14 (*1869) 159-, 242-. , considered in relation to hypothetical table of mortality. Ryan, G. H. I. Act. J. 24 (1884) 19-. owing to hereditary phthisis. Manly, H. W. I. Act. J. 30 (1893) 97-, 338-. . Macaulay, T. B. I. Act. J. 30 (1893) 335-. Facts, mode of tabulating, to show risks. Meikle, J. I. Act. J. 31 (1895) 229-. Family annuities. King, G. I. Act. J. 30 (1893) 291-. Farr's healthy life table. Davis, W. Assur. Mg. 10 (*1863) 59-. Fatal accidents in various occupations. Whittall, W. J. H. I. Act. J. 23 (*1882) 188-. Fecundity in Europe at beginning of nineteenth century. Benoiston de Chdteauneuf, . Par. Mm. Sav. Etr. 3 (1832) 608-. of marriages in interior of Cuba. La Sagra, E. de. C. E. 58 (1864) 524-. Female compared with male lives. Walford, C. I. Act. J. 19 (*1876) 174-. Females of British peerage, adjusted mortality table. Thiele, T. M. I. Act. J. 16 (*1872) 43-, 118- . Finite differences calculus, and its applications. Otter, W. C. Assur. Mg. 7 (*1858) 333-; 8 (*1860) 19-. Finlaison and English life table. Bailey, A. H. Assur. Mg. 9 (*1861) 357-. . Porter, H. W. Assur. Mg. 10 (*1863) 31-. Finlaison 's report on mortality of government life annuitants. Porter, H. W. Assur. Mg. 9 (*1861) 277-. tables. Smith, H. A. Assur. Mg. 13 (*1867) 58-. Fire assurance. Barrois, T. Lille Mm. S. (1834) 85-. , apportionment of loss in. Miller, T. Assur. Mg. 6 (*1857) 202-. , divisions of risk in. Miller, T. Assur. Mg. 8 (*1860) 232-. in Great Britain. Brown, S. Assur. Mg. 7 (*1858) 259-, 341- ; 8 (*1860) 27-. , scientific application of data. Walford, C. I. Act. J. 21 (*1879) 1-. , specific and average. Miller, T. Assur. Mg. 8 (*1860) 140-. statistics. Miller, T. Assur. Mg. 6 (*1857) 333-; I. Act. J. 22 (*1881) 103-. , settlement under average policies. Atkins, R. Assur. Mg. 3 (*1853) 147-. . . Lazarus, W. Assur. Mg. 4 (*1854) 73-. . . Atkins, R. Assur. Mg. 7 (*1858) 24-; 8 ('I860) 1-. , specific and average policies. Christie, D. Assur. Mg. 8 (*1860) 146-. Fires in London from 1833 to 1849. Broicn, S. Assur. Mg. [1 (*1851)] 31 bis. 1856, causes. Fothergill, C. G. Assur. Mg. 7 (*1858) 91-. First and subsequent marriage in peerage. Day, A. Assur. Mg. 10 (*1863) 181-. Force of mortality, formula. De Forest, E. L. I. Act. J. 22 (*1881) 231-. , general expression. Sheppard, H. N. I. Act. J. 32 (1896) 295. , . Lidstone, G. J. I. Act. J. 32 (1896) 390-. , Makeham's formula. Calderon, H. P. I. Act. J. 35 (1901) 157-. Foreign loans, rate of interest on, theory. Sutton, W. I. Act. J. 19 (*1876) 77-. Formula a x = vp (1 + a,,.^) and other matters. Gray, P. Assur. Mg. 12 (*1866) 176-. Foundlings, historical and statistical notice. Lepasquier, . Eouen Tr. Ac. (1830) 73-. French law, succession of illegitimate children. Cournot, A. A. (vi Adds.) Ferussac Bll. Sc. Mth. 16 (1831) 3-. , . Quillet, . N. A. Mth. 4 (1845) 249-. Friendly societies, application of actuarial principles, &G. Hardy, G. F. I. Act. J. 27 (1889) 245-. , experiences. Hardy, R. P. I. Act. J. 31 (1895) 86-. , present position. Brown, S. Assur. Mg. 11 (*1864) 333-. , sickness and mortality experienced in. Tompkins, H. Assur. Mg. 5 (*1855) 6-. , tabulation of results. Watson, A. W. I. Act. J. 35 (1901) 268-. society levies. King, G. I. Act. J. 26 (1887) 389-. . Hardy, G. F. I. Act. J. 26 (1887) 478-. Fruitful marriages, probability. Sprague, T. B. I. Act. J. 25 (1886) 160-. , relative age of husband and wife. Chatham, J. I. Act. J. 27 (1889) 37-. Gain or loss from mortality in annuity company. Ryan, G. H. I. Act. J. 30 (1893) 189-. Geographic statistics, correction of anterior elements. Luca, F. de. Nap. Ed. 5 (1866) 183-. Gold and silver, consumption in Britain 1810 to 1829. Jacob, W. Edinb. N. Ph. J. 12 (1832) 346-. Gompertz's formula, and application to duration of life. Simon, C. Bll. Sc. Mth. As. 2 (1871) 282-. law. De Morgan, A. Assur. Mg. 8 (*1860) 181- ; 9 (*1861) 214-. . Sprague, T. B. Assur. Mg. 9 (*1861) 288-. . Govan, J. I. Act. J. 34 (1899) 147-. , consideration of alleged inaccuracies. Sprague, T. B. Assur. Mg. 10 (*1863) 32-. , exposition, new. Edmonds, T. R. Assur. Mg. 9 (*1861) 327-. , further development Makeham, W. M. I. Act. J. 28 (1890) 152-, 185-, 316-, 416. , Makeham's extension. Woolhouse, W. S. B. I. Act. J. 28 (1890) 481-. 110 1635 Theory of Statistics. Actuarial Mathematics 1635 Gompertz's law, Makeham's modification, application. Colenso, F. E. I. Act. J. 31 (1895) 337-. , value. Edmonds, T. R. Assur. Mg. 10 (*1863) 104-. Gotha life assurance, mortality experience. Hopf, G. Assur. Mg. 5 (*1855) 324-; 6 (*1857) 1-. Government annuity and assurance rates. Adler, M. N. Assur. Mg. 12 (*1866) 265-. life annuities. Adler, M. N. Assur. Mg. 12 (*1866) 3-. Graduated mortality table from limited experi- ence. Chandler, S. C. (jun.). I. Act. J. 17 (*1873) 161-. , graphic method. Lidstone, G. J. I. Act. J. 30 (1893) 212-. Graduation. Higham, J. A. I. Act. J. 31 (1895) 319-. of experience, Jellicoe's formula. Suttcm, W. I. Act. J. 18 (*1875) 375-. formulae. Higham, J. A. I. Act. J. 27 (1889) 427-. . Hardy, G. F. I. Act. J. 32 (1896) 371-. , Higham's theory. Levine, A. I. Act. J. 32 (1896) 290-. of mortality table, application of Makeham's method. Sorley, J. I. Act. J. 20 (*1878) 340-. , methods. Button, W. I. Act. J. 20 (*1878) 170-, 192-. tables. Sorley, J. I. Act. J. 22 (*1881) 309-. . Ackland, T. G. I. Act. J. 23 (*1882) 352-. . Higham, J. A. I. Act. J. 25 (1886) 15-, 245-. , Makeham's law. King, G., & Hardy, G. F. I. Act. J. 22 (*1881) 191-. , . Sorley, J. I. Act. J. 22 (*1881) 230-. , Woolhouse's method, Hardy's plan for computation. Garment, D. I. Act. J. 24 (1884) 224-. peerage mortality. Berridge, G. W. Assur. Mg. 12 (*1866) 220-. by summation. Todhunter, E. I. Act. J. 32 (1896) 378-. , Woolhouse's method. Sprague, T. B. I. Act. J. 29 (1892) 59-, 232-. Grant of policies without further payments (after certain time). Sprague, T. B. Assur. Mg. 7 (*1858) 58-. Great powers (of Europe). Farr, W. Assur. Mg. 6 (*1857) 147-. Growth of children. Boas, F. Science 19 (1892) 256-, 281-. , human, mathematical theory. Lehmann, J. W. H. D. Nf. Vsm. B. (1843) 299-. Height and weight of man, relation between. Macaulay, T. B. I. Act. J. 23 (*1882) 62-. Human will and chances, comparison. Guy, W. A. Assur. Mg. 5 (*1855) 315-. probability theory. Adan, H. I. Act. J. 33 (1898) 523-, 527-. . Young, T. E. I. Act. J. 33 524-, 529-. Human will, uniform action. Lazarus, Assur. Mg. 3 (*1853) 169-. -- , -- . Hopf, G. Assur. Mg. 3 (*1853) Immediate annuities, series for expressing. Merrifield, C. W. Assur. Mg. 3 (*1853) 324-. Importation of animal, vegetable and mineral substances into the Clyde, 1853-1857. Connal, M., & Keddie, . B. A. Ep. 28 (1858) 185-. --- and vegetable substances (Liverpool, 1851-55). Archer, T. C. B. A. Ep. (1857) Incomplete years of exposure, treatment. Lidstone, G. J. I. Act. J. 31 (1895) 304-. Increasing annuities and assurances, approxi- mate calculation. Lidstone, G. J. I. Act. J. 31 (1895) 68-. and decreasing premium scales. Laundy, S. L. Assur. Mg. 10 (*1863) 286-. reversionary charges. Paterson, W. B. I. Act. J. 35 (1901) 245-. Index, best means of constructing. Curtis, F. A. Assur. Mg. 8 (*1860) 54-. numbers, use in investigation of trade statistics. Bourne, S. B. A. Ep. (1885) 859-. Industrial assurances, observations. Eea, C. H. E. I. Act. J. 34 (1899) 105-. Inordinate mortality, how far controllable by human agency. Porter, H. W. Assur. Mg. 9 (*1861) 12-, 89-, 149-. Insanity in Scotland. Mitchell, (Sir) A. I. Act. J. 28 (1890) 426-, 432-. Insolvency in life assurance companies. Bailey, A. H. I. Act. J. 16 (*1872) 389-. Insolvent companies, liquidation and recon- struction. Sprague, T. B. I. Act. J. 16 (*1872) 229-. Integral for compound survivorship annuity, approximate evaluation. Todhunter, R. I. Act. J. 33 (1898) 311-. Intellectual and moral statistics of France. Fayet, . C. E. 11 (1840) 565-. Interest, abridged multiplication applied to. Burnier, F. [1872] Laus. Bll. S. Vd. 11 (1873) 362-. accounts. De Morgan, A. Assur. Mg. 10 (*1863) 281-. -- . T.,A.H. Assur. Mg. 10 (*1863) 357-. and annuities, demonstration of formulae. De Morgan, . Assur. Mg. 4 (*1854) 277-. , calculation, method. Perkins, G. R. Silliman J. 47 (1844) 51-. , capitalising in course of year. Radell, . Grunert Arch. 2 (1842) 68-. , compound. Adams, Jas. Thomson A. Ph. 4 (1822) 450-. , . Lobatto, R. Quetelet Cor. Mth. 6 (1830) 195-. , . Brassinne, E. N. A. Mth. 9 (1850) 403-. , , and annuities : to find p from relation Z. Mth. Ps. 22 (1877) 190-. Ill 1635 Theory of Statistics. Actuarial Mathematics 1635 Interest, compound, average amount of sum at. Smith, H. A. I. Act. J. 14 (*1869) 158-. , , calculated by trigonometrical functions. Vervaet, P. Casopis 13 (1884) 264-; Fschr. Mth. (1884) 211. , , continuously reckoned. Studnicka, F. J. Casopis 2 (*1873) 85-; Fschr. Mth. (*1873) 609. , , and current accounts. Starkov, A. N. Es. S. Nt. Mm. (Mth.) 3 (*1881). , , curve. Carpenter, F. D. Y. V. Nost. Eng. Mg. 25 (1881) 385-. , , formulae. Scherling, C. Grunert Arch. 2 (1842) 213-. , , infinitesimal. Ferriot, . N. A. Mth. 2 (1843) 461. , , . Dutoit, A. L. Laus. Bll. S. Vd. 5 (1857) 169-. , , new formula for calculating. Catalan, E. C. As. Fr. C. E. 1 (1872) 141-. , , payable "momently." Young, (Dr) T. Ph. Mg. 2 (1827) 332-. _, _, powers, &c. Berdelle, C. As. Fr. C. E. 8 (1879) 170-. , , a result of. Bienaymg, J. Par. S. Phlm. PV. (1839) 60-. t and simple, equivalence. De Morgan, A. Assur. Mg. [1 (*1861)] 335-. , , solution of problems, without tables. Adler, M. N, I. Act. J. 23 (*1882) 359-. , , theory. Borgogelli, M. Em. N. Line. At. 47 (1894) 68-. , , time required to attain given amount. Studnicka, F. J. Casopis 13 (1884) 238-; Fschr. Mth. (1884) 210-. , equation of payment. Schlechter, . Grunert Arch. 34 (I860) 291-. , , extension of term, company's accounts. Schlechter, . Schlomilch Z. 5 (1860) 214-. , fall in rate. Burn, J. I. Act. J. 34 (1899) 474-. , laws of. Montel, E. de. C. E. 124 (1897) 224-. payment, theory. Orlov, A. Kazan S. Nt. (Ps.-Mth.) P. 4 (1886) 250-. , question of. Orchard, W. Assur. Mg. 4 (*1854) 61-. , . Gompertz, B. Assur. Mg. 4 (*1854) 245-. , . I. Assur. Mg. 4 (*1854) 253-. rate, approximations in calculating reduc- tion. Stewart, J. I. Act. J. 32 (1896) 261-. in certain investments, approximation. M. Assur. Mg. 6 (*1857) 54-. , short periods. Kelland, P. Edinb. E. S. P. 3 (1857) 274-. , simple or compound. Wittstein, T. Grunert Arch. 40 (1863) 240-. , . Oettinger, L. Grunert Arch. 40 (1863) 243-. at 6 per cent., method of computing. Philomath. [Pseudon.] (vi Adds.) Bost. Mm. Am. Ac. 1 (1785) 174-. , theoretical and practical. Crelle, A. L. [1852] Crelle J. 49 (1855) 349-. Interpolation from irregular series. De Forest, E. L. Smiths. Ep. (1871) 275-; (1873) 319-. Invalid lives, practice of certain assurance companies. Pinckard, G. H. Assur. Mg. [1 (*1861)] 273-. Invalidity probabilities, formulas. Samot, D. J. A. I. Act. J. 21 (*1879) 288-. Investments of funds of assurance companies. Brown, S. Assur. Mg. 7 (*1858) 241- . Isolated reversions. Shaw, C. G. Assur. Mg. 2 (*1852) 295-. Issue of (hitherto) childless marriage, pro- bability. Sprague, T. B. I. Act. J. 22 (*1881) 117-. Joint life annuity tables, method of finding D numbers. Henry, J. I. Act. J. 14 (*1869) 212-. lives' assurance, table. Braid, W. Assur. Mg. 5 (*1855) 350, 363-. Land, price of, and purchase by the State, mathematical theory. Walras, L. [1880] Laus. S. Vd. Bll. 17 (1881) 189-. Landed property, theory of yield of soil applied to valuation. Pasini, V. Yen. At. 4 (1845) 388-. Last r survivors of m lives, value of annuity. King, (?. I. Act. J. 22 (*1881) 293-. survivor of three lives, annuity found by actuaries' tables. Smyth, E. I. Act. J. 17 (*1873) 379-. , value of annuity. Godicard, W. I. Act. J. 17 (*1873) 266-. Law of large numbers. Poisson, S. D. C. E. 2 (1836) 377-. (Quetelet's) in intellectual mani- festations of man. Bitnicz, L. (xn) Mag. Ak. Ets. (1851) 241-. Leaseholds, renewals. Bunyon, C. J. Assur. Mg. 3 (*1853) 280-. Leases; perpetual renewal. Adams, Jas. Thomson A. Ph. 2 (1821) 307- ; 3 (1822) 12-. , Sir Isaac Newton's table. De Morgan, A. Assur. Mg. 9 (*1861) 185-. , value. B., W. (vi Adds.) Thomson A. Ph. 10 (1825) 335-. " Leipzieger Krankencasse." Lazarus, W. Assur. Mg. 8 (*1860) 351-. Liabilities of assurance companies, estimate. Tucker, B. Assur. Mg. 10 (*1863) 312-. , . Bailey, A. H. Assur. Mg. 11 (*1864) 111-. , , Bailey's. Sprague, T. B. Assur. Mg. 12 (*1866) 113-. . , valuation. Sprague, T. B. Assur. Mg. 11 (*1864) 90-. connected with buildings, estimation of value, with view to settlement by sum of money or annual payment. Eminich, W. Z. Bauw. 1 (1851) 195-. Liability of assurance company, estimation. Sprague, T. B. I. Act. J. 15 (*1870) 411-. Life assurance. Littrow, J. J. von. Baum- gartner Z. 9 (1831) 257-. . Sang, E. Edinb. T. Sc. S. Arts 1 (1841) 339-, 349-, 360-, 394-. 112 1635 Theory of Statistics. Actuarial Mathematics 1635 Life assurance. Rieu, A. [1872] Laus. Bll. S. Vd. 11 (1873) 269-. . Landre, C. L. N. Arch. Wisk. 12 (1886) 182- ; Fschr. Mth. (1885) 194-. , applications suggested by 1851 census. Scott, G. Assur. Mg. 6 (*1857) 47-. , average risk. Kanner, M. I. Act. J. 14 (*1869) 439-. calculations, modern methods of com- putation. Laundy, S. L. Assur. Mg. 8 (*1860) 58-. , . "Milne, J." Assur. Mg. 8 (*1860) 118-. companies in Austria. Daninos, S. A. Assur. Mg. 3 (*1853) 121-. Germany. Hopf, G. Assur. Mg. 3 (*1853) 134-, 220- ; 9 (*1861) 42-. , solvency. Macfadyen, J. R. [1870] Glasg. Ph. S. P. 7 (1871) 221-. compositions, notation for use in. Lazarus, W. Assur. Mg. 12 (*1866) 48-. contracts, reservations. Paraira, M. C. N. Arch. Wisk. 16 (1889) 215- ; Fschr. Mth. (1889) 229. , defects in system (1831). Fraser, W. Edinb. N. Ph. J. 11 (1831) 118-. , formulae connected with. Rubenson, R. Stockh. Ofv. (1891) 463-. in France. Curtis, F. A. I. Act. J. 19 (*1876) 414-. fund, progress of profit. Searle, T. J. I. Act. J. 30 (1893) 493-. , Hillman's tables, errors in. Laundy, S. L. Assur. Mg. 9 (*1861) 239-. , mathematical theory. Samot, D. J. A. N. Arch. Wisk. 5 (*1879) 35-; 6 (*1880) 124-. , , use of mortality statistics. Landre, C. L. N. Arch. Wisk. 10 (1884) 18-. , mean gain or loss for whole duration of assurance. Dienger, J. [1877] Epm. Mth. 2 (1879) 89-. , premium, basis of determination. Weninger, V. (xn) [1861] Mag. Ak. Ets. (Mth. Term.) 2 (1861-62) 193-. , reserve. Gobbi, U. Mil. I. Lomb. Ed. 29 (1896) 612-. premiums, calculation. Stamkart, F. J. N. Arch. Wisk. 2 (*1876) 97-. , renunciation of payment. Rieu, A. Laus. Bll. S. Vd. 9 (1866-68) 361-. ; present worth, and annual payment. Stamkart, F. J. Arch. Wisk. Gn. 2 (1860- 63) 314-. , theory. Malmsten, C. J. Crelle J. Mth. 83 (1877) 245-; Acta Mth. 1 (1882) 63-. assurances, mean error in. Gram, J. P. Ts. Mth. 6 (1889) 97-, 184- ; Fsch. Mth. (1888) 232-. contingencies, old and new methods for finding. Gray, P. Assur. Mg. [1 (*1851)] 96 bis-. , problem. Gray, P. I. Act. J. 21 (*1879) 67-. , valuation. Gompertz, B. Phil. Trans. (1820) 214-. Life contingencies, valuation. Hargreave, C. J. Ph. Mg. 5 (1853) 39-. contingency calculation, improvement. Farren, E. J. Assur. Mg. 5 (*1855) 185- ; 8 ("I860) 121-. , solution of problems concerning. F., E. J. Assur. Mg. [1 (*1851)] 92 bis, 355-. expectation. Willich, C. M. Assur. Mg. 7 (*1858) 181-. . Biden, W. D. Assur. Mg. 7 (*1858) 352-. . Willich, C. A. M. Assur. Mg. 8 (*1860) 139-. , meaning of phrase. Sprague, T. B. Assur. Mg. 13 (*1867) 381-. expectations. Landre, C. L, N. Arch. Wisk. 15 (1888) 22-; Fschr. Mth. (1888) 237-. , formulae connected with. Geer, P. van. N. Arch. Wisk. 12 (1886) 60- ; Fschr. Mth. (1885) 194. experience in 1863, remarks. Tilt, R. R. I. Act. J. 32 (1896) 1-. interests and reversions, formula and tables. Nightingale, H. E. I. Act. J. 30 (1893) 9-. office, estimation of liability. Garment, D. I. Act. J. 21 (*1879) 74-. , tabulation of facts in records. Sprague, T. B. I. Act. J. 31 (1895) 205-. valuations, principles. McCandlish, J. M. I. Act. J. 20 (*1878) 12-. , probability and mean duration, Geneva 1761-1813. Odier, L. Bb. Brit. 55 (1814) 213-. risks, assessment. Chisholm, J. I. Act. J. 25 (1886) 408-. tables, construction. Farr, W. Phil. Trans. (1859) 837-. , Gray's method of construction. Hannyngton, J. C. I. Act. J. 21 (*1879) 455-; 22 (*1881) 136-. of Institute of Actuaries. Pell, M. B. I. Act. J. 21 (*1879) 137-. . Sprague, T. B. I. Act. J. 21 (*1879) 452-. , Philadelphia. Chase, P. E. [1869] Am. Ph. S. P. 11 (1871) 17-. tenant and reversioner, apportionment between. Baden, A. I. Act. J. 16 (*1872) 269-. , . Sprague, T. B. I. Act. J. 18 (* 1875) 77-. Limitation of risks. Sprague, T. B. Assur. Mg. 13 (*1867) 20-. Limited and contingent interests in property. Bunyon, C. J. I. Act. J. 18 (*1875) 1-. Lives, and joint lives, values. Young, (Dr) T. Tilloch Ph. Mg. 47 (1816) 3-. Loading premiums. Makeham, W. M. I. Act. J. 15 (* 1870) 354-. . Smith, H. A. I. Act. J. 20 (*1878) 145-. . Crisford, G. S. I. Act. J. 25 (1886) 184-, 387-. Loans, arithmetical theory. Ronchetti, F. G. Mt. 24 (1886) 242-. , railway. Hultman, F. W. Ts. Mt. Fys. 3 (1870) 278-. 113 1635 Theory of Statistics. Actuarial Mathematics Morrison, R. Assur. Mg. 11 Loans repayable by instalments. Makeham, W. M. I. Act. J. 18 (*1875) 132-. , interest rate. Gray, P. I. Act. J. 14 (*1869) 91-, 182-, 397-. Longevity and expectation of life in United States. Worcester, J. E. [1825] Bost. Mm. Am. Ac. 1 (1833) 1-. , human. Ponzi, G. Tortolini A. 7 (1856) 417-. , increase in France, 1770 to 1845. Dupin, G. C. E. 26 (1848) 585-, 613-. of inhabitants of Ipswich and Hingham. Wiggles-worth, E. [1782] Bost. Mm. Am. Ac. 1 (1785) 565-. Lubbock's formula of summation. Sprague, T. B. I. Act. J. 22 (*1881) 55-. Machinery, effect on public wealth, and wages fund. Tozer, J. Camb. Ph. S. T. 6 (1838) 507- . Makeham's law. Quiquet, A. C. E. 106 (1888) 1465-, 1569. , generalisation. Quiquet, A. C. B. 109 (1889) 794-. Marine assurance. Flemmich, . Assur. Mg. 2 (*1852) 102-. . Lance, W. Assur. Mg. 2 (*1852) 362-. at Hamburg. Lazarus, W. Assur. Mg. 5 (* 1855) 221-. , principles. (* 1864) 285-. risks, statement of collisions. Rucker, J. A. Assur. Mg. [1 (* 1851)] 60-. Marriage, duration and number of surviving partners. Trembley, J. Berl. Mm. Ac. (1799-1800) 110-. , influence on death-rate in Scotland. Stark, J. [1866] Edinb. E. S. P. 6 (1869) 49-. and mortality tables combined. Chatham, J. I. Act. J. 28 (1890) 384-. Marriages, births, and deaths at St. Petersburg, tables. Krafft, W. L. [1798] St Pet. Ac. Sc. N. Acta 12 (1801) 246-. at different ages, proportion. Brown, S. Assur. Mg. 7 (*1858) 188-. Married woman's right of dower in real estate if she survives husband, table showing. Bowditch, J. I. Am. Ac. Mm. 1 (1833) 331-. Mean age, determination. Broca, P. Par. S. Ap. Bll. 2 (1879) 298-. duration of life. Ivernois, d\ Quetelet Cor. Mth. 8 (1834) 232-. rations of rural population of France. Hervt-Mangon, C. F. C. E. 79 (1874) 931-. Meaning of expression (1 - v). Gray, P. Assur. Mg. 10 (*1863) 117-. Median estimates. Galton, F. B. A. Ep. (1899) 638-. Medical statistics of life assurance companies. Porter, H. W. Assur. Mg. 4 (* 1854) 256-. Metals considered as money. Gergonne, J. D. Gergonne A. Mth. 21 (1830-31) 189-. Military assurance. Fischer, F. Hamb. Mth. Gs. Mt. 1 (1889) 4-. Monetary standard, variations in value. Brit. Ass. Comm. B. A. Ep. (1887) 247-; (1888) 181- ; (1889)133-. 1635 Money and coin, theory. Murhard, F. W. A. Hermbstadt Ms. 1 (1814) 374-; 2 (1814) 1-. -lending. Weninger, V. (xn) Mag. Ak. Ets. (Mth. Term.) 4 (1863) 147-. , moral importance, two views. Czuber, (or Cuber) E. Arch. Mth. Ps. 62 (1878) 267-. Moral statistics. Quetelet, L. A. J. Brux. Ac. Sc. Mm. 21 (1848) 112 pp. values. Henderson, R. [1895] N. Y. Am. Mth. S. Bll. 2 .(1896) 46-. Mortality. ages 75 to 81. Thomson, W. T. Assur. Mg. [1 (*1851)] 29 Ms-. American assured lives. Brown, S. Assur. Mg. 8 (*1860) 184-. and annuity tables. Brown, S. I. Act. J. 16 (* 1872) 428-. arising from military operations. Hodge, W.B. Assur. Mg. 7 (*1858) 80-, 151-, 201-, 275-. naval operations. Hodge, W. B. Assur. Mg. 6 (*1857) 254-. assured lives. King, G. I. Act. J. 19 (*1876) 381- ; 20 (* 1878) 233-. in France. Hardy, G. F. I. Act. J. 33 (1898) 485-. West Africa. L., A. I. Act. J. 33 (1898) 516-. Indies. Hardy, G. F., d' Rothery, H. J. I. Act. J. 27 (1889) 161-. British Army and Navy. McLauchlan, J. J. I. Act. J. 34 (1899) 251-. canton of Geneva. Espine, M. d\ Bb. Un. 26 (1840) 255- ; 55 (1845) 124-. children, Tasmania. Nowell, E. C. Tasm. E. S. P. (1875) 108-. the clergy. Stiissi, H. I. Act. J. 18 (*1875) 343-. Danish clergy, 1650-1878. Westergaard, H. I. Act. J. 23 (*1882) 29-. depositors in Norwegian widows fund (1846- 1872). Broch, O. J. N. Mg. Ntvd. 21 (1876) 167-. determination from population statistics. Knapp, (Dr) [G. F.] II Polit. 4 (1867) (Tech.) 17-, 91-, 181-, 313-, 452-. from different diseases, influence of selection. Dovey, W. R. I. Act. J. 23 (*1882) 285-. English clergy, Hodgson's tables. Bowser, W. A. I. Act. J. 17 (*1873) 328-. , . Brown, S. I. Act. J. 17 (* 1873) 339. equations and curves. Isely, L. Neuch. S. Sc. Bll. 17 (1889) 38-. estimation of, with migrant elements. Kiittner, W. Z. Mth. Ps. 38 (1893) 148-. experience of Scottish Amicable Assurance Society. Spens, W. Assur. Mg. 10 (*1863) 61-, 197-. Finland. Lindelof, L. Helsingf. Bd. 49 (1890) 115-. and some other countries. Lindelof, L. L. Helsingf. Ofv. 15 (1873) 129-. France between 20 and 25, and 25 and 30, compared with other countries. Espine, M. d'. Bb. Un. Arch. 5 (1859) 220-. 114 1635 Mortality Glasgow and other cities. Macfadyen, J. R, Glasg. Ph. S. P. 7 (1871) 425-. government officials on West Coast of Africa. Hart, J. R. I. Act. J. 33 (1898) 307-. hourly distribution. Lawson, R. I. Act. J. 19 (*1876) 110-. of infants in Scotland. Robertson, W. I. Act. J. 22 (* 1881) 140-. influence of high and low prices on rate. Barton, J. (n Adds.) Ph. Mg. 5 (1834) 278-. innkeepers, publicans, etc. Stott, J. I. Act. J. 20 (*1878) 35-. law. Edmonds, T. R. Assur. Mg. 9 (*1861) 170- . . Lazarus, W. Assur. Mg. 10 (* 1863) 283-. . Makeham. W. M. Assur. Mg. 13 (*1867) 325-. , answer to De Morgan's criticism. Gompertz, B. Assur. Mg. 9 (* 1861) 296-. , British India. Henderson, H. B. As. Kesearches 20 (1836) 190-. and construction of annuity tables. Makeham, W. M. Assur. Mg. 8 (*1860) 301-. , empirical expression. Buniakovskij, V. Ja. St Pet. Ac. Sc. Mm. (Rs.) 15 (*1869) (Suppl.) No. 4, 20 pp. , expectation of life and values of annuities in India and England. Hannyngton, J. C. Beng. J. As. S. 12 (1842) 1057-. , expression by geometrical series. Wool- house, W. S. B. Assur. Mg. 11 (* 1864) 150-. , Makeham's modification of Gompertz's theory. Gray, P. Assur. Mg. 11 (*1864) 236-. , mathematical. Young, T. E. I. Act. J. 22 (*1881) 139-. _ , _. Wittstein, T. I. Act. J. 24 (1884) ' 153- ; 33 (1898) 399-. , mode of determining life contingencies. Gompertz, B. Phil. Trans. (1825) 513-. from observations. Hattendorff, K. Gott. Nr. (1871) 382-. laws. Jellicoe, C. B. S. P. 4 (1839) 115. of Gompertz and Makeham. Bertrand, J. C. B. 106 (1888) 1042-. Manchester and Salford. Henry, T. [1786] Manch. Ph. S. Mm. 3 (1790) 159-. and marriage of Europeans in India. Brown, S. [1862] Assur. Mg. 11 (* 1864) 1-. experiences, certain. Hewat, A., & Chatham, J. I. Act. J. 31 (1895) 428-. Massachusetts. Elliott, E. B. Am. As. P. (1857) 51-. miners in district of Marazion. Couch, R. Q. Cornwall Pol. S. T. (1860) 18-. St Ives and of agricultural popula- tion of St Buryan. Couch, R. Q. Cornwall Pol. S. T. (1859) 30-. St Just. Couch, R. Q. Cornwall Pol. S. T. (1857) 1- ; (1858) 1-. officers in the army, 1824-1873. Alvord, B. Am. As. P. 23 (1874) (pt. i.) 57-. Paris and London. Tite, W. B. A. Bp. 34 (1864) (Sect.) 177-. periodical observation, scheme for. Curtis, F. A. I. Act. J. 19 (*1876) 229-. Mortality 1635 from peritonitis (males and females) between fifteen and fifty. Robertson, W. I. Act. J. 19 (* 1876) 118-. Philadelphia, 1809-1816. Vaughan,J. [1817] Am. Ph. S. T. 1 (1818) 430-. and population laws in France. Demon/errand, F. Par. EC. Pol. J. 16 (26 e cah.) (1838) 249-. probabilities. Landre, C. L. N. Arch. Wisk. 10 (1884) 26-. progressive. Meikle, J. Assur. Mg. 3 (* 1853) province of Bologna. Predieri, P. N. A. Sc. Nt. 7 (1847?) 396- ; 8 (1847) 31-. rate on assured lives, effect of withdrawals. Gray, W. T. I. Act. J. 24 (1884) 256-. , . Sprague, T. B. I. Act. J. 24 (1884) 293-. in West Indies. Stott, J. I. Act. J. 21 (*1879) 153-. in certain parts of Africa. Sprague, A. E. I. Act. J. 33 (1898) 285-. , De Wit's theory. Orchard, W. Assur. Mg. 2 (*1852) 393-. deduced from experience of assured lives. Whittall, W. J. H. I. Act. J. 31 (1895) 161-. in early manhood. Bailey, A.H. I. Act. J. 14 (* 1869) 247. among emigrants. McLauchlan, J. J. I. Act. J. 18 (*1875) 381-. Europeans in India. Sprague, T. B. I. Act. J. 19 (*1876) 295-. Jamaica. Marshall, J. Assur. Mg. 4 (*1854) 39-. , experience of an assurance company. Jellicoe, C. Assur. Mag. 4 (*1854) 199-. and its increase. Chatham, J. I. Act. J. 29 (1892) 81-. in India. Finlaison, A. J. I. Act. J. 18 (*1875) 153-. infancy and childhood. Bowser, W. A. I. Act. J. 17 (* 1873) 26-. as influenced by time of assurance. Sprague, T. B. I. Act. J. 15 (* 1870) 328-. among Jews. Sprague, T. B. I. Act. J. 27 (1889) 155- . in navy. Sprague, T. B. I. Act. J. 25 (1886) 49-. New South Wales. Pell, M. B. N. S. W. B. S. T. 1 (1867) 66-; I. Act. J. 21 (*1879) 257-. peerage during nineteenth century. Bailey, A. H., & Day, A. Assur. Mg. 9 (*1861) 305-. select lives. Spens, W. Assur. Mg. 12 (*1866) 304. throughout life, formula. Thiele, T. N. I. Act. J. 16 (*1872) 313-, 464. , Zeuner's method for determining. Danielewicz, B. Prace Mt.-Fiz. 6 (1895) 183-. rates in Australia. Burridge,A. F. I. Act. J. 24 (1884) 333-. as disturbed by migrations. Welton, T. A. I. Act. J. 16 (* 1872) 153-. , methods for deducing. Ackland, T. G. I. Act. J. 33 (1898) 68-, 131-. 115 H 2 1635 Mortality rates, methods for deducing discontinuances. Todhunter, R. I. Act. J. 33 (1898) 273-. of natives and Europeans in India. Brown, S. I. Act. J. 16 (* 1872) 187-. of India. Hardy, G. F. I. Act. J. 25 (1886) 217-. and their causes. Lazarus, W. I. Act. J. 18 (* 1875) 54-, 212-. in United States. Gill, . Assur. Mg. 3 (*1853)300-. Victoria. Burridge, A. F, I. Act. J. 23 (*1882) 309-. ratio in Europe. Dieterici, G. F. W. Berl. Ab. (1851) (Ph.) 721-. in relation to age and sex. Bertillon, (Dr) . Fr. Cg. Md. Chir. 3 (1866) 663-. Salem, 1782 and 1783. Holyoke, E. A. Bost. Mm. Am. Ac. 1 (1785) 546-. among Sardinian soldiers in times of peace. Bonino, J. J. Tor. Mm. Ac. 35 (1831) 233-. selected classes. Demonferrand, F. C. R. 9 (1839) 244-. lives in Germany. Brune, . Assur. Mg. 3 (* 1853) 29-. , inadequacy of data. Spens, W. Assur. Mg. 4 (* 1854) 1-, 139-. shifting population. Wittstein, T. Grunert' Arch. 39 (1862) 67-. and sickness among labouringclasses. Edmonds, T. R. Assur. Mg. 5 (*1855) 127-. statistics, collection. Sang, E. I. Act. J. 15 (* 1870) 257-. , formula. Westergaard, H. (xn) Ts. Mth. 1 (1877) 1-. , . Enestrom, G. Stockh. Ofv. (1891) 185- ; Fschr. Mth. (1891) 241-. supplement to two papers in Phil. Trans. 1820, 1825. Gompertz, B. [1861] Phil. Trans. (1862) 511-. and survival at different ages in Geneva, 1838- 1845. Espine, M. d'. Gen. Mm. S. Ps. 11 (1846) 457-. table for Brabant. Quetelet, L. A. J. Brux. Ac. fill. 6 (1859) 345-. deduced from new experience. Bowser, W. A. I. Act. J. 16 (*1872) 146-. whole life premium table. Wylie, W. Assur. Mg. 2 (*1852) 391-. , Northampton, mode of construction. Sutton, W. I. Act. J. 18 (*1875) 107-. tables. Weninger, V. (xn) [1860] Mag. Ak. Ets. (Mth. Term.) 2 (1861-62) 51-. in America. McCay, C. F. I. Act. J. 16 (* 1872) 20-. , arrangement of data. Meikle, J. Assur. Mg. 13 (*1867) 261-. , Bengal Civil Service. Hannyngton, J. C. Beng. J. As. S. 19 (1850) 250-. , calculation. Quetelet, L. A. J. Brux. Ac. Bll. 19 (1852) (pte. 3) 289-. , comparison. Young, (Dr) T. QJ. Sc. 2 (1828) 342-. , computation. Thiele, T. N. Kj0b Ov (1900) 139-. , construction. Woolhouse, W. S. B. Assur. Mg. 13 (* 1867) 75-. Theory of Statistics, etc. 1635 Mortality, continued. tables, construction. Baumhauer, M. M. von. I. Act. J. 16 (* 1872) 34-. , from imperfect data. Makeham, W. M. I. Act. J. 16 (* 1872) 344-. , , principles of. Makeham, W. M. Assur. Mg. 12 (* 1866) 305- . for Finland. Lindelof, L. Helsingf. Ofv. 38 (1896) 113- . , formulae. Enestrom, G. Stockh. Ofv. (1891) 251-, 343-; Fschr. Mth. (1891) 242-, 243-. , for calculating. Enestrom, G. Stockh. Ofv. (1891) 441- ; Fschr. Mth. (1891) 244-. and policy values. Meikle, J. I. Act. J. 23 (*1882) 385-. , select, construction and use. Sprague, T. B. I. Act. J. 21 (* 1879) 229- ; 22 (* 1881) 391-. , theory- Samot, D. J. A. N. Arch. Wisk. 3 (*1877) 145-. United States of America. Brown, S. Assur. Mg. 13 ( 1867) 272-. various .classes. Hardy, G. F. I. Act. J. 23 (* 1882)1-. countries of Europe. Anon, (vi 354) Edinb. N. Ph. J. 16 (1834) 259-. Movement of population, diocese of Maurienne, 1810-1830. Anon, (vi 1123) Sav. Mm. Ac. 5 (1831) 255-. , France. Demon/errand, F. C. E. 2 (1836) 437-. , Macon and environs, 1740-1828. Benon des Chanes, . (xn) Macon S. C. E. (1828) 65-. , Palermo. Villot, . A. Sc. Nt. 10 (1827) 442-. , Paris. Villerme, L. A. Sc. Nt. 8 (1826) 423-. -, Portugal. Silva, D. A. da. Lisb. J. Sc. Mth. 2 (1870) 255-. , Spain, 1866. La Sagra, R. de. C. B. 67 (1868) 460-. Multiple annuities, formation of values and amounts. Gray, P. I. Act. J. 24 (1884) 129-. Mutual life assurance, theory. Templeton, J. M. I. Act. J. 20 (*1878) 77-. Negative policy values. Deuchar, J. J. W. I. Act. J. 19 (*1876) 97-. . Gray, W. T. I. Act. J. 20 (*1878) 73-, 150-. . King, G. I. Act. J. 20 (*1878) 148-. Net premium and interest rate. Sutton., W. I. Act. J. 17 (*1873) 446-. mode of valuation, objections to. Makeham, W. M. I. Act. J. 15 (*1870) 449-. present value of instruments of credit, etc. Ronchetti, F. G. Mt. 26 (1888) 133-. profits, possible methods of dividing. Wilbraham, H. Assur. Mg. 6 (*1857) 278-. Number of white persons in United States, tables. Bowditch, J. I. Am. Ac. Mm. 1 (1333) 345-. 116 1635 Theory of Statistics, etc. Obligations, contracting and discharge, theory. Boudsot, A. A. (xn) Doubs S. Mm. 2 (1858) 340-. Obligatory assurance, theory. Enestrom, G. Stockh. Ofv. (1895) 807- ; Fschr. Mth. (1895) 251-. Occupation, influence on health. Nelson, F. G. P. I. Act. J. 17 (*1873) 95-. Old age and invalid assurance, theory. Lazarus, W. Hamb. Mth. Gs. Mt. 2 (1890) (Festschr. Tl. 2) 158-. Orchard's tables and a theoretical mortality table. Gray, P. Assur. Mg. 6 (*1857) 181-. "Paid up" policy as equivalent to policy payable. Macfadyen, J. E. I. Act. J. 15 (*1870) 297-. Partial commutation of premium. Gray, P. I. Act. J. 17 (*1873) 224-. Payment of yearly premium. Wiegand, A. Assur. Mg. 12 (* 1866) 54-. . Laundy, S. L. Assur. Mg. 12 (* 1866) 55-. Pension calculation, formula. Enestrom, G. Stockh. Ofv. (1893) 405- ; Fschr. Mth. (1893-94) 360-. Pensions. Lindelof, L. Helsingf. Ofv. 26 (1884) 67-; Fschr. Mth. (1884) 189; Helsingf. Acta 14 (1885) 1-; 15 (1888) 209- ; Helsingf. Bd. 51 (1892) 43-; 54 (1894) 1-; Acta Mth. 18 (1894) 89-. . Widows and orphans fund of Leipzig University, 1882. Drobisch, M. W. Leip. Mth. Ps. B. 34 (1882) 51-. Perpetuity contingent on surviving four lives. M., T. Assur. Mg. 12 (*1866) 301. Philosophy of statistics. Woolhouse, W. S. B. I. Act. J. 17 (*1873) 37-. Policies of assurance and life interests. Sprague, T. B. Assur. Mg. 8 (*1860) , valuation. Anon. Assur. Mg. 10 (*1863) 252-, 322-. on recently assured lives, reserve on. Teece, E. I. Act. J. 22 (*1881) 250-. , . Sprague, T. B. I. Act. J. 22 (*1881) 264-. Policy on longer of two lives, value. Carr, T. I. Act. J. 14 (*1869) 415-. valuation, subject to half yearly or quarterly premiums. Sprague, T. B., & King, G. I. Act. J. 23 (* 1882) 256-, 264-. value. Meikle, J. Assur. Mg. 11 (*1864) 241-. and rate of interest. Sutton, W. I. Act. J. 17 (*1873) 227-. . Walton, W. G. I. Act. J. 22 (*1881) 135-. tables, modes of constructing. Manly, H. W. I. Act. J. 15 (* 1870) 169-. in terms of premium. Wilding, R. I. Act. J. 22 (* 1881) 131. values, constructing models of. Chisholm, J. I. Act. J. 25 (1886) 141-. , effect of increased mortality. Sprague, T. B. I. Act. J. 21 (* 1879) 77-. Population 1635 Policy values found by various methods, com- pared. Manly, H. W. I. Act. J. 14 (* 1869) 249-. Political arithmetic. Czornig, K. J. Prag Jb. Bohm. Ms. 2 (1831) 22-. Whewell, W. \ economy. [1850] Camb. Ph. S. T. 9 (1856) 128- ; (pt. 2) [1]-. and arithmetic, laws. Balbo, P. Tor. Mm. Ac. 34 (1830) 47-. assurance, mathematical theory. Wittstein, T. I. Act. J. 17 (*1873) 178-, 355-, 417-. , mathematics applied to. Eau, . Brux. Ac. S. Bll. 8 (1841) (pte. 2) 148-. , . Helm, G. Dresden Isis Sb. (1887) (Ab.) 3-. , . Edgeworth, F. Y. B. A. Rp. (1889) 671-. Population. Benares. Prinsep, J. As. Researches 17 (1832) 470-. births, etc., in the Netherlands. Quetelet, L. A. J. Brux. Ac. Sc. Mm. 4 (1827) 117-. Bombay. Macintosh, J. Thomson A. Ph. 14 (1819) 429-. British Isles. Coquebert de Montbret, (Baron). Par. S. Gg. Bll. 1 (1822) 191-. density, proposed units of areas. Steinhauser, A. Wien Mt. Gg. Gs. 7 (1863) 129-. France. Dupin, C. C. R. 28 (1849) 369-. in 18th and 19th centuries. Dupin, C. C. R. 1 (1835) 103-. , influence of price of corn. Dupin, C. C. R. 2 (1836) 585-. general considerations. Anon, (vi 703) J. de Ps. 93 (1821) 320-, 414-. general laws. Quetelet, L. A. J. Brux. Ac. Bll. 9 (1842) (pte. 2) 550-. . Pouillet, C. S. M. C. R. 15 (1842) 861-. (Pouillet). Demonf errand, F. C. R. 15 (1842) 1096-. Geneva. Mallet, E. Gen. Mm. S. Ps. 7 (1836) 321-. increase, 1770-1848. Carnot, H. C. R. 31 (1850) 288-. , apprehensions caused by. Hoffmann, J. G. Berl. Ab. (1835) (Ph.) 121-. and decrease in Danish states. Callisen, . Dn. Vd. Selsk. Skr. 6 (1809-12) (2'. hceft.) 169-. dependent on abundance of cereals and other foods. Predieri, P. N. A. Sc. Nt. 1 (1850) 58-. in Europe and America, decrease of rate. Storer, H. E. [1858] Am. J. Sc. 43 (1867) 141-. kingdom of Naples, decrease of rate. Augustinis, M. de. Ranuzzi An. Gg. (1845) 35-. Prussia in town and country. Dieterici, C. F. W. Berl. Ab. (1857) (Ph.) 99-. relation to means of subsistence. Bourne, S. B. A. Rp. (1877) (Sect.) 165-. law of increase. Verhulst, P. F. Quetelet Cor. Mth. 10 (1838) 113- ; Brux. Ac. Sc. Mm. 18 (1845) 38 pp. ; 20 (1847) 32 pp. 117 1635 Population law of increase and decrease. Besnard, G. Caen Mm. Ac. (1856) 203-. duration of families. Bienayme, J. Par. S. Phlm. PV. (1845) 37-. during last 100 years. Edmonds, T. E. Assur. Mg. 2 (*1852) 57-. mathematical theory. Quetelet, L. A. J. L'l. 2 (1834) 311-. Mexico, 1840. Galeotti, H. G. Brux. Ac. Bll. 8 (1841) (pte. 2) 95-. and mortality statistics, emigration and immi- gration being taken into account. Enestrom, G. Stockh. Ofv. (1896) 225- ; Fschr. Mth. (1896) 187-. movement, European birth- and death-rate. Benoiston de Chateauneuf, . A. Sc. Nt. 7 (1826) 314-. Oceania, Penes Is. Cagnazzi, L. de S. Nap. Ed. 2 (1843) 118-. statistics, Austria, etc. Ficker, A. [1859] Wien Mt. Gg. Gs. 4 (1860) (Ab.) 71-. , generalisation from. Enestrom, G. Stockh. Ofv. (1896) 403- ; Fschr. Mth. (1896) 188. , theory of pensions. Enestrom, G. Stockh. Ofv. (1893) 361- ; Fschr. Mth. (1893-94) 358-. theory. Vinde, . Eure J. d'Ag. 6 (1829) 331-. white and slaves, increase in United States. Harvey, G. Edinb. Ph. J. 8 (1823) 41-, 328-; 9 (1823) 63-. world, ancient and modern. Merritt, . Par. S. Gg. Bll. 9 (1828) 1-. , races, religions, etc. Dieterici, C. F. W. Peterm. Mt. (1859) 1-. Piemature death among select lives. Barren, E.J. Assur. Mg. 4 (* 1854) 66-, 141. Premium-obligation, worth for any period. Hultman, F. W. Ts. Mt. Fys. 4 (1871) 251-. payable for life or for term, difference between. Meikle, J. Assur. Mg. 5 (* 1855) 154-. rates and benefits accruing, incongruities. S., H. A. Assur. Mg. 8 (*1860) 167-. for Bengal. Jellicoe, C. Assur. Mg. [1 (*1851)] 166-. to provide certain periodical returns. Tucker, E. Assur. Mg. 9 (*1861) 245-. scales, use of reduced. Hardy, G. F. I. Act. J. 31 (1895) 261-. values, method of interpolating. Samot, D. J. A. I. Act. J. 20 (*1878) 347-. , . Garment, D. I. Act. J. 21 (*1879) 213-. Premiums, calculation by Gompertz's law. Makeham, W. M. Assur. Mg. 9 (* 1861) 361. , half yearly and quarterly, obtained from annual. Laundy, S. L. Assur. Mg. 11 (*1864) 232-. Present value of payments (when reproductive differs from remunerative rate). Lidstone, G. J. I. Act. J. 33 (1898) 412-. Theory of Statistics, etc. 1635 Present values, calculation. Heegmann, A. Lille Mm. S. (1827-28) 1-. Price of annuity for n years of life of a person aged x. Macfadyen, J. E. I. Act. J. 19 (*1876) 141-. Probabilities, application to judicial statistics. Couniot, A. A. Liouv. J. Mth. 3 (1838) 257-. , therapeutic statistics and statistics generally. Hagenbach-Bischoff ', E. [1877] Basel Vh. 6 (1878) 516-. involved in widows' fund. Gruson, J. P. Berl. Ab. (1812-13) (Mth.) 1-. . Values of contingent reversions, 3 lives involved in survivorship. Morgan, W. [1799] Phil. Trans. (1800) 22-. Probability of duration, decrement and ex- pectation of life in New Hampshire and Massachusetts. Wigglesworth, E. Bost. Mm. Am. Ac. 2 (1793) 131-. election results. Korteiceg, D. J. [1875] Arch. Neerl. 12 (1877) 65-. , local. Enestrom, G. Stockh. Ofv. (1893) 623-; Fschr. Mth. (1893-94) 363. of success of candidates for three- or four- cornered constituencies. Hayward, E. B. B. A. Rp. 38 (1868) (Sect.) 9-. Profit, determination and distribution. Meikle, J. Assur. Mg. 11 (* 1864) 251-. distribution in mutual assurance societies. Pell, . I. Act. J. 14 (*1869) 382-. Profits, formulas and tables for applying. Manly, H. W. I. Act. J. 27 (1889) 362-; 28 (1890) 333-. , , note on Manly's paper. Calderon, H. P. I. Act. J. 28 (1890) 394-. Progressive taxation, theory. Gobbi, U. Mil. I. Lomb. Ed. 30 (1897) 452-. Property, division between children according to civil code. Catalan, E. C. L'l. 30 (1862) 139-. held for life and in reversion, valuation. Jellicoe, C. Assur. Mg. 6 (*1857) 61-. , . Wilbraham, H. Assur. Mg. 6 (*1857) 211-. and income tax, inequitable distribution. Jellicoe, C. Assur. Mg. 2 (*1852) 213-. Proportional parliamentary representation. Wasteels, J. Mathesis 20 (1900) 81-. representation at elections. Curie, J. As. Fr. C. E. (1889) (Ft. 2) 993-. , methods. Enestrom, G. Stockh. Ofv. (1896) 543-; Fschr. Mth. (1896) 189. of minorities. Bayssellance, A. Bordeaux S. Sc. Mm. 3 (1880) 285-. Prosperity in nations, relation to ethical con- ditions and mental development. Dieterici, C. F. W. Berl. Ab. (1855) (Ph.) 433-. Purchasing life assurances as an investment. Day, A. Assur. Mg. 8 (*1860) 326-. Pure premium method of valuation. Bailey, A. H. I. Act. J. 21 (*1879) 115-. as affected by expense and selection. Sorley, J. I. Act. J. 21 (*1879) 192-. Quinquennial census, deduction of yearly alterations. Enestrom, G. Stockh. Ofv. (1893) 541- ; Fschr. Mth. (1893-94) 362. 118 1635 Theory of Statistics. Actuarial Mathematics 1635 Kailway production and renewal, mathematical theory of cost. Mohr, . Giving. 37 (1891) 39-. travelling, economics. Schneidewind, A. Arg. S. Ci. A. 39 (1895) 5-; 40 (1895) 77-; 41 (1896) 185-, 209-. , . Soulages, E. Arg. S. Ci. A. 42 (1896) 337-. Eatio of newly contracted marriages to con- temporary population. Dieterici, C. F. W, Berl. Ab. (1856) (Ph.) 559-. Rationale of some actuarial estimates. Jellicoe, C. Assur. Mg. 8 (*1860) 310-. Eejection of fractions of 1 in extensive valuations. De Morgan, A. Assur. Mg. 10 (*1863) 247-. Belation of births to deaths, effects of migration and affluence. Benon des Chdnes, . (xn) Macon S. C. E. (1824) 89-. Eelationship, human, algebra of. Macfarlane, A. [1879-81] Edinb. E. S. P. 10 (1880) 224-; 11 (1882) 5-, 162- ; Ph. Mg. 11 1881 436-. ', , . Brunei, . Bordeaux S. Sc. Mm. 4 (1894) xxv-. , a problem in. Macfarlane, A. Edinb. E. S. P. 15 (1889) 116- . Eelationships (kinship), Gal ton's arithmetical notation. Macfarlane, A. Nt. 28 (1883) 588. Eeliability of data, as tested by conclusions. Farren, E. J. Assur. Mg. 3 (* 1853) 204-. Eemarriage rate of widowers. Sprague, T. B. I. Act. J. 22 (*1881) 77-. Eents and value of ground in Borne. Armellini, T. Em. N. Line. At. 27 (1874) 111-. Eeproduction table, English. Farr, W. Phil. Trans. 171 (1880) 281 -. Eeserves of assurances, Keuchel's formula. Danielewicz, B. Wiad. Mt. 2 (1898) 150-. with return of premium. Danielewicz, B. Wiad. Mt. 3 (1899) 275-. fixed term assurances, calculation. Danielewicz, B. Wiad. Mt. 2 (1898) 216-. on life assurances. Danielewicz, B. Wiad. Mt. 4 (1900) 60-. owing to different data of valuation. Valentine, J. I. Act. J. 18 (*1875) 229-. Eetirants in eighth year. Macaulay, T. B. I. Act. J. 22 (*1881) 134-. Eeturnable premiums, calculation. Makeliam, W. M. Assur. Mg. 12 (*1866) 233-. Eeversionary annuities. King, G. I. Act. J. 22 (*1881) 296-. payable half yearly or quarterly. Carr, T. Assur. Mg. 7 (*1858) 109-. , etc. Sprague, T. B. I. Act. J. 15 (*1870) 126-; 18 (*1875) 76. , value of second nomination. Hardy, G. F. I. Act. J. 25 (1886) 134-. annuity, formula for. M., A. (Editor). Assur. Mg. 5 (*1855) 52. , . Sprague, T. B. Assur. Mg. 7 (*1858) 174-. payable oftener than once a year, value. Evans, W. I. Act. J. 19 (*1876) 12-. bonus attaching to each policy, method of calculation. Hardy, E. P. I. Act. J. 31 (1895) 330- . Eeversionary interests, certain contingent. Warner, S. G. I. Act. J. 31 (1895) 280-. life interests as securities for loans. Sprague, T. B. I. Act. J. 17 (*1873) 229-. , valuation. Sprague, T. B. I. Act. J. 14 (*1869) 417-. , . Davies, G. I. Act. J. 15 (* 1870) 138-. , . Sprague, T. B. I. Act. J. 27 (1889) 107-. , . Sunderland, A.W. I. Act. J. 27 (1389) 220-. transactions by lifei. assurance companies. Macfadyen, J. E. I. Act. J. 20 (*1878) 385-. Eisk premiums for survivorship assurances. Sunderland, A. W. I. Act. J. 27 (1889) 81-. Eisks of assurance, mode of tabulation. Chatham, J. I. Act. J. 32 (1896) 393-. in granting life assurance. Bremiker, C. I. Act. J. 16 (*1872) 216-, 285-. , valuation, and division of surplus. Jellicoe, C. Assur. Mg. 10 (* 1863) 328-. Eoyal Society, effect of limitation of number of fellows elected each year on eventual total. Strachey, (Lt.-Gen.) E. E. S. P. 51 (1892) 463-. Savings banks. Dupin, C. C. E. 4 (1337) 597. . Crelle, A. L. Crelle J. 39 (1850) 133-. , progressive development. Dupin, C. C. E. 17 (1843) 1008-. , statistics. Dupin, C. C. E. 20 (1845) 1593-. Second marriages in the peerage. Day, A. Assur. Mg. 12 (*1866) 185-. Selection among assured lives. Higham, J. A. I. Act. J. 20 (* 1878)1-. of lives for assurance'. Thomson, S. C. I. Act. J. 21 (*1879) 170-. Sickness and invalidism, in relation to mor- tality. Makeham, W. M. I. Act. J. 16 (*1872) 408-. mortality among native and European troops in India. Scales, W. H. Assur. Mg. 5 (* 1855) 245-. , present state of information. Tomp- kins, H. Assur. Mg. 3 (* 1853) 7-. rate in France. Brown, S. Assur. Mg. 5 (*1855) 208-. tables, Eatcliffe's. Frankland, F. W. I. Act. J. 21 (*1879) 380. Simpson's rule, generalisation for joint life annuities. Allen, J. M. I. Act. J. 33 (1898) 535-. Single and annual premium, proof of formulae. Gray, P. Assur. Mg. 10 (*1863) 238-. , . Wiegard,D.A. Assur. Mg. 10 (*1863) 286. life contingencies, calculation. De Morgan, A. Assur. Mg. 12 (*1866) 328-; 13 (*1867) 129-. Sinking fund assurance. Faulks, J. E. I. Act. J. 34 (1899) 562-. Smoothing of mortality observations, and Oppermann's formula. Gram, J. P. Ts. Mth. 2 (1884) 113- ; Fschr. Mth. (1884) 187-. 119 Theory of Statistics. Actuarial Mathematics 1635 Social statistics and uniform action of human will. Broicn, S. Assur. Mg. 2 (*1852) 341-. Stability of results with reference to number of transactions. Campbell, R. Assur. Mg. 9 (*1861) 216-. Standard weight of Cologne mark. Eytelwein, J. A. Berl. Ab. (1816-17) 42-. Statistical investigation of evolution. Cun- ningham, J. T. Nt. 51 (1894-95) 510. method for determining the vitality of a population, that of Sweden in particular. Enestrmn, G. Stoefch. Ofv. (1893) 481- ; Fschr. Mth. (1893-94) 361. rejection of extreme variations. Sheppard, W. F. L. Mth. S. P. 31 (1900) 70-. researches in the Netherlands. Quetelet, L. A. J. Brux. Ac. Sc. Mm. 5 (1829) vi + 57 pp. science, new step in. Gallon, F. Nt. 51 (1894-95) 319. study on recruiting for the Polytechnic School. Montessus de Bailor e, F. de. Par. EC. Pol. J. 63 (1893) 27-. tables, errors of the argument. Kleiber, J. B. A. Ep. (1888) 618. , probability of uniformity. Campbell, E. Ph. Mg. 18 (1859) 359-. Statistics. Wittstein, T. D. Nf. B. 40 (1865) 112-. . Geddes, P. Nt. 24 (1881) 523-. , application of binomial law. Brown, S. I. Act. J. 17 (*1873) 340-. , association of attributes. Yule, G. U. [1899] Phil. Trans. (A) 194 (1900) 257-. , Bengal. Bedford, J. JR. Beng. As. S. J. 22 (1853) 387-. , Camborne. Lanyon, E. Cornwall Pol. S. T. (1841) 99-. , classification. Geddes, P. [1881] Edinb. E. S. P. 11 (1882) 295-. of conditions of life among working classes. Bodio, L. Em. E. Ac. Line. T. 6 (1882) 317-. , correction of some documents. Demon- ferrand, F. Par. EC. Pol. J. 16 (27 e cah.) (1839) 75-. of effect of cholera, 1832 and 1849. Dupin, C. C. E. 34 (1852) 65-. Egypt, 1837. Duhamel, A. Es. Gg. Gs. D. 1 (1849) 93-. , formula for reduction. Enestrom,G. Stockh. Ofv. (1893) 397-; Fschr. Mth. (1893-94) 359-. of fuels. Morville, N. [1798] Kiob. Dn. Vd. Selsk. Skr. 1 (1800) 205-. general, of different countries. Quetelet, L.A.J. Brux. Ac. Bll. 12 (1861) 99-. , judiciary. Liouville, E. Liouv. J. Mth. 18 (1873) 145-. and last census, British Empire. Merritt, . QJ. Sc. 2 (1827) 283-. , Lot (dept.). Cordier,L. J. Mines 21 (1807) 445-; 22(1807)5-. , mathematical. Ktittner, W. Z. Mth. Ps. 25 (1880) 11-; 26 (1881) 297-; 32 (1887) 1635 Statistics, States of the world. Babbi, A. Mil. G. I. Lomb. 3 (1842) 387-. , use. Delauney, . C. E. 108 (1889) 909-. Still-born children, estimation of number. Demonferrand, F. C. E. 3 (1836) 615-. Suicide in Finland. Westerlund, F. W. Hel- singf. Bd. 58 (1900) 111-. , statistics. Jopling, R. T. Assur. Mg. [1 (*1851)] 308- ; 2 (1852) 32-. Summation, approximate, of numerical tables. Buniakovskij, V. Ja. [1867] St Pet. Ac. Sc. Mm. (Es.) 12 (1868) (Suppl.) No. 4, 41 pp. Sums of drawings, final law. De Morgan, A. I. Act. J. 14 (*1869) 175- . Surplus, conditions giving rise to. Jellicoe, C. Assur. Mg. 2 (*1852) 333-. determination. Gill, C. Assur. Mg. [1 (*1851)] 357-. . Younger, S. Assur. Mg. 4 (*1854) 249-. and distribution. M . , A . (Editor.) Assur. Mg. [1 (*1851)] 22 bis-, 159-, 359-. , certain objectionable methods. Jellicoe, C. Assur. Mg. 3 (*1853) 185-. distribution. Romans, S. Assur. Mg. 11 (*1864) 121-. . Manly, H. W. I. Act. J. 23 (M882) 233-. . Lidstone, G. J. I. Act. J. 32 (1896) 73-. , as defeating object of life assurance. Terry, J. Assur. Mg. 10 (*1863) 130-. , methods. Sprague, T. B. Assur. Mg. 6 (*1857) 290- , 344-; 7 (*1858) 61-. , , various. Pattison, W. P. Assur. Mg. 9 (*1861) 341-. , formula for returning to assured their con- tributions. Harvey, C. J. I. Act. J. 24 (1884) 173-. premiums and interest, bonuses arising from. Sunderland, A. W. I. Act. J. 26 (1887) 357-. Surrender of policies, allowance. Jellicoe, C, Assur. Mg. [1 (*1851)] 279-. , . Crisford, G. S. I. Act. J. 21 (*1879) 301-. values, solution of problem. Marr, T. I. Act. J. 14 (*1869) 156-. , and their calculation. Fulford, F. W. I. Act. J. 35 (1901) 199-. Survivorship annuities, calculation by columnar method. Meikle, J. Assur. Mg. 11 (*1864) 40-. assurance tables, construction. Gray, P. Assur. Mg. 5 (*1855) 107-. premiums, calculation. Sunderland, A. W. I. Act. J. 27 (1889) 509-. between two lives, probabilities. Gray, P. Assur. Mg. [1 (*1851)] 137-. , . Ackland, T. G. I. Act. J. 27 (1889) 157-. , , method for computing. Chisholm, D. Assur. Mg. 2 (*1852) 305-. Survivorships, general problems concerning. Makeham, W. M. Assur. Mg. 12 (*1866) 61-. , three life. Makeham, W. M. Assur. Mg. 12 (*1866) 118-. 120 1635 Theory of Statistics, etc. Calculus of Differences, etc. 1640 Symbol d x , origin. Cherriman, J. B. I. Act. J. 21 (*1879) 295-. Taxation of temporary versus perpetual an- nuities. Hardy, P, Assur. Mg. 3 (*1853) 195-. and wealth, relation between. Gobbi, U. Mil. I. Lomb. Kd. 32 (1899) 756-. Taxes, principles regulating incidence. Jenkin, F. Edinb. B. S. P. 7 (1872) 618. Ten year non-forfeiture policies. Younger, S. I. Act. J. 14 (*1869) 476-; 15 (*1870) 151-. , American. Cherriman, J. B. I. Act. J. 16 (*1872) 384-. , . McClintock, E. I. Act. J. 17 (*1873) 301-. Term assurance, life against life, value. Smith, H. A. Assur. Mg. 9 (*1861) 295. for two joint lives. Otter, W. C. Assur. Mg. 8 (*1860) 113-. Terms of imprisonment, statistics and curves. Gallon, F. Nt. 52 (1895) 174-. Tontine, American, and mutual assessment schemes. Manly, H. W. I. Act. J. 26 (1887) 182-. , perpetual. Navier, C. L. M. H. A. C. 12 (1819) 103-. Tontines, early history. Hendriks, F. Assur. Mg. 10 (*1863) 205-. Two life assurances, extra premiums in respect of. Ryan, G. H. I. Act. J. 24 (1884) 305-. Under-average lives, treatment of. Sunderland, A. W. I. Act. J. 29 (1892) 419-. Uniform mortality law from birth to old age. Gompertz, B. I. Act. J. 16 (1872) 329-. seniority, law. Henderson, R. I. Act. J. 32 (1896) 293-. United States, eighth census. Brown, S. Assur. Mg. 13 (* 1867) 226-. , formula for predicting population. Prit- chett, H. S. St Louis Ac. T. 5 (1892) 599-. Universal notation , necessity for a. Begault, A . I. Act. J. 33 (1898) 1-. Utility, theory. Gobbi, U. Mil. I. Lomb. Bd. 81 (1898) 237-, 325-, 1300-. Vaccination, longevity in France before and after its introduction. Dupin, C. C. B. 27 (1848) 565-. and small pox, statistics. Sprague, T. B. I. Act. J. 20 (*1878) 2 16-. Valuation of absolute reversions. Mount- castle, H. I. Act. J. 15 (*1870) 148-. , certain methods. Gray, W. T. I. Act. J. 20 (*1878) 309-. methods, notes. Dovey, W. R. I. Act. J. 34 (1899) 346-. producing definite reserve. Hardy, JR. P. I. Act. J. 1 (1895) 325-. reserves, approximate calculation. Chis- holm, J. I. Act. J. 27 (1889) 442-. returns, with reference to Life Assurance Companies Act. Todhunter, R. I. Act. J. 35 (1901) 1-. of whole term policies, tables. Hume, A. S., (& Stott, W. I. Act. J. 34 (1899) 396-. Value of benefits, on principle of collective assurance. Hardy, R. P. I. Act. J. 30 (1893) 79-. Value of contingent reversion under certain limitations. Hardy, P. Assur. Mg. 2 (*1852) 91-, 298. options. Stephewon, J. W. Assur. Mg. 12 (*1866) 302-. . Younger, S. Assur. Mg. 13 (*1867) 55. . Gray, P. Assur. Mg. 13 (*1867) 104-. . Younger, S. Assur. Mg. 13 (*1867) 118. , Stephenson's theory. Makeham, W. M. Assur. Mg. 12 (*1866) 363-. , . Smith, H. A. Assur. Mg. 13 (*1867) 103. and prices, mathematical investigations. Fisher, I. Conn. Ac. T. 9 (1892-95) 1-. of property and conditions which determine it. Buquoy, G. von. Oken Isis (1824) 28-. reversions payable at instant of death. Chisholm, D. Assur. Mg. 4 (*1854) 70-. selection as exercised by policy holder. Higham, J. A. .Assur. Mg. [1 (*1851)] 179-. Values, elementary, special arrangement. Oakes, W. H. Assur. Mg. 12 (*1866) 57-. Vital statistics ; geometric mean. Gallon, F. B. S. P. 29 (1879) 365-. of Glasgow, 1869 and 1870. Scott, A. Glasg. Ph. S. P. 8 (1873) 78-. Society of Friends. Fryer, A. Manch. Ph. S. Mm. 14 (1857) 153-. towns in Scotland. Sykes, W. H. B. A. Bp. (1842) 121-. Wealth, social, mathematical theory. Walrus, L. [1876] Laus. S. Vd. Bll. 14 (1877) 365-, 525-. Whole-life assurance business, approximate valuation. Woolhouse, W. S. B. I. Act. J. 27 (1889) 433-. assurances, a certain conversion of. M., T. Assur. Mg. 13 (*1867) 246-. time assurances on single lives, valuation. Coles, J. Assur. Mg. 7 (*1858) 179-. Widows' pensions, statistics. Enestrom, G. Stockh. Ofv. (1894) 479-; Fschr. Mth. (1893-94) 363-. and spinsters, marriage rates. Sprague, T. B. I. Act. J. 22 (*181) 352-. Yearly into fractional yearly payment, trans- formation. McKenzie, D. J. McG. I. Act. J. 23 (*1882) 162-. 1640 Calculus of differences ; interpolation. Adjunct of a given linear finite difference form. Bortolotti, E. Em. B. Ac. Line. Bd. 5 (1896) (Sem. 1) 349-. Algebra of difference-tables. Everett, J. D. QJ. Mth. 31 (1900) 357-. Approximate summation, formulae. Hardy, G. F. I. Act. J. 24 (1884) 95-. of series. Korteweg, D. J. N. Arch. Wisk. 2 (*1876) 161-. Approximation methods of former times. Giin- ther, S. D. Nf. B. (*1877) 95-. 121 Calculus of Differences ; Interpolation 1640 Boole's finite differences, notation. Wilding, E. I. Act. J. 22 (*1881) 60-. Calculus of differences, algebraic transformation by. Cesdro, E. N. A. Mth. 5 (1886) 489-. , finite and infinitesimal. Gregory, D. F. Camb. Mth. J. 1 (1339) 212-. , point in. Tisserand, F. C. E. 70 (1870) 678-. , theorem. Hamilton, (Sir) W. E. B. A. Ep. (1843) (pt. 2) 2-. enlargement (extension of calculus of finite differences). MeClintock, E. Am. J. Mth. 2 (1879) 101 -. , theorems. MeClintock, E. Am. J. Mth. 17 (1895) 69-. functions derived from limiting ratios. Echols, W. H. [1395] A. Mth. 10 (1895-96) 50-. generating functions. L., E. J. Camb. Mth. J. 2 (1841) 94-. Central-difference formulas. Sheppard, W. F. [1399] L. Mth. S. P. 31 (1900) 449-. interpolation formula* Everett, J. D. B. A. Ep. (1900) 648-. formulas. Spencer, J. I. Act. J. 33 (1898) 349-. Constant coefficients in finite differences, theory. Naylor, J. I. Act. J. 23 (*1382) 420-. Convergents applied to Cauchy's interpolation fractions. Pade, H. C. E. 130 (1900) 697-. Cotes' numbers, calculation by approximate quadratures. Blazek, G. Prag Sb. (1879) 167-. Cotton spinning problem. Woolhouse, W. S. B. Assur. Mg. 11 (*1364) 224-. Cyclical equations, connection with method of least squares. Nekrasov, P. A. Eec. Mth. (Moscou) 12 (1385) 377-. Deduction of increase rates from tables. Everett, J. D. Nt. 60 (1899) 271. . Eunge, C. Nt. 60 (1399) 365-. A m O n applied to integration. Herapath, J. Tilloch Ph. Mg. 65 (1325) 321-. and Hargreave's form of it. Sums of powers of numbers etc. Scott, G. [1865] QJ. Mth. 8 (1867) 21-. Determination of functions from special values of variable. Chebuishev, P. L. (xn) [1369] Eec. Mth. (Moscou) 4 (1369-70) (Pt. 1) 231-. Difference, ith, deduced from nth differential. Picart, A. N. A. Mth. 12 (1373) 413-. Differences and differential coefficients, con- nexion between. Nielsen, H. P. N. Ts. Mth. 8 (B) (1397) 86-; Fschr. Mth. (1897)246. differentials of functions of zero. Hamilton, (Sir) W. E. [1831] Ir. Ac. T. 17 (1337) 235-. nth order. Spitzer, S. Grunert Arch. 33 (1359) 115-. method, application to series summed by help of impossible quantities. Gompertz, B. Phil. Trans. (1806) 174-. , construction of tables by. Gray, P. Assur. M ? . 13 (*1367) 61-, 149-, 293-; I. Act. J. 14 (*1869) 307-. , , 2 theorems. Knight, T. Phil. Trans. (1317) 234-. ,mth,ofO. Scott, G. QJ. Mth. 5 (1362) 323-. 1640 Differences, relation between various orders. Harvey, G. Thomson A. Ph. 7 (1816) 475-. , series of, theory. Werner, 0. Grunert Arch. 23 (1354) 231- ; 24 (1855) 90-. , , . Wallentin, I. G. Arch. Mth. Ps. 63 (1879) 56-. and Taylor's theorem. Adams, Jas. Thom- son A. Ph. 11 (1818) 275. , theory. Sharpe, S. Ph. Mg. 19 (1831) 137-. of 0. Alberti, V. G. Mt. 38 (1900) 11 1-. Differentiation by aid of finite differences and vice-versa. Lobatto, E. Crelle J. 16 (1837) 11-. , arithmetical. Philippoic, M. M. Fschr. Mth. (1899) 267. under sign of integration, extension to finite differences, with applications. Chio, F. Tor. At. Ac. Sc. 6 (1870-71) 194-. Empirical formulae. Kleiber, I. A. Es. Ps.-C. S. J. 22 (Ps.) (1890) 11-; J. de Ps. 10 (1891) 433-. , comparison. Runge, C. Z. Mbh. Ps. 45 (1900) 78-. Equation of finite differences, giving sums and differences of Ex. Gaudin, (Col.) . Les Mondes 3 (1863) 567-. Equations, algebraic, theorems. Mathieu, E. N. A. Mth. 15 (1856) 409-. of differences for approximating to differential equation. Coriolis, G. Liouv. J. Mih. 2 (1837) 230-. Error of linear interpolation. Bary, . Ger- gonne A. Mih. 21 (1830-31) 281-. Errors by interpolation from logarithmic and other tables. Woodward, E. S. A. Mth. 2 (1885-86) 54-. , logarithmic. Howe, H. A. A. Mth. 1 (1884-85) 126- ; 2 (1885-86) 39-, 77-; 3 (1887) 74. , theory, formula in. Makeham, W. M. I. Act. J. 28 (1890) 393-. Euler's summation formula. Landre, C. L. N. Arch. Wisk. 6 (*1880) 212-. Exponential functions and finite differences. Herschel, (Sir) J. F. W. [1815] Phil. Trans. (1816) 25-. Factorial notation. Elphinstone, H. W. QJ. Mth. 2 (1858) 122-, 254-. Faculties expressed by A m O n . Jeffery, H. M. QJ. Mth. 4 (1861) 364-. Figurate numbers and interpolation, short proofs of theorems. Soufflet, . Les Mandes 13 (1867) 336-. series. Smyth, B. B. Kan. Ac. Sc. T. 14 (1896) 29-. Finite difference equations, partial. Paoli, P. Mod. S. It. Mm. 8 (1794) 575-. integrals expressed by definite integrals. Tortolini, B. Tortolini A. 4 (1853) 209-. differences calculus. Eussell, W. H. L. M3ss. Mth. 11 (1882) 33-. applied to definite integrals. Paoli, P. Mod. S. It. Mm. 20 (1828) 255-. , A n (x m ), factor in general term of. Kohlau, E. Crelle J. 6 (1830) 255-. , determinants of functions in the. Bortolotti, E. Em. E. Ac. Line. Ed. 5 (1896) (Sem. 1) 254-. 122 1640 Calculus of Differences, etc. Finite differences calculus, formula in. Carr, T. I. Act. J. 14 (* 1869) 479-. Malmsten, C. J. Stockh. Ak. Hndl. (1844) 363-; Crelle J. 35 (1847) 55-. --- , formulae. Tortolini, P. A. Mt. 5 (1863) 181-. --- , , useful. Hill, G. W. Des MMnes Anal. 1 (1874) 141-. --- , methods. Carmichael, R. NH. Ev. 7 (1360) (P.) 78-, 170-. --- -, theorem, new. Booth, J. Ph. Mg. 19 (1841) 125-. , theorems, new, based chiefly on re- searches on complex variability. Casorati, F. A. Mt. 10 (1880-82) 10-. -- , and theory of numbers, contribution to. Busche, E. Mth. A. 53 (1900) 243-. -- , direct method. Adams, Jas. Thomson A. Ph. 16 (1820) 281-. -- , formula. Cayley, A. QJ. Mth. 2 (1858) 198-. -- , formulae. Cauchy, A. L. C. E. 19 (1344) 1179-. -- , integrals. Cauchy, A. L. C. E. 39 (1854) 21 i-. -- , . Bmmiakowtky, V. St Pet. Ac. Sc. Mm. 1 (1859) No. 9, 13 pp. , E. Les 701-. -- , . Barbier, Mondes 3 (1863) -- , . Thoman,F. C. E. 57 (1863) 778-. -- of powers, conversion to definite integrals. Cauchy, A. L. [1815] Par. EC. Pol. J. 17 (28 cah.) (1841) 147-. -- , return to original series from nth term of with series of differences. Mtiller, G. W. Grunert Arch. 1 (1341) 211-. , series connected with. Lukas, F. C. Wien Az. 32 (1395) 269-. -- , and Taylor's series. Carvallo, E. N. A. Mth. 10 (1391) 24-. or mixed differences, uestions depending on. Combescure, E. Par. EC. Norm. A. 3 (1874) 305-. Forms A" O x and congeners. Horner, J. QJ. Mth. 4 (1861) 111-, 204-. Formula for A0 i / i when and i are very large numbers. Cayley, A. [1387] Edinb. E. S. P. li (1388) 119-. Functions, approximate forms for representa- tion. Pincherle, S. Bologna Ac. Sc. Mm. 10 (1889) 77-. Harmonic series, representation by sequences of factors. Eogel, F. Arch. Mth. Ps. 9 (1890) 297-. Illusory interpolations, new examples. Meray, C. Ell. Sc. Mih. 20 (1396) 266-. Indeterminate analysis, application of calculus of finite differences. Piola, G. A. Sc. Lomb. Ven. 1 (1831) 101- ; Tortolini A. 1 (1350) 263-. Integrability of differential and finite difference equations, conditions. Mmich, S. R. Tor- tolini A. 1 (1350) 321-. Integral, finite, 2 n #a; expressed as a definite integral. Abel, N. H. Mg. Ntvd. 6 (1325) 182-. Interpolation 1640 Integral, finite, 2e*7/. Malmsten, C. J. Ups. N. Acta S. Sc. 12 (1844) 293-. Integration of difference equations reducible to linear form. Oltramare, G. As. Fr. C. E. (1893) (Pt. 2) 106-. in finite differences. Trembley, J. Berl. Mm. Ac. (1799-1300) 18-. of linear difference-equations. Oltramare, G. As. Fr. C. E. (1891) (Pt. 2) 66-; (1895) (Pt. 2) 175-. , use in arithmetical calculations. Cesdro, E. Palermo Cir. Mt. Ed. 1 (1387) 293-. Interpolated multiples. Soule, . Bordeaux S. Sc. Mm. 4 (1894) iii-. Interpolating functions. Cauchy, A. L. C. E. 11 (1840) 775-. . Genocchi, A. Tor. Ac. Sc. At. 13 (1877) 716- ; 15 (1880) 269-; C. E. 86 (1878) 466-. . Peano, G. Tor. Ac. Sc. At. 18 (1882) 573-. , elementary proof of a fundamental property. Schwarz, C. H. A. Tor. Ac. Sc. At. 17 (1381) 740-. INTEEPOLATION. Legendre, A. M. Con. des Temps (1819) 302-. Degen, C. F. As. Nr. 1 (1823) 231-. Clausen, T. Crelle J. 5 (1829) 305-. Encke, J. F. Berl. As. Jb. (1830) 265-. Cauchy, A. L. [1835] Mod. Mm. S. It. 21 (1836) 374-. Biot, J. B. C. E. 13 (1344) 545. Brassinne, E. Liouv. J. Mth. 11 (1846) 177-. Cauchy, A. L. Con. des Temps (1852) 129- ; C. E. 37 (1853) 64-. Lehmann, J. W. H. As. Nr. 39 (1855) 257-. Grunert, J. A. Grunert Arch. 32 (1859) H9-. Hermite, C. C. E. 48 (1359) 62-. Rouche, E. N. A. Mth. 13 (1859) 26-. Cebysev, P. St Pet. Ac. Sc. Mn. (Rs.) 4 (* 1864) (Suppl.) No. 5, 23 pp. Tisserand, F. C. E. 68 (1869) 1101-. Nell, A. M. Arch. Mth. Ps. 61 (1877) 135-. Bartl, C. [1877] Arch. Mth. Ps. 62 (1378) 202-. Laisant, C. A. As. Fr. C. E. (1891) (Pt. 2) 222-. Pincherle, S. Bologna Ac. Sc. Maa. 3 (1392) 293-. Borel, E. C. E. 124 (1397) 673-. Laisant, . As. Fr. C. E. (1897) (Pt. 2) 86-. Markov, A. St Pet. Ac. Sc. Mm. 6 (1398) No. 5, 69 pp. ; Fschr. Mih. (1898) 252-. Veltmann, W. Z. Mth. Ps. 44 (1899) 303- ; 45 (1900) 337. and application to cubic equations. Vieille, J. N. A. Mth. 10 (1851) 48-. applied to extraction of square and cube roots. Dolguschin, P. A. Fschr. Mth. (1898) 139. approximate. Bellavitis, G. Ven. At. (1858- 59) 1109-. fractions. Gyory, S. (xn) Mag. Ak. Ets. (1854) 165-. 123 1640 Interpolation Interpolation 1640 astronomical formulae. Le Verrier, U. J. J. (x) Par. Obs. A. 1 (1855) 73-; 2 (1856) 301+ [167] pp. ; 3 (1857) 199-, and [1]- ; 4 (1858) 1-, and [1]-; 5 (1859) 1- ; 6 (1861) 435pp.; 10 (1874) 1-, and [1]-; 11(1876) 560+ [136] pp. ; 12 (1876) 76+ [178] +A. 80 + [A. 286] pp. ; 13 (1876) 1-, and [1]- ; 14 (1877) Pt. 1, A. 1-, and [A. 1]-; Pt. 2, 1-, and [1]-. and Bernoulli's numbers. Lipschitz, R. 0. S. C. E. 86 (1878) 119-. Briggs' Log. Tables. Legendre, A. M. Con. des Temps (1817) 219-. and calculus of differences finite or infinite, principles. Ampere, A. M. Gergonne A. Mth. 16 (1825-26) 329-. certain forms. Bartlett, W. P. G, Bost. Mm. Am. Ac. 8 (1863) 435-. chemical and physical observations. Bartlett, W. P. G. Silliman J. 34 (1862) 27-. choice of best ordinate. Kleiber, I. A, Kazan S. Nt. (Ps.-Mth.) P. 8(1890) 232-; Fschr. Ps. (1890) (Ab. 1) 13. by circular functions. Gomes Teixeira, F. N. A. Mth. 4 (1885) 351-. curves of 3rd and 5th order. Dupin, C. C. E. 25 (1847) 769-. differences. Burckhardt, J. K. Zach M. Cor. 12 (1805) 332-. at equidistant values. Cebysev, P. St Pet. Ac. Sc. Mm. (Rs.) 25 (*' 1874-75) (Suppl.) No. 5, 30 pp. formula. Tchebicheff, P. [1854] St Pet. Ac. Sc. fill. 13 (1855) 210-. . Brioschi, F. Tortolini A. 2 (1859) 132-. . Gomes-Teixeira, F. Liege S. Sc. Mm. 10 (1883) (No. 10) 7 pp. , Cauchy. Schott, C. A. U. S. Coast Sv. Ep. (1860) 392-. , Gauss, infinitely extended. Bendixson, I. Acta Mth. 9 (1887) 1-. , Hermite, expressed algebraically. Schering, E. C. E. 92 (1881) 510-. , Lagrange. Dirksen, E. H. Crelle J. 1 (1826) 22 1-. , . Gamier, J. G. Brux. Ac. Bll. 6 (1839) (pte. 2) 19-. , . StudnicJca, F. J. Casopis 2 (* 1873) 82-; Fschr. Mth. (1873) 122-. , . Hermite, C. [1877] Crelle J. Mth. 84 (1878) 70-. , . Stieltjes, T. J. (jun.\. [1881] Amst. Ak. Vs. M. 17 (1882) 239-. , . Bendixson, I. C. E. 101 (1885) 1050-, 1129-. , . Echols, W. H. A. Mth. 8 (1893-94) 22-. , . Lewicki, W. Wiad. Mt. 3 (1899) 264-; Fschr. Mth. (1899) 244. . , generalisation. Williot, V. Bll. Sc. Mth. 14 (1890) 218-. . , remainder in. Sonin, N. J. Fschr. Mth. (1899) 242. , (mortality-tables). Selling, E. Crelle J. Mth. 106 (1890) 193-. , new. Alexander, J. H. Silliman J 7 (1849) 14-. formula, Newton. Oppermann, L. I. Act. J. 15 (*1870) 145-, 177-. , . Jensen, J. L. W. V. Kjrfb. Ov. (1894) 246-. , Pictet. SzUy, K. [1879] J. de Ps. 9 (1880) 303-. , Prony. Minich, S. R. Ven. At. 13 (1867- 68) 1153-. , Tchebicheff's (Cebysev). Tchebicheff, P. N. A. Mth. 18 (1859) 193-. formulae. Ferrel, W. [1865] Am. Ac. P. 7 (1868) 2-. . Carvallo, J. C. E. 106 (1888) 346-. . Radau, R. Bll. As. 8 (1891) 273-, 325-, 376-, 425-. , application. Ackland, T. G. I. Act. J. 32 (1896) 286-. and application to solution of equations. Olivier, L. Crelle J. 2 (1827) 197-. for functions of several variables. Kronecker, L. Berl. Mb. (1865) 686-. , Gauss, for n=7, 8 and 9. Perott, J. QJ. Mth. 25 (1891) 200-. , Hansen. Mollweide, C. As. Nr. 2 (1824) 259-. , Lagrange. Lewicky, W. Arch. Mth. Ps. 17 (1900) 214-. , , generalised, applications. Puzyna, J. Krk. Ak. (Mt.-Prz.) Pam. 14 (1888) 1-; Fschr. Mth. (1888) 251-. , and Newton. Gerono, G. C. N. A. Mth. 16 (1857) 317-, 358-. , . Transon, A. (Prof.). N. A. Mth. 19 (I860) 248-. , new. Astrand,J.J. ChristianiaF. (1890) (Ov.) 5-. , various. Cauchy, A. L. C. E. 12 (1841) 283-. by fractional rational function. Jacobi, C. G. J. Crelle J. 30 (1846) 127-. functions of a variable. Sarrus, F. (vi Adds.) Strasb. S. Sc. N. Mm. 2 (1832) 203-. in heat. Bary, E. C. E. 3 (1836) 378. of implicit functions, and calculation of roots. Maksimovich, V. P. (xn) [1881] Kazan S. Nt. (Ps.-Mth.) P. 1 (1883) [No. 5] (Suppl.) 23 pp. and integration in series. Buchwaldt, F. Ts. Mth. 5 (1887) 79-, 97-; Fschr. Mth. (1887) 281-. of irrational and logarithmic functions by numerical tables. Lefort, P. A. F. [1857] Edinb. E. S. P. 8 (1875) 602-. inverse. Siebdrat, (Dr.). Dresden Sb. Isis (1871) 233-. kinematic. Laisant, C. A. As. Fr. C. E. (1890) (Pt. 2) 70-. Lagrange's theorem, generalisation. Sylvester, J. J. Ph. Mg. 6 (1853) 374-. by least squares. Gergonne, J. D. Gergonne A. Mth. 6 (1815-16) 242-. . Ceby'sev, P. L. (Tchebicheff). St Pet. Ac. Sc. Mm. 1 (1859) No. 15, 24 pp. . Borchardt, C. W. [1860] Crelle J. 58 (1861) 270-. 124 1640 Interpolation by least squares, formulae. CebySev, P. L. (Chebuishev). (xn) Bee. Mth. (Moscou) 3 1868) (Ft. 1) 157-. , (Tchebychef). Schols, C. M. [1375] Amst. Ak. Vs. M. 9 (1876) 301- ; Arch. Neerl. 12 (1877) 102-. and new method, comparison. Cauchy, A. L. C. B. 37 (1853) 100-. , legitimacy. Meray, C. Par. EC. Norm. A. 1 (1884) 165-. as means of calculating irrationals. Opper- mann, L. Kjob. Ov. (1375) 18-. and mechanical quadrature. Grunert, J. A. Grunert Arch. 20 (1853) 361-. quadratures in astronomy. Santini, G. Yen. Mm. I. 13 (1S66) 177-. method. Uylenbroek, P. J. Leijd. A. Ac. (1820-21) 77 pp. , Cauchy. Grunert, J. A. Grunert Arch. 2 (1342) 41-. , . Seeliger, H. As. Nr. 96 (1880) 235-. , , modified by method of least squares. Despeyrous, C. Toul. Mm. Ac. 5 (1867) 251-. , general. Buquoy, G. von. Oken Isis (1827) 889-. , geometrical. Genge, C. Ziir. Vjschr. 31 (1886) 268-. , new. Astrand, J. J. Leip. As. Gs. Vjschr. 10 (1875) 279-. , general. McGlintock, E. Am. J. Mth. 2 (1879) 307-. , Tschebyschew. Backlund, 0. St Pet. Ac. Sc. Bll. 29 (1384) 477-. methods. Le Verrier, U. J. J. (vi Adds.) Par. A. Obs. 1 (1855) 73-; 2 (1856) 1-; 3 (1857) 199- ; 4 (1358) 1- ; 5 (1359) 1-; 6 1861).!-. , special. Niven, W. D. [1377] Mess. Mth. 7 (1878) 35-. mixed. Cousinery, B. E. Ing. 1 (1848) 125-. new formula. Sprague, T. B. I. Act. J. 22 (* 1881)270-. between observed values. Tchebiche/, P. St Pet. Ac. Sc. Bll. 16 (1858) 353- ; St Pet. Ac. Sc. Mm. 1 (1859) No. 5, 81 pp. by parabolic curve. Michal, A. C. B. 37 (1853) 185-. curves. Michal, A. A. Pon. Chauss. 10 (1865) 59-. parabolic, by least squares method. Degen, C. F. [1823] St Pet. Mm. Sav. 6tr. 1 (1831) 12-. , . Grave, P. Kazan S. Nt. (Ps.-Mth.) P. 7 (1889) 270- ; Fschr. Mth. (1889) 220. periodic functions. Weyer, G. D. E. Leip. As. Gs. Vjschr. 22 (1887) 292-. philosophy. Gergonne, J. D. Gergonne A. Mth. 5 (1814-15) 252-. in physics. Mollweide, C. Gilbert A. 62 (1819) 422-. problem. Korkine, A. Bll. Se. Mth. As. 6 (1882) 228-. . Badau, R. Bll. As. 4 (1887) 515-. . Bohlmann, G. Gott. Nr. (1899) 260-. . Pincherle, S. Palermo Cir. Mt. Bd. 14 (1900) 142-. Calculus of Differences, etc. 1640 Interpolation, continued. problem, Cauchy. Netto, E. Mth. A. 42 (1893) 453-. , general. Greatheed, S. S. (vi Adds.) Camb. Mth. J. 3 (1843) 97-. problems, mechanical solution. Jung, G. Mil. I. Lomb Bd. 13 (1380) 226-. proof of formula. Berridge, G. W. I. Act. J. 14 ( 1869) 244-. with proofs of methods of Briggs, Mouton, and Newton. Maurice, F. Con. des Temps (1847) 181-. questions of fluid motion. Barre de Saint- Venant, . C. B. 17 (1343) 1108-. recurring series. Maillet, E. N. A. Mth. 14 (1395) 473-. with reference to development and differentia- tion. Roberts, S. [1865] QJ. Mth. 7 (1866) 184- ; 8(1867)52-, 139-. results. Thiele, T. N. (xn) Ts. Mth. 1 (1883) 183- ; Fschr. Mth. (1884) 198-. to second order without change of sign. Airy, (Sir) G. B. As. S. M. Not. 26 (1866) 236-. series. Bangma, 0. S.,(& J. F. Keyser. Amst. Mengelwerk 2 (1316) 42-. A v A,A 2 ,...A 1 A n A 3 ...A n . Svanberg, A. F. Stockh. Ak. Hndl. (1828) 1-. in statistics. Westergaard, H. I. Act. J. 32 (1896) 276-. by successive differences, exactness. Mollweide, C. Zach M. Cor. 21 (1810) 331-. summation and adjustment of tables. Wool- house, W. S. B. Assur. Mg. 11 (*1864) 61-, 301- ; 12 (*1866) 136-. and summation of numerical series, method. Donald, J. Glasg. Ph. S. P. 9 (1875) 272-. symmetric function. Borchardt, C. W. Berl. Ab. (1860) (Mth.) 1-. table, general. Amante, F. (xn) [1844] Nap. Ac. Pont. At. 4 (1351) 251-. tables. Heawood, P. J. Mess. Mth. 27 (1898) 121-. , adequacy. Jelinek, V. Casopis 21 (1892) 31- ; Fschr. Mth. (1392) 227. theorem of Hermite. Gomes-Teixeira, F. Crelle J. Mth. 100 (1887) 83-. theory. Lerch, M. Prag Ceske Ak. Fr. Jos. Bz. (Trida 2) 1 (1892) Art. 32, 15 pp. ; Fschr. Mth. (1892) 227. . loachimescu, A. G. Bucarest S. Sc. Bll. 6 (1897) 472-. of values depending on two variables. Seydler, A. Prag Sb. (1889) 2 (Mth.-Nt.) 337-; Fschr. Mth. (1889) 253-. Inverse calculus of differences, formula. Prouhet, E. N. A. Mth. 10 (1851) 186-. , notes. Combescure, E. C. B. 74 (1872) 454-. method of finite differences. Brinkley, J. Phil. Trans. (1807) 114-. Least squares method, case where number of unknowns is large. Nekrasov, P. A. Bee. Mth. (Moscou) 12 (1885) 189-. Legendre's functions, use of, for interpolation. Seeliger, H. Munch. Ak. Sb. 20 (1891) 499-. 125 1640 Calculus of Differences, etc. Linear difference forms with commutative factors of prime degree. Torelli, G. Nap. Ed. 35 (1896) 238-. Measurement of cylinder. Rasch, J. W. N. Arch. Wisk. 7 (*1881) 117-. Mortality tables, theory. Samot, D.J. A. N. Arch. Wisk. 3 (M877) 145- . Numerical solution and reversion of series, remark on Woolhouse's paper. De Morgan, A. I. Act. J. 15 (*1870) 327-. Partial differences. Euler, L. [1777] St Pet. Ac. Sc. N. Acta 15 (1806) 3-. summation. Helm, G. Z. Mth. Ps. 22 (1877) 400-. Plurality of integrals in calculus of differences. Poisson, S. D. [1800] Par. EC. Pol. J. 4 (1801-2) (11 cah.) 173-. Progression, arithmetical, successive differences of powers of terms. Lhuilier, S. Gergonne A. Mth. 2 (1811-12) 185-. Progressions, arithmetical, whose terms are only approximately known. Lucas, F. C. B. 96 (1883) 1026-. with 2nd and 3rd differences. Jadanza, N. G. Mt. 6 (1868) 375-; 7 (1869) 17-. Quadrature, arithmetical. Danielsson Hill, C. J. Stockh. Ak. Hndl. (1854) 405-. , Gauss's formula, and Hermite's formula of interpolation. Mansion, P. C. E. 104 (1887) 488-. , , remainder in. Mansion, P. Brux. Ac. Bll. 11 (1886) 293-. Quadratures and interpolation, report on present knowledge of actual application. Merrifteld, C. W. B. A. Bp. (1880) 321-. Eational approximation to non-rational homo- geneous function. Eesal, H. A. C. E. 80 (1875) 1185-. fractions, integration by finite differences of. Beyer, E. I. von. (xn) Bee. Mth. (Moscou) 4 (1869-70) (Pt. 1) 297-; 5 (1870) (Pt. 1) 64-, 145-. Beduction of higher differences when the constant increment is changed. Gudermann, C. As. Nr. 26 (1848) 315-. Beversion of series and its application to solution of numerical equations. Hagen, J. G. Am. Ph. S. P. 21 (1884) 93-. Second differences in interpolation, easy method of taking into account. Nell, . Arch. Mth. Ps. 70 (1884) 302-. Series of numbers and their interpolation. Ruff, H. Arch. Mth. Ps. 17 (1900) 426-. , summation by differences. Fuss, N. St Pet. Ac. Sc. Mm. 11 (1830) 246-. Simpson's formula and another. Catalan, E. C. N. A. Mth. 10 (1851) 412-. for approximation to areas. Daug, H. T. Stockh. Ofv. 17 (1860) 429-. method of interpolation, simplification. Parmentier, T. N. A. Mth. 15 (1876) 241-. rule, history. Heinrich, G. Bb. Mth. 1 (1900) 90-. Solution of linear equations with large number of unknowns by successive approximation, Seidel's process. Mehmke, R. Eec. Mth. (Moscou) 16 (1893) 342-. Linear Substitutions 2000 Solution of system of linear equations by succes- sive approximation. Mehmke, R. , & Nekrasov, P. A. [1892] Eec. Mth. (Moscou) 16 (1893) 437-. Stirling's formula. Serret, J. A. C. E. 50 (1860) 662-. . Bonnet, 0. C. E. 50 (1860) 862-. , elementary proof. Cesdro, E. N. A. Mth. 2 (1883) 43-. , extension. Hermite, C. Mth. A. 41 (1893) 581-. and other interpolation formulae. Stuart, G. H. Mess. Mth. 19 (1890) 19-. series generalised, applications. Raabe, J. L. Zur. Mt. 3 (1853-55) 332-. for interpolation by symmetric differences. Merrifield, C. W. [1866] Mess. Mth. 4 (1868) 110-. Successive differences of observations. Breger, . C. B. 93 (1881) 1119-. Sums and differences, theory, higher examples. Vdllas, A. Mag. Ak. Ets. (1850) 381-. ,. (Schlomilch.) Oettinger, L. Grunert Arch. 13 (1849) 36-. Table of A TO O n -7-n(m). Cayley, A. [1879] Camb. Ph. S. T. 13 (1883) 1-. Time growth equation of trees. Blonay, H. de. Laus. S. Vd. Bll. 28 (1892) 207-. of vegetation in forests, interpolation applied to. Regneault, E. E. Nancy Mm. S. Sc. (1838) 169-. observations, interpolation. Thiele, T. N. Leip. As. Gs. Vjschr. 22 (1887) 302-. Trigonometrical formulae of interpolation. Fouret, G. C. E. 99 (1884) 963-, 1011-, 1044, 1062-, 1140-. Trigonometry, finite differences in. Cagnoli, A. Mod. Mm. S. It. 8 (1799) 2 14-. Value of A'HV'VM" 1 where m and n are very large. Tait, P. G. Edinb. Mth. S. P. 5 (1887) 83-. Values of A'H), tables. Herschel, J. \attributed to Lacroix in (m)]. N. A. Mth. 13 (1854) 272-. Variations, finite, analytical. Lorgna, A. M. Verona S. It. Mm. 4 (1788) 156-. Linear Substitutions. 2000 General. Distributive operations and homography. Pin- cherle, S. Mil. I. Lomb. Ed. 29 (1896) 397-. Extraction of roots by substitutions. Levanen, S. Helsingf. Ofv. 34 (1892) 28-; Fschr. Mth. (1892) 135. Formulas given by Eisenstein and Hesse, note on two. Cayley, A. Crelle J. 29 (1845) 54-. Integral functions of a system of mn variables which form m rows and n columns. Mertens, F. Krk. Ak. (Mt.-Prz.) Pam. 17 (1890) 143- ; Fschr. Mth. (1890) 186-. Linear substitutions. Stephanos, C. Liouv. J. Mth. 6 (1900) 73-. 126 2010 Determinants Linear substitutions applied to theory of groups. Laurent, H. N. A. Mth. 16 (1897) 149-. , reduction. Jordan, C. Par. EC. Pol. J. 29 ( = Cah. 48) (1880) 15 1-. transformations of quadratic form. Boole, G. Camb. Mth. J. 4 (1845) 167-. , general theory. Boole, G. Camb. Mth. J. 3 (1843) 1-, 106-. , points in theory. Philippot, I. H. Liege S. Sc. Mm. 20 (1898) No. 9, 14 pp. Eational reversible substitutions. Kraus, L. Prag Sb. (1882) 338-; (1883) (Mth.-Vortr.) 187-. Representation of binary homographies by points of space, with application to spherical rotations. Stephanos, C. Mth. A. 22 (1883) 299-. Salmon's "Higher Algebra." Bellavitis, G. Yen. Mm. I. 9 (1860) 237-. Space, division by linear transformation groups. Dyck, W. Leip. Mth. Ps. B. 35 (1883) 61-. Theorem on two contrary substitutions. Lip- schitz, E. 0. S. Am. J. Mth. 1 (1878) 336-. . Sylvester, J. J. Am. J. Mth. 1 (1878) 341-. Umbral notation. Barley, (Rev.) E. B. A. Bp. (1887) 600-. 2010 Determinants. Adjoint minors. Landsberg, G. Crelle J. Mth. 109 (1892) 225-. Adjugate determinant. Huir, T. [1894] Edinb. B. S. P. 20 (1895) 323-. Algebraic determinant, a special. Weihrauch, K. Z. Mth. Ps. 36 (1891) 34-. determinants, reduction. Eussell, W. H. L. E. A. Bp. (1885) 910-. expressions unaltered by certain substitu- tions. Glaisher, J. W. L. [1880] Mess. Mth. 10 (1861) 60-. reversibility, new theorem. Andre, D. Par. S. Mth. Bll. 24 (1896) 136-. Alternant of even number of letters, Jacobi's form. Muir, T. [1900] Edinb. B. S. P. 23 (1902) 133- . , multiplication by symmetric function. Muir, T. [1899] Edinb. B. S. P. 22 (1900) 539-. Alternants. Muir, T. [1879] Edinb. B. S. P. 10 (1880) 102-. . Starkov. A. N. Es. S. Nt. Mm. (Mth.) 10 (1889) 143-. . Muth, P. Crelle J. Mth. 122 (1900) 89-. , a class of. Loria, G. Prag Sb. (1897) (Mth.-Nt.) No. 57, 13 pp. , constant multiples of the difference product of the variables. Anglin, . [1888] Edinb. B. S. P. 15 (1889) 468-. and elliptic functions, identities connected with. Muir, T. [1900] Edinb. B. S. T. 40 (1905) 187-. , history. Muir, T. [1900] Edinb. B. S. P. 23 (1902) 93-. , summation of certain series. Anglin, A. H. [1887] Edinb. E. S. P. 14 (1888) 194-. Determinants 2010 Alternants, theorems connected with. Anglin, A. H. [1886-88] Edinb. E. S. P. 13 (1886) 823- ; 15 (1889) 381-. of the 3rd order, a formula of reduction for. Johnson, W. W. Am. J. Mth. 7 (1885) 347-. , geometrical representation. Johnson, W. W. QJ. Mth. 21 (1886) 2 17-. 4th order, calculation of co-factors. Johnson, W. W. Am. J. Mth. 7 (1885) 380-. Alternating functions. Cauchy, A. L. [1812] Par. EC. Pol. J. 10 (1815) (17 e cah.) 29-. . Jacobi, C. G. J. Crelle J. 22 (1841) 360-. and applications. Cauchy, A. L. C. E. 10 (1840) 178- ; 13 (1841) 939-. , a class of. Muir, T. [1887] Edinb. B. S. T. 33 (1888) 309-. of n variables. Scott, E. F. [1881] Mess. Mth. 11 (1882) 98-. , reduction to alternants. Johnson, W. W. Am. J. Mth. 7 (1885) 345-. used in elimination. Cauchy, A. L. C. B. 12 (1841) 414-. sums. Cauchy, A. L. C. B. 13 (1841) 939-. Arithmetical determinant, value of certain. Smith, H. J. S. L. Mth. S. P. 7 (1875-76) 208-. determinants, certain. Cesdro, E. G. Mt. 23 (1885) 182-, 367 ; Em. B. Ac. Line. Ed. 1 (1885) 709-, 711-. of higher dimension. Gegenbauer, L. Wien Ak. Sb. 101 (1892) (Ab. 2 a) 425-. Bernoullian number expressed as a combina- tional determinant. Dupuis, N. F. Cn. B. S. P. & T. 7 (1890) (Sect. 3) 20-. Bernoulli's and Euler's numbers, representa- tion by determinants. Sachse, A. Arch. Mth. Ps. 68 (1882) 427-. Bibliography. Muir, T. QJ. Mth. 18 (1882) 110- ; 21 (1886) 299-. Binary determinants. Brioschi, F. Crelle J. 52 (1856) 133-. Bipartite functions. Muir, T. Edinb. E. S. T. 32 (1887) 461-. and determinants, new relations between. Muir, T. L. Mth. S. P. 16 (1884-85) 276-. Bordered determinants. Le Paige, C. M. M. H. H. Par. S. Mth. Bll. 8 (1880) 128-. . Arnaldi, M. G. Mt. 34 (1896) 209-. skew determinant. Muir, T. Edinb. E. S. P. 21 (1897) 342-. symmetric determinant, Cayley's theorem. Muir, T. QJ. Mth. 18 (1882) 46-. Cauchy 's determinant and aleph functions. Crocchi, L. [1878] G. Mt. 17 (1879) 2 18-, 380. Circle touching three circles on a sphere. Trzaska, W. Par. T. Nauk. Sc. Pam. 1 (*1871) 123- ; Bll. Sc. Mth. As. 6 (*1874) 153. Circulant determinants. Torelli, G. Nap. Ed. 21 (1882) 83-. Circulants. Studnicka, F. J. Mh. Mth. Ps. 10 (1899) 193-. , detached theorems on. Muir, T. Edinb. E. S. T. 32 (1887) 639-. 127 2010 Determinants Circulants, final expansion. Muir, T. Mess. Mth. 14 (1885) 169-. of odd order. Muir, T. QJ. Mth. 18 (1882) 261-. , resolution. Muir, T. Edinb. E. S. P. 21 (1897) 369-. Co-axial minors of determinant of the fourth order. Muir, T. Edinb. B. S. T. 39 (1900) 323-. , relations between. Nanson, E. J. Ph. Mg. 44 (1897) 362-. , expressibility of determinant in terms of its. Muir, T. Ph. Mg. 38 (1894) 537-. Combination of determinants, theorem. Syl- vester, J. J. Camb. and Dubl. Mth. J. 8 (1853) 60-. Combinations applied to theory of determinants. Picquet, H. Par. EC. Pol. J. 28 (= Cah. 45) (1878) 201-. Combinatory analysis of determinants. Deruyts, J. Liege S. Sc. Mm. 11 (1885) No. 13, 11 pp. determinants. Zmurko, L. [1866] Wien D. 27 (1867) (2 te Ab.) 63-. Commutants, property. Cayley, A. Ph. Mg. 30 (1865) 411-. Composite determinants. Sylvester, J. J. Crelle J. Mth. 88(1880)49-. , Sylvester's memoir, remark. Borchardt, C. W. Crelle J. Mth. 89 (1880) 82-. Composition of determinants and bilinear forms, a mode of. Stephanos, C. G. Mt. 36 (1898) 376-. systems of n 2 magnitudes with them- selves. Kronecker, L. Berl. Ak. Sb. (1890) 1081-. Compound determinants. Scott, E. F. [1881] Mess. Mth. 11 (1882) 96-. . Van Velzer, C. A. (xn) [1881] J. H. Un. Cir. [1] (1882) 132. . Scott, E. F. [1882] L. Mth. S. P. 14 (1882-83) 91-. . Van Velzer, C. A. Am. J. Mth. 6 (1884) 164-. . Metzler, W. H. Am. J. Mth. 16 (1894) 131- ; 20(1898)253-. , theorems on a special kind. Hunyady, J. (xn) Mag. Tud. Ak. Etk. (Mth.) 7 (1881) (No. 21) 15 pp. Concomitant determinants, certain. Eussell, J. W. L. Mth. S. P. 28 (1897) 430-. Constituents, elements, and minors of deter- minant, number of. Cunningham, A. J. Sc. 4 (1874) 212-. Continuant, condensation of. Muir, T. Edinb. Mth. S. P. 2 (1884) 16-. , development. Mollame, V. Kv. Mt. 3 (1893) 47-. , the discriminant of a special quadratic form. Cesdro, E. Mathesis 12 (1892) 5-. Continuants. Muir, T. [1874] Edinb. E. S. P. 8 (1875) 229-, 380-. , note. Sylvester, J. J. Mess. Mth. 8 (1879) 187. , properties. Studnicka, F. J. Prag Sb. (1886) (Mth.-Nt.) 3-. , roots of certain. Segar, H. W. Mess. Mth. 22 (1893) 171-. Determinants 2010 Continuants, theorem. Muir, T. Ph. Mj. 3 (1877) 137-, 360- . Convergence of infinite determinants. Koch, H. von [1896] Stockh. Ak. Hndl. Bh. 22 (Afd. 1) (1897) No. 4, 31 pp. Coordinates, curvilinear. Combescure, E. Par. EC. Norm. A. 4 (1867) 93-. Cramer's determinants. Cantor, M. N. A. Mbh. 14 (1855) 113-. , or algebraic resultants. Le Cointe, J. L. A. (vra) A. Mt. 4 (1861) 233-. Cubic determinants. Armenante, A. [1866] G. Mt. 6 (1868) 175-. . Padova, E. G. Mt. 6 (1368) 182-. , forms of. Scott, E. F. [1881] L. Mth. S. P. 13 (1881-82) 33-. of infinite order. Cazzaniga, T. Mth. A. 53 (1900) 272-. and others of higher class, and deter- minants of alternate numbers. Scott, E. F. [1879] L. Mth. S. P. 11 (1879-80) !7-- , theory. Sziits, M. Mth. Termt. Ets. 8 (1890) 220- ; Mth. Nt. B. Ung. 8 (1891) 199-. Decomposition of systems of n 2 magnitudes, and application to invariants. Kronecker, L. Berl. Ak. Sb. (1889) 479-, 603-. Deduction of certain determinants from others. Segar, H. W. Mess. Mth. 22 (1893) 57-. Determinant, a. Stern, M. A. Crelle J. 73 (1871) 374-. , . Glaisher, J. W. L. As. Fr. C. E. 6 (1877) 177-. , . Minozzi, A. G. Mt. 16 (1878) 148-. analysis. Cauchy, A. L. C. R. 42 (1856) 366-. of any degree, development. Cadenat, A. As. Fr. C. E. (1900) (Pt. 2) 241-. , application of a. Zehfuss, G. Schlomilch Z. 7 (1862) 439-. , . Cherriman, J. B. (xn) [1882] Cn. B. S. P. & T. 1 (1883) (Sect. 3) , . Legoux, A. QJ. Mth. 19 (1883) 41-. with binomial elements, development. Al- beggiani, M. G. Mt. 10 (1872) 279-. , calculation of a. Lemonnier, H. N. A. Mth. 18 (1879) 518-. , a certain. Zehfuss, G. Schlomilch Z. 3 (1858) 298-. , . Gegenbauer, L. Wien Ak. Sb. 96 (1888) (Ab. 2) 489-. , . Laisant, C. A. Par. S. Mth. Bll. 17 (1889) 104-. , . Bourlet, G. N. A. Mth. 16 (1897) 369-. , , theorem on. Mansion, P. Brux. Ac. Bll. 46 (1878) 892-. , development of a. Brioschi, F. [1857] Tortolini A. 1 (1858) 9-. , . Humbert, E. N. A. Mth. 4 (1885) 289-. , particular. Caldarera, G. Catania Ac. Gioen. At. 7 (1894) Mm. 8, 15 pp. 128 2010 Determinants Determinant, evaluation of a certain. Cayley, A. QJ. Mth. 2 (1358) 163-. , . Dostor, G. Arch. Mth. Ps. 64 (1879) 57-. , . Berry, A. Camb. Ph. S. P. 10 (1900) 2-. , . Crawford, L. Edinb. Mth. S. P. 18 (1900) 25-. expressions for Laplace's coefficients, Ber- noulli's and Euler's numbers, etc. Glaisher, J. W. L. [1876-79] Mess. Mth. 6 (1877) 49-; 7 (1378) 160- ; 8 (1879) 158-. the sum of a harmonical progression. Muir, T. [1877] Edinb. B. S. P. 9 (1878) 361-. formed bv bordering product of two deter- minants. Muir, T. Mess. Mth. 11 (1882) 161-. forms and their applications. Echols, W. H. A. Mth. 6 (1891-92) 135- ; 7 (1892-93) 11-, 93-, 109-. of four points in plane, with respect to a 5th. Laisant, C. A. Lisb. J. Sc. Mth. 5 (1898) 205-. , geometrical and statical interpretation of a. Walton, W. QJ. Mth. 4 (1861) 252-. identities in study of perspective triangles. Caspary, F. [1881] Crelle J. Mth. 95 (1883) 36-. , maximum value of a certain. Davis, E. W. (xn) [1882] J. H. Un. Cir. [2] (1883) 22. minors, certain aggregates of. Muir, T. [1900] Edinb. K. S. P. 23 (1902) 142-. , new development. Shaw, J. B. [1900] N. Y. Am. Mth. S. fill. 7 (1901) 244, 256-. notations and evaluation of a special deter- minant. Moore, E. H. A. Mth. 1 (1900) 177-. occurring in calculation of symmetric functions. Gubler, E. Ziir. Vjschr. 35 (1890) 79-. with polynomial elements, development. Albeggiani, M. [1874] G. MS. 13 (1875) 1-. , properties of. Wisselink, D. B. N. Arch. Wisk. 3 (* 1877) 84-. , property of a. Sardi, C. G. Mt. 2 (1864) of sines and cosines. Scott, E. F. Mess. Mth. 10 (1381) 142-. Smith and Mansion. Cesar -o, E. Par. EC. Norm. A. 2 (1885) 425-. , special form. Glaisher, J. W. L. [1878] QJ. Mth. 16 (1879) 15-. of system compounded of two others, repre- sentation. Hensel, K. Acta Mth. 14 (1890- 91) 317-. of functions. Bertrand, J. Liouv. J. Mth. 16 (1851) 212-. , vanishing. Heawood, P. J. Mth. Gz. 1 (1900) 344. with vanishing diagonal, Baltzer on number of terms. Monro, C. J. [1872] Mess. Mth. 2 (1873) 38-. | a rs | , number of terms when a rr = 0. Hansted, B. G. Teix. J. Sc. 2 (1880) 154-. of 4th order, unique property of an axi- symmetric. Muir, T. A. Mth. 1 (1884- 85) 31-. Determinants 2010 Determinant of 6th order, peculiar. Muir, T. Ph. Mg. 31 (1391) 429-. , development of a special. Muir, T. Mess. Mth. 13 (1884) 95-. mnth order, development. Muir, T. Edinb. R. S. T. 39 (1900) 623-. Determinantal form of highest common factor. Scheibner, W. Leip. Mth. Ps. B. 40 (1888) Determinants. Jacobi, C. G. J. Crelle J. 22 (1841) 285-. . Catalan, E. C. Brux. Ac. Bll. 13 (1846) 534-. . Bazin, . Liouv. J. Mth. 18 (1851) 145-. . Terquem, 0. N. A. Mth. 10 (1851) 124-. . Bazin, . Liouv. J. Mth. 19 (1854) 209-. . Newman, F. W. B. S. P. 8 (1856-57) 426-. . Oliver, J. E. Camb. (M.) Mth. M. 3 (1860) 86-. . Battaglini, G. Nap. Ed. 1 (1862) 101-. . Tait, P. G. Mess. Mth. 1 (1862) 25-. . Homer, J. [1865] QJ. Mth. 8 (1867) . Battaglini, G. G. Mt. 9 (1871) 136-. . Baltzer, R. (ix) Leip. Mth. Ps. B. 25 (1873) 523-. . Scott, R. F. Mess. Mth. 8 (1879) 182- ; 12 (1883) 105-. . Lindelof, L. L. Helsingf. Ofv. 22 (1880) 123-. . Young, T. E. I. Act. J. 26 (1887) 149-. . Gegenbauer, L. Wien Ak. Sb. 96 (1888) (Ab. 2) 5-; 97 (1889) (Ab. 2a) 154-. . Ponte Horta, F. da. [1890] Lisb. J. Sc. Mth. 2 (1892) 67-. . Campbell, J. E. [1892] L. M,h. S. P. 24 (1893) 67-. . Schendel, L. Z. Mth. Ps. 38 (1893) 84-. of alternate numbers. Spottiswoode, W* L. Mth. S. P. 7 (1875-76) 100-. in analytical mechanics. Cohen, A. Phil. Trans. (1862) 469-. , application. Jamet, V. N. A. Mth. 1& (1877) 372-. , to contact of circles and spheres. Bauer > C. W. N. A. Mth. 19 (1860) 440-. , curves of 4th order, etc. Hesse, L. 0. [1853] Crelle J. 49 (1855) 243-. , moments. Durrande, H. N. A.. Mth. 12 (1873) 265-. , , note. Gerono, G. C. N. A. Mth. 9 1 (1870) 392-. , applications. Rubini, R. Tortolini A. 8: (1857) 179-. , . Neuberg, J. J. B. (xn) Mathesis 3 (1383) 29-. , to algebra and geometry. Versluys, J. Arch. Mth. Ps. 50 (1869) 157-, 210- ; 51 (1870) 49-; 53 (1871) 137-. , geometry. Joachimstluil, F. Crelle J. 40 (1850) 21-. , . Mertens, F. (vm) [1873] Crelle J. 77 (1874) 102-. , 6 metrical). Frobenius, G. [1874] Crelle J. Mth. 79 (1875) 185-. VOL. I. 129 2010 Determinants Determinants, applications to geometry of n dimensions. Rahusen, A. E. Delft 6c. Pol. A. 4 (1888) 104-. arising in cyclotomy. Hermes, J. Arch. Mth. Ps. 6 (1888) 276-. of binomial coefficients. Zeipel, E. V. von. Lund Acta Un. 2 (1865) (Mth.) No. 2, 68 pp. ; 8 (1871) (Mth.) No. 3, 36 pp. . Gilnther, S. Z. Mth. Ps. 24 (1879) 96-. . Studnitka, F. J. Prag Sb. (1879) 292-. , Borchardt's theory. Lerch, M. Casopis 23 (1894) 76-; Fschr. Mth. (1893-94) 213. , calculation. Jiirgens, E. D. Nf. Vh. (1900) (Th. 2, Htilfte 1) 7-. , certain. Eugenio, V. G. Mt. 8 (1870) 285-. , . Glaisher, J. W. L. B. A. Bp. (1876) (Sect.) 13-. , . Voss, A. E. Mth. A. 13 (1878) 161-. , . Escherich, G. von. Mh. Mth. Ps. 3 (1892) 68-. of certain class (Grevy's), theorems on. Bottcher, L. E. Krk. Ak. (Mt.-Prz.) Bz. 18 (1901) 382-; Crc. Ac. Sc. Bll. (1900) 227-. , class. Zeipel, E. V. von. Stockh. Ofv. 19 (1862) 439-. , . Dickson, J. D. H. [1878] Edinb. B. S. T. 28 (1879) 625-. , . Loria, G. G. Mt. 26 (1888) 329-. , condensation (a method of calculation). Dodgson, C. L. [1866] B. S. P. 15 (1867) 150-. in conies. Pasch, M. Crelle J. Mth. 89 (1880) 247-. connected with algebraical expressions having same form as component factors. Roberts, S. Mess. Mth. 8 (1879) 138-. and continued fractions, convergence. Koch, H. von. C. B. 120 (1895) 144-. of co-variants. Scott, R. F. Mess. Mth. 10 (1881) 142-. derived from differential coefficients, new species. Studnicka, F. J. Mh. Mth. Ps. 10 (1899) 338-. of determinants. Ovidio, E. d\ [1876-90] Tor. Ac. Sc. At. 11 (1875-76) 949-; 12 (1876) 331- ; 26 (1891) 131-. , development, new method, and application to finding resultant of any two equations. Bonolis, A. [1882] G. Mt. 21 (1883) 336-. , developments. Bonolis, A. [1876] G. Mt. 15 (1877) 113-. , . Capelli, A. Nap. Ed. 28 (1889) 58-. of differences. Raimondi, R. G. Mt. 26 (1888) 185-. , divisors. Frobenius, G. Berl. Ak. Sb. (1894) 31-. and duadic disynthemes, note. Sylvester, J. J. Am. J. Mth. 2 (1879) 89-, 214-. , elementary divisors of, application to linear differential equations. Sauvage, L. Mars. Fac. Sc. A. 6 (1897) Fasc. 5, 9 pp. , treatment. Hesse, L. O. G. Mt. 10 (1872) 217-, 325-. Determinants 2010 Determinants whose elements are composed of elements of adjoint series, theorems. Hunyady, J. (xn) Mag. Tud. Ak. Etk. (Mth.) 1 (1881) (No. 19) 27 pp. --- rational fractions. Scott, R. F. Mess. Mth. 11 (1882) 165-. ---- squares of lines joining vertices of two tetrahedrons. Siebeck, H. Crelle J. 62 (1863) 151-. --- sums of powers. Studnicka, F. J. Prag Ceske Ak. Fr. Jos. Bz. (Tfida 2) 9 (1900) Art. 4, 8 pp. ; Fschr. Mth. (1900) 156. of even order. Brioschi, F. Crelle J. 52 (1856) 133-. figurate numbers. Gaidar era, F. G. Mt. 9 (1871) 223-. -- form Zehfuss, G. Schlomilch Z. 4 (1859) 232-. formed of elements with any number of indices. Garbieri, G. [1876] G. Mt. 15 (1877) 89-. from the same elements, relations between. Pascal, E. Mil. I. Lomb. Bd. 29 (1896) 436-. ?i 2 elements. Bagnera, G. G. Mt. 25 (1887) 228-. , formula in. Pincherle, S. Bologna Bd. (1883) 105-. , -- . Mertens, F. Wien Ak. Sb. 91 (1885) (Ab. 2) 622-. , -- . Nielsen, N. N. Ts. Mth. 7 (B) 1896) 59-; Fschr. Mth. (1896) 111. , -- . Schulze, E. Z. Mth. Ps. 42 (1897) 313-. , -- . Valentiner, H. N. Ts. Mth. 9 (B) 1898) 15-. , -- . Schulze, E. Z. Mth. Ps. 44 (1899) 167-. , formula. Rubini, R. Nap. Bd. 5 (1866) 109-. , . Starkov, A. P. Kazan S. Nt. (Ps.- Mth.) P. 3 (1885) 75-. and functions, fractional. Dietrich, M. Crelle J. 69 (1868) 190-. of functions and their differentials. Pasch, M. Crelle J. Mth. 80 (1875) 177-. , a general property of two. Kretkowski, W. xn) [1882] Krk. Ak. (Mt.-Prz.) Pam. 9 (1884) 45-; Fschr. Mth. (1884) 122-. , generalisation of a theorem. Marcolongo, R. G. Mt. 25 (1887) 298-. , geometrical application. IVhite, H. S. A. Mth. 1 (1900) 103-. , interpretation. [Barrl de'] Saint-Venant, . C. B. 36 (1853) 582-. , significance of certain. Bauer, G. Munch. Sb. (1872) 345-. , treatment. Waelsch, E. Mh. Mth. Ps. 9 (1898) 207-. of higher dimensions. Gegenbauer, L. [1880-90] Wien Ak. D. 43 (1882) (Ab. 2) 17- ; 46 (1883) (Ab. 2) 291- ; 49 (1885) (Ab. 2) 225-; 50 (1885) (Ab. 1) 145- ; 57 1890) 735-. 130 2010 Determinants Determinants of higher dimensions and applica- tion to formation of invariants. Escherich, G. von. [1880] Wien Ak. D. 43 (1882) (Ab. 2) 1-. , history and theory. Giinther, , D. Nf. Tbl. (*1874) 78-. , identical relation between certain. Pain- leve, P. Par. S. Mth. Bll. 22 (1894) 116-. involving powers of circles. Johnson, W. W. J. H. Un. Cir. [10 (1890-91)] 46. whose leading elements are in A.P. Cazzaniga, T. Mil. I. Lomb. Ed. 29 (1896) 541- with magic matrix. Caldarera, F. Palermo G. Sc. Nt. 1 (1866) 173-. of a matrix, relations between. Valilen, K. T. Crelle J. Mth. 112 (1893) 306-. , . Ahrens, W. Z. Mth. Ps. 42 (1897) 65-. maximum value. Scarpis, U. Mil. I. Lomb. Ed. 31 (1898) 1441-. , method of solving. Macloskie, G. A. Mth. 1 (1900) 74-. of minor determinants. Franke, E. [1862] Crelle J. 61 (1863) 350-. and minors of original and adjoint systems. Studnicka, F. J. Casopis 1 (* 1872) 6-; Fschr. Mth. (1872) 59. modern algebra in lower mathematics. Prange, A. J. A. N. Arch. Wisk. 17 (1890) 158- ; Fschr. Mth. (1890) 181-. , multiplication. Gomes Teixeira, F. G. Teix. J. Sc. 3 (1881) 185-. , . Loria, G. G. Teix. J. Sc. 7 (1886) 101-. , . Cazzaniga, T. A. Mt. 2 (1899) 229-. , , application. Souillart, C. N. A. Mth. 19 (1860) 320-. , and multiplication of complex quantities. Palmstrftm, A. Arch. Mth. Ntvd. 13 (1890) 128- ; Fschr. Mth. (1890) 409. , multiplication, elementary proof of theorem. Falk, M. B. A. Ep. (1878) 473-. , formula, verification. Weyr, E. Prag Sb. (1880) 55-. , of minors. Zancevskij, I. M. [1894] Eec. Mth. (Moscou) 17 (1395) 587-; Fschr. Mth. (1893-94) 217. ; (n - l)th power of a symmetric deter- minant by 2nd power of another. Muir, T. Am. J. Mth. 4 (1881) 273-. , rule. Le Paige, C. M. M. H. H. Par. S. Mth. Bll. 9 (1381) 67-. , , geometric application. Mertens, F. Prag Sb. (1880) 207-. , theorem, proof. Kdnig, J. Mth. A. 14 (1879) 507-. , , . Falk, M. Gott. Nr. (1885) 181-. , of two of nih degree. Hoza, F. Arch. Mth. Ps. 59 (1376) 403-. , of two of same degree. Presle, A. de. Par. S. Mth. Bll. 14 (1886) 157-. of n dimensions. Tanner, H. W. L. L f Mth. S. P. 10 (1878-79) 167-. Determinants 2010 Determinants, new formula, with reference to theory of linear differential equations. Anissimoff, W. Mth. A. 51 (1899) 388-. , numerical calculation of class, and con- tinued fractions. Dickson, J. D. H. L. Mth. S. P. 10 (1878-79) 226-. of p dimensions, theory. Zajqczkowski, W. (xn) [1880] Krk. Ak. (Mt.-Prz.) Pam. 6 (1881) (1 ter)-. with polynomial elements. Muir, T. Mess. Mth. 13 (1884) 135-. of powers. Mitchell, 0. H. Am. J. Mth. 4 (1381) 341-. , powers of. Weltzien, C. Mth. A. 50 (1898) 282-. , product of two. Hunyady, E. von. Z. Mth. Ps. 11 (1366) 359-. , . Weltzien, C. Mth. A. 42 (1893) 598-. , proof of formula due to Cauchy. Echols, W. H. A. Mth. 8 (1893-94) 21-. , geometric theorem by. Mansion, P. [1875] Mess. Mth. 5 (1876) 158-. , properties. Dostor, G. (vn) Arch. Mth. Ps. 56 (1374) 238-. , property of certain. Burnside, W. Mess. Mth. 23 (1894) 112-. , deduced from substitution theory. Scarpis, U. G. Mt. 37 (1899) 73-. , , new. Studnicka, F. J. Prag Sb. (1380) 50-. , a proposition in theory of. Drude, P. Gott. Nr. (1887) 118-. , . Baltzer, E. Gott. Nr. (1887) 389-. , . Voss, A. Munch. Ak. Sb. 19 (1390) 329-. of quadratic forms. Brioschi, F. N. A. Mth. 11 (1852) 307-. , rational proof of first principles. Laquiere, E. M. As. Fr. C. E. (1381) 226-. of a rectangular matrix. Pascal, E. A. Mt. 24 (1896) 241-. , relation between two. Del Ee, A. G. Mt. 19 (1881) 116-. , relations between certain. Deruyts, J. Brux. Ac. Bll. 23 (1392) 507-. of several functions of one variable. Frobenius, G. (vn) [1873] Crelle J. 77 (1874) 245-. sines and cosines, method of evaluation. Scott, R. F. Mess. Mth. 8 (1379) 155-. Smith. Mansion, P. (xn) Brux. S. Sc. A. 2 (1878) (Pt. 2) 211- . , solution of question relating to. Hadamard, J. Bll. Sc. Mth. 17 (1893) 240-. , some forms. Lerch, M. Prag Ceske Ak. Fr. Jos. Ez. (Tflda 2) 8 (1899) Art. 12, 16 pp. ; Fschr. Mth. (1899) 151-. , hitherto unproved theorems. Muir, T. Edinb. E. S. P. 18 (1892) 73-. , whose terms are distances of point from 3 points. Cayley, A. QJ. Mth. 5 (1862) 381-. , squared distances of points. Cayley, A. QJ. Mth. 3 (1860) 275-. , theorem. Mirza-Nizam, . N. A. Mth. 4 (1865) 500-. 131 i2 2010 Determinants Determinants, theorem. Stern, M. A. [1865] Crelle J. 66 (1866) 285-. , . Hesse, L. 0. Crelle J. 69 (1868) 319-. _, _. Sardi, C. G. Mt. 6 (1868) 357-. , . Gundelfinger, S. Z. Mth. Ps. 18 1873) 312-. _, __. Cayley, A. QJ. Mth. 15 (1878) 55-. , . Glaisher, J. W. L. Mess. Mth. 8 (1879) 140-. , . Fact de Bruno, F. Am. J. Mth. 3 (1880) 154-. __, _. Daublebsky von Sterneck, R. Mh. Mth. Ps. 6 (1895) 205-. , . Metzler, W. H. Am. J. Mth. 20 (1898) 273-. , . Lovett, E. O. A. Mth. 12 (1898-99) 161-. , . Kantor, S. Gott. Nr. (1899) 272-. , and applications. Siacci, F. Tor. At. Ac. Sc. 7 (1871-72) 772-. , its converse. Puchta, A. [1877] Wien Ak. D. 38 (1878) (Ab. 2) 215-. , , fundamental. Sylvester, J. J. Ph. Mg. 2 (1851) 142-. , , . Becker, J. C. Z. Mth. Ps. 16 1871) 526-. , of Glaisher's. Scott, R. F. QJ. Mth. 17 (1881) 129-. _, Jacobi's. Segar, H. W. Mess. Mth. 21 (1892) 148-. , Netto relating to. Pascal, E. Em. E. Ac. Line. Bd. 5 ^1896) (Son. 1) 188-. , , new. Studnicka, F. J. Prag Sb. (1873) 342-; Casopis 9 (*1880) 97-; Fschr. Mth. (1880) 114. , , . Puchta, A. [1881] Wien Ak. D. 44 (1882) (Ab. 2) 277-. , of Sylvester's, proof. Igel, B. Mh. Mth. Ps. 9 (1898) 47-. , theorems. Cauchy, A. L. C. E. SI (1845) 407-. , . Mogni, A. G. Mt. 3 (1865) 10-. , . Giinther, S. Erlang. Sb. Ps. Md. S. 5 (1873) 88-. , . Scott, R. F. [1878] Mess. Mth. 8 (1879) 33-. , . Studnicka, F. J. Prag Sb. (1896) (Mth.-Nt.) No. 6, 5 pp. , and application to proofs of Pascal's and Brianchon's theorems. Mertens, F. Crelle J. Mth. 84 (1878) 355-. , , certain. Nanson, . Crelle J. Mth. 122 (1900) '179-. , , elementary. Spottiswoode, W. [1853] Crelle J. 51 (1856) 209-, 328-. , , . Ferrers, N. M. QJ. Mth. 1 (1857) 364. , , general. Muir, T. [1879] Edinb. E. S. T. 29 (1880) 47-. , , homoloidal law, theory of orders, etc. Sylvester, J. J. Ph. Mg. 37 (1850) 363-. , of Mainardi. Bellavitis, G. Ven. At. (1857-58) 623-; (1858-59) 334-. , , new. Studnicka, F. J. Casopis 1 (*1872) 203-. , , proofs. Janni, V. G. Mt. 12 (1874) Determinants 2010 Determinants, theorems, two. Zehfuss, G. Schlomilch Z. 7 (1862) 436-. , _, _. Ovidio, E. d\ G. Mt. 1 (1863) 135-. , , . Kapteyn, W. N. Arch. Wisk. 10 (1884) 180-. , , . Netto, E. Acta Mth. 17 (1893) 199-. , theory. Cayley, A. [1843] Camb. Ph. S. T. 8 (1849) 50-. , . Bellavitis, G. Ven. Mm. I. 7 (1857) 67-. _, _. Cayley, A. Ph. Mg. 21 (1861) 180-. , . Dietrich, M. Arch. Mth. Ps. 44 (1865) 344-. , . Kronecker, L. [1869] Crelle J. 72 1870) 152-. __, __. Veltmann, W. Z. Mth. Ps. 16 (1871) 5 16-. _, _. Wey ranch, J. J. [1871] Crelle J. 74 (1872) 273-. , . Studnicka, F. J. Prag Sb. (1872) 78-; Casopis 2 (*1873) 282-; Fschr. Mth. (1873) 90. _, _. Weihrauch, K. Z. Mth. Ps. 19 (1874) 354-. , . Studnicka, F. J. Prag Sb. (1877) 120-. , . Fiirstenau, E. [1879] Crelle J. Mth. 89 (1880) 86-. , . Igel, B. [1880] Wien Ak. Sb. 82 (1881) (Ab. 2) 1288-. , . Landre, C. L. N. Arch. Wisk. 6 (1880) 208-. , . Rados, G. Mth. Termt. Ets. 4 (1886) 268-; Mth. Nt. B. Ung. 8 (1891) 60-. , . SzOts, N. von. Mth. A. 33 (1889) 477-. , . Igel, B. Mh. Mth. Ps. 3 (1892) 55-. , . Miller, E. [1892] Kan. Un. Q. 1 (1893) 133-. , . StudniSka, F. J. Casopis 25 (1896) 241- ; Fschr. Mth. (1896) 108; Mh. Mth. Ps. 10 (1899) 1-. , , analytical. Schering, E. Gott. Ab. 22 (1877) (Mth.) 41 pp. ; C. E. 86 (1878) 1387-. , , application of modular systems to. Netto, E. Crelle J. Mth. 108 (1891) 144-. , based on solution of groups of general linear equations. Matzka, W. [1877] Prag Ab. 9 (1878) (Mth.) (No. 1) 61 pp. ^, (chess problem). Serdobinskil, V. E. (xn) [1877] Eec. " 83) (Pt. 1) 74-. Mth. (Moscou) 10 (1882- , , elementary. Gomes Teixeira, F. G. Teix. J. Sc. 1 (1878) 138-. , , elements. Hoppe, R. Arch. Mth. Ps. 65 (1880) 65-. , , in the historical order of its develop- ment. Muir, T. [1886-89] Edinb. E. S. P. 13 (1886) 547-; 14 (1888) 452-; 15 (1889) 481- ; 16 (1890) 207-, 389-, 748-. , , method of stating. Kneser, A. Dorpat Sb. 9 (1892) 522-. 132 2010 Determinants Determinants, theory, origin and develop- ment. Studnicka, F. J. Casopis 5 (* 1876) 1-, 88-, 193-, 279-. t ) third fundamental proposition, new method of deduction. Studnicka, F. J. Casopis 17 (1888) 193- ; Fschr. Mth. (1888) 143. of 3rd order, note. Tait, P. G. [1866] Edinb. B. S. P. 6 (1869) 59-. 4th order, development. Guimaraes, R. As. Fr. C. K. (1897) (Pt. 2) 129-. 3 ; 4 and 5 rows, rules for their calcula- tion by inspection. Teixeira, J. P. G. Teix. J. Sc. 11 (1892) 88-; Fschr. Mth. (1392) 141. with triple indices, product expressed as one ordinary determinant. Gassparis, A. de. Rm. R. Ac. Line. T. 3 (1379) 44-. and m indices, theorems. Gasparis, A. de. Nap. Rd. 7 (1368) 118-. vector constituents. Chapman, C. H. J. H. Un. Cir. [9 (1889-90)] 77. Determination of four functions to satisfy a certain equation. Vivanti, G. Palermo Cir. Mt. Rd. 6 (1892) 100-. Dialytic determinant, evaluation of a certain. Taylor, W. W. [1895] L. Mfch. S. P. 27 (1396) 60-. Discriminating symmetrical determinant. Syl- vester, J. J. Crelle J. Mth. 88 (1380) 6-. Distances between points, equation connecting mutual. Muir, T. Edinb. Mth. S. P. 3 (1885) 34-. , relations. Brioschi, F. N. A. Mth. 14 (1855) 172-. Double series, two, arising from number of terms in determinants. Dickson, J. D. H. L. Mth. S. P. 10 (1878-79) 120- . Doubly orthosymmetrical determinants. Weih- rauch, K. Z. Mth. Ps. 26 (1381) 64-, 132-. skew determinants. Sylvester, J. J. C. R. 89 (1879) 24-. Equation for max. and min. of ?i-ary quadric. Gravelaar, N. L. W. A. N. Arch. Wisk. 4 (*1878) 113-. Equations, symmetrical linear, with indeter- minate coefficients, solution by determinants. Stockwell, J. N. Gould As. J. 6 (1361) 145-. Factorisation of characteristic equation of in- duced substitutions. Rados, (?. Mth. Termt. Ets. 17 (1899) 66-; Mth. Nt. B. Ung. 17 (1901) 248-. Factors of a special form of determinant. Glaisher, J. W. L. QJ. Mth. 15 (1878) 347-. Formula of Lagrange generalised by Cauchy. Barbier, E. C. R. 96 (1383) 1345-. Liouville's. Lucas, F. C. R. 70 (1370) 1167-. Lucas, connecting n quantities and their sum, proof. Albeggiani, M. G. Mt. 13 (1875) 107-. Formulas, fundamental, of spherical trigono- metry derived from theorem in determinants. Studnicka, F. J. Prag Sb. (1375) 1-. Determinants 2010 Formulae relative to polar operations. Capelli, A. G. Mt. 32 (1894) 376-. Functional determinant, property. Brioschi, F. QJ. Mth. 1 (1857) 365-. determinants. Jacobi, C. G. J. Crelle J. 22 (1341) 319-. . Combescure, E. Par. EC. Norm. A. 4 (1367) 93-. . Casorati, F. [1874] Mil. I. Lomb. Mm. 13 (1377) 181 -. . Faa de Bruno, F. Mth. A. 18 (1881) 280-. . Kraus, L. Wien Ak. Sb. 90 (1885) (Ab. 2) 813-. _ __. Torelli, G. Palermo Cir. Mt. Rd. 7 (1893) 75-. , application. Trzaska, W. Par. T. Nauk Sc. Pam. 1 (*1871) 113- ; Bll. Sc. Mth. As. 6 (*1874) 153. , elementary proof of theorem. Nanson, E. J. [1877] Mess. Mth. 7 (1878) 120-. and Jacobian matrices. Giordano, G. G. Mt. 38 (1900) 210-. , property. Clebsch, R. F. A. Crelle J. 69 (1868) 355-; 70 (1869) 175-. , and application to implicit functions. Gilbert, P. [1869] Brux. Ac. Sc. Mm. 38 (1871) 12 pp. , rational. Mansion, P. [1879] Mess. Mth. 9 (1880) 30-. , , generalisation. Mertens, F. Krk. Ak. (Mt.-Prz.) Pam. 16 (1889) 60-; Fschr. Mth. (1889) 151-. , _, theorems. Anglin, A. H. [1886] Ir. Ac. P. 4 (1884-88) 645-. , theorem (Hesse). Zmurko, W. Par. T. Nauk Sc. Pam. 1 (*1871) 89-. , (Jacobi). Donkin, W. F. Camb. and Dubl. Mth. J. 9 (1854) 161-. t theory. Neumann, C. Mth. A. 1 (1869) 208-. Functions analogous to determinants. Dah- lander, G. R. Stockh. Ofv. 20 (1863) 295-. Fundamental operations of arithmetic, genera- lisation. Schapira, H. D. Nf. Tbl. (*1882) 128-. Geometrical series of 2nd order. Hochheim, A. Magdeb. Nt. Vr. Jbr. u. Ab. (1886) 127- ; (1887) 25-. Goniometrical determinants, some. Weihrauch, K. Z. Mth. Ps. 36 (1891) 71-. Grassmann's method applied to prove theorems in determinants. Miiller, E. Z. Mth. Ps. 44 (1899) 28-. Harmonic determinant, condition for equal roots. Tarleton, F. A. [1887] Ir. Ac. P. 1 (1889-91) 10-. Hesse, theorem (Hessian determinant). Gor- dan, P. Erlang. Ps. Md. S. Sb. 8 (1876) 89-. Hessian. Segre, C. Rm. R. Ac. Line. Rd. 4 (1895) (Sem. 2) 143-. , application of exponential polygon. Eddy, H. T. Des Moines Anal. 2 (1875) 104-. , change of independent variables in a. Nanson, F. J. Mess. Mth. 25 (1896) 139-. determinant. Voss, A. Mth. A. 30 (1887) 418-. 133 2010 Determinants Hessian determinant of Hessian of surface of 3rd order. Bauer, G. Munch. Ak. Ab. 14 (1883) (Ab. 3) 77-. determinants, theory. Pasch, M. Crelle J. Mth. 80 (1875) 169-. Higher algebra, and theorems in determinants. Zeipel, E. V. von. Stockh. Ak. Hndl. 3 (1859-60) 32 pp. Identities, composition, and relation to deter- minants. RensJuiio, A. Mess. Mth. 4 (1868) 34-. in determinants. Hunyady, J. [1882] (xn) Mag. Tud. Ak. Etk. (Mth.) 9, No. 10 (1882) 19 pp. ; (x) Crelle J. Mth. 94 (1883) 171- rational functional determinants. Prouhet, E. N. A. Mth. 15 (1856) 86-. Infinite determinants. Poincare, H. Par. S. Mth. fill. 14 (1886) 77-. . Roberts, E. H. A. Mt. 10 (1895-96) 35-. . Cazzaniga, T. A. Mt. 26 (1897) 143-. applied to functional equations. Koch, H. von. Stockh. Ak. Hndl. Bh. 25 (1900) (Afd. 1) No. 5, 24 pp. linear homogeneous differential equations. Koch, H. von. Stockh. Ofv. (1890) 225-, 499-; Fschr. Mth. (1890) 316. , convergence. Koch, H. von. [1896] Stockh. Ak. Hndl. Bh. 22 (Afd. 1) (1897) No. 4, 31 pp. , theory. Koch, H. von. Stockh. Ofv. (1890) 411- ; Fschr. Mth. (1890) 185. Integral elements, reduction of systems of n a . Kronecker, L. Crelle J. Mth. 107 (1891) 135-. Irregular determinants, formation. Perott, J. Crelle J. Mth. 95 (1883) 232-; 96 (1884) 327-. and subtriplicate forms. Mathews, G. B. Mess. Mth. 20 (1891) 70-. Jacobian of functions, extension of theory. Gomes Teixeira, F. Mh. Mth. Ps. 1 (1890) 481-. , property of the. Faa de Bruno, F. Am. J. Mth. 3 (1880) 154-. Jacobians, a proposition relating to. Peano, G. G. Mt. 27 (1889) 226-. Kronecker's linear relation among minors of symmetric determinant. White, H. S. N. Y. Am. Mth. S. Bll. 2 (1896) 136-. Lame", proof of theorem of. Trzaska, W. Par. T. Nauk. Sc. Pam. 10 (*1878) Art. 9, 3 pp. Laplace's theorem for the distribution of a determinant, extension. Netto, E. Crelle J. Mth. 114 (1895) 345-. Linear forms and determinants. Mtray, C. Liouv. J. Mth. 10 (1884) 181-. , theorem on system of. Frattini, G. Km. B. Ac. Line. Bd. 2 (1886) (Sent. 1) Magic squares, relation to theory of deter- minants. Studnicka, F. J. Prag Sb. (1876) 269-. Maximum modulus which a determinant can have. Hadamard, . C. B. 116 (1893) 1500-. Determinants 2010 Minor determinants. Janni, G. G. Mt. 1 (1863) 270-. of an adjoint determinant. Hoza, F. Arch. Mth. Ps. 59 (1876) 401-. linearly equivalent quadratic functions, relation. Sylvester, J. J. Ph. Mg. 1 (1851) 295-. symmetrical determinant, property. Hazzidakis, J. N. [1880] Crelle J. Mth. 91 (1881) 238-. systems. Kronecker, L. Berl. Ak. Sb. (1882) 821-. . Mehmke, R. Mth. A. 26 (1886) 209-. , linear relations. Runge, C. Crelle J. Mth. 93 (1882) 319-. , theory. Hoza, F. Arch. Mth. Ps. 59 (1876) 387-. Minors of a determinant of Hankel, relations between. Cazzaniga, T. Mil. I. Lomb. Bd. 31 (1898) 610-. , relation between. Vanecek, M. N. Prag Sb. (1886) (Mth.-Nt.) 21-. , law of extensible. Muir, T. [1881] Edinb. B. S. T. 30 (1883) 1-. , principal, of a symmetrical determinant. Schultze, E. Bll. Sc. Mth. As. 10 (1876) 242-. Multinomial determinant. Segar, H. W. Mess. Mth. 21 (1892) 177-. Multiple determinants, properties. Deruyts, J. Liege S. Sc. Mm. 9 (1882) (No. 7) 11 pp. Multiplication of trigonometric and hyperbolic functions. Trzaska, W. Par. T. Nauk Sc. Pam. 10 (*1878) Art. 8, 7 pp. Normal determinants, reciprocals of. Caz- zaniga, T. Tor. Ac. Sc. At. 34 (1898) 351- or 495-. Numerical determinant, calculation. Cazza- niga, T. G. Mt. 36 (1898) 362-. , sign of term, rule. Mainardi, G. Mil. At. I. Lomb. 1 (1858) 105-. 0, a certain differential, determinantal operator. Vivanti, G. Palermo Cir. Mt. Bd. 4 (1890) 261-. Orders of conditions that minors of different degree of given matrix may vanish. Segre, C. Bm. B. Ac. Line. Bd. 9 (1900) (Sem. 2) 253-. Orthogonal substitutions. Jacobi, C. G. J. Crelle J. 15 (1836) 309-. Orthogonants, property. Loria, G. G. Teix. J. Sc. 7 (1886) 129-. , theory. Netto, E. Acta Mth. 19 (1895) 105-. Orthosymmetric determinants. Brunn, H. Z. Mth. Ps. 37 (1892) 291-. Partial compounds. Nansoii, E. J. Mess. Mth. 28 (1899) 17-. determinants, theorem of Jacobi, generali- sation. Barbier, E. C. B. 97 (1883) 82-. differentials as determinants. Gubkin, I. S. [1880] Mosc. S. Sc. Bll. 73 (No. 1) (1891) 29-. Pascal's triangle, certain of whose elements constitute a determinant, properties of latter. Pincherle, S. G. Mt. 19 (1881) 385-. 134 2010 Determinants Permanents and determinants, relation. Muir, T. [1897] Edinb. B. S. P. 22 (1900) 134-. Persymmetric determinants. Muir, T. [1881] Mess. Mth. 11 (1882) 65-. Pfaffian having a vacant minor, development. Muir, T. [1900] Edinb. E. S. T. 40 (1905) 49-. Poisson's determinants. Albeggiani, M. L. Palermo Cir. Mt. Ed. 1 (1887) 6-. generalised. Albefigiani, M. L. Palermo Cir. Mt. Ed. 1 (1887) 9-, 12-, 314-. Potential determinants, mutual relations. Jonquieres, de. C. E. 120 (1895) 408-, 580. "Power" and "combination" determinants. Studnicka, F. J. Prag Sb. (1897) (Mth.-Nt.) No. 1, 20pp., No. 16, 16pp. determinants. Stiidnicka, F. J. Casopis 26 (1897) 105- ; Fschr. Mth. (1897) 142-. [SMi 2 - )] StudnUka, F. J. Prag Sb. (1896) (Mth.-Nt.) No. 22, 8 pp. Powers and determinants, relation. Giinther, S. Z. Mth. Ps. 24 (1879) 244-. Product of 2 determinants, decomposed. Solution of theorem of Sylvester's. Faa de Bruno, F. [1851] Liouv. J. Mth. 17 (1852) 190-. Eeciprocal determinants. Gruzincev, A. P. T, Determinants 2010 Simple alternants. Muir, T. [1887] Edinb. E. S. P. 14 (1888) 433-. , class. Muir, T. [1888] Edinb. E. S. P. 15 (1889) 298-. Skew determinant of infinite order, mean value of numerical coefficients. Sylvester, J. J. C. E. 89 (1879) 497-. determinants. Cayley, A. Crelle J. 32 (1846) 119- ; 38 (1849) 93-. . Brioschi, F. Liouv. J. Mth. 19 (1854) 253-. . Hermite, C. Camb. and Dubl. Mth. J. 9 (1854) 63-. . Cayley, A. Crelle J. 50 (1855) 277-; 55 (1858) 277-. . Muir, T. Ph. Mg. 12 (1881) 391-. . Mertens, F. Wien Ak. Sb. 96 (1888) (Ab. 2) 1245-. . Cazzaniga, T. Mil. I. Lomb. Ed. 30 (1897) 1303-. , Cayley's theory applied to displacement of rigid system about a fixed origin. Schl&fti, L. [1865] QJ. Mth. 8 (1867) 167-. of higher dimensions. Gegenbauer, L. Wien Ak. D. 55 (1889) (Ab. 1) 39-. , history. Muir, T. [1900] Edinb. E. S. P. 23 (1902) 181-. partly zero-axial, condensation. Muir, L. Mth. - Kharkov Mth. S. Com. 3 (1893) 94-; Fschr. Mth. (1891) 147. . Musso, G. G. Mt. 31 (1893) 201- ; 32 (1894) 81-. Eegular determinants and derived systems. Hensel, K. Crelle J. Mth. 114 (1895) 25-. Eesolution of a determinant into quadratic factors. Muir, T. [1881] Mess. Mth. 11 (1882) 105-. determinants, general problem. Giinther, S. Arch. Mth. Ps. 59 (1876) 130-. , new method. Teixeira, J. P. G. Teix. J. Sc. 11 (1892) 173-. linear equations, method, and application to determinants. Clasen, (I'abbe) B. I. Brux. S. Sc. A. 12 (1888) (Pt. 2) 251-. Screws, theory of, note on determinant in. Ball, (Sir) E. S. [1890] Ir. Ac. P. 1 (1889-91) 375-. Semi-determinants. Scheibner, W. Leip. B. 11 (1859) 151-. -symmetrical determinants of even order. Le Paige, C. M. M. H. H. Prag Sb. (1880) 125-. Sets of integers whose determinant is > or <1. Hermite, C. Liouv. J. Mth. 14 (1849) 21-. Sign of any term of a determinant. Zehfuss, G. (xi) Z. Mth. Ps. 3 (1858) 247-, 249-. . Johnson, W. W. [1876] Mess. Mth. 7 (1878) 59. . Tanner, H. W. L. [1879] Mess. Mth. 9 (1880) 51-. . Perry, H. M. (xn) [1881] J. H. Un. Cir. [1] (1882) 151. . Muir, T. [1899] Edinb. E. S. P. 22 (1900) 441-. , short rule for. Jenkins, M. Mess. Mth. 25 (1896) 60-. i. S. P. 13 (1881-82) 161-. , property. Hill, M. J. M. L. Mth. S. P. 26 (1895) 341-. or symmetric determinant of infinite order, mean value of coefficients in development of. Sylvester, J. J. C. E. 89 (1879) 24-. symmetric determinants. Janni, G. G. Mt. 1 (1863) 275-. , property. Deruyts, F. Liege S. Sc. Mm. 17 (1892) No. 2, 6 pp. Stieltjes' theorem, twofold generalisation. Taber, H. L. Mth. S. P. 27 (1896) 613-. Sturm's functions, coefficients expressed as determinants. Heal, W. E. A. Mth. 2 (1885-86) 85-. , determination of a series from a single determinant. Vleck, E. B. van. [1399] A. Mth. 1 (1900) 1-. Sums of determinants. Deruyts, J. Liege S. Sc. Mm. 10 (1883) (No. 4) 11 pp. powers of natural numbers, representation by determinants. Siacci, F. A. Mt. 7 (1865) 19-. Sylvester's determinant. Hesse, L. O. Tor- tolini A. 2 (1359) 5-. , transformation to Cauchy's. Biehler, C. N. A. Mth. 19 (1880) 110- . Symbol analogous to determinants. Ferber, . Par. S. Mth. Bll. 27 (1399) 285-. of positive determinants, application. Ferber, . Par. S. Mth. Bll. 28 (1900) 128-. Symbolic determinants. Rusjan, C. K. Fschr. Mth. (1899) 315. Symmetric determinant connected with Lagrange's interpolation problem. Muir, T. L. Mth. S. P. 13 (1881-82) 156-. , number of terms. Cayley, A. As. S. M. Not. 34 (1874) 303-, 335. 135 2010 Determinants Symmetric determinant, special. Geftenbauer, L. [1880] Wien Ak. Sb. 82 (1881) (Ab. 2 938-. , , note. Muir, T. Des Moines Anal. 8 (1881) 169-. determinants. Mertens, F. Crelle J. Mth. 82 (1877) 207-. , class. Noether, M. Mth. A. 16 (1880) 551-. , , and its use in continued fractions. Studnicka, F. Prag Sb. (1872) 74-. , new properties. Muir, T. QJ. Mth. 18 (1882) 166-. , theorem relating to. Buchheim, A. Mess. Mth. 14 (1885) 143-. forms of determinants. Scott, R. F. Mess. Mth. 8 (1879) 131-, 145-. , skew-symmetric, and orthogonal matrices, properties. Taber, H. L. Mth. S. P. 22 (1891) 449-. , , , . Metzler,W.H. Am. J. Mth. 15 (1893) 274-. Systems of moduli applied to theory of deter- minants. Kronecker, L. Crelle J. Mth. 107 (1891) 254-. Three-dimensional determinants. Hedrick, E, E. [1899] A. Mth. 1 (1900) 49-. Transformable system, determinant of a. Deruyts, J. Brux. Ac. Bll. 32 (1896) 433-. Transformation of determinant into any equi- valent determinant. Van Velzer, C. A. Des Moines Anal. 9 (1882) 116-. . Muir, T. Des Moines Anal. 10 (1883) 8-. continuant. Muir, T. [1881] Edinb. E. S. T. 30 (1883) 5-. . Chrystal, G. [1881] Edinb. E. S. T. 30 (1883) 13-. determinants in conic sections. Caspary, F. [1880] Crelle J. Mth. 92 (1882) 123-. , mode of. Fouret, G. Par. S. Mth. Bll. 14 (1886) 146- . , proof. Fiore, V. G. Mt. 10 (1872) 170. formulae. Rubini, R. G. Mt. 16 (1878) 198-, 344. , a new. Studnicka, F. J. Prag Sb. (1879) 489-. of product of determinants, Sylvester's theorem, note. Sperling, J. F. de. Liouv. J. Mth. 5 (1860) 121-. rectangular coordinates. Janse, L. (Bz.) N. Arch. Wisk. 12 (1886) 109- ; Fschr. Mth. (1885) 676.' Transformations of determinants. Siacci, F. [1872] A. Mt. 5 (1871-73) 296-. Triangle, application of determinants to geo- metry of. Avillez, J. F. d' (Visconde de Reguengo). [1897] Lisb. J. Sc. Mth. 5 (1898) 14-. , determinant giving area in terms of sides. Ritsert, E. Z. Mth. Ps. 17 (1872) 513-. Two-step determinants, theorem. Torelli, G. . G. Mt. 4 (1866) 294-. Uni-modular determinant, construction. Weihrauch, K. Z. Mth. Ps. 21 (1876) 134-. Vanishing aggregates of determinants. Muir, T. [1888] Edinb. E. S. P. 15 (1889) 96-. Discriminants and Resultants 2020 Wronskian determinant. Peano, G. Mathesis 9 (1889) 75-, 110- ; Em. E. Ac. Line. Ed. 6 (1397) (Sem. 1) 413-. . Vivanti, G. Em. E. Ac. Line. Ed. 7 (1898) (Sem. 1) 194-. , generalisation of properties. Pincherle, S. Em. E. Ac. Line. Ed. 6 (1897) (Sem. 1) 301-. , . Bortolotti, E. Em. E. Ac. Line. Ed. 7 (1898) (Sem. 1) 45-. determinants. Lecornu, L. As. Fr. C. E. (1894) (Pt. 2) 282-. , properties and some applications. Dickstein, S. Prace Mt.-Fiz. 1 (1888) 5-; Fschr. Mth. (1888) 151. Wronskians, fundamental property. Demoulin, A. Mathesis 17 (1897) 62-. Zero-determinants of infinite order. Cazzaniga, T. A. Mt. 1 (1898) 83-. 2020 Discriminants and resultants. Algebraic aggregate, discriminant. Hensel, K. Gott. Nr. (1897) 247-. , fundamental equation, and non-essential factors of the discriminant. Hensel, K. Gott. Nr. (1897) 254-. equivalences and elimination. Laurent, H. N. A. Mth. 5 (1886) 432-, 456-. resolvents. Rados, G. Mth. Termt. Ets. 17 (1899) 421- ; Mth. Nt. B. Ung. 18 (1903) 236-. C-discriminant as an envelope. Macdonald, J. A. Edinb. E. S. T. 39 (1900) 27-. Continuant, the discriminant of a special quadratic form. Cesdro, E. Mathesis 12 (1892) 5-. Determinantal equation connected with small oscillations, etc. Muir, T. Mess. Mth. 14 (1885) 141-. , Lagrange's. Muir, T. Ph. Mg. 43 (1897) 220-. , , Sylvester's proof of the reality of the roots of. Muir, T. Am. J. Mth. 19 (1897) 31 2-. , reciprocity of roots (Brioschi's theorem). Faa de Bruno, F. Liouv. J. Mth. 19 (1854) 304. Determination of common roots of given equa- tions by means of their eliminant. Baraniecki, M. A. Par. T. Nauk Sc. Pam. 10 (*1878) Art. 5, 7 pp. Discriminant of cubic giving axes of a quadric surface, decomposition into sum of squares. Bauer, G. [1868] Crelle J. Mth. 71 (1870) 40-. , degree with respect to a parameter. Roberts, E. [1869] QJ. Mth. 10 (1870) 204-. of finite hypergeometric Beries. Hilbert, D. Crelle J. Mth. 103 (1888) 337-. general quadratic. Dupuis, N. F. Cn. E. S. P. & T. 7 (1890) (Sect. 3) 15-. Jacobians of three quadratic forms. Igel, B. [1876] Wien Ak. Sb. 74 (1877) (Ab. 2) 355-; 75 (1877) (Ab. 2) 830-. 136 2020 Discriminants and Resultants Discriminant of quadratic forms, property. Picquet, H. As. Fr. C. E. (1875) 225-. quartic equation. Weill, M. N. A. Mth. 2 (1883) 265-. quaternary quadratic form, properties. Walker, J. J. [1865] Mess. Mth. 4 (1868) 25-, 187-. , special. Cayley, A. QJ. Mth. 11 (1871) 211-. surfaces of algebraic equations, coefficients being coordinates. Czuber, E. Mh. Mth. Ps. 5 (1894) 62-. , transformation of characteristic equations by. BrioscM, F. Tortolini A. 7 (1856) 64-. Discriminantal equation, roots. Brisse, C. N. A. Mth. 9 (1890) 367-. Discriminants. Bellavitis, G. Yen. At. (1858- 59) 65-, 82-. . Brill, A. Mth. A. 12 (1877) 87-. . Casorati, E. Mil. I. Lomb. Ed. 18 (1885) 108-. of algebraic equations, factorisation of. Netto, E. Mth. A. 24 (1884) 579-. functions of one variable. Kronecker, L. Crelle J. Mth. 91 (1881) 301-. , arithmetical researches on non-essential factors of. Kernel, K. Crelle J. Mth. 113 (1894) 128-. , forms of, theorem. Kronecker, L. Crelle J. Mth. 100 (1887) 79-. , formulae, and application to discriminants of discriminants, and to polar curves. Henrici, 0. [1868] L. Mth. S. P. 2 (1869) 104-. , invariants, covariants, and hyperdetenni- nants, elementary notions. Terquem, O. N. A. Mth. 18 (1859) 249-, 299-, 446-. of resultants. Brill, A. Gott.' Nr. (1889) 89-. and resultants of equations for singularities. Meyer, F. Gott. Nr. (1888) 74-; (1890) 366-, 493-; (1891) 14-. , theorem. Sylvester, J. J. E. S. P. 14 (1865) 336-. , theory. Netto, E. Crelle J. Mth. 90 (1881) 164- ; Acta Mth. 1 (1882) 371-. Discriminating cubic ; representation of dis- criminant as sum of squares. Tannery, J. Bll. Sc. Mth. As. 7 (1883) (Pt. 1) 103-. determinant equations, reality of roots of. Elliott, E. B. QJ. Mth. 31 (1900) 233-. equation (formed from symmetric deter- minant), properties. Kaiser, . D. Nf. Tbl. (1887) 228. equation, reality of roots of, when deter- minant is not symmetric. Bendixson, I. Stockh. Ofv. (1900) 1099-. Divisibility, notion including, and general theory of elimination. Molk, J. Acta Mth. 6 (1885) 1-. Eliminant of ). Muir, T. [1896] Edinb. E. S. P. 21 (1897) 360-. , persymmetric. Muir, T. [1899] Edinb. B. S. P. 22 (1900) 543-. of set of quaternary quadrics. Muir, T. [1896] Edinb. E. S. P. 21 (1897) 328-. Elimination 2020 Eliminant of set of ternary quadrics. Muir, T. [1896] Edinb. E. S. P. 21 (1897) 220-. or quaternary quadrics. Nanson, E. J. [1899] Edinb. E. S. P. 22 (1900 353-. , Sylvester's, form. Gordon, P. D. Nf. Vh. (1893) (Th. 2, Halfte 1) 4. of system of equations, degree. Vahlen, K. T. Crelle J. Mth. 113 (1894) 348-. , formation. Biermann, 0. Mh. Mth. Ps. 5 (1894) 17-. two equations. Mansion, P. Brux. S. Sc. A. 18 (1894) (Pt. 1) 5-. , degree. Janni, V. G. Mt. 12 (1874) 27. , . Lachlan, R. Camb. Ph. S. P. 9 (1898) 313-. , dialytic form. Sylvester, J. J. Ph. Mg. 16 (1840) 132-. Eliminants and associated roots. Elliott, E. B. L. Mth. S. P. 16 (1884-85) 224-. , Sylvester's and Cauchy's, equality. Mansion, P. [1879] Mess. Mth. 9 (1880) 60-. Elimination. Poisson, S. D. Par. EC. Pol. J. 4 (1801-2) (11" cah.) 199-. Badano, . Geneva Mm. Ac. 3 (1814) 125-. Babbage, C. QJ. Sc. 3 (1817) 355-. Poncelet, J. V. Gergonne A. Mth. 8 (1817-18) 201-. Gergonne, J. D. Gergonne A. Mth. 21 (1830- 31) 41-. Murphy, E. [1832] Camb. Ph. S. T. 5 (1835) 65-. Sylvester, J. J. Ph. Mg. 17 (1840) 379-. Liouville, J. Liouv. J. Mth. 6 (1841) 345-. Terquem, 0. N. A. Mth. 1 (1842) 125-. Moon, E. Camb. Mth. J. 3 (1843) 183-. Finck, B. Liouv. J. Mth. 9 (1844) 334-. Bouillon, . N. A. Mth. 4 (1845) 120-. Cockle, Jas. Ph. Mg. 27 (1845) 292-. Cayley, A. Crelle J. 34 (1847) 30-. Crelle, A. L. Crelle J. 43 (1852) 37-. Genocchi, A. N. A. Mth. 14 (1855) 259-. Cayley, A. [1861] Crelle J. 60 (1862) 373-. Lemonnier, H. C. E. 80 (1875) 252-. Eouche-, E. N. A. Mth. 16 (1877) 105-. Lemonnier, H. Par. EC. Norm. A. 7 (1878) 77-, 151-. Mansion, P. C. E. 87 (1878) 975- ; Brux. Ac. Bll. 46 (1878) 899-; 47 (1879) 532-; 48 (1879) 463-, 473-. Le Paige, C. M. M. H. H. C. E. 90 (1880) 1210-. Biehler, C. N. A. Mth. 1 (1882) 529-. Laurent, H. N. A. Mth. 11 (1892) 5-. Worontzoff, . N. A. Mth. 11 (1892) 291-. Laurent, H. N. A. Mth. 12 (1893) 355-. Hadamard, J. C. E. 119 (1894) 995-, 1243 ; Acta Mth. 20 (1897) 201-. Netto, E. Giessen Oberh. Gs. B. 32 (1897-99) 78-. Bezout-Cauchy method. Falk, M. Ups. S. Sc. N. Acta 10 (1879) No. 15, 36 pp. . Hioux, V. N. A. Mth. 18 (1879) 289-. 137 2020 Elimination Bezout's method. Cayley, A. Crelle J. 53 (1857) 366-; Mess. Mth. 2 (1863) 88-. . Painvin, L. N. A. Mth. 13 (1874) 278-. . Stickelberger, . D. Nf. B. (* 1883) 58. and continued fractions. Borchardt, C. W. Berl. Ak. Ab. (1878) (Mth. 1-). by cross-multiplication. Smith, Arch. Camb. Mth. J. 1 (1839) 46. degree of equation resulting from. Minding, E. F. A. Crelle J. 22 (1841) 178-. . Magnus, L. J. Crelle J. 26 (1843) 365-. - , expression for. Minding, E. F. A. [1843] St Pet. Ac. Sc. Bll. 2 (1844) 273-. dialytic method. Sylvester, J. J. Ir. Ac. P. 2 (1840-44) 130-. applied to ternary systems of equations. Sylvester, J. J. Camb. Mth. J. 2 (1841) 232-. , extensions. Sylvester, J. J. Ph. Mg. 2 (1851) 221-. , Sylvester's. Madsen, (Kapt.) V. H. 0. (xn) Ts. Mth. 5 (1875) 144-. in equations of 1st order. Gergonne, J. D. Gergonne A. Mth. 4 (1813-14) 148- ; 12 (1821-22) 281- ; 20 (1829-30) 109-. . Majo, L. de. Nap. Mm. Ac. Sc. 1 (1852-54) 101-. by finding common divisor. Raymond, G. M. Gergonne A. Mth. 2 (1811-12) 38-. higher orders. Kramp, C. Gergonne A. Mth. 1 (1810-11) 321-. Euler's method. Biehler, C. N. A. Mth. 6 (1887) 67-. example. Anderson, Jas. Mathematician 1 (1845) 181-. formation of eliminant. Bes, K. Amst. Ak. Vs. 8 (1900) 173- ; Amst. Ak. P. 2 (1900) 85-. formula. Cayley, A. L. Mth. S. P. 11 (1879- 80) 139-. Gauss's method. Hall, A. A. Mth. 8 (1893- 94) 64. general theory. Voizot, . L'l. 3 (1835) .' Bes, K. Amst. Ak. Vh. (Sect. 1) 6 (1899) No. 7, 121 pp. by H.C.F. Bret, . Par. EC. Pol. J. 8 (1809) (15 e cah.) 162- ; Gergonne A. Mth. 3 (1812-13) 13-. . Sarrus,F. (vi Adds.) Strasb. S. Sc. N. Mm. 2 (1832) 190-. indeterminate multipliers. Smith, Arch. Camb. Mth. J. 1 (1839) 49-. introduction to a theory. Garbieri, G. G. Mt. 30 (1892] 41-. linear method for algebraic equations. Sylvester, J. J. Ph. Mg. 18 (1841) 425-. and linear simultaneous equations, algebraic or differential. Mainardi, G. Mil. At. I. Lomb. 1(1858) 95-. Liouville's theorem, demonstration and ap- plication. Fouret, G. N. A. Mth. 9 (1890) Elimination 2020 method. Craufitrd, A. Q. G. Camb. Mth. J. 2 (1841) 276-. . Cayley, A. N. A. Mth. 18 (1859) 397-. of obtaining final equation without irrelevant factors. Araldi, A. (ix) [1858] Mod. Ac. Sc. Mm. 3 (1861) 3-. methods. Cossali, P. [1812] Mod. Mm. S. It. 16 (1813) 272-. of Fontaine, Vandermonde, and Laplace. Terquem, O. N. A. Mth. 5 (1846) 153-. new and extended theory. Sylvester, J. J. Ph. Mg. 15 (1839) 428-; 16 (1840) 37-. and polars. Clebsch, A. [1860] Crelle J. 58 (1861) 273-. problem. Borchardt, C. W. Berl. Mb. (1859) 376-. . Krey, H. Mth. A. 12 (1877) 476-. . Stephanos, C. C. B. 97 (1883) 1050-. comprising theory. Ventejols, . C. B. 84 (1877) 546-. of Sylvester's. Tait, . [1892] Edinb. B. S. P. 19 (1893) 131-. . Muir, T. [1892] Edinb. B. S. P. 20 (1895) 300-. . Cayley,. [1894] Edinb. B. S. P. 20 (1895) 306-. . Muir, T. Edinb. B. S. P. 20 (1895) 371-. . Nanson, E. J. [1897] Edinb. B. S. P. 22 (1900) 150-. problems. Cayley, A. QJ. Mth. 8 (1867) 183- ; 11 (1871) 99-. on curves satisfying given conditions. Brill, A. Gott. Nr. (1870) 525-. a process in. Biehler, C. N. A. Mth. 19 (1880) 202-, 332-. between quadratic and set of linear functions (reply to Boole). Sylvester, J. J. Camb. and Dubl. Mth. J. 6 (1851) 171-. quadratics, theorems in. Sylvester, J. 7. Ph. Mg. 37 (1850) 213-. result. Salmon, G. Camb. and Dubl. Mth. J. 3 (1848) 169-. . Cayley, A. Tortolini A. 7 (1856) 454-. resultants, comparison. Borchardt, C. W. Crelle J. 57 (1860) 183-. simple. Drinkwater, J. E. Ph. Mg. 10 (1831) 24-. by special algorithm. Gruson, J. P. Berl. Ab. (1816-17) 1-. symbolic process. Clebsch, A. Berl. Mb. (1860) 536-. by symmetric functions. Liouville, J. N. A. Mth. 6 (1847) 295-. symmetrical method in simple equations by determinants. Warner, J. D. Am. As. P. (1881) 50-. from a system of equations. Brill, A. [1871] Mth. A. 5 (1872) 378-. theorem. Cayley, A. QJ. Mth. 12 (1873) 5-. theory. Eichelot, F. J. Crelle J. 21 (1840) 226-. . Grunert, J. A. Grunert Arch. 2 (1842) 76-, 337-, 345-. . Crelle, A. L. Berl. Ab. (1844) 1- ; (1845) 1-. . Chio, F. Palomba Bac. 3 (1847) 193-. 138 2020 Elimination Discriminants and Resultants 2020 theory. Cayley, A. Camb. and Dubl. Mth. J. 3 (1848) 116-. . Salmon, G. QJ. Mth. 7 (1866) 327-. . ArzeKi, C. [1876] G. Mt. 15 (1877) 62-, . Noether, M. [1876] Erlang. Ps. Md. S. Sb. 9 (1877) 66-. . Toeplitz, J. Z. Mth. Ps. 23 (1878) 61-. . Laurent, H. N. A. Mth. 2 (1883) 145-. . Netto, E. Acta Mth. 7 (1885) 101-. . Schmidt, C. Z. Mth. Ps. 31 (1886) 214-. . Schendel, L. Z. Mth. Ps. 32 (1887) 46-. . Laurent, H. N. A. Mth. 7 (1888) 60-, 116-. . Loria, G. G. Teix. J. Sc. 9 (1889) 33-. . Kretkmvski, W. Prace Mt.-Fiz. 2 (1890) 21- ; Fschr. Mth. (1890) 174-. . Pleskot, A. Prag Ceske Ak. Fr. Jos. Ez. (Tfida 2) 5 (1896) Art. 25, 13 pp.; Fschr. Mth. (1896) 91. . Mertens, F. Wien Ak. Sb. 108 (1899) (Ab. 2 a) 1173-, 1344-. , and algebraic curves. Brill, A. Mth. A. 4 (1871) 510-. , principle of projection in. Meyer, F. D. Nf. Vh. (1890) (Th. 2) 9-. transformation and canonical forms . Sylvester, J. J. Camb. and Dubl. Mth. J. 6 (1851) 186-. of two arbitrary functions. Tanner, H. W. L. [1875] Mess. Mth. 5 (1876) 98-. between two equations. Valerius, B. Liege A. Ac. (1826-27) 75 pp. . Meyer, A. Quetelet Cor. Mth. 5 (1829) 100-. . Jacobi, C. G. J. Crelle J. 15 (1836) 101-. . Petersen, J. P. C. (xn) [1873] Ts. Mth. 4 (1874) 83-. . Zeuthen, H. G. (xn) Ts. Mth. 4 (1874) 165-. . Cauchy, . N. A. Mth. 15 (1876) 385-, 433-. . Forestier, C. Toul. Ac. Sc. Mm. 9 (1877) 142-. . Mansion, P. Brux. Ac. Bll. 48 (1879) 491-. . Kronecker, L. Berl. Ak. Mb. (1881) 535-. , application of indeterminate analysis. Wallace, (Prof.) W. Edinb. Ph. J. 1 (1819) 29-. , final equation. Hesse, L. 0. Crelle J. 27 (1844) 1-. in one unknown. Darboux, G. (rx) Bll. Sc. Mth. As. 10 (1867) 56- ; 1 (1877) 54-. two unknowns. Vene, A. A. Brux. Mm. Cour. 4 (1824) (No. 2) 26 pp. . Finck, B. N. A. Mth. 4 (1845) 198-. . of 2nd order. Horner, W. G. Ph. Mg. 5 (1834) 188-. ,2 unknowns. Merlieux, E, N. A. Mth. 2 (1843) 34-. 3rd order. Clebsch, E. F. A. [1864] Crelle J. 64 (1865) 95-. three equations of 2nd order. Hesse, L. 0. Crelle J. 28 (1844) 68-. Eliminations arising in trigonometry. Meyer, F. D. Mth. Vr. Jbr. 4 (1897) 92-. , equations resulting from various. Bezout, . N. A. Mth. 18 (1859) 120-. Equation for max. and min. of n-ary quadric. Gravelaar, N. L. W. A. N. Arch. Wisk. 4 (*1878) 113-. of squares of differences. Brioschi, F. Tortolini A. 7 (1856) 5-. Equations whose discriminant is a square. Netto, E. Crelle J. Mth. 95 (1883) 237-. Gauss's method of substitution. Johnson, W. W. A. Mth. 7 (1892-93) 65-. Order of manifoldness arising from several linear systems. Fieri, M. Palermo Cir. Mt. Ed. 11 (1897) 58-. ^-discriminant of differential equation of first order, and its connection with general theory of envelopes. Chrystal, . Edinb. E. S. T. 38 (1897) 803-. Power determinant, a. Jung, V. Casopis 29 (1900) 41- ; Fschr. Mth. (1900) 157. determinants. Jung, V. Prag Ceske" Ak. Fr. Jos. Ez. (Tfida 2) 8 (1899) Art. 38, 22 pp. ; Fschr. Mth. (1899) 153. Product of differences of roots of equations, relation of discriminants and resultants to. Sylvester, J. J. Mess. Mth. 9 (1880) 164-. Property of product of linear factors. Brioschi, F. Camb. and Dubl. Mth. J. 9 (1854) 137-. Eesultant of a covariant and a fundamental form. Kohn, G. Wien Ak. Sb. 100 (1891) (Ab. 2a) 1013-. , degree. Ise, E. G. Mt. 11 (1873) 253-. of elimination of z from algebraic equations f(z) = Q and (z) = Q, new representation. Rosenhain, G. Crelle J. 30 (1846) 157-. , Euler's. Hesse, L. 0. Tortolini A. 2 (1859) 5-. and G.C.M. of integral algebraical ex- pressions. Sziits, N. von. Mh. Mth. Ps. 9 (1898) 34-. of integral functions. Biermann, 0. Mh. Mth. Ps. 2 (1891) 143-. multipartite system of equations, degree and "weight." Sylvester, J. J. E. S. P. 12 (1862-63) 674-. n equations. Faa de Bruno, F. Tortolini A. 7 (1856) 222-. an jj-ic and of a cubic. Pascal, E. G. Mt. 25 (1887) 257-. , order of, case of evaporation. Clifford, W. K. [1870] L. Mth. S. P. 3 (1869-71) 80-. R of two forms of the mth and nth degrees, P.D.E. satisfied by. Gordon, P. Gott. Nr. (1870) 427-. , the reduced. Brill, A. Munch. Ak. Ab. 17 (1892) 89-. of several algebraic equations. Laurent, H. N. A. Mth. 12 (1893) 305-. system of equations. Schlafli, L. Wien D. 4 (1852) (Ab. 2) 1-. (Schlafli). Cayley, A. Crelle J. 50 (1855) 277-. two equations. Cayley, A. [1857- 67] Phil. Trans. 147 (1857) 703- ; 158 (1868) 173-. 139 2020 Discriminants and Resultants Eesultant of two congruences. Sylvester, J. J. (xn) [1881] J. H. Un. Cir. [1] (1882) 131. equations. Brioschi, F. Creile J. 53 (1857) 372-. . Ise,E.[1868] G. Mt. 8 (1870) 1-. . Gordon, P. Mth. A. 3 (1871) 355-; C. E. 127 (1898) 539-. , application to twisted cubic curve. Waelsch, E. Mh. Mth. Ps. 2 (1891) 421-. , Brioschi's theorem. Fad de Bruno, F. Creile J. 64 (1857) 283-. of 4th order. Faa de Bruno, F. Tortolini A. 1 (1858) 362-. forms of the same order, new repre- sentation. Stahl, W. Mth. A. 35 (1890) 395-. Eesultants. Vanecek, M. N. Prag Sb. (1888) (Mth.-Nt.) 152-. . Gordan, P. Mth. A. 45 (1894) 405-. , anastrophic. Cauchy, A. L. C. E. 44 (1857) 370. corresponding to interpolation. Borchardt, C. W. Creile J. 57 (1860) 111-. of covariants. Gordan, P. Mth. A. 4 (1871) 169-. , formation. Reuschle, C. Z. Mth. Ps. 30 (1885) 106-. , note. Haskell, M. W. N. Y. Mth. S. Bll. 1 (1892) 223-. , property. Brill, A. [1879] Mth. A. 16 (1880) 345-. of quantics, determination of their numerical coefficients. Vanecek, M. N. Prag Sb. (1888) (Mth.-Nt.) 286-. , theorems. Igel, B. [1877] Wien Ak. Sb. 76 (1878) (Ab. 2) 145-. .theory. Netto,E. Creile J. Mth. 116 (1896) 33-; 117 (1897)57-. , , extension. Sylvester, J. C.E.58(1864) 1074-, 1130, 1178-. , , note. Netto, E. Giessen Oberh. Gs. B. 32 (1897-99) 37-. of two integral functions. Nagelsbach, H. Z. Mth. Ps. 17 (1872) 333-. , remark on property. Kowalew- ski, G. Arch. Mth. Ps. 17 (1900) 202-. , two, in theory of rectifiable compound logarithmic waves. Sylvester, J. J. Ph. Mg. 37 (1869) 375-. Eoots common to two complete equations. Hioux, V. Par. EC. Norm. A. 10 (1881) 383-. equations. Abel, N. H. Gergonne A. Mth. 17 (1826-27) 204-. . Brioschi, F. N. A. Mth. 14 (1855) 81-. . Garbieri, G. Catania Ac. Gioen. At. 6 (1893) Mm. 16, 9 pp. Sub-discriminant. Perrin, R. Liouv. J. Mth. 10 (1894) 129-. Sylvester's problem of the three ternary equa- tions, new solution. McLaren, (Lord). Edinb. E. S. P. 19 (1893) 264-. Vanishing discriminant, application. Azza- relli, M. Tortolini A. 3 (1860) 16-. , applications to geometry. Azzarelli, M. Em. N. Line. At. 30 (1877) 290-. Linear Substitutions 2030 2030 Characteristic properties of linear substitutions; types of linear substitutions. (See also 1210.) "Adjoint" substitutions, theory. Rados, G. [1891] Mth. Termt. Ets. 10 (1892) 34-; Mth. Nt. B. Ung. 10 (1893) 98-. Analytic functions suitable to represent sub- stitutions. Dickson, L. E. Am. J. Mth. 18 (1896) 210-. Bipartite quadric, automorphic linear trans- formation. Cayley, A. Phil. Trans. (1858) 39-. , . Muir, T. Am. J. Mth. 20 (1898) 215-. Canonical form, bi-orthogonal reduction of a lineo-linear form to. Sylvester, J. J. C. E. 108 (1889) 651-. of linear homogeneous substitution in a Galois field. Dickson, L. E. Am. J. Mth. 22 (1900) 121-. substitutions. Maschke, H. Mth. A. 50 (1898) 220-. a linear substitution of finite order. Burnside, W. Mess. Mth. 28 (1899) 111-. quadric, proof of reduction by real orthogonal substitution. Sylvester, J. J. Mess. Mth. 19 (1890) 1-. , reduction of a linear substitution to. Burnside, W. L. Mth. S. P. 30 (1899) 180-. , . Dixon, A. C. [1899] L. Mth. S. P. 31 (1900) 170-. , .. Bromwich,T.J.PA. [1899] L. Mth. S. P. 31 (1900) 289-. Characteristic functions of substitutions. Cauchy, A. L. C. E. 21 (1845) 1254-. Equations containing linear functions of recti- linear coordinates. Dieu, T. N. A. Mth. 4 (1865) 165-. Exchangeable linear substitutions. Amaldi, U. Mil. I. Lomb. Ed. 33 (1900) 731-. . Schlesinger, L. Creile J. Mth. 121 (1900) 177-. substitutions, theory. Demeczky, . C. E. 120 (1895) 39-. Induced linear substitutions. Franklin, F. Am. J. Mth. 16 (1894) 205-. . Rados, G. Mth. Termt. Ets. 16 (1898) 378-; Mth. Nt. B. Ung. 16 (1899) 241-. . Visnya, A. Mth. Nt. B. Ung. 16 (1899) 187-. , groups. Rados, G. Mth. Termt. Ets. 17 (1899) 44-; Mth. Nt. B. Ung. 17 (1901) 227-. substitutions, rank of determinant in. Kurschdk, J. Mth. Termt. Ets. 18 (1900) 44-; Mth. Nt. B. Ung. 18 (1903) 229-. Integral linear substitutions. Pick, G. Mth. A. 24 (1884) 588-. of determinant 1, composition of. Krazer, A. A. Mt. 12 (* 1883-84) 283-. 140 2030 Linear Substitutions 2030 Integral linear substitutions not given by alge- braical congruences. Pick, G. Mth. A. 28 (1887) 119-. Invariants of orthogonal substitutions (regarded as rotations). Schlafli, L. [1865] Crelle J. 65 (1866) 185-. Inverse orthogonal matrices. Sylvester, J. J. Ph. Mg. 34 (1867) 461-. Kummer's configuration, and orthogonal sub- stitutions. Study, E. Leip. Mth. Ps. B. 44 (1892) 122-. Linear orthogonal substitutions, special case. Stern, M. A. Crelle J. 67 (1867) 293-. substitutions. Polignac, C. de. Par. S. Mth. Bll. 4 (1876) 120- ; 5 (1877) 69-. . Poincare, H. C. E. 98 (1884) 349-. . Picard, E. C. E. 98 (1884) 416-. . Burnside, W. Mess. Mth. 22 " 190-. , certain. Deruyts, J. Brux. Ac. Bll. 24 (1892) 102-. , a property. Burnside, W. Mess. Mth. 20 (1891) 163-. of a single variable with periodic coefficients. Ocagne, M. d'. Par. S. Mth. Bll. 19 (1891) 37-. , special groups of. Fricke, R. Mth. A. 38 (1891) 50-, 461-. , theory. Netto, E. Acta Mth. 17 (1893) 265-. , . Loeicy, A. Mth. A. 48 (1897) 97-; 49 (1897) 448-. , , new. Laurent, H. N. A. Mth. 15 (1896) 345-. by which a quadratic ternary form re- produces itself. Tannery, J. Bll. Sc. Mth. As. 11 (1876) 221-. transformations. Cayley, A. Camb. Mth. J. 4 (1845) 193- ; Camb. and Dubl. Mth. J. 1 (1846) 104-. , some properties. Deruyts, J. Brux. Ac. Bll. 16 (1888) 576-. Orthogonal and Boolian invariant systems, . symbolism for. MacMahon, (Maj.) P. A. B. A. Ep. (1900) 647. , related, substitutions. Igel, B. [1877] Wien Ak. D. 39 (1879) (Ab. 2) 29-. substitution, properties of coefficients. Bar- delli, G. Mil. I. Lomb. Ed. 9 (1876) 167-. substitutions. Grunert, J. A. Arch. Mth. Ps. 44 (1865) 478-. . Gundelftnger, S. Crelle J. Mth. 99 (1886) 147-. . Netto, E. Acta Mth. 9 (1887) 295-. . Catalan, E. As. Fr. C. E. (1893) (Pt. 2) 19-. . Taber, H. N. Y. Mth. S. Bll. 3 (1894) 251-. . Rados, G. Mth. Termt. Ets. 18 (1900) 128- ; Mth. Nt. B. Ung. 18 (1903) 231-. , conditions among coefficients. Rados, G. Mth. Termt. Ets. 16 (1898) 123- ; Mth. Nt. B. Ung. 16 (1899) 236-. with determinant ( - 1). Weill, G. Par. EC. Norm. A. 11 (1894) 23-. , equation connected with. Borsch, A. Z. Mth. Ps. 24 (1879) 391-. Orthogonal substitutions in involution. Prym, F. Gott. Ab. 38 (1892) (Mth.) 42 pp. , parametric expression of coefficients. Hunyady, J. Mag. Tud. Ak. Etk. (Mth.) 14 (1892) No. 2, 43 pp., No. 3, 14 pp.; Mth. Nt. B. Ung. 7 (1890) 233-. and regular divisions of space. Goursat f E. C. E. 106 (1888) 1786- ; Par. EC. Norm. A. 6 (1889) 9-. (single alternate substitution). Taber, H. Am. J. Mth. 16 (1894) 123-. , theory. Voss, A. E. Mth. A. 13 (1878) 320-. , . Rados, G. Mth. Termt. Ets. 10 (1892) 16-; Mth. Nt. B. Ung. 10 (1893) 95-. system in parallel arrangement. Voss, A. Mth. A. 24 (1884) 48-. systems. Kronecker, L. Berl. Ak. Sb. (1890) 525-, 601-, 691-, 873-, 1063-. Powers and roots of linear substitutions. Syl- vester, J.J. C. E. 94 (1882) 55-. Quadratic form, automorphic transformation. Rosanes, J. Crelle J. Mth. 80 (1875) 52-. , transformations. Jordan, C. Liouv. J. Mth. 4 (1888) 349-. , . Loewy, A. Ac. Nt. C. N. Acta 65 (1896) 1-. . , orthogonal transformation to sum of squares. Kneser, A. Arch. Mth. Ps. 15 (1897) 225-. , reduction. Timerding, H. E. Crelle J. Mth. 122 (1900) 172-. , to squares : equation | a^k + S \ = 0. Walecki, . N. A. Mth. 1 (1882) 401-, 556-. , transformation. Enneper, A. Z. Mth. Ps. 9 (1864) 358-. , to sum of squares. Hesse, L. 0. Crelle J. 45 (1853) 93-; 57 (1860) 175-. , . Gundelfinger, S. [1880] Crelle J. Mth. 91 (1881) 221-. , . Zajaczkowski, W. (xn) [1883] Krk. Ak. (Mt.-Prz.) Pam. 9 (1884) 1- ; Fschr. Mth. (1884) 165-. , . . Presle, de. Par. S. Mth. Bll. 14 (1886) 98-. , , Gauss's method. Somoff, J. St. Pet. Ac. Sc. Bll. 18 (1873) 442-. forms and the discriminantal equation. Laurent, H. N. A. Mth. 10 (1891) 60S-. , reduction to least number of terms, Jacobi, C. G. J. Berl. B. (1848) 414-. , and transformation. Jordan, C. Liouv. J. Mth. 19 (1874) 397-. , removal of product terms from two. Jacobi, C. G. J. Crelle J. 12 (1834) 1-. , simultaneous transformation. Cayley, A . Camb. and Dubl. Mth. J. 4 (1849) 47-; QJ. Mth. 2 (1858) 192-. , . Siacci, F. Tor. At. Ac. Sc. 7 (1871-72) 758-. , theory. Jamet, V. Mars. Fac. Sc. A. 10 (1900) Fasc. 5, 18 pp. , transformation. Smith, Arch. Camb. Mth. J. 1 (1839) 272-. , . Cayley, A. Crelle J. 50 (1855) 277-. 141 2030 Linear Substitutions Quadratic forms, transformation. Trudi, N. Nap. Kd. 1 (1862) 113-. , . Brioschi, F. G. Mt. 1 (1863) 26-. , . Lemonnier, H. [1874] Par. S. Mth. Bll. 3 (1875) 48-. , transformations. Kretkmvski, W. (xn) Krk. Ak. (Mt.-Prz.) Pam. 7 (1882) 69-. Quadratics, reduction to canonical form. Sardi, C. G. Mt. 5 (1867) 35-. Quadric automorphic linear transformation. Muir, T. [1897] Edinb. E. S. T. 39 (1900) 209-. Quaternary quadratic form, automorphic linear transformations. Bachmann, . D. Nf. Tbl. (*1871) 138-. Eeal linear transformations whose invariant is an alternant. Taber, H. Am. Ac. P. 31 (1896) 336-. . a real quadratic form. Taber, H. Am. Ac. P. 32 (1897) 75-. orthogonal substitution. Taber, H. Am. Ac. P. 28 (1893) 212-. Boots of substitutions. Kirkman, T. P. B. A. Ep. (1860) (pt. 2) 4-; (1861) (pt. 2) 4-. Solutions of the equation SX=XS. Voss, A. Munch. Ak. Sb. 26 (1897) 273-. Special automorphic groups and quadratic forms. Fricke, R. Mth. A. 39 (1891) 62-. Structure of linear groups with quadratic in- variants. Dickson, L. E. L. Mth. S. P. 30 (1899) 70-. Substitutions permutable inter se. Cauchy, A, L. C. E. 21 (1845) 1188-. . , and other systems of substitutions. Cauchy, A. L. C. E. 21 (1845) 1238-. , product of two. Miller, G. A. Am. J. Mth. 22 (1900) 185-. , regular, irregular, and conjugate. Cauchy, A. L. C. E. 21 (1845) 835-, 895-, 931-, 972-, 1025-. of six letters. Betti, E. C. E. 63 (1866) 878. which do not alter value of a function. Cauchy, A. L. C. E. 21 (1845) 1234- . of several variables, number and form. Cauchy, A. L. C. E. 21 (1845) 1287-. Ternary substitutions of determinant unity with integral coefficients. Burn-side, W. Mess. Mth. 24 (1895) 109-. Type of successive linear substitutions. Velt- mann, W. Arch. Mth. Ps. 58 (1876) 342-. 2040 General theory of quantics. Absolute invariants of integral linear trans- formation groups. Hurwitz, A. Mth. A. 20 (1882) 125-. Algebraic forms. Cambescure, E. Par. EC. Norm. A. 1 (1864) 269-. in any number of variables. Deruyts, J. Brux. Ac. Bll. 15 (1888) 951-. , geometrical applications. Le Paige, C. M. M. H. H. [1878] Brux. Mm. Cour. 4 C , 42 (1879) (No. 4) 71 pp. with Hessian determinant vanishing identically. Noether, M. Erlang. Ps. Md. S. Sb. 8 (1876) 51-. Quantics 2040 Algebraic forms with Hessian determinant vanishing identically. Gordan, P. , (& Noether, M. Mth. A. 10 (1876) 547-. , invariants of a class. Deruyts, J. Brux. Ac. Bll. 23 (1892) 152-. with p variables, theory of. Perrin, R. C. E. 104 (1887) 108-. , points in theory. Le Paige, C. M. M. H. H. Liege S. Sc. Mm. 9 (1882) (No. 4) 23pp. , principle of classification. Rosanes, J. Crelle J. 76 (1873) 313-. , property. Le Paige, C. M. M. H. H. Mth. A. 15 (1879) 206- . with several series of variables. Le Paige, C. M. M. H. H. C. E. 94 (1882) 31-, 69-. variables, development by polars. Capelli, A. Em. E. Ac. Line. Ed. 7 (1891) (Sem. 1) 161- ; 1 (1892) (Sem. 1) 3-. , symbolic representation. Clebsch, A. [1860] Berl. Mb. (1860) 536-; Crelle J. 59 (1861) 1-. , theorem, with application to binary sextic. Ovidio, E. d'. Palermo Cir. Mt. Ed. 6 (1892) 225-. , theory. Sylvester, J. J. C. E. 101 (1885) 1110-. , , arithmetical. Kronecker, L. Crelle J. Mth. 93 (1882) 365-. , , general. Capelli, A. Em. E. Ac. Line. Mm. 12 (1882) 529-. , , . Deruyts, J. Liege S. Sc. Mm. 17 (1892) No. 4, vi + 157 pp. , , new. Sylvester, J. J. C. E. 101 (1885) 1042-, 1225-, 1461. . , , operations in. Capelli, A. Mth. A. 37 (1890) 1-. Anharmonic ratio and invariant theory of com- plex quantities. Schick, J. [1900] Munch. Ak. Sb. 30 (1901) 249-. Associated forms. Kohn, G. Wien Ak. Sb. 100 (1891) (Ab. 2a) 865-. , general theory. Sylvester, J. J. Camb. and Dubl. Mth. J. 6 (1851) 289-. , theory of Clebsch and Gordan. Syl- vester, J. J. C. E. 86 (1878) 448-. Asyzygetic invariants, extension of a theorem. Sylvester, J. J. C. E. 102 (1886) 1430-. Boolian trinomial. Cockle, (Sir) J. [1868] Mess. Mth. 5 (1871) 33-. Canonical forms, a certain class. Roberts, R. A. [1894] N. Y. Am. Mth. S. Bll. 1 (1895) and hyperdeterminants, discovery in. Sylvester, J. J. Ph. Mg. 2 (1851) 391-. Chemico-algebraical theory. Frankland, E. Am. J. Mth. 1 (1878) 345-. Chemistry and algebra. Sylvester, J. J. Nt. 17 (1878) 284. Combinants. Betti, E. Tortolini A. 1 (1858) 344-. _. Gordan, P. [1871] Mth. A. 5 (1872) 95-. , theory. Stroh, E. Mth. A. 22 (1883) 393-. Contrariants, a new species of invariants. Sylvester, J. J. Ph. Mg. 18 (1884) 374-. 142 2040 General Theory of Quantics Contravariants, invariants and covariants. Janni, G. G. Mt. 1 (1863) 174-, 194-, 240-, 340-. Covariant, a definition. Keyser, C. J. N. Y. Am. Mth. S. Bll. 5 (1899) 468-. of system of n quantics. Newson, H. B. N. Y. Am. Mth. S. Bll. 2 (1896) 272-. Covariants. Cayley, A. [1852] Crelle J. 47 (1854) 109-. . Jordan, C. C. E. 80 (1875) 875-. , algebra applied to theory. Roberts, M. [1861] QJ. Mth. 5 (1862) 18-. , composition, theorem. Jordan, C. C. E. 81 (1875) 495-. , computation by transvection. McClintock, E. Am. J. Mth. 14 (1892) 222-. , etc. Bellavitis, G. Yen. At. (1858-59) 65-, 82-. , formula for number of, extension to in- dependent linear transformations. Capelli, A. Em. E. Ac. Line. Mm. 15 (1883) 233-. of 4th order and 4th class of two conies, simultaneous. Bjorling, C. F. E. Stockh. Ofv. 33 (1876) No. 3, 21-. given degree, number of, for forms of any species. Capelli, A. G. Mt. 20 (1882) 293-. , lists. McClintock, E. N. Y. Mth. S. Bll. 1 (1892) 85-, 236. , note on certain. Le Paige, C. M. M. H. H. Brux. Ac. Bll. 1 (1881) 490-. , primary. Deruyts, J. Brux. Ac. Bll. 20 (1890) 116-. , property of invariants of. Kohn, G. Wien Ak. Sb. 102 (1893) (Ab. 2o) 801-. of a system of quantics. Story, W. E. Mth. A. 41 (1893) 469-. .theorem. Cayley, A. Mth. A. 5 (1872) 625-. , . Jordan, C. C. E. 80 (1875) 1160-. _, _. Cayley, A. [1878] Crelle J. Mth. 87 (1879) 82-. , , general. Faa de Bruno, F. C. E. 90 (1880) 1203-. , theorems. Faa de Bruno, F. [1880] Crelle J. Mth. 90 (1881) 186-. , theory, etc. Brioschi, F. Mil. G. I. Lomb. 8 (1856) 329-; Tortolini A. 1 (1858) 158-, 296-, 549-; 2 (1859) 82-, 265-; 3 (I860) 160-. Critical functions. Cockle, Jas. QJ. Mth. 4 (1861) 97-, 265-. . Warren, J. W. [1863-64] QJ. Mth. 6 (1864) 231-, 372-. Cubic forms with any number of variables, property. Voss, A. Mth. A. 27 (1886) 515-. Decomposable forms, especially cubics. Meyer, A. Ziir. Vjschr. 42 (1897) 149-. Decomposition of forms, conditions. Hada- mard, . Par. S. Mth. Bll. 27 (1899) 34-. Differences, products. Cayley, A. QJ. Mth. 15 (1878) 174-. Differential equations satisfied by concomitants of quantics. Forsyth, A. R. L. Mth. S. P. 19 (1889) 24-. Equation in S, applications of properties of quantics to. Brisse, C. N. A. Mth. 1 (1882) 193-. 2040 Equivalence of forms. Jordan, C. C. E. 88 (1879) 906- ; 90 (1880) 1422- ; Par. EC. Pol. J. 29 ( = eah. 48) (1880) 111-. Factorisable forms, theory. Rados, G. Mth. Termt. Ets. 4 (1886) 198-. Forms, calculus. Sylvester, J. J. Camb. and Dubl. Mth. J. 7 (1852) 52-, 179- ; 8 (1853) 62-, 256-; 9 (1854)85-. of higher grade. Kronecker, L. Berl. Ak. Sb. (1883) 957-. , theorem. Voss, A. Munch. Ak. Sb. 18 (1889) 15-. , theorems. Cazzaniga, T. G. Mt. 38 (1900) 321-. Functions, 2nd order. Cayley, A. Crelle J. 38 (1849) 105-. , similar, of several groups of functions or elements. Reiss, M. Quetelet Cor. Mth. 5 (1829) 201-. Fundamental law, algebraic. Jacobi, C. G. J. Crelle J. 53 (1857) 275-. , , remarks on Jacobi' s paper. Bor- chardt, C. W. Crelle J. 53 (1857) 281-. Geometric significance of algebraic forms in theory of cubic curves. Gundelftnger, S. Mth. A. 3 (1875) 136-. Ground-forms and syzygants. (Extracts from :.) Hammond, J. ( (1883) 59, 84-, 150. J. H. Un. Cir. letters to Sylvester.) Hammond, J. (xn) . [2] ' theory. Gravelaar, N. L. W. A. N. Arch. Wisk. 3 (*1877) 193-. Higher algebra, note on a theorem. Dawson, H. G. Mess. Mth. 17 (1888) 69-. Homogeneity, principle, note. Serret, P. N. A. Mth. 14 (1855) 312-. Homogeneous isobaric functions, some pro- perties. Elliott, E. B. [1892] L. Mth. S. P. 24 (1893) 21-. Homology of sets. Cayley, A. QJ. Mth. 1 (1857) 178. Hyperdeterminant calculus, exercises. Salmon, G. Camb. and Dubl. Mth. J. 9 (1854) 19-. identity. Cayley, A. Mess. Mth. 21 (1892) 131-. Hyperdeterminants. Cayley, A. [1846-51] Crelle J. 30 (1846) 1-; 34 (1847) 148- ; 42 (1852) 368-. and canonical forms, discovery. Sylvester, J. J. Ph. Mg. 2 (1851) 391-. , conditional , scholium on theorem . Sylvester, J. J. Am. J. Mth. 1 (1878) 370-. Independent concomitants of simultaneous forms, number of. Sylvester, J. J. C. E. 84 (1877) 1359-, 1427-. Inertia of quadratic form. Hermite, C. Crelle J. 53 (1857) 271-. . Borchardt, C. W. Crelle J. 53 (1857) 281-. . Buchwald, E. (xn) Ts. Mth. 2 (1872) 20-. forms, demonstration of law. Presle, de. Par. S. Mth. Bll. 15 (1887) 179-. , law. Frobenius, G. Berl. Ak. Sb. (1894) 241-, 407-. Invariant, definition. Elliott, E. B. Mess. Mth. 16 (1887) 5-. 143 2040 General Theory of Quantics 2040 Invariant formation by operator Q. Note on Macmahon's theory. Sylvester, J. J. Mess. Mth. 13 (1884) 163-. functions associated with a transformable system. Deruyts, J. Brux. Ac. Bll. 32 (1896) 82-. , law of formation. Deruyts, J. Brux. Mm. Cour. 4, 51 (1889) No. 10, 16 pp. , mutual differentiation. Deruyts, J. Brux. Ac. Bll. 16 (1888) 207-. , number. Deruyts, J. Brux. Ac. Bll. 21 (1891) 437-. , property. Deruyts, J. Brux. Ac. Bll. 26 (1893) 258-. , reduction. Deruyts, J. Brux. Ac. Bll. 20 (1890) 265-; 24 (1892) 286-. , in the system of geometric variables. Deruyts, J. Brux. Ac. Bll. 24 (1892) 558-. operations, a law of reciprocity. Capelli, A. Nap. Bd. 27 (1888) 189-. . which permit permutation. Capelli, A. Nap. Ed. 26 (1887) 236-; 27 (1888) 45-. , permutative character. Capelli, A. Nap. Rd. 25 (1886) 135-. properties of system of a linear form and an alternate bilinear form. Morera, G. [1883] Tor. Ac. Sc. At. 18 (1882) 383-. sum of products of coefficients of pair of quantics. Eankine, W. J. M. QJ. Mth. 1 (1857) 357-. systems, finite character. Maurer, L. Munch. Ak. Sb. 29 (1900) 147-. , general. Maurer, L. Miinch. Ak. Sb. 18 (1889) 103-. theory. Meyer, F. D. Mth. Vr. Jbr. 1 (1892) 79-. and chemistry, relation of formulae. Gordan, P., & Alexejeff, W. [1900] Erlang. Ps. Md. S. Sb. 32 (1901) 107-. Invariantive differential operators. Capelli, A. Nap. Ed. 26 (1887) 110-. Invariants. Brioschi, F. Tortolini A. 5 (1854) 207-, 409-. . Bdlavitit, G. Yen. At. (1858-59) 65-, 82-; Tortolini A. 2 (1859) 137-. . Blerzy, . N. A. Mth. 17 (1858) 301- ; 18 (1859) 420-. . Sylvester, J. J. C. E. 85 (1877) 992-, 1035-, 1091-. of algebraic forms, conception of. Study, E. Leip. Mth. Ps. B. 39 (1887) 137-. functions of forms. Wolffing, E. Mth. A. 43 (1893) 26-. , arithmetical. Poincare, H. As. Fr. C. E. (1881) 109-. calculus, test operators, properties; and application to development of any power of a logarithm. Sylvester, J. J. Ph. Mg. 32 (1866) 461-. of a certain orthogonal transformation with reference to theory of strain and stress. Sharp, W. J. C. L. Mth. S. P. 13 (1881-82) 216-. common to two quadratic forms in 2, 3, or 4 variables. Picquet, H. As. Fr. C. E. (1874) 1205-. Invariants, complete systems of. Hilbert, D. D. Mth. Vr. Jbr. 1 (1892) 61- ; Mth. A. 42 (1893) 313-. , covariants, and contravariants. Foglini, G. Em. N. Line. At. 31 (1878) 249-. , , , theory. Janni, G. G. Mt. 2, (1864) 135-, 161-, 211-. and covariants as criteria for roots of an equation. Schramm, H. A. Mt. 1 (1867-68) 259-; 3(1869-70)41-. , definition. Mansion, P. Brux. S. Sc. A. 12 (1888) (Pt. 1) 47-. of function transformed by a quadratic substitution. Spottiswoode, W. Em. E. Ac. Line. T. 7 (1883) 218-. system of three conies. Burnside* W. S. [1869] QJ. Mth. 10 (1870) 239-. , generation by integration. Hurwitz, A. Gott. Nr. (1897) 71-. of homogeneous forms. Mainardi, G, [1857} Mil. G. I. Lomb. 9 (1856) 208-. , identical equation in theory. Cayley, A* QJ. Mth. 12 (1873) 115-. of integral polynomials, certain. Laisant* C. A. Palermo Cir. Mt. Ed. 8 (1894) 249-. , principles deduced from theory of linear formations. Gram, J. P. (vn) [1873] Mth. A. 7 (1874) 230-. of quadratic forms. Kleiber, J. Z. Mth. Ps. 37 (1892) 79-. and higher geometry. Study, E* Leip. Mth. Ps. B. 49 (1897) 443-. quantics. Deruyts, J. Brux. Mm. Cour. 4, 52 (1890-93) No. 2, 28 pp. in special mathematics. Campoux, P. de. N. A. Mth. 8 (1869) 395-; 9 (1870) 113-. , theorem, fundamental. Ovidio, E. d'. G. Mt. 15 (1877) 187-. , , , proof. Christoffel, E. B. [1867] Crelle J. 68 (1868) 246-. , , , . Aronhold, . Crelle J. 69 (1868) 185-. , of Hilbert. Gordan, P. Mth. A. 42 (1893) 132-. , , proof. Dahl, C. Ts. Mth. 4 U880V 154-. , theory. Spottiswoode, W. E. S. P. 7 (1854-55) 204-. , . Combescure, E. Liouv. J. Mth. 20 (1855) 337-. , . Faa de Bruno, F. Tortolini A. 6 (1855) 328-. , . Aronhold, S. H. Crelle J. 62 (1863} 281-. , . Cayley, A. [1870] Mth. A. 3 (1871V 268-. , . Christoffel, E. B. [1881] Mth. A. 19 (1882) 280-. _ _. Veltmann, . Z. Mth. Ps. 34 (1889) 321-. , . Maurer, L. Crelle J. Mth. 107 (1891V 89-. , . Hilbert, D. Gott. Nr. (1891) 232-; (1892) 6-, 439-. _, _. Weber, H. Gott. Nr. (1893) 109-. , . Hurwitz, A. Mth. A. 45 (1894) 381-. 144 2040 General Theory of Quantics 2040 Invariants, theory, elementary development of principles. Gram, J. P. (xii) Ts. Mth. 3 (1873) 1-. , , foundations. Veltmann, W. Z. Mth. Ps. 22 (1877) 277-. , , fundamental problem. Clebsch, R. F. A . Gott. Ab. 17 (1872) 62 pp. ; Mth. A. 5 (1872) 427-. , , new application. Fuchs, L. Crelle J. Mth. 81 (1876) 97-. Involution, theory. Cayley, A. [1864] Camb. Ph. S. T. 11 (1871) 21-. , , in connection with algebraic forms. Deruyts, F. Brux. Ac. Bll. 14 (1887) 650-. Involutions. Gerbaldi, F. Palermo Cir. Mt. Ed. 9 (1895) 167-. with ordinary singularities. Guccia, G. B. Palermo Cir. Mt. Ed. 7 (1893) 49-; 8 (1894) 227-. Irrational covariants. Klein, F. Gott. Nr. (1888) 191-. Isobaric function, properties. Amanzio, D. Nap. Ac. Pont. At. 17 (1887) 85-. functions. Fan de Bruno, F. Tortolini A. 7 (1856) 76-. . Stassano, P. G. Mt. 24 (1886) 57-. homogeneous functions. Cesaro, E. G. Mt. 22 (1884) 33-, 166. Jacobians and polar opposites. Clifford, W. K. (i) Mess. Mth. 2 (1864) 229-. Joachimsthal, theorem, remark. Lebesgue, V. A. N. A. Mth. 7 (1848) 225-. Kronecker's modular systems, and inertia of quadratic forms. Netto, E. Crelle J. Mth. 110 (1892) 184-. Linear forms, theory. Hensel, K. Crelle J. Mth. 107 (1891) 241-. transformations and theory of covariants. Deruyts, J. Brux. Mm. Cour. 4, 51 (1889) No. 9, 22 pp. Multiple forms, theorem. Peano, G. Tor. Ac. So. At. 17 (1881) 73-. involutions. Valyi, G. Mth. Termt. Ets. 12 (1894) 394-; 13 (1895) 408- ; 14 (1896) 381- ; Mth. Nt. B. Ung. 13 (1897) 244-, 259-, 343-. points in systems of equations and of their Jacobian. Barcroft, D. [1886] J. H. Un. Cir. [6] (1886-87)' 30. Multiplicity, new theory. Sylvester, J. J. Ph. Mg. 3 (1852) 460-. Operator that produces all the covariants and invariants of any system of quantics. Story, W. E. L. Mth. S. P. 23 (1892) 265-. Perpetuants, applications to theory of finite quantics. Hammond, J. Am. J. Mth. 8 (1886) 104-. Projective invariants, report. White, H. S. [1898] N. Y. Am. Mth. S. Bll. 5 (1899) 161-. Quadratic forms, formulae. Day, E. H. G. QJ. Mth. 14 (1877) 184-. , geometric interpretation of invariants. Hesse, L. 0. Crelle J. 45 (1853) 82-. , non-homogeneous. Cremona, L. Mil. I. Lomb. Ed. 4 (1867) 199-. Quadratic forms, physical theories involving. Maxwell, J. C. L. Mth. S. P. 4 (1871-73) 334-. in n variables. Cotterill, J. H. (n) Mess. Mth. 2 (1864) 210-. , reduction. Jordan, C. C. E. 93 (1881) 113-, 181-, 234-. , theorem, and application to small oscil- lations. Weierstrass, C. T. W. Berl. Mb. (1858) 207- ; Berl. Ak. Mb. (1879) 430-. , of Weierstrass. Brioschi, F. (vi Adds.) Tortolini A. 1 (1858) 250-. Quadrilinear forms. Le Paige, C. M. M. H. H. [1882] Tor. Ac. Sc. At. 17 (1881) 299-. Quadriquadric function, two invariants of a. Cayley, A. Mess. Mth. 20 (1891) 68-. Quantic. Catalan, E. C. C. E. 47 (1858) 1073-. with n variables, reduction to one with n - 1. Hesse, L. O. Crelle J. 42 (1851) 117-. Quantics. Lejeune-Dirichlet, G. Berl. B. (1841) 280-. . Genoechi, A. N. A. Mth. 12 (1853) 393-. . Cayley, A. Phil. Trans. 144 (1854) 245-; 146 (1856) 101-, 627-; 148 (1859) 415-, 429-; 149 (1860) 61-; 151 (1862) 277-; 157 (1867) 513- ; 161 (1871) 17-; 169 (1878) 603- ; E. S. P. 23 (1875) 496 (corr.). . Laisant, . Par. S. Mth. Bll. 20 (1892) 117-. of alternate numbers used for determining invariants and covariants of quantics in general. Clifford, W. K. L. Mth. S. P. 10 (1878-79) 124-. , decomposition into linear factors. Painvin, L. C. E. 50 (1860) 84-; Liouv. J. Mth. 6 (1861) 209-. , general properties. Sylvester, J. J. Camb. and Dubl. Mth. J. 6 (1851) 1-. , theorems. Gundelfinger, S. Mth. A. 4 (1871) 164-. , theory and application to curves and surfaces. Reiss, M. Mth. A. 2 (1870) 385-. , index symbol. Carmicliael, E. Camb. and Dubl. Mth. J. 6 (1851) 277-; 8 (1853) 81-. , method. Terquem, 0. N. A. Mth. 7 (1848) 5-. , note. Aelt, . N. A. Mth. 3 (1864) 289-. , properties. Malmsten, C. J. Ts. Mt. Fys. 4 (1871) 225-. , property. Gukovskij, A. N. Es. S. Nt. Mm. (Mth.) 11 (1890) 145-. , theory. Hesse, L. O. Crelle J. 56 (1859) 263-. Eeciprocity, Hermite's law. Gordan, P. Gott. Nr. (1897) 182-. , , extension. Deruyts, J. Brux. Ac. Bll. 22 (1891) 11-. laws. Igel, B. Wien Ak. D. 54 (1888) (Ab. 2) 75-. in theory of algebraic forms. Brioschi, F. Tortolini A. 7 (1856) 303-. Eeduced invariant functions, characteristic equations. Deruyts, J. Brux. Ac. Bll. 25 (1893) 450-. 145 2050 Quantics Eeducibility of every quadratic form to sum of n squares. Sylvester, J. J. (vi Adds.) Ph. Mg. 4 (1852) 138-. Representation of definite forms as sums of squares. Hilbert, D. Mth. A. 32 (1888) 342-. Reproduction of forms. Poincare, H. C. R. 97 (1883) 949-. Resolution into linear factors. Walker, J. J. QJ. Mth. 6 (1864) 328-. of a form of mth order and nth degree. Schendel, L. Z. Mth. Ps. 32 (1887) 83-. Roots of solvable quantics as symmetric functions of homologues. Kummell, C. H. Am. J. Mth. 18 (1896) 74-. Secondary invariants. Rogers, L. J. L. Mth. S. P. 20 (1889) 161-. Semi-combinants as concomitants of affiliants. White, H. S. Am. J. Mth. 17 (1895) 235-. Seminvariants. Petr, K. Prag Cesk6 Ak. Fr. Jos. Rz. (Tftda 2) 6 (1897) Art. 38, 22 pp.; Fschr. Mth. (1897) 108. , generalisation. Deruyts, J. Brux. Mm. Cour. 4, 51 (1889) No. 8, 20 pp. Symbolic invariants in general theory of algebraic forms, relations. Pascal, E. Rm. R. Ac. Line. Mm. 5 (1888) 375-. Systems, finite, of forms in theory of rational functions. Kdnig, J. [1880?] Mth. A. 18 (1881) 69-. Syzygetic relations among powers of linear quantics. Clifford, W. K. [1869] L. Mth. S. P. 3 (1869-71) 9-. , Clifford's paper. Cayley, A. [1891] L. Mth. S. P. 23 (1892) 99-. Syzygies. Cayley, A. (xn) J. H. Un. Cir. [2] (1883) 150. , irreducible, of quantics, tables. Sylvester, J. J. Am. J. Mth. 4 (1881) 41-. among perpetuants, expression by means of n"tions. MacMahon, (Capt.) P. A. Am. th. 10 (1888) 149-. Transformation principle of Clebsch, and appli- cation. Gundelfinger, S. [1872] Mth. A. 6 (1873) 16-. Trilinear forms. Le Paige, C. M. M. H. H. [1881] C. R. 92 (1881) 1048-, 1103- ; 93 (1881) 264-, 509- ; Rm. N. Line. At. 35 (1882) 140-. , a system of two. Le Paige, C. M. M. H. H. Rm. N. Line. At. 35 (1882) 54-. Universal invariants. Eabut, . C. R. 115 (1892) 926-, 1028. Variables of different kinds, forms in several series. Deruyts, J. Brux. Ac. fill. 28 (1894) 157-. 2050 Binary Forms. Absolute invariant and covariants of a bi- quadratic form, geometrical representation. Lacour, . N. A. Mth. 17 (1898) 341-. Algebraic forms, theorem, with application to binary sextic. Ovidio, E. d'. Palermo Cir. Mt. Rd. 6 (1892) 225-. Anharmonic ratio and absolute invariant of biquadratic. Wedekind, L. Mth. A. 17 (1880) 1-. Binary Forms 2050 Apolarity, applications of principle. Igel, B. Wien Ak. Sb. 92 (18861 (Ab. 2) 1153-. Asyzygetic covariants of the third degree of a binary form. Stroh, E. Mth. A. 31 (1888) 444-. Binary cubic form, automorphic transforma- tion. Cayley, A. L. Mth. S. P. 14 (1883) 103-. , geometrical representation. Ger- baldi, . Palermo Cir. Mt. Rd. 3 (1889) 276-. , relations between roots and those of first covariants. Mainardi, G. Rm. At. N. Line. 26 (1873) 391-. forms. Eisenstein, G. Crelle J. 27 (1844) 89*. . Arndt, F. Grunert Arch. 17 (1851) 1- ; Crelle J. 53 (1857) 309-. . Hermite, C. QJ. Mth. 1 (1857) 20-. . Cayley, A., d- Hermite, . QJ. Mth. 1 (1857) 85-. . Battaglini, G. Nap. Rd. 3 (1864) 109-, 163-. . Hermite, C. Crelle J. 63 (1864) 30-. . Beltrami, E. Bologna Ac. Sc. Mm. . Brioschi, F. [1876] Mth. A. 11 (1877) 111-. . Tannery, J. Bll. Sc. Mth. As. 1 (1877) (Pt. 1) 260-. . Pepin, T. Rm. N. Line. At. 37 (1884) 227-. , classification. Pepin, T. Rm. N. Line. At. 39 (1886) 23-. , complete form-system for three. Gall, (Frhr.) von. Mth. A. 45 (1894) 207-. , geometrical application. Torelli, G. G. Mt. 24 (1886) 270-. , interpretation. Retail, V. Palermo Cir. Mt. Rd. 2 (1888) 25-. , representation of system of two, and their associated forms. Roberts, W. R. W. [1881] L. Mth. S. P. 13 (1881-82) 16-. , imaginary elements. Pittarelli, G. G. Mt. 23 (1885) 368-. , reduction. Hermite, C. C. R. 48 (1859) 351-. , simultaneous. Clebsch, R. F. A. Crelle J. 67 (1867) 360-. , system of two. Gundelfinger, S. Mth. A. 7 (1874) 452-. , . Gerbaldi, F. G. Mt. 17 (1879) 373-. , three. Igel, B. Wien Ak. D. 49 (1885) (Ab. 2) 277-. , system of two. Caporali, E. Nap. Rd. 22 (1883) 95-. , . Mollo, C. G. Mt. 28 (1890) 330-. , several. Torelli, G. Nap. Rd. 24 (1885) 258-. , geometrical representation. Gerbaldi, . Palermo Cir. Mt. Rd. 3 (1889) 276-. and quartic forms, geometry. Newson, H. B. [1893] Kan. Un. Q. 2 (1894) 85-. 146 2050 Binary Forms Binary cubics, geometric interpretation. Re- B tali, V. [1887] Palermo Cir. Mt. Ed. 2 (1888) 25-. decimic, reduction to canonical form. Spottiswoode, W. B. A. Ep. (1861) (pt. 2) 11 . evectant form. Roberts, S. QJ. Mth. 11 (1871) 42-. expressions, canonical form. Lerch, M. Prag Sb. (1883) (Mth. Vortr.) 447-. form, determination from initial members of covariants. Gordan, P. Mth. A. 40 (1892) 503-. of 7(th order with a sum of powers n e . Paolis, R. de. Em. E. Ac. Line. Mm. 12 (1882) 405-. , conditions for factor of mth order. Igel, B. [1880] Wien Ak. Sb. 82 (1881) (Ab. 2) 943-. forms. Betti, E. Tortolini A. 7 (1856) 60-. . Brioschi, F. Tortolini A. 7 (1856) - 231-. . Gundelfinger, S. [1871] Crelle J. 74 - (1872) 87-. . Cassani, P. G. Mt. 10 (1872) 230-. . Motto, C. G. Mt. 27 (1889) 327-. , absolute invariants, partial differential equations satisfying, in higher transforma- tions. Clebsch, R. F. A. Gott. Ab. 15 (1870) (Mth.) 65-. of alternate variables. Clifford, W. K. L. Mth. S. P. 10 (1878-79) 214-. , any degree. Battaglini, G. [1866] Nap. At. Ac. Sc. 3 (1866-68) (No. 10) 34 pp. , application to differential equation. Battaglini, G. G. Mt. 24 (1886) 128-. , plane geometry. Laguerre, E. C. E. 78 (1874) 744-. , canonical form. Cayley, A. Crelle J. 54 (1857) 48-. , , new, of groups of. Meyer, F. W. Mth. A. 21 (1883) 434-. , class. Klein, [C.] F. [1874] (vm) Er- lang. Sb. Ps. Md. S. 6 (1874) 160- ; (x) 7 (1875) 49-. , . Brioschi, F. [1876] A. Mt. 8 (1877) 24-. , , properties. Brioschi, F. Em. E. Ac. Line. Ed. 2 (1886) (Sem. 1) 302-. , classes, two. Maisano, G. Em. E. Ac. Line. Mm. 15 (1883) 44-. , Clebsch's theory, two points in. Ovidio, E. d'. Tor. Ac. Sc. At. 22 (1886-87) 427-. , combined, invariants. Bessel, A. St. Pet. Ac. Sc. Mm. (Rs.) 14 (1868) (Suppl.) ' No. 3, 88 pp. , common factor of two. Noether, M. Erlang. Ps. Md. S. Sb. 27 (1896) 110-. , , Noether's paper. Liiroth, J. Erlang. Ps. Md. S. Sb. 27 (1896) 119. , construction. Thieme, H. Mth. A. 23 (1884) 597-. and correlations in n dimension space. Waelsch, E. Mh. Mth. Ps. 6 (1895) 261-, 375-. forms and elimination. Steplianos, C. Par. EC. Norm. A. 1 (1884) 329-. 147 Binary Forms 2050 inary forms of equal degree, systems of, and complete system of cubics. Peano, G. Tor. Ac. Sc. At. 17 (1881) 580-. and equation of 6th degree. Brill, A. Mth. A. 20 (1882) 330-. - , 1st four degrees. Battaglini, G. G. Mt. 2 (1864) 170-, 193-, 243-, 340 ; 3 (1865) 24-, 51-, 218-. - , pertaining to a ternary quad- ratic form. Battaglini, G. Nap. Ed. 4 (1865) 351- ; 5 (1866) 35-, 141-. , and 2nd degrees. Battaglini, G. Nap. Ed. 3 (1864) 76-. - , 3rd and 4th degrees. Bessel, A. A. (xn) Eec. Mth. (Moscou) 3 (1868) (Pt. 1) 123-. , geometric interpretation. Wedekind, L. Mth. A. 9 (1876) 209-. - , . Lindemann, F. Mth. A. 23 (1884) 111-. - with integral coefficients, reduction. Hermite, C. Crelle J. 36 (1848) 357-. linear automorphic transformations. Klein, C. F. [1875] Mth. A. 9 (1876) 183-. - , mixed, 3rd and 4th degrees. Battaglini, G. Nap. Ed. 3 (1864) 282-. - , mutual transformation, possibility of. Clebsch, R. F. A. [1869] Mth. A. 2 (1870) 373-. - with prescribed discriminant. Hilbert, D. Mth. A. 31 (1888) 482-. - i . functional determinant. Hilbert, D. Mth. A. 33 (1889) 227-. - , representation. Clebsch, R. F. A., & Gordan, P. [1867] A. Mt. 1 (1867-68) 23-. , , application to hyperelliptic integrals. Gordan, P. A. Mt. 2 (1868-69) 346-. - , in plane and space. Laguerre, E. (x) Par. S. Phlm. Bll. 8 (1872) 83-. , as sums of powers. Rosanes, J. [1872] Crelle J. 75 (1873) 172-. , , especially 2n-ic by n + 1 powers. Bauer, G. Munch. Ak. Sb. 22 (1893) 3-. - represented on curve of genus zero. Maisano, G. Palermo Cir. Mt. Ed. 1 (1887) 14-. with several series of variables. Le Paige, C. M. M. H. H. Brux. Ac. Bll. 2 (1881) 40-; Lisb. J. Sc. Mth. 9 (1883) 211-. , simultaneous, system of four. Perrin, R. Par. S. Mth. Bll. 15 (1887) 45-. , some. Brioschi, F. [1876] Erlang. Ps. Md. S. Sb. 9 (1877) 30-. _ t system derivable from a single form. Astis, F de. G. Mt. 36 (1898) 161-. _ , systems, simultaneous. Gordan, P. Mth. A. 2 (1870) 227-. , theorem. Arcais, F. d'. Bologna Ac, Sc. Mm. 9 (1878) 173-. , _. Qcagne, M. d'. C. E. 102 (1886) 916-. - , and application to biquadratic. Arndt, F. Grunert Arch. 17 (1851) 409-. , of Hilbert's. Brioschi, F. Palermo Cir. Mt. Ed. 10 (1896) 153-. - , , and Jacobian. Faa de Bruno, F. Mth. A. 18 (1881) 280-. E2 2050 Binary Forms Binary forms, theorem on fnth differentials. Dawson, H. G. Mess. Mth. 18 (1889) 145- . , theorems, and application to multiple roots of sextic equation. Maisano, G. Km. B. Ac. Line. T. 7 (1883) 231-. , , extensions. Ovidio, E. d\ Tor. Ac. Sc. At. 14 (1878) 963-. , theory. Hermite, C. Camb. and Dubl. Mth. J. 9 (1854) 172-. , . Clebsch, R.F. A. Gott. Nr. (1870) 405-. , . Noether, M. Erlang. Ps. Md. S. Sb. 13 (1881) 41-. , . Gundelfinger, S. Gott. Nr. (1883) 115- ; Crelle J. Mth. 100 (1887) 413-. , . Deruyts, J. Brux. Ac. Bll. 14 (1887) 53-. , . Waelsch, E. Wien Ak. Sb. 100 (1891) (Ab. 2o) 574-. , , notes. Pittarelli, G. G. Mt. 15 (1877) 362-. , _, point in. Capelli, A. G. Mt. 16 (1878) 217-. , transformations/ geometric interpreta- tion. Clebsch, R.F. A. Gott. Nr. (1871) 335-. with two series of variables, new symbol in theory. Pascal, E. Nap. Bd. 26 (1887) 200-. vanishing covariants. Gordan, P. Mth. A. 12 (1877) 147-. which are complete powers, conditions for. Hilbert, D. Mth. A. 27 (1886) 158-. polars of a form. Wiederhold, P. Mth. A. 8 (1875) 444-. whose variables are fundamental integrals of a linear differential equation. Vernier, P. Par. S. Mth. Bll. 22 (1894) 133-. groups with special property of combinants. Hilbert, D. Mth. A. 30 (1887) 561-. nonic, complete system of groundforms. Sylvester, J. J. Am. J. Mth. 2 (1879) 98-. octavic, complete system of forms. Gall, (Frhr.) von. Mth. A. 17 (1880) 31-, 139-, 456. , proof of impossibility of its possessing groundform of deg.-order 10 . 4. Sylvester, J. J. Am. J. Mth. 4 (1881) 62-. , properties. Brioschi, F. C. B. 96 (1883) 1689-. quadratic forms, system of three. Astis, F. de. G. Mt. 37 (1899) 148-. quantic, a, b, c form. Hammond, J. L. Mth. S. P. 27 (1896) 393-. quantics. Brioschi, F. Tortolini A. 7 (1856) 5-, 15-, 69-. , derivatives. Cayley, A. QJ. Mth. 15 (1878) 157-. , reduction to canonical form. Faa de Bruno, F. Liouv. J. Mth. 17 (1852) 193-. quartic form, Legendre's reduction to canoni- cal form. Warren J. [1865] QJ. Mth. 7 (1866) 338-. , properties of a. Brioschi, F. G. Mt. 22 (1884) 130-. , theory. Igel, B. Mh. Mth. Ps. 10 (1899) 102-. forms. Arndt, F. Grunert Arch. 18 (1852) 111-. Binary Forms 2050 Binary quartic forms. Battaglini, G. Nap. Bd. 3 (1864) 201-, 234-. . Clebsch, R. F. A. Crelle J. 67 (1867) 371-. and elliptic functions, connection between. Study, E. Am. J. Mth. 17 (1895) 216-. in involution. Battaglini, G. Nap. Bd. 3 (1864) 263-. , simultaneous system of two. Bertini, E. [1875] G. Mt. 14 (1876) 1-. , system derivable from same quintic. Stasi, F. G. Mt. 37 (1899) 160-. and ternary cubics. Brioschi, F. [1875J A. Mt. 7 (1876) 52-. quintic. Battaglini, G. Bm. B. Ac. Line. At. 2 (1875) 582-. , application of theorem to. Ovidio, E. d\ Palermo Cir. Mt. Bd. 7 (1893) 1-. , canonical form. MacMahon, P. A* QJ. Mth. 19 (1883) 337-. and sextic, new canonical forms. Groice t B. E. Kan. Un. Q. 6 (1897) 201-. quintics. Ovidio, E. d'. Tor. Ac. Sc. At. 15 (1879) 591-. . Maisano, G. Bm. B. Ac. Line. Mm. 14 (1883) 97-. septic. Sylvester, J. J. C. B. 87 (1878) 899-. , complete form-system of the. Gall> (Frhr.) von. Mth. A. 31 (1888) 318-. sextic. Perrin, R. C. E. 96 (1883) 1717-, 1776-, 1842-. . Maisano, G. Em. B. Ac. Line. Mm. 19 (1884) 9-. , formulae relative to. Ovidio, E. d\ Tor. Ac. Sc. At. 27 (1892) 535-; 28 (1893) 133. , its syzygies and their relations. Ham- mond, J. Am. J. Mth. 7 (1885) 327-. sextics. Clebsch, R. F. A., & Gordan, P. C. B. 64 (1867) 582-. . Clebsch, R.F. A. Gott. Ab. 14 (1869) 75pp. . Bolza, 0. Mth. A. 30 (1887) 546-. -with linear transformations into them- selves. Bolza, 0. Am. J. Mth. 10 (1888) 47-. Canonic roots of binary quantic of odd order. Cayley, A. Ph. Mg. 25 (1863) 206-. , sequel to Cayley's theorems. Sylvester^ J. J. Ph. Mg. 25 (1863) 453-. Canonical forms, point in Sylvester's theory. Morley, F. [1900] N. Y. Am. Mth. S. Bll. 7 (1901) 202, 206. Catalecticant is an invariant, proof. Humbert, . As. Fr. C. B. (1888) (Pt. 2) 163-. Combinants of algebraic binary forms. Le Paige, C. M. M. H. H. Brux. Ac. Bll. 48 (1879) 530-. binary associated with curves of 4th order. Berzolari, L. A. Mt. 20 (1892-93) 101-. form-systems, etc. Gross, W. Mth. A. 32 (1888) 136-. linear system of quintics. Berzolari, L^ Palermo Cir. Mt. Ed. 7 (1893) 5-. 148 2050 Binary Forms Combinants of two binary biquadratic forms, complete system. Stephanos, C. C. E. 97 (1883) 27-. - forms, complete system. Stroh, E. Mth. A. 34 (1889) 321-. Concomitant binary forms in terms of the roots. MacKinnon, (Miss) A. L. A. Mth. 9 (1894- 95) 95- ; 12 (1898-99) 95-. Concomitants, irreducible, of any number of binary quartics. Young, A. L. Mth. S. P. 30 (1899) 290- . Condition for involution of two quartics. Binary Forms 2050 Covariants, elementary, of system of two bi- quadratic binary forms, number. Sylvester, J. J. C. B. 84 (1877) 1285-. , fundamental, of binary cubo-quadratic system. Sylvester, J. J. C. E. 87 (1878) 242-, 287-. , , system of 2 cubics, true number. Sylvester, J. J. C. E. 89 (1879) 828-. , independent, of binary forms, number. Stroh, E. Mth. A. 31 (1888) 441-. and invariants of binary form, theorem. Gordan, P. Crelle J. 69 (1868) 323-. Alagna, E. Palermo Cir. Mt. Ed. 5 (1891) . Brioschi, F. Tortolini A. 1 301-. (1858) 296-, 549-; 2 (1859) 82-, 265-; 3- Conditions that binary octavic should have (I860) 160-. 4 double points. Alagna, R. Palermo Cir. . Bonsdor/, E. [E. J.] [1876] Mt. Ed. 4 (1890) 25-. Helsingf. Acta 11 (1880) 167-. form F(x 2 , y-) maybe expressed in terms , relations. Perrin, E. C. E. of an exact square. Giordano, G. G. Mt. 96 (1883) 426-. , residues. Perrin, R. Par. 37 (1899) 367-. Conjugate binary forms and their geometric construction. Schlesinger, 0. [1882] Mth. A. 22 (1883) 520-. Covariant formula. Caijley, A. QJ. Mth. 16 (1879) 224-. Covariants of binary cubic, geometrical inter- pretation. Cayley, A. QJ. Mth. 10 (1870) 148-. forms. Cayley, A. Crelle J. 50 (1855) 277-. . Betti, -E. Tortolini A. 1 (1858) 129-. . Jordan, C. C. E. 82 (1876) 269- ; 87 (1878) 202- ; Liouv. J. Mth. 2 (1876) 177- ; 5 (1879) 345-. . Le Paige, C. M. M. H. H. Brux. Ac. Bll. 49 (1880) 113-. . Petersen, J. P. C. (xn) Ts. Mth. 4 (1880) 177- ; 5 (1881) 33- ; 6 (1889) 152- ; Fschr. Mth. (1888) 117-. , development. Bonsdorff, E. [E. J.] [1877] St. Pet. Ac. Sc. Bll. 24 (1878) , geometric representation. Linde- mann, C. L. F. Par. S. Mth. Bll. 5 (1877) 113- ; 6 (1878) 195-. , reduction of two. Stroh, E. Mth. A. 22 (1883) 290-. quantic. Cayley, A. B. A. Ep. 41 (1871) (Sect.) 9-. , finite number of. Cayley, A. QJ. Mth. 17 (1881) 137- ; Mth. A. 34 (1889) 319-. of rith degree. Roberts, M. QJ. Mth. 4 (1861) 168-, 324- ; 5 (1862) quintic. Roberts, M. Tortolini A. 3 (1860) 340-. , biquadratic (Cayley's formula). Brioschi, F. Crelle J. 53 (1857) 377-. of a biquadratic form, geometrical repre- sentation. Le Paige, C. M. M. H. H. Brux. Ac. Bll. 50 (1880) 115-. cubo-biquadratic system. Le Paige, C. M. M. H. H. Brux. Ac. Bll. 46 (1878) 765-. , double, of binary forms. Laguerre, E. (x) Par. S. Phlm. Bll. 8 (1872) 48-. S. Mth. Bll. 11 (1883) 88-. , theory. Dienger, J. [1870] Prag Ab. 4 (1871) 47 pp. even character of binary sextic relations. Stephanos, C. C. E. 96 (1883) 232-, 1564-. same binary form, relations be- tween. Brioschi, F. A. Mt. 11 (1882-83) 291-. , irrational, of certain binary forms. Study, E. Am. J. Mth. 17 (1895) 185-. , irreducible, of binary octavic. Sylvester, J. J. C. E. 93 (1881) 192-, 365-. , , septic. Sylvester, J. J. C. E. 87 (1878) 505-. , , cubo-biquadratic binary system, exact number. Sylvester, J. J. C. E. 87 (1878) 477-. , linear fundamental, of two binary cubics. Ovidio, E. d'. Tor. Ac. Sc. At. 15 (1879) 267-. , simultaneous, of two quartics and of two quintics. Pascal, E. A. Mt. 16 (1888-89) 85-. , special mode of forming. Wiltheiss, E. Mth. A. 35 (1890) 433-; 36 (1890) 134- ; 37 (1890) 229-. of system of three binary forms, principle for obtaining. Igel, B. [1882] Wien Ak. D. 46(1883) (^6.2)350-. , two simultaneous, of two biquadratic binary forms. Ovidio, E. d\ Tor. Ac. Sc. At. 15 (1879) 301-. Cubi-quadric system. Hammond, J. Am. J. Mth. 8 (1886) 138-. Cubic resolvent of a binary quartic. White, H. S. N. Y. Am. Mth. S. Bll. 3 (1897) 250-. transformation of a binary cubic form. Torelli, G. Nap. Ac. Pont. At. 18 (1888) 215-. Determinant, functional, of any number of binary forms. Le Paige, C. M. M. H. H. C. E. 92 (1881) 688-. Differential operators, certain linear, inter- change of variables in. Elliott, E. B. [1889] Phil. Trans. (A) 181 (1891) 19-. Discriminant of binary cubic, arithmetical theorem connected with. Sylvester, J. J. C. E. 60 (1865) 1011-, 1121-. 149 2050 Binary Forms Binary Forms 2050 Discriminant of binary cubic, theorem. Eisen- stein, G. Crelle J. 27 (1844) 105-. quantic. Cay ley, A. QJ. Mth. 10 (1870) 23. , calculation. Bauer, G. Munch. Ak. Sb. 16 (1887) 183-. septic f=a x 7 . Gordan, P. Mth. A. 31 (1888) 566-. septics. BrioscM, F. A. Mt. 26 (1897) 255-. sextic. Maimno, G. [1884-85] Palermo Cir. Mt. Bd. 1 (1887) 7-; Mth. A. 30 (1887) 442-. . Gordan, . Erlang. Ps. Md. S. Sb. 17 (1885) 40-. sextics. Brioschi, F. A. Mt. 1 (1867- 68) 159. Discriminants of binary forms. Bonsdorff, E. [E. J.] [1876] Helsingf. Acta 11 (1880) 191-. and resultant of two binary quantics, geo- metrical relation. Cayley, A. L. Mth. S. P. 5 (1874) 31-. resultants of binary forms. Meyer, F. Gott. Nr. (1895) 119-, 155- ; Acta Mth. 19 (1895) 385-. Divisors of binary cubics. Pepin, T. C. B. 92 (1881) 173-. Eliminant of two binary quantics. MacMahon, (Capt.) P. A. QJ. Mth. 23 (1889) 139-. Factions, definition. Cayley, A. B. A. Bp. (1875) (Sect.) 10. Finite character of number of invariants of binary forms. Mertens, F. Crelle J. Mth. 100 (1887) 223-. system of binary forms. Study, Erlang. Ps. Md. S. Sb. . G Md. S. Sb. [19] (1888) 35-. invariants of a binary funda- mental form. Hilbert, D. Mth. A. 33 (1889) 223-. . Petersen, J. Mth. A. 35 (1890) 110-. Form system of binary forms. Gordan, P. Bpm. Mth. 1 (1877) 12-. Generating functions and tables of groundforms for binary quantics, calculation. Franklin, F. Am. J. Mth. 3 (1880) 128-. Geometrical property of invariant II (ah) 2 of a binary 2n-ic, etc. Schoute, P. H. Amst. Ak. Vs. 7 (1899) 379-; Amst. Ak. P. 1 (1899) 313-. Graphical representation of invariants and co- variants, application of new atomic theory. Sylvester, J. J. Am. J. Mth. 1 (1878) Graphs, Clifford's. Spottiswoode, W. L. Mth. S. P. 10 (1878-79) 204-. , . Buchheim, A. L. Mth. S. P. 17 (1887) , , application to ordinary binary quanti Kempe, A. B. [1885] L. Mth. S. P. 17 (18! 107-, 419. , Sylvester-Clifford, application to ordinary binary quantics. Kempe, A. B. L. Mth. S. P. 24 (1893) 97-. Harmonic and equianharmonic polars of binary forms. Sclwute, P. H. Amst Ak. Vs. M. 6 (1889) 277-; As. Fr. C. B. (1889) (Pt. 2) 100-. Hessian of a binary cubic. Fontene, G. N. A. Mth. 18 (1899) 459-. form, geometrical illustration. Schoute, P. H. Amst. Ak. Vs. M. 6 (1889) 190-. , theorem on. Gerbaldi, F. Palermo Cir. Mt. Bd. 3 (1889) 22-. , . Schoute, P. H. Palermo Cir. Mt. Bd. 3 (1889) 160-. quantic in terms of the roots. McMahon, J. A. Mth. 5 (1889-90) 17-. sextic, and discriminant of octavic. Maisano, G. Palermo Cir. Mt. Bd. 3 (1889) 53- ; 4 (1890) 1- ; 5 (1891) 152-. covariant of a binary quintic. Lindemann, C. L. F. [1882] Mth. A. 21 (1883) 71-. covariants of binary biquadratic forms, geometric hints for construction. Wedekind, L. Erlang. Ps. Md. S. Sb. 7 (1875) 93-. , Jacobian, etc., in one dimension geometry. Newson, H. B. [1894] Kan. Un. Q. 3 (1895) 103-. of a product of linear functions. Franklin, F. J. H. Un. Cir. [9 (1889-90)] 31; A. Mth. 5 (1889-90) 103-. , identical vanishing of. Franklin, F. J. H. Un. Cir. [9 (1889-90)] 77-. Higher algebra, theorem in (binary quantics) . Russell, E. QJ. Mth. 21 (1886) 373-. Identities, linear, between squares of binary forms. Meyer, F. W. Mth. A. 21 (1883) b. [19] (1888) 31-. 441-. Jordan, . Erlang. Ps. Indefinite binary forms with conjugate in- determinates. Picard, E. C. B. 97 (1883) 745-. Invariant of binary cubic, expression in terms of complete system. Pascal, E. Nap. Bd. 26 (1887) 245- ; 27 (1888) 67-. conditions that two binary quintics may have four common roots. Berzolari, L. A. Mt. 19 (1891-92) 269-. configuration of a binary form. Mertens, F. Wien Ak. Sb. 98 (1890) (Ab. 2a) 73.- derived forms, mutual actions. Sylvester, J. J. Crelle J. Mth. 85 (1878) 89-. , direct computation. CJuipman, C. H. [1891] J. H. Un. Cir. [11 (1891-92)] 41-. forms of a system of binary biquadratics. Pascal, E. Nap. Ed. 27 (1888) 402-. operations admitting of permutation. Capelli, A. Nap. Ac. At. 1 (1888) No. 1, 17 pp. processes, on binary fields of higher genus. Pick, G. Gott. Nr. (1894) 311-. , simultaneous, of a binary quadratic and binary biquadratic form. Clebsch, R. F. A. [1870] Mth. A. 3 (1871) 262-. , quadratic , of two binary forms. Le Paige, C. M. M. H. H. Brux. Ac. Bll. 44 (1877 365-. of 10th order of a sextic. Le Paige, C. M. M. H. H. Liege S. Sc. Mm. 9 (1882) (No. 3) 12 pp. cs. 17 (1887) 150 2050 Binary Forms Invariant of 12th order of quintic. Fact, de Bruno, F. QJ. Mth. 1 (1857) 361-. 18th order of quintic. Le Paige, G. M. M. H. H. C. E. 92 (1881) 241-. and solution of quintic equa- tion. Hermite, C. Crelle J. 59 (1861) 304-. theory applied to solution of algebraic equations. Berger, A. Stockh. Ofv. (1890) 165- ; Fschr. Mth. (1890) 140. of binary forms. Schapira, H. D. Mth. Vr. Jbr. 2 (1893) 63-. , generating functions. MacMahon, (Capt.) P. A. Am. J. Mth. 9 (1887) 189-. Invariants of binary biquadratic. Jamet, V. N. A. Mth. 19 (1900) 419-. forms. Capelli, A. Palermo G. Sc. Nt. 15 (1882) 59-. . Hilbert, D. Leip. Mth. Ps. B. 37 (1885) 427-; Mth. A. 28 (1887) 381-. . Gordan, . Liouv. J. Mth. 6 (1900) 141-. , etc. (letter to M. Hermite). Hilbert, D. Liouv. J. Mth. 4 (1888) 249-. for higher transformations. Gordan, P. [1868] Crelle J. 71 (1870) 164-. , representation. Hilbert, D. Mth. A. 30 (1887) 15-. in two series of variables, geo- metrical interpretation. Kohn, G. D. Nf. Vh. (1896) (Th. 2, Hdlfte 1) 5-; D. Mth. Vr. Jbr. 5 (1901) (Heft 1) 58-. octavic, relation between. Alagna, R. Palermo Cir. Mt. Ed. 6 (1892) 77-. sextic. Richmond, H. W. QJ. Mth. 31 (1900) 57-. system, relations between. Torelli, G. Nap. Ed. 25 (1886) 126-. , complete systems, application of symmetric functions. Mertens, F. Krk. Ak. (Mt.-Prz.) Ez. 2 (1892) 141- ; Fschr. Mth. (1892) and covariants of binary forms. Novi, G. G. Mt. 2 (1864) 306-, 321-. , especially of 3rd and 4th degree. Siacci, F. A. Mt. 7 (1865) 73-. , law of reciprocity. Sylvester, J. J. C. E. 86 (1878) 446-. quartic considered as two ternary quadrics. Burnside, W. S. [1869] QJ. Mth. 10 (1870) 211-. , fundamental, of binary forms, method for obtaining ensemble of. Sylvester, J. J. C. E. 84 (1877) 1113-, 1211-. , irreducible, of binary forms, higher limit to number. Sylvester, J. J. C. E. 86 (1878) 1437-, 1491-, 1519-. , to binary quintic, synoptical table. Sylvester, J. J. Am. J. Mth. 1 (1878) 370-. , rule for calculating number. Sylvester, J. J. [1878] Mess. Mth. 8 (1879) 1-. of equation of nth degree. Studnicka, F. U, Casopis 10 (*1881) 208- ; Fschr. Mth. (*1881) 130. , fundamental, of binary octavic. Sylvester, J. J. C. E. 84 (1877) 240-, 532-. Binary Forms 2050 Invariants, fundamental, of binary quantics, limits to order and degree. Sylvester, J. J. E. S. P. 27 (1878) 11-. of given order and degree of binary decimic, table of numbers of. Sylvester, J. J. C. E. 89 (1879) 395-. hyperelliptic equations. Brun, F. de. Stockh. Ofv. (1897) 173- ; Fschr. Mth. (1897) 400-. integral polynomials, certain. Laisant, C. A. Palermo Cir. Mt. Ed. 8 (1894) 249-. , irreducible, of the binary quintic and sextic. Elliott, E. B. Mess. Mth. 26 (1897) 105-. , , degrees 20 and 30 for binary septic. Hammond, J. Mth. A. 36 (1890) 255-. , method of research for. Elliott, E. B. B. A. Ep. (1891) 568. of octavic, irreducible relations among. Alagna, R. Palermo Cir. Mt. Ed. 10 (1896) 41-. , Prof. Malet's, identification with Sir James Cockle's criticoids. Harley, (Rev.) R. E. S. P. 38 (1885) 45-. , , . Malet, J. C. E. S. P. 38 (1885) 211-. , rational, of binary form. Alexejeic, W. G. Fschr. Mth. (1899) 110-. , representation by B functions. Bolza, O, Gott. Nr. (1887) 418- ; Mth. A. 30 (1887) 478-. of simplest simultaneous binary forms. Bessel, A. [1868] Mth. A. 1 (1869) 173-. simultaneous, of binary forms. Gordan, P. Mth. A. 5 (1872) 595-. , , and resultant of two binary forms of orders 6 and 3. Ovidio, E. d'. Tor. Ac. Sc. At. 28 (1893) 20-. , , of two binary forms of orders 5 and 4, and their resultant. Ovidio, E. d'. Em. E. Ac. Line. Mm. 4 (1887) 607-. , theorem concerning number of. Sylvester, J. J. Ph. Mg. 5 (1878) 178-. of two binary forms having a common factor. Brioschi, F. Erlang. Ps. Md. S. Sb. 27 (1896) 116-. of orders 5 and 2, or 5 and 3. Ovidio, E. d\ [1881] Em. S. It. Mm. 4 (1882) No. 2, 19 pp. quartic forms, relations between. Ovidio, E. d\ Tor. Ac. Sc. At. 15 (1879) 471-. Involution. Hesse, L. 0. [1863] Crelle J. 63 (1864) 179-. Involutions determined by quartic.corresponding points. Walker, J. J. [1870] L. Mth. S. P. 3 (1869-71) 69-. equations, recurring formula in. Weyr, E. Casopis 9 (*1880) 279- ; Fschr. Mth. (*1880) 544-. , cubic. Le Paige, C. M. M. H. H. Wien Ak. Sb. 81 (1880) (Ab. 2) 845-; 83 (1881) (Ab. 2) 351-. , higher, and invariants of binary forms. Le Paige, C. M. M. H. H. Brux. Ac. Bll. 44 (1877) 231-. , , multiple points. Le Paige, C. M. M. H. H. Brux. Ac. Bll. 46 (1878) 247-. 151 2050 Binary Forms Irreducible forms of cubo-biquadratic system, true number of. Sylvester, J. J, C. B. 87 (1878) 445-. Isobaric functions , binary and modified . Trudi , N. G. Mt. 12 (1874) 110-. Jacobian of a biquadratic and a quadratic. Giordano, G. G. Mt. 35 (1897) 349-. Jerrard's transformation and combinants. Igel, B. Wien Ak. D. 53 (1887) (Ab. 2) 155-. Linear substitution, formulae for. Elliott, E. B. B. A. Kp. (1894) 581. transformations, theory of. Boole, G. Camb. and Dubl. Mth. J. 6 (1851) 87-. Modular transformation of 5th order. Invariant theory of its sextic equation. Gordan, P. A. Mt. 1 (1867-68) 367-. Octahedron, forms connected with. Pittarelli, G. Bm. B. Ac. Line. Ed. 4 (1888) (Sem. 1) 509-. Orthogonal concomitants. Vries, J. de. Amst. Ak. Vs. 8 (1900) 562- ; Amst. Ak. P. 2 (1900) 485-. Pencils (faisceaux) of binary cubic forms. Berzolari, L. Mth. A. 51 (1899) 473-. ( ) forms with same functional determinant. Hilbert, D. Leip. Mth. Ps. B. 39 (1887) 112-. ( ) having same Jacobian. StepJianos, C. C. B. 93 (1881) 994- ; Par. Mm. Sav. Etr. 27 (1883) No. 7, 138 pp. Peninvariants of binary forms. Ocagne, M. d'. Brux. S. Sc. A. 11 (1887) (Pt. 2) 314- ; C. B. 104 (1887) 961-. . Perrin, R. C. E. 104 (1887) 1097-, 1258-. . Ocagne, M. d'. C. E. 104 (1887) 1364-. . Perrin, E. [1887] Par. S. Mth. Bll. 16 (1888) 82-. , principal. Ocagne, M. d'. Par. S. Mth. Bll. 16 (1888) 183- . Perpetuants. MacMahon, P. A. Am. J. Mth. 7 (1885) 26-, 259-; L. Mth. S. P. 26 (1895) Point covariant with five points, construction by ruler. Morley, F. Mth. A. 49 (1897) 596-. Polynomials with 2 variables. Collins, E. St. Pet. Ac. Sc. Bll. 2 (1837) 321-. Projective involutions. Paolis, E. de. Bm. E. Ac. Line. Ed. 2 (1886) (Sem. 2) 335-. Quadratic and biquadratic forms, system. Harbordt, F. Mth. A. 1 (1869) 210-. Quadric transformations. Spottiswoode, W. L. Mth. S. P. 16 (1884-85) 148-. Quartic, aU+&pH=0, solution. Cayley, A. [1868] Mth. A. 1 (1869) 54-. Quintic, the B. G. F. for the. Hammond, J. J. H. Un. Cir. [5] (1885-86) 37. Begular graphs, theory. Petersen, J. Acta Mth. 15 (1891) 193-. Besultant of a binary quadric and n-ic. White, H. S. N. Y. Am. Mth. S. Bll. 1 (1895) 11-. two binary biquadratic forms, reduction to fundamental invariants. Warren, J. [1865] QJ. Mth. 7 (1866) 67-. Binary Forms 2050 Besultant of two binary biquadratic forms, reduction to fundamental invariants. Ovidio, E. d\ Tor. Ac. Sc. At. 15 (1879) 385-. , and relation between their simultaneous invariants. Brioschi, F. Tor. Ac. Sc. At. 31 (1895) 299- or 441-. cubics. Cayley, A. QJ. Mth. 6 (1864) 380-. forms, various representations. Schendel, L. Z. Mth. Ps. 33 (1888) 1-, 65-. Besultants. Vanetek, M. N. Prag Sb. (1888) (Mth.-Nt.) 152-. of binary forms, structure. Netto, E. Gott. Nr. (1895) 209-. Semi-invariant tables. Cayley, A. Am. J. Mth. 7 (1885) 59-. for the weights 11 and 12, corrected. Cayley, A. Am. J. Mth. 14 (1892) 195-. Seminvariants. Cayley, A. QJ. Mth. 19 (1883) 131- ; 20 (1885) 212-. . MacMahon, P. A. Am. J. Mth. 8 (1886) . Ocagne, M. d\ Brux. S. Sc. A. 10 (1886) (Pt. 2) 75-. . Deruyts, J. Liege S. Sc. Mm. 15 (1888) No. 4, 11 pp., No. 6, 8 pp. ; Brux. Ac. Bll. 19 (1890) 255-. . Roberts, S. [1890] L. Mth. S. P. 21 (1891) 219-. . Cayley, . QJ. Mth. 26 (1893) 66-. of a binary quantic, etc. Elliott, E. B. Mess. Mth. 23 (1894) 91-. , exact differential expressions connected with. Elliott, E. B. Mess. Mth. 25 (1896) 173-. of a given type, Cayley 's number of. Elliott, E. B. L. Mth. S. P. 23 (1892) 298-. of infinite form, memoir. Cayley, A. Am. J. Mth. 7 (1885) 1-. , operators in theory. MacMahon, P. A. QJ. Mth. 20 (1885) 362-. , some properties. Deruyts, J. Brux. Ac. Bll. 13 (1887) 226-. and symmetric functions. MacMahon, P. A. Am. J. Mth. 6 (1884) 131-. , theory. Cayley, A. QJ. Mth. 21 (1886) 92-. Semi- (sub) invariants to binary quantics of unlimited order. Sylvester, J. J. Am. J. Mth. 5 (1882) 79-. Sextic perpetuants for any weight w, proof that number of=0. Cayley, A. [1883] (xii) J. H. Un. Cir. [3] (1884) 13. Steiner's covariant of binary sextic. Maisano, G. Mth. A. 31 (1888) 493-. . Ovidio, E. d'. Tor. Ac. Sc. At. 24 (1889) 164-. Sylvester's canonical form of binary quantics, degree 2n - 1. Booth, W. QJ. Mth. 20 (1885 270-. Symbol of operations in the theory of forms. Brioschi, F. Mil. I. Lomb. Ed. 22 (1889) 117-. Symbolic calculus in theory of binary forms. Pascal, E. G. Mt. 26 (1888) 33-, 102-. notation of Aronhold and Clebsch. Osgood, W. E. Am. J. Mth. 14 (1892) 251-. 152 2050 Binary Forms Symbolic notation of binary forms. Kluyver, J. C. N. Arch. Wisk. 20 (1893) 97-. representation of syzygants and covariants. Stroh, E. Mth. A. 36 (1890) 262-. Syzygants of binary sextic forms. Stroh, E. Mth. A. 34 (1889) 306-. , fundamental, of binary quintic. Stroh, E. Mth. A. 34 (1889) 354-. , irreducible, of binary sextic. Gall, (Frhr.) von. Mth. A. 35 (1890) 63-. , , two simultaneous cubic forms. Gall, (Frhr.) von. Mth. A. 31 (1888) of two simultaneous binary biquadratic forms. Gall, (Frhr.) von. Mth. A. 33 (1889) 197- ; 34 (1889) 332-. . Stroh, E. Mth. A. 36 (1890) 154-. . Gall, (Frhr.) von. Mth. A. 43 (1893) 550-. Syzygies for the binary sextic, new, obtained by Aronhold's operation. Ovidio, E. d'. Tor. Ac. Sc. At. 28 (1893) 118-. , some classes of binary. Ovidio, E. d'. Tor. Ac. Sc. At. 28 (1893) 447-. Tables of binary cubic forms. Cayley, A. QJ. Mth. 11 (1871) 246-. generating functions and groundforms of binary duodecimic. Sylvester, J. J. Am, J. Mth. 4 (1881) 41-. for binary quantics of first Ternary Forms 2060 Transformation and reduction of binary forms. Silva, D. A. da. Lisb. Mm. 3 (1851) (Sc. Ex.) 1-. Transvectants from powers of quadratic. Berzolari, L. Palermo Cir. Mt. Ed. 12 (1898) 258-. Transvection of binary forms, fundamental property. Stroh, E. Mth. A. 33 (1889) 61-. and its geometric significance. Pittarelli, G. G. Mt. 16 (1878) 225-; 17 (1879) 160-. Twisted cubic curve, representation of binary cubic and quartic forms by. Berzolari, L. Palermo Cir. Mt. Ed. 5 (1891) 9-, 33-. , forms of fifth order on. Waelsch, E. Wien Ak. Sb. 100 (1891) (Ab. 2a) 803-. Weierstrassian formulae applied to binary quartic and ternary cubic. Baker, H. F. QJ. Mth. 24 (1890) 1-. 2060 Ternary forms. Aronhold, theorems, generalisation. Gundel- finger, S. [1870] Crelle J. 73 (1871) 175-. Biternary forms with contragredient variables. Clebsch, R. F. A., d- Gordan, P. [1868] Mth. A. 1 (1869) 359-. Characteristics of conic sections. Study, E. Mth. A. 27 (1886) 58-. ten orders. Sylvester, J. J. Am. J. Mth. , Cremona's formula for. Study, E. Mth. 2 (1879) 223-. simultaneous binary quantics of first four orders, taken two and two together. Sylvester, J. J. Am. J. Mth. 2 (1879) 293-, 324-. , reduced and representative for certain ternary systems of binary forms. Sylvester, J. J. Am. J. Mth. 5 (1882) 241-. , syzygy, for binary quantic. Hammond, J. Am. J. Mth. 8 (1886) 19-. Tamisage, case of failure in Sylvester's theory. Hammond, J. [1882] L. Mth. S. P. 14 (1882-83) 85-. , . Cayley, A. L. Mth. S. P. 14 (1883) 88- ; (xn) J. H. Un. Cir. [2] (1883) 85-. Transformation of algebraic equations by co- variants. Brioschi, F. Mth. A. 29 (1887) 327-. binary biquadratic form to form con- taining only even powers. Hesse, L. O. Crelle J. 41 (1851) 243-. cubic form into sum of cubes. Hesse, L. 0. Crelle J. 38 (1849) 262-. form, higher. Cayley, A. Mth. A. 4 (1871) 359-. forms and of the corresponding in- tegrals. Brioschi,. Em. E. Ac. Line. Ed. 4 (1895) (Sem. 1) 363-. of order not prime. Meyer, F. W. Mth. A. 21 (1883) 528-. quantics. Grow, (Miss) B. E. [1900] N. Y. Am. Mth. S. Bll. 7 (1901) 3, 12-. elliptic integral to Weierstrass's form. Pittarelli, G. Em. E. Ac. Line. Ed. 4 (1888) (Sem. 1) 703-. A. 27 (1886) 102-. , revision of Study's theory. Zeuthen, H. G. Mth. A. 37 (1890) 461-. Combinant N of the ternary cubic. Maisano, G. Palermo Cir. Mt. Ed. 4 (1890) 153-. Combinants of three ternary quadrics. Gerbaldi, F. Tor. Ac. Sc. At. 25 (1890) 390-. Common tangents of Hessian and Cayleyan of curve. Maisano, G. [1885] Palermo Cir. Mt. Ed. 1 (1887) 46-. Concomitants of ternary cubic, the 34. Cayley, A. Am. J. Mth. 4 (1881) 1-. cubics. Dingeldey, F. Mth. A. 31 (1888) 157-. forms, the associated. Brunyate, W. E. QJ. Mth. 25 (1891) 155-. three ternary quadrics. Baker, H. F. Camb. Ph. S. T. 15 (1894) 62-. Conditions that a quadric may be one-signed. Nanson, E. J. Mess. Mth. 25 (1896) 157- ; 26 (1897) 57-. Covariant conies and cubics of plane cubic. White, H. S. N. Y. Am. Mth. S. T. 1 (1900) of 9th order of ternary cubic form, geometric significance. Gerbaldi, F. Tor. Ac. Sc. At. 15 (1879) 707-. S of plane quartic, theorem on. Giant, E. Palermo Cir. Mt. Ed. 14 (1900) 16-. Covariants of plane cubic. Gordan, P. N. Y. Am. Mth. S. T. 1 (1900) 9-. sixth degree of ternary quartic. Maisano, G. Palermo Cir. Mt. Ed. 1 (1887) 54-. Curves analogous to Hessians, Steinerians, and Cayleyans. Maisano, G. Palermo Cir. Mt. Ed. 1 (1887) 66-. 153 2060 Ternary Forms Cyclicants or ternary reciprocants, and allied functions. Elliott, E. B. L. Mth. S. P. 19 (1889) 377-. Differential equations of pure ternary re- ciprocants. Elliott, E. B. L. Mth. S. P. 18 (1886-87) 142-. operators, certain linear, interchange of variables in. Elliott, E. B. [1889] Phil. Trans. (A) 181 (1891) 19-. Discriminant of ternary cubic forms. Mangeot, S. N. A. Mth. 12 (1893) 421-. form. Gordan, P. Munch. Ak. Sb. 17 (1888) 477-. , order of. Roberts, S. [1869] L. Mth. S. P. 3 (1869-71) 14-. Eliminant of set of general ternary quadrics. Muir, T. [1899] Edinb. B. S. T. 39 (1900) 667-; 40 (1905) 23-. Generating functions of ternariants, essay to- wards the. Forsyth, A. R. L. Mth. S. P. 29 (1898) 487-. Hessian covariant of a ternary form. Wolffing, E. Mth. A. 36 (1890) 97-. of product of two ternary forms. Gerbaldi, F. Palermo Cir. Mt. Ed. 3 (1889) 60-, 280. Hessians and Cayleyans of cubic curves. Gor- dan, P. N. Y. Am. Mth. S. T. 1 (1900) 402-. of ternary quantics, identical vanishing. Manning, H. P. J. H. Un. Cir. [9 (1889-90)] 78. Invariant of a quadratic bi-ternary form. Cayley, A. Crelle J. 57 (1860) 139-. system of a ternary cubic form. Mertens, F. Wien. Ak. Sb. 97 (1889) (Ab. 2a) 437-. ternary forms. Mertens, F. Wien Ak. Sb. 95 (1887) (Ab. 2) 942-. of trilinear ternary form. Pasch, M. Mth. A. 52 (1899) 127-. Invariants and covariants of ternary forms, properties. Bernardi, G. [1880] G. Mt. 19 (1881) 136-, 258-. , simultaneous, from which resultant of 3 ternary quadratic forms is composed. Igel, B. Wien Ak. Sb. 77 (1878) (Ab. 2) 783-. of 3 ternary quadratic forms. Mertens, F. Wien Ak. Sb. 93 (1886) (Ab. 2) 62-. Involution AF+BG + GH=0, where the letters denote ternary quadrics. Cayley, A. Mess. Mth. 22 (1893) 182-. Irrational covariant cubics of plane cubics. White, H. S. N. Y. Am. Mth. S. T. 1 (1900) 170-, 508. Jacobian of net of quadrics. Schoute, P. H. Kazan S. Ps.-Mth. Bll. 9 (1900) 1-, (Prot.) 33. 3 ternary forms. Gerbaldi, F. G. Mt. 27 (1889) 33-. Number of proper ternary n-ics. Cunningham, (Lt.-Col.) A. Mess. Mth. 23 (1894) 1-. Plane cubic curves. Gerbaldi, F. Palermo Cir. Mt. Ed. 7 (1893) 19-. Quadratic forms, reduction, with proof of See- ber's theorem on reduced ternary forms. Lebesgue, V. A. Liouv. J. Mth. 1 (1856) Eesultant of three ternary quadratic forms. Cayley, A. Crelle J. 57 (1860) 139-. Ternary Forms 2060 Eesultant of three ternary quadratic forms. Hermite, C. Crelle J. 57 (1860) 371-. . Radan, R. N. A. Mth. 8 (1869) 358-. . Gordan, P. Liouv. J. Mth. 3 (1897) 195-. Eesultants of ternary forms. Gordan, P. Mth. A. 50 (1898) 113-. Ternariants, algebraically complete systems. Forsyth, A. R. Am. J. Mth. 12 (1890) 1-, 115-. Ternary cubic form. Tanner, H. W. L. L. Mth. S. P. 27 (1896) 187 -. , resolution into factors, conditions. Walker, J. J. [1865] QJ. Mth. 7 (1866) 272-. , linear factors. Spottiswoode, W. QJ. Mth. 7 (1866) 281-. , 3 linear factors, conditions. Brioschi, F. A. Mt. 7 (1876) 189-. , transformation to canonic form. Spottiswoode, W. [1863] Crelle J. 63 (1864) 244-. , . Brioschi, F. C. E. 81 (1875) 590-. forms. Cayley, A. Camb. and Dubl. Mth. J. 1 (1846) 97-. . Aronhold, S. H. Crelle J. 39 (1850) 140- ; 55 (1858) 97-. . Hermite, C. Liouv. J. Mth. 3 (1858) 37-. . Brioschi, F. C. E. 56 (1863) 304-. . Clebsch, R. F. A., <& Gordan, P. [1868] Mth. A. 1 (1869) 56-. . Gundelfinger, S. Mth. A. 4 (1871) 144-. . Clebsch, R. F. A., & Gordan, P. Mth. A. 6 (1873) 436-. arising from division of circle. Eisen- stein, G. Crelle J. 28 (1844) 289- ; 29 (1845) 19-. , canonic transformation, property of parameter. Brioschi, F. L. Mth. S. P. 12 (1880-81) 58-. , relation between 2. Cayley, A. Ph. Mg. 20 (1860) 512-. , theory. Harnack, A. Mth. A. 9 (1876) 218-. and quartic forms. Hesse, L. 0. Crelle J. 41 (1851) 285-. definite forms. Hilbert, D. Acta Mth. 17 form, conditions of decomposition. Hada- mard, . Bordeaux S. Sc. PV. (1896-97) 100-. , representation by linear combination of 2 others. Bertini, E. Mil. I. Lomb. Ed. 24 (1891) 1095-. forms. Eisenstein, G. Liouv. J. Mth. 17 (1852) 473-. . Simerka, W. Wien SB. 38 (1859) 390-. . Brioschi, F. A. Mt. 15 (1887-88) 235-. , any degree. Battaglini, G. Nap. At. Ac. 4 (1869) (No. 3) 38 pp. and Aronhold's invariants. Sylvester, J. J. Ph. Mg. 5 (1853) 299-, 367-. 154 2060 Ternary Forms Ternary forms, conditions for linear factors of. Brill, A. [1896-97] D. Mth. Vr. Jbr. 5 (1901) (Heft 1) 52- ; Mth. A. 50 (1898) 157-. of 4th, 5th, 6th, 7th orders, equations arising from certain. Brioschi, F. Mil. I. Lomb. Ed. 17 (1884) 401-. , Hilbert's method of deriving invariants, etc. White, H. S. Am. J. Mth. 14 (1892) 283-. , property. Brioschi, F. G. Mt. 1 (1863) 65-. , reduced systems. Bonsdorff, E. Hel- singf. Acta 14 (1885) 397-. with several series of variables. Capelli, A. G. Mt. 18 (1880) 17-. , simultaneous, equivalent to given ternary. Forsyth, A. E. QJ. Mth. 23 (1889) 102-. linear forms. Study, E. Mth. A. 30 (1887) 120-. quadratic forms. Battaglini, G. [1867] Nap. At. Ac. Sc. 3 (1866-68) (No. 17 e No. 26) 26 + 32 pp. , automorphic transformation. Her- mite, C. Crelle J. Mth. 78 (1874) 325-. , complete invariantive system of three. Fischer, E., d- Mumelter, K. Mh. Mth. Ps. 8 (1897) 97-. .indefinite. Picard,E. C.E.97(1883) 845-. , linear syzygetic relations between coefficients. Genese, E. W. B. A. Ep. (1882) 452-. , positive, reduction. Charve, L. Par. c. Norm. A. 9 (1880) (Suppl.) 156 pp. , representation by squares. Igel, B. [1874] A. Mt. 6 (1875) 141-. , simultaneous reduction of 2, to canonical form. Vogt, H. N. A. Mth. 15 (1896) 441-. , system of 2. Perrin, E. Par. S. Mth. Bll. 18 (1890) 1-. , simultaneous. Osgood, W, F. Am. J. Mth. 14 (1892) 262-. , 3. Ciamberlini, C. G. Mt. 24 (1886) 141-. , simultaneous. Gundel- finger, S. [1874] Crelle J. Mth. 80 (1875) 73-. , theory. Meyer, A. Crelle J. Mth. 98 (1885) 177- ; 112 (1893) 87-. quartic f=x^x 2 + x./x 3 + x/x 1 , complete form system of. Gordan, P. Mth. A. 17 (1880) 217- ; 20 (1882) 487-. f=x l s x 2 + x-\(/) of quadratic forms. Weier- strass's theorem on multiple roots of dis- criminant. Hovestadt, H. Z. Mth. Ps. 26 (1881) 392-. Ternary and quaternary linear transformations, generating systems. White, H. S. Am. J. Mth. 14 (1892) 274-. Theory of Equations 2400 Theory of Equations. 2400 General. Addition-substitutions in algebraic equations. Bertelsen, B. P. N. Ts. Mth. 9 (B) (1898) 84-; Fschr. Mth. (1898) 73. Algebraic equation, depression of, owing to linear relation between roots. Sayre, H. A. [1891] A. Mth. 6 (1891-92) 45-. and its derivative, relation between roots. Berg, F. J. van den. N. Arch. Wisk. 9 (*1882) 1-, 60; 11 (1884) 153- ; 15 (1888) 100- ; Fschr. Mth. (*1882) 51; (1884) 76; (1888) equations. Siebel, A. Arch. Mth. Ps. 56 (1874) 422-; 57 (1875) 73-, 350- ; 58 (1876) 127- ; 60 (1877) 138- ; 61 (1877) 122- ; 65 (1880) 394-; 67 (1882) 375-. . w es t, E. C. E. 91 (1880) 598-, 664-. , application of form theory to. Szabo, P. Orv.-Termt. Ets. (Termt. Szak) (1894) 153-. and an arithmetical progression, theorem. Volpicelli, P. N. A. Mth. 14 (1855) 120-. Cotes's theorem. Beltrami, E. G. Mt. 1 (1863) 123- . , rationalisation. Forstemann, W. A. Crelle J. 14 (1835) 236-. , . Cayley, A. Camb. and Dubl. Mth. J. 8 (1853) 97-. , theorems. Mathieu, E. N. A. Mth. 15 (1856) 409-. , . Lucas, F. C. E. 78 (1874) 431-. , , general. Laguerre, E. N. A. Mth. 19 (1880) 241-. , theory. Brassinne, E. Toul. Mm. Ac. 1 (1863) 23-. , . Konig, J. Mth. A. 21 (1883) 424-. , , developments. Kronecker, L. Berl. Ak. Mb. (1879) 205-. , , an extension. Sylvester, J. J. C. E. 58 (1864) 689-. whose highest powers have small coeffi- cients. Morton, W. B. QJ. Mth. 31 (1900) 247-. (repeated) transformation, principle of. Schapira, H. Heidi. Nt. Md. Vh. 4 (1892) 25-. transformation. Homer, W. G. [1823] Mathematician 1 (1845) 108-, 136-, 311- ; 2 (1847) 32-, 129-. Alternate functions and cyclotomy. Cauchy, A. L. C. E. 10 (1840) 178-. Coefficients in a transformed equation. Lodge, A. Mess. Mth. 12 (1883) 178-. Complex magnitudes applied to solution of equations. Lerch, M. Casopis 20 (1891) 265-, 302- ; Fschr. Mth. (1891) 83. Construction of equations, historico-critical notice. Favaro, A. Mod. Ac. Sc. Mm. 18 (1878) 127- ; 19 (1879) 233-. ubic Cubic and higher involutions. Fries, J. de. Mh. Mth. Ps. 5 (1894) 1-. involutions. Fries, J. de. Palermo Cir. Mt. Ed. 5 (1891) 289-; Amst. Ak. Vs. 3 (1895) 64-. 156 2400 Theory of Equations. General Differential operators, use in theory of equa- tions. Hammond, J. L. Mth. S. P. 14 (1882-83) 119-. Equation whose roots are functions of roots of 2 equations. Mackenzie, J. L. Edinb. Mth. S. P. 5 (1887) 59-. 2 roots of given equation. Amigues, E, N. A. Mth. 11 (1892) 245-. Equations. Adder, . Bordeaux Ac. Se. Pbl. (1834) 50-. , algebraic theory. Pellet, A. E. Par. S. Mth. Bll. 15 (1887) 61-. , clearing surds from. All-man, W. [1813] Phil. Trans. (1814) 23-. , congeneric surd. Horner, W. G. Ph. Mg. 8 (1836) 43-. , identities in theory. Brioschi, F. Tortolini A. 5 (1854) 301-. , impossible. Cockle, Jas. Ph. Mg. 36 (1850) 281-. , . Finlay, R. Manch. Ph. S. Mm. 9 (1851) 236-. , and surd. Harley, R. Manch. Ph. S. Mm. 9 (1851) 207-. , infinitesimal theory of, and implicit func- tions. Pellet, A. Par. S. Mth. Bll. 23 (1895) 7-. with integral coefficients. Kronecker, L. Crelle J. 53 (1857) 173-. . Runge, C. Gott. Nr. (1899) 89-. , metaphysics of. Cossali, P. Padova N. Sag. 1 (1817) 159-. with rational coefficients. Dedekind, R. D. Mth. Vr. Jbr. 1 (1892) 33-. , rationalisation, direct method. Scheffler, H. Grunert Arch. 13 (1849) 389-. in tenth book of Euclid's elements. Christen- sen, S. A. Ts. Mth. 6 (1889) 161- ; Fschr. Mth. (1888) 5. , theory. Abbati, P. Mod. Mm. S. It. 10 (1803) 385-. , . Cockle, Jas. Ph. Mg. 27 (1845) 292-. _ t _. Catalan, E. C. C. B. 47 (1858) 797-. , . Becker, J. K. Grunert Arch. 34 (1860) 288-. , . Buys-Ballot, C. H. D. Amst. Vs. Ak. 13 (1862) 430-. , . Cockle, (Sir) J. [1863] Manch. Lt. Ph. S. P. 3 (1864) 171-. , . Brassinne, E. [1864] Toul. Mm. Ac. Sc. 3 (1865) 190-. , . Petersen, J. P. C. (xn) Ts. Mth. 1 (1865) 57-; 3 (1867) 46; 5 (1869) 174- ; 3 (1879) 170- ; 6 (1882) 3-. , . Laurent, H. [Herm.] C. R. 62 (1866) 140-. , , algebraic transformation, and quad- ruple algebra. Cockle, Jas. Ph. Mg. 3 (1852) 436-. , , analysis. Cockle, Jas., & Davies, . Ph. Mg. 32 (1848) 351-. _, _, _. Cockle, Jas. Ph. Mg. 37 (1850) 493-. t ; application of rational mechanics to. Lucas, F. C. B. 89 (1879) 224-. 2400 QJ. Equations, theory, formulae. Hudson, C. Mth. 18 (1882) 74-. , , general, note. Vieille, J. N. A. Mth. 6 (1847) 174-. , , historical note. Bertrand, J. N. A. Mth. 14 (1855) 30-. , , . Tortolini, B. C. E. 47 (1858) 598-. , , illustration of theorem. Cayley, A- Mess. Mth. 11 (1882) 111-. , , synopsis. Cayley, A. Mess. Mth. 5 (1876) 39-. whose roots have their real parts negative. Hurwitz, A. Mth. A. 46 (1895) 273-. Equivalent equations, some. Taylor, C. QJ. Mth. 10 (1870) 93-. Fourier's "Analysis," error in. Murphy, R. Ph. Mg. 11 (1837) 38-. Homogeneous functions of 2 polynomials of the same degree. Mirman, L. N. A. Mth. 4 (1885) 173-. "Invention nouvelle en Palgebre " (reproduc- tion of 16th century work by Albert Girard). Bierens de Haan, D. N. Arch. Wisk. 11 (1884) 83-. Irrationality in solution of f(xyz) = 0.j Enriques > F. Em. E. Ac. Line. Ed. 4 (1895) (Sem. 2) 311- ; Mth. A. 49 (1897) 1-. Irrationals, algebraic, theory. Capelli, A . Nap. Ed. 30 (1891) 61-. Isobaric calculus. Cesdro, E. N. A. Mth. 3 (1884) 561- ; 4 (1885) 59-, 578. Logic of characteristics. Cayley, A. QJ. Mth. I (1857) 257-. Number of variations of a polynomial whose coefficients depend on a parameter. Andre", D. Par. EC. Norm. A. 2 (1885) 75-. Progressions, geometrical, Euler-Pfaff theorem. Grunert, J. A. Grunert Arch. 2 (1842) 220-. , , . Luchterhandt, A. R. Grunert Arch. 3 (1843) 305-. , , . Gopel, A. Grunert Arch. 3 (1843) 394-. Quadri-quadric porism. Mathews, G. B. QJ. Mth. 30 (1899) 16-. Quintic surds, 2, rationalisation of expression. Tortolini, B. (vm) A. Mt. 4 (1861) 285-. Badicals, removal from equations. Mooney, D. [1796] Ir. Ac. T. 6 (1797) 221-. , . Hultman, F. W. Ts. Mt. Fys. 4 (1871) 1-. Eational functions of n elements, and general theory of equations. Konig, J. [1878] Mth. A. 14 (1879) 212-. Eeviews of books by Fourier and Drobisch. Gartz, . Grunert Arch. 1 (1841) 225-. Boots of derived equation, property. Legebeke, G. J. [1881] N. Arch. Wisk. 8 (*1882) 75-; Arch. Neerl. 16 (1881) 273-. System of trigonometrical equations, with applications to porisms of coaxal conies. Wolstenholme, J. QJ. Mth. 10 (1870) 356-; II (1871) 66-. Systems of equations with peculiarities. Krey,H. [1881] Mth. A. 19 (1882) 497-. 157 2410 Algebraic Equations 2410 Elements of the theory of algebraic equations; existence of roots, symmetric functions; rational fractions; partial frac- tions. Algebra, fundamental theorem, proof. Mertens, F. Wien Ak. Sb. 97 (1889) (Ab. 2 a) 1505-. , , and solution of equations by square roots. Vahlen, K. T. Acta Mth. 21 (1897) 287-. Algebraic equation, roots of derivative. Berloty, . C. E. 99 (1884) 745-, 832. , . Cayley,A. C. R. 110 (1890) 174-, 215-. equations, a law of inertia for. Meyer, F. Gott. Nr. (1891) 279-. Cauchy's theorem. Zbikovskij, A. K. Kazan S. Nt. (Ps.-Mth.) P. 6 (1888) 240-. Coefficients in expansions invol /ing symmetric expressions. Anglin, A. H. Liouv. J. Mth. 2 (1886) 139-. expressed as functions of sums of like powers of roots. Janni, V. Nap. Ed. 18 (1879) 199-. and sums of powers of roots, relations between. Cauchy, A. L. C. E. 12 (1841) 698-, 813- . Decomposition of certain expressions into 2 squares. Realis, S. N. A. Mth. 4 (1865) 451-. Development of an algebraic fraction. Mac- Mahon, P. A. Am. J. Mth. 6 (1884) 287-. Diaphoric functions. Tanner, H. W. L. Mth. Gz. 1 (1900) 152- . (Tanner). Genese, E. W. Mth. Gz. 1 (1900) 184. Differences of roots, analytic use. Minich, S. E. [1875] Em. E. Ac. Line. At. 3 (1876) (Pt. 2) 303. Differential equations in theory of asym- metric algebra. Forsyth, A. E. Camb. Ph. S. T. 16 (1898) 291-. Discriminant of cubic equation and quadratic expression Q 2 - 3PP', relation. Young, J. E. [1876] Ir. Ac. P. 2 (1875-77) 744-. Equation of differences. Cayley, A, Tortolini A. 2 (1859) 365-; Phil. Trans. (1860) 93-. 7ith order, and system of n equations satisfied by roots. Antaev, S. N. [1897-1900] Eec. Mth. (Moscou) 20 (1899) 33-; 21 (1901) 1-; Fschr. Mth. (1897) 92; (1900) 94. for product of differences of (all but one) roots of an equation. Cayley, A. Phil. Trans. (1861) 45-. of quotients of roots. Terquem, O. N. A. Mth. 4 (1845) 516-. with roots in arithmetical progression. Viaggi, F. G. Mt. 15 (1877) 376-. roots of which are diminished by p, re- lations between coefficients. Lobatto, E. Amst. Vs. Ak. 4 (1856) 208-. , similar functions of roots. Kronecker, L. Liouv. J. Mth. 19 (1854) 279-. 2410 Equation of squares of differences. Desmarcst, E. N. A. Mth. 1 (1842) 169-. . Tarnier, . N. A. Mth. 3 (1844) 410-. . Pegado, M. Lisb. A. 2 (1858) 19-. , progressive development. Minich, S. E. Yen. At. (1858-59) 343-. of roots, existence of relation among coefficients. Ellis, E. L. (\iAdds.) Gamb. Mth. J. 1 (1839) 256-. , and geometrical applica- tion to tangents to a curve from one point. Joachimsthal, F. N. A. Mth. 9 (1850) 98-. sums of roots taken 2 together. Gerono, G. C. N. A. Mth. 2 (1843) 128-. with 2 roots rationally expressible in terms of others (Brioschi). Genocchi, A. N. A. Mth. 16 (1857) 95-. Equations with equal differences of roots. Gatti, S. [1876] G. Mt. 15 (1877) 28-. . Viaggi, F. G. Mt. 15 (1877) 376-. , formulae in theory. Sharp, W. J. C. Mess. Mth. 9 (1880) 188-. , irreducible, relations of roots. Schane- mann, T. [1852] Wien D. 5 (1853) 143-. , , (Schonemann) . Ettingshausen, A. von. Wien SB. 8 (1852) 442-. , proof that roots have form p + q>J-I. Crelle, A. L. Berl. Ab. (1829) 1-. , resolvability into factors of 1st or 2nd degree, Laplace's proof, etc. Hube, K. (vi Adds.) Krk. Eoczn. Uniwers. 3 (1818) 92-. with roots expressible as linear functions of one root. Pellet, A. E. Bll. Sc. Mth. As. 4 (1880) (Pt. 1) 262-. in geometrical progression. Terquem, O. N. A. Mth. 5 (1846) 210-. progression. Guilmin, A. N. A. Mth. 8 (1849) 242-. ; theorem that complex roots occur in pairs, Gauss's proof. Combette, E. (xn) Brest S. Ac. Bll. 7 (1871) 28 (bis)-. , theory. Gauss, C. F. Gott. Ab. 4 (1848- 50) 3-. , , fundamental principle. Du Bourguet, . Gergonne A. Mth. 2 (1811-12) 338-. , , . Bret, . Gergonne A. Mth. 3 (1812-13) 31-. , , . Du Bourguet, . Gergonne A. Mth. 3 (1812-13) 94-. , , . Bret, -. Gergonne A. Mth. 3 (1812-13) 369-. , , . Encontre, D. Gergonne A. Mth. 4 (1813-14) 201-. , , . Liouville, J. Liouv. J. Mth. 4 (1839) 501- ; 5 (1840)31-. , , . Woodtouse, L. G. Teix. J. Sc. 6 (1885) 177-. , , theorem. Hocks, H. Z. Mth. Ps. 28 (1883) 123-. , , , Cauchy's. Eudio, F. Ziir. Vjschr. 39 (1894) 345-. , , , Mourey's proof. Grunert, J. A. Grunert Arch. 1 (1841) 81-. 158 2410 Existence of Roots Equations, theory, fundamental theorem, proof. Stern, M. A. Crelle J. 23 (1842) 370-. , , -- , . Wittstein, T. Grunert Arch. 11 (1848) 218-. Every equation in one unknown can be satisfied by a complex value of this unknown, geo- metric proof. Wittstein, T. Grunert Arch. 6 (1845) 225-. rational integral function vanishes for some real or complex value of variable, proof. Stieltjes, T. J. (jun.) N. Arch. Wisk. 9 (*1882) 196-. EXISTENCE OF BOOTS. Murphy, E. Ph. Mg. 2 (1833) 60-, 220-. Henry, J. (vi Adds.) Ph. Mg. 3 (1833) 417. (proof.) Clausen, T. As. Nr. 17 (1840) 325-. (__.) Deahna, F. Crelle J. 20 (1840) 337-. ( .) Eisenstein, G. Crelle J. 29 (1845) 96. (_.) Staudt, G. K. C. von. Crelle J. 29 (1845) 97-. ( .) Goodwin, H. Camb. and Dubl. Mth. J. 2 (1847) 224-. (_.) Airy, (Sir) G. B. [1858] Camb. Ph. S. T. 10 (1864) 283-, 327-. Cayley, A. Ph. Mg. 18 (1859) 436-. Challis, J. Ph. Mg. 19 (I860) 46-. Foscolo, G. G. Mt. 2 (1864) 13-. Young, G. P. Cn. J. 9 (1864) 26-. De Morgan, A. L. Mth. S. P. 1 (1866) No. 6, 1 p.; Camb. Ph. S. P. 1 (1866-67), 24-. ( .) Laurent, H. Les Mondes 11 (1866) 438-. (_.) Walton, W. [1869] QJ. Mth. 11 (1871) 178-. ( .) Clifford, W. K. [1870] Camb. Ph. S. P. 2 (1876) 156-. (_.) Gordan, P. Erlang. Ps. Md. S. Sb. 8 (1876) 138-. ( .) Macnie, J. 80-. Des Moines Anal. 5 (1878) ( .) Malet, J. C. [1878] Ir. Ac. T. 26 (1879) 453-. ( .) Mansion, P. (xn) Brux. S. Sc. A. 4 (1880) (Pt. 2) 99-. ( .) Netto, E. Crelle J. Mth. 88 (1880) 16-. ( .) Dutordoir, H. [1882] (xn) Brux. S. Sc. A. 7 (1883) (Pt. 1) 52-; (Pt. 2) 437-. ( .) Mansion, P. (xn) Mathesis 3 (1883) (Suppl.) 14 (bis)-. (.) Maksimovich, V. P. (xn) Kazan S. Nt. (Ps.-Mth.) P. 1 (1883) [No. 7] 49-. (_.) WalecM, . N. A. Mth. 2 (1883) 241-. ( .) Hoist, E. Christiania F. (1886) No. 1, 8 pp. ; Acta Mth. 8 (1886) 155-. (_.) juel, C. Ts. Mth. 5 (1887) 161- ; Fschr. Mth. (1887) 70. ( .) Laisant, C. A. Par. S. Mth. Bll. 15 (1887) 42-. ( .) Zeuthen, H. G. N. Ts. Mth. 1 (B) (1890) 65-; Fschr. Mth. (1890) 105-. ( .) Amigues, E. C. B. 112 (1891) 212-. ( .) Brunei, . Bordeaux S. Sc. Mm. 2 (1891) i-. (_.) Carvallo, E. N. A. Mth. 10 (1891) 109-. Existence of Roots 2410 (proof.) Loria, G. Bb. Mth. (1891) 99- ; (1893) 47-; Bv. Mt. 1 (1891) 185- ; 2 (1892) 37-; 3 (1893) 105-. ( .) Meray, C. Bll. Sc. Mth. 15 (1891) 236-. ( .) Phragmen, E. Stockh. Ofv. (1891) 113-. (_.) Weierstrass, K. Berl. Ak. Sb. (1891) 1085-. ( .) Mertens, F. Mh. Mth. Ps. 3 (1892) 293-. ( .) Mansion, P. Brux. S. Sc. A. 17 (1893) (Pt. 1) 65-. ( .) Bocher, M. N. Y. Am. Mth. S. Bll. 1 (1895) 205-. (_.) Carnoy, J. Em. N. Line. At. 53 (1900) 167-. ( .) Lamb, H. Manch. Lt. Ph. S. Mm. & P. 43 (1900) A T o. 7, 4 pp. d'Alembert's theorem. Brisse, C. Par. EC. Pol. J. 56 (1886) 163-. . Catalan, E. Bv. Sc. 43 (1889) 265-. . Durand-Greville, E. Bv. Sc. 43 (1889) 266-. . Vivanti, J. Bv. Sc. 43 (1889) 267. . Lougraire, L. de. Bv. Sc. 43 (1889) 267-. . L. Ev. Sc. 43 (1889) 268-. . Delbceuf, J. Ev. Sc. 43 (1889) 269-. . Jamet, V. Mathesis 14 (1894) 5-; N. A. Mth. 14 (1895) 437-. Cauchy's proof, simplification. Sussmann, J. Crelle J. 44 (1852) 57-. (proof) and Cauchy's theorem on imaginary roots. De Morgan, A. [1857] Camb. Ph. S. T. 10 (1864) 261-. Gordan's proof. Dalwigk, F. von. Mth. A. 34 (1889) 158-. of higher equations. Burg, A. Wien Jb. Pol. I. 17 (1832) 141- ; 19 (1837) 155-. ,2 proofs. Ullherr, J. C. Crelle J. 31 (1846) 231-. proof that every equation has root of form p + gv /^I. Ullherr, J. C. D. Nf. Vsm. B. (1845) 85-. of a rational integral equation. Elliott, E. B. L. Mth. S. P. 25 (1894) 173-. n roots in every equation of nth degree. Scheffler, H. Grunert Arch. 15 (1850) 390-. . Valeriani, V. G. Mt. [13 (1875)] 33-. I . Mandl, M. Wien Az. 24 (1887) 257-. .proof. Challis, J. Ph. Mg. 17 (1859) 112-. I __ _ , (Challis). Airy, G. B. Ph. Mg. 17 (1859) 176-. (reply to Airy). Challis, J. Ph. Mg. 17 (1859) 283-. trinomial and quadrinomial equations. Pineto, Z. St. Pet. Ac. Sc. Mm. (Rs.) 9 (*1866) 254-, 306. Expression of x n + y n in powers of (x + y) and xy. Gruson, J. P. Berl. Ab. (1814-15) (Mth.) 30-. 159 2410 Factors, resolution of integral functions into. Bretschtieider, C. A, Arch. Mth. Ps. 46 (1866) 422-. , ----- . Mandl, M. D. Nf. Vh. (1894) (Th. 2, Hdlfte 1) 7. , resolvability into real simple and quadratic. Gauss, C. F. Gott. Cm. 3 (1814-15) 107-, 135-. , ------ . Collins, E. St. Pet. Ac. Sc. Bll. 2 (1837) 377-; Crelle J. 18 (1838) 119-. , ------ . Perott, J. Crelle J. Mth. 99 (1886) 141-. Formula (x-a)(x-b)f l (x)=-F(x)+f(x) IF (a) x - b F(b)x-a\ \f(a)a-b + f(b)b-a\- Lemonnier, H. Par. S. Phlm. Bll. 2 (1878) 97-. Fractions, a theorem on. Catalan, E. Brux. Ac. Bll. 19 (1890) 742-. Function aleph, its properties. Dickstein, S. Krk. Ak. (Mt.-Prz.) Pam. 12 (1886) (35)-; Fschr. Mth. (1886) 121. -- , some theorems. Dickstein, S. Krk. Ak. (Mt.-Prz.) Pam. 16 (1889) 53-. Functions of differences, fundamental fact as to. Elliott, E. B. Mess. Mth. 29 (1900) 180-. --- of roots of given equation. Selivanov, D. T. Eec. Mth. (Moscou) 15 (1891) 581-. -- roots, and coefficients, relations. Dickson, J. D. H. Edinb. K. S. T. 29 (1880) 535-. of an equation, Newton's identities concerning. Saalschvtz, . Konigsb. Schr. 36 (1895) [5]-. Generating function, Borchardt's. Faa de Bruno, F. [1875] Crelle J. Mth. 81 (1876) 217-. -- , . Kostka, C. [1876] Crelle J. Mth. 82 (1877) 212-. -- , . ReJwfovsky, V. Casopis 11 (* 1882) 111-; Fschr. Mth. (*1882) 105-. -- of symmetric functions. Meier, F. Brux. Ac. Bll. 10 (1860) 608-. , symmetric relations found by. Bor- chardt, C. W. [1855] Crelle J. 53 (1857) 193-. Homogeneous products, sums, and roots of equations, theorems. Horner, J. QJ. Mth. 6 (1864) 360-. Infinite partial fractions, reduction of ex- ponential fractions to. Cisa de Gresi, . Tor. Mm. Ac. 31 (1827) 225-. Integral symmetric functions. Mertens, F. Krk. Ak. (Mt.-Prz.) Rz. 1 (1891) 333-; Fschr. Mth. (1891) 154-. Irrational fractions, rationalisation of denomi- nator. Sommer, B. Grunert Arch. 18 (1852) Isobaric functions, use in recurring series. Pietrocola, C. Nap. Ac. At. 9 (1899) No. 8, 23pp. Jacob! , theorem, general analysis of formula Algebraic Equations 2410 *& Liouville, J. C. R. 13 (1841) Logarithmic derived function of an integral polynomial. Laisant, C. A. As. Fr. C. R. (1894) (Ft. 2) 208-. MacMahon's generalisation of the Newton- Girard formulae. Gegenbauer, L. [1900] Amst. Ak. Vs. 9 (1901) 332-; Amst. Ak. P. 3 (1901) 347-. Newton, a formula of. Cesdro, E. Mathesis 6 (1886) 172-. Number of roots, theorem of d'Alembert. Delbceuf, J. Rv. S. 42 (1888) 823-. Numerical equations, theory. Anon, (xi 53) N. A. Mth. 17 (1878) 104-. Partial fractions. Franchini, P. [1829] Lucca At. Ac. 6 (1830) 173-. . Eisenstein, G. Crelle J. 32 (1846) 71-. . Meyer, U. H. Grunert Arch. 7 (1846) 316-, 386-. , identities concerning. Schmidt, J. JR. Amst. Mengelwerk 2 (1816) 153-. , reduction to. Serooskerken, R. C. van T. Utr. A. Ac. (1817-18) 67 pp. , . De Wit, Ae. Utr. A. Ac. (1817- 18) 54 pp. , . Dirksen, E. H. Crelle J. 1 (1826) 53-. , . Crelle, A. L. Berl. Ab. (1831) , . Oettinger, L. Crelle J. 32 (1841) 63-, 148-. , . Amstein, H. Laus. S. Vd. B1L 17 (1881) 145-. , . Hoepflingen-Bergendorf, H. von. Arch. Mth. Ps. 66 (1881) 314-. , . Steplmnos, C. [1883] Bll. Sc. Mth. As. 8 (1884) 120-. , , with application to trigonometry. Saalschiitz, L. Mh. Mth. Ps. 9 (1898) 132-, 268. , , geometrical meaning of. Eeuschle, C. Z. Mth. Ps. 41 (1896) 103-. , and their integration, theory. Sauvage,. L. Mars. Fac. Sc. A. 2 (1892) 3-. , theorems. Mainardi, G. Tortolini A. 1 (1850) 342- ; 2 (1851) 505-. Polynomials approximating to fractional func- tions. Cebysev, P. [1892] St. Pet. Ac. Sc. Mm. (Rs.) 72 (1893) (Suppl.) No. 7, 13 pp. , Jacobi's. Stieltjes, . C. R. 100 (1885) 620-. Power of polynomial, formula for. Piarron de Mondesir, E. S. As. Fr. C. R. (1877) 154-. Powers of roots, Newton's theorems combined in one formula. Torriani, J. E. Lisb. Ac. Sc. Mm. (Cor.) 3 (pte. 2) (1814) 1-. Rational fraction , \f o2in' decomposition. Gerono, G. C. N. A. Mth. 18 (1859) 346. , formula of Laplace's. Y 467-. Jilrgensen, C. Crelle J. 11 (1834) 136-. 160 2410 Algebraic Equations 2410 Eational fractions. Stainville, de. Gergonne A. Mth. 3 (1812-13) 279-. , algebraic integrals in. Bouniakmvsky, N. St Pet. Ac. Sc. Mm. 3 (1833) 205-. , approximation to functions of one variable by. Laguerre, E. Par. S. Mth. Bll. 5 (1877) 78-. , decomposition. Stainville, de. Par. EC. Pol. Cor. 3 (1814-16) 25-. , . Crelle, A. L. Crelle J. 9 (1832) 231- ; 10 (1833) 42-. , . Finck, B. N. A. Mth. 4 (1845) 295-. , . Rawson, R. Mathematician 1 (1845) 265-; 2 (1847) 6-. , . Liouville, J. Liouv. J. Mth. 11 (1846) 462-. , . Terquem, 0. N. A. Mth. 6 (1847) , . Montojo, S. Cadiz Period. M. Ci. 1 (1848) 54-. , . Brioschi, F. Crelle J. 50 (1855) 239-, 318-. , . Rouche, E. C. E. 46 (1858) 540-. , . Vieille, J. C. K. 49 (1859) 746-. , . Horner, J. [1860] QJ. Mth. 5 (1862) 39-. , . Moch, S. N. A. Mth. 1 (1862) 339-. , . Realis, S. N. A. Mth. 3 (1864) 438-. , . Trudi, N. G. Mi. 2 (1864) 225-, 257-. , . St Germain, A. de. N. A. Mth. 8 , . Hermite, C. (xn) Brux. S. Sc. A. 2 (1878) (Pt. 2) 157-. , . Gomes Teixeira, F. G. Teix. J. Sc. 1 (1878) 5-, 17-, 33-, 49-, 97-, 113- ; 2 (1880) 33-. , . Jamet, V. N. A. Mth. 8 (1889) 228-. , , and theory of residues. Rouche, E. C. E. 49 (1859) 863-. , transformation. Rangoni, L. [1834] Mod. Mm. S. It. 21 (1836) 65-. , development in partial fractions. Anon. (vi_536) Gergonne A. Mth. 10 (1819-20) 255. , expansion. Trudi, N. (xi) Nap. Ac. At. 2 (1865) No. 17, 68 pp. ; (vm) Nap. Ed. 5 (1866) 441-. . Factorials, ratios of reciprocal. Cauchy, A. L. C. E. 17 (1843) 921-, 1159-. . , , and transformation of logarithms into definite integrals. Cauchy, A. L. C. E. 17 (1843) 779-. , geometric properties. Lucas, F. C. E. 78 (1874) 140-, 180-, 271-. , index. Hermite, C. Par. S. Mth. Bll. 7 (1879) 128- . , theorems by Hermite. Laguerre, E. Crelle J. Mth. 89 (1880) 339-. , theory. Picquet, H. Par. 6c. Pol. J. Cah. 53 (1883) 79-. functions of the cubic. Sawin, A. A. Mth. 9 (1894-95) 158-. VOL. I. 161 Eational functions of roots, form. Realis. S. G. Mt. 9 (1871) 206-. of integral equation. Blutel, . Par. S. Mth. Bll. 20 (1892) 92-. Eationalisation of algebraic expressions, Scheibner, W. Leip. B. 15 (1863) 63-. consisting of sums of square roots (Fermat's problem). Lehmann, J. W. H Grunert Arch. 35 (I860) 207-. any binomial denominator. Decher, G. Grunert Arch. 20 (1853) 245-. fractions. Unferdinger, F. Grunert Arch. 33 (1859) 104-. a sum of 2 th roots. Rychlicki, S. Arch. Mth. Ps. 68 (1882) 180-. Eecurring functions of 3rd degree. Gohierre de Longcliamps, . As. Fr. C. E. 9 (1880) 115-. Eegular difference terms. Kempe, A. B. L. Mth. S. P. 25 (1894) 343-, 396. Eoots of equations, dimensions. De Morgan, A . QJ. Mth. 1 (1857) 1-, 80, 232-. , expression by transcendental functions impossible. Wantzel, L. Par. S. Phlm. PV. (1845) 114-. , form of. Olivier, L. Crelle J. 1 (1826) 97-. , forms of. Casinelli, A. (s. L.) Bologna N. Cm. 8 (1846) 317 , geometrical representation. Goodwin, H. [1846] Camb. Ph. S. T. 8 (1849) 342-. , properties. Cockle, Jos. QJ. Mth. 5 (1862) 291-. , property. Brioschi, F. N. A. Mth. 15(1856)366-. , , with geometrical applications. Laisant, C. A. As. Fr. C. E. (1888) (Pt. 2) , square roots of. Terquem, 0. N. A. Mth. 6 (1847) 456-. , transformation. Gelder, J. de. Amst. Mengelwerk 1 (1816) 46-. quadratic with complex coefficients. Johnson, . N. Y. Mth. S. Bll. 2 (1893) 171. Successive powers of root, Anglin's formula. Cayley, A. QJ. Mth. 19 (1883) 223-. Sum of like powers of roots. Faure, H. N. A. Mth. 14 (1855) 94-. , as function of coefficients. Pellet, A. E. N. A. Mth. 14 (1875) 259-. , and vice-versa. Farkas, J. Arch. Mth. Ps. 65 (1880) 433-. Sums of like powers of roots of equations, and Brioschi's theorem. Genocchi, A. N. A. Mth. 12 (1853) 260- ; 14 (1855) 246-. powers of roots of general quadratic. Kapteyn, W. N. Arch. Wisk. 3 <*1877) 203-. , Newton's formula, generalisation. Lachlan, R. Mess. Mth. 13 (1884) 155-. , theorem. Collins, E. [1837] St Pet. Ac. Sc. Bll. 3 (1838) 52-. , , and application to infinite products. Stockier, F. de B. G. Lisb. Mm. Ac. Sc. 2 (1799) 1-. , , proof. Schubert, F. T. [1808] St Pet. Ac. Sc. Mm. 2 (1810) 124-. 2410 Symmetric Functions Sums of powers of roots to (2n-l) th power, to construct polynomial from. Laguerre, E. Par. S. Mth. Bll. 5 (1877) 26-. , Waring' s formula. Janni, V. Nap. Ed. 17 (1878) 27-. , . Cayley,A. Mess. Mth. 21 (1892) 133-. , . Almeida Arez, J. B. d\ G. Teix. J. Sc. 14 (1900) 117- ; Fschr. Mth. (1900) 162. , and Wronski's aleph function, Crocchi's proposition. Dickstein, S. Krk. Ak. (Mt.-Prz.) Pam. 12 (1886) (41)-; Fschr. Mth. (1886) 121. SYMMETEIC FUNCTIONS. Cauchy, A. L. [1824] Par. Mm. Ac. Sc. 9 (1830) 104-. Conti, C. Padova N. Sag. 3 (1831) 90-. Pioch, A. Brux. Ac. Bll. 2 (1835) 252-. Transon, (Prof.) A. N. A. Mth. 9 (1850) 75-. Dienger, J. Grunert Arch. 16 (1851) 471-. Jaufroid, B. N. A. Mth. 12 (1853) 116-. Brioschi, F. Tortolini A. 5 (1854) 313-, 422-. Borchardt, C. W. N. A. Mth. 14 (1855) 26-. Faa de Bruno, F. Tortolini A. 6 (1855) 338-, 412-, 476-. Kronecker, L. Liouv. J. Mth. 1 (1856) 385-. Betti, E. Crelle J. 54 (1857) 98-. Cayley, A. Phil. Trans. (1857) 489-. Roberts, M. Tortolini A. 3 (1860) 172- ; QJ. Mth. 4 (1861) 57-; (vm) A. Mt. 4 (1861) 50-. Matthiessen, L. Z. Mth. Ps. 12 (1867) 322-. Bonolis, A. G. Mt. 11 (1873) 321-. Faa de Bruno, F. C. B. 76 (1873) 163-. Isely, J. P. [1874] Neuch. S. Sc. Bll. 10 (1876) 59-. Malet, J. C. [1874] Ir. Ac. T. 25 (1875) 337-. Kapteyn, W. N. Arch. Wisk. 2 (* 1876) 73-. Boschi, P. Bologna Ac. Sc. Mm. 7 (1876) 569-. Kronecker, L. Berl. Ak. Mb. (1880) 936-. Forsyth, A. R. Mess. Mth. 10 (1881) 44-. Durfee, W. P. [1882] (xn) J. H. Un. Cir. [2] (1883) 23. Biehler, C. N. A. Mth. 3 (1884) 218-. Ocagne, M. d\ G. Teix. J. Sc. 7 (1886) 133-. Allardice, R. E. Edinb. Mth. S. P. 7 (1889) Worontzoff, . N. A. Mth. 10 (1891) 325-; 12 (1893) 116-. Kohn, G. Wien Ak. Sb. 102 (1893) (Ab. 2a) 199-. Brunei, . Bordeaux S. Sc. Mm. 4 (1894) Junker, F. Mth. A. 45 (1894) 1-. Metzler, W. H. L. Mth. S. P. 28 (1897) 390-. Gordan, P. Mth. A. 52 (1899) 501- ; D. Mth. Vr. Jbr. 8 (1900) (Heft 1) 178-. algebraic theorem. Mertens, F. Wien Ak. Sb. 106 (1897) (Ab. 2 a) 422-. and alternating functions, genesis. Frisiani, P. Mil. Effem. As. (1846) 97-. application to tetrahedra. Reiss, M. Quetelet Cor. Mth. 10 (1838) 229-. Symmetric Functions 2410 associated quantities. Sylvester, J. J. C. E. 98 (1884) 779-. and binomial theorem. Bret, . Gergonne A. Mth. 4 (1813-14) 25-. Brioschi's theorem. Sylvester, J. J. C. E. 98 (1884) 858-. calculation. Abbati, P. [1804] Mod. Mm. S. It. 12 (1805) 8-. . Hammond, J. L. Mth. S. P. 13 (1881-82) 79-. . Laurent, H. N. A. Mth. 6 (1887) 416-. Cayley 's proof of rule for degree. Sylvester, J. J. Ph. Mg. 5 (1853) 199-. class. Kapteijn, W. [1881] Arch. Mth. Ps. 67 (1882) 102-. of permanent. Muir, T. Edinb. E. S. P. 11 (1882) 409-. classes. Garbieri, G. [1882] Yen. I. At. 1 (1883) 33-. classification. Mathews, G. B. QJ. Mth. 25 (1891) 127-. complete. Torelli, G, G. Mt. 5 (1867) 110-. , and coefficients of an equation, corre- spondence between. Crocchi, L. G. Mt. 20 (1882) 301-. "critical" and Spencian. Cockle, Jas. QJ. Mth. 4 (1861) 97-. and a determinant of common occurrence. Trudi, N. Nap. Ed. 3 (1864) 121-. determinant occurring in calculation. Gubler, E. Ziir. Vjschr. 35 (1890) 79-. determination from coefficients. Kostka, C. [1875] Crelle J. Mth. 81 (1876) 281-. development by sums of like powers. Janni, V. G. Mt. 23 (1885) 34-. of differences of roots of an equation. Tannery, J. C. E. 98 (1884) 1420-. elementary, relations between. Junker, F. Mth. A. 38 (1891) 91-. , and sums of powers. Junker, F. Z. Mth. Ps. 41 (1896) 199-. form (x, - X^P, #,, x 2 being roots. Roberts, M. N. A. Mth. 19 (1860) 23-. formula concerning. Pesci, G. Bologna Ed. (1886-87) 49-. formulae, 2. Petersen, J. P. C. (xn) Ts. Mth. 6 (1876) 9-. fundamental principle. Gergonne, J. D. Gergonne A. Mth. 3 (1812-13) 238-. . Bret, . Gergonne A. Mth. 4 (1813- 14) 25-. systems for. Vahlen, K. T. Acta Mth. 23 (1900) 91-. geometric representation. Spitzer, S. [1850] Haidinger Ab. 4 (1851) (Ab. 3) 1-. and inverse operators. MacMahon, P. A. L. Mth. S. P. 15 (* 1883-84) 20-. maximal and minimal values, proof of theorems of Cauchy. Terquem, 0. Liouv. J. Mth. 5 (1840) 37. multiplication. Babczynski, T. Z. Mth. Ps. 17 (1872) 147-. . MacMahon, P. A. Mess. Mth. 14 (1885) 164-. of n variables, general theorem. Jung, G. C. E. 82 (1876) 988-. new theorem. MacMahon, P. A. QJ. Mth. 20 (1885) 365-. 162 2410 Symmetric Functions new theory. MacMahon, P. A. Am. J. Mth. 11 (1889) 1- ; 12 (1890) 61- ; 13 (1891) 193- ; 14 (1892) 15-. Newton's propositions, proof. Steen, A. Ts. Mth. 3 (1885) 30-; Fschr. Mth. (1885) 111. theorem. Tweedie, C. Edinb. Mth. S. P. 11 (1893) 61. operators. Lachlan, B. L. Mth. S. P. 18 (1886-87) 39- ; 19 (1889) 294-. of pairs of variables. Brill, A. Gott. Nr. (1893) 757-. partition problem. Lemoyne, G. G. Mt. 10 (1872) 93-. problem. Holmboe, B. M. Mg. Ntvd. 2 (1836) 279-. properties of a complete table. MacMahon, P. A. Am. J. Mth. 10 (1888) 42-. rational, calculation with. Hoppe,R. Schlomilch Z. 4 (1859) 353-. relation between forms. Kostka, C. [1881] Crelle J. Mth. 93 (1882) 89-. , and invariants. Gambioli, D. G. Mt. 29 (1891) 41-. representation by sums of powers. Hankel, H. [1865] Crelle J. 67 (1867) 90-. of roots and combinations of coefficients of weight 11 and 12. Rehofovsky, W. [1882] Wien Ak. D. 46 (1883) (Ab. 2) 53-. common to several equations. Betti, E. Tortolini A. 1 (1858) 193-. 2 equations. Dewulf, . N. A. Mth. 19 (I860) 18-. 2 equations, representation. Hess, E. Z. Mth. Ps. 15 (1870) 325-. of a pair of cubic equations. Walker, J. J. [1865] Mess. Mth. 3 (1866) 107-. and resultants of simultaneous equations. Escherich, G. von. Wien Ak. D. 36 (1876) (Ab. 2) 251-. of systems of equations. MacMahon, P. A. [1890] Phil. Trans. (A) 181 (1891) 481-. 2 equations. Cayley, A. Phil. Trans. (1857) 717-. , tables to the degree 10. Cayley, A. Am. J. Mth. 7 (1885) 47-. satisfying partial differential equations. Netto, E. Z. Mth. Ps. 38 (1893) 357-. and seminvariants. Cayley, A. Am. J. Mth. 15 (1893) 1-. of several series of variables. Junker, F. Mth. A. 43 (1893) 225-. simple and complete, relation. Crocchi, L. [1879] G. Mt. 18 (1880) 377-. of sines of angles ( a -\ ] . Johnson, W. W. Des Moines Anal. 6 (1879) 105-. specified number of roots. Murphy, E. (vi Adds.) Ph. Mg. 10 (1831) 413-. symbolical representation and invariant charac- ters. Gambioli, D. G. Mt. 30 (1892) 192-. tabulation. Durfee, W. P. Am. J. Mth. 5 (1882) 348-. in terms of coefficients. Mainardi, G. Em. At. N. Line. 20 (1867) 163-. theorem. Cockle, Jas. Ph. Mg. 28 (1846) 132-. Symmetric Functions 2420 theorem of Cauchy. Bertrand, J. C. B. 45 (1857) 1-. theorems. Analin, A. H. H. [1880-83] Ir. Ac. P. 3 (1883) 289-; Edinb. K. S. P. 12 (1884) 236-. theory. Schonemann, T. Crelle J. 19 (1839) 231-, 289-. . Cockle, Jas. Mathematician 3 (1850) 176-, 246-; Suppl. 27-. . Borchardt, C. W. Berl. B. (1855) 165-. . Mertens, F. Crelle J. 69 (1868) 289-; Wien Ak. Sb. 81 (1880) (Ab. 2) 988-. . Anglin, A. H. Crelle J. Mth. 98 (1885) 175-. . Konig, G. Mth. Termt. Ets. 8 (1890) 9-; Mth. Nt. B. Ung. 8 (1891) 83-. . Mertens, F. Wien Ak. Sb. 108 (1899) (Ab. 2a) 473-. and theory of distributions. MacMahon, P. A. L. Mth. S. P. 19 (1889) 220-. of 12 ic , tables. Durfee, W. P. Am. J. Mth. 5 (1882) 45-. 13 ic . MacMahon, P. A. Am. J. Mth. 6 (1884) 289-. 14 ic . Durfee, W. P. Am. J. Mth. 9 (1887) 278-. two ternary forms. Junker, F. Wien Ak. D. 64 (1897) 439-. Waring's formula. MacMahon, P. A. Mess. Mth. 14 (1885) 8-. formulae. Terquem, 0. N. A. Mth. 8 (1849) 76-. Vieille, J. C. B. 44 (1857) 1311-. ,. . . and differentials of sums of kth powers of roots. Serret, J. A. C. B. 45 (1857) 60-. Symmetry of a certain algebraic expression, proof. Allardice, R. E. Edinb. Mth. S. P. 4 (1886) 37. , law. MacMahon, P. A. QJ. Mth. 22 (1887) Teleological method. Dickstein, S. Prace Mt.-Fiz. 3 (1892) 126- ; Fschr. Mth. (1892) 97-. ', Wronski's. Krauze, W. Prace Mt.-Fiz. 3 (1892) 110- ; Fschr. Mth. (1892) 97. Wronski's formulae, proof of two. Dickstein, S. Krk. Ak. (Mt.-Prz.) Pam. 12 (1886) (87)-; Fschr. Mth. (1886) 121. 2420 Reality, multiplicity, separation, of roots. Algebra, fundamental theorem and Newton's approximation method. Mertens, F. Wien Ak. Sb. 101 (1892) (Ab. 2a) 415-. Algebraic equations and Campbell's quadratic functions. Poulain, A. C. B. 106 (1888) 470-, 562, 1479. , general theorems. Poulain, A. N. A. Mth. 6 (1867) 21-. 163 L2 2420 Reality, multiplicity, separation, of roots 2420 Algebraic equations, theorems, 2. Myasoyedov, A. N. [1883] (xn) Bee. Mth. (Moscou) 11 (1883-84) 616-. Budan and Fourier's theorem. Fouret, G. N. A. Mth. 11 (1892) 82-, 352. Sudan's method of extracting roots. Nicholson, P. Tilloch Ph. Mg. 60 (1822) 173-. theorem concerning roots of an algebraic equation. Genocchi, A. Tortolini A. 7 (1856) 462-. Complex roots. Canterzani, S. Verona Mm. S. It. 2 (1784) 720-. of equations, existence, Hermite's proposition, generalisation. Studnicka, F. J. Casopis 26 (1897) 95- ; Fschr. Mth. (1897) 95. , , proof. Kinkelin, H. Mth. A. 1 (1869) 502-. , , theorem. Kolbe, J. Wien Sb. 67 (1873) (Ab. 2) 188-. , and factors of polynomials. Wantzel, L. N. A. Mth. 3 (1844) 325-. numerical equations. Deladereere, A., <& Finck, . N. A. Mth. 3 (1844) 41-. Conditions that an equation of degree n may have only p distinct roots. Antomari, X. N. A. Mth. 16 (1897) 63-. Continuity of roots. Rouquet, V. N. A. Mth. 15 (1876) 154-. , proof. Bonnet, 0. Bll. Sc. Mth. As. 2 (1871) 215-. Derived equations. Merlieux, E. N. A. Mth. 3 (1844) 178-. polynomials, property. Young, J. R. Ph. Mg. 31 (1866) 369-. Descartes, Eolle, Fourier, Sturm, and Cauchy, theorems of, deduced from one principle. Terquem, 0. N. A. Mth. 3 (1844) 188-, 209-, 555-, 577-. Descartes's rule, proof. Tebay, S. (vi Adds.) Ph. Mg. 24 (1844) 24-. of signs. Terquem, 0. N. A. Mth. 2 (1843) 248-. . Finck, B. N. A. Mth. 3 (1844) 316-. . . Guilmin, A. N. A. Mth. 5 (1846) 239-, 334-. Descartes's rule of signs, generalisation. Gegen- bauer, L. Wien Ak. Sb. 83 (1881) (Ab. 2) 321-. , lowering limits furnished by. Andre, D. C. E. 98 (1884) 212-. , theorem in complete algebraic polynomials applied to. Jonquieres, E. de. C. E. 99 (1884) 1143- . theorem, extension. Lucas, E. Par. S. Mth. Bll. 8 (1880) 187-. Distribution of roots, theorems of Laguerre and Myasoyedov. Klark, N. I. (xn) Kazan S. Nt. (Ps.-Mth.) P. 1 (1883) \_(No. 8)] 11-. Double roots. Guldberg, A. S. Ts. Mth. 4 (1886) 97-; Fschr. Mth. (1887) 86. Electrical determination of roots. Lucas, F. C. E. 106 (1888) 195-, 268-, 645-, 1072- ; 111 (1890) 965-. Equal roots. Vachette, A. N. A. Mth. 6 (1847) 213-. . Ostrogradsky, M. A. [1849 j St Pet. Ac. Sc. Bll. 8 (1850) 193-. . Hudson, G. QJ. Mth. 18 (1882) 215-, 327-. , condition for. Salmon, G, Camb. and Dubl. Mth. J. 5 (1850) 159-. , theory. Rouche, E. N. A. Mth. 16 (1857) 66-. , . Mansion, . Brux. S. Sc. A. 16 1892 (Ft. 1) 54-. , , and application to tangents to curves. Roger, E. N. A. Mth. 3 (1844) 51-. Equalities among roots, conditions for existence of. Cayley, A. Phil. Trans. (1857) 727-. Equality of a given number of roots. Gerono, G. C. N. A. Mth. 1 (1842) 90- ; 6 (1847) 75-, 2 roots, condition. Joachimsthal, F. Crelle J. 33 (1846) 371-. Equation / 2 (z) + F/ 2 (z) = has all roots imagi- nary. Cesdro, E. N. A. Mth. 4 (1885) 321-. Equations with all roots imaginary. Catalan, E. C. Liege S. Sc. Mm. 15 (1888) No. 1, 224-. real. Gegenbauer, L. [1881] Wien Ak. Sb. 84 (1882) (Ab. 2) 1102-. . Zehfuss, G. Grunert Arch. 34 _. Kraus, L. Casopis 15 (1886) (1860) 400-. 63 ; Fschr. Mth (1886) 70 -. Walton, W. QJ. Mth. 5 (1862) J* _L __' _ fcagne M. d\ C. E. 106 153 -- (1888) 731-. Parpaite,-. N. A. Mth. 8 (1869) _ V ! . Selivanov, D. Th. Kazan S. . Laguerre, E. N. A. Mth. 18 (1879) 5-. . Poincare, H. C. E. 97 (1883) 1418-. . Jonquieres, E. de. Em. N. Line. At. 38 (1885) 55-. . Borel,E. Bll. Sc. Mth. 20 (1896) 327-. (Harriot's) rule of signs, demonstration. Grunert, J. A. Crelle J. 2 (1827) 335-. rule of signs, demonstration (Gauss). Ps. -Mth. Bll. 9 (1900) 51-; Fschr. Mth. (1900) 94-. , class. Clebsch, A. [1861] Crelle J. 62 (1863) 232-. , . Biehler, C. N. A. Mth. 19 (1880) 149- ; 6 (1887) 9-. , properties. Laguerre, E. N. A. Mth. 19 (1880) 224-. , of coefficients. Florow, P. S. Fschr. Mth. (1898) 70. , property. Sylvester, J. J. Crelle J. Mth. 87 (1879) 217-. signs, demonstration (Gauss). Ger- gonne, J. D. Gergonne A. Mth. 18 (1827-28) , source. Fouret, G. C. E. 106 352-. (1888) 1135-. " , extension. Laguerre, E. C. E. , types. Fouret, G. C. E. 106 92(1881)230-. (1888)1220-. 164 2420 Reality, multiplicity, separation, of roots Equations, difference product in case of equal roots. Franke, E. [1876] Crelle J. Mth. 83 (1877) 65-. with a given number of complex roots. Gegenbauer, L. [1882] Wien Ak. Sb. 87 (1883) (Ab. 2) 264-. no positive root, conditions. Andre, D. C. E. 98 (1884) 561-. , 3rd & 4th orders, nature of roots, criteria. Gaines, R. E. A. Mth. 1 (1900) 111-. Fourier and Sturm, theorems. Crelle, A. L. Crelle J. 13 (1835) 119-. , , graphic representation. Tanner, .H.W.L. Mess. Mth. 18 (1889) 95-. , theorem on roots of equations and re- curring series. Stern, M. A. Crelle J. 9 (1832) 305-. Fourier's rule, proof. Young, J. R. Ph. Mg. 23 (1843) 6-. for real roots. Hill, C. J. D. Lund Acta Un. 8 (1871) (Mth.) No. 2, 8 pp. theorem on equations. O'Kinealy, J. Ph. Mg. 48 (1874) 95-. . Sharp, W. J. C. QJ. Mth. 17 (1881) 132-. Functions whose roots separate each other. Markow, W. A. Fschr. Mth. (1899) 100. . Iwanow, J. J. Fschr. Mth. (1899) 100. . Markow, W. A. Fschr. Mth. (1899) 100- . Imaginary and equal roots, criteria. Homer, J. [1867] QJ. Mth. 9 (1868) 57-. roots. Cauchy, A. L. , Par. S. Phlm. Bll. 2420 Imaginary roots, Newton's rule. Sylvester, J. J. [1865] L. Mth. S. P. 1 (1866) No. 2, 16 pp. ; Ir. Ac. T. 24 (1871) 247-. , , invariantive criteria for quintic. Sylvester, J. J. Phil. Trans. 154 (1864) 579-. , for number. Lobatto, B. Amst. Vs. Ak. 9 (1859) 92-. , . Minich, S. R. Yen. At. 11 (1865-66) 127-. , , proof. Young, J. R. Ph. Mg. 30 (1865) 113-, 289-, 363-; 31 (1866) 369-; 32 (1866) 121- ; Ir. Ac. P. 10 (1870) 343-. , , Sylvester's theorem. Poulain, A. Les Mondes 11 (1866) 16-. , , . Genocchi, A. N. A. Mth. 6 (1867) 5-. , and Sylvester's theorem. Jon- quieres, E. de. C. B. 99 (1884) 62-, 111-, 165-, 269-. , , theorem connected with. Syl- vester, J. J. [1879] Mess. Mth. 9 (1880) , number. Berard, . Gergonne A. Mth. 9 (1818-19) 345-. , between limits. Dienger, J. Grunert Arch. 21 (1853) 361-. , , test for. Lobatto, R. Liouv. J. Mth. 9 (1844) 295-. of numerical equations, number. Anon. (vi 529) Gergonne A. Mth. 9 (1818-19) 60-. , test for. Lavernede, T. Gard. Not. Tr. Ac. (1809) 195-. (1817) 5-, 161- ; Par. EC. Pol. J. 11(1820) , . Faade Bruno, F. Liouv. (18e cah.) 411-. J. Mth. 15 (1850) 363-. , Eolle's theorem extended to. Liou- ville, J. Liouv. J. Mth. 9 (1864) 84-. Brux. S. . Bonnet, O. N. A. Mth. 4 (1845) 236-. . Terquem, 0. N. A. Mth. 5 (1846) 141-. . Prouhet, E. N. A. Mth. 14 (1855) 199-. . Buys-Ballot, C. H. D. Amst. Vs. Ak. 7 (1858) 316-. . Heaivood,P.J. Mth. Gz. No. 4 (1895) 31-. . Vaes, F. J. N. Arch. Wisk. 3 (1898) 117-. , Budan's criterion. Christie, J. R. Ph. Mg. 21 (1842) 96-. , Cauchy's theorem, extension. De Mor- gan, A. [1857] (DC) Camb. Ph. S. P. 1 (1866) 194. , , proof. Liouville, J., & Sturm, C. Liouv. J. Mth. 1 (1836) 278-. , , proofs. Stiirm, J. C. F. Liouv. J. Mth. 1 (1836) 290-. , criteria. Young, J. R. Ph. Mg. 22 (1843) 186-, 252-; 23 (1843) 450- ; 29 (1846) 32-. , . Colombier, P. A. G. N. A. Mth. 7 (1868) 308-, 501-. of equations of order higher than 4th. Carandino, G. G. Arcad. 32 (1826) 90-. , existence. Dupre, A. Gergonne A. Mth. 18 (1827-28) 68-. , . Gergonne, J. D. Gergonne A. Mth. 19 (1828-29) 124-. , _, test. Serret, P. N. A. Mth. 6 (1847) 200-. , limiting values of coefficients for given number. Schramm, H. (1866) 325-. , . Mansion, . Sc. A. 13 (1889) (Pt. 1) 42-. , rules analogous to Newton's. Minich, S. R. Ven. At. 11 (1865-66) 211-. , for inferior limit to number. Minich, S. R. Ven. At. 11 (1865-66) 309-. , theorem. Sturm, J. C. F. N. A. Mth. 5 (1846) 115-. Infinite roots of equations. Gerono, G. C. N. A. Mth. 3 (1844) 32-, 85-. , and asymptotes. Gerono, G. C. N. A. Mth. 4 (1845) 33-. Integral functions Y n = II ( y - 2 cos -r |i and = 1\ n + l/ Sturm's functions. Vries, J. de. Amst. Ak. Vs. 4 (1896) 133-. Interlacing of functions. Sylvester, J. J. Crelle J. Mth. 88 (1880) 1-. Laguerre's polynomials, property. Laurent, H. N. A. Mth. 19 (1880) 380-. Least root, general expression for. Konig, J. Mth. A. 9 (1876) 530-. roots. Dickson, J. D. H. [1877] Edinb. E. S. T. 28 (1879) 119-. Limit, inferior, of roots of an algebraic equation. Mlasojedov, A. N. Eec. Mth. (Moscou) 12 (1885) 22-. Arch. Mth. Ps. 45 , , . Fouret, . Par. S. Mth. Bll. 20 (1892) 4-. 165 2420 Reality, multiplicity, separation, of roots commensurable roots. N. A. Mth. 18 (1859) Limit, superior, of Montebello, G. de. 356-. , , negative roots deduced from Newton's formula. Vitasse, . N. A. Mth. 18 (1859) 213-. , , positive roots. Mourgues, . N. A. Mth. 9 (1850) 108-. , , . Gisclard, L. N. A. Mth. 11 (1852) 107-. , , . Gohierre de Longchamps, . N. A. Mth. 19 (1880) 71-. , , in numerical equations. Len- theric, . Gergonne A. Mth. 20 (1829-30) 297-. , , roots, determination. Candeze, G. N. A. Mth. 20 (1881) 49-. , , , Laguerre's rule. Candeze, G. N. A. Mth. 19 (1880) 307-. , , of numerical equations, Petersen's criterion proved erroneous. Giudice, . Palermo Cir. Mt. Bd. 6 (1892) 157-. Limits of equations. Piani, D. Bologna N. Cm. 3 (1839) 237-. to moduli of roots. Dim, U. A. Mt. 1 (1898) 77-. . Petrovitch, M. C. B. 129 (1899) 583-, 873-. of roots. Vene, A. A. Bouen Tr . Ac. (1821) 97-; Ferussac Bll. Sc. Mth. 4 (1825) 206-. . Tillot, J. N. A. Mth. 4 (1845) 224-. . Toussaint, . N. A. Mth. 18 (1859) 310-. . Realis, S. N. A. Mth. 3 (1864) 477-. . Gyory, S. (xn) Mag. Tud. Ak. Ets. 1 (No. 6) (1867) 125-. . Tranton,A. N. A. Mth. 11 (1872) 254-. . Fouret, . Par. S. Mth. Bll. 20 (1892) 35-. , Descartes' s theorem applied. Fourier, J. B. J. Par. S. Phlm. Bll. (1820) 156-, 181-. , and determination of commen- surable roots. Thibault, . N. A. Mth. 2 (1843) 517-. of equation with only real roots. Pleskot, A. Prag Sb. (1897) (Mth.-Nt.) No. 37, 9 pp. ; N. A. Mth. 18 (1899) 301- ; 19 (1900) 144. literal equations. Bouvier, L. C. Gergonne A. Mth. 16 (1825-26) 54-. numerical equations. Bret, . Gergonne A. Mth. 6 (1815-16) 112-. , approximate. Maiziere, de. Gergonne A. Mth. 3 (1812-13) 41-. of higher order. Row- botham, J. Tilloch Ph. Mg. 63 (1824) 369-. , superior, of roots. Rachmanov, P. N. Kharkov Mth. S. Com. 4 (1895) 141- ; Fschr. Mth. (1893-94) 140. , variation, in algebraic equations. Chevil- lard, A. J. N. A. Mth. 2 (1843) 343-. Multiple roots of algebraic equations. Brioschi, . C. B. 121 (1895) 582-. , theorem. Sondat, P. N. A. Mth. 19 (1900) 25-. 2420 Ph. N. A. Multiple roots, theory. Sylvester, J. J. Mg. 18 (1841) 249-. Multiplicity of roots. Beynac, F. A. Mth. 9 (1850) 317-. , amount and distribution. Sylvester, J. J. Ph. Mg. 18 (1841) 136-. in the octavic equation. Alagna, R. Palermo Cir. Mt. Bd. 4 (1890) 287-. Nature of roots, Borchardt's proposition. Studnicka, F. J. Casopis 27 (1898) 237-; Fschr. Mth. (1898) 73. Newton and Descartes, rule of signs. Orzabal, A. Arg. S. Ci. A. 22 (1886) 39-, 97-. 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Crelle J. 50 (1855) 263-. , geometrical investigation of number. Pincherle, S. Bv. Mt. 3 (1893) 54-. between given limits. Realis, S. G. Mt. 9 (1871) 355-. , limit. Gohierre de LongcJiamps, . N. A. Mth 18 (1879) 49-. , , superior. Paque, A. Liege Mm. S. Sc. 10 (1855) 199-. , of number. Olivier, L. Crelle J. 1 (1826) 223-. , . Sylvester, J. J. C. B. 60 (1865) 1261-. , in a class of equations. Syl- vester, J. J. C. B. 58 (1864) 494-. , limits, method of finding. Sylvester, J. J. N. A. Mth. 12 (1853) 329-. , , theorem. Sylvester, J. J. N. A. Mth. 12 (1853) 286-. , number. Cauchy, A. L. [1813] Par. S. Phlm. Bll. (1814) 95- ; Par. EC. Pol. J. 10 (1815) (17 e cah.) 457-. , between limits. Bonnet, 0. N. A. Mth. 6 (1847) 464-. 166 2420 Reality, multiplicity, separation, of roots 2420 Eeal roots, number between limits. Petr, K. Prag Ceske Ak. Fr. Jos. Kz. (Tfida 2) 6 (1897) Art. 8, 11 pp. ; Prag Fr. Jos. Ac. Sc. Bll. (Mth.-Nt.) 4 (1897) (Pt. 2) 61-. -- , --- , Sturm's theorem. 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(Tfida 2) 8 (1899) Art. 23, 12 pp. ; Fschr. Mth. (1899) 103. ---- secular equation, proof by Fourier's method. Sylvester, J. J. Crelle J. Mth. 88 (1880) 4-. Laisant, Eoots of cubic, criteria. Vyvyan, T. G. Mth. Gz. 1 (1900) 39. , equal numerically, of opposite sign, cri- teria. Mtiller, F. Arch. Mth. Ps. 46 (1866) 32-. of an equation, limits to. Gob, A. Mathesis 9 (1889) 269-. , multiplicitity of, and a limit to. Demoulin, A. Mathesis 9 (1889) 268-. equations. Spitzer, S. Grunert Arch. 25 (1855) 137-. , analysis. Murphy, E. Phil. Trans. (1837) 161-. with complex coefficients. Hermite, C. Crelle J. 53 (1857) 182-. , elementary representation and inspec- tion. Bjnrling, C. F. E. Ts. Mt. Fys. 1 (1868) 119-. with integral coefficients. Cauchy, A.L. C. E. 24 (1847) 407-. , nth degree, with application to bi- quadratic. Mossbrugger, L. Grunert Arch. 28 (1857) 205-. , surd. Moon, E. Camb. Mth. J. 3 (1843) 20-. , theorems. Fleury, L. L. Cherb. Mm. S. Sc. 8 (1861) 373-. , new formulae relative to. Silva, J. A. M. da. [1882] G. Teix. J. Sc. 4 (1883) 3-. , number satisfying a given condition. Cauchy, A. L. G. E. 40 (1855) 1329-. , theory. Bourget, J. N. A. Mth. 9 (1870) 505-. _, _. Bjorling, C. F. E. Stockh. Ak. Hndl. 10 (1871) (No. 3) 53 pp. Eule of signs, new theory. Meech, L. W. Des Moines Anal. 2 (1875) 81-, 97-. Eules of signs, combinations of four. Sylvester, J. J. C. E. 61 (1865) 282-. Separation of nearly equal roots. Christie, J. E. Ph. Mg. 21 (1842) 96-. real roots of equation. Schramm, Ht Grunert Arch. 36 (1861) 420-. , theorem. Korteweg, D. J. N. Arch. equations. Ovejero, J. Arg. S Wisk. 4 (1899) 46-. and signs of roots. Du Bourguet, . Par. S. Phlm. Bll. (1815) 14-. Eolle's theorem. Collin, J. N. A. Mth. 20 (1881) 132-. , and an application. Biehler, C. N. A. Mth. 6 (1887) 75-, 190-. , extension. Boer, F. de. Amst. Ak. Vs. M. 19 (1884) 384-; Arch. Neerl. 19 (1884) 207-. , . McCulloch, J. F. A. 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N. A. Mth. 19 (1880) 145-. , . Levy, L. N. A. Mth. 19 (1880) 148. 167 2420 Reality, multiplicity, separation, of roots Separation of roots by method of differences. Desboves, A. H. N. A. Mth. 13 (1854) 60-. . Capelli, A. Nap. Ed. 33 (1894) 191-, 214-. , method, and Lagrange's formula. Pellet, A. E. Bll. Sc. Mth. As. 5 (1881) (Pt. 1) 393-. of numerical equations. Zinna, A. G. Mt. 6 (1868) 344-. . Laguerre, E. C. E. 89 (1879) 635- ; 92 (1881) 1146-. some equations. Markov, A. A. Kharkov Mth. S. Com. (1885) 89- ; Fschr. Mth. (1886) 161. , and superior limit. Laguerre, E. N. A. Mth. 19 (1880) 49-, 97-. Signs and nature of roots. Anon, (vi 537) Gergonne A. Mth. 10 (1819-20) 341-. Sturmians, class of. Chrystal, G. [1881] Edinb. E. S. T. 30 (1883) 161-. Sturm's functions. Cayley, A. Liouv. J. Mth. 11 (1846) 297- ; 13 (1848) 269-. . Brioschi, F. N. A. Mth, 13 (1854) 71-. . Gilbert, P. Liouv. J. Mth. 12 (1867) -87-. . Brioschi, F. C. E. 68 (1869) 1318-. -T- . Kronecker, L. Berl. Ak. Mb. (1878) 95-. , auxiliary, relation to roots of an equation. Sylvester, J.J. B. A. Ep. (1841) (pt. 2) 23-. , calculation by determinants. Lemonnier, H. C. E. 80 (1875) 252-. , coefficients expressed as determinants. Heal, W. E. A. Mth. 2 (1885-86) 85-. , determination of a series from a single determinant. Vleck, E. B. van. [1899] A. Mth. 1 (1900) 1-. , formulae for equations of 2nd, 3rd, and 4th orders. Koralek, P. N. A. Mth. 11 (1852) 333-. , functions analagous to. Lemonnier, H. Par. S. Mth. Bll. 6 (1878) 149-. , . Miasojedov, A. N. Eec. Mth. (Moscou) 12 (1885) 461- ; Mth. (1885) 63-. and their reciprocal relations. Kronecker, L. Berl. Mb. (1873) 117-. of 2nd species. Wendlandt, H. Arch. Mth. Ps. 62 (1878) 1-. , series of. Gegenbauer, L. Amst. Ak. Vs. 5 (1897) 185-. , . Vries, J. de. N. Arch. Wisk. 3 (1898) 40-. ' , substitutes for. Mollanie, V. Nap. Ed. 22 (1883) 256-. , Sylvester's theorem on. Sturm, J. C. F. Liouv. J. Mth. 7 (1842) 356-. , . Turazza, D. Yen. At. 11 (1865-66) 419-. and syzygies between rational integral functions. Sylvester, J. J. Phil. Trans. (1853) 407-. , tables for equations of 2nd, 3rd, 4th, and 5th orders. Cayley, A. Phil. Trans. (1857) 733-. , theorem of Cayley's, proof. Zeipel, E. V. von. QJ. Mth. 3 (1860) 108-. 2420 N. Sturm's method, application. Biehler, C. A. Mth. 19 (1880) 76- ; 6 (1887) 421-. of finding real roots, expressions for ?uotients in. Sylvester, J. J. B. A. Ep. L853) (pt. 2) 1-. separation of roots, modification. Nazimov, P. [1885] Eec. Mth. (Moscou) 13 (1886) 119- ; Fschr. Mth. (1886) 71-. quotients, explicit values. Sylvester, J. J. Ph. Mg. 6 (1853) 293-. remainders. Heilermann, . Crelle J. 43 (1852) 43- ; 48 (1854) 190-. . Faa de Bruno, F. Tortolini A. 7 (1856) 313-. series. Gegenbauer, L. Wien Ak. Sb. 81 (1880) (Ab. 2) 576-. theorem. Terquem, 0. N. A. Mth. 2 (1843) 97-. . Badon-Ghijben, J. (vn) Amst. N. Ws. Ntk. Vh. 1 (1844) 89-. . Mourgues, . N. A. Mth. 9 (1850) 278-. . Borchardt, C. W. N. A. Mth. 11 (1852) 403-. . Hermite, C. C. E. 36 (1853) 294-. . Bertrand, J. C. E. 43 (1856) 1108-. . Kronecker, L. C. E. 68 (1869) 1078-. . Hattendorff, K. Gott. Nr. (1873) 779-. . Darboux, G. Bll. Sc. Mth. As. 8 (1875) 56-, 92-. . Janni, V. G. Mt. 20 (1882) 166-. . Nieivenglowski, B. N. A. Mth. 9 (1890) 181-. . Willotte, H. Toul. Ac. Sc. Mm. 3 (1891) 44-. , application. Spiller, . Mth. Misc. 2 (1839) 133-. , . Candioti, M. R. Arg. S. Ci. A. 27 (1889) 201- . , to transformations of binomial equations. Sturm, J. C. F. Liouv. J. Mth. 7 (1842) 132-. , determination of X n . Young, J. R. Ph. Mg. 7 (1835) 454-. , extension. Phragmgn, E. C. E. 114 (1892) 205-, 440. , to system of equations. Bendixson, I. [1892] Stockh. Ak. Hndl. Bh. 18 (Afd. 1) (1893) No. 2, 18 pp. and limits of roots of equation. Sylvester, J. J. Ph. Mg. 5 (1853) 446- ; 6 (1853) 14-, 138-, 210-. , Ostrogradski's. Desboves, A. H. N. A. Mth. 6 (1867) 348-. , proof. Ettingshausen, A. von. Baum- gartner Z. 7 (1830) 444-. , . Trudi , N. G. Mt. 1 (1863) 59-. , remarks. Candeze, G. N. A. Mth. 20 (1881) 193-. , simplification of calculations. Housel, . N. A. Mth. 6 (1867) 351-. theorems of comparison. BocJier, M. N. Y. Am. Mth. S. T. 1 (1900) 414-. Variations, number gained or lost in multiplica- tion of f(x) by x n a. Andre, D. C. E. 99 (1884) 182-. , in multiplication by x - a. Andrt, D. C. B. 98 (1884) 292-. 168 2430 Equations of the lower orders, etc. 2430 Equations of the second, third and fourth orders : other particular equations. Abelian cubics and symmetrical equations. Cockle, Jas. QJ. Mth. 5 (1862) 237-. Algebraic equations. Jonquieres, E. de. C.E. 99 (1884) 345-, 450, 469-, 483-, 546. . Lalanne, L. C. E. 99 (1884) 463-. , especially irreducible case of Cardan's formula. Gegenbauer, L. Liege S. Sc. Mm. 2 (1900) A r o. 10, 6 pp. , pair. Mansion, P. Mess. Mth. 7 (1878) 57-. , solution by radicals and by series. Favero, G. B. Em. E. Ac. Line. Mm. 6 (1889) 415-. whose first member satisfies a linear differential equation of the 2nd order. Laguerre, E. C. E. 90 (1880) 809- ; 94 (1882) 412-, 508- . resolution of equations, theory. Giudice, F. Ev. Mt. 2 (1892) 193-. Anharmonic-ratio sextic. Cayley, A. QJ. Mth. 10 (1870) 56-. Binary equations of 4th or 5th order, conditions for 3 equal roots, etc. Cayley, A. Phil. Trans. 158 (1868) 577-. Continued roots. Dixon, T. S. E. Des Moines Anal. 5 (1878) 20-. De Moivre, class of equations solved by, and their derivatives. Realis, S. N. A. Mth. 4 (1865) 209-, 289-. De Moivre' s formula. Barsotti, G. Lucca At. Ac. 17 (1860) 149-. theorem and Cardan's formula applied to equations. Barsotti, G. Pisa A. Un. Tosc. Sc. Cosm. 5 (1858-61) 99-. Equation, roots of which are products in pairs of roots of another equation. Malet, J. C. QJ. Mth. 13 (1875) 30-. , 2 given equations. Malet, J. C. [1878] A. Mt. (1878-79) 306-. , (-); r=0, 1, 2...n. Siacci, F. G. Arcad. 18 (1860) 166-. for secular inequalities of planetary motion, Cauchy's theorem that roots are essentially real. Hermite, C. C. E. 41 (1855) 181-. with variable parameter, roots. Cauchy, A. L. C. E. 12 (1841) 1133-. Equations, certain reducible types. Spitzer, S. Wien SB. 8 (1852) 422- ; Grunert Arch. 22 (1854) 1-. depending on numerical functions. [Bou- gaieff, N. F.] Bugaev, N. V. [1874] (xn) Eec. Mth. (Moscou) 7 (1874-75) (Pt. 1) 424-; (rx) Bll. Sc. Mth. As. 10 (1876) 102-. determining functions of double argument. Hill, C. J. D. Crelle J. 11 (1834) 241-. , first 4 orders, solution. Eisenstein, G. Crelle J. 27 (1844) 81-. of form x 3 -bx = l. Lockhart, J. Mathe- matician 2 (1847) 253-, 319-. x* + ax 2 + bx + c = 0, solution. Leire, P. (xn) Ts. Mth. 1 (1877) 117-. + &c. = 0. Laguerre, E. C. E. 93 (1881) 890-. 2430 Equations of form (xP - a") \f/(x) = Q. Berloty . N. A. Mth. 1 (1882) 173-. (a, b, c, d) = (a 2 , b 2 , c 2 , d*), solution. Cayley, A. Mess. Mth. 15 (1886) 59-. (a, b, c - ) = (aP, IP, c" - ), solution. Vivanti, G. G. Mt. 27 (1889) 229-. () = (), . Tanner, H. W. L. Mess. Mth. 19 (1890) 118-. _ , theorem. Cauchy, A. L. Par. S. Phlm. Bll. (1816) 133-. , irreducible, periods ; theorems of Lionnet's. Meyer, G. F. [1868] Arch. Mth. Ps. 49 (1869) 168-. , , series. Ocagne, M. d\ Brux. S. Sc. A. 10 (1886) (Pt. 2) 90-. , number of, in any "Farey series" of which limiting number is given. Sylvester, J. J. Ph. Mg. 15 (1883) 251- ; 16 (1883) 230-. , denominators and numerators of which do not exceed certain number. Sylvester, J. J. (xn) J. H. Un. Cir. [2] (1883) 44-. , , formed with integers less than given number. Sylvester, J. J. C. E. 96 (1883) 409- ; Mess. Mth. 27 (1898) 1-. , property. Farey, J. Tilloch Ph. Mg. 47 (1816) 385-. , . Glaisher, J. W. L. Ph. Mg. 7 (1879) 321-. , tabulation of all, between given limits. Sang, E. [1878] Edinb. E. S. T. 28 (1879) 287-. Function E (f (x)) , modification ; quadratic and non quadratic residues of form 4/c + 1. Bunlakovskij, V. Ja. [1885] St Pet. Ac. Sc. Mm. (Rs.) 52 (1886) 124- ; Fschr. Mth. (1886) Functions arising in theory of numbers. Berger, A. Stockh. Ofv. (1898) 579-. of even numbers, Laplace's method. Jacobi, C. G. J. As. Nr. 28 (1849) 257-. in theory of numbers, general questions. Bervi, N. V. [1894] Eec. Mth. (Moscou) 18 (1896) 519- ; Fschr. Mth. (1896) 156-. , present state of knowledge. Bervi, N. V. Eec. Mth. (Moscou) 19 (1897) 182- ; Fschr. Mth. (1896) 156. General aspects of theory of numbers. Bourdat, . Grenoble Ac. Delph. Bll. 3 (1850) 37-. remarks. Sludskti, T. A. [1864] (xn) Eec. Mth. (Moscou) 2 (1867) (Ft. 2) 195-. Geometrical illustrations of some theorems. Davis, E. W. Am. J. Mth. 15 (1893) 84-. Geometry, application to theory of numbers. Busche, E. Crelle J. Mth. 104 (1889) 32-. of numbers. Minkowski, H. D. Mth. Vr. Jbr. 1 (1892) 64-. Goldbach's odd numbers. Stern, M. A. N. A. Mth. 15 (1856) 23-. theorem, Euler's extension. Terquem, O. N. A. Mth. 11 (1852) 326-. Hamiltonian differences, Sylvester's tables. Glashan, J. C. QJ. Mth. 27 (1895) 242-. History, remarks. Henry, C. As. Fr. C. E. (1880) 201-. Identities, numerical, etc. Serdobinskil, V. E. [1872] (xn) Eec. Mth. (Moscou) 6 (1872-73) (Pt. 1) 107-. , , method of finding and application to theory of numerical functions. Baskakov, S. I. [1881] (xn) Eec. Mth. (Moscou) 10 (1882-83) (Pt. 1) 313-. 189 2800 Theory of Numbers. General Inequalities relating to prime numbers. Syl- vester, J. J. Nt. 38 (1888) 259-. Infinite series, use in theory of numbers. Lejeune-Dirichlet, G. Crelle J. 18 (1838) 259-. Infinitesimal analysis, applications to theory of numbers. Lejeune-Dirichlet, G. Crelle J. 19 (1839) 324-; 21 (1840) 1-, 134-. Integers of certain form. Rocquigny, F. [G.~] de. Les Mondes 53 (1880) 410-. , property of a certain series. Sylvester, J. J. C. E. 88 (1879) 1297-. , solutions by. Liouville, J. Liouv. J. Mth. 4 (1859) 271-. , successions that cannot be indefinitely continued. Sylvester, J. J. [1882] (xn) J. H. Un. Cir. [2] (1883) 2-. , systems. Cesdro, E. Km. E. Ac. Line. Ed. 4 (1888) (Sem. 1) 457-. , theorem. Serret, J. A. Liouv. J. Mth. 13 (1848) 12-. , . Durege, . Grunert Arch. 30 (1858) 163-. , (Durege). Lottner, C. L. E. Grunert Arch. 32 (1859) 111-. , theory. Binet, J. P. M. Liouv. J. Mth. 6 (1841) 449-. Integral calculus, applications. Lucas, E. C. E. 82 (1876) 1303-. , . Weber, H. Gott. Nr. (1896) 275-. Latent integer, formal property. Sylvester, J. J. QJ. Mth. 1 (1857) 185. Laws of numbers, problems. Marchand, D. Les Mondes 38 (1875) 460- ; 39 (1876) 26-, 346-, 693-. Legendre's functions, application. Bauer, G. Munch. Ak. Sb. 24 (1895) 343-. Linear modular system, reduction. Fund, 0. Hamb. Mth. Gs. Mt. 3 (1900) 423-. with root of product of MI numbers. Boutroux, . N. A. Mth. 1 (1842) 368-. Maximum product of components. Derbes, J. N. A. Mth. 19 (I860) 117-. Mean values in theory of numbers. Lejeune- Dirichlet, G. Berl. Ab. (1849) (Mth.) 69-. . Berger,A. Ups. S. Sc. N. Acta 14 (1891) No. 2, 130 pp. Miscellaneous notes. Catalan, E. As. Fr. C. E. (1891) (Ft. 2) 194-. nth powers that are sums of nth powers. Martin, A. Smiths. Misc. Col. 33 (1888) Art. 4, 107-. (Wash. Ph. S. Bll. 10 (1888).) Nasik cubes. Trost, A. H. QJ. Mth. 8 (1867) 74. , general properties. Frost, A. H. [1877] QJ. Mth. 15 (1878) 93-, 366-. squares, general properties. Frost, A. H. [1877] QJ. Mth. 15 (1878) 34-. Natural system of numbers. Pujols de la Bas- tida, V. Madrid Ev. 4 (1854) 567-; 5 (1855) 436-. Nature and properties of numbers. Jamard, . Eouen Tr. Ac. (1804) 75-. New basis for a general theory. Kraus, J. Z. Mth. Ps. 37 (1892) 321- ; 39 (1894) 11-. Number equal to sum of p squares, certain. Catalan, E. C. (xn) Mathesis 3 (1883) 199. 2800 Number, prime or composite, test. Koch, H. von. Stockh. Ofv. (1900) 789-. of times a prime is contained in product of n primes. Andre, D. N. A. Mth. 13 (1874) 185-. 3, properties. Gegenbauer, L. Wien Ak. 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(Moscou) 2 (1867) (Ft. 1) 10-. , succession of 2n + l, so that sum of squares of first n + l = that of last n. Dostor,G. Arch. Mth. Ps. 64 (1879) 350-. , three, such that sum or difference of any two is square. Euler, L. [1780] St. Pet. Ac. Sc. Mm. 6 (1818) 54-. Odd and even numbers. Martin, Art. Des Moines Anal. 2 (1875) 20-. , theorem of Gauss. Stern, M. A. Gott. Nr. (1869) 330-. numbers, theorems on two. Catalan, E. C. Liege S. Sc. Mm. 6 (1877) No. 4, 4 pp. , Turner's property. Grunert, J. A. Grunert Arch. 1 (1841) 59-. . , . Hellerung, . Grunert Arch. 1 (1841) 318-. Origin and development of theory of numbers. StudniSka, F. J. Casopis 4 (*1874) 1-. Partition analysis applied to properties of any system of consecutive integers. MacMahon, P. A. [1899] Camb. Ph. S. T. 18 (1900) 12-. of even number into two primes. Sylvester, J. J. [1871] L. Mth. S. P. 4 (1871-73) 4-. (P n ) an< l 0(2p n ) P being a prime. Sancery, L. [1875] Par. S. Mth. Bll. 4 (1876) 17-. Perfect cubes. Henkle, W. D. 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Mac- Mahon, P. A. L. Mth. S. P. 23 (1892) 305-. Powers, difference of consecutive. Tronsens, A. N. A. Mth. 19 (1860) 310-. of number. Anon, (vi 886) N. A. Mth. 13 (1854) 424-. Primes. Laurent, H. N. A. Mth. 18 (1899)234-. , characteristic relations for. Hacks, J. Acta Mth. 17 (1893) 205-. of definite forms. Bang, A. S. N. Ts. Mth. 2 (B) (1891) 73-; Fschr. Mth. (1891) 183. , determination of certain. Lawrence, F. W. L. Mth. S. P. 28 (1897) 465-. equal to sums or differences of squares or multiples of squares. Voiles, F. (xi) Par. S. Phlm. Bll. 7 (1871) 191-. , expression in terms of preceding primes. Isenkrahe, C. Mth. A. 53J1900) 42-. , forms arising from (a + v/6) k. Genocchi, A. A. Mt. 2 (1868-69) 256-. , general formula. Dormoy, E. C. E. 63 (1866) 178- . , geometrical interpretation. Cassani, P. Yen. I. At. (1896-97) 103-. , roots. Oltramare, G. Crelle J. 45 (1853) 303-. , theorems. Levi, G. Yen. I. At. (1892- 93) 1816-. , , enunciation. Proth, F. C. E. 83 (1876) 1288-. , theory. La Vallee-Poussin, C. J. de. Brux. S. Sc. A. 20 (1896) (Pt. 1) 100-, (Pt. 2) 183-, 281- ; 21 (1897) (Pt. 1) xxiii, 1-, 60-, (Pt. 2) 251-. Primitive numbers (not containing square fac- tors), distribution. Bougaev, . C. E. 74 (1872) 449-. , properties. Maria, P. C. G. Mt. 4 (1866) 345-. new laws. Coninck, G. de. N. A. Principles, first. Skrivan, G. Wien Jbr. Ober-Eealsch. Inn. Stadt 2 (1860) 1-. , fundamental. Poinsot, L. C. E. 12 (1841) 803- ; Liouv. J. Mth. 10 (1845) 1-; Grunert Arch. 7 (1846) 367-. , (Poinsot). Grunert, J. A. Grunert Arch. 7 (1846) 168-. Problem : to find several consecutive cubes whose sum is a square. Catalan, E. C. Brux. Ac. Bll. 22 (1866) 339-. Problems, numerous. Gelin, E. Mathesis 16 (1896) (Suppl.) 34 pp. Products of digits of certain numbers. Sanctts, P. de. Em. N. Line. At. 52 (1899) 58-; 53 (1900) 57-. and series involving prime numbers only. Glaisher, J. W. L. QJ. Mth. 27 (1895) 270- ; 28 (1896) 1-. Progressions, application. Volpicelli, P. Tor- tolini A. 5 (1854) 449-. Properties of integers derived from space in- tuitions. Minkowski, H. N. A. Mth. 15 (1896) 394-. numbers. Vince, S. [1810] Ir. Ac. T. 12 (1815) 34-. . Anon. (vi319) Dingier 34 (1829) 68. . Perkins, G. B. Silliman J. 40 (1841) 112-. . Whitworth, W. A. QJ. Mth. 6 (1863) 163-. . Valat , J. P. F. Bordeaux Ac. Act. 38 (1877) 73-. . Torelli, G. G. Mt. 16 (1878) 152-. Mth. 13 (1874) 568-. , properties. 0., D. QJ. Sc. 5 (1818) 55-. , relations. Prouhet, E. C. E. 33 (1851) 225. , remainders. Buttel, P. Grunert Arch. 26 (1856) 241-. , theorem. Fregier, . Gergonne A. Mth. 9 (1818-19) 211-. , theorems. Terquem, 0. N. A. Mth. 5 (1846) 70-. , uses in mechanics. Noel, J. N. Quetelet Cor. Mth. 1 (1825) 124-, 199- whose sum is same power of some number. Martin, Art. QJ. Mth. 26 (1893) 225-. , relations. Marchand, D. Les Mondes 44 (1877) 424-. and their differences. Kirwan, B. [1809] Ir. Ac. T. 11 (1810) 131-. of 2 and 3. Sludskii, T. A. [1869] (xn) Eec. Mth. (Moscou) 4 (1869-70) (Pt. 2) 171-. 5 and their multiples. Berdelle, C. As. Fr. C. E. 8 (1879) 176-. Prime of form 8n + l, theorem. Stern, M. A. Crelle J. 32 (1846) 89-. , formation from lower primes. Scherk, H. F. Crelle J. 10 (1833) 201-. greater than given prime. Boije af Gennds, C. 0. Stockh. Ofv. (1893) 469-. , theorem of Legendre. Pepin (le pere), T. Liouv. J. Mth. 5 (1879) 21-. 191 2800 Theory of Numbers. General Properties of numbers. Dostor,G. Arch. Mth. Ps. 63 (1879) 221-. . Pepin, T. C. E. 120 (1895) 1254-. , theorem. Ingleby, C. M. Mess. Mth. 3 (1866) 30-. Property of numbers. H 'eider, M. Haidinger B. 1 (1846) 186. . Midy, . N. A. Mth. 5 (1846) 640-. . Dostor, G. J. N. A. Mth. 8 (1849) 101-. . Volpicelli, P. Em. At. 6 (1852-53) 631. . Genocchi, A. Brux. Ac. Bll. 20 (1853) (pte. 2) 397-. . Booth, J. E. S. P. 7 (1854-55) 42-. . Anon, (vi 314) Crelle J. 50 (1855) 187-. . Boncompagni, B. Tortolini A. 6 (1855) 371-. - in theory of numbers. Earnshaw, S. (vi Adds.) Camb. Mth. J. 2 (1841) 79-. Propositions, miscellaneous. Cauchy, A. L. C. E. 24 (1847) 996-; 25 (1847) 132-, 177-, 242-. , . Catalan, E. C. N. A. Mth. 13 (1874) 518-. , . Thue,A. Arch. Mth. Ntvd. 16 (1893) 255-; 19 (1897) No. 4, 27 pp.; Fschr. Mth. (1897) 165. Eeal representation of imaginary configurations, and application to theory of numbers. Busche, E. Crelle J. Mth. 122 (1900) 227-. Eeciprocals of primes. Goodwyn, H. Nichol- son J. 4 (1801) 402-. Eemainder as function of divisor and dividend. Schli'milch, 0. Grunert Arch. 10 (1847) 424-. Eepresentation of number in form Sa r n . Lemoine, E. As. Fr. C. E. (1900) (Pt. 2) 72-. Eesidues of recurring series with regard to prime modulus. Mantel, W. N. Arch. Wisk. 1 (1895) 172- ; Fschr. Mth. (1893-94) 423-. Ehythmomachy, an ancient game. Terquem, 0. N. A. Mth. 5 (1846) 662-. Boots of numbers. Sanchis-Barrachina, E. N. A. Mth. 8 (1869) 265-. Series involving inverse even powers of subeven and supereven numbers. Glaisher, J. W. L. Mess. Mth. 23 (1894) 176-. , of natural numbers. Collignon, E. As. Fr. C. E. (1896) (Pt. 2) 17-. , number occupying given place. Barbier, E. C. E. 105 (1887) 795-, 1238-. , . Ocagne, M . d' . C. E. 106 (1888) 190-. , problems. Bertrand, J. N. A. Mth. 8 (1849) 354-. odd numbers, theorems. Selle, (le vicomte) de. Aix Ac. Mm. 18 (1900) 75-. primes, and resolution of number into factors. Gough, J. Nicholson J. 1 (1802) 1-. in which coefficients are sums of divisors of exponents, powers of. Glaisher, J. W. L. Mess. Mth. 15 (1886) 33-. , square of. Glaisher, J. W. L. Mess. Mth. 14 (1885) 156-. 2800 Series + ! + i4-f + T 1 r+"- (primes) . Glaisher, J. W. L. QJ. Mth. 25 (1891) 369-. i- t + f + T*r-Tir - (primes). Glaisher, . W. L. QJ. . . Mth. 25 (1891) 375-. Glaisher > W. L. QJ. Mth. 26 (1893) 33-. i+i-t + A -A + . Glaisher, J. W. L. QJ. Mth. 26 (1893) 48-. Sets of integers, determinant of which is > or < 1. Hermite, C. Liouv. J. Mth. 14 (1849) 21-. Square array of 16 numbers satisfying 22 con- ditions. Lebesgue, V. A. N. A. Mth. 15 (1856) 403-. numbers, analysis. Wheeler, A. D. Silli- man J. 25 (1834) 87-. -- , and biquadrates, ascertaining by inspec- tion. Saint, W. Nicholson J. 22 (1809) 291-. -- , properties. Hoyt, D. H. Camb. (M.) Mth. M. 3 (1860) 2-. -- , tables giving sets of final digits. Schady, . Crelle J. Mth. 84 (1878) 85-. that are sums of squares. Martin, A. Wash. Ph. S. Bll. 11 (1892) 580-. Squares, theorems of 2, 4, 8, 16. Cayley, A. QJ. Mth. 17 (1881) 258-. Sum and difference of two squares. Claude, R. P. L. N. A. Mth. 2 (1863) 88-. of Jith powers of numbers < and prime to n. Thatker, A. Crelle J. 40 (1850) 89-. reciprocals of like powers of primes, theorem. GlaisJier, J. W. L. As. Fr. C. E. 6 (1877) 172-. series 2 z , p being prime number. Glaisher, J. W. L. Mess. Mth. 7 (1878) 190-. Sums of inverse powers of primes. Glaisher, J. W. L. QJ. Mth. 25 (1891) 347-. . like powers of prime numbers. Bour, J. E. E. Par. S. Phlm. Bll. 2 (1865) 8-. series of reciprocals of primes and of their powers. Merrifield, C. W. [1881] E. S. P. 33 (1882) 4-. squares, relations. Dostor, G. Arch. Mth. Ps. 67 (1882) 265-. --- , theorems. Realis, S. N. A. Mth. 12 (1873) 212- ; 18 (1879) 500-. three squares. Dainelli, U. G. Mt. 15 (1877) 378-. , value of some. Stern, M. A. Gott. Ab. 17 (1872) (Mth.) 63-. Tables of uniform reversible functions of two variables on the simplest number-domains. Schroder, E. Mth. A. 29 (1887) 299-. Theorems, Berger's, proof and generalisation. Cesdro, E. (xn) Mathesis 1 (1881) 99-. , various. Talbot, W. H. F. [1862] Edinb. E. S. T. 23 (1864) 45-. , . Gegenbauer, L. Wien Ak. D. 49 (1885) (Ab. 1) 1-, (Ab. 2) 105- ; 57 (1890) 497-; 60 (1893) 25-. , . Catalan, E. Brux. Ac. Mm. 46 (1886) No. 1, 16 pp. Theory of numbers and automorphic functions, relations. Fricke, R. D. Mth. Vr. Jbr. (1899) (Heft 1) 94-. 192 2810 Divisibility; linear congruences Divisibility 2810 Transformation, numerical. Davies, T. S. (n Adds.) Ph. Mg. 34 (1849) 347-. Treatise on numbers. Crelle, A. L. Crelle J. 27 (1844) 6-, 107-, 330- ; 28 (1844) 111-; 29 (1845) 58-, 103-. Triads of once-paired elements. Homer, J. [1867] QJ. Mth. 9 (1868) 15-. Wronski's theory of numbers. Dickstein, S. Krk. Ak. (Mt.-Prz.) Ez. 4 (1893) 73-, 396; Crc. Ac. Sc. Bll. (1892) 64-. 2810 Divisibility; linear con- gruences. Arithmetical "lines" (series), arrangement. Tarry, G. N. A. Mth. 18 (1899) 149-, 292. Bugaev's "order", i.e. sum of exponents of prime factors of a number. Cistuikov, I. I. Bee. Mth. (Moscou) 20 (1899) 595-; Fschr. Mth. (1899) 194. Chinese problem of remainders. Matthiessen, H.F.L. [1880-81] Crelle J. Mth. 91 (1881) 254-; Z. Mth. Ps. 26 (1881) (H.-lt. Ab.) 33-. , solution. liocquigny, F. [G.] de. Les Mondes 54 (1881) 304-. , . Domingues, A. Les Mondes 55 (1881) 62-. Congruence 487 2 = 486. Shanks, W. E. S. P. 25 (1877) 551-. theorem relating to sums of binomial co- efficients. GMsher, J. W. L. QJ. Mth. 30 (1899) 361-. Congruences. Lebesgue, V. A. N. A. Mth. 9 (1850) 436-; C. E. 51 (1860) 9-. . Hathaway, A. S. Am. J. Mth. 6 (1884) 316-. relating to sums of products of first n numbers, etc. Glaisher, J. W. L. QJ. Mth. 31 (1900) 1-. Congruent numbers. Woepcke, F. Torto- lini A. 3 (1860) 206-. . Gitnther, S. Prag Sb. (1878) 289-. , multiplication. Woepcke, F. (vm) A. Mt. 4 (1861) 247-. Decomposition of fraction into quantiemes (fractions with unity for numerator). Boby- nin, V. V. Z. Mth. Ps. 44 (1899) (Suppl.) 1-. Difference of two powers divisible by power of prime. Bouniakowsky , V. [1829] St. Pet. Ac. Sc. Mm. 1 (1831) 139-. product, divisibility. Segar, H. W. Mess. Mth. 23 (1894) 31-. DIVISIBILITY. Gergonne, J. D. Gergonne A. Mth. 5 (1814-15) 170-. Joubin, . L'l. 2 (1834) 230. Transon, A. (Prof.) N. A. Mth. 4 (1845) 173-. Elliott, E. B. Camb. (M.) Mth. M. 1 (1859) 45-. Zbikowski, A. [1860] St. Pet. Ac. Sc. Bll. 3 (1861) 151-. Niederrist, . Grunert Arch. 38 (1862) 384-. Folie, F. Lie"ge Mm. S. Sc. 3 (1873) 85-. B-utjaev, N. V. (xn) Eec. Mth. (Moscou) 8 (1876) (Pt. 1) 501-. Ponte Horta, F. da. G. Teix. J. Sc. 1 (1878) 57. Broda, K. Arch. Mth. Ps. 63 (1879) 413-. Kessler, 0. Z. Mth. Ps. 28 (1883) 60-. Ihirfee, W. P. J. H. Un. Cir. [4] (1885) 23. Catalan, E. Ev. Sc. 38 (1886) 508. Bougon, . Ev. Sc. 38 (1886) 508. Dickstein, S. Fschr. Mth. (1886) 136. Perrin, R. As. Fr. C. E. (1889) (Pt. 2) 24-. Geliii, E. Mathesis 12 (1892) 65-, 93-. Levanen, S. Helsingf. Ofv. 34 (1892) 109- ; Fschr. Mth. (1892) 164-. Fontes, . As. Fr. C. E. (1893) (Pt. 2) 240-. Belohldvek, F. Casopis (23) (1894) 59- ; Fschr. Mth. (1893-94) 273. Bourgerel, G. L. Ev. Sc. 1 (1894) 411-. by any number, criterion. Fontes, . C. E. 115 (1892) 1259-. prime. Matter, C. F. C., d- Holten, C. (xn) Ts. Mth. 5 (1875) 177-. . Hommel, L. L. (xn) Ts. Mth. 6 (1876) 15-. , character. Loir, . C. E. 106 (1888) 1070-, 1194. , criterion. Christie, R. W. D. Nt. 40 (1889) 247. of certain expressions by p, proof. Cauchy, A. L. Par. EC. Pol. J. 9 (1813) (16 e cah.) 99-. numbers. Simerka, V. Casopis 8 (*1879) 187- ; Fschr. Mth. (*1879) 125-. . Mackay, J. S. Edinb. Mth. S. P. 4 (1886) 55-. numerical expression. Weyr, E. Casopis 11 (*1882) 39-; Fschr. Mth. (*1882) 120. quotient by powers of factorial. Andre, D. C. E. 94 (1882) 426-. criteria. Unferdinger, F. Wien Sb. 59 (1869) (Ab. 2) 465-. . Delbceuf, J. Ev. Sc. 38 (1886) 377-. . Noel, . Ev. Sc. 38 (1886) 378-. . Greenstreet, W. J. Mth. Gz. 1 (1900) 186-. , history. Fontes, . Toul. Ac. Sc. Mm. 5 (1893) 459-. criterion. Dupain, J. C. N. A. Mth. 6 (1867) 368-. . Burgess, H. T. [1897] Nt. 57 (1897-98) 8-. , and Eratosthenes' sieve. Valeria, P. Ev. Sc. 52 (1893) 764. for prime numbers and all odd numbers (except multiples of 5). Riess, L. T. Fschr. Mth. (1898) 137. of faculties, geometric analogy. Pincherle, S. Bologna Ed. (1892-93) 17-. Fergola's theorem. Torelli, G. G. Mt. 5 (1867) 250-. of fraction with denominator k ! Time, A. Arch. Mth. Ntvd. 14 (1890) 247-. general characters. Lubin, J. N. A. Mth. 13 (1874) 528-. of integers. Lapparent, de. Cherb. Mm. S. Sc. 4 (1856) 235-. . Jonquieres, de. C. E. 120 (1895) 534-. law. Burgess, H. T. [1897] Nt. 57 (1897-98) 30, 55. . Bdrgen, C. [1897] Nt. 57 (1897-98) 54-. VOL. I. 193 2810 Divisibility Divisibility; linear congruences 2810 by number 7. Brooks, E. Des Moines Anal. 2 (1875) 129-. . Evans, A. Des Moines Anal. 10 (1883) 134-. , and conversion of fractions. Reyer, A. P. Grunert Arch. 25 (1855) 176-. , criteria. Heilinann, J. Ev. Sc. 38 (1886) 187. 9, criterion, generalisation. Wilbra- Jiam, H. Camb. and Dubl. Mth. J. 6 (1851) 32-. 13. Breton, P. Ev. Sc. 38 (1886) 444-. . Sclwbbens, . Ev. Sc. i 445. 19. Badoureau, A. N. A. Mth. 18 (1879) 35-. numbers 7, 11, 13, tests. Grunert, J. A. Arch. Mth. Ps. 42 (1864) 478-. and 13 especially. Pinaud, A. Toul. Mm. Ac. 1 (1844) 341-, 347-. , tests. Lindman, C. F. Grunert Arch. 26 (1856) 467-. of numbers of form 2 2 " + l, new case (Pervou- chine). Bouniakowsky, V. St Pe"t. Ac. Sc. Bll. 24 (1878) 559 ; 25 (1879) 63-. one given number by another. Schlegel, S. F. V. Z. Mth. Ps. 21 (1876) 365-. product of factorials by another, con- ditions. Landau, E. N. A. Mth. 19 (1900) 344-. by power of 2, theorem. Fregier, . Ger- gonne A. Mth. 9 (1818-19) 285-. prime numbers. Sensenig, D. M. Des Moines Anal. 3 (1876) 25-. a prime to 10. Dietz, , & Vincenot, . Metz Mm. Ac. 33 (1851-52) 37-. problem. Elefanti, F. E. S. P. 10 (1859-60) 208-. of product, distribution of primes. Catalan, E. C. Liege S. Sc. Mm. 12 (1885) No. 2, 20-, 119-. question; Polignac, C. (Prince) de. C. E. 96 (1883) 485-. simple properties. Elliott, E. B. Mess. Mth. 27 (1898) 12-. of sum of products of first n numbers taken r together. Glaisher, J. W. L. Mess. Mth. 28 (1899) 184-. theorem. Crelle, A. L. Crelle J. 5 (1830) 296 ; Gergonne A. Mth. 20 (1829-30) 349-. . R., O. , N. A. Mth. 4 (1845) 81-. . Bauer, M. Mth. Nt. B. Ung. 15 (1899) of Euler, correction. Chevillier, I. N. A. Mth. 14 (1855) 433-; 15 (1856) 260-. theorems. Catalan, E. Brux. Ac. Bll. 7 (1884) 448-. . Allardice, R. E. Edinb. Mth. S. P. 8 (1890) 16-. oi n(r + i) by n!, proof. Greatheed, S. S. (n Adds.) Camb. Mth. J. 1 (1839) 102-, n!, theorem. Gomes Teixeira, F. Arch. Mth. Ps. 2 (1885) 265-. Division of a number by 9 or 11. Dodgson, C. L. Nt. 56 (1897) 565-. numbers, general law. Bugaev, N. Eec. Mth. (Moscou) 12 (1885) 283-j Fschr. Mth. (1885) 145-. by a prime, theorem. Mantel, W. N. Arch. Wisk. 4 (*1878) 57-. , . Oskamp, G. A. N. Arch. Wisk. 4 (*1878) 83-. , theorem. Lejeune-Dirichlet, G, Berl. B. (1851) 20-. , on remainders (Lejeune-Dirichlet). Ter- quem, 0. N. A. Mth. 13 (1854) 396-. Divisors in case of two arbitrary functions, theorem. Bugaev, N. V. [1867] (xn) Eec. Mth. (Moscou) 3 (1868) (Pt. 1) 69-. , construction of tables. Davis, W. B. L. Mth. S. P. 4 (1873) 416-. of first n numbers, relations. Glaisher, J. W. L. L. Mth. S. P. 22 (1891) 359-. an integer. Lionnet, E. N. A. Mth. 7 (1868) 68-. , which exceed an assigned limit. Gegenbauer, L. Wien Ak. Sb. 98 (1890) (Ab. 2a) 28-. integers. Gegenbauer, L. Wien Ak. Sb. 91 (1885) (Ab. 2) 600-. . Eaorov, D. T. [1891] Eec. Mth. "loscou) 16 (1893) 236-; Fschr. Mth. (1892) (Me iss number. Crelle, A. L. Berl. Ab. (1832) 33-. . Hain, E. Arch. Mth. Ps. 55 (1873) 290-. , formula of Euler, proof. Lebesgue, V. A. N. A. Mth. 12 (1853) 232-. , method of finding. Canterzani, S. Bologna Mm. I. It. 2 (pte. 2) (1810) 445-. , ^ functions. Gegenbauer, L. Wien Ak. Sb. 93 (1886) (Ab. 2) 447-. , . Lerch, M. Prag Sb. (1887) (Mth.-Nt.) 683-; (1894) (Mth.-Nt.) No. 11, 11 pp., No. 32, 20 pp. , theorems. Glaisher, J. W. L. [1875] Mess. Mth. 5 (1876) 52. , . Gegenbauer, L. Wien Ak. Sb. 91 (1885) (Ab. 2) 1194-. , . Cesdro, E. G. Teix. J. Sc. 7 (1886) 3-. , . Lerch, M. Bll. Sc. Mth. 12 (1888) 100- ; Casopis 21 (1892) 90-, 185- ; Fschr. Mth. (1892) 186-. , . Bugaev, N. V. [1893-1898] Eec. Mth. (Moscou) 17 (1895) 720- ; 20 (1899) 549-; Fschr. Mth. (1895) 218- ; (1899) 194. , , two. Nachtikal, F. Casopis 25 (1896) 344-; Fschr. Mth. (1896) 139. numbers, graphic representation. Saint- Loup, L. C. E. 107 (1888) 24-; Par. EC. Norm. A. 7 (1890) 89-. and products of factors. Roberts, S. QJ. Mth. 20 (1885) 370-. prime, theorem concerning remainders. Gauss, C. F. Gott. Cm. 16 (1804-08) 69-. , reduction of systems. Hensel, K. Crelle J. Mth. 118 (1897) 234-; 119 (1898) 114-. 194 2810 Divisibility; linear congruences 2810 Divisors of series of natural numbers. Busche, E. [1896] Hamb. Mth. Gs. Mt. 3 (1900) 234-. 2w, Liouville's theorem. Mathews, G. B. [1893] L. Mth. S. P. 25 (1894) 85-. E () m (<) n = (. Laguerre, E. Liouv. J. Mth. 6 (1880) 99-. gtan- 1 !;). Laguerre, E. Par. S. Mth. Bll. 5 (1877) 95-. Eisenstein's. Muir, T. [1877] Edinb. E. S. T. 28 (1879) 135-. equal. Heilermann, . Schlomilch Z. 5 (1860) 362-. Euler's ; and Wallis's theorem. Bauer, G. [1872] Munch. Ab. 11 (1873) (Ab. 2) 97-. expansion in series by means of. Chebuishev, P. L. [1865] (xn) Eec. Mth. (Moscou) [1] (1866) 291-. forms of numbers determined by. Roberts, S. [1878] L. Mth. S. P. 10 (1878-79) 29-. Continued fractions 2815 for fraction 1 .3 .5 ... (2-l)/2 .4 .6 ... 2n. Stieltjes, . QJ. Mth. 25 (1891) 198-. function (l + x) n . Fade, H. C. E. 129 (1899) 753-, 875-. Gauss's theorems, proof. Clausen, T. Crelle J. 3 (1828) 311. generalisation. Lejeune-Dirichlet, G. Berl. B. (1842) 93-. . Jacobi, C. G. J. Crelle J. 69 (1868) 29-. (Jacobi). Bachmann, P. [1872] Crelle J. 75 (1873) 25-. . Poincare,H. C. E. 99 (1884) 1014-. . Minkowski, H. Par. EC. Norm. A. 13 (1896) 41-. geometric application. Leger, E. Quetelet Cor. Mth. 9 (1837) 483-. interpretation. Lieblein, J. Z. Mth. Ps. 12 (1867) 185-. indefinite, transformation to vulgar fractions. Pezzi, F. Mod. S. It. Mm. 11 (1804) 410-. infinite. Schulten, N. G. af (fil.) [1844-47] Helsingf. Acta 2 (1847) 661- ; 3 (1852) 427-. . Spitzer, S. Grunert Arch. 33 (1859) 418-. . Studnicka, F. J. Prag Sb. (1892) (Mth. Nt.) 254- ; Casopis 22 (1893) 15-. Convergence. Pringsheim, A. Munch. Ak. Sb. 28 (1899) 295- ; 30 (1901) 233. , evaluation. Spitzer, S. Grunert Arch. 30 (1858) 331- ; 33 (1859) 474-. for infinite series. Buzengeiger, K. H. J. Zach M. Cor. 22 (1810) 513-. . Muir,T. Edinb. E. S. T. 27 (1876) 467-. irrational numbers. Gylden, H. C. E. 106 (1888) 1584-, 1777-. ("e-^dx. Jacobi, G. G. J. Crelle J. 12 (1834) 346-. Jacobi 's formula ^~ = C sin [ (n + 1) cos" 1 x\. Hermite, C. [1875] Liege S. Sc. Mm. 6 (1877) No. 7, 7 pp. law of formation. Giinther, S. [1875] Wien Ak. D. 36 (1876) (,16. 2) 187-. for nth roots. Seeling, P. Arch. Mth. Ps. 46 (1866) 80-. for nth surds. Hoffmann, K. E. Arch. Mth. Ps. 64 (1879) 9-. with negative quotients. Saalschiitz, L. Crelle J. Mth. 120 (1899) 132-, 242-, 354. negative, for square roots. Stern, M. A. [1864] Gott. Ab. 12 (1866) 48 pp. numerical values. Glaisher, J. W. L. [1874] QJ. Mth. 13 (1875) 255-. periodic. Kramp, C. Gergonne A. Mth. 1 (1810-11) 261-, 319-. . Galois, E. Gergonne A. Mth. 19 (1828- 29) 294-. . Lebesgue, V. A. Ferussac Bll. Sc. Mth. 15 (1831) 155-. . Ramus, C. Kiob. Ov. (1836-37) 1-; Kjtfb. Dn. Vd. Selsk. Afh. 7 (1838) 197- ; Crelle J. 20 (1840) 13-. . Broager, P. D. Kjtfb. Dn. Vd. Selsk. Afh. 8 (1841) 145-. 199 2815 Continued fractions periodic. Bianchi, G. [1844] Mod. Mm. S. It. 23 (1846) 219-. . Rodrigues, O. N. A. Mth. 4 (1845) 109-. . Catalan, E. C. N. A. Mth. 4 (1845) . Galois, E. Liouv. J. Mth. 11 (1846) 385-. . Siebeck, H. Crelle J. 33 (1846) 68-. . Le Francois, (Prof.) . Brux. Ac. Bll. 16 (1849) 338-. . Stern, M. A. [1854] Crelle J. 53 (1857) . ' Simon, 0. E. Grunert Arch. 33 (1859) 448-. . Gunther, S. Z. Mth. Ps. 22 (1877) 31-. . Le Paige, C. M. M. H. H. Brux. Ac. Bll. 43 (1877) 337-. . Appell, P. Arch. Mth. Ps. 62 (1878) 183-. . Jonquieres, E. de. C. E. 96 (1883) 568-, 1297- . Muir, T. Edinb. E. S. P. 12 (1884) 389-, 578-. . Selivanov, D. T. Eec. Mth. (Moscou) 15 (1891) 635-; Fschr. Mth. (1891) 221-. . Landsberg, G. Crelle J. Mth. 109 (1892) 231-. (Landsberg). Netto, E. Crelle J. Mth. 110 (1892) 349-. , algebraic. Bortolotti, E. Palermo Cir. Mt. Ed. 9 (1895) 136-. , , generalisation. Bortolotti, E. Palermo Cir. Mt. Ed. 6 (1892) 1-. , arithmetical theorem. Sylvester, J. J. Mess. Mth. 19 (1890) 63-. , calculation. Anon, (vi 552) Gergonne A. Mth. 14 (1823-24) 337-. , closed form. Hoffmann, K. E. Arch. Mth. Ps. 62 (1878) 310-. , composition of periods. Jonquieres, E. de. C. E. 96 (1883) 694-, 832-. , convergence. Thiele, T. N. (xn) Ts. Mth. 3 (1879) 70-. , . Stolz, . Innsb. Nt. Md. B. 15 (1886) xxi-; 17 (1887) xviii-. , . Pringsheim, A. [1900] Miinch. Ak. Sb. 30 (1901) 463-. , convergents. Oettinger, L. Arch. Mth. Ps. 43 (1865) 301-. , (Oettinger). Strehlke, F. Arch. Mth. Ps. 48 (1868) 1-. , . Ocagne, M. d\ C. E. 108 (1889) 499-. , equivalent to square-roots of integers. Ely, G. S. A. Mth. 1 (1884-85) 72. , given number of quotients. Seeling, P. Arch. Mth. Ps. 49 (1869) 4-. , identities between convergents. Jonquieres, E. de. C. E. 96 (1883) 1351-, 1420-, 1490-, 1571-, 1721-. and indeterminate analysis. Tedenat, . Gard Not. Tr. Ac. (1811) 254-. , nih convergent. Strehlke, F. Arch. Mth. Ps. 42 (1864) 343-. , with one term. Clausen, T. Crelle J. 3 (1828) 87- ; N. A. Mth. 5 (1846) 203-. , symmetric period. Musso, G. G. Mt. 33 (1895) 1-, Continued fractions 2815 for TT. Sylvester, J. J. Ph. Mg. 37 (1869) 373-. . Jung, V. Casopis 9 (*1880) 157- ; Fschr. Mth. (*1880) 149-. , Sylvester's and other forms. Muir, T. Ph. Mg. 47 (1874) 331-. primes of form 4\ + l, as sum of 2 squares. Smith, H. J. S. Crelle J. 50 (1855) 91-. probability of large quotients. Gylden, H. Stockh. Ofv. (1888) 349-. problem. Clausen, T. St Pet. Ac. Sc. Bll. 9 (1851) 353-. for quadratic surds. Anon, (n 551) Ger- gonne A. Mth. 14 (1823-24) 324-. . Schulten, N. G. af (fil.) [1846-47] 52) 435-. ' ;- Helsingf. Acta 2 (1847) 1023- ; 3 (1855 . Schaar, M. Brux. Ac. Bll. 19 (1852) 16-. . Hoffmann, K. E. Arch. Mth. Ps. 64 (1879) 1-. . Hansted, B. G. Teix. J. Sc. 2 (1880) 154-. . Sylvester, J. J. C. E. 108 (1889) 1037-, 1084-. . Dolbnia, J. N. A. Mth. 10 (1891) 134-. . Boutin, A. Mathesis 17 (1897) 8-. . Crelier, . C. E. 128 (1899) 229-. . , convergence. Falk, M. Ts. Mt. Fys. 4 (1871) 249-. , periodicity. Schulten, N. G. af (fil.) [1846] Helsingf. Acta 2 (1847) 1063-. , . Charves, . Bll. Sc. Mth. As. 1 (1877) 41-. quotients in arithmetical progression. Hur- u-itz, A. Ziir. Vjschr. 41 (1896) (Festschr., Th. 2) 34-. for rational fractional power. Matthiessen, L. Z. Mth. Ps. 10 (1865) 315- . reduction to series. De Morgan, A. Ph. Mg. 24 (1844) 15-. . Heilermann, . Crelle J. 46 (1853) 88-. regular. Gegenbauer, L. Wien Ak. D. 58 (1891) 177- ; Mh. Mth. Ps. 6 (1895) 209-. relation to astronomical cycles. Brunei, . Bordeaux S. Sc. Mm. 5 (1890) xxv-. . Tannery, P. Bordeaux S. Sc. Mm. 5 (1890) xxix-. representation as definite integrals. Spitzer, S. Grunert Arch. 30 (1858) 81-. of numbers. Schulten, N. G. af (fil.) [1845] Helsingf. Acta 2 (1847) 1009-. for roots. Collins, E. (vn) St. P6t. Ac. Sc. Bll. 7 (1840) 357-. of equations. Montessus, E. de. Brux. S. Sc. A. 21 (1897) (Pt. 2) 71-. numbers. C. Camb. Mth. J. 3 (1843) 120-. roots and their convergence. Doppler, C. Wien Jb. Pol. I. 17 (1832) 175-. for series. Kausler, C. F. [1802] St Pe"t. Ac. Sc. Mm. 1 (1809) 156-. . Viscovatoff, B. [1802] St Pet. Ac. Sc. N. Acta 15 (1806) 181-. . Schubert, F. T. [1815] St Pet. Ac. Sc. Mm. 7 (1820) 139-. 200 UNIVERSITY OF CALIFORNfit SANTA BARBARA COLLEGE 2815 Continued fractions for series. Eytelwein, J. A. Berl. Ab. (1820- 21) 15-. . Le Barbier, . Gergonne A. Mth. 21 (1830-31) 262-. . Heine, H. E. Crelle J. 32 (1846) 205-. . Heilermann, . Crelle J. 33 (1846) 174-. -. Hankel, H. Leip. B. 14 (1862) 17-. . Broman, K. E. Ts. Mt. Fys. 4 (1871) 139-. . Mollame, V. [1882] Catania Ac. Gioen. At. 17 (1883) 129-. . Gambioli, D. Bologna Ed. (1890-91) 43-. . . Cebysev, P. St. Pet. Ac. Sc. Mm. (Es.) 64 (1891) (Suppl.) No. 7, 67 pp. and similar functions. Meyer, F. Konigsb. Schr. 38 (1897) [57]-. simple, convergence. Fade, H. C. E. 112 (1891) 988-. solution of continued fraction equation. Mol- lame, V. Bv. Mt. 2 (1892) 212-. for solution of linear differential equation. Laguerre, E. [1878-79] C. E. 87 (1878) 923- ; Par. S. Mth. Bll. 8 (1880) 21-. square roots. Ivory, J. [1800] Edinb. E. S. T. 5 (1805) 20-. . Booth, J. Camb. Mth. J. 2 (1841) 226-. . De Morgan, A. (n Adds.) Camb. Mth. J. 2 (1841) 239-. . Schlomilch, 0. Grunert Arch. 6 (1845) 147-. (Schlomilch). Miiller, E. Grunert Arch. 6 (1845) 151-. . Arndt, F. Crelle J. 31 (1846) 343- ; Grunert Arch. 12 (1849) 211-. . Serret, J. A. Liouv. J. Mth. 12 (1847) 518-. . Badon-Ghijben, J. [1864] Amst. Vs. Ak. 17 (1865) (Ntk.) 121-. . Weyr, Ernil. Prag Sb. (1869) 18-. . Schlomilch, O. Z. Mth. Ps. 17 (1872) 70-. . Weyr, Ed. Prag Sb. (1877) 65-. . Hermes, J. [1879] Arch. Mth. Ps. 65 (1880) 438-. . Gunther, S. [1882] Bordeaux S. Sc. Mm. 5 (1883) 91-. . Muir, T. Mess. Mth. 14 (1885) 115-. . Thaarup, F. N. TS. Mth. 11 (A) (1900) 33-. , and factors. Kausler, C. F. [1807] St Pet. Ac. Sc. Mm. 2 (1810) 95-. , periodicity. Crelier, L. Arch. Sc. Ps. Nt. 6 (1898) 366-. "step" fractions. Gohierre de Longchamps, . G. Mt. 15 (1877) 299-. successive approximations. De Morgan, A. Camb. Mth. J. 4 (1845) 97-. summation. Stern, M. A. Crelle J. 8 (1832) 42-. of series by. Homer, W. G. Thomson A. Ph. 11 (1826) 416- ; 12 (1826) 48-. symmetric, convergents. Musso, G. N. A. Mth. 14 (1895) 70-. in " Theoria Motus," pp. 97-98. Eamus, C. Mth. Ts. 1 (1859) 129-. Indeterminate equations 2815 and theory of numbers. Catalan, E. Brux. Ac. Mm. 52 (1893-94) No. 1, 28 pp. Wallis's. Euler, L. [1780] St Pet. Ac. Sc. Mm. 5 (1815) 24-. for f|^Y^ Laguerre, E. [1879] Par. S. Mth. Bll. 8 (1880) 36-. (x-a) a (x-bf. Fuchs, L. Acta Mth. 4 (1884) 89-. (x-a) a (jr-b)0. Hermite, C. Acta Mth. 4 (1884) 91-. in r m m + 1 - m ~w" + l-etc. Schulten, N. G. af (fil.) [1844] Helsingf. Acta 2 (1847) 629-. :p7-s o etc - Amoretti, F. N. A. Mth. 14 JL -t- & -f- o + (1855) 40-. fT2+~ 3T 6tC ' S P itzer > s - Grunert Arch. 25 (1855) 137-. ^r (Legendre). Cata- z+ z + l+ z + 2 + Ian, E. C. Liege S. Sc. Mm. 13 (1886) 205-. INDETEEMINATE EQUATIONS. Bouniakowsky, V. [1829] St Pet. Ac. Sc. Mm. 1 (1831) 139-. Poselger, F. T. Berl. Ab. (1832) 1-. Cauchy, A.L. C. E. 12 (1841) 698-, 813-. Bouniakowsky, V. [18471 St Pet. Ac. Sc. Bll. 6 (1848) 196-. Lebesgue, V. A. N. A. Mth. 8 (1849) 81-, 347-. Wheeler, A. D. Camb. (M.) Mth. M. 2 (1860) v 21-, 55-, 193-, 398-. Simerky [Simerka], V. Prag Ab. 12 (1863) 193-. Lebesgue, V. A. N. A. Mth. 8 (1869) 452-. Catalan, E. Brux. Ac. Bll. 9 (1885) 531-. Weill, . N. A. Mth. 4 (1885) 189-. Jonquieres, E. de. G. Mt. 24 (1886) 1-. Cordone, G. G. Mt. 33 (1895) 106-, 218-. Maillet, E. Toul. Ac. Sc. Mm. 7 (1895) 179-. binary form, least integral value. Paoli, P. [1801] Mod. S. It. Mm. 9 (1802) 85-. class. Palmstrpm, A. Christiania Skr. (Mth.- NH. Kl.) (1899) No. 7, 10 pp. Diophantine analysis, advance. Euler, L. St Pet. Ac. Sc. Mm. 11 (1830) 1-. , elliptic and Abelian integrals, use in. Jacobi, C. G. J. Crelle J. 13 (1835) 353-. , geometric problem solved by. Euler, L. [1782] St Pet. Ac. Sc. Mm. 7 (1820) 3-. by integral calculus. Bouniakowsky, V. [1852J St Pet. Ac. Sc. Bll. 11 (1853) 65-. , problems. Fuss, N. [1820] St Pet. Ac. Sc. Mm. 9 (1824) 151-. equation. Davis, E. F. Edinb. Mth. S. P. 3 (1895) 179-. equations. Hilbert, D. Gott. Nr. (1897) 48-. , irreducible, limit of roots. Wirtinger, W. Mb.. Mth. Ps. 1 (1890) 47-. 201 2815 Indeterminate equations Diophantine equations, solutions. Catalan, E. C. Li6ge S. Sc. Mm. 15 (1888) No. 1, 73-, 75-, 120-, 211-, 259. problem before Diophantus. Biot, J. B. C. K. 28 (1849) 576-. , Fermat's solution. Roberts, S. As. Fr. C. E. (1886) (Pt. 2) 43-. , solution. Kausler, C. F. [1806] St Pet. Ac. Sc. Mm. 1 (1809) 271-. , . Strong, T. Silliman J. 1 (1818) 426-. , . Wlieeler, A. D. Silliman J. 20 (1831) 295-. , . Bouniakowsky, V. [1841] St Pet. Ac. Sc. Mm. 5 (1844) 1-. , . Boije af Gennds, C. 0. N. A. Mth. 19 (1880) 278-. problems. Eider, L. St Pet. Ac. Sc. Mm. 9 (1824) 3- ; 11 (1830) 46-. , solution. Eider, L. [1778] St Pet. Ac. Sc. Mm. 4 (1813) 3-. 5 solutions of two. Strong, T. Silliman J. 31 (1837) 156-. speculations. Lenhardt, W. Mth. Misc. 1 (1838) 263-, 330- ; 2 (1839) 127-. examples. Steen, A. (xn) Ts. Mth. 1 (1865) 20-. impossibility of some. Lebesgue, V. A. N. A. Mth. 2 (1863) 68-. mechanical solution. Tarry, H. As. Fr. C. E. (1897) (Pt. 1) 177. Paoli's formula for number of solutions. Lucas, E. Mathesis 10 (1890) 129-. (Lucas). Catalan, E. Ma- thesis 10 (1890) 197-. power of polynomial, law of coefficients. Teixeira, J. P. G. Teix. J. Sc. 13 (1897) 65- ; Fschr. Mth. (1897) 167. problems, class, with application to decomposi- tion of fractions. Gergonne, J. D. Gergonne A. Mth. 10 (1819-20) 271-. , solution. Leslie, J. [1788] Edinb. E. S. T. 2 (1790) 193-. , . Kausler, C. F. [1800] St Pet. Ac. Sc. N. Acta 15 (1806) 116-. , . Perkins, G. R. Des Moines Anal. 1 (1874) 40-, 101-, 151 -. , two. Bouniakowsky, V. [1864] St Pe"t. Ac. Sc. Bll. 8 (1865) 163-. quadratic duplicate equalities. Collins, M. B. A. Ep. (1855) (Pt. 2) 2-. series derived from. Oliveira Ramos, J. C. d\ d- Faria, C. -J. de. G. Teix. J. Sc. 7 (1886) 41- ; Fschr. Mth. (1885) 129-. solution. Libri, G. Crelle J. 9 (1832) 277-. . Pepin, T. Em. N. Line. At. 38 (1885) 20-. . Thue, A. Thrond. Skr. (1896) No. 7, 14 pp. of problem of Beha-Eddin's. Liicas, E. N. A. Mth. 15 (1876) 359-. by series. Libri, G. Crelle J. 9 (1832) 313-. system. Lebesgue, V. A. N. A. Mth. 9 (1850) 46-. tetrahedron problem. Chabanel, C. N. A. Mth. 13 (1874) 289-. theorem. Liouville, J. C. E. 10 (1840) 381-. Indeterminate equations 2815 theorem of Liouville. Gegenbauer, L. Wien 606-. 78 (1874) 144- ; Ak. Sb. 95 (1887) (Ab. 2) ( rin, T. C. E. 88 (1879) 1255-. theorems. Pepin, and transformation of plane figures. Ruffini, F. P. Bologna Ac. Sc. Mm. 9 (1878) 199-. triangle with bisectors of sides rational. Euler, L. [1779] St Pet. Ac. Sc. Mm. 2 (1810) 10-. distances of centre of gravity from ver- tices rational. Eider, L. [1778] St Pet. Ac. Sc. N. Acta 12 (1801) 101-. 2 or 3 variables. Maillet, E. Liouv. J. Mth. 6 (1900) 261-. 3 variables, solution by slide-rule. San Roberto, P. di. Tor. At. Ac. Sc. 2 (1866-67) 454-. several variables. Bianchi, G. Mod. Mm. S. It. 24 (1850) 265-. , number of solutions. Berger,A. Stockh. Ofv (1886) 355-; Fschr. Mth. (1886) 149-. virgins problem. Sylvester, J. J. Ph. Mg. 16 (1858) 371-. 1st order. Pilatte, . Gergonne A. Mth. 2 (1811-12) 230-. Gergonne, J. D. Gergonne A. Mth. 3 (1812- 13) 147-. Libri, G. Gergonne A. Mth. 16 (1825-26) 297-. Crelle, A. L. Berl. Ab. (1836) 1-. Brassinne, E. Toul. Mm. Ac. 4 (1837) 189-. Crelle, A. L. Berl. Ab. (1840) 1-. Chevillard, A. J. N. A. Mth. 2 (1843) 471-. Grunert, J. A. Grunert Arch. 3 (1843) 203-. Catalan,- E. C. N. A. Mth. 3 (1844) 97-. Midy, . N. A. Mth. 4 (1845) 146-. Grunert, J. A. Grunert Arch. 7 (1846) 162-. Jau/roid, B. N. A. Mth. 11 (1852) 158-. Reuschle, C. (jun.) Z. Mth. Ps. 19 (1874) 272. Meray, C. C. E. 94 (1882) 1167- ; Par. EC. Norm. A. 12 (1883) 89-. Catalan, E. Mathesis 10 (1890) 220-, 241-, 275-. Brocard, H. Mntp. Ac. Mm. 11 (1892) 139-. Gram, J. P. N. Ts. Mth. 3 (B) (1892) 57-, 73- ; Fschr. Mth. (1892) 174. Giudice, F. G. Mt. 36 (1898) 225-. approximate solution. Kronecker, L. Berl. Ak. Sb. 1884) 1179-, 1271- solution. Pezzi, F. Mod. S. It. Mm. 11 (1804) 410-. . Casterman, L. Liege A. Ac. (1819-20) 36 pp. . Binet, J. P. M. [1827] Par. EC. Pol. J. 13 (1831) (20 e cah.) 289-. . Paoli, G. de. Opusc. Mt. Fis. 1 (1832) 262-, 327-. . Binet, J. P. M. Liouv. J. Mth. 6 (1841) 449-. . Heger, I. Wien SB. 21 (1856) 550- ; Wien D. 14 (1858) 1-. . Slavlk, J. Casopis 14 (1885) 137- ; Fschr. Mth. (1885) 122. 202 2815 Indeterminate equations solution. Ayza, E. Fschr. Mth. (1897) 168. , forms. Weihrauch, K. Z. Mth. Ps. 19 (1874) 53-. systems. Smith, H. J. S. B. A. Rp. (1860) (pt. 2) 6; Phil. Trans. (1861) 293-. . Ovidio, E. d'. Tor. Ac. Sc. At. 12 (1876) 334-. 2 or more variables, solution by continued fractions. Pleskot, A. Casopis 22 (1893) 71- ; Fschr. Mth. (1893-94) 278. 4 variables, solutions, number. Weihrauch, K. Z. Mth. Ps. 22 (1877) 234-. n variables. Sylvester, J. J. Ph. Mg. 16 (1858) 369-. 2nd order. Minding, E. F. A. Crelle J. 7 (1831) 140-. Arndt, F. Grunert Arch. 12 (1849) 211-. Gopel, A. Crelle J. 45 (1853) 1-. Realis, S. N. A. Mth. 3 (1884) 305-, 448. Desboves, A. N. A. Mth. 5 (1886) 226-. Marcolongo, E. G. Mt. 25 (1887) 161- ; 26 (1888) 65-. Jonquieres, de. C. R. 126 (1898) 863-, 990, 991. Euler's formula. Chasles, M. Liouv. J. Mth. 2 (1837) 37-. and fourth. Pepin, T. Em. N. Line. At. 32 (1879) 79-. Lagrange's and Gauss's methods. Jonquieres, de. C. R. 127 (1898) 694-. solution. Kunerth, A. Wien Ak. Sb. 75 (1877) (Ab. 2) 7-. by continued fraction, criteria. Kunerth, A. [1880] Wien. Ak. Sb. 82 (1881) (Ab. 2) 342-. in rational numbers. Kunerth, A. [18781 Wien Ak. Sb. 78 (1879) (Ab. 2) 327-. 3rd order. Realis, S. N. A. Mth. 17 (1878) 454-. Jonquieres, E. de. Em. N. Line. At. 37 (1884) 183-. or higher. Desboves, A. C. R. 87 (1878) 159-, 321-. impossibility. Lucas, E. Par. S. Mth. Bll. 8 (1880) 173-. and problem of Sylvester's. Lucas, E. N. A. Mth. 17 (1878) 507- ; Am. J. Mth. 2 (1879) 178. reduction to 2nd order. Le Besgue, V.A. C. E. 64 (1867) 1267-. 4th order. Pepin, T. Em. N. Line. At. 31 (1878) 397- ; Em. N. Line. Mm. 4 (1888) 227-; 9 (1893) (Pt. 1) 247-; 11 (1895) 223-. with 4 variables, solution. Eisenstein, G. Crelle J. 27 (1844) 75-. 5th order. impossibility of some equations. Lejeune- Dirichlet, G. Crelle J. 3 (1828) 354-. Indeterminate equations 2815 ab (ma? + nV*)=cd (me 2 -Mid 2 ). Eider, L. [1778] St Pet. Ac. Sc. N. Acta 13 (1802) 45-. = 1 + bx + cy + dx* + cxy + fy* + gx-y + hxy* + ix z y z . Euler, L. St Pet. Ac. Sc. Mm. 11 (1830) 58-. 4 (x* - 1) = (x - 1) (F 2 nZ 2 ) . Legendre, A . M. [1830] Par. Mm. Ac. Sc. 11 (1832) 81-. 4 (XP - 1) = ( x - 1) (i/ 2 pz 2 ). Germain, (Mile.) S. Crelle J. 7 (1831) 201-. 2 2 "-?/=2.r. Liouville,J. Liouv. J. Mth. 5 (1840) 360. x m -y*=l. Catalan, E. C. Crelle J. 27 (1844) 192. j/ = cos7r.r, solution in rational numbers. Schulten, N. G. af (fil.) [1844] Helsingf. Acta 2 (1847) 617-. ax + by + cz = Q. Cauchy, A. L. Grunert Arch. 7 (1846) 305-. Ax 3 + By 3 +Cz 3 = Dxyz. Sylvester, J. J. Ph. Mg. 31 (1847) 189- , 293-. ax + by = c. Dieu, T. N. A. Mth. 9 (1850) 67-. ax (a + b) y = m. Grebe, E. W. Grunert Arch. 14 (1850) 333-. a a x 2 = a,Xj + 1, smallest integers satisfying. Crelle, A. L. Berl. B. (1850) 141- ; Crelle J. 42 (1852) 299-. x m =y'^+l, impossibility of solution in inte- gers. Lebesgue, V. A. N. A. Mth. 9 (1850) 178-. a x - bv= 1, a and b being primes. Gerono, G. C. N. A. Mth. 16 (1857) 394-. ax + by + cz + , etc. = n, number of solutions in positive integers. Hermite, C. QJ. Mth. 1 (1857) 370-. aj + 2a 2 + 3a 3 +...+na n =n. Fergola, E. Nap. Ed. 2 (1863) 262-. ns [s 2 + (n- - 1) r 2 ] = 8y 3 . Le Besgue, V. A . (vm) A. Mt. 5 (1863) 328. _ = _. Genocchi,A. (vn) A. Mt. 5 (1863) 329-. (x + h) n -x n = z n , impossibility. Gaudin, . C. E. 59 (1864) 1036-. Ax + By = S. Clasen, B. I. Par. EC. Norm. A. 4 (1867) 347-. 4 (xf>-l) = (x-l) (y-pz"*). Staudt, G. K. C. von. Crelle J. 67 (1867) 205-. S(Cx + D). Serdoblnskii, V. E. [1878] (xn) Eec. Mth. (Moscou) 9 (1878-81) (Pt. 1) 557 . 'iTx^r; *2;"iX f = n. Sadun, E. A. Mt. 15 t " 1 (1887-88) t 209-. = t/ 3 , posthumous problem of Leibnitz. Studnicka, F. J. Casopis 18 (1889) 97-; Fschr. Mth. (1889) 30-. 203 2820 Quadratic residues 125-. = c. Spelta, C. G. Mt. 33 (1895) ^*^. Stfrmer, C. Christiania Skr. (Mth.-Nt. Kl.) (1895) No. 11, 21pp. u m + v n =iv k . Ermakov, V. [1898] Kec. Mth. (Moscou) 20 (1899) 293-. A 1/1*1 MA . . . BN } Vi NJ>* ... = C. Stormer, C. C. B. 127 (1898) 752-. 2/ 2 = ax + 6. Besant, W. H. Mth. Gz. 1 (1900) 130. 2820 Quadratic residues. Gauss, C. F. Gott. Cm. 4 (1816-18) 3-. Lebesgue, V. A. Liouv. J. Mth. 5 (1840) 186-. Stern, M. A. Brux. Mm. Cour. 15 (1840-41) 38 pp. Scliaar, M. [1849-50] Brux. Ac. Sc. Mm. (1850) xxiv-. Genocchi, A. Brux. Mm. Cour. 25 (1851-53) 55 pp. Stern, M. A. [1862] Crelle J. 61 (1863) 334-. Candida, Z. A. G. Teix. J. Sc. 1 (*1878) 171-. Schering, E. Acta Mth. 1 (1882) 153-. Nazimow, P. Bll. Sc. Mth. 9 (1885) 15-. Buniakovskij, V. Ja. [1886] St Pet. Ac. Sc. Mm. (Rs.) 55 (1887) (SiijppZ.) No. 5, 16 pp. ; Fschr. Mth. (1887) 173. Lipschitz, E. C. B. 108 (1889) 489-. Matrot, A. As. Fr. C. B. (1890) (Pt. 2) 82-. Frolov, M. As. Fr. C. B. (1892) (Pt. 2) 149-. Eeich, K. Arch. Mth. Ps. 11 (1892) 176-, vn-. Pepin, . C. B. 128 (1899) 1553-. Congruence 2 = c (mod. p*), resolution. To- nelli, A. Em. B. Ac. Line. Ed. 1 (1892) (Sem. 1) 116- ; 2 (1893) (Sent. 1) 259-. Congruent division and quadratic residues. Stern, M. A. [1861] Crelle J. 61 (1863) 66-. Continued fractions and quadratic forms. La Vallee-Poussin, C. J. de. Brux. S. Sc. A. 19 (1895) (Pt. 1) 111-. Density of series of prime numbers. Konig, J. Mth. Nt. B. Ung. 14 (1898) 322. Eisenstein's geometrical proof. Cayley,A. QJ. Mth. 1 (1857)- 186-. Historical study. Pepin, T. Bm. N. Line. Mm. 16 (1899) 229-. Prime modulus, square roots of unity for a. Tanner, H. W. L. Mess. Mth. 21 (1892) 139-. Problems on numbers (Fermat). Pepin, T. Bm. N. Line. Mm. 8 (1892) 85-. Quadratic and cubic residues. Meyer, G. Arch. Mth. Ps. 63 (1879) 1-. reciprocity, formula of Gauss, extension. Le Besgue, V. A. C. B. 59 (1864) 1067-. residue-character. Gegenbauer, L. Wien Ak. Sb. 100 (1891) (Ab. 2a) 1072-. , determination. Schering, E. Gott. Ab. 24 (1879) (Mth.) 47 pp. Quadratic residues 2820 Quadratic residue-character, determination by continued fractions. Zeller, C. Gott. Nr. (1879) 197-. , Gauss's criterion. Schering, E. Berl. Ak. Mb. (1876) 330-. , . Kronecker, L. Berl. Ak. Mb. (1876) 331-. of number 2. Stieltjes, T. J. (jun.) N. Arch. Wisk. 9 (1882) 193- ; Toul. Fac. Sc. A. 11 (1897) A, 4 pp. 3 for any prime. Bricard, E. N. A. Mth. 16 (1897) 546-. residues and non-residues. Cauchy, A. L. C. E. 10 (1840) 437-. . Lebesgue, V. A. Liouv. J. Mth. 7 (1842) 137-. of a prime, distribution. Aladov, N. S. Bee. Mth. (Moscou) 18 (1896) 61-; Fschr. Mth. (1896) 146. , . Sterneck, E. von. [1898] Eec. Mth. (Moscou) 20 (1899) 269- ; Fschr. Mth. (1898) 153-. primes, combination. Bounia- kowsky, V. [18681 St Pet. Ac. Sc. Bll. 13 (1869) 25-. , periods and primitive roots. Arndt, C. F. Grunert Arch. 2 (1842) 1-. of primes of form 4m + 3. Woronoj, G. F. Fschr. Mth. (1899) 184. , trigonal, and bitrigonal residues. Stern, M. A. [1869] Crelle J. 71 (1870) 137-. Eeciprocity law. Eisenstein, G. Crelle J. 28 (1844) 41-. . Liouville, J. Liouv. J. Mth. 12 (1847) 95-. . Mansion, P. Mess. Mth. 5 (1876) 140-. . Petersen, J. P. C. (xn) Ts. Mth. 3 (1879) 86- ; (x) Am. J. Mth. 2 (1879) 285-. . Kronecker, L. Bll. Sc. Mth. As. 4 (1880) 182-. . Sylvester, J. J. C. B. 90 (1880) 1053-, 1104-. _ _. Gegenbauer, L. Wien Ak. Sb. 90 (1885) (Ab. 2) 1026- ; 97 (1889) (Ab. 2a) 427-. . Kronecker, L. Crelle J. Mth. 104 (1889) 348-. . Lucas, E. St Pet. Ac. Sc. Bll. 33 (1890) 495-. . Gegenbauer, L. Wien Ak. Sb. 103 (1894) (Ab. 2a) 285-. . Uasenohrl, F. Wien Az. 31 (1894) 74-. . Kmiig, G. Mth. Term. Ets. 13 (1895) 429- ; Acta Mth. 22 (1899) 181-. for complex numbers. Lejeune-Dirich- let, G. Crelle J. 9 (1832) 379-. and Gauss's sums. Mertens, F. Wien Ak. Sb. 103 (1894) (Ab. 2o) 1005-. , history. Kronecker, L. Berl. Ak. Mb. (1875) 267-. , Legendre's, extension. Jenkins, M. [1867] L. Mth. S. P. 2 (1869) 29-. , , to numbers not prime. Genocchi, A. C. E. 90 (1880) 300-. , , and Gauss's sums. Sochocki, J. Par. T. Nauk Sc. Pam. 10 (*1878) Art. 1, 37 pp. ; Fschr. Mth. (*1878) 315. 204 2820 Quadratic residues Reciprocity law, Legendre's, proof. Eisenstein, G. Crelle J. 27 (1844) 322-. , , . Lebesgue, V. A. Liouv. J. Mth. 12 (1847) 457-. , , . Zolotareff, G. N. A. Mth. 11 (1872) 354-. , , . Pepin, T. Em. N. Line. At. 43 (1890) 192- ; 51 (1897) 123-. . , , , Gauss's 3rd. Lerch, . G. Teix. J. Sc. 8 (1887) 137- , , theorem of Zolotarev. Lerch, M. Prag Cesk( Ak. Fr. Jos. Bz. (Trida 2) 5 (1896) Art. 17, 8 pp. ; Fschr. Mth. (1896) 141. , proof. Schaar, M. Brux. Ac. Bll. 14 (1847) 79-. , . Eisenstein, G. Berl. B. (1850) 189-. , . Stern, M. A. Gott. Nr. (1870) 237-. , . Schering, E. Gott. Nr. (1879) 217-. , . Kronecker, L. Berl. Ak. Sb. (1884) 519-, 641. , _. Genocchi, A. C. R. 101 (1885) 425-. , . Busche, E. Crelle J. Mth. 99 (1886) 261-. , . Gegenbauer, L. Mh. Mth. Ps. 4 (1893) 190-. , . Busche, E. [1896] Hamb. Mth. Gs. Mt. 3 (1900) 233-. , , Eisenstein's. Fischer, E. Mh. Mth. Ps. 11 (1900) 176-. , , elementary. Lange, . Leip. Mth. Ps. B. 48 (1896) 629- ; 49 (1897) 607-. , , Gauss's 1st. Lejeune-Dirichlet, G. Crelle J. 47 (1854) 139- ; Liouv. J. Mth. 4 (1859) 401-. , , 3rd. Thomae, C. J. Z. Mth. Ps. 26 (1881) 134-. , , . Kronecker, L. Berl. Ak. Sb. (1884) 645-; Crelle J. Mth. 97 (1884) 93-. , , . Schering, E. Berl. Ak. Sb. (1885) 113-. , , (Schering). Kronecker, L. Berl. Ak. Sb. (1885) 117-. , , 4th. Kronecker, L. Berl. Ak. Mb. (1880) 686-, 854-. - , by inversion. Hermes, J. Arch. Mth. Ps. 5 (1887) 190-, vni. , , Schering's. Hacks, J. Acta Mth. 12 (1889) 109-. , proofs. Hermite, C. Crelle J. Mth. 100 (1887) 51-. , _. Schmidt, H. Crelle J. Mth. Ill (1893) 107-. for Zeller's quadratic residues. Kronecker, L. Berl. Mb. (1872) 846-. laws. Eisenstein, G. Crelle J. 28 (1844) 53-. . Kummer, E. E. Crelle J. Mth. 100 (1887) 10-. . Stou/, X. C. R. 123 (1896) 486-; 126 (1898) 812-. theorem in field (1, t\/2). Bonaventura, P. G. Mt. 30 (1892) 221-. Quadratic binary forms 2830 Residues and non-residues of primes of form 4n + 3, distribution. Gotting, E. [1868] Crelle J. Mth. 70 (1869) 363-. of polynomials, form. Collins, E. St Pet. Ac. Sc. Bll. 1 (1836) 113-. primes, properties. Osborn, G. Mess.. Mth. 25 (1896) 45-. Series of quadratic residues. Weihrauch, K. Z. Mth. Ps. 32 (1887) 1-. Sums of quadratic residues and non-residues, numbers. Fields, J. C. Crelle J. Mth. 112 (1893) 247-. Symbol, Legendre's, ( - ) Bouniakowsky, V.. St Pet. Ac. Sc. Bll. 14 (1870) 497-. , ( ), new propositions. Bouniakowsky, V. [1876] St Pet. Ac. Sc. Bll. 22 (1877) 358-. , , generalised. Gegenbauer, L. [1881J Wien Ak. Sb. 84 (1882) (Ab. 2) 1089-. , , , algorithms for determination. Gegenbauer, L. [1880] Wien Ak. Sb. 82 (1881) (Ab. 2) 931-. , , and new function in number theory. Bock, W. Hamb. Mth. Gs. Mt. 1 (1889V 187-, 201-. , , series involving. Berger, A. Ups. S. Sc. N. Acta 12 (1885) No. 7, 31 pp. , Legendre-Jacobi. Lebesgue, V. A. Liouv^ J. Mth. 12 (1847) 497- ; 15 (1850) 215- ; C. B. 59 (1864) 940- ; 60 (1865) 377-. , . Gegenbauer, L. Wien Ak. Sb. 91 (1885) (Ab. 2) 11- ; 96 (1888) (Ab. 2) 491- ; 100 (1891) (Ab. 2a) 855-. , . Scheibner, W. Leip. Mth. Ps. Ab. 25 (1899) 367-. , , theorem involving. Mandl, M. QJ_ Mth. 25 (1891) 227-. (\ . Dedekind, E. C. B. 91 (1880) 154-. _ (_). Gegenbauer, L. Wien Ak. Sb. 92 (1886) (Ab. 2) 876-. ( ), evaluation. Eisenstein, G. Crelle J. 27 (1844) 317-. Theory, and Euler's summation formula. Genocchi, A. Tortolini A. 3 (1852) 406-. of Gauss, generalisation. Mandl, M. Mh Mth. Ps. 1 (1890) 465-. 2830 Quadratic binary forms. Kausler, C. F. [1801] St Pet. Ac. Sc. N. Acta. 15 (1806) 156-. Wezel, J. L. Liege A. Ac. (1821-22) 48 pp. Lejeune-Dirichlet, G. Berl. Ab. (1833) 101- ; Berl. B. (1840) 49-. Seheffiar, H. Crelle J. 45 (1853) 349-. Mainardi, G. Mil. At. I. Lomb. 1 (1858) 106-. Cantor, G. Z. Mth. Ps. 13 (1868) 259-. Grube, F. Z. Mth. Ps. 19 (1874) 492-. Jonquieres, E. de. C. K. 98 (1884) 1358-, 1515. Pellet, A. E. C. R. 98 (1884) 1482. Cajiiri, F. J. H. Un. Cir. [4] (1885) 122. Gegenbauer, L. Wien Ak. Sb. 96 (1888) (Ab.2\ 476-. Lachtin, L. K. [1889-91] Rec. Mth. (Moscou) 14 (1890) 487-; 15 (1891) 573-; Fschr. Mth. (1890) 215. 205 2830 Quadratic binary forms Preobrazenskij, P. V. [1890] Eec. Mth. (Moscou) 15 (1891) 118- ; Mosc. S. Sc. Bll. 65 (No. 1) (1890) 62-; Fschr. Mth. (1890) 215, 215-. Tano, F. Bll. Sc. Mth. 14 (1890) 215-. Pepin, . C. E. 119 (1894) 397-; 120 (1895) 494. Bauer, M. Mth. Termt. Ets. 13 (1895) 316- ; Mth. Nt. B. Ung. 13 (1897) 37-. Jonquieres, de. C. E. 126 (1898) 991-, 1077-, 1177. Automorphs. Tanner, H. W. L. Mess. Mth. 24 (1895) 180-. Classification. Pepin, T. Em. N. Line. At. 33 (1880) 354-. Class-number. Arndt, F. Crelle J. 56 (1859) 72-. . Liouville,J. Liouv. J. Mth. 14 (1869) 1-. (Liouville). Stieltjes, T. J. (jun.) C. E. 97 (1883) 1358-, 1415-. , Dirichlet's formula. Arndt, F. Crelle J. 56 (1859) 100. , . Lerch, M. Prag CesfoS Ak. Fr. Jos. Ez. (Trida 2) 7 (1898) Art. 5, 51 pp. ; Prag Fr. Jos. Ac. Sc. Bll. (Mth. Nt.) 5 (1898) 33-. , formulae. Lerch, . Bll. Sc. Mth. 21 (1897) 290-. , Legendre's symbol and Moebius's numbers. Lerch, M. Prag Ceske Ak. Fr. Jos. Ez. (Trida 2) 7 (1898) Art. 6, 12 pp. ; Prag Fr. Jos. Ac. Sc. Bll. (Mth. Nt.) 5 (1898) 33-. relations. Gegenbauer, L. Wien Ak. Sb. 93 (1886) (Ab. 2) 288-. and modular correspondences of prime degree. Hunvitz, A. Leip. Mth. Ps. B. 37 (1885) 222-. -numbers, Dirichlet's functions F(s) = 2 (^] . i. Hurwitz,A. Z. Mth. Ps. 27 (1882) (H. -It. Ab.) 86-. Complex forms. Smith, H. J. S. E. S. P. 13 (1864) 278-. Composition. Lejeune-Dirichlet, G. Crelle J. 47 (1854) 155- ; Liouv. J. Mth. 4 (1859) 389-. . Schlfijli, L. Crelle J. 57 (1860) 170-. . Pepin, T. [1879] Em. N. Line. At. 33 (1880) 6-. . Klein, F. Gritt. Nr. (1893) 106-. . Mertens, F. Wien Ak. Sb. 104 (1895) (Ab. 2a) 103-,. . La Vallee Poussin, C. J. de. Brux. Mm. Cour. 8, 53 (1895-96) No. B, 59 , Dirichlet's metho Ofv. 35 1893 57-. 9 pp. , Dirichlet's method. Levdnen, S. Helsingf. Concordant and discordant numbers. Euler, L. [1780] St Pet. Ac. Sc. Mm. 8 (1822) 3-. ---- , Euler's view. Degen, C. F. [1823] St Pet. Mm. Sav. Etr. 1 (1831) 29-. Congruence x 2 + y*=l (mod. p). Schonemann, T. Crelle J. 19 (1839) 93-. x*-Dy'* = \ (mod. p). Frattini, G. [1884] Em. E. Ac. Line. Ed. 1 (1885) 136-. Congruence, 2nd degree, number of roots. Berger, A. Stockh. Ofv. (1887) 127- ; Fschr. Mth. (1887) 181-. Quadratic binary forms 2830 Congruences, 2nd degree, numerical solution for every simple modulus. Kunerth, A. [1878] Wien Ak. Sb. 78 (1879) (Ab. 2) 338-. , with prime modulus, resolution. Bugaev, N. V. [1881] (xn) Eec. Mth. (Moscou) 10 (1882-83) (Pt. 1) 103-. Conjugate forms, arithmetical theory. Mathews, G. B. QJ. Mth. 27 (1895) 230-. Connexion. Cunningliam, (Lt.-Col.) A. [1896] L. Mth. S. P. 28 (1897) 289-. Decomposition of number into difference of 2 squares. Sahara, A. Arch. Mth. Ps. 61 (1877) 446-. numbers into quadratic binary forms. Jonquieres, E. de. As. Fr. C. E. (1878) 40-; C. E. 87 (1878) 399- ; N. A. Mth. 17 (1878) 419-, 433-. sum of 2 squares. Kausler, C. F. [1800] St Pet. Ac. Sc. N. Acta 14 (1805) 232-. . Volpicelli, P. Palomba Eac. 5 (1849) 263, 313, 392, 402 ; Em. At. 4 (1850-51) 508-. . Muir, T. L. Mth. S. P. 8 (1876-77) 215-. . Roberts, S. L. Mth. S. P. 9 (1877-78) 187-. . Glaisher, J. W. L. QJ. Mth. 19 (1883) 212-. . McDonald, J. H. Cn. E. S. P. & T. 6 (1900) (Sect. 3) 77-. , Gauss's formulae. Volpicelli, P. Tortolini A. 1 (1850) 527-; Em. At. 4 (1850-51) 22-, 71-. , . Genocchi, A. N. A. Mth. 13 (1854) 158-. , rectification of formulae. Volpicelli, P. [1853] Tortolini A. 5 (1854) 176. product into sum of two squares. Unfer- dinger, F. Arch. Mth. Ps. 49 (1869) 116-. square into two others. Votyicelli, P. N. A. Mth. 9 (1850) 305-. Determinants, irregular, formation. Perott, J. Crelle J. Mth. 95 (1883) 232-; 96 (1884) 327-. , and subtriplicate forms. Mathews, G. B. Mess. Mth. 20 (1891) 70-. , irregularity. Lejeune-Dirichlet, G. Berl. B. (1853) 194-. Diophantine relation j/ 2 + y'~ = square. Cayley, A. L. Mth. S. P. 20 (1889) 122-. Diophantus and quadratic forms. Jacobi, C. G. J. Berl. B. (1847) 265-. Dirichlet's investigation, extension. Bach- mann, P. Mth. A. 16 (1880) 537-. " Disquisitiones Arithmeticae " (Gauss) No. 162, extension. Jonquieres, de. C. E. 127 (1898) 596-. Divisor of sum of two squares. Euler, L. [1778] St Pet. Ac. Sc. N. Acta 15 (1806) 29-. , Fermat's theorem. Catalan, E. C. Liege S. Sc. Mm. 15 (1888) No. 1, 11-. of same form. Eugenia, V. G. Mt. 8 (1870) 162-. Divisors of form y^ + Az 2 , number of classes. Jacobi, C. G. J. Crelle J. 9 (1832) 189-. numbers of form mx- + ny 2 . Euler, L. [1778] St Pe"t. Ac. Sc. Mm. 5 (1815) 3-. 206 2830 Quadratic binary forms Divisors of numbers of form x- cy 2 . Wantzel, L. Par. S. Phlm. PV. (1848) 19-. 2 . Hermite, C. Liouv. J. Mth. = d*, solution. Quadratic binary forms 2830 14 (1849) 451-. Equations. Euler, L. [1780] St Pet. Ac. Sc. Mm. 10 (1826) 3-. . x* + y* = z*. Saorgio, (le pere) . [1799] Turin Mm. Ac. 6 (1800) 239-. . x*-Ay*=l. Pezzi, F. [1806] Mod. S. It. Mm. 13 (1807) 342-. . y^-Ax' 2 -B. Tedenat,. Gergonne A. Mth. 1 (1810-11) 349-. . ay* + b = x*. Kramp, C. (vi Adds.) Ger- gonne A. Mth. 1 (1810-11) 351-. . ax-+b = y 2 , x' 2 + y' 2 = z*. Du Hays, . Liouv. J. Mth. 7 (1842) 325-. . x 2 + if=z 2 , property. Binet, J. P. M. C. E. 28 (1849) 686-, 755. . o; 2 -r7/ 2 =z 2 . Liouville,J. C. B. 28 (1849) 687. . x 2 + y*=z orz 2 . Volpicelli, P. G. Arcad. 119 (1849-50) 20-. . x* + y*=N. Bellavitis, G. Tortolini A. 1 (1850) 422-. . x* + y*=z*. Volpicelli, P. Tortolini A. 1 (1850) 369-, 443-; Em. At. 4 (1850-51) 124-, . x*-Dy z =-N. Tchebicheff, P. Liouv. J. Mth. 16 (1851) 257-. . :r 2 + 7/2=(a 2 + fc*)*. Volpicelli, P. Em. At. 5 (1851-52) 315-. . x* + y*=N. Chelini, D. Tortolini A. 3 (1852) 126-. . x*-y*=e. Volpicelli, P. Em. At. 6 (1852-53) 77-. . x z -y* = c. Volpicelli, P. Tortolini A. 6 (1855) 120-. . x 2 + h = y*, x"*-h = z 2 . Boncompagni, B. Tortolini A. 6 (1855) 135-. . x 2 -i>7/ 2 =4, Z> = 5 (mod. 8). Cayley, A. Crelle J. 53 (1857) 369-. . ax' 2 + px + y = ty 2 +ny+0. Cayley, A. N. A. Mth. 16 (1857) 161-. . x*-ny* = l. Gerono, G. C. N. A. Mth. 18 (1859) 122-, 153-. . x 2 -fgy z -l, factors. Konig, J. F. Grunert Arch. 33 (1859) 1-. . x* + y* = z 2 . Sturm, J. B. Grunert Arch. 33 (1859) 92-. . AP=x* + ay*. Besge, V. A. Liouv. J. Mth. 6 (1861) 239-. . x' 2 -Ny*=-I. Eichaud, C. Liouv. J. Mth. 9 (1864) 384-; 10 (1865) 235-; 11 (1866) 145-. . a;-4y=l. Eichaud, C. Em. At. N. Line. 19 (1866) 177-. . Ax*By*=vP. Calzolari, L. [1869] G. Mt. 8 (1870) 28-. . A+Bt* = u*. Khanikof, N. de. C. E. 69 (1869) 185-. . x' 2 + cy*=A. Petersen, J. P. C. (xn) Ts. Mth. 1 (1871) 76-. . x*-Ay*=l. Seeling, P. Arch. Mth. Ps. 52 (1871) 40-. Zolotareff, G. . . N. A. Mth. 11 (1872) 539-. Equations. ax 2 l = y*. Matthiessen, L. Z. Mth. Ps. 18 (1873) 426. . x 2 + y 2 = z' 2 , x 2 -y' 2 = u-. Destournelles, G. Le S. Fr. Cg. Sc. 40 (1874, v. 1) 167-. . t 2 -Du a =4 where D is a positive odd number and not square, solution. Schmidt, W. Z. Mth. Ps. 19 (1874) 92-. . x-+y' 2 N. Chavannes, F. L. F. Laus. S. Vd. Bll. 13 (1874-75) 474-. . x* + y z = z*. Genocchi, A. Tor. Ac. Sc. At. 11 (1876) 811-. . y*-ax* = bz. G anther, S. Liouv. J. Mth. 2 (1876) 331-. . x*-fy* = u*, x^ + &y 2 = v 2 . Lucas, E. N. A. Mth. 15 (1876) 466-. . 2 2 - 2 = w 8 and 2t> 2 + w 2 = 3z 2 . Lucas, E. N. A. Mth. 16 (1877) 409-. . x*-Ay* = u*, x* + Ay*=v*. Lucas, E. N. A. Mth. 17 (1878) 446-. . x' 2 +k = y 2 . Motta Pegado, L. P. da. G. Teix. J. Sc. 1 (1878) 150-. x . x 2 + Py* = w 2 , a; 2 - Py 2 = v-, solution. Eoberts, S. [1879] L. Mth. S. P. 11 (1879- 80) 35-. . x* - 2P;/ 2 = - z 2 or 2z 2 . Eoberts, S. L. Mth. S. P. 11 (1879-80) 83-. . ax*-y' i =-l, properties of numerical solutions. Durfee, W. P. [1881] (xn) J. H. Un. Cir. [1] (1882) 178. . X 2 +r 2 =^. Piuma, C. M. G. Mt. 19 (1881) 31 1-. . 2^-1=3;, 22 2 -l = x 2 , Fermat's theorem. Pepin, T. [1882] Em. N. Line. At. 36 (1883) 23-. . Ax"* = y 2 + l, etc. Catalan, E.G. As. Fr. C. E. 12 (1883) 98-. . 2y 2 - 1 = x, 2z 2 - 1 = z 2 , Fermat's theorem. Genocchi, A. N. A. Mth. 2 (1883) 306-. . x* + k = y*. Eealis, S. N. A. Mth. 2 (1883) 289-. . x* + nxy-ny*=l. Eealis, S. N. A. Mth. 2 (1883) 494-. . (n + 4)ar*-n0 s = 4. Eealis, S. N. A. Mth. 2 (1883) 535-. . n = x 2 + y' 2 , number of solutions, theorems on function expressing. Stieltjes, T. J. (jun.) C. E. 97 (1883) 889-. . x*-Ky* = z*. Ocagne, M. d'. C. E. 99 (1884) 1112. . a* + y=z*. Airy, G. B. Nt. 33 (1886) 532. . a; 2 + 2/ 2 =71, solutions and formulae. Berger, A. Acta Mth. 9 (1887) 301-. . t 2 - Dw 2 = - 1. Perott, J. Crelle J. Mth. 102 (1888) 185-. . axy + x*-y*=l. Studnicka, F. J. Prag Sb. (1888) (Mth.-Nt.) 92-. . a; 2 - Dy* = N, geometrical meaning. Frattini, G. Em. E. Ac. Line. Ed. 1 (1892) (Sem. 1) 51-. . x*-Dy*=-N (Cebysev). Frattini, G. Em. E. Ac. Line. Ed. 1 (1892) (Sem. 2) 85-. . a; 2 + l = 2i/ 2 , z 2 -l = 2i/ 2 . Lemoine, E. G. Teix. J. Sc. 11 (1892) 68-. . (a + 2)x^-(a- 2) y' 2 = 4, solutions. Palm- strftm, A. Berg. Ms. Aarb. (1896) No. 14, 11 pp. 207 2830 Quadratic binary forms Equations, t' 2 - Du- = - 1 . Jonquieres, de. C. B. 126 (1898) 1837-. . x 2 + y* = z*. Graeber, . Arch. Mth. Ps. 17 (1900) 36-. Equivalent substitutions. Hermes, J. [1882] Crelle J. Mth. 95 (1883) 165-. Extended system of forms. Gordan,P. Mth. A. 33 (1889) 372-. Factorisation by binary quadratic forms. LevSnen, S. Helsingf . Ofv. 34 (1892) 334- ; Fschr. Mth. (1892) 165. Form x 2 + y a . Genocchi, A. Tortolini A. 5 (1854) 491-. Forms, binary and ternary. Selling, E. [1874] Liouv. J. Mth. 3 (1877) 21-, 153-. with complex coefficients. Mathews, G. B. QJ. Mth. 25 (1891) 289-. determinant, class-number, Dirichlet's formula. Mathews, G. B. L. Mth. S. P. 23 (1892) 159-. conjugate indeterminates. Picard, E. C. B. 96 (1883) 1567-. , indefinite. Picard, E. Par. EC. Norm. A. 1 (1884) 9-. given determinant, class-number. Pepin, T. Par. EC. Norm. A. 3 (1874) 165-. negative determinant. Kronecker, L. Berl. Ak. Mb. (1875) 223-. . Hermite, . Bll. Sc. Mth. 10 (1886) 23-. . Aladow, J. S. Fschr. Mth. (1899) 201-. , class-number. Kronecker, L. Crelle J. 57 (1860) 248-. , . Gegenbauer, L. Wien Ak. Sb. 93 (1886) (Ab. 2) 54-. , . Hurwitz, A. Crelle J. Mth. 99 (1886) 165- ; Acta Mth. 19 (1895) 351-. , . Lerch, M. C. B. 121 (1895) 878-. - , , function expressing. Limtville, J. C. B. 62 (1866) 1350-. , of primitive. Hacks, J. Acta Mth. 14 (1890-91) 321-. , relations. Gierster, J. [1879-83] Mth. A. 17 (1880) 71-, 74-; 21 (1883) 1-; 22 (1883) 190-. , . Hurwitz, A, Leip. Mth. Ps. B. 36 (1884) 193- ; Mth. A. 25 (1885) 157-. , numbers. Berger, A. [1882] Ups. S. Sc. N. Acta 11 (1883) Wo. 7) 22 pp. , mean class-number. Gegen- bauer, L. Wien Ak. Sb. 92 (1886) (Ab. 2) 1807-. , property. Kronecker, L. Berl. Mb. (1862) 302-. determinants, periods. Simerka, W. Wien SB. 31 (1858) 33-. discriminant, class-number. Lerch, M. Prag Ceske Ak. Fr. Jos. Bz. (Tfida 2) 7 (1898) Art. 4, 16 pp. ; Fschr. Mth. (1898) , representation of whole numbers by. Gegenbauer, L. Wien Ak. Sb. 92 (1886) (Ab. 2) 380-. Quadratic binary forms 2830 Forms with negative discriminant, theory of elliptic functions applied to. Petr, K. Prag Ceskf* Ak. Fr. Jos. Bz. (Tfida 2) 9 (1900) Art. 38, 17 pp. ; Fschr. Mth. (1900) 212-. positive determinant. Lejeune-Dirichlet, G. Berl. Ab. (1854) (Mth.) 99-; Liouv. J. Mth. 2 (1857) 353-. . Mertens, F. Crelle J. Mth. 89 (1880) 332-. . Vivanti, J. Z. Mth. Ps. 31 (1886) 273-; 32 (1887) 287-. , equivalence of reduced. Mertens, F. Wien Ak. Sb. 103 (1894) (Ab. 2a) 995. , property. Lejeune-Dirichlet, G. Berl. B. (1855) 493-. reducible to (a, o, c), (a, ia, c), and (a, b, a). Fields, J. C. [1885] J. H. Un. Cir. [5] (1885- 86) 38. representing same numbers. Schering, E. Liouv. J. Mth. 4 (1859) 253-. , semi-definite. Rados, G. Mth. Termt. Ets. 13 (1895) 191-. Functions Y and Z satisfying 4 (XP - l)l(x - 1) = Y 2 i^Z 2 , p prime of form 4fc=tl. Rogers, L. J. L. Mth. S. P. 30 (1899) 23-. Gauss's sums. Seguier, de. C. B. 123 (1896) 166-. . Lerch, M. Casopis 28 (1899) 1- ; Fschr. Mth. (1898) 168. , analogues. Lerch, M. Prag Sb. (1897) (Mth.-Nt.) No. 43, 16 pp. n-l 22irt . Series S e~iT . Mertens, F. Berl. Ak. Sb. (1896) 217-. with several variables. Jordan, C. C. B. 73 (1871) 1316-. Geometrical representation of results. Bara- niecki, M. A. Par. T. Nauk So. Pam. 8 (*1876) Art. 9, 8 pp. Indefinite forms. Mark-off, A. Mth. A. 15 (1879) 381- ; 17 (1880) 379-. Large primes, criteria ; application of binary quadratic forms. Seelhoff, P. Am. J. Mth. 7 (1885) 264-; 8 (1886) 26-. Linear forms of the divisors of x z A. Pepin, . C. B. 123 (1896) 683-, 737-. Malmsten's functions and invariants of quad- ratic forms. Lerch, M. Prag Ceske Ak. Fr. Jos. Bz. (Tfida 2) 2 (1893) Art. 4, 12pp.; Fschr. Mth. (1893-94) 790-. Number with prime factors 4n + l, as sum of two squares. Jonquieres, E. de* N. A. Mth. 17 (1878) 241-, 289-. Numbers whose squares are sums of two squares. Egidi, G. Bm. N. Line. At. 50 (1897) 103-, 126-. Numerical forms. Thiele, T. N. (xn) Sk. Nt. Mrfd. F. (1873) 192-. Odd forms. Liouville, J. Liouv. J. Mth. 11 (1866) 221-. Pellian equation. Frischauf, J. Wien Sb. 55 (1867) (Ab. 2) 121-. . Roberts, S. L. Mth. S. P. 15 ( 1883- 84) 247-. 208 2830 Quadratic binary forms Pellian equation. Aubel, H. van. As. Fr. C. E. (1885) (Pt. 2) 135-. . Frattini, G. G. Mt. 33 (1895) 371- ; 34 (1896) 98-. and connected problems. Oettinger, L. Arch. Mth. Ps. 49 (1869) 193-. , and connexion with rational triangles. Muller, E. Arch. Mth. Ps. 5 (1887) 111-. , correction of error in Barlow. Martin, Art. Des Moines Anal. 2 (1875) 140-. , extension of Degen's tables. Brit. Ass. Comm. (Cayley). B. A. Bp. (1893) 73-. , property. Meyer, A. Zur. Vjschr. 32 (1887) 363-. , . Stflrmer, C. N. Ts. Mth. 7 (B) (1896) 49-; Fschr. Mth. (1896) 166. and quadratic binary forms of positive determinant. Smith, H. J. S. L. Mth. S. P. 7 (1875-76) 196-. , solution. Luce, J. B. Silliman J. 8 (1849) 55-. , . Minnigerode, B. Gott. Nr. (1873) 619-. , by elliptic functions. Kronecker, L. Berl. Mb. (1863) 44-. , theorems. Lejeune-Dirichlet, G. Berl. Ab. (1834) 648-. , - (Dirichlet). Tarn, F. Crelle J. Mth. 105 (1889) 160-. , . Stfrmer, C. Christiania Skr. (Mth. Nt. Kl.) (1897) No. 2, 48 pp. Pell's equation y*-3x z = l. Boutin, A. Mathesis 18 (1898) 159-. Prime factors of numbers of form l + 4x 2 . Markov, A. St Pet. Ac. Sc. Bll. 3 (1895) 55-; Fschr. Mth. (1895) 201. A+x*. Ivanov, I. St Pet. Ac. Sc. Bll. 3 (1895) 361- ; Fschr. Mth. (1895) 201. number 4n + 1 is sum of two squares. Her- mite, C. N. A. Mth. 12 (1853) 45-. . Gopel,A. N. A. Mth. 12 (1853) 136-, 170-. . Harmuth, T. Arch. Mth. Ps. 66 (1881) 327-. . Realis, S. N. A. Mth. 4 (1885) 367-. . Bock,J.W. Hamb. Mth. Gs. Mt. 1 (1889) 101-. numbers expressible by mx 2 + ny*. Euler, L. [1778] St Pet. Ac. Sc. N. Acta 12 (1801) of form A 2 + \B* (B being odd). Liou- ville, J. Liouv. J. Mth. 10 (1865) 281-, 285-, 289-, 293-, 295-. 4^ 2 + 5 2 (A being odd). Liouville, J. Liouv. J. Mth. 11 (1866) 41-. v m 2 +fcn 2 . Kiipper, K. Casopis 10 (*1881) 10-; Fschr. Mth. (1881) 135. as sum of two squares. Harmuth, T. [1881] Arch. Mth. Ps. 67 (1882) 215-. Primitive forms, whose minus determinant is >0 and =3 (mod. 8). Liouville, J. Liouv. J. Mth. 11 (1866) 191-. Problem. Miller, T. H. Edinb. Mth. S. P. 9 (1891) 23-. Pythagorean numbers, generalisation. Ziige, . Arch. Mth. Ps. 17 (1900) 354-. Other quadratic forms 2840 Quadratic forms and certain ' ' greatest integer ' ' formulae. Lerch, M. Prag fieske" Ak. Fr. Jos. Bz. (Tfida 2) 7 (1898) Art. 7, 8 pp. ; Prag Fr. Jos. Ac. Sc. Bll. (Mth. Nt.) 5 (1898) 33-. -- giving primes. Oltramare, G. Gen. I. Nt. Mm. 5 (1857) (pte. 2) 1-. -- and special automorphic groups. Fricke, R. Mth. A. 39 (1891) 62-. moduli, forms. Young, J. R. Ph. Mg. 33 (1848) 45-. Eeduction. Hurwitz, A. Mth. A. 45 (1894) 85-. , continual. Picard, E. C. K. 96 (1883) 1779-. SerieS Z Asymptotic value. Mertens, F. Wien Ak. Sb. 106 (1897) (Ab. 2a) Sums of squares equal to a square, method of finding. Euler, L. St P6t. Ac. Sc. Mm. 11 (1830) 12-. Theorem of Dirichlet. Cayley, A. Camb. and Dubl. Mth. J. 9 (1854) 163-. --- . Meyer, A. Crelle J. Mth. 103 (1888) 98-. --- . Mertens, F. Wien Ak. Sb. 109 (1900) (Ab. 2a) 415-. Gauss, demonstration. Perott, J. de. [1889] J. H. Un. Cir. [9 (1889-90)] 30. 2840 Quadratic forms of three or more variables; bilinear forms. (See also 2060, 2070.) Bouniakowsky, V. [1849] St Pet. Ac. Sc. Mm. 7 (1853) 303 -. Arndt, F. Grunert Arch. 13 (1849) 105-, 410- ; 15 (1850) 429-. Hermite, C. Crelle J. 47 (1854) 313-, 343-. Liouville, J. Liouv. J. Mth. 2 (1857) 424. Lipschitz, R. Crelle J. 54 (1857) 193-. Hermite, C. C. B. 55 (1862) 684-. Skrivan, G. Grunert Arch. 38 (1862) 259-. Bachmann, P. Z. Mth. Ps. 16 (1871) 181-. Siacci, F. Em. At. E. Ac. 25 (1872) 339-. Bachmann, P. Crelle J. Mth. 76 (1873) 331-. Korkine, A.,<& Zolotareff, G. Mth. A. 6 (1873) 366-. Frobenius, G. C. E. 85 (1877) 131-. Jordan, C. Par. EC. Pol. J. Cah. 51 (1882) 1-. Minkowski, H. Acta Mth. 7 (1885) 201-. Algorithm for treatment. Hermes, J. [1881] Arch. Mth. Ps. 68 (1882) 432-. Application to complex variables. Preobra- schensky, P. W. Fschr. Mth. (1898) 176. Composition. Bazin, . Liouv. J. Mth. 16 (1851) 161-. . Arndt, F. Crelle J. 56 (1859) 64-. Decomposition of numbers into sums of squares. Malfatti, G. F. Mod. S. It. Mm. 12 (1805) 296-. Plana, G. [1859] Tor. Mm. Ac. 20 (1863) 113-. 209 2840 Quadratic Forms of three or more Variables ; Bilinear Forms 2840 Liouville, J. Liouv. J. Mth. 6 (1861) 233-; 13 (1868) 1-. Glaisher, J. W. L. Ph. Mg. 1 (1876) 44-. Catalan, E. C. Em. N. Line. At. 35 (1882) 103- ; 37 (1884) 49-. Glaisher, J. W. L. QJ. Mth. 20 (1885) 80-. Matrot, A. As. Fr. C. E. (1890) (Ft. 2) 79-. Catalan, E. Brux. Ac. Bll. 27 (1894) 10-. Lemoine, E. As. Fr. C. E. (1896) (Pt. 2) 73-. 2 or 3 squares. Catalan, E. C. As. Fr. C. E. 12 (1883) 98-. 3 squares. Lejeune-Dirichlet, G. Liouv. J. Mth. 4 (1859) 233-. -- . Liouville, J. Liouv. J. Mth. 5 (1860) 141-. 4 squares. Jacobi, C. G. J. Crelle J. 3 (1828) 191 ; 12 (1834) 167-. -- . Hermite, C. C. E. 37 (1853) 133-. -- . Lebesgue, V. A. Liouv. J. Mth. 2 (1857) 149-. -- . Sylvester, J. J. QJ. Mth. 1 (1857) 196-. -- . Liouville, J. Liouv. J. Mth. 3 (1858) 357-. -- . Zehfuss, G. Grunert Arch. 30 (1858) 465-. -- . Liouville, J. Liouv. J. Mth. 13 (1868) 1-. . Glaisher, J. W. L. QJ. Mth. 19 (1883) 212-. -- . Weill, . Par. S. Mth. Bll. 13 (1885) 28-. -- , Liouville' s theorem (1858). Stern, M. A. Crelle J. Mth. 105 (1889) 250-. -- , relations among roots. Pollock, (Sir) F. [1858] (vi Adds.) Phil. Trans. (1859) 49-. 5 squares. Eisenstein, G. Crelle J. 35 (1847) 368-. . Pepin, T. [1883] Em. N. Line. At. 37 (1884) 9-. -- . Stieltjes, T. J. (jun.) C. E. 97 (1883) 981-, 1545-. -- . Hurwitz, A. C. E. 98 (1884) 504-. -- . Smith, H. J. S. Par. Mm. Sav. Etr. 29 (1887) No. 1, 72 pp. 10 squares. Liouville, J. Liouv. J. Mth. 11 (1866) 1-. 12 squares. Liouville, J. Liouv. J. Mth. 5 (1860) 143- ; 9 (1864) 296-. Catalan's theorem, corollary. Moureaux, M. C. E. 118 (1894) 700-. Dirichlet's method. Kronecker, L. Berl. Mb. (1864) 285-. Division of prime mn + l into m squares. Lebesgue, V. A. C. E. 39 (1854) 593-. Equations, 2nd order, 3 variables. Poletti, G. [1826] Tor. Mm. Ac. 31 (1827) 409-. , , . Gerono, G. C. N. A. Mth. 15 (1856) 322-, 386-. , . , 5 variables. Cockle, Jas. Ph. Mg. 1 (1851) 413-. , , n variables. Desboves, . N. A. Mth. 3 (1884) 225-. . a; 2 + y* = z* + t*. Lebesgue, V. A. N. A Mth. 7 (1848) 37-. . t 2 + w 4 + t> 2 + w 2 =4m. Lejeune-Dirichlet, G. Liouv. J. Mth. 1 (1856) 210-. Equations. /= ax* + a'x' 2 + a"x" 2 + 2bx'x" + 2b'xx" + 2b"x'x = 0. Smith, H. J. S. E. S. P. 13 (1864) 110-. _. t t = x* + y* + z*. Le Besgue, V. A. C. E. 66 (1868) 396-. . t/ 2 = x^ + x 2 z + . . . + x n 2 . Calzolari, L. G. Mt. 7 (1869) 313-. . y = x 2 + (x + 1) 2 and t/ 2 = z 2 + (z + 1) 2 . Jon- quieres, E. de. N. A. Mth. 17 (1878) 241-, 289-. . 2t> 2 =M 2 + t 2 , 2w? 2 =* 2 + 2u 2 . Pepin, T. Em. N. Line. At. 32 (1879) 281-. . ax* + by* + cz z + dv?=0. Meyer, A. Ziir. Vjschr. 29 (1884) 209-. . ax* + by- + cz' 2 + du> + ev' 2 = 0. Meyer, A . Ziir. Vjschr. 29 (1884) 220-. Every primitive quadratic form can represent infinite number of prime numbers. Weber, H. Mth. A. 20 (1882) 301-. Forms with complex coefficients and variables. Lejeune-Dirichlet, G. Crelle J. 24 (1842) 291-. . Bianchi, L. Em. E. Ac. Line. Ed. 5 (1889) (San. 1) 589-. , classification. Minnigerode, B. Gott. Nr. (1873) 160-. conjugate variables. Picard, E. Mth. A. 39 (1891) 142-. integral coefficients. Minkowski, . Par. Mm. Sav. Etr. 29 (1887) No. 2, 180 pp. more than 3 variables, orders and genera. Smith, H. J. S. E. S. P. 13 (1864) 199- ; 16 (1868) 197-. Formulae. Liouville, J. Liouv. J. Mth. 18 (1873) 142-. .system. Cayley,A. Crelle J. 39 (1850) 14-. Gauss's formula 4 =Y 2 -pZ 2 , theorem derived from. Catalan, E. C. Brux. Ac. Bll. 6 (1883) 264. Genus, extension of arithmetical notion. Pom- care, H. C. E. 94 (1882) 67-, 124-. Geometric representation, new method. Pom- care, H. Par. EC. Pol. J. 28 (=Cah. 47) (1880) 177-. Hermite's forms. Loewy, A. C. E. 123 (1896) 168-. (Loewy). Fuchs, L. C. E. 123 (1896) 289-. Integral power of sum of 3 squares is sum of 3 squares. Catalan, E. C. Brux. Ac. Bll. 6 (1883) 264. Multiplication. Genocchi, A. Tortolini A. 3 (1860) 202-. n-ary forms, indefinite. Meyer, A. Ziir. Vjschr. 36 (1891) 241-. Non-homogeneous forms and quadratic surfaces, classification. Hensel, K. Crelle J. Mth. 113 (1894) 303-. Partitions of any integer into sum of 10 tri- angular numbers. Liouville, J. C. E. 62 (1866) 771-. Positive forms. Korkine, A., d- Zolotareff, G. Mth. A. 11 (1877) 242-. . Minkowski, H. Crelle J. Mth. 99 (1886) 1- ; 101 (1887) 196- ; 107 (1891) 278-. 210 2840 Quadratic Forms of three or more Variables ; Bilinear Forms 2840 Prime number, sum of four squares, demon- strations of theorem. Catalan, E. C. Li^ge S. Sc. Mm. 15 (1888) No. 1, 159-, 259. Primes of form 8n + 3, In + 2, 7 + 4, quadratic resolution. Eisenstein, G. Crelle J. 37 (1848) 371 + 1, quadratic resolution. Stieltjes, T. J. (jun.) Amst. Ak. Vs. M. 19 (1884) 105- ; Toul. Fac. Sc. A. 11 (1897) D, 6 pp. Products of sums of squares. Degen, C. F. St Pet. Ac. Sc. Mm. 8 (1817-18) 207-. . Graves, J. T. Ir. Ac. P. 3 (1847) 527-. . Cayley,A. Ph. Mg. 4 (1852) 515-. . Liouville, J. Liouv. J. Mth. 2 (1857) 351-. . Souillart, C. N. A. Mth. 19 (1860) 320-. . Mathieu, E. [1861] Crelle J. 60 (1862) 351-. . Genocchi,A. G. Mt. 2 (1864) 47-. . Unferdinger, F. Wien Sb. 59 (1869) (Ab. 2) 455-. . Thomson, J. J. [1877] Mess. Mth. 7 (1878) 73-. . Studnicka, F. J. Prag. Sb. (1883) (Mth. Vortr.) 475-. . Antomari,X. C. E. 104 (1887) 566-. , Euler's theorem. Young, J. E. Ph. Mg. 30 (1847) 424-; 31 (1847) 123; Ir. Ac. T. 21 (1848) 311- ; Ir. Ac. P. 4 (1850) 19-. , . Roberts, S. QJ. Mth. 16 (1879) 159- ; 17 (1881) 276-. Properties. Poincare, H. C. E. 89 (1879) 344-, 897-. , with tables. Pollock, (Sir) F. [1853] Phil. Trans. (1854) 311-. Quadratic or bilinear forms, simultaneous transformation of two. Lipschitz, R. Berl. Ak. Sb. (1890) 485-. form of irreducible factors of binomial equations. Trudi,N. A. Mt. 2 (1868-.69) 150-. and linear form, simultaneous reduction. Poincare, H. C. E. 91 (1880) 844-; Par. EC. Pol. J. 56 (1886) 79-. forms and group-theory. Lipschitz, R. Acta Mth. 10 (1887) 137-. of prime numbers and their powers. Cauchy, A. L. C. E. 9 (1839) 473-, 519- ; 10 (1840) 51-, 85-, 181-, 229-. and linear forms of prime numbers. Oltra- mare, G. Crelle J. 49 (1855) 151-. representations, theorem of Euler, Brioschi and Genocchi. Puchta, A. Wien Ak. Sb. 96 (1888) (Ab. 2) 110-. Quaternary forms. Bazin, . Liouv. J. Mth. 19 (1854) 215-. . Pepin, . Liouv. J. Mth. 6 (1890) 5-. , positive. Korkine, A., & Zolotareff, G. Mth. A. 5 (1872) 581-. , reduced, determinant <20, table. Charve, L. C. E. 96 (1883) 773-. , , reduction. Charve, L. Par. EC. Norm. A. 11 (1882) 119-. Eeduction, with proof of Seeber's theorem on reduced ternary forms. Lebesgue, V. A. Liouv. J. Mth. 1 (1856) 401-. Eepresentations of integer by certain forms, number. Gegenbauer, L. Wien Ak. Sb. 103 (1894) (Ab. 2a) 115-. sum of definite multiples of squares, mean number. Gegenbauer, L. Wien Ak. Sb. 93 (1886) (Ab. 2) 215-. S(a,--a,-) 2 expressed as sum of n squares. Glaisher, J. W. L. [1878] Mess. Mth. 8 (1879) 48. Square numbers, problem. Hart, D. S. Des Moines Anal. 3 (1876) 81-. as sums of squares. Avillez, J. F. tf (Visconde de Reguengo). Lisb. J. Sc. Mth. 5 (1898) 90-. , sums, theorem. Studnicka, F. J. Prag Sb. (1894) (Mth.-Nt.) No. 15, 8 pp. of odd number is difference of two triangular numbers, generalisations. Henry, C. N. A. Mth. 19 (1880) 517-. Sums of three squares. Catalan, E. C. Em. N. Line. At. 34 (1881) 63-, 135-. Ternary forms. Hermite, C. Crelle J. 40 (1850) 173-. . Bachmann, P. Crelle J. 70 (1869) 365-. . Picard, a. C. E. 94 (1882) 1241-. with complex coefficients. Fricke, R. Gott. Nr. (1895) 11-. different determinants, comparison. Eisenstein, G. Berl. B. (1852) 350-. , indefinite. Hermite, C. Crelle J. 47 (1854) 307-. , . Meyer, A. Crelle J. Mth. 108 (1891) 125- ; 113 (1894) 186- ; 114 (1895) 233-; 115 (1895) 150- ; 116 (1896) 307-. , orders and genera. Smith, H. J. S. Phil. Trans. 157 (1867) 255-. , positive. Markov, V. A. Kharkov Mth. S. Com. 4 (1895) 1-; Fschr. Mth. (1893-94) 321-. , , criticism of Seeber. Anon, (vi 311) Crelle J. 20 (1840) 312-. , , reduction. Lejeune-Dirichlet, G. Berl. B. (1848) 285-; Crelle J. 40 (1850) 209-. , , . Charve, L. Par. EC. Norm. A. 9 (1880) (Suppl.) 156 pp. , , . Minkowski, H. C. E. 96 (1883) 1205-. , , theorems. Eisenstein, G. Crelle J. 35 (1847) 117-. , reduction. Hermite, C. [1874] Crelle J. Mth. 79 (1875) 17-. , transformation. Bachmann, P. [1869] Crelle J. 71 (1870) 296-. (3) = squares, problem. Matteson, J. Des Moines Anal. 2 (1875) 46-. and quaternary forms, indefinite. Fricke, R. Gott. Nr. (1893) 705-. Theorem. Brioschi, F. Tortolini A. 5 (1854) 201-. . Lejeune-Dirichlet, G. Liouv. J. Mth. 2 (1857) 273-. . Andre, D. Par. S. Mth. Bll. 15 (1887) 188-. Theorems. Lipschitz, R. Crelle J. 53 (1857) 238-. Theory of numbers, especially quadratic forms. Hermite, C. Crelle J. 40 (1850) 261-. 211 o2 2840 Quadratic Forms of three or more Variables; Bilinear Forms 2840 Transformation of linear forms of primes into quadratic forms. Oltramare, G. Gen. I. Nt. Mm. 14 (1879) 66 pp. sum of 2 or 3 squares into itself by real substitution. Lipschitz, . Liouv. J. Mth. 2 (1886) 373-. two quadratic forms into each other. Minkowski, H. Crelle J. Mth. 106 (1890) 5-. Triangles, right-angled, series. Pagnini, G. M. Firenze Opusc. Sc. 3 (1807) 3-. o;2 + 3,2 + 2 2 + (2, a; 2 + 2t/ 2 + 3z- + 6t 2 . Liouville, J. Liouv. J. Mth. 10 (1845) 169-. a; 2 + ay 2 + fcz 2 + abt z . Liouville, J. Liouv. J. Mth. 1 (1856) 230. a;2 + 2,2 + 5(22 + (2). Liouville, J. Liouv. J. Mth. 4 (1859) 47-. x 2 + 2,2 + 3(22 + (2). Liouville, J. Liouv. J. Mth. 5 (I860) 147-. a; 2 + 7/2 + 2(22 + (2). Liouville, J. Liouv. J. Mth. 5 (I860) 269-, 305-. x 2 + 3t/ 2 + 4z 2 + 12t 2 . Liouville, J. Liouv. J. Mth. 6 (1861) 135-. #2 + y2 + 2 2 + 2t 2 , x 2 + 2 (y 2 + z 2 + 1 2 ) . Liouville, J. Liouv. J. Mth. 6 (1861) 225-. X 2 + Y 2 + Z 2 + 8T 2 . Liouville, J. Liouv. J. Mth. 6 (1861) 324-. z 2 + 2j/ 2 + 4z 2 + 8t 2 . Liouville, J. Liouv. J. Mth. 6 (1861) 409-. 40-. Liouv. J. ville, J. Liouv. J. Mth. 6 (1861) 440-. X* + 2Y' 2 + 22T 2 + 4T 2 . Liouville, J. Mth. 7 (1862) 1-. X* + 8 (Y 2 + Z 2 + T 2 ) . Liouville, J. Liouv. J. Mth. 7 (1862) 5-. X^ + 4 Y 2 + 4.Z 2 + 8 T 2 . Liouville, J. Liouv. J. Mth. 7 (1862) 9-. X 2 + 8r 2 + 8. 2 + 16T 2 . Liouville, J. Liouv. J. Mth. 7 (1862) 13-. a; 2 + 2y- + 4z 2 + 4t 2 . Liouville, J. Liouv. J. Mth. 7 (1862) 62-. x 2 + 22,2 + 82 2 + 8 (2. Liouville, J. Liouv. J. Mth. 7 (1862) 65-. x 2 + 82/ 2 + 16z 2 + 16t 2 . Liouville, J. Liouv. J. Mth. 7 (1862) 69-. a; 2 + 4z/ 2 + 4z 2 + 16t 2 . Liouville, J. Liouv. J. Mth. 7 (1862) 73-. Liouville, J. Liouv. J. Mth. 7 (1862) 77-. z' 2 + 4t 2 . 7 (1862) 99-. +fl*+4*+8& 7 (1862) 103 T . Liouville, J. Liouv. J. Mth. Liouville, J. Liouv. J. Mth. Liouville, J. Liouv. J. Mth. 7 (1862) 105-. a;2+2,2 + 8 22 + 8 (2. Liouville, J. Liouv. J. Mth. 7 (1862) 109-. x 2 + 4j,2 + 8 22 + 8 (2. Liouville, J. Liouv. J. Mth. 7 (1862) 113-. a;2+2,2 + i62 2 + 16f' ! . Liouville, J. Liouv. J. Mth. 7 (1862) 117-. 2 +4i/ 2 + 8z 2 + 16 2 . Liouville, J. Liouv. J. Mth. 7 (1862) 143-. x 2 + 2t/ 2 -t-16z 2 + 16f 2 Liouville, J. Liouv. J. Mth. 7 (1862) 145-. a; 2 + 2j/ 2 + 2z' 2 + 8f 2 . Liouville, J. Liouv. J. Mth. 7 (1862) 148-. x 2 + 2t/ 2 + 4z 2 + 16t 2 . Liouville, J. Liouv. J Mth. 7 (1862) 150-. 2 . Liouville, J. Liouv. J. Mth. 7 (1862) 153-. 2 + 2,2 + 2z 2 + 8 2 . Liouville, J, Liouv. J. Mth. 7 (1862) 155-. a; 2 +t/ 2 + 4z 2 + 16t 2 . Liouville, J. Liouv. J. Mth. 7 (1862) 157-. a; 2 + 2t/ 2 -t-2z 2 + 16t 2 . Liouville, J. Liouv. J. Mth. 7 (1862) 161-. a; 2 + t/ 2 + z' 2 + 16t 2 . Liouville, J. Liouv. J. Mth. 7 (1862) 165-. a: 2 + j/ 2 + 8z 2 + 16i 2 . Liouville, J. Liouv. J. Mth. 7 (1862) 201-. x 2 -+ y' 2 + 2z 2 + 16( 2 . Liouville, J. Liouv. J. Mth. 7 (1862) 205-. a: 2 + 8z/ 2 + 8z 2 + 64t 2 . Liouville, J. Liouv. J. Mth. 7 (1862) 246-. a; 2 + 8y 2 -l-16z 2 + 64t 2 . Liouville, J. Liouv. J. Mth. 7 (1862) 249-. x2 + 2,2 + 2 2 + 3(2. Liouville, J. Liouv. J. Mth. 8 (1863) 105-. a; 2 + t/ 2 + 2z 2 + 2z< + 2t 2 . Liouville, J. Liouv. J. Mth. 8 (1863) 115-. a; 2 + 7/ 2 + z 2 + zt + t 2 . Liouville, J. Liouv. J. Mth. 8 (1863) 120-. a; 2 + y 2 + 2z 2 + 6t !! . Liouville, J. Liouv. J. Mth. 8 (1863) 124-. x 2 + y' z + 2z 2 + 3f 2 . Liouville, J. Liouv. J. Mth. 8 (1863) 129-. a; 2 + 2y 2 + 4z 2 + 6t 2 . Liouville, J. Liouv. J. Mth. 8 (1863) 134-. a; 2 + a;i/ + 3/ 2 + z 2 + zt + t 2 . Liouville, J. Liouv. J. Mth. 8 (1863) 141-. a; 2 + 2,2 + 22 + 12 (2. Liouville, J. Liouv. J. Mth. 8 (1863) 161-. x 2 + 2j/ 2 + 2z 2 +12t 2 . Liouville, J. Liouv. J. Mth. 8 (1863) 169-. a; 2 + 1/ 2 + 4z 2 + 12t 2 . Liouville, J. Liouv. J. Mth. 8 (1863) 173-. .2 + 42,2 + 422 + 12(2. Liouville, J. Liouv. J. Mth. 8 (1863) 177-. 3a; 2 + 4y 2 + 4z 2 + 4t 2 . Liouville, J. Liouv. J. Mth. 8 (1863) 179-. x 2 + 1/ 2 + 3z 2 + 4( 2 . Liouville, J. Liouv. J. Mth. 8 (1863) 182-. x 2 + 3z/ 2 + 4z 2 + 4t 2 . Li&uville, J. Liouv. J. Mth. 8 (1863) 185-. 2a; 2 + 2z/ 2 + 3z 2 + 4' 2 - Liouville, J. Liouv. J. Mth. 8 (1863) 189-. x 2 + 2,2 + Z 2 + 3(2. Liouville, J. Liouv. J. Mth. 8 (1863) 193-. x 2 + 4 2,2 + i22-J+i6( 2 . Liouville, J. Liouv. J. Mth. 8 (1863) 205-. 2 . Liouville, J. Liouv. J. Mth. 8 (1863) 209-. c- + 3w 2 + 3z 2 + 6< a . Liouville, J. Liouv. J. Mth. 8 (1863) 214-. ). Liouville, J. Liouv. J. Mth. 8 (1863) 219-. ~ ' "" "). Liouville, J. Liouv. J. Mth. 8 (1863) 225-. a; 2 + xj/ + y 2 + 3 (z 2 + 1 2 ) . Liouville, J". Liouv. J. Mth. 8 (1863) 227-. 3x 2 + 3?/ 2 + 3z 2 + 4t 2 . Liouville, J. Liouv. J. Mth. 8 (1863) 229-. 3x 2 + 3z/ 2 + 4z 2 + 12 2 . Liouville, J. Liouv. J. Mth. 8 (1863) 239-. 3^2 + 4^2 + 12 2 2 + 12(2. Liouville, J. Liouv. J. Mth. 8 (1863) 241-. 212 2840 Quadratic Forms of three or more Variables ; Bilinear Forms 2840 12t 2 . Limiville, J. Liouv. J. Mth. 8 (1863) 243-. x* + 3y 2 +12z 2 +l2t*. Liouville, J. Liouv. J. Mth. 8 (1863) 249-. x 2 + 12y 2 + I2z 2 + I2t 2 . Limiville, J. Liouv. J. Mth. 8 (1863) 253-. 3x 2 + 4i/ 2 + 12z 2 + 48f 2 . Limiville, J. Liouv. J. Mth. 8 (1863) 255-. x* + y* + 3(z* + tft. Limiville, J. Liouv. J. Mth. 8 (1863) 296. x 2 + xy + y 2 + 2z 2 + 2zt + 2t 2 . Liouville, J. Liouv. J. Mth. 8 (1863) 308-. x z + y* + z- + 5t 2 . Liouville, J. Liouv. J. Mth. 9 (1864) 1-. x 2 + y 2 + 2z 2 + 2zt + 3t 2 . Liouville, J. Liouv. J. Mth. 9 (1864) 13-. o;2 + 5 fa* + Z 2 + 4 i) . Liouville, J. Liouv. J. Mth. 9 (1864) 17-. 2z 2 + 2zj/ + 3j/ 2 + 5z 2 + 5t 2 . Liouville, J. Liouv. J. Mth. 9 (1864) 23-. x 2 + y 2 + z 2 + t 2 + u 2 + 3v 2 . Liouville, J. Liouv. J. Mth. 9 (1864) 89-. x 2 + 3(y- + z 2 + t 2 + u 2 + v 2 ). Liouville, J. Liouv. J. Mth. 9 (1864) 105-. x 2 + y 2 + z* + t 2 + 2u* + 2uv + 2v*. Liouville, J. Liouv. J. Mth. 9 (1864) 115-. 2a; 2 + 2xy + 2y* + 3 (z 2 + t 2 + u 2 + v 2 ) . Liouville, J. Liouv. J. Mth. 9 (1864) 119-. x 2 + y* + 2z' 2 + 2zt + 2t 2 +3u i + 3v 2 . Liouville, J. Liouv. J. Mth. 9 (1864) 123-. x* + 2y 2 +2yz + 2z 2 + 3t 2 . Liouville, J. Liouv. J. Mth. 9 (1864) 160. x 2 + y 2 + z 2 + t 2 + u 2 + 2v 2 . Liouville, J. Liouv. J. Mth. 9 (1864) 161-. x 2 + 2 (y 2 + z 2 + t 2 + u 2 + v 2 ) . Liouville, J. Liouv. J. Mth. 9 (1864) 175-. x 2 + xy + y 2 + 6z 2 + 6zt + et 2 . Liouville, J. Liouv. J. Mth. 9 (1864) 181-. 2.-c 2 + 2xy + 2y 2 + 32 2 + 3zt + 3t 2 . Liouville, J, Liouv. J. Mth. 9 (1864) 183-. x 2 + xy + y 2 + 3z 2 + 3zt + 3t 2 . Liouville, J. Liouv. J. Mth. 9 (1864) 223-. x 2 + y 2 + z 2 + t 2 + 2(u 2 + v 2 ). Liouville, J. Liouv. J. Mth. 9 (1864) 257-. x' 2 + y 2 + 2(z 2 + t 2 + u 2 + v 2 ). Liouville, J. Liouv. J. Mth. 9 (1864) 273-. **+aw+8**+6l. Liouville, J. Liouv. J. Mth. 9 (1864) 299-. x 2 + 2(?/ 2 + 2 2 + t 2 + w 2 ) + 4t? 2 . Liouville, J. Liouv. J. Mth. 9 (1864) 421-. x' 2 + y 2 + bz 2 + 5t 2 . Liouville, J. Liouv. J. Mth. 10 (1865) 1-. 2a; 2 + 2xy + 3y 2 + 22 2 + 2zt + Bt 2 . Liouville, J. Liouv. J. Mth. 10 (1865) 9-. z 2 + ?/ 2 + 9z 2 + 9t 2 . Liouville, J. Liouv. J. Mth. 10 (1865) 14-. 2x' 2 + 2xy + 5y 2 + 2* 2 + 2zt + 5t 2 . Liouville, J. Liouv. J. Mth. 10 (1865) 21-. X 2 + y 2 + a i z i + t 2) ^ Liouville, J. Liouv. J. Mth. 10 (1865) 43-. x 2 -f 2y' 2 + az 2 + 2at 2 . Liouville, J. Liouv. J. Mth. 10 (1865) 49-. ar + 4W 2 + 4z 2 + 4t 2 + 4u 2 + 4u 2 . Liouville, J. Liouv. J. Mth. 10 (1865) 65-. 2-1.^2-)- 4^2 + 4f^ + 4w 2 + 4w 2 . Liouville, J. Liouv. J. Mth. 10 (1865) 71-. X 2 + 2 y 2 + 2 2 2 + 4^2 a. 4 M 2 + 4^2. Liouville, J. Liouv. J. Mth. 10 (1865) 73-. 4w 2 + 4t; 2 . Liouville, J. Liouv. J. Mth. 10 (1865) 77-. x 2 + y 2 + 2z* + 2t 2 + 4u 2 + 4v 2 . Liouville, J. Liouv. J. Mth. 10 (1865) 145-. x 2 + 1/ 2 + z 2 + 1 2 + 4w 2 + 4v 2 . Liouville , J. Liouv. J. Mth. 10 (1865) 151-. x 2 + i/ 2 + 2 2 + 2( 2 + 2w 2 + 4v 2 . Liouville, J. Liouv. J. Mth. 10 (1865) 155-. 2 +2/ 2 + 2 2 + 2 + w 2 + 4i> 2 . Liouville, J. Liouv. J. Mth. 10 (1865) 161-. xZ + lyZ+W + W + iut + lGv*. Liouville, J. Liouv. J. Mth. 10 (1865) 203-. * 2 +2/ 2 + 6z 2 + 6t 2 , 2x- + 2y'* + 3z* + St 2 . Liou- ville, J. Liouv. J. Mth. 10 (1865) 359-. Liouville, J. Liouv. J. Mth. 11 (1866) 39-. 2x 2 + 2xy + 3y 2 +15z Liouville, J. Liouv. J. Mth. 11 (1866) 103-. x 2 + 2y 2 + 2yz + 2z 2 + 6t 2 , x 2 + 2y 2 + Bz 2 + 3t 2 . Liouville, J. Liouv. J. Mth. 11 (1866) 131-. x 2 + By 2 + az 2 + Bat 2 . Liouville, J. Liouv. J. Mth. 11 (1866) 211-. 2x 2 + By 2 + 4-z 2 + Sylvester, J. J. Ph. Mg. 31 (1847) 467-. Higher Irresolvable Forms 2860 Equations, x 3 + y 3 + z 3 + u 3 = 0. Grassmann, H . Arch. Mth. Ps. 49 (1869) 49-. x*+y*=z 3 + u s . Hermite,C. N. A. Mth. 11 (1872) 5-. . x 3 =i/ 2 + 17. Gerono, C. C. N. A. Mth. 16 (1877) 325-. . ax* + by*=cz 2 . Desboves, A. C. E. 87 (1878) 522-, 598-, 925. . x s a = 2/ 2 . Jonquie.res,E. de. N. A. Mth. 17 (1878) 374-, 514-. - - . . Lucas, E. As. , -. - ( \ P Fr. C. B. (1878) 164-. . X 3 +Y 3 =AZ 3 . Lucas, E. N. A. Mth. 17 (1878) 425-. Desboves, A. C. B. 88 (1879) 638-. . Ax* + By*=Cz*. Lucas, E. N. A. Mth. 18 (1879) 67-. m Desboves, A. N. A. Mth. 18 (1879) 265-, 398-, 433-, 481-. . 7.r 4 -5i/*=2 Z 2 . Pepin,T. Liouv. J. Mth. 5 (1879) 405-. _. U n V n = S n +W n , solution in integers, real or complex. Desboves, A. As. Fr. C. B. 9 (1880) 239-. . x 3 + y s =Az 3 . Lucas, E. N. A. Mth. 19 (1880) 206-. . ax* + by* = z- 2 . Pepin, T. C. E. 91 (1880) 100- ; 94 (1882) 122-. . x n + y n + z n =U n . Schier,0. [1880] Wien Ak. Sb. 82 (1881) (Ab. 2) 883-. . x* + y 3 = Az 3 . Pepin, T. Em. N. Line. At. 34 (1881) 73-. . ax t + by t =cz*. Pepin, T. [1882] Em. N. Line. At. 36 (1883) 34-. . aX 4 -6Y 4 = 2Z 2 . Desboves, A. As. Fr. C. B. (1887) (Pt. 1) 175. . aX* + bY i = cZ 2 . Desboves, A. C. B. 104 (1887) 846-, 1602-. . aX* + bY*=cZ*, andaX 4 + bY* + dX 2 Y*= cZ*. Desboves, A. C. B. 104 (1887) 1832-. . U* + F 4 = S 4 + W*. Fauquembergue, E. Mathesis 9 (1889) 241-. . z 2 + cF 2 = 2. Pepin, T. Em. N. Line. Mm. 8 (1892) 41-. . x t >+y 5 = 2 m z*. Levdnen, S. Helsingf. Ofv. 35 (1893) 69-. . X* + 35Y* = Z*. Pepin, . Liouv. J. Mth. 1 (1895) 351--. V. A. x* 2 i/ 4 and 2*z* = x* y 4 . Lebesgue, ,.. ax + by* = cz*. Maillet, E. As. Fr. C. Liouv. J. Mth. 18 (1853) 73-. . ax + bky = z(x* + ky*). Oltramare,G. Crelle J. 49 (1855) 142-. . x 3 + y 3 =z 2 , solution in relative primes. Hoppe, R. Schlomilch Z. 4 (1859) 304-. . a; 3 + y 3 = x - y , solution in rational numbers . Hoppe, R. Schlomilch Z. 4 (1859) 359-. . x 3 + y 3 + z 3 =v 3 . Bunlakovskij, V. [1864] StPet. Ac. Sc.Mm. (Rs.) 6 (*1865) 142-. . x 3 + y 3 + z 3 = w 2 , solution by toroid. Catalan, . E. C. Brux. Ac. Bll. 22 (1866) 29-. . x 3 + y 3 + z 3 =u 3 . Richaud,C. Em. At. N. Line. 19 (1866) 183-. Rickaud, C. Em. At'. N. Line. 20 (1867) 91-.' B. (1897) (Pt. 2) 156-. . a; 4 -8.rV + 8t/ 4 = 2 2 . Pepin, T. Bm. N. Line. Mm. 14 (1898) 71-. . x 4 + 4^V+(2/z-l)V=2 2 > where 4ft -1 and 2/i - 1 are primes. Pietrocola, C. G. Mt. 36 (1898) 77-. . * A + w*=cA Maillet, E. C. E. 129 198-. ' FermaVs theorem on x n + y n =z n . Barlow, P. Nicholson J. 27 (1810) 193-. Legendre, A. M. Par. Mm. Ac. Sc. 6 (1823) Lam6, G. [1839] C. E. 9 (1839) 45-; Par. Mm. Sav. Etr. 8 (1843) 421-. 218 2860 (Lamp's memoir.) Cauchy, A. L. C (1839) 359-. Paulet, F. Quetelet Cor. Mth. 11 (1839) 307-. Drach, S. M. Ph. Mg. 27 (1845) 286-. Lame, G. (m & vra). C. E. 24 (1847) 310-, 352, 569-; Liouv. J. Mth. 12 (1847) 172-. (Lame's demonstration.) Terquem, 0. N. A. Mth. 6 (1847) 132-. Cauchy, A. L. C. E. 24 (1847) 469-, 516-, 578-, 633-, 661-. Kummer, E. E. [1847-49] Berl. B. (1847) 132-, 140-, 305- ; Crelle J. 40 (1850) 130-. Talbot, W. H. F. [1856] Edinb. E. S. T. 21 (1857) 403-. Calzolari, L. Tortolini A. 8 (1857) 339-. Tait, P. G. [1864] Edinb. E. S. P. 5 (1866) 181. Calzolari, L. [1865] A. Mt. 6 (1864) 280-. Lukas, F. C. Arch. Mth. Ps. 58 (1876) 109-. Liouville, E. C. E. 89 (1879) 1108-. Korkine, A. C. E. 90 (1880) 303-. Lefebure, A. C. B. 90 (1880) 1406-. Pepin, T. C. E. 91 (1880) 366-. Schier, 0. Wien. Ak. Sb. 81 (1880) (Ab. 2) 392. Jonquieres, E. de. Em. N. Line. At. 37 (1884) 146-. Catalan, E. C. Liege S. Sc. Mm. 12 (1885) No. 2, 179-, 403; 13 (1886) 387-. Mathews, G. B. Mess. Mth. 15 (1886) 68-. Borletti, E. Mil. I. Lomb. Ed. 20 (1887) 222-. Mansion, P. Brux. Ac. Bll. 13 (1887) 16-, 225. Varisco, D. G. Mt. 27 (1889) 371-. Tafdmacher, A. Santiago de Chile Un. A. 82 (1892) 271-, 415-. Dutordoir, . Brux. S. Sc. A. 17 (1893) (Ft. 1) 81-. Wendt, E. Crelle J. Mth. 113 (1894) 335-. Korneck, G. Arch. Mth. Ps. 13 (1895) 1-, 263-. Mathews, G. B. Mess. Mth. 24 (1895) 97-. Gram, I. P. Sk. Nf. F. (1898) 182. n = 2X. Kummer, E. E. Crelle J. 17 (1837) 203-. = 3. Tait, P. G. [1870] Edinb. E. S. P. 7 (1872) 144. n=3. Gunther, S. Prag Sb. (1878) 112-. 7i = 3. Perrin, E. Par. S. Mth. Bll. 13 (1885) 194-. 7i=4. Tafdmacher, A. Santiago de Chile Un. A. 84 (1893) 307-. n = 5. Lame, G. Liouv. J. Mth. 12 (1847) 137-. n = 6. Tafelmacher, A. Santiago de Chile Un. A. 97 (1897) 63-. n = 7. Lame, G. Liouv. J. Mth. 5 (1840) 195-. = 7. Lebesgue, V.A. Liouv. J. Mth. 5 (1840) 276-, 348-. A. Mt. 6 (1864) 287-. C. E. 82 (1876) 676-, Higher Irresolvable Forms E 2860 Genocchi, A n=l. Pepin, T. 743-. n = 7. Genocchi, A n=14. Lejeune-Dirichlet, G. Crelle J. 9 (1832) 390-. 7i = 37. Mirimanoff, D. Crelle J. Mth. Ill (1893) 26-. application of complex numbers. Kummer, E. E. Berl. Ab. (1857) (Mth.) 41-. and Arabian MS. Kummer, E. E. N. A. Mth. 9 (1850) 386-. equation connected with. Jonquieres, de. C. E. 120 (1895) 1139-, 1236. x^ + y** = z 2 ". Lebesgue, V. A. Liouv. J. Mth. 5 (1840) 184-. C. E. 82 (1876) 910-. Form x 3 + ny 3 + n?z 3 - Snxyz, arithmetical theory. Mathews, G. B. L. Mth. S. P. 21 (1891) 280-. Eeduction of biquadratic form to a square. Pepin, T. Em. N. Line. At. 32 (1879) quartic to a square, formulae for. Pepin, T. Em. N. Line. At. 30 (1877) 211-. Square formed by adding cube to given number. Pepin, (le rev. pere) J. F. T. (xn) Brux. S. Sc. A. 6 (1882) (Pt. 2) 86-. values of quartic form x* + mx i ij i + y*. Euler, L. [1782] St Pet. Ac. Sc. Mm. 7 (1820) 10-. Sum o'f cubes, decomposition of integer into. Maillet, E. As. Fr. C. E. (1895) (Pt. 2) 242-. , number into. Elefanti, F. QJ. Mth. 4 (1861) 339-. (2), decomposition of number into. Pepin, . Liouv. J. Mth. 15 (1870) 217-. or difference of 2 biquadratics cannot be biquadratic. Kausler, C. F. [1799] St Pet. Ac. Sc. N. Acta 13 (1802) 237-. cubes cannot be cube. Kausler, C. F. [1799] St Pet. Ac. Sc. N. Acta 13 (1802) 245-. cubo-cubes cannot be a cubo- cube. Kausler, C. F. [1801] St Pet. Ac. Sc. N. Acta 15 (1806) 146-. of 4th powers, decomposition of numbers into. Lucas, E. N. A. Mth. 17 (1878) 536-. r fP^yb ^^,5 ( X = 2, 3, 4) not di- V * / visible byjp. Gegenbauer, L. Wien Ak. Sb. 95 (1887) (Ab. 2) 838-. Sums of cubes, numbers as. Zornow, . Crelle J. 14 (1835) 276-. , . Jacobi, C. G. J. Crelle J. 42 (1851) 41-. powers, representation of integers as. Matheics, G. B. Mess. Mth. 25 (1896) 69-. Ternary cubic form. Sylvester, J. J. Tortolini A. 7 (1856) 398-; Am. J. Mth. 2 (1879) 280-, 357-; 3 (1880) 58-, 179-. . Sawin, A. M. [1884] A. Mth. 1 (1884-85) 58-. , resolution in integers. Desboves, A. N. A. Mth. 5 (1886) 545-. Values of x, y, v, z for which uvz* (ax* +by*)* + Ax*y* (av 2 + bz 2 )* is a square. Euler, L. St P6t. Ac. Sc. Mm. 9 (1824) 14-. 219 2870 Algebraic numbers; Ideals; Resolvable Forms 2870 2870 Forms of higher degree which can be considered as products of linear factors; algebraic numbers; ideals. (See also 0820.) Abelian number- fields (Zahlkorper) . Weber, H. Acta Mth. 8 (1886) 193- ; 9 (1887) 105-. , Kronecker's theorem. Hilbert, D. Gott. Nr. (1896) 29-. Algebraic composite fields. Hensel, K. Crelle J. Mth. 120 (1899) 99-. congruences and universal arithmetic. Kronecker, L. Crelle J. Mth. 100 (1887) 490-. fields, groups of numbers in. Weber, H. Mth. A. 48 (1897) 433-; 49 (1897) 83-; 50 (1898) 1-. number-field, unimodular substitutions in. Hurwitz, A. Gott. Nr. (1895) 332-. fields, historical and bibliographical account. Hilbert, D. D. Mth. Vr. Jbr. 4 (1897) 175-. , units in. Minkowski, H. Gott. Nr. (1900) 90-. numbers. Dedekind, R. Bll. Sc. Mth. As. 11 (1876) 278-; 1 (1877) 17-, 69-, 144-, 207-. . Hurwitz, A. Gott. Nr. (1895) 324-. , analogue of Biemann-Boch theorem in theory of. Landsberg, G. Mth. A. 50 (1898) 577-. , arithmetical theory. Hurwitz, A. Gott. Nr. (1895) 230-. , class of lowest order containing given classes. Kneser, A. Mth. A. 30 (1887) 179-. , criterion. Minkowski, H. Gott. Nr. (1899) 64-. depending on cube root of given number. Markoff, A. St Pet. Ac. Sc. Mm. 38 (1892) No. 9, 37 pp. derived from irreducible cubic equation. Mathews, G. B. L. Mth. S. P. 24 (1894) 327-. and function theory. Granwall, H. Stockh. Ofv. (1897) 199-. , fundamental equation of a class-domain (Gattungsbereich). Mertens, F. Wien Ak. Sb. 103 (1894) (Ab. 2a) 5-. of given class-domain. Hensel, K. Crelle J. Mth. 101 (1887) 99-. , greatest common divisor. Sochocki, J. Prace Mt.-Fiz. 4 (1893) 95-; Fschr. Mth. (1893-94) 297-. , logarithms. Stormer, C. C. B. 130 (1900) 1603-. , real, property of whole system. Cantor, G. [non M.} [1873] (vn) Crelle J. Mth. 77 (1874) 258-. Analytical loci. Cauchy,A.L. C. B. 24 (1847) 885-. Arithmetical demonstration of algebraic theo- rems. Kneser, A. Crelle J. Mth. 102 (1888) Aurifeuillians. Cunningham, (Lt.-Col.) A. L. Mth. S. P. 29 (1898) 381-. Binary cubic forms, calculation and classifica- tion for negative determinants. Arndt, F. [1851] Grunert Arch. 31 (1858) 335-. forms, which cannot represent powers of primes. Gegenbauer, L. Wien Ak. Sb. 97 (1889) (Ab. 2a) 368-. Biquadratic forms, class, prime divisors. Le- jeune-Dirichlet, G. Crelle J. 3 (1828) 35-. Class-number divisible by prime X. Kummer, E. E. Berl. Ak. Mb. (1874) 239-. for numbers formed from llth and 13th roots of unity. Wolfskehl, P. Crelle J. Mth. 99 (1886) 173-. Classes of resolvable forms. Schering, E. [1868] Gott. Ab. 14 (1869) 16 pp. numbers and Hilbert's theorem. Capelli, A. Nap. Bd. 35 (1896) 198-. Complex factorisation of numbers. Cauchy, A. L. C. B. 24 (1847) 1022- ; 25 (1847) 46-. factors of prime numbers. Zignago, I. Bv. Mt. 4 (1894) 151-. number systems, irreducible. Formenti, C. Mil. I. Lomb. Bd. 17 (1884) 6-. numbers. Schwering, K. Crelle J. Mth. 102 (1888) 56-. , advantages of geometric representation. Vivanti, G. Bv. Mt. 2 (1892) 167-. , class. Markoff, A. C. B. 112 (1891) 780-, 1049-, 1123-, 1288. connected with equation 6 3 - 2 = 0. Mathews, G. B. L. Mth. S. P. 24 (1893) 319-. , extension. Vahlen, T. Konigsb. Schr. 38 (1897) [72]-. of form a + bi. Gegenbauer, L. Wien Ak. Sb. 91 (1885) (Ab. 2) 1047-. formed from fourth roots of unity. Gegenbauer, L. Wien Ak. D. 50 (1885) (Ab. 1) 153-. , probabilities in con- nexion with. Gegenbauer, L. Wien Ak. Sb. 98 (1890) (Ab. 2a) 635-. , quadratic reciprocity law for. Busche, E. Hamb. Mth. Gs. Mt. 2 (1890) (Festschr., Tl. 2) 80-. , ideal prime factors. Kummer, E. E. Berl. Ab. (1856) (Mth.) 1-. , norms. Schwering, K. Acta Mth. 11 (1887-88) 265-. , systems of, and bilinear groups. Cartan, E. Toul. Fac. Sc. A. 12 (1898) B, 99 pp. , , transformation groups. Study, E. Mh. Mth. Ps. 1 (1890) 283-. , theory. Ivanov, I. St Pet. Ac. Sc. Mm. (Es.) 72 (1893) (Suppl.) No. 9, 14 pp.; Fschr. Mth. (1893-94) 302. primes. Gegenbauer, L. Wien Ak. Sb. 98 (1890) (Ab. 2a) 1036-. formed from fifth roots of unity. Tanner, H. W. L. L. Mth. S. P. 24 (1893) 223-. Cubic number-fields, number of ideal classes in. Dedekind, R. Crelle J. Mth. 121 (1900) 40-. , table of class numbers. Reid, L. W. N. Y. Am. Mth. S. Bll. 6 (1900) 268, 277-. 220 Algebraic numbers; Ideals; Resolvable Forms 2870 Decomposition of numbers / 12 - 9# 12 into two cubes. Henry, C. N. A. Mth. 20 (1881) 418-. Dirichlet's biquadratic number-field. Hilbert, D. Mth. A. 45 (1894) 309-. Divisors of certain polynomials and existence of certain primes. Genocchi, A. C. B. 98 (1884) 411-. complex numbers, theorems. Gegenbauer, L. Mh. Mth. Ps. 1 (1890) 39-. , elementary, of compounded matrices. Hen- sel, K. Crelle J. Mth. 114 (1895) 109-. , , theory and applications. Sauvage, L. Par. lie. Norm. A. 8 (1891) 285-; 10 (1893) 9-. , , of two systems. Hensel, K. Crelle J. Mth. 117 (1897) 346-. , fundamental, of algebraic systems. Hensel, K. Crelle J. Mth. 117 (1897) 333-. , , system with reference to two diffe- rent rational fields. Hensel, K. Crelle J. Mth. 118 (1897) 173-. Dual group generated by three moduli. Dede- kind, R. Mth. A. 53 (1900) 371-. Equation F (x, y, z) = $ (a) (ft) . . . = 1. Lejeune- Dirichlet, G. C. E. 10 (1840) 285-. F (x, y, z) =

(ft) . . . = 1 (Lejeune-Dirich- let). Libri, G. C. E. 10 (1840) 311-. Equivalences, theory. Grunert, J. A. Arch. Mth. Ps. 44 (1865) 443-; 45 (1866) 454-. Factorisation of function with respect to prime modulus p. Pellet, A. E. C. E. 86 (1878) 1071-. , complex, of primes, theorem. Cellerier, C. Gen. S. Ps. Mm. 32 (1894-97) No. 7, 61 pp. x n _ a i n arbitrary field. Wendt, E. Mth. A. 53 (1900) 450-. Functions of several variables, irreducibility. Bendixson, I. Stockh. 6fv. (1892) 189-. , matrices. Kronecker, L. Berl. Mb. (1869) 159-, 688-. , two biquadratic, of 4 variables, whose pro- duct is a similar function. Tortolini, B. Tortolini A. 2 (1859) 9-. Fundamental equation of a class for a real prime as modulus. Hensel, K. Crelle J. Mth. 113 (1894) 61-. system and discriminant of classes of alge- braic numbers. Landsberg, G. Crelle J. Mth. 117 (1897) 140-. Galoisian number-fields. Hilbert, D. Gott. Nr. (1894) 224-. Galois's imaginaries, operation symbols involv- ing. Le Vavasseur, . Toul. Ac. Sc, Mm. 9 (1897) 247-. Greatest common divisor. Gegenbauer, L. Wien Ak. Sb. 101 (1892) (Ab. 2a) 1143-. , theorem. Mertens, F. Wien Ak. Sb. 101 (1892) (Ab. 2a) 1560-. Groups of functions, representation by a basis. Ritter, E. Gott. Nr. (1895) 158-. Ideal complex numbers, properties of class- number. Kronecker, L. Berl. Mb. (1870) 881-. division of algebraic quantities by real primes. Hathaway, A. S. J. H. Un. Cir. [4] (1885) 79. 2870 Ideals. Hathaway, A. S. Am. J. Mth. 9 (1887) 162-. in algebraic number-field, existence of primitive numbers. Wiman, A. Stockh. Ofv. (1899) 879-. , Dedekind's. Baur, L. Mth. A. 32 (1888) 151-. , Dedekind-Weber, in a hyper-elliptic field. Baur, L. Mth. A. 41 (1893) 491-. and higher congruences, connection of theories. Dedekind, R. Gott. Ab. 23 (1878) 5-. , theory. Dedekind, R. Gott. Nr. (1894) 272-; (1895) 106-. , . Hurwitz, A. Gott. Nr. (1894) 291-. , . Furtwdngler, P. Gott. Nr. (1895) 381-. Integral functions. Heine, H. E. [1852] Crelle J. 48 (1854) 237-, 243-. . with binomial divisors. Hill, C. J. D. Crelle J. 70 (1869) 103-. of integers, irreducible factors. Runge, C. Crelle J. Mth. 99 (1886) 89-. , . Mandl, M. Crelle J. Mth. 113 (1894) 252-. , modular factors. Cauchy, A. L. C. E. 24 (1847) 1116-. polynomials with prime divisors of assigned form. Lefebure, A. C. E. 98 (1884) 293-, 413-, 567-, 613- ; Par. EC. Norm. A. 1 (1884) 389-; 2 (1885) 113-. several variables, resolution into factors. Hancock, H. Par. EC. Norm. A. 17 (1900) 89-. , linear elements. Meray, C. Par. EC. Norm. A. 2 (1885) 289-. Modular systems, decomposition. Moore, E. H. N. Y. Am. Mth. S. Bll. 3 (1897) 372-. and general complex numbers, theory. Kronecker, L. Berl. Ak. Sb. (1888) 429-, 447-, 557-, 595-, 983-. , Kronecker's, canonical forms for. Han- cock, H. Crelle J. Mth. 119 (1898) 148-. , , reduction. Hancock, H. QJ. Mth. 27 (1895) 147- ; Crelle J. Mth. 122 (1900) 265-. , relations between. Fund, O. Hamb. Mth. Gs. Mt. 3 (1900) 325-. of second rank, elementary properties of pure. Hensel, K. Crelle J. Mth. 119 (1898) 175-. , and rings of numbers. Lands- berg, G. Gott. Nr. (1897) 277-. Moduli, extension of symbol (a, b) in theory. Dedekind, R. Gott. Nr. (1895) 183-. , theory. Steinitz, E. Mth. A. 52 (1899) 1-. Numbers, as sum of 2 cubes. Lenhart, W. [1836] Mth. Misc. 1 (1838) 114-. Pairs of forms xy (x* - y 4 ) whose product or quotient is square. Euler, L. St P6t. Ac. Sc. Mm. 11 (1830) 31-. Period equation of numbers formed with Xth roots of unity. Stouff, X. C. E. 125 (1897) 859-. Polynomial of several variables, test for facto- risability. Laurent, H. N. A. Mth. 12 (1893) 315-. ' 221 2880 Application of Trigonometry ; Cyclotomy 2880 Primary prime functions. Hancock, H. [1900] N. Y. Am. Mth. S. Bll. 7 (1901) 202, 206-. Prime factor of class-number H, divisibility by X. Kronecker, L. Liouv. J. Mth. 1 (1856) 39&-. factors of .certain forms, Lejeune-Dirichlet's theorems. Kronecker, L. Berl. Ak. Sb. (1888) 417-. ideals in relation to substitution groups. Frobenius, G. Berl. Ak. Sb. (1896) 689-. , separation of number-field into. Hilbert, D. Mth. A. 44 (1894) 1-. integral complex numbers formed from fourth roots of unity. Gegenbauer, L. Wien Ak. Sb. 101 (1892) (Ab. 2a) 984-. Primes and prime ideals in domain of fifth roots of unity. Gmeiner, J. A. Mh. Mth. Ps. 11 (1900) 1-. Primitive factors of 4 (1 -a-)/(l -x), reduc- tion of product to form Y-^nZ 2 , Cauchy's theorem. Genocchi,A. C. E. 67 (1868) 1035-. Quadratic complex numbers, certain systems. Western, A. E. [1898] Camb. Ph. S. T. 17 (1899) 109-. Kepresentation of numbers of a class-domain. Hensel, K. Crelle J. Mth. 103 (1888) 230-. by forms. Poincare, H. C. E. 92 (1881) 777- ; Par. S. Mth. Bll. 13 (1885) 162-. . Gegenbauer, L. Wien Ak. Sb. 95 (1887) (Ab. 2) 618-. Symmetric matrices. Kronecker, L. Berl. Ak. Sb. (1889) 349-. 2880 Application of trigonometri- cal functions to arithmetic; cyclotomy. Absolutely smallest residues of real magnitudes. Kronecker, L. Berl. Ak. Sb, (1885) 383-, 1045-. Arithmetical series, Dirichlet's theorem. Syl- vester, J. J. As. Fr. C. B. (1888) (Pt. 2) 118. Binomial equations. Liouville, J. Liouv. J. Mth. 2 (1857) 413-. Trudi, N. [1866] Nap. At. Ac. 3 (1866-68) No. 6, 49 pp. Rubini, R. G. Mt. 5 (1867) 184-. and algebraic .radicals. Valat, . Bordeaux Act. Ac. Sc. (1843) 321-. alternating sums formed with primitive roots. Cauchy, A. L. C. K. 10 (1840) 560-. application of Sturm's theorem. Gascheau, G. Liouv. J. Mth. 7 (1842) 126-. to summation of series. Berger,A. Ups. S. Sc. N. Acta 13 (1887) No. 7, 36 pp. depression of degree. Lobatschewsky, N. (vi Adds.) Kazan Mm. Un. (1834) 1-. existence and characteristics of roots. Meray, C. Par. EC. Norm. A. 2 (1885) 337-. functions of roots. Schlmemann, T. Crelle J. 17 (1837) 372-. irreducibility. Mertens, F. Wien Ak. Sb. 99 (1891) (Ab. 2a) 907-. , (1832) -M -1 = reducibility. Bucca, F. Palermo Cir. Mt. Ed. 14 (1900) 136-. reducible. Vahlen, K. Acta Mth. 19 (1895) 195-. roots. Tardy, P. A. Mt. 3 (1869-70) 331-. , extraction. Berkhout, J. J. T. van (\i Adds.) Arch. Wisk. Gn. 1 (1859) 54-. solution. Gamier, J. G. Brux. Ac. Bll. 6 (1839) 474-. , cases. Lentheric, . Gergonne A. Mth. 21 (1830-31) 101-. , trigonometric. Tetmajer, J. (XH) Krk. Ak. (Mt.-Prz.) Pam. 5 (1880) 117-. sums of powers of a primitive root. Cauchy, A. L. C. E. 10 (1840) 594-. transformations. Gram, J. P. Ts. Mth. 5 (1887) 44-; Fschr. Mth. (1887) 72-. z 257 =l, solution. Richelot, F. J. Crelle J. 9 1-, 146-, 209-, 337-. = 0, solution. Fischer, P. A. Crelle J. 11 (1834) 201-. xP = l. Lebesgue, V. A. C. E. 5 (1837) 722-. A B =C. Pagani, G. M. Brux. Ac. Bll. 4 (1837) 387; Brux. Ac. Sc. Mm. 11 (1838) 11 pp. ,-c m -l = 0, trigonometric series derived from. Cauchy, A. L. C. E. 10 (1840) 719-. xP-l = 0, p prime, solution. Realis, S. N. A. Mth. 2 (1843) 5-, 147-. xP=l, auxiliary equation of mth order (p=mTr + l). Lebesgue, V. A. C. E. 18 (1844) 696-. x p -1 = 0, modular indices of polynomials con- nected with. Cauchy, A. L. C. E. 25 (1847) 93-. XP - 1 = 0, p prime, solution. Plana, G. [1850] Tor. Mm. Ac. 11 (1851) 413-. a: 257 -1 = 0. Cayley, A. Crelle J. 41 (1851) 81. x n -l = 0, primitive roots. Lebesgue, V. A. N. A. Mth. 11 (1852) 417-. zP=l, Lagrange's resolvent, expression ana- logous to. Kummer, E. E. Em. At. 6 (1852-53) 237-. x*>=l, p prime, solution. Lebesgue, V. A. C. E. 38 (1854) 914-. w n =l, periods formed by roots when n is not prime. Fuchs, L. [1862] Crelle J. 61 (1863) 374-. x m -l = Q. Catalan, E. C. Brux. Ac. Bll. 29 (1870) 182-. x p - 1 = 0, trisection and quartisection. Cayley, A. L. Mth. S. P. 11 (1879-80) 4-. a;**- 1 = 0, quinquisection. Cayley, A. L. Mth. S. P. 12 (1880-81) 15- ; 16 (1884-85) xf- 1 = 0. Scott, C. A. Am. J. Mth. 8 (1886) 261-. a^" 1 = 0, quinquisection. Tanner, H. W. L. L. Mth. S. P. 18 (1886-87) 214-. a;** -1 = 0, p prime, solution. Butters, J. W. Edinb. Mth. S. P. 7 (1889) 10-. a; 17 -1=0. Cayley, A. Mess. Mth. 19 (1890) 184-. a;P-l = 0, solution. Pierpont, J. N. Y. Am. Mth. S. Bll. 2 (1896) 77-. a: 71 1 = 0, solution. Jarollmek, V. fiasopis 27 (1898) 209- ; Fschr. Mth. (1898) 75. 222 2880 Application of Trigonometry ; Cyclotomy 2880 Congruences, application of cyclotomy. Moesta, C. W. Grunert. Arch. 10 (1847) 98-. , to divisors of cyclotomic functions. Hathaway, A. S. [1880] (xn) J. H. Un. Cir. [1] (1882) 67. Congruent roots. Rummer, E. E. Crelle J. 53 (1857) 142- . Cyclotomic equations, irreducibility. Dedekind, E. Crelle J. 54 (1857) 27-. , m prime, m-fold section. Glashan, J. C. Am. J. Mth. 21 (1899) 276-. , quinquisection. Glashan, J. C. [1897] Am. J. Mth. 21 (1899) 270-. , theory. Migotti, A. [1882] Wien Ak. Sb. 87 (1883) (Ab. 2) 7-. functions. Lucas, E. C. E. 90 (1880) 855-. . Tanner, H. W. L. L. Mth. S. P. 20 (1889) 63-, 258-. , considered with respect to prime modulus p. Hathaway, A. S. [1881] (xn) J. H. Un. Cir. [1] (1882) 131. , divisors. Sylvester, J. J. C. B. 90 (1880) 287-, 345-. , , Sylvester's theorem, proof. Pepin, T. C. E. 90 (1880) 526-. Cyclotomy. Gauss, C. F. Par. S. Phlm. Bll. 3 (1802) 102-. . Frischauf, J. Wien Sb. 55 (1867) (Ab. 2) 113-. , application to theory of numbers. Jacobi, C. G. J. Berl. B. (1837) 127- ; Crelle J. 30 (1846) 166- . , (Jacobi). Lebesgue, V. A. Liouv. J. Mth. 19 (1854) 289-. derived from linear equations (for primes of form 2 m + l). Hermes, J. [1878] Crelle J. Mth. 87 (1879) 84-. , generalisation. Stickelberger, L. Mth. A. 37 (1890) 321-. , , algebraic. Berger, A. Stockh. 6fv. (1891) 683-; Fschr. Mth. (1891) 190-. , new method. Quincke, G. A. Ps. C. 149 (1873) 270-. , primes of form 2 2 " + l. Lucas, E. C. E. 85 (1877) 136-. , proofs of theorems related to. Lanavicensis (Sylvester, J. J.). Ph. Mg. 18 (1859) 281-. by purely geometrical means ; possible cases. Hessel, J. F. C. Grunert Arch. 37 (1861) 269-. Determinants arising in cyclotomy. Hermes, J. Arch. Mth. Ps. 6 (1888) 276-. Divisibility of prime factor of class-number H by X. Kronecker, L. Liouv. J. Mth. 1 (1856) 396-. Division of angle into 2 n + l equal parts. Kempe, A. N. Arch. Wisk. 1 (1895) 163- ; Fschr. Mth. (1893-94) 1152-. any number of equal parts. Kempe, A. N. Arch. Wisk. 1 (1895) 215- ; Fschr. Mth. (1893-94) 1153. Divisors of Fermatians. Sylvester, J, J. C. E. 106 (1888) 446-; Nt. 37 (1888) 417-. form Ax 2 + By 2 + Cz 2 , proof of theorem of Lagrange, and divisors of functions which multisect the roots of unity. Sylvester, J. J. Am. J. Mth. 3 (1880) 390-. Divisors of functions of primitive roots of unity. Sylvester, J. J. C. E. 92 (1881) 1084-. resulting from cyclotomy. Kummer, E. E. Crelle J. 30 (1846) 107-. Equation for primitive roots of unity, irreduci- bility. Lebesgue, V. A. Liouv. J. Mth. 4 (1859) 105-. nth roots of unity. Bang, A. S. N. Ts. Mth. 6 (B) (1895) 6-; Fschr. Mth. (1895) 121. t 2 -pu* = l, p prime, solution by circular functions. Lejeune-Dirichlet, G. Crelle J. 17 (1837) 286-. a;p-i + p- 2 + . . . + 1 = is irreducible when is prime. Kronecker, L. Crelle J. 280; Liouv. J. Mth. 1 (1856) 399-. T- X=l + x+ ... + xP~ l = 0, p prime, irreduci- bility. Prouhet, E. N. A. Mth. 9 (1850) 348-. Equations, resolubility, especially a; 71 -1=0. Allegret, A. C. E. 43 (1856) 860-. Eulerian numbers, residues with respect to prime modulus. Lucas, E . [1877] Par. S. Mth. Bll. 6 (1878) 49-. Gauss, formula where Kronecker, L. Liouv. J. Mth. 1 (1856) 392-. , theorem on 4 - , extension. Lucas, E. Tor. Ac. Sc. At. 13 (1877) 271-. Gauss's sums. Seguier, de. C. E. 123 (1896) 166-. -- . Lerch, M. Casopis 28 (1899) 1- ; Fschr. Mth. (1898) 168. -- , analogues. Lerch, M. Prag Sb. (1897) (Mth.-Nt.) No. 43, 16 pp. , series S e - . Mertens, F. Berl. Ak. Sb. (1896) 217-. -- with several variables. Jordan, C. C. E. 73 (1871) 1316-. Imaginary cube roots. Horner,W.G. Thomson A. Ph. 8 (1816) 279-, 388-. --- , every number has four. Lockhart, J. Thomson A. Ph. 5 (1815) 439-. Eegular polygon, 17 sides, inscription. James, S. Ir. Ac. T. 13 (1818) 175-. -- , -- , . Grunert, J. A. Arch. Mth. Ps. 42 (1864) 361-. -- , -- , . Collignon, E. As. Fr. C. E. 8 (1879) 162-. s, number of inscriptible. Dickson, N. Y. Mth. S. Bll. 3 (1894) 123-. -- ,7 and 13 sides, construction. Affolter, F. G. Mth. A. 6 (1873) 592-. -- , 15 and 17 sides, inscription. Lebesgue, V. A. N. A. Mth. 5 (1846) 683-. -- , 17 sides, construction. Staudt, G. K. C. von. Crelle J. 24 (1842) 251. -- , -- , . Breton [de Champ], P. N. A. Mth. 5 (1846) 226-, 340. -- , -- , . Houel, J. N. A. Mth. 16 (1857) 310-. 223 2880 Application of Trigonometry Begular polygons, 17 sides, construction (von Staudt). Schroter, H. [1872] Crelle J. 75 (1873) 13-. _ , _, . Richmond, H. W. QJ. Mth. 26 (1893) 206-. \ , _. Gerard, L. Mth. A. 48 (1897) 390-. , , _. Fontene, G. Mathesis 19 (1899) 179-. , 19 and 37 sides, construction. Amaldi, I. G. Mt. 30 (1892) 141-. 1 257 sides. Baden- Ghijben, J. Amst. 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G.] Berl. Mb. (1859) 488-. , 7th, llth, 13th, 17th, 19th, 23rd, and 29th roots of unity of primes <1000. Reuschle, [C. G.] Berl. Mb. (1859) 694-; (1860) 190-, 194-, 714-, 719-, 725-, 729-. Trigonometric equations. Guilmin, A. N. A. Mth. 5 (1846) 49-. functions deduced from equation x n =l. Svanberg, J. Stockh. Ak. Hndl. 33 (1812) series, a special. PreobraZenskij , P. V. Kazan S. Nt. (Ps.-Mth.) P. 6 (1888) 10-; Fschr. Mth. (1888) 172-. sin ~ , arithmetical role. Cesdro, E. A. Mt. 13 (1885) 315-. cos p= , quadratic radical expression for. Steg- gall, . Edinb. Mth. S. P. 7 (1889) 4-. XP-A, irreducibility. Mertens, F. Mh. Mth. Ps. 2 (1891) 291-. x n -l, irreducible divisors. Gravelaar, N. L. W. A. N. Arch. Wisk. 20 (1893) 7-; Fschr. Mth. (1893-94) 158-. Transcendental functions 2890 2890 Application of other tran- scendental functions to arith- metic. Asymptotic formulae, note. Gegenbauer, L. Wien Ak. Sb. 93 (1887) (^6. 2) 35-. laws. Lejeune-Dirichlet, C. Berl. B. (1838) 13-. . Gegenbauer, L. Wien Ak. D. 49 (1885) (Ab. 1) 37-. -value of series 2 -_-L_ 5 . Mertens, F. Wien Ak. Sb. 106 (1897) (Ab. 2o) 411-. values of some numerical functions. Her- mite, C. Crelle J. Mth. 99 (1886) 324-. (Hermite). Gegenbauer, L. Wien Ak. Sb. 99 (1891) (Ab. la) 387-. Combinations of special kind. Bumakovskij , V. Ja. [1871] St Pet. Ac. Sc. Mm. (Rs.) 20 (*1872) (Suppl.) No. 2, 71pp. Continuous variables in theory of numbers. Hermite, C. Crelle J. 41 (1851) 191-. Decomposition of number into sum of squares. Lorenz, L. V. (xn) Ts. Mth. 1 (1871) 97-. . Mathews, G. B. L. Mth. S. P. 27 (1896) 55-. ; four squares. Pepin, T. Bm. N. Line. At. 38 (1885) 139-. Discontinuous factor, case. Lorenz, L. N. Ts. Mth. 1 (B) (1890) 76-; Fschr. Mth. (1890) 296-. Division, theorem on remainders. Lejeune- Dirichlet, G. Berl. B. (1851) 20-. , (Lejeune-Dirichlet). Terquem, 0. N. A. Mth. 13 (1854) 396-. Divisors of a number, theorems. Cesdro, E. G. Teix. J. Sc. 7 (1886) 3-. 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Mt. 21 (1893) 47-. . Speckmann, G. Arch. Mth. Ps. 12 (1894) 439-. G. Teix. J. Sc. . Wendt, E. Crelle J. Mth. 115 (1895) 85-. (Dirichlet). La Vallee- Poussin, C. J. de. [1895] Brux. Mm. Cour. 8, 53 (1895-96) No. 6, 32 pp. ( ). Mertens, F. Wien Ak. Sb. 104 (1895) (Ab. 2a) 1093-, 1158- ; 106 (1897) (Ab. 2a) 254-; 108 (1899) (Ab. 2a) 32-. progressions, with difference and first term mutually prime. Gegenbauer, L. Wien Ak. Sb. 100 (1891) (Ab. 2a) 1018-. series. Sylvester, . Mess. Mth. 21 (1892) 1-, 87-, 192. Complex primes, Dirichlet' s theorem that linear complex function contains infinite number of. Mertens, F. Wien Ak. Sb. 108 (1899) (Ab. 2a) 517-. , Tchebicheff's theorems extended to. Poincare, H. Liouv. J. Mth. 8 (1892) 25-. Composite numbers, long successions. Glaisher, J. W.L. [1877] Mess.Mth.,7 (1878) 102-, 171-. , . Lucas, E. Mess. Mth. 8 (1879) 81. Determination of primes. Bouniakowsky, V. [1839] St Pet. Ac. Sc. Mm. 4 (1841) 447-. Divisibility of product, distribution of primes. 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B. 96 (1883) 463-. between given limits. Drach, S. M. Ph. Mg. 24 (1844) 192-. . Meissel, E. [1869] Mth. A. 2 (1870) 636-. . Oppermann, L. Kjtfb. Ov. (1882) 169- ; (Resume, 9-). . Voile, G. Em. N. Line. Mm. 14 (1898) 143-. , analytical expression for. Levi-Civita, T. Em. E. Ac. Line. Ed. 4 (1895) (Sem. 1) 303-. . , Tchebycheff 's theory. Syl- vester, J. J. Am. J. Mth. 4 (1881) 230-. , theorem of Sylvester's. Gegenbauer, L. Mh. Mth. Ps. 10 (1899) 370-. 100,000,001 and 100,001,699. Davis, W. Liouv. J. Mth. 11 (1866) 188-. and 2N, formulae for. Piarron de Hondesir, E. S. As. Fr. C. E. (1877) 79-. 7i + l and 2n, etc. Catalan, E. C. [1882] Liege S. Sc. Mm. 10 (1883) (No. 1), 12 pp. . x and (1 + h) x, theorem of Stieltjes. Cahen, . C. E. 116 (1893) 490. Numbers prime to and less than product of first 7i primes, properties. Walla, F. Arch. Mth. Ps. 66 (1881) 353-. Prime-pairs, enumeration. Glaisher, J. W. L. [1878] B. A. Ep. (1878) 470- ; Mess. Mth. 8 (1879) 28-. Primes in Burckhardt's and Dase's tables, enumeration. Glaisher, J. W. L. [1876-77] Camb. Ph. S. P. 3 (1880) 17-, 47-; B. A. Ep. (1877) (Sect.) 20-. , construction of a table. Laisant, C. A, As. Fr. C. E. (1891) (Ft. 2) 165-. , criteria. Montferrier, A. S. de. Quetelet Cor. Mth. 5 (1829) 94-. , . Warner, J. D. Am. As. P. (1890) 54-. , , and factors. Seelhoff, P. Am. J. Mth. 8 (1886) 39-. , distribution problems. Pandau, E. Par. S. Mth. Bll. 28 (1900) 25-. of form 4w + l and 4n + 3, distribution. Tcheblcheff, P. St Pet. Ac. Sc. Bll. 11 (1853) 208. , separate enumeration. Glaisher, J. W. L. B. A. Ep. (1878) 470- ; E. S. P. 29 (1879) 192-. 100n + 1, enumeration. Daublebsky von Sterneck, E. Wien Az. 31 (1894) 2-. Primes of forms 4s + 1, 4s -1, theorems. Ge- geubauer, L. Mh. Mth. Ps. 4 (1893) 89-, 98, 184. , formulae, general. Polignac, A. (Prince de). Toul. Mm. Ac. 1 (1857) 308-. , properties and distribution. Polignac, A. (Prince de). C. E. 29 (1849) 397, 738-. in relation to an arithmetical progression. Polignac, A. (Prince de). C. E. 54 (1862) 158. , report on calculations made by Bertelsen. Gram, J. P. Acta Mth. 17 (1893) 301-. , theorem. Laurent, H. C. E. 126 (1898) 809-. Eecurring series, application. Lucas, E. C. E. 82 (1876) 165- ; 83 (1876) 1286-. Eiemann's function, maximum value. Preo- braiemkij, P. V. Mosc. S. Sc. Bll. 78 (No. 1) (1892) 27-; Fschr. Mth. (1892) 170. f(), Bourget,. C. E. 101 (1885) 304- . . Jensen, J. L. W. V. C. E. 104 (1887) 1156-. . Gram, J. P. Kjtfb. Ov. (1895) 303-. . Landau, E. C. E. 129 (1899) 812-. , and analogous functions. Cahen, E. Par. EC. Norm. A. 11 (1894) 75-. , number of primes below given limit. La Vallee-Poussin, C. J. de. Brux. Mm. Cour. 8, 59 (1899-1900) No. 1, 74 pp. for a odd and positive, values. Kluyver, . Bll. Sc. Mth. 20 (1896) 116-. , zeros. Hadamard, . C. E. 122 (1896) 1470-; 123 (1896) 93; Par. S. Mth. Bll. 24 (1896) 199-. f(x) and logarithmic integral. Phrag- m4n, E. Stockh. Ofv. (1891) 599-. (t), arithmetical application. Franel, J. Zur. Vjschr. 41 (1896) (Festschr., Th. 2) 7-. f, property. Mertens, F. Wien Ak. Sb. 107 (1898) (Ab. 2a) 1429-. prime number formula, calculation of some terms. Phragmen, E. Stockh. Ofv. (1891) 721-. Series of primes. Smith, H. J. S. Ashmol. S. P. 35 (1857) 128-. . Lorenz, L. V. (xn) Ts. Mth. 2 (1378) 1-. Sieve of Eratosthenes. Colombier, P. A. G. N. A. Mth. 2 (1843) 408-. , extension. Bunlakovskij , V. Ja. St Pet. Ac. Sc. Mm. (Rs.) 41 (*1882) (Suppl.) No. 3, 32 pp. , historical notice. Polignac, A. (Prince de). N. A. Mth. 12 (1853) 429-. , six arithmological propositions de- duced from. Polignac, A. (Prince de). N. A. Mth. 8 (1849) 423-. for primes. Speckmann, G. Arch. Mth. Ps. 11 (1892) 439-; 12 (1894) 431-. . Schapira, H. [1896] D. Mth. Vr. Jbr. 5 (1901) (Heft 1) 69-. . Christie, R. W. D. Nt. 56 (1897) 10-. ; number of primes; series of odd numbers. Terquem, O. N. A. Mth. 5 (1846) 607-. Sum of logarithms of primes not exceeding x. Cahen . C. E. 116 (1893) 86-. 227 p2 2910 Special Numbers, etc. Sum of i>th powers of primes less than n. Gegenbauer, L. Wien Ak. Sb. 94 (1887) (Ab. 2) 903-. Tchebychef-de Polignac theorem. Cesdro, E. N. A. Mth. 4 (1885) 418-. . Gegenbauer, L. Mh. Mth. Ps. 3 (1892) 319-. Tchebychef's theory of primes. Daublebsky van Sterneck, JR. Wien Ak. Sb. 109 (1900) (^6. 2a) 1137-. 2910 Special numbers and numerical functions. Arithmetical theorems. Gegenbauer, L. Wien Ak. Sb. 92 (1886) (^6. 2) 1055-. Asymptotic calculus, foundations. Cesdro, E. C. E. 106 (1888) 1651- ; 107 (1888) 426-. , . Jensen, J. L. W. V. C. E. 107 (1888) 81-. , proposition. Cesdro, E. A. Mt. 16 (1888-89) 178-. law in theory of numbers. Stieltjes, . C. E. 101 (1885) 368-. laws of functions in theory of numbers. Lipschitz, E. Berl. Mb. (1865) 174-. numbers. Cesdro, E. Em. E. Ac. Line. Ed. 4 (1888) (Sem. 1) 452-. values, formula relating to. Franel, J. Mth. A. 51 (1899) 369-. Bernoulli's numbers. Ohm, M. Crelle J. 20 (1840) 11-. Eisenlohr, 0. Crelle J. 28 (1844) 193-. Bellavitis, G. Tortolini A. 4 (1853) 108-. Fergola, E. Nap. Ac. Sc. Mm. 2 (1855-57) 315-. Kronecker, L. Liouv. J. Mth. 1 (1856) 385-. Bjorling, E. G. Stockh. Ofv. 14 (1857) 107-. Catalan, E. C. C. E. 54 (1862) 1030-, 1059-. Le Besgue, V. A. C. E. 58 (1864) 853-, 937-. Catalan, E. C. C. E. 58 (1864) 902-, 1105- ; 81 (1875) 441-. Cayley, A. Mess. Mth. 4 (1875) 157-. Hermite, C. [1875] Crelle J. Mth. 81 (1876) 93-. Le Paige, C. M. M. H. H. C. E. 81 (1875) 966-; Brux. Ac. Bll. 41 (1876) 1017. (Le Paige.) Catalan, E. C. Brux. Ac. Bll. 41 (1876) 1018-. Vorontsov, M. M. N. A. Mth. 15 (1876) 12-. Berg, F. J. van den [1878] (xn) Amst. Ak. Wet. P. (1878-79) (No. 5) 3-. Stern, M. A. Crelle J. Mth. 84 (1878) 267-. Gohierre de Longchamps, . Par. EC. Norm A. 8 (1879) 55-. Kuttner, W. Z. Mth. Ps. 24 (1879) 250-. Stern, M. A. Crelle J. Mth. 88 (1880) 85-: 92 (1882) 349-. Zhbikovskii, A. K. [1882] (xn) Eec. Mth. (Moscou) 10 (1882-83) (Ft. 1) 127-. Kronecker, L. Crelle J. Mth. 94 (1883) 268-. Lipschitz, R. Crelle J. Mth. 96 (1884) 1-. Gomes Teixeira, F. Am. J. Mth. 7 (1885) 288-. Genacchi, A. Crelle J. Mth. 99 (1886) 315-. Studnicka, F. J. Casopis 15 (1886) 97-; Fschr. Mth. (1886) 224-. Bernoulli's Numbers 2910 Gegenbauer, L. Wien Ak. Sb. 96 (1888) (Ab. 2) 491-. Ocagne, M. d\ Par. S. Mth. Bll. 17 (1889) 107-. Voronoj, G. T. Kharkov Mth. S. Com. 2 (1891) 129- ; Fschr. Mth. (1890) 268. Mangeot, S. Mars. Fac. Sc. A. 2 (1892) 63-. Saalschiitz, L. Konigsb. Schr. 33 (1892) [44]-. Carda, K. Mh. Mth. Ps. 5 (1894) 185-. Tschistiakow, J. J. Fschr. Mth. (1895) 283-. Sincov, D. [1898] Kazan S. Ps.-Mth. Bll. 8 (1899) 104- ; Fschr. Mth. (1898) 221. Petr, K. Casopis 28 (1899) 24-; Fschr. Mth. Schwering, K. Mth. A. 52 (1899) 171-. analytic expression. Plana, G. Tor. Mm. Ac. 25 (1820) 403-. arithmetical developments concerning. Rogel, F. Arch. Mth. Ps. 11 (1892) 77-. asymptotic representation of integral part. Lipschitz, . Bll. Sc. Mth. 10 (1886) 135-. bibliography. Ely, G. S. Am. J. Mth. 5 (1882) 228-. and binomial coefficients, relation. Hammond, J. [1875] L. Mth. S. P. 7 (1875-76) 9-. calculation. Catalan, E. C. Lifcge S. Sc. Mm. 12 (1885) No. 2, 86-, 91-, 97-, 103-, 320-. up to B 62 by Staudt's theorem. Adams, J. C. B. A. Ep. (1877) (Sect.) 8-. , simplest method. Williot, . Par. S. Mth. Bll. 16 (1888) 144-. congruence theorem relating to. Glaisher, J. W. L. QJ. Mth. 31 (1900) 253-. connected with fundamental discriminant, generalisation. Gegenbauer, L. Wien Ak. Sb. 102 (1893) (Ab. 2a) 1059- . connection of difference series with. Saal- schutz, . Konigsb. Schr. 41 (1900) [14]-. deduction from von Staudt's property. Glaisher, J. W. L. [1872] L. Mth. S. P. 4 (1871-73) 212-. definite integrals connected with. Carda, K. Mh. Mth. Ps. 6 (1895) 121-. determination. Presle, de. Par. S. Mth. Bll. 14 (1886) 100-. and differences of IP. Catalan, E. C. Torto- lini A. 2 (1859) 239-. errata in formulae. Blissard, J. QJ. Mth. 9 J.868) 71-. Euler's. Catalan, E. C. [1867] Brux. Ac. Sc. Mm. 37 (1869) 19 pp. . Ely, G. S. Am. J. Mth. 5 (1882) 337-. . Worpitzky, J. Crelle J. Mth. 94 (1883) 203-. . Cesdro, E. N. A. Mth. 5 (1886) 305-, 496. . Berg, F. J. van den. Amst. Ak. Vs. M. 5 (1889) 358-; 6 (1889) 265-; Arch. Neerl. 24 (1891) 99-. . Hermite, C. Prag Sb. (1894) (Mth.- Nt.) No. 37, 4 pp. , and coefficients of power-series. Rogel, F. Arch. Mth. Ps. 10 (1891) 318-. , new theorem on primes. Sylvester, J. J. Ph. Mg. 21 (1861) 127-. , new theory. Lucas, E. C. E. 83 (1876) 539-; A. Mt. 8 (1877) 56-. 228 2910 Bernoulli's Numbers and Euler's, related expressions. Fields, J. C. Am. J. Mth. 13 (1891) 191-. , relations between. Sheppard, W. F. QJ. Mth. 30 (1899) 18-. -, theory. Stern, M. A. Gott. Ab. 23 (1878) (Mth.) 44 pp.; 26 (1880) (Mth.) 45 pp. , . Haussner, E. Gott. Nr. (1893) 777-. expression by definite integral. Hamilton, (Sir) W. R. Ph. Mg. 23 (1843) 360-., formulae of Seidel and Stern. Lucas, E. Par. S.Mth. Bll. 8 (1880) 169-. fractional part. Cesdro, E. A. Mt. 14 (1886- 87) 221-. function related to them, as Tn to n ! Glaisher, J. W. L. B. A. Bp. 42 (1872) (Sect.) 17-. and functions, generalisation. Berger, A. Stockh. Ak. Hndl. Bh. 13 (Afd. 1) (1888) No. 9, 43 pp.; Stockh. Ofv. (1888) 433-; Acta Mth. 14 (1890-91) 249-. fundamental theorems relating to. Glaisher, J.W.L. Mess. Mth. 29 (1900) 49-, 129-. general expression. Genocchi, A. Tortolini A. 3 (1852) 395-. of higher orders. Grigorjev, E. Kazan S. Ps.-Mth. Bll. 7 (1898) 146- ; Fschr. Mth. (1898) 221-. independent expression for. Berger, A. Stockh. Ofv. (1889) 129- ; Fschr. Mth. (1889) 247. representation. Ndgelsbach, H. Z. Mth. Ps. 19 (1874) 219-. logarithms of first 40. Thoman, F. C. B. 50 (1860) 905-. method of finding. Canterzani, S. Mod. Mm. S. It. 11 (1804) 173-. new expression for. Teixeira, J. P. Lisb. J. Sc. Mth. 3 (1895) 73-. recursion formula. Eogel, F. Prag Sb. (1895) (Mth.-Nt.) No. 26, 4 pp. numbers a n< m in proof of Staudt's theorem. Ely, G. S. (xn) J. H. Un. Cir. [2] (1883) analogous to. Matter, K. Ziir. Vjschr. 45 (1900) 238-. and Pascal's arithmetical triangle, relations of. Lucas, E. N. A. Mth. 15 (1876) 497-. properties. Stern, M. A. Crelle J. Mth. 81 (1876) 290- ; Bll. Sc. Mth. 10 (1886) 280-. . Lipschitz, . D. Nf. B. (*1883) 56-. property, proof. Glaisher, J. W. L. Mess. Mth. 2 (1873) 190-. recurring determination. Schlomilch, O. Gru- nert Arch. 3 (1843) 9-. (Schlomilch). Gopel, A. Grunert Arch. 3 (1843) 64-. formulas involving. Glaisher, J. W. L. Mess. Mth. 28 (1899) 36-. and related functions. Le Paige, C.M.M.H.H. (xn) Brux. S. Sc. A. 1 (1877) (Pt. 2) 43-. series, simple method of deducing. Seidel, P. L. von. Munch. Ak. Sb. 7 (1877) 157-. secant coefficients. Schlomilch, 0. Grunert Arch. 1 (1841) 360-. . Meyer, G. F. Grunert Arch. 35 (1860) 449-. .formula for. Schlomilch, 0. Crelle J. 32 (1846) 360-. Special Numbers, etc. 2910 Bernoulli's Numbers, continued. and secant coefficients, general expression for both. Scherk,H.F. Crelle J. 4 (1829) 299-. sum and difference formulae. Bauer, G. [1860] Crelle J. 58 (1861) 292-. table of first 62. Adams, J. C. [1877] Crelle J. Mth. 85 (1878) 269-. tables of 250, and their logarithms. Glaisher, J. W. L. [1871-72] Camb. Ph. S. T. 12 J.873) 384-. their composition from integers and re- ciprocal primes. Zeller, C. Bll. Sc. Mth. As. 5 (1881) (Pt. 1) 195-. theorem. Staudt, G. K. C. von. Crelle J. 21 (1840) 372-. (Staudt). Schlafli,L. QJ. Mth. 6 (1863) 75-. , Staudt-Clausen. Catalan, E. C. Bll. Sc. Mth. As. 4 (1880) 77-. , . Lucas, E. Par. S. Mth. Bll. 11 (1883) 69-. , . Cesdro, E. Brux. Ac. Bll. 20 (1890) 280-. , . Kluyver, J. C. Mth. A. 53 (1900) 591-. , , and Binet's theorem. Lucas, E. N. A. Mth. 16 (1877) 157-. ultra-Bernoullian numbers. Trudi, N. Nap. At. I. Inc. 4 (1867) 105-. Bernoulli's series. Sonin, N. St Pet. Ac. Sc. Bll. 7 (1897) 337-; Fschr. Mth. (1897) 224. , condition for first n terms being a given function. Hill, C. J. D. Stockh. Ofv. 14 (1857) 259-. , constants occurring in summations by. Glaisher, J. W. L. [1872] L. Mth. S. P. 4 (1871-73) 48-. , series of differences of terms of. Markoff, A. Mth. A. 19 (1882) 27-. Cauchy's numbers. Chessin, A. S. A. Mth. 10 (1895-96) 1-. , and their application to astronomy. Bourget, J. Liouv. J. Mth. 6 (1861) 33-. , Bessel's functions, relation between. Chessin, A. S. A. Mth. 12 (1898-99) 170-. Clapeyron's numbers, properties. olin, J. M. Prag Sb. (1878) 146-. Constant which occurs in formula for I 1 . 2 s . 3 s ... n". Glaisher, J. W. L. Mess. Mth. 24 (1895) 1-. Dirichlet's series. Hadamard, . Bordeaux S. Sc. PV. (1896-97) 41-. Discontinuity factors, arithmetical. Eogel, F. Arch. Mth. Ps. 17 (1900) 147-. Discontinuous function, Biemann's. Frischauf, J. Z. Mth. Ps. 34 (1889) 193-. functions. Sokolov, D. N. Bee. Mth. (Mos- cou) 16 (1893) 282-; Fschr. Mth. (1892) 187. logarithmic functions. Bougalef, . C. B. 106 (1888) 1067-. Divisors of number. Hermite, C., & Lipschitz, R. Acta Mth. 2 (1883) 299-. , formula of Hermite. Busche, E. Crelle J. Mth. 100 (1887) 459- ; 103 (1888) 125. , theorems. Bougaief, N. C. B. 120 (1895) 432-. 229 2910 Special Numbers and Numerical Functions 2910 Divisors of number, theorems (Bougaief). Gegenbauer, L. Mb. Mth. Ps. 7 (1896) 26. numbers, theorem of Bugaief. Gegen- bauer, L. Wien Ak. Sb. 95 (1887) (Ab. 2) 219-. Enumerative functions. Cesdro, E. A. Mt. 14 (1886-87) 141-. Equation residues and coefficients, analogy. Roselli, E. Em. Cor. Sc. 1 (1848) 373-. *2 W _ Mangoldt, H. von. Berl. Ak7sb. (1897) 835-. Eulerian and other coefficients, congruence theorem relating to. Glaisher, J. W. L. L. Mth. S. P. 31 (1900) 193-. (/.-function. Busche, E. Mth. A. 31 (1888) 70-. --- . Baker, H. F. [1889] L. Mth. S. P. 21 (1891) 30-. Euler's constant, calculation. Glaisher, . J. W. L. E. S. P. 19 (1871) 514-. -- , expression for. Franklin, F. (xn) J. H. Un. Cir. [2] (1883) 143. -- , history. Glaisher, J. W. L. [1871] Mess. Mth. 1 (1872) 25- ; 2 (1873) 64. -- , logs e 2, 3, 5, 7, 10, to 260 places. Adams, J. C. E. S. P. 27 (1878) 88-. -- , relation connecting constants analogous to. Glaisher, J. W. L. Mess. Mth. 24 (1895) 24-. - -- , series for. Nielsen, N. N. Ts. Mth. 8 (B) (1897) 10- ; Fschr. Mth. (1897) 235. , value to 260 places. Adams, J. C. B. A. Ep. (1877) (Sect.) 14-. functions. Rogel, F. Prag Sb. (1893) (Mth.-Nt.) No. 23, 52 pp. ; (1896) (Mth.-Nt.) No. 2, 45 pp. numbers. Catalan, E. C. C. E. 66 (1868) 415-. - -- . Stern,M.A. CrelleJ.Mth. 79 (1875) 67-. - -- . Lucas, E. [1877] Mess. Mth. 7 (1878) 139-. -- . Eadicke,A. CrelleJ.Mth. 89 (1880) 257-. -- , an analogue of. Studnicka, F. J. Prag Sb. (1900) (Mth.-Nt.) No. 9, 8 pp. ^, coefficients analogous to. Glaisher, J. W. L. L. Mth. S. P. 31 (1900) 216-. Expansion of numerical functions in series. Lerch, . C. E. 108 (1889) 171-. ------ . Rogel, F. Prag Sb. (1897) (Mth.-Nt.) No. 46 [44], 26 pp.; (1900) (Mth.-Nt.) No. 30, 7 pp. -- ce x -l ' Fer 9 la ' E ' Na P- Ac - Sc - Mm - 2 (1855-57) 315-. - -- (f~ ^)" serie s- Lucas, E. [1877] Mess. Mth. 7 (1878) 82-. -- ^ I in powers of x, series resulting from. Oltramare,G. As. Fr.C.E. (1881)117-. Expansions , numerical . Cayley , A. Q J . Mth . 3 (1860) 366-. Factorials, numerical, of 2nd order. Collins, E. [1835] St Pet. Ac. Sc. Mm. 3 (1838) 225-. Formula of Gauss, generalisation. Busche, E. C. E. 129 (1899) 952-. Function connected with divisors of a number. Glaisher, J. W. L. [1890] L. Mth. S. P. 21 (1891) 395-. expressing 1 n - ' Minln, A. P. [1874] (xn) Eec. Mth. (Moscou) 7 (1874-75) (Pt. 1) 485-. relating to (4?n + l) and (4m + 3) divisors of number. Glaisher, J. W. L. L. Mth. S. P. 15 (*1883-84) 104- . Functions, numerical. Liouville, J. Liouv. J. Mth. 2 (1857) 141-, 244-, 377-, 408, 409-, 425-. , . Bellavitis,G. Ven. At. (1858-59) 536-. , . Bouniakowsky, V. [1861] St Pet. Ac. Sc. Mm. 4 (1862) No. 2, 35 pp. >-, . Liouville, J. Liouv. J. Mth. 8 (1863) 347-. , . Bugaev, N. V. [1867] (xn) Eec. Mth. (Moscou) 3 (1868) (Pt. 1) 69-. , . Gegenbauer, L. Wien Ak. Sb. 89 (1884) (Ab. 2) 37-. , , class. Cesdro, E. G. Mt. 25 (1887) 14-. , , geometrical proof. Vries, J. de. Amst. Ak. Vs. 5 (1897) 218-. , , obtained by differentiation. Rogel, F. Arch. Mth. Ps. 10 (1891) 209-. , , problems. Bougaieff, N. V. N. A. Mth. 13 (1874) 381-. T , ., series giving relation between. Minin, A. P. Mosc. S. Sc. Bll. 83 (No. 1) (1893) 23-. ,, theory. Cantor, G. Gott. Nr. (1880) 161-. Generating Junction, numerical, of binary seventhic. Cayley, A. Am. J. Mth. 2 (1879) 71-. functions in theory of numbers, class. MacMahon, P. A. [1893] Phil. Trans. (A) . 185 (1895) 111-. Greatest integer in certain series. Cesdro, E. [1882] Liege S. Sc. Mm. 10 (1883) (No. 6) 355 pp. integers. Gegenbauer, L. Wien Ak. Sb. 94 (1887) (Ab. 2) 611-. . Busche,E. CrelleJ. Mth. 103 (1888) 118-. , sums. Hacks, J. ActaMth. 10 (1887) 1-. , . Mathews, G. B. L. Mth. S. P. 31 (1900) 355-. , , in arithmetical progressions. Zeller, C. Gott. Nr. (1879) 243-. , theorem. Schubert, H. [1895] Hamb. Mth. Gs. Mt. 3 (1900) 223-. , (Schubert). Busche, E. [1895] Hamb. Mth. Gs. Mt. 3 (1900) 225-. Identities, numerical, application to theory of numerical functions. Baskakov, S. I. [1881] (xn) Eec. Mth. (Moscou) 10 (1882- 83) (Pt. 1) 313-. , , Liouville's, deduced from Bugaev's theorem. Sokolov, D. N. [1891] Eec. Mth. (Moscou) 16 (1893) 89- ; Fschr. Mth. (1891) 197. , a source of. Ocagne, M. d'. Par. S. Mth. Bll. 15 (1887) 133-. 230 2910 Special Numbers and Numerical Functions 2910 Inverse arithmetical functions. Cesdro, E. G. Mt. 23 (1885) 168-. Laisant's function Z(l+\/2), calculation. Seydler, A. Casopis 20 (1891) 89-; Fschr. Mth. (1891) 442. , , new formula. Studnicka, F. J. Casopis 20 (1891) 66- ; Fschr. Mth. (1891) 441. Mean and asymptotic expressions in arithmetic. Cesdro, E. G. Mt. 25 (1887) 1-. values of certain numerical functions. La Vallee-Poussin, de. Brux. S. Sc. A. 22 (1898) (Ft. 1) 84-. Mobius-Mertens function. Lerch's theorem. Dintzl, E. Mh. Mth. Ps. 11 (1900) 67-. Number and sum of divisors of number. Hammond, J. Mess. Mth. 20 (1891) 158-, 182-. Numbers with no square factors. Gegenbauer, L. Wien Ak. Sb. 100 (1891) (Ab. 2a) 1054-. representable by S/ r (x r ). Pascal, E. G. Mt. 25 (1887) 45-. * =5 (mod. 12), theorem. Liouville, J. Liouv. J. Mth. 14 (1869) 7-. Q n given by n?Q n -2 a (3n 2 -3ra+l) Q n _^ + 2 2 numbers. Perott, J. Bll. Sc. Mth. As. 5 (1881) (Pt. 1) 37-. Symbol . Rocquigny, F. [G.] de. Les Mondes 48 (1879) 327. Tangent and secant coefficients, independent representation. Rogel, F. Arch. Mth. Ps. 8 (1890) 295-. Totient sums, asymptotic evaluation of certain. Lehmer, D. N. Am. J. Mth. 22 (1900) 293-. Totients, two kinds of fcth, and notation for. Story, W. E. [1881] (xn) J. H. Un. Cir. [1] (1882) 132, 151. Transcendental numerical functions, new. Tagliaferro, N. As. Fr. C. B. (1878) 140-. A and T (Kramp). Argand, . Gergonne A. Mth. 5 (1814-15) 236-. (n), theorems. Catalan, E. C. Liouv. J. Mth. 4 (1839) 7-. . Grunert, J. A. Grunert Arch. 3 (1843) 196-. . Prouhet, E. N. A. Mth. 4 (1845) 75-. . Arndt, F. Crelle J. 31 (1846) 246-. . Oltramare, G. Gen. I. Nt. Mm. 4 (1856) 1-. . Liouville, J. Liouv. J. Mth. 2 (1857) 110- . . Sturm, J. B. Grunert Arch. 29 (1857) 448-. , theorem, proof. Crone, C. (xn) Ts. Mth. 4 (1880) 158-. . Rocquigny, F. [G.] de. Les Mondes 51 (1880) 335-. , and notation of numbers. Eocquigny, F. [(?.] de. Les Mondes 54 (1881) 160-. . Catalan, E. C. Li6ge S. Sc. Mm. 15 (1888) No. 1, 21-. = number of divisors of N, value of N. Minin, A. P. [1893] Bee. Mth. (Moscou) 17 (1895) 538- ; Fschr. Mth. (1893-94) 277. , generalisation. Zsigmondy, K. Crelle J. Mth. Ill (1893) 344-. . .Steggall, J. E. A. Edijib. Mth. S. P. 12 (1894) 23-. , theorems. Moreau, C. N. A. Math. 17 . Seliwanow, D. F. Fschr. Mth. (1899) 172. E (x), and reversion of series. Meissel, E. Crelle J. 48 (1854) 301-. . Liouville, J. Liouv. J. Mth. 2 (1857) 231 2910 Special Numbers, etc. E(x), properties. Stern, M. A. [1860] Crelle J. 59 (1861) 146-. . Sylvester, J. J. C. E. 50 (1860) 732-. , theorems. Bouniakowsky, V. C. E. 94 (1882) 1459- ; St Pet. Ac. Sc. Bll. 28 (1883) 257-, 411- ; 29 (1884) 250-, 503-. . Hermite, . Am. J. Mth. 6 (1884) 173-. , representation by an infinite series. Prings- heim, A. Mth. A. 26 (1886) 193-. . Stern, M. A. Acta Mth. 8 (1886) 93-; 10 (1887) 53-; Crelle J. Mth. 102 (1888) 9-; 106 (1890) 337-. . Mansion, . Brux. S. Sc. A. 13 (Pt. 1) 55-. . Giuliani, . Palermo Cir. Mt. Ed. 5 (1891) 285-. with complex argument. Busche, E. [1892- 93] Crelle J. Mth. 110 (1892) 338- ; Hamb. Mth. Gs. Mt. 3 (1900) 103-. , formulae concerning. Lerch, . G. Teix. J. Sc. 12 (1895) 129-. . Bertolani, G. G. Mt. 34 (1896) 1-. 1) ... . Cistlakov, I. I. [1894] Eec. Mth. (Moscou) 17 (1895) 530- ; Fschr. Mth. (1893-94) 269-. Liouv. J. 2km h . 5 Seguier, P. de. 55-. 2920. Irrationality and transcend- ence of particular numbers, such as and T. (See also 6810.) Eisenstein's theorem on series for root of alge- braic equation. Hermite, C. L. Mth. S. P. 7 (1875-76) 173-. . Gomes Teixeira, F. Arch. Mth. Ps. 3 (1886) 315-. Exponential function. Hermite, C. C. E. 77 (1873) 18-, 74-, 226-, 285-. , irrationality and transcendence. Stieltjes, . C. E. 110 (1890) 267-. Irrational quantities, algebraic, theorem. Liou- ville, J. C. E. 18 (1844) 910-. Transcendental numbers deduced from proba- bilities. Bouniakowski, F. St Pet. Ac. Sc. Bll. 1 (1836) 177 ; St Pe"t. Ac. Sc. Mm. 3 (1838) 457-, 517-. , two theorems. Gronwall, H. Stockh. Ofv. (1897) 623-; (1898) 153-. e. Leclerc, . N. A. Mth. 14 (1855) 225-. Paque, A. Liege Mm. S. Sc. 10 (1855) 199-. Eealis, S. N. A. Mth. 6 (1867) 541- ; 7 (1868) 16-, 158-, 191-. arithmetical character. Borel, E. C. E. 128 (1899) 596-. calculation. Bretschneider, C. A. Grunert Arch. 3 (1843) 27-. . Studnicka, F. J. Casopis 20 (1891) 61- ; Fschr. Mth. (1891) 440-. from continued fraction. Glaisher, J. W. L. B. A. Ep. 41 (1871) (Sect.) 16-. and continued fraction, relation between. Cailler, . As. Fr. C. E. (1889) (Pt. 2) 158-. irrationality. Gruson, J. P. Berl. Ab. (1816- 17) 18-. . Liouville, J. Liouv. J. Mth. 5 (1840) 192-. 232 2920 Transcendental Numbers irrationality. Thue, A. [1888] Arch. Mth. Ntvd. 14 (1890) 251- ; Fschr. Mth. (1888) 263. and transcendence. Jamet, V. N. A. Mth. 10 (1891) 215-. limits for. Ligowski, W. Arch. Mth. Ps. 57 (1875) 220-. new series for. Eados, G. Mth. Termt. Ets. 11 (1893) 358-. transcendence. Cailler, C. As. Fr. C. R. (1891) (Pt. 2) 83-. . Hurwitz,A. Gott. Nr. (1893) 153- ; C. B. 116 (1893) 788-. . Gordan, . C. E. 116 (1893) 1040-. , change in Hermite's first proof. Venske, 0. Gott. Nr. (1890) 335-. value to 205 places. Shanks, W. B. S. P. 6 (1854) 397- ; 20 (1872) 27-. / l\ w e=lim. (1 + -) . Liouville,J. Liouv. J. Mth. V n J 5 (1840) 280. = , expansion when n = oo . Spitzer, S. Grunert Arch. 23 (1854) 127-. = , n an integer. Dcmkin, W. F. [1862] QJ. Mth. 6 (1863) 1-. = , demonstration. Fort, 0. Schlomilch Z. 7 (1862) 46-. i e = lim. (! + #)*, value when x = 0. Schulten, N. G. of (fil.) [1842] Helsingf . Acta 2 (1847) 81. = , expansion. Wittstein, T. Grunert Arch. 3 (1843) 327-. = ,. Frost, P. QJ. Mth. 7 (1866) 381-. e=lim. (1 + ax)*, expansion. Appell, P. Arch. =lim. (1 + ax)*, expansio Mth. Ps. 65 (1880) 171-. e*, irrationality. Hermite, C. B. A. Bp. 43 (1873) (Sect.) 22-. e and IT, approximate numerical theorem. Glaisher, J. W. L. [1877] QJ. Mth. 15 (1878) 125-. , irrationality. Pads, H. Bll. Sc. Mth. 12 (1888) 144-. , transcendence. Gordan, P. Mth. A. 43 (1893) 222-. , . Hilbert, D. Gott. Nr. (1893) 113-. , . Mertens, F. Wien Ak. Sb. 105 (1896) (Ab. 2a) 839-. , . Jamet, V. Ens. Mth. 2 (1900) 355-. Lindemann, C. L. F. Berl. Ak. Sb. (1882) 679- ; Mth. A. 20 (1882) 213-. (Lindemann). Weierstrass, K. Berl. Ak. Sb. (1885) 1067-. approximation. Belli, G. Brugnatelli G. 3 (1820) 409-. . Catalan, E. C. N. A. Mth. 1 (1842) 190-. . Caminati,P. As.Fr.C.B. (1887) (Pt.l) 173. approximations. Barrois, T. Gergonne A. Mth. 4 (1813-14) 360-. calculation. Fanien,A. N. A.Mth.9(1850) 190-. . Dupain, J. C. N. A. Mth. 15 (1856) 82-. . Frisby, E. [1872] Mess. Mth. 2 (1873) 114-. . Glaisher, J. W. L. [1872] Mess. Mth. 2 (1873) 119-. Analysis. General 3190 calculation, elementary. Farcy, A. N. A. Mth. 3 (1844) 582-. construction. Efimov, M. Kazan S. Ps.-Mth. Bll. 5 (1896) (Prot.) 66. convergency of Euler's developments. Gra- bowski, E. Prace Mt.-Fiz. 3 (1892) 130- ; Fschr. Mth. (1892) 232. error by method of isoperimeters. Hermann, A . N. A. Mth. 5 (1866) 509-. formula for. Grebe, E. W. Grunert Arch. 12 (1849) 181-. geometric construction impossible. Saalschiitz, . Konigsb. Schr. 35 (1895) [23]-. hyperbolic analogue. Studnicka, F. J. Liege . S. Sc. Mm. 14 (1888) No. 6, 12 pp. ; Prag Sb. (1892) (Mth.-Nt.) 250-. , Laisant's, calculation. Studnicka, F. J. Liege S. Sc. Mm. 17 (1892) No. 6, 7 pp. logarithm of. Glaisher, J. W. L. L. Mth. S. P. 14 (1882-83) 134-. and its integral powers are incommensurable. Collignon, E. Par. S. Phlm. Bll. 3 (1879) 195-. irrationality. Valat, . Bordeaux Act. Ac. Sc. 26 (1864) 269-. , Lambert's proof, and irrationality of other quantities. Glaisher, J. W. L. B. A. Bp. 41 (1871) (Sect.) 12-. and transcendence, proof. Sylvester, . C. E. Ill (1890) 866-. . (Sylvester). Markow,A.A. Fschr. Mth. (1899) 386. and powers of TT, series and products for. Glaisher, J. W. L. [1877] Mess. Mth. 7 (1878) 75-. series for. Glaisher, J. W. L. [1872] QJ. Mth. 12 (1873) 232-. of Schwab for. Virieu, J. de. N. A. Mth. 18 (1859) 234-. . Tannery, J. Bll. Sc. Mth. As. 5 (1881) 454-. transcendence. Tognoli, 0. G. Mt. 24 (1886) 354-. . Brunei, . Bordeaux S. Sc. Mm. 2 (1891) xi-. transformation of Leibnitz's series for. Polster, F. Arch. Mth. Ps. 63 (1879) 447-. - , Nicholson's formula for, discussion. Strnad, A. fiasopis 19 (1890) 144- ; Fschr. Mth. (1890) 261-. M (Briggs' modulus), value to 205 places. Shanks, W. E. S. P. 6 (1854) 397-; 20 (1872) 27-. (For applications of arithmetical methods to algebraic functions see 4010.) ANALYSIS 3190 General. La Place, P. S. (le marquis) de. Par. EC. Pol. J. 8 (1809) (15 cah.) 229-. Herschel, (Sir) J. F. W. Phil. Trans. (1814) 440-. 233 3190 Analysis. General Libri, G. [1822-30] Tor. Mm. Ac. 28 (1824) 251- ; Ferussac Bll. Sc. Mth. 14 (1830) 1-. Cauchy, A. L. [1827] Par. Mm. Ac. Sc. 8 (1829) 97-. Ostrogradsky, M. A. Ferussac Bll. Sc. Mth. 14 (1830) 249-. Jacobi, C. G. J. C. E. 3 (1836) 59-. Ostrogradsky, . St P6t. Ac. Sc. Bll. a (1838) 209-. Cauchy, A. L. C. B. 11 (1840) 933-. Analysis, algebraic. Bellavitis, G. [1867] Yen. Mm. I. 14 (1868) 1- ; 17 (1872) 189. , application to problems. Quillet, . N. A. Mth. 4 (1845) 249-. , applications. Gors, . Eouen Tr. Ac. (1835) 63-. , controls. Cauchy, A. L. C. E. 44 (1857) 849-. , correlations. Cockle, Jas. Ph. Mg. 24 (1862) 531-. , elementary. Winckler, A. Wien Sb. 59 (1869) (Ab. 2) 356-. , general theorems and their applications. Prouhet, E. C. E. 34 (1852) 359-. , higher. Textor, von. Zach M. Cor. 22 (1810) 202-. , . Janni, G. G. Mt. 14 (1876) 321-. , , method, new. Schell, W. Grunert Arch. 24 (1855) 489 ; 25 (1855) 1-. , methods. Cauchy, A. L. Bb. It. 60 (1830) 202-. , for development. Sigma (S). (vi Adds.) Tilloch Ph. Mg. 65 (1825) 344-. , proofs. Stern, M. A. Grunert Arch. 1 (1841) 57-. , pure, metaphysics. Piola, G. Bb. It. 97 (1840) 315-. ' , report on memoir by Stouvenel. Valat, . Bordeaux Act. Ac. Sc. (1839) 639-. , memoirs by Burmann. Lagrange, J. L., (& Legendre, . Par. Mm. de PI. 2 (1799) (H.) 13-. , researches. Franchini, P. Mod. S. It. Mm. 11 (1804) 254- ; 12 (1805) 160- ; Lucca At. Ac. 5 (1829) 289- ; 7 (1831) 97-. , . JBrioschi, F. C. E. 56 (1863) 659-. , simplification. Eyres, E. S. Tilloch Ph. Mg. 43 (1814) 177-. and synthesis. Toplis, J. Tilloch Ph. Mg. 20 (1805) 193-. . Gergonne, J. D. Gergonne A. Mth. 7 (1816-17) 345-. . Franceschinis, F. M. Padova N. Sag. 1 (1817) 317-. . Zinelli, F. M. Poligrafo 12 (1832) 468. , theory. Murphy, R. [1836] Phil. Trans. (1837) 179-. , transcendental. Galois, E. Gergonne A. Mth. 21 (1830-31) 182-. Analytical aphorisms. Schlomilch, O. Grunert Arch. 5 (1844) 90-. Approximation, methods. Trembley, J. Berl. Mm. Ac. (1803) 50-. Approximations. Schulze, N. W. Crelle J. 13 (1835) 250-. Foundations of Analysis 3200 Approximations. Winckler, A. [1875] Wien Ak. Sb. 72 (1876) (Ab. 2) 623-. Conditional coefficients. Davies, T. S. Mathe- matician 1 (1845) 5-. Functions, notation. Enestrom, G. Bb. Mth. (1891) 89-. Generic equations (expressing a general law of relation). Blissard, J. QJ. Mth. 4 (1861) 279-; 5 (1862) 58-, 185-. Higher mathematics. Magistrini, G. B. Mod. S. It. Mm. 17 (1815) 445-. Multiplicity of values. Cockle, Jas. Ph. Mg. 6 (1853) 444-. Notation. Bjorling, E. G. Stockh. Ak. Hndl. (1845) 75-; Grunert Arch. 9 (1847) 383-; 11 (1848) 39-. , a universal analytical. Mansion, P. Mess. Mth. 4 (1875) 166-. Polynomials, two, approximate linear expres- sions. Tchebitchef, P. L. Bll. Sc. Mth. As. 1 (1877) (Pt. 1) 289-. Pure mathematics, analysis, synthesis, defini- tions, bibliography. Bellavitis, G. (ix) Ven. I. Mm. 15 (1870) 375- ; Ven. I. At. 5 (1879) 673-. Series, etc. Catalan, E. C. [1877-89] Brux. Ac. Sc. Mm. 42 (1878) (No. 3) 32 pp. ; 48 (1892) (No. 1) 98 pp. Synthesis, algebraic. Cauchy, A. L. C. E. 16 (1843) 867-, 967-, 1039- ; 18 (1844) 803-. Transcendental calculations. Paque, A. Liege Mm. S. Sc. 16 (1861) 145-. Foundations of Analysis 3200 General. Analysis, Lagrange 's elementary course . Prony, R. de. Par. EC. Pol. J. 1 (1794) (2 e cah.) 206-. , pure and applied. Prony, R. de. Par. EC. Pol. J. 2 (1795) (6 e cah.) 213-. and theory of numbers, applications. Bervi, N. V. Mosc. S. Sc. Bll. 92 (No. 1) (1896) 1-; Fschr. Mth. (1896) 157-. Analytical expression, general, nature and signification. Raymond, G. M. Chambery Mm. Ac. Sav. 1 (1825) 170-. " Calcul d'exposition." Gruson, J. P. Berl. Mm. Ac. (1799-1800) 157-. Calculus, finite and infinitesimal. Lorgna, A. M. Turin Mm. Ac. 3 (1786-87) 409-. of principal relations. Hamilton, (Sir) W. R. B. A. Ep. (1836) (pt. 2) 41-. Function y = -^~\ - Catalan, E. C. Liege S. Sc. Mm. 15 (1888) No. 1, 230-. Functional systems, characteristics. Kronecker, L. Berl. Ak. Mb. (1878) 145-. , , Kronecker's theory. Dyck, W. [1894] D. Mth. Vr. Jbr. 4 (1897) 94-. Functions, algebraic, of one variable. Hensel, K. D. Mth. Vr. Jbr. 1 (1892) 56-. , classification. H., C. Skandia 10 (1837) 5. 234 Theory of Functions of Real Variables 3210 Functions, transcendental, Liouville's classi- fication. Konigsberger, L. Mth. A. 28 (1887) 483-. which vanish with their variables. Hamil- ton, {Sir) W. R. Ir. Ac. T. 16 (Pt. 1) (1830) 63- ; 16 (Pt. 2) (1831) 129-. . Moore, A. A. [1837] Camb. Ph. S. T. 6 (1838) 317-. Indeterminate expressions, equivalents. Har- J reave, C. J. B. A. Ep. (1857) 184-. uction, andsymbolic formulas. Cauchy,A.L. G. B. 39 (1854) 169-. Infinitely small and infinitely great, two types. Stohs, 0. Mth. A. 31 (1888) 601-. Leibnitz's calculus. Arnoux, E. As. Fr. C. E. (1900) (Pt. 2) 1089-. Limiting values, generalisation and application to curves. Schwarz, C. H. A. A. Mt. 10 (1880-82) 129-. Line functions. Arzela, C. Em. E. Ac. Line. Ed. 5 (1889) (Sem. 1) 342-. Powers, theory. Crelle, A. L. Berl. Ab. (1829) 13-. Principles. Lutter, N. [1860] (xn) Mag. Ak. Ets. (Mth. Term.) 2 (1861-62) 3-. , fundamental. Mansion, P. (xn) Brux. S. Sc. A. 3 (1879) (Pt. 2) 259-. Products and functions, symbolic. Cauchy, A. L. C. E. 43 (1856) 169-. Quantity, idea of, successive generalisation. Hotel, J. [1882] Bordeaux S. Sc. Mm. 5 (1883) 149-. Theorem on product II (x + a*z). Gruson, J. P. Berl. Ab. (1814-15) 1 (M th.) 36-. Theories, various, from one principle. Steen, A. Mth. Ts. 1 (1859) 145-. Transcendental analysis, principles. Lamarle, E. Liege Mm. S. Sc. 2 (1845-46) 221-. 3210 Theory of functions of real variables. Cauchy, A. L. G. E. 43 (1856) 69-. Bing, F. Mth. Ts. 4 (1862) 129-, 154-, 171-. Schendel, L. Crelle J. Mth. 84 (1878) 80-. Baire, R. C. E. 125 (1897) 691- ; A. Mt. 3 (1899) 1-. Stefano, A. de. G. Mt. 38 (1900) 178-. Analytic functions. Lagrange, J. L. Par. EC. Pol. J. 2 (1798) 232-; 3 (1800) Suppl., 277 pp. , development for real value of the variables. Painleve, . G. E. 126 (1898) 385-. , principles and application. Ferroni, P. [1823] Mod. S. It. Mm. 20 (1828) 1-. , problem. Collins, E. [1827] St Pet. Ac. Sc. Mm. 1 (1831) 345-. , rational development. Kummer, E. E. Crelle J. 41 (1851) 368-. Approximation of functions. Lebesgue, . fill. So. Mth. 22 (1898) 278-. Ascending and descending functions. Ceroni, G. A. Sc. Lomb. Yen. 11 (1841) 149-, 197- ; 12 (1842) 123-, 137-. 3210 Calculus of functions. Lagrange, J. L. [1799] Par. 86. EC. Norm. 10 (1801) 6- ; Par. EC. Pol. J. 7 (1808) (14 e cah.) Suppl., 90 pp. indices of functions. Cauchy, A. L. Par. EC. Pol. J. 15 (1837) (25 e cah.) 176-. Classification of functions. Schmidten, H. G. Kiob. Ov. (1827-28) 2-. Coefficients, limiting or restricting (which re- duce to unity when certain conditions are fulfilled, and to zero when not). Cauchy, A. L. G. E. 37 (1853) 150-. Constants of expression, conditions for inde- pendence. Maximovitch, W. de. Liouv. J. Mth. 6 (1880) 167-. Continuity. Luroth, J. [1872] Mth. A. 6 (1873) 319-. . Lamb, H. Manch. Lt. Ph. S. Mm. & P. 41 (1897) No. 10, 9 pp. and discontinuity, distribution of points in pointwise discontinuous functions of a real variable. Schlmflies, A. Gott. Nr. (1899) 187-. of functions. Schlomilch, 0. Grunert Arch. 12 (1849) 430-. , criterion. Stringham, I. As. S. Pac. Pb. 2 (1890) 100- ; 3 (1891) 112. , method of defining. Ruffini, F. P. (xi) Mod. Ac. Sc. Mm. 12 (1871) 3-. , theorem. Darboux, G. Bll. Sc. Mth, As. 3 (1872) 307-. principle, application. Young, J. R. Camb. Phi S. T. 8 (1849) 429-. Continuous and discontinuous functions, dif- ferentiation and representation. Cauchy, A. L. G. E. 40 (1855) 382-. discrete quantities, difference. Toschi, A. Palomba Eac. 3 (1847) 216-. function with discontinuous differential coefficient. Lerch, M. G. Teix. J. Sc. 9 (1889) 97-. not admitting of differentiation, a. Schwarz, H. A. Arch. Sc. Ps. Nt. 48 (1873) 33-, 242 ; Sch. Nf. Gs. Vh. 56 (1873) 252-. with oscillations in every interval. Kopcke, A. Mth. A. 34 (1889) 161; 35 (1890) 104-. functions. Cauchy, A. L. C. E. 18 (1844) 116- ; 20 (1845) 375-. with extreme values in every interval. Konig, J. Mh. Mth. Ps. 1 (1890) 7-. of one real variable. Schee/er, L. Acta Mth. 5 (1884) 183-, 279-. . Broden, T. Crelle J. Mth. . 118 (1897) 1-. . . Kowalewski, G. Leip. Mth. Ps. B. 52 (1900) 214-. two real variables. Townsend, E. J. [1900] N. Y. Am. Mth. S. Bll. 7 (1901) 244, 251-. n variables. Riquier, . Par. EC. Norm. A. 7 (1890) 265-. Demonstration of a lemma. Chavannes, F. Laus. Bll. S. Vd. 2 (1846-48) 284-. Derived functions, class. Formenti, C. Mil. I. Lomb. Ed. 26 (1893) 330-, 382-, 482-. 235 3210 Theory of Functions of Real Variables 3210 Discontinuity, curious examples. Plateau, J. A. F. Brux. Ac. Bll. 43 (1877) 84-, 255-. of functions, theorem. Casorati, F. Mil. I. Lomb. Ed. 1 (1868) 123-. Discontinuous function. Lerch, M. G. Mt. 26 (1888) 375-. functions. Libri, G. Crelle J. 7 (1831) 224-; 10 (1833) 303-. . Boole, G. [1846] Ir. Ac. T. 21 (1848) 124-. . Cauchy, A. L. C. E. 28 (1849) 277-. . Darboux, G. [1874] Par. EC. Norm. A. 4 (1875) 57- ; 8 (1879) 195-. , application to analytic formulae. Libri, G. C. E. 15 (1842) 401-. connected with continuous. Baire, R. C. E. 126 (1898) 1621-. developable by continuous. Baire, R. C. E. 126 (1898) 884-. , real, of real variable, analytical represen- tation. Severini, C. Tor. Ac. Sc. At. 33 (1897) 762- or 1002- ; 34 (1898) 199-, 374- or 325-, 518-. , representation. Rawson, R. [1846] Manch. Ph. S. Mm. 8 (1848) 235-. , sum, theorem of Abel's. Catalan, E. C. Liege S. Sc. Mm. 15 (1888) No. 1, 153-. , theorem, proof. Baire, R. Par. S. Mth. Bll. 28 (1900) 173-. , theory. Piola, G. [1830] Mod. Mm. S. It. 20 (1828) 573-. , . Haan, D. Bierens de. Amst. Vh. 7 (1859) 52 pp. , . Baire, R. C. E. 129 (1899) 1010-. and indeterminate functions. Wead, C. K. [1900] Wash. Ph. S. Bll. 14 (1906) 65-. punctuated functions. Volterra, V. G. Mt. 19 (1881) 76-. Double functions, changes. Decker, G. Grunert Arch. 21 (1853) 423- ; 27 (1856) 471-. Elimination, a functional. Russell, W. H. L. Mess. Mth. 1 (1872) 102. Empiric curves, deduction of true function from. Pucci, E. Em. E. Ac. Line. Mm. 6 (1889) 316-. Functional powers. Collins, E. [1835] St Pet. Ac. So. Mm. 3 (1838) 313-. problems. Abel, N. H. Mg. Ntvd. 1 (1823) 216- ; Grelle J. 2 (1827) 386-. . Decsy, S. (xn) Mag. Tud. Ak. Ets. 7 (No. 6) (1873) 80-. symbols. Cayley, A. QJ. Mth. 4 (1861) 225-. Functions analogous to sine and cosine. Olivier, L. Crelle J. 2 (1827) 243-. , application to pure geometry. Hoist, E. [1879] Par. S. Mth. Bll. 8 (1880) 52-. , approximate representation, application of method of least squares. Schlomilch, O. Schlomilch Z. 3 (1858) 124-. dependent on lines. Volterra, V. Em. E. Ac. Line. Ed. 3 (1887) (Sem. 2) 225-, 274-. determined by certain addition theorem. Zignago, I. Em. N. Line. At. 53 (1900) 139-. , differentiate, with infinitely close (dicht) maxima and minima. Broden, T. Stockh. Ofv. (1900) 423-, 743-. expressible rationally in terms of one another. Woepche, C. Liouv. J. Mth. 4 (1859) 339-. , having differential coefficient at every point, and oscillating infinitely often in every interval. Pereno, I. G. Mt. 35 (1897) 132-. no derivative throughout one interval. Dini, U. A. Mt. 8 (1877) 121-. not expansible by Taylor's theorem. Pringsheim, A. Mth. A. 44 (1894) 41-. oscillating infinitely often, and discontinuous functions. Hankel, H. Mth. A. 20 (1882) 63-. of real arguments, classification by changes in least intervals. Du Bois-Reymond, P. [1874] Crelle J. Mth. 79 (1875) 21-. one real variable. Burnside, W. Mess. Mth. 23 (1894) 39-. . Somigliana, C. Em. E. Ac. Line. Ed. 8 (1899) (-Sem. 1) 4-, 618. . Straneo, P. Em. E. Ac. Line. Ed. 8 (1899) (Sem. 1) 438-, 618. functions, differentiation. Runge, C. Z. , analytic representation. Weier- Mth. Ps. 42 (1897) 205-. strass, K. Berl. Ak. Sb. (1885) 633-, Evaluation of functions in indeterminate form. 789-. Unferdinger, F. Grunert Arch. 26 (1856) , . Mittag-Leffler, G. 224-. Palermo Cir. Mt. Ed. 14 (1900) 217-. Finite and continuous functions having no with derivatives of every order. derivative. Dini, U. Em. E. Ac. Line. T. Maurer, L. Mth. A. 41 (1893) 377-. 1 (1877) 70-, 130-. Fluctuating functions. Hamilton, (Sir) W. R. [1840] Ir. Ac. T. 19 (1843) 264-. Fractional functions, decomposition and trans- formation. Rangoni, L. [non Gabba, A.] and their differential coefficients. Harnack, A. Mth. A. 23 (1884) 244-; 24 (1884) 217-. , discontinuities. Bettazzi, R. Palermo Cir. Mt. Ed. 6 (1892) 173-. , geometric mean value. Lemoyne, G. G. Mt. 16 (1878) 209-. , special class. Study, E. Mth. A. 46 (1896) 298-. Brescia Cm. (1830) 150-. Function satisfying no algebraic functional equa- tion. Holder, O. Gott. Nr. (1887) 662-. Functional analysis with application to paral- x , . lelogram of forces. Anon, (vi 553) Gergonne , theory. Echols, W. H. A. A. Mth. 14 (1823-24) 350-. Mth. 8 (1893-94) 65-. indices. Liventsov, A. L (xn) Eec. Mth. two real variables. Borel, E. Par. EC. (Moscou) 8 (1876) (Pt. 1) 277-. Norm. A. 13 (1896) 79-. 236 3210 Theory of Functions of Real Variables 3210 Functions of two real variables. Biermann, 0. Mth. A. 48 (1897) 393-. , and the idea of arbitrary function. Borel, E. G. R. 121 (1895) 811-. , within a finite range. Ascoli, G. Mil. I. Lomb. Ed. 22 (1889) 317-, 438-, 686-, 804-. several real variables, differentiation and integration. Bettazzi, R. G. Mt. 22 (1884) 133-, 200. with recurring derivatives. Sang, E. [1866] Edinb. R. S. T. 24 (1867) 523-. , representation by a curve (strip). Klein, F. (vm) Erlang. Sb. Ps. Md. S. 6 (1874) 52-. Harmonic functions of two real variables and functions of a complex argument, oscillations in values. Schottky, E. Crelle J. Mth. 117' (1897) 225-. Homogeneous functions, extension of Euler's theorem. Philip, W. E. Edinb. Mth. S. P. 18 (1900) 101-. Implicit functions of a real variable. Ingrami, G. Bologna Rd. (1888-89) 108-. , transformation into explicit. Cauchy, A. L. C. R. 38 (1854) 944-. , monodromic and monogenous. Cauchy, A. L. C. R. 40 (1855) 878-. Integral polynomials of one variable. Hermite, C. N. Ts. Mth. 5 (B) (1894) 1-. Invariability of form. Maiziere, de. Ger- gonne A. Mth. 1 (1810-11) 368-. Irrational functions. Cauchy, A. L. C. R. 32 (1851) 68-, 162-. Isobaric calculus. Cesdro, E. N. A. Mth. 3 (1884) 561- ; 4 (1885) 59-, 578. Lagrange's "Theorie des Fonctions Analy- tiques," note. Poissan, S. D. Liouv. J. Mth. 2 (1837) 140-, 189-. , , Poisson's. Poinsot, L. C. R. 4 (1837) 559-. Limits of functions. Matzka, W. Prag Sb. (1878) 262-. , superior, of functions, calculation. Cauchy, A. L. C. R. 15 (1842) 138-. Mean value, theorem. Stringham, I. As. S. Pac. Pb. 2 (1890) 100- ; 3 (1891) 112. Ordered functions. Somigliana, C. Rm. R. Ac. Line. Rd. 8 (1899) (Son. 2) 125-. Principle, fundamental, elementary proof. Mansion, P. B. A. Rp. (1876) (Sect.) 26-. Properties, behaviour, etc. Scheffler, H. Grunert Arch. 28 (1857) 121-. , general. Ruffini, P. [1806] Mod. S. It. Mm. 13 (1807) 292-. Rational functions, approximate square roots. Bugaev, N. V. [1881] (XH) Rec. Mth. (Moscou) 10 (1882-83) (Ft. 1) 524-. , elements of a theory. Kraus, L. Casopis 14 (1885) 49-; Fschr. Mth. (1885) 369-. Real non-analytic functions, development. Painleve, P. C. R. 126 (1898) 459-. Reciprocal functions. Cauchy, A. L. Par. S, Phlm. Bll. (1818) 178-. Reciprocal relation of functions. Raabe, J. L. [1852] Ztir. Mt. 3 (1853-55) 198-. Simple functions, gradation. Oettinger, L. Crelle J. 11 (1834) 75-, 173- ; 12 (1834) 295-; 13 (1835) 292-. Singularities, new general condensation prin- ciple. Cantor, G. Mth. A. 19 (1882) 588-. Summatorial functions, properties. Hill, C. J. D. Sk. Nf. F. 8 (I860) 518-. Sums. (Bjorling.) Folk, M. Ts. Mt. Fys. 4 (1871) 71-. Systems. Eicci, G. Rm. R. Ac. Line. Rd. 5 (1889) (Sem. 1) 112-. . Stdckel, P. Crelle J. Mth. 112 (1893) 311-. , inversion. Kneser, A. Mth. A. 45 (1894) 446-. Theorem. Grolous, J. Par. S. Phlm. Bll. 12 (1875) 95-. . Tonelli, A. A. Mt. 9 (1878-79) 173-. . Poincare, H. Par. S. Mth. Bll. 11 (1883) 112-. , applications. Krey, H. Z. Mth. Ps. 26 (1881) 357-. on determination of u, v to satisfy uf(x) + vF(x)=E. Cauchy, A. L. Baum- gartner Z. 1 (1826) 88-. Theorems. Prym, F. E. Crelle J. 71 (1870) 223-. . Thomae, C. J. Gott. Nr. (1878) 466-. , two, of Tchebitchef's, generalisation. Minine, A. Les Mondes 55 (1881) 185-. Transformation of functions. Deflers, . Par. S. Phlm. Bll. (1819) 161-. --- . Nicholson, P. Tilloch Ph. Mg. 62 (1823) 168-, 244-, 348-, 433-. --- . Buquoy, G. von. Oken Isis (1827) 130-. --- . Collins, E. St Pet. Ac. Sc. Mm. 1 (1831) 583-. --- . Novi, G. Tortolini A. 1 (1850) 517-. --- . Schlomilch, 0. Liouv. J. Mth. 8 (1863) 99-. , general. Boole, G. Camb. and Dubl. Mth. J. 3 (1848) 112-. --- into isotropic means. Cauchy, A. L. C. R. 34 (1852) 159- ; 43 (1856) 261-. --- by substitution. Collins, E. [1826] St Pet. Ac. Sc. Mm. 1 (1831) 181-. and simplification, methods by interpolation. Eichelmy, P. [1855] Tor. Mm. Ac. 17 (1858) 245-. Weierstrass's function, /(#) = S n & B cos (a n xir). Wiener, L. C. Crelle J. Mth. 90 (1881) 221-. z =f (x,y), graphic representation. Lalanne, L. Par. S. Phlm. PV. (1843) 71-. f(x)f(x + l)...f(x + h), value. Gasparis, A. de. Nap. Rd. 1 (1852) 199-. F(x, y), when a function of (x, y), con- ditions. Dienger, J. Grunert Arch. 21 (1853) 219-. fix), Wronski's expansion. Echols, W. H. N. Y. Mth. S. Bll. 2 - 178-. 237 3220 Infinite Series, etc. 3220 Infinite series; infinite pro- ducts and other infinite pro- cesses. (See also 5610, 5620.) INFINITE SEEIES. Fontana, G. Verona S. It. Mm. 2 (1784) 386-. Paoli, P. Verona S. It. Mm. 4 (1788) 429-. Parseval, M. A. [1799] Par. Mm. Sav. Etr. 1 (1806) 638-. Bianchi, G. [1844] Mod. Mm. S. It. 23 (1846) 239-. Malmsten, C. J. Crelle J. 38 (1849) 1-. Dienger, J. Crelle J. 38 (1849) 266-, 331-. Tremoil, . N. A. Mth. 9 (1850) 66. Bjorling, E. G. Stockh. Ofv. 10 (1853) 147-. Schlomilch, 0. Schlomilch Z. 1 (1856) 180- ; 5 (I860) 132-. Kinkelin, H. Bern Mt. (1858) 88-. Greer, H. R. QJ. Mth. 3 (1860) 262-, 370-. Isely, M. Neuch. Bll. 6 (1861-63) 356-. Cockle, Jas. [1863] (n) Mess. Mth. 2 (1864) 196-. Lindman, C. F. Arch. Mth. Ps. 50 (1869) 219-. Enneper, A. Z. Mth. Ps. 15 (1870) 47-. Schlomilch, 0. X. Leip. Mth. Ps. B. 29 (1877) 101-. Barbour, L. G. Des Moines Anal. 8 (1881) 191-. West, E. Liouv. J. Mth. 7 (1881) 111-. Catalan, E. C. Liege S. Sc. Mm. 12 (1885) No. 2, 148-, 152-, 206-, 282-, 294-. Cesdro, E. G. Teix. J. Sc. 7 (1886) 171- ; 8 (1887) 15-. Biehler, C. N. A. Mth. 6 (1887) 243-. Gutzmer, A. G. Teix. J. Sc. 8 (1887) 81-. Studnicka, F. J. Casopis 16 (1887) 81-; Fschr. Mth. (1887) 227. Weyr, E. G. Teix. J. Sc. 8 (1887) 97-. Catalan, E. Brux. Ac. Bll. 18 (1889) 41-. Tauber, A. Mh. Mth. Ps. 8 (1897) 273-. Absolutely summable series, series t (M) and analytical continuation. Borel, E. C. E. 131 (1900) 830-. Algebraic series. Sidler, G. Ziir. Vjschr. 1 (1856) 180- ; Bern Mt. (1899) 13-. , new. Wallace, J. Silliman J. 7 (1824) 278-; 9 (1825) 98-. . (Wallace). B. (viAdds.) Silliman J. 8 (1824) 131- ; 9 (1825) 293-. Analytic series. Svanberg, J. Stockh. Ak. Hndl. 22 (1801) 30-. , genetic explanation. Barfuss, F, W. Grunert Arch. 8 (1846) 387-. Applications to arithmetic. Sardi, C. G. Mt. 7 (1869) 257-. Approximation to geometric means by series of arithmetical and harmonic means. Mathieu, J. J. A. N. A. Mth. 18 (1879) 529-. series. Meyer, A. N. Ts. Mth. 5 (B) (1894) 81-; 6 (B) (1895) 41-, 73-; Fschr. Mth. (1893-94) 241. Approximations and series. Houzeau, J. C. Brux. Ac. Bll. 41 (1876) 961-. Binomial Theorem 3220 Asymptotic values of certain series. Le Roy, E. Bll. Sc. Mth. 24 (1900) 245-. , infinite approximations, and order of infinity in solution of equations. Du Boia- Reymond, P. Mth. A. 8 (1875) 363-, 574-. Bernoulli's functions. Baraniecki, M. A. Krk. Ak. (Mt.-Prz.) Kz. 13 (1886) 183- ; Fschr. Mth. (1885) 416. Binet's series, and products of a large number of factors. Gilbert, P. Brux. S. Sc. A. 10 (1886) (Pt. 2) 191-. Binomial theorem. Ampere, A. M. Gergonne A. Mth. 15 (1824-25) 369-. Burg, A. Crelle J. 1 (1826) 367-; Wien Jb. Pol. I. 16 (1830) 171-. De Morgan, A. Camb. Mth. J. 3 (1843) 61-. Barfuss, F. W. Grunert Arch. 5 (1844) 155- ; 7 (1846) 3-; 8 (1846) 387-. (Barfuss.) Schlomilch, O. Grunert Arch. 5 (1844) 437-; 10 (1847) 321-. Bianchi, G. [1844] Mod. Mm. S. It. 23 (1846) 239-. Terquem, O. N. A. Mth. 9 (1850) 459-. I'"_ Heine, H. E. Crelle J. 55 (1858) 279-. Hinricks, G. Mth. Ts. 2 (1860) 38-. Janni, V. G. Mt. 12 (1874) 229-, 312. Lanza, G. Des Moines Anal. 1 (1874) 177-. Gohierre de Longchamps, . N. A. Mth. 17 (1878) 101-. applications. Ferroni, P. [1801] Mod. S. It. Mm. 9 (1802) 291-. demonstrations. Little, J. [1810] Ir. Ac. T. 12 (1815) 31-. . Knight, T. Phil. Trans. (1816) 331- ; (1817) 245-. . Herapath, J. Thomson A. Ph. 13 (1819) 365-. (Herapath). Ward, L. T. Tilloch Ph. Mg. 65 (1825) 432. . Colburn, W. Bost. J. Ph. 2 (1825) 474-. . Strong, T. Silliman J. 12 (1827) 132-. . Crelle, A. L. Crelle J. 4 (1829) 305- ; 5 (1830) 187-. . Cerquero, J. S. Quetelet Cor. Mth. 10 (1838) 202-. . Roche, . N. A. Mth. 1 (1842) 42-, 118-. . Cox, H. Camb. and Dubl. Mth. J. 7 (1852) 98-. . Paque, A. Liege Mm. S. Sc. 10 (1855) 1-. . Macnie, J. Des Moines Anal. 7 (1880) 190-. , direct. Gould, D. Silliman J. 19 (1831) 50-. by fluxions. Evans, L. Tilloch Ph. Mg. 64 (1824) 270-. and generalisation. Eisenstein, G. Crelle J. 28 (1844) 44-; N. A. Mth. 8 (1849) 344-. expansion. .Andre, D. N. A. Mth. 10 (1871) 74-. 3220 Infinite Series Binomial Theorem, continued. expansion. Kluyver, J. C. Amst. Ak. Vs. 6 (1898) 421-. , approximate formulas. Astrand, J. J. Grunert Arch. 12 (1849) 420-. , properties derivable from. Young, J. E. Ph. Mg. 33 (1848) 268-. expression in form of a factorial which is always convergent. Kummell, C. H. A. Mth. 12 (1898-99) 60-. fractional index, demonstration. Robertson, A. Phil. Trans. (1806) 305-. or negative index, demonstration. Anon. (vi 1170) Thomson A. Ph. 7 (1816) 346-. , . Eodrigues, O. Liouv. J. Mth. 3 (1838) 550-. , Euler's demonstration. Muirhead, E. F. Edinb. Mth. S. P. 17 (1899) 38-. , extension. Davidov, A. Y. (xn) Eec. Mth. (Moscou) 3 (1868) (Pt. 2) 1-. , historical note. Terquem, 0. N. A. Mth. 6 (1847) 85-. generalised, expansion of factorial by. Cayley, A. Ph. Mg. 6 (1853) 182- ; Mess. Mth. 5 (1871) 102-. geometrical treatment. Koppe, M. Arch. Mth. Ps. 61 (1877) 113-. and logarithmic theorem. Catlin, M. Mth. Misc. 1 (1838) 113-. methods of demonstration. Davies, T. S. Tilloch Ph. Mg. 66 (1825) 115-. negative index, demonstration. Trowbridge, D. Des Moines Anal. 2 (1875) 71-. integral index, demonstration. Grunert, J. A. Grunert Arch. 30 (1858) 336-. particular case. Catalan, E. C. C. E. 45 (1857) 641-. series analogous to binomial series. Jacobi, C. G. J. Crelle J. 32 (1846) 197-. , two groups. Weiss, E. Wien Ak. Sb. 91 (1885) (Ab. 2) 587-. Calculation of series. Poncelet, J. V. [1833] Par. Mm. Sav. Etr. 6 (1835) 641-. Class. Schlomilch, O. Schlomilch Z. 1 (1856) 21-. Classification. Chessin, A. S. J. H. Un. Cir. [15 (1895-96)] 39. Coefficients in a certain series. Bfdrling, E. G. Crelle J. 28 (1844) 284-. , especially factorial coefficients, determina- tion. SchlSmilch, 0. Grunert Arch. 18 (1852) 306-. Conditionally convergent series. Schlomilch, 0. Leip. B. 24 (1872) 327-. , multiplication. Pringsheim, A. Mth. A. 21 (1883) 327-. : , . Voss, A. Mth. A. 24 (1884) 42-. and products. Pringsheim, A. Mth. A. 22 (1883) 455-. Continued identity, method. Kummell, C. H. A. Mth. 5 (1889-90) 85-, iv. Continuity of sums, theorem of Cauchy. I>u Bois-Reymond, P. Mth. A. 4 (1871) 135-. . , two theorems of Abel. Pringsheim, A. Munch. Ak. Sb. 27 (1898) 343-. Convergence 3220 Convergence. Olivier, L. Crelle J. 2 (1827) 31-. (Olivier.) Abel, N. H. Crelle J. 3 (1828) 79-. Cauchy, A. L. G. E. 9 (1839) 587-; 10 (1840) 640- ; 11 (1840) 639-, 667-. Binet, 'J. P. M. Liouv. J. Mth. 6 (1841) 495-. Cauchy, A. L. C. E. 19 (1844) 141-, 157. Malmsten, C. J. Ups. N. Acta S. Sc. 12 (1844) 255-. Schulten, N. G. af (fil.) [1844] Helsingf. Acta 2 (1847) 733-. Cauchy, A. L. C. E. 20 (1845) 126-. Lebesgue, V. A. N. A. Mth. 4 (1845) 66-. Malmsten, C. J. Grunert Arch. 8 (1846) 419-. Terquem, 0. N. A. Mth. 5 (1846) 101-. Timmermans, J. A. Brux. Ac. Bll. 13 (1846) 140-. ' Schaar, M. Brux. Mm. Cour. 4", 22 (1846-47) 9 pp. Schlomilch, 0. Grunert Arch. 14 (1850) 105-. Arndt, F. Grunert Arch. 20 (1853) 43-, 461- ; 21 (1853) 78-. Kinkelin, H. Bern Mt. (1858) 57-. Ermakof, V. Bll. Sc. Mth. As. 2 (1871) 250- ; 7 (1883) 142-. Petersen, J. P. C. (xn) Ts. Mth. 3 (1873) 49-. Grolous, J. As. Fr. C. E. 6 (1877) 209-. Kohn, G. Arch. Mth. Ps. 67 (1882) 63-. Jensen, J. L. W. V. Ts. Mth. 2 (1884) 63- ; Fschr. Mth. (1884) 192-. Mansion, P. Mathesis 5 (1885) 270-. Bugaev, N. V. [1888] Eec. Mth. (Moscou) 14 (1890) 279- ; Fschr. Mth. (1889) 232. Cesdro, E. C. E. 106 (1888) 1142- ; N. A. Mth. 7 (1888) 49-, 190. (Cesaro.) Jensen, J. L. W. V. C. E. 106 (1888) 1520-. Lerch, M. Prag Sb. (1890) 1 (Mth. Nt.) 219-. Cohen, E. N. A. Mth. 10 (1891) 453-. Gilbert, P. Brux. S. Sc. A. 15 (1891) (Pt. 1) 69-. Bryant, E. [1895] L. Mth. S. P. 27 (1896) 69-. Lerch, M. D. Nf. Vh. (1899) (Th. 2, Halfte 1) Sincov, D. M. Eec. Mth. (Moscou) 20 (1899) 616- ; Fschr. Mth. (1899) 228. and approximate value of functions of high numbers. Lobatschewsky , N, (vi Adds.) Kazan Mm. Un. (1835) (cah. 2) 211-. characteristics. Andries, F. Brux. A. Un. (1846) 99-. . Doucet, . N. A. Mth. 8 (1869) 266-. . Hadamard, . C. E. 117 (1893) 844-. for series with positive terms ; and functions increasing indefinitely. Hadamard , J, Acta Mth. 18 (1894) 319-, 421. circle of convergence. Leau, . C. E. 127 (1898) 711-, 794. 239 3220 Convergence Infinite Series Convergence 3220 circle of convergence, convergence and di- vergence on. Thome, L. W. Crelle J. Mth. 100 (1887) 167-. for reverted series. Stolz, O. Ak. Sb. 104 (1895) (Ab. 2a) 485-. , singular points on, and summation of divergent series. Lean, . C. E. 127 (1898) 607-. condensation test, Cauchy's. Hill, M. J. M. Mess. Mth. 26 (1897) 102-. conditions. Liouville, J. C. E. 11 (1840) 615-. . Dirksen, E. H. Berl. B. (1845) 349-. . Fouret, G. N. A. Mth. 9 (1890) 222-. (Fouret). Giudice, . Palermo Cir. Mt. Ed. 4 (1890) 284-. . Giudice, . Palermo Cir. Mt. Ed. 4 (1890) 275-, 276-, 278-, 279, 279-, 280-, 281-, 283. of continued fractions. Markov, A. St Pet. Ac. Sc. Mm. (Es.) 72 (1893) 8- ; Fschr. Mth. (1893-94) 329-. criteria. Schulz von Strasznitzki, L. C. Baumgartner Z. 5 (1829) 10-. . Schlomilch, 0. Z. Mth. Ps. 11 (1866) 354-. . Cajori, F. [1892] N. Y. Mth. S. Bll. 2 (1893) 1-. . Saint-Germain, A. de. C. E. 115 (1892) 1258-. , Bertrand's, generalisation. Petrini, H. Stockh. Ofv. (1899) 691-. , connection. Zahradnik, K. Casopis 7 (*1878) 91-; Fschr. Mth. (*1878) 176. dating from beginning of eighteenth century. Enestrim, G. Stockh. Ofv. 36 (1879) No. 9, 71-. , Gauss's. Hornstein, K. Prag Sb. (1864) 52-. , , simple proof. Pranghofer, J. Wien Sb. 53 (1866) (Ab. 2) 503-. , general. Lerch, M. fiasopis 20 (1891) 285-; Fschr. Mth. (1891) 248-. , Eiemann's. Hurwitz, A. Mth. A. 44 (1894) 83-. , , extension. Arzela, C. [1900] Bologna Ed. 5 (1901) 25-. and divergence. Burg, A. Wien Jb. Pol. I. 17 (1832) 112-. . Raabe, J. L. Baumgartner Z. 10 (1832) 41- ; Crelle-J. 11 (1834) 309-. . Rummer, E. E. Crelle J. 13 (1835) 171-. . Bonnet, 0. [Bennet erroneously inC.S.P.] Liouv. J. Mth. 8 (1843) 73-. . Schlomilch, O. Z. Mth. Ps. 10 (1865) . Aller, C. van. N. Arch. Wisk. 12 (1886) 204-. , conditions. Dirksen, E. H. Berl. Ab. (1832) 77-. , criterion. Hoppe, B. Grunert Arch. 26 (1856) 217-. , limiting regions and so-called limit. Pringsheim, A. Munch. Ak. Sb. 26 (1897) 605-. l ' and divergence of series with positive terms. Du Bois-Reymond, P. C. E. 107 (1888) 941-. Wien . Pringsheim, A. Mth. A. 35 (1890) 297- ; 39 (1891) 125-. . Petrini, H. Stockh. Ofv. (1900) 895-. , limits. Thomae, J. [1871] A. Mt. 5 (1871-73) 121-. , new theory. Du Bois- Reymond, P. [1872] Crelle J. 76 (1873) 61-. or divergence, simultaneous, of two series. Schlomilch, O. Z. Mth. Ps. 18 (1873) 425-. of double series of Lame. Escary, . C. E. 88 (1879) 558-. in theory of heat. Escary, . As. Fr. C. E. (1886) (Pt. 2) 149- ; (1887) (Pt. 2) 1092. in finite difference and in differential series. Crelle, A. L. [1839] Crelle J. 22 (1841) 249-. of hypergeometric series. Horn, J. Mth. A. 34 (1889) 544-. Lambert's series. Stolz, 0. Wien Ak. Sb. 95 (1887) (Ab. 2) 659-. limit of convergence, behaviour of infinite series at. Lasker, E. [1900] Phil. Trans. (A) 196 (1901) 431-. , Du Bois-Eeymond's, and a special form of conditions of convergence. Prings- heim, A. Munch. Ak. Sb. 27 (1898) 303-, 356-. of series representing integrals of differential equations. Picard, E. C. E. 106 (1888) 1466-. limits of convergence, theorem. Cauchy, A. L. C. 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Mth. 14 (1877) 83-. new infinite series from given series by change of sign. Mildner, E. [1882] Wien Ak. Sb. 86 (1883) (Ab. 2) 999-. Derived series, theory. Werner, 0. Grunert Arch. 22 (1854) 264-. Differentiation of products of terms of a series. Collins, E. [1826] St Pet. Ac. Sc. Mm. 1 (1831) 3-. series. Schlomilch, 0. Grunert Arch. 10 (1847) 74-. . Dienger, J. Grunert Arch. 11 (1848) 38. . Lerch, . G. Teix. J. Sc. 11 (1892) 107-. Diminishing series with alternating signs. Oltramare, G. Crelle J. 45 (1853) 345-. Dirichlet's paradox. Unferdinger, F. [1868-69] Wien Sb. 57 (1868) (Ab. 2) 621- ; 60 (1870) (Ab. 2) 591-. . Schlomilch, 0. Z. Mth. Ps. 15 (1870) 134-. series. Lerch, . C. E. 128 (1899) 1310-. , multiplication and non-vanishing. Mertens, F. Crelle J. Mth. 117 (1897) 169-. , theory. Pringsheim, A. Mth. A. 37 (1890) 38-. Discontinuity. Schlomilch, O. Schlomilch Z. 4 (1859) 161-. of values, tests. Duhamel, J. M. C. Liouv. J. Mth. 19 (1854) 112-. Discontinuous series, case. 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(xn) Krk. Ak. (Mt.-Prz.) Pam. 3 (1877) 155- ; 4 (1878) 1-. of transcendental functions, cases. Trudi, N. Nap. At. I. Inc. 4 (1867) 105-. trigonometric. Rogel, F. Prag Sb. (1892) (Mth.-Nt.) 273-. in two different series of powers, comparison. Gopel, A. Grunert Arch. 6 (1845) 25-. uniform, of all positive quantities less than unity. Lilroth, J. [1882] Mth. A. 21 (1883) use of complementary variables. Cauchy, A. L. C. E. 20 (1845) 280-. without use of derivatives. [1895] Echols, W. H. A. Mth. 10 (1895-96) 17-. of (l + x) m , Euler's. Catalan, E. C. Liouv. J. Mth. 9 (1844) 161-. - {(x - xj* +(y- ytf + ('- *!) 2 } -*. Plana, G. G. Arcad. 104 (1845) 182-. f(y) -f( x )> i n powers of y -x. Levanen, S. Helsingf. Ofv. 17 (1875) 1-. F(x), expansion of derivatives in powers of index of derivative. Griinwald, A. C. Prag Sb. (1880) 276-; Prag Ab. (1882) No. 2, 63pp. of (Sa r r )~ 1 (a =l), in powers of x. Gam- bioli, D. [1891] Bologna Ed. (1891-92) 39-. Finite expressions for infinite series. Dienger, J. Crelle J. 34 (1847) 209-. Formula. Glaisher, J. W. L. Mess. Mth. 5 (1876) 164-. Functions ' ' differing as little as possible from zero." Tchebitchef, P. L. Liouv. J. Mth. 19 (1874) 319-. General problem, solution, Andre, D. C. E. 92 (1881) 697-. solution of algebraic problems by series. Marron, F. [1850] Madrid Mm. 2 (1853) 233-. Geodetic positions on spheroid, formulae, ex- pansion. Andrce, . Kjob. Ov. (1859) 27- ; As. Nr. 50 (1859) 161- ; 53 (1860) 369-. Geometric representation. Popper, J. Wien Az. 1 (1864) 34-. series of second order. Hochheim, A. Magdeb. Nt. Vr. Jbr. u. Ab. (1886) 127- ; (1887) 25-. Harmonic series. Kinkelin, H. [1861] Basel Vh. 3 (1863) 18-. . Knar, J. Arch. Mth. Ps. 41 (1864) 297-; 43 (1865) 134-. . Simon, H. Arch. Mth. Ps. 8 (1890) 113- ; 9 (1890) vn. , properties, with table. Hill, C. J. D. Lund. Acta Un. 3 (1866) (Mth.) No. 1, 16 pp. and Stirling's formula. Mansion, P. (xn) Mathesis 1 (1881) 169-. Higher series, new forms. Lukas, F. C. [1881] Arch. Mth. Ps. 67 (1882) 327-. Identities. Glaisher, J. W. L. [1875] Mess. Mth. 5 (1876) 111-. Indeterminate coefficients. Wittstein, T. Grunert Arch. 3 (1843) 300-. . Schlomilch, O. Grunert Arch. 3 (1843) 269-. (Schlomilch). Barfuss, F. W. Grunert Arch. 4 (1844) 225-. (Barfuss). Schlomilch, 0. Grunert Arch. 5 (1844) 374-. (Schlomilch). Barfuss, F. W. Grunert Arch. 7 (1846) 29-. , extension of theory. Pexider, J. Casopis 28 (1899) 277-; Fschr. Mth. (1899) 240. Integrability, case. Arzela, C. Em. E. Ac. Line. Ed. 1 (1885) 321-. Integral and integro-geometric series. Roe, E. D. [1896] A. Mth. 11 (1896-97) 184-. series. Hadamard, J. [1897] Acta Mth. 22 (1899) 55-. with several variables. Lemaire, E. Bll. Sc. Mth. 20 (1896) 286-. Integration. Schlomilch, 0. Grunert Arch. 3 (1843) 278-. . Du Bois-Reymond, P. Berl. Ak. Sb. (1886) 359-. . Fiske, T. S. N. Y. Am. Mth. S. Bll. 3 (1897) 223-. Inversion. Cesdro, E. A. Mt. 13 (1885) 339-. . JeHek, 0. Wien Ak. Sb. 99 (1891) (Ab. 2a) 191-. . Zorawski, K. Prace Mt.-Fiz. 5 (1894) 146- ; Fschr. Mth. (1893-94) 393-. Inversions of terms of given infinite series, and analytic representation of result. Pringsheim, A. Mth. A. 38 (1891) 153- . Irrationality. Stern, M. A. Crelle J. Mth. 95 (1883) 197-. of value, cases. Stern, M. A. Crelle J. 37 (1848) 95-. 2m 1 Limit of S - for m =00. Unferdinger, F. Arch. Mth 1 . Ps. 47 (1867) 231- ; Wien Ak. Sb. 55 (1867) (Ab. 2) 93-. r=m (n-1) S (m+rS) e for m=oo. Unfer- dinger, T F] Wien Sb. 57 (1868) (Ab. 2) 621-. Limited series, remainders. Cauchy, A. L. C. E. 34 (1852) 156-. 244 3220 Infinite Series Limiting values. Hoppe, E. Arch. Mth. Ps. 68 (1882) 37-. of series, Appell's theorem. Cesdro, E. Mathesis 13 (1893) 241-. on circle of convergence. Holder, 0. Mth. A. 20 (1882) 535-. for infinite values of variables, cases. Denzler, W. Ziir. Vjschr. 1 (1856) 333-. Mixed periodic series. Oltramare, G. Gen. I. Nt. Mm. 5 (1857) 1-. series. Studnicka, F. J. Prag Sb. (1892) (Mth.-Nt.) 98-. Moduli of series. Cauchy, A. L. C. B. 17 (1843) 1220-. Multiple and divergent series, formulas result- ing from. Cauchy, A. L. C. B. 20 (1845) 329-. modular series. Cauchy, A. L. C. B. 19 (1844) 1375-. -valued series. De Morgan, A. [1836] Camb. Ph. S. T. 6 (1838) 185-. Multiplication. Cesdro, E. Bll. Sc. Mth. 14 (1890) 114-. . Cajori, F. N. Y. Mth. S. Bll. 1 (1892) 184-. of two series. Stieltjes, T. J. N. A. Mth. 6 (1887) 210-. , rule for. Mertens, F. Crelle J. Mth. 79 (1875) 182-. , . Jensen, J. L. W. V. (xn) Ts. Mth. 3 (1879) 95-. Neutral series. Marrat, W. Thomson A. Ph. 3 (1822) 417-. , theorem. De Morgan, A. [1864-68] Camb. Ph. S. T. 11 (1871) 190-, 239-, 447-. New class. Babbage, C. [1824] Camb. Ph. S. T. 2 (1827) 218-. Numerical calculation. Lerch, M. D. Mth. Vr. Jbr. 8 (1900) (Heft 1) 217-. series and their geometrical applications. Collignon, E. As. Fr. C. B. (1888) (Pt. 2) value of a series. Glaisher, J. W. L. L. Mth. S. P. 8 (1876-77) 200- ; Mess. Mth. 7 (1878) 190-. Paradox. Glaisher, J. W. L. [1871] Mess. Mth. 1 (1872) 97-. Periodic aggregates, transformation. Gylden, H. St Pet. Ac. Sc. Bll. 1 (1894) 379-. series, multiple, theorem.' Enneper, A. QJ. Mth. 1 (1857) 280-. Permanence of equivalent forms. Anon. A. Mth. 6 (1891-92) 81-. Power of a series. Caldarera, G. Acireale Ac. At. 5 (1894) (Sc.) 1-. Power series. Appell, P. C. B. 87 (1878) 689-. Kronecker, L. Berl. Ak. Mb. (1878) 53-. Stieltjes, . C. B. 103 (1886) 1243-. Tognoli, 0. G. Mt. 25 (1887) 155-. Pincherle, S. A. Mt. 21 (1893) 138-. Vivanti, G. A. Mt. 21 (1893) 193-. Fleuri, G. Aust. As. Bp. (1895) 217-. Power Series 3220 Puzyna, J. Krk. Ak. (Mt.-Prz.) Bz. 11 (1897) 270- ; Crc. Ac. Sc. Bll. (1896) 295-. abbreviated, expansion. Jordan, W. As. Nr. 85 (1875) 73-. always divergent. Pincherle, S. C. B. 128 (1899) 407-. and arithmetical series of rth rank, combina- tion. Mildner, E. Wien Ak. Sb. 87 (1883) (Ab. 2) 532-. ascending. Andre, D. Par. EC. Norm. A. 12 (1883) 287-. asymptotic determination. Cesdro, E. Nap. Bd. 32 (1893) 187-. behaviour on circle of convergence. Pringsheim, A. [1885-1900] Mth. A. 25 (1885) 419- ; Munch. Ak. Sb. 30 (1901) 37-. on circle of convergence, and Fourier's series. Pringsheim, A. [1895] Munch. Ak. Sb. 25 (1896) 337-. with coefficients depending on one variable. Vivanti, G. A. Mt. 21 (1893) 25-. coefficients of mth power. Hansted, B. (xn) Ts. Mth. 5 (1881) 12-. convergence and continuity. Arndt,F. Grunert Arch. 25 (1855) 211-. -- limit (Abel's theorem). Catalan, E. Mathesis 4 (1884) 25-. converging, with rational numerical coefficients and prescribed zeros. Hurwitz, A. Acta Mth. 14 (1890-91) 211-. differentiation. Schlomilch, 0. Schlomilch Z. 5 (1860) 292-. generalisation. Mittag-Leffler, G. Stockh. 6fv. (1898) 135- ; Fschr. Mth. (1898) 358-. of a theorem. Seidel, L. Munch. Sb. 2 (1862) 91-. of given function. Gomes Teixeira, F. Crelle J. Mth. 122 (1900) 97-. integers and their reciprocals, transforma- tions. Eogel, F. Arch. Mth. Ps. 10 (1891) 169-. linear criteria of convergence. Van Vleck, E. B. N. Y. Am. Mth. S. T. 1 (1900) 293-, 509. and 'majorant' functions. Delassus, E. Bll. Sc. Mth. 20 (1896) 73-. of a number. Hennessy, J. P. B. A. Bp. (1858) (pt. 2) 4-. property. Konig, G. [1882] (xn) Mth. ; Mth. Nt 1 (1882-83) 73-. ' t. B. Ung. . , . Term. Ets. 1 (1883) 60- ' radius of convergence. Hadamard, . C. B. 106 (1888) 259-. relations between approximate convergents. Frobenius, G. [1879] Crelle J. Mth. 90 (1881) 1-. representing algebraic functions, Eisenstein's criterion. Sutdk, J. Mth. Termt. Ets. 11 (1893) 362- ; Mth. Nt. B. Ung. 12 (1895) 1-. and their remainders. Schlomilch, 0. Schlo- milch Z. 1 (1856) 129-. of two variables. Vivanti, G. Mh. Mth. Ps. 5 (1894) 99-. --- (Vivanti). Zindler, K. Mh. Mth. Ps. 5 (1894) 287-. --- , regions of convergence. Phragme"n, E. Stockh. Ofv. 40 (1883) No. 10, 17-; Fschr. Mth. (1884) 203. 245 Infinite Series 3220 Power Series, continued. of two variables, regions of convergence. Zindler, K. Mh. Mth. Ps. 4 (1893) 115-. unique development of function. Mourner, G. J. D. N. Arch. Wisk. 19 (1892) 100- ; Fschr. Mth. (1892) 360. Problem. Petrovitch, M. N. A. Mth. 15 (1896) 58-. Product of two series of form 1 + a - +a(a + k) j-^ +&c. is of same form. Stainville, de. Ger- gonne A. Mth. 9 (1818-19) 229-. Products and sums, determination of real numbers by. Giudice, F. Tor. Ac. Sc. At. 29 (1894) 188-. Progression, geometrical, theorem. Meech, L. W. Gould As. J. 4 (1856) 65-. Progressions. Toledo, L. 0. de. Fschr. Mth. (1899) 244. , geometrical. Heis, E. Grunert Arch. 6 (1845) 104-. , , of higher order. Eetali, V. G. Mt. 11 (1873) 349-. , , solution of problem supposed inde- terminate. Rotheram, J. Manch. Ph. S. Mm. 3 (1790) 330-. , , theorems. Blackburne, C. Ph. Mg. 6 (1835) 196- ; B. A. Ep. (1837) (ft. 2) 2. of higher order. Bonolis, A. G. Mt. 12 (1874) 179-, 231-. various orders. Cauchy, A. L. C. E. 20 (1845) 2-. . Le Cointe, J. L. A. [1863] A. Mt. 6 (1864) 124-. Properties. Nielsen, N. N. Ts. Mth. 3 (B) (1892) 91- ; Fschr. Mth. (1892) 231-. of a class of series. Lipschitz, R. Crelle J. Mth. 105 (1889) 127- ; 106 (1890) 27-. Property. Schlomilch, 0. Schlomilch Z. 3 (1858) 130-. . Kdnigsberger, L. Mth. A. 27 (1886) 397-. Eational relations between certain series. Nielsen, N. N. Ts. Mth. 8 (B) (1897) 7-; Fschr. Mth. (1897) 228-. Eecurring relations in coefficients of series, especially Bernoulli's numbers. Berg, F. J. van den. [1880?] Amst. Ak. Vs. M. 16 (1881) 74-; Arch. Neerl. 16 (1881) 387-. Eeduced forms (reduites) of exponential function , convergence. Pade,H. C. E. 127 (1898) 444-. Eeduction, general method. Baton de la Goupilliere, J. N. C. E. 44 (1857) 1145- ; Par. FJC. Pol. J. 37 cah. (1858) 97-. Eemarkable class. Bjorling, E. G. Stockh. Ak. Hndl. (1846) 17-; Liouv. J. Mth. 17 (1852) 454-. series. Hessel, J. F. C. Grunert Arch. 5 (1844) 287-. Eepresentation by integrals. Kronecker, L. Crelle J. Mth. 105 (1889) 157-, 345-. Eesidue, Jacobi's method for determination. Scheibner, W. Leip. Mth. Ps. B. 45 (1893) 432-. Eeversion. Canterzani, S. Verona Mm. S It 5 (1790) 88-. Eeversion. Brinkley, J. [1798] Ir. Ac. T. 7 (1800) 321-. . Degen, C. F. As. Nr. 2 (1824) 243-. . Collins, E. [1831] St Pet. Ac. Sc. Mm. 2 (1833) 31-. . Mobius, A. F. Crelle J. 9 (1832) 105-. . Sylvester, J. J. E. S. P. 7 (1854-55) 219-. . Mainardi, G. Mil. At. I. Lomb. 1 (1858) 90-. (Mainardi). Bellavitis, G. Ven. At. (1857- 58) 623- ; (1858-59) 334-. . Glaisher, J. W. L. Mess. Mth. 6 (1877) 142- ; Ph. Mg. 18 (1884) 518-. . McMahon, J. N. Y. Mth. S. Bll. 3 (1894) 170-. , formula. Cayley, A. [1855-57] Crelle J. 52(1856)276-; 54(1857)156-. , general coefficient. Martynowski, J. Brux. Ac. Bll. 4 (1837) 135. Secant series, coefficients. Stern, M. A. Crelle J. 26 (1843) 88-. Semi-convergent series. Glaisher, J. W. L. [1872] QJ. Mth. 12 (1873) 52-. . Stieltjes, T. Par. EC. Norm. A. 3 (1886) 201-. , arbitrary constants as multipliers. Stokes, (Sir) G. G. Camb. Ph. S. P. 6 (1889) 362-. , multiplication. Cajori, F. Am. J. Mth. 15 (1893) 339-; N. Y. Am. Mth. S. Bll. 1 (1895) 180-. , and involution. Cajori, F. Am. J. Mth. 18 (1896) 195-. , order of terms. Manning, H. B. Am. J. Mth. 16 (1894) 151-. , , change. Borel, E. Bll. Sc. Mth. 14 (1890) 97-. , use of Bernoulli's function in ex- pansion. Schlomilch, O. Schlomilch Z. 1 (1856) 193-. Series for any binomial power. Euler, L. [1779] St Pet. Ac. Sc. Mm. 4 (1813) 75. , coefficients of which are in arithmetical progression. Grunert, J. A. Grunert Arch. 9 (1847) 322-. , inverses of binomial coefficients. Roberts, S. Mess. Mth. 9 (1880) 166-. , involve powers of Bernoullian numbers. Glaisher, J. W. L. Mess. Mth. 19 (1890) 138-. ' , proceed by sines and cosines of multiple angles. Tralles, J. G. Berl. Ab. (1820-21) 137-. of continuous functions, continuity of sum. Arzela, C. Bologna Ed. (* 1883-84) 79-. cubes. Simon, 0. E. Grunert Arch. 27 (1856) 313-. and definite integrals. Kummer, E. E. Crelle J. 17 (1837) 210-, 228-. (Kummer). Giuliani, G. G. Mt. 26 (1888) 234-. . Glaisher, J. W. L. Mess. Mth. 22 (1893) 97-. , theory. Pringsheim, A. Mth. A. 37 (1890) 591-. 246 3220 Infinite Series 3220 Series derived from indeterminate equation. Oliveira Ramos, J. C. d\ & Faria, C. J. de. G. Teix. J. Sc. 7 (1886) 41-; Fschr. Mth. (1885) 129-. , each term = sum of two preceding. Clausen, P. A. Mth. Ts. 2 (1860) 168-. , exponents of which are contained in two quadratic forms. Jacobi, C. G. J. Crelle J. 37 (1848) 61-, 221-. of factorials. Schlomilch, O. Leip. B. 11 (1859) 109-. , finite or infinite, and series for inverted functions, powers. Buchwaldt, F. Ts. Mth. 3 (1885) 65-, 192. of functions. Arzela, C. Bologna Bd. (1883) 142-. . Pincherle, S. G. Teix. J. Sc. 11 (1892) 129-. . Arzela, C, Bologna Ac. Sc. Mm. 8 (1899-1900) 701-. , theorem. Arzela, C. Rm. B. Ac. Line. Bd. 1 term A n (x - a^ (x-aj) ... (x- a n ) , historical note. Enestrom, G. C. R. 103 523-. of geometric means, case. Jarrige, A. [1874] (xn) Isere S. Bll. 5 (1876) 409-. integral polynomials representing real functions. Severini, C. Palermo Cir. Mt. Bd. 14 (1900) 157-. involving Bernoulli's numbers. Ohm, M. [1825] St Pet. Mm. Sav. Etr. 1 (1831) 109-. of irreducible fractions. Ocagne, M. d'. Par. S. Mth. Bll. 14 (1886) 93- ; 15 (1887) 33-. _ _ Kronecker : SS (am 2 + 2bmn + era 2 )-". Franel, . Sch. Nf. Gs. Vh. (1896) 58-. and limits. Cockle, (Sir) J. [1875] Mess. Mth. 5 (1876) 25-. for log x, a certain. McClintock, E. [1891] Am. J. Mth. 14 (1892) 67-. of polynomials. Poincare, H. C. B. 96 (1883) 637-. , case. Catalan, E. C. [1880] Brux. Ac. Sc. Mm. 43 (1882) (No. 10) 40 pp. positive terms. Dini, U. G. Mt. 6 (1868) 166-. . Baillaud, B. Par. EC. Norm. A. 6 (1869) 185-. . Giudice, F. G. Mt. 28 (1890) 283-. . Jamet, V. C. E. 114 (1892) 57-. . Sparre, K. E. Arch. Mth. Ntvd. 16 (1893) 195-. , including Euler's. Giudice, F. G. Mt. 27 (1889) 345-. , sums of coefficients. Tchebycheff, P. Acta Mth. 9 (1887) 182-. , . Sonin, N. Ja. [1894] St Pet. Ac. Sc. Bll. 2 (1895) 15-; Fschr. Mth. (1895) 275-. , two theorems. Giudice, . Palermo Cir. Mt. Bd. 4 (1890) 64-. proceeding by inverse factorials. Genocchi, A. Tortolini A. 2 (1859) 367-. Series and products. Liouville, J. Liouv J. Mth. 2 (1857) 438-. , association law. HoSevar, F. Mh. Mth. Ps. 5 (1894) 307-. , (Hocevar). Peano, . Mh. Mth. Ps. 6 (1895) 204. , theorems and proofs. Jensen, J. L. W. V. (xn) Ts. Mth. 5 (1881) 65-. of quantities representing number of values of function of n variables, first terms. Cauchy, A. L. C. B. 21 (1845) 1093-. reciprocal binomial coefficients. Eogel, F. Arch. Mth. Ps. 11 (1892) 412-. representing continuous function without derived function (Weierstrass). La Vallee- Poussin, C. de. Brux. S. Sc. A. 16 (1892) (Pt. 1) 57-. discontinuous functions, property. Seidel, L. Munch. Ab. 5 (1847) 379-. real quantity in given interval. Mollame, V. Catania Ac. Gioen. At. 19 (1886) 129-. reciprocal of polynomial of nth degree, coefficients. Studnicka, F. J. Casopis 15 (1886) 170- ; Fschr. Mth. (1886) 215-. simultaneously convergent and divergent. Schlomilch, O. X. Z. Mth. Ps. 26 (1881) 63-. solving differential equations. Picart, A. N. A. Mth. 2 (1883) 109-. of successive derived functions of function. Pincherle, S. Palermo Cir. Mt. Bd. 11 (1897) 165-. summable by definite integrals. Schlomilch, O. Grunert Arch. 4 (1844) 23-. , terms of which are products of corresponding terms of two given series. Durdn Loriga, J. Fschr. Mth. (1893-94) 409. , contain binomial coefficients as factors. Werner, 0. Grunert Arch. 9 (1847) 219-. for - . Forsyth, A. R. Mess. Mth. 12 (1883) 142-. Slowly converging series, transformation into rapidly converging series. Markoff, A. St Pet. Ac. Sc. Mm. 37 (1890) No. 9, 18 pp. Special type of series. Eogel, F. Arch. Mth. Ps. 7 (1889) 372-. . Ocagne, M. d'. N. A. Mth. 11 (1892) 526-. Sum of all negative powers of all integers except 1 = 1 (Steiner), etc., etc. Stern, M. A. N. A. Mth. 8 (1849) 250-. fractions of form . r^r , . Stern, (2 + x)2+y + l M.A. Crelle J. 25 (1843) 280. geometrical progression. Young, J. R. Ph. Mg. 28 (1846) 10-. . Euzet,. N. A. Mth. 13 (1854) 301-. reciprocals of square numbers. Enestrom, G. Bb. Mth. (1890) 22-. series with terms of unlike sign. Studnicka, F. J. fiasopis 17 (1888) 256-; Fschr. Mth. (1888) 256. Summation of numerical functions, approxi- mate. Halphen, G. H. C. B. 96 (1883) 634-. 247 3220 Summation Infinite Series Summation 3220 Summation* Kausler, C. F. [1808] St P6t. Ac. Sc. Mm. 3 (1811) 137-. Littrow, J. J. von. [1814] St Pet. Ac. Sc. Mm. 7 (1815-16) 110-. Poisson, S. D. Par. S. Phlm. BU. (1822) 134- ; Par. EC. Pol. J. 19 eah. (1823) 404-. Schubert, F. T. [1824] St Pet. Ac. Sc. Mm. 11 (1830) 158-. Phillips, G. Ph. Mg. 11 (1832) 339-, 435-. Bessel, F. W. As. Nr. 16 (1839) 1-. Bawson, B. [1844-47] Manch. Ph. S. Mm. 7 (1846) 464- ; Mathematician 1 (1845) 265- ; 2 (1847) 6-. Oettinger, L. Grunert Arch. 26 (1856) 1-, 212-. Frisby, E. Des Moines Anal. 1 (1874) 121-. Buchwaldt, F. (xn) Ts. Mth. 2 (1878) 76-. Andre, D. C. E. 86 (1878) 1017- ; 87 (1878) 973- ; Par. EC. Norm. A. 8 (1879) 239- ; 9 (1880) 209- ; 12 (1883) 191-. by approximative fractions. Adcock, B. J. Am. J. Sc. 4 (1872) 505. functions of series. Oettinger, L. Crelle J. 16 (1837) 131-. Bernoulli's method. Arndt, F. Crelle J. 31 (1846) 253-. cases. Gauss, C. F. Gott. Cm. 1 (1808-11) 40pp. . Tralles, J. G. Berl. Ab. (1814-15) (Mth.) 179-. . Fuss, N. [1816-21] St Pet. Ac. Sc. Mm. 7 (1820) 194- ; 10 (1826) 115- ; 11 (1830) 230-. . Lamia, K. Baumgartner Z. 3 (1827) 27-, 312-. . Stern, M. A. [1831] Crelle J. 10 (1833) 209-. . Noel, J. N. Quetelet Cor. Mth. 7 (1832) 184-. . Liouville, J. Liouv. J. Mth. 2 (1837) 107-. . Lebesgue, V. A. Liouv. J. Mth. 5 (1840) 42-. . Stern, M. A. Crelle J. 20 (1840) 321-. . Catalan, E. C. Liouv. J. Mth. 7 (1842) . Arndt, F. Grunert Arch. 6 (1845) 187- ; Crelle J. 31 (1846) 235-. . Wolfers, J. P. Grunert Arch. 11 (1848) 419-. (Gauss, 1808). Heine, H. E. N. A. Mth. 9 (1850) 414-. . Catalan, E. C. N. A. Mth. 15 (1856) 293-. . Cayley, A. QJ. Mth. 2 (1858) 167-. . Zehfuss, G. (xi) Z. Mth. Ps. 3 (1858) . Besge, V. A. Liouv. J. Mth. 5 (1860) 367-. . Lindman, C. F. Stockh. Ofv. 20 (1863) 467-. . Unferdinger, F. Arch. Mth. Ps. 41 (1864) cases. Escher, P. Arch. Mth. Ps. 44 (1865) 374-. . Grunert, J. A. Arch. Mth. Ps. 47 (1867) 359-. . Andre, D. N. A. Mth. 10 (1871) 368-. . Wolstetiholme, J. L. Mth. S. P. 4 (1871- 73) 283-. . Falk, M. BU. Sc. Mth. As. 10 (1876) 204-. . Hoppe, B. Arch. Mth. Ps. 61 (1877) 224. . Sgkora, A. Arch. Mth. Ps. 61 (1877) 445-. . Croker, J. M. Mess. Mth. 7 (1878) 60-. . Hoppe, B. Arch. Mth. Ps. 62 (1878) 165-. . Trowbridge, D. Des Moines Anal. 5 (1878) 168-. . Ocagne, M. d'. N. A. Mth. 1 (1882) 171-. . Mandl, M. Wien Ak. Sb. 94 (1887) (Ab. 2) 947-. . Lindman, C. F. Stockh. Ofv. (1888) 69-; Fschr. Mth. 20 (1888) 425-. . Cayley, A. Mess. Mth. 19 (1890) 29-. . Segar,H.W. Mess. Mth. 20 (1891) 142- ; 21 (1892) 145-. . Petersen, J. Sk. Nf. P. (1892) 354-. . Weyr, E. Casopis 21 (1892) 161- ; Fschr. Mth. (1892) 230-. . Schau, E. N. Ts. Mth. 8 (B) (1897) 1- ; Fschr. Mth. (1897) 228. . Her mite, . Kazan S. Ps.-Mth. BU. 8 107-. Heine, H. E. Crelle J. Binet, J. P. M. C. E. - I + T + 9+I6+- *"- L 'V1 St Pet. Ac. Sc. Mm. 3 (1811) 26-. . S -3 and S ^ ^ Clausen, T. Crelle J. 5 (1829) 380- . 31 (1846/133-. . S. 27 (1848) 199-. Schlomilch, O. Schlomilch Z. 3 (1858) 180-. . S Am Ende, . Grunert - . a oP + + a n Arch. 35 (1860) 220-. _. S fi^" ) Du Bois-Beymond, P. Mth. A. 5 (1872) 399-. . S ^ . Ligowski, W. [1877] Arch. Mth. Ps. 62 (1878) 334-. . Sz r . Ligowski, W. [1879] Arch. Mth. Ps. 65 (1880) 329-. i=oo 2* X . S -T '- 5-. Gomes Teixeira, F. G. i=o e* . x + 1 Teix. J. Sc.5 (*1883) 185-. of class of series. Querret, . Gergonne A. Mth. 13 (1822-23) 361-. 248 3220 Summation Infinite Series Summation 3220 of class of series. Frenet, F. [1856] Lyon Ac. So. Mm. 7 (1857) 59-. . Andre, D. G. E. 88 (1879) 740-. compound series and application to proba- bilities. Terrot, C. H. Edinb. E. S. T. 20 (1853) 541-. by continued fractions. Homer, W. G. Thom- son A. Ph. 11 (1826) 416- ; 12 (1826) 48-. of convergent series, cases. Verniory, . Mathesis 12 (1892) 265-. by development of (1 - ax cx z \~ m . Dienger, J. [1847] Crelle J. 43 (1852) 88-. differential formulae. Euler, L. [1780] St Pet. Ac. Sc. Mm. 5 (1815) 45-. of divergent series. De Morgan, A. Assur. Mg. 12 (*1866) 245-. . Borel, E. C. E. 121 (1895) 1125-. elementary. Bode, J. Grunert Arch. 34 (1860) 397-. of elementary series, cases. Nielsen, N. N. Ts. Mth. 7 (B) (1896) 63-; Fschr. Mth. (1896) 208. equation of coefficients occurring in summation. Eytelwein, J. A. Berl. Ab. (1820-21) 1-. and expansion. Dienger, J. Grunert Arch. 7 (1846) 430- ; 8 (1846) 450-. formula. Frenet, F. C. E. 44 (1857) 1274-. . Perrin, E. [1876] Par. S. Mth. Bll. 5 (1877) 47-. . Laisant, C. A. As. Fr. C. E. (1878) 179-. , Dorna's. Curtze, M. Arch. Mth. Ps. 46 (1866) 357- ; 47 (1867) 238-. , Euler's. Franel, J. Mth. A. 47 (1896) 433-. , or Maclaurin's. Malmsten, C. J. Acta Mth. 5 (1884) 1-. involving Bernoulli's numbers. Spitzer, S. Grunert Arch. 23 (1854) 457-. for 3rd power of terms. Hellerung, . Grunert Arch. 2 (1842) 198-. J reciprocal power series. Meyer, G. F. Arch. Mth, Ps. 41 (1864) 220-. , universal. Meech, L. W. Gould As. J. 2 (1852) 121-. formulas. Young, J. E. Ph. Mg. 10 (1837) 121- ; 11 (1837) 41-. . Genocchi,A. Tortolini A. 6 (1855) 70-. , and Bernoulli's numbers. Bauer, G. [1860] Crelle J. 58 (1861) 292-. of Euler and Stirling, complementary terms. Sonin, N. C. E. 108 (1889) 725-. , new. Lucas, E. N. A. Mth. 14 (1875) 487-. of four algebraic series. Glaisher, J. W. L. [1877] Mess. Mth. 7 (1878) 119-. geometric series, 2nd and higher orders, by definite integrals. Glaisher, J. W. L. QJ. Mth. 11 (1871) 328-. harmonic series. Malfatti, G. F. Turin Mm. Ac. 4 (1788-89) 53-. investigations. Lorgna, A. M. Verona S. It. Mm. 1 (1782) 268-. Kummer' s method, auxiliary functions. Bresse, . C. E. 64 (1867) 1023-, 1138-. memoir by Goldbach, 1718. Enestrom, G. Bb. Mth. (1887) 23-. method. Schlomilch, 0. Grunert Arch. 11 (1849) 130-. , -. oon, E. Ph. Mg. 28 (1846) 136-. s. Babbage, C. Phil. Trans. (1819) of negative powers of natural numbers. Schlomilch, 0. X. Leip. Mth. Ps. B. 29 (1877) 106-. neutral series. Young, J. E. Ph. Mg. 27 (1845) 362-, 437-. --- . Moon, E. new methods. 249-. peculiarity. Schlomilch, 0. Grunert Arch. 10 (1847) 45-. of power series. Steen, A. (xn) Ts. Mth. 2 (1878) 183- . powers of reciprocals of numbers. Warming, E. Kjtfb. Dn. Vd. Selsk. Skr. 8 (1895-98) 395-. series of arcs. Dobinski, G. [1878] Arch. Mth. Ps. 63 (1879) 393-. --- circular arcs. Grunert, J. A. Arch. Mth. Ps. 47 (1867) 361-. -- with simple generating functions. Oet- tinger, L. Crelle J. 14 (1835) 262-, 330- ; 15 (1836) 264-, 317-. of sines or cosines of multiple arcs. Clausen, T. Crelle J. 4 (1829) 281-. whose law is known. Gruson, J. P. Berl. Mm. Ac. (1804) 83-. slowly converging and diverging series. Young, J. E. Ph. Mg. 6 (1835) 348-; 7 (1835) 25-. --- series. Kummer, E. E. Crelle J. 16 (1837) 206-. ---- . Lubbock, J. W. As. S. M. Not. 10 (1849-50) 180-. ---- . Fabian, 0. (xn) Krk. Ak. (Mt.- Prz.) Pam. 2 (1876) 37-. ---- (alternate harmonic series), rapid. Janet, A. C. E. 118 (1894) 239-. solution of two problems. Trowbridge, D. Des Moines Anal. 3 (1876) 154-. of terms very far on. Cauchy, A. L. C. E. 32 (1851) 389-. trigonometric and other series. Eogel, F. Arch. Mth. Ps. 12 (1894) 37-. -- series. Adams, Jas. Thomson A. Ph. 15 (1820) 335- ; 4 (1822) 21-. --- . Macully, J. F. Mth. Misc. 2 (1839) 118-. --- . Dirksen, E. H. Berl. B. (1842) 20-; (1843) 83-. --- . Miiller, J. H. T. Grunert Arch. 11 (1848) 439-. --- . Blissard, J. Mess. Mth. 2 (1863) 50-. --- , cases. Schulten, N. G. of (fil.) Stockh. Ak. Hndl. 40 (1819) 184-. --- , . Talbot, W. H. F. Gergonne A. Mth. 14 (1823) 88-, 187-. --- , . Catalan, E. C. N. A. Mth. 3 (1844) 570-. --- , . Dienger, J. Crelle J. 41 (1851) 48-. --- , . Eeinemund, F. Brux. Ac. Bll. 40 (1875) 801- ; 43 (1877) 356-. ,. Stprmer, C. Christiania F. (1892) No. 17, 21 pp. --- , . Nielsen, N. Kjjzfo. Ov. (1896) 348-. --- , class. Glaisher, J. W. L. B. A. Ep. (1879) 269-. 249 3220 Infinite Series Trigonometric Series 3220 Summation, continued. of trigonometric series, class. Eogel, F. Arch. Mth. Ps. 15 (1897) 255-. --- , geometrical. Croker, J. M. QJ. Mth. 3 (1860) 181. Alextief, . Bll. Sc. Mth. As. 1 (1877) 44. two series, by " representative notation." Blissard, J. QJ. Mth. 6 (1863) 167-. Syntagmatic and power series. Cauchy, A. L. C. E. 20 (1845) 463-. Terms deduced from given products of n con- tiguous terms. Fuss, N. [1813] St P6t. Ac. Sc. Mm. 6 (1818) 118- ; 7 (1820) 214- . Theorem. Lejeune-Dirichlet, G. Liouv. J. Mth. 1 (1856) 80-. -. Laurent, H. N. A. Mth. 1 (1862) 126-. . Lerch, M. Acta Mth. 10 (1887) 87-. . Gutzner, A. Mth. A. 32 (1888) 596-. of Abel. Saint-Germain, A. de. N. A. Mth. 4 (1885) 159-. , generalisation. La Maestra, A. C. E. Ill (1890) 782-. Baschwitz. Catalan, E. Brux. Ac. Bll. 18 (1889) 770. Dirichlet. Phragmtn, E. Stockh. Ofv. (1892) 199-. Gauss, proof. RoucM, E. N. A. Mth. 5 (1866) 10-. , . Brisse, C. N. A. Mth. 9 (1870) 36-. Kummer. Cesdro, E. C. E. 106 (1888) 1791-. Theorems. Schlomilch, 0. Leip. B. (1854) 127-. . Winckler, A. Wien SB. 41 (1860) 675-. . Sinram, T. Arch. Mth. Ps. 63 (1879) 103-. . Stieltjes, T. J. (jun.) N. Arch. Wisk. 9 (*1882) 98-. . Jackson, F. H. Edinb. Mth. S. P. 13 (1895) 115-, 160-. , two. Saalschutz, L. Konigsb. Schr. 36 (1895) 67-. Theory. Clausen, T. Crelle J. 3 (1828) 92-. . Bjorling, E. G. Ups. N. Acta S. Sc. 13 (1847) 61-, 143-. . Bonnet, O. Brux. Mm. Cour. 4, 23 (1848-50) 116 pp. . Schlomilch, O. Grunert Arch. 14 (1850) 146-. . Catalan, E. C. C. E. 43 (1856) 626-. (Catalan). Cauchy, A. L. C. E. 43 (1856) . Catalan, E. C. L'l. 26 (1858) 121. . Schendel, L. Z. Mth. Ps. 16 (1871) 211-. . Meissel, E. [1875] Arch. Mth. Ps. 60 (1877) 337-. . Lerch, M. Prag Sb. (1885) (Mth.-Nt.) 174- ; G. Teix. J. Sc. 7 (1886) 79-. . Cesdro, E. N. A. Mth. 9 (1890) 353-, 528. . Lerch, M. Prag Sb. (1891) (Mth.-Nt.) 250-. . Franel, J. Mth. A. 52 (1899) 529-. , application of numerical identities. Sonine, N. J. [1885] Fschr. Mth. (1886) 214-. Theory, elementary. Grunert, J. A. Grunert Arch. 23 (1854) 1-. , general principle. Schmidten, H. G. Kiob. Ov. (1828-29) 2-; Crelle J. 5 (1830) 388-. , new mode of analysis. Herschel, (Sir) J. F. W. Edinb. Ph. J. 2 (1820) 23-. Transformation. Gruson, J. P. Berl. Mm. Ac. (1804) 121-. . Cauchy, A. L. C. E. 11 (1840) 639-, 667-. . Catalan, E. C. [1864-65] C. E. 59 (1864) 618- ; Brux. Mm. Cour. 4, 33 (1867) 50 pp. . Winckler, A. Wien Sb. 51 (1865) (Ab. 2) 291-. (Baschwitz). Catalan, E. Brux. Ac. Bll. 18 (1889) 666-. , case. Euler, L. [1778] St Pet. Ac. Sc. N. Acta 12 (1801) 58-. , . Eisenstein, G. Crelle J. 27 (1844) 193-. , . Schlomilch, 0. Leip. B. 13 (1861) 120-. , . Walton, W. [1862] QJ. Mth. 6 (1863) 22-. , . Catalan, E. C. N. A. Mth. 2 (1863) 107-. , . Pincherle, S. Palermo Cir. Mt. Ed. 2 (1888) 225-. , , ax + bx'* + cx? etc. Strong, T. Am. Ph. S. P. 3 (1843) 139-. Grunert, J. A. Grunert Arch. 26 (1856) 107-. of functions by continued fractions. CebySev , P. St Pet. Ac. Sc. Mm. (Rs.) 9 (*1866) (Suppl.) No. 1, 26 pp. , Leclert's method. Catalan, E. C. Bll. Sc. Mth. As. 1 (1877) 356-. by Stirling. Dietrich, M. [1860] Crelle J. 59 (1861) 163-. Trigonometric series. Herschel, (Sir) J. F. W. Camb. Mm. Anal. S. (1813) 33-. Dippe, M. C. Grunert Arch. 7 (1846) 108-. Heine, E. Crelle J. 71 (1870) 353-. Cantor, G. [1871] Mth. A. 4 (1871) 139- ; 5 (1872) 123- ; Acta Mth. 2 (1883) 329-. Ascoli, G. [1872] Mth. A. 6 (1873) 231-. Cantor, G. [1879] Mth. A. 16 (1880) 113-, 267-. Hoppe, R. Arch. Mth. Ps. 64 (1879) 435-. Poincare, H. C. E. 95 (1882) 766-; 97 (1883) 1471- ; 101 (1885) 1131-. Holder, 0. Mth. A. 24 (1884) 181-. for arbitrary function between limits, con- vergence. Lejeune-Dirichlet, G. Crelle J. 4 (1829) 157- ; 17 (1837) 35-. cases. Frisiani, P. Mil. G. I. Lomb. 5 (1853) 446-. . Trowbridge, D. Des Moines Anal. 4 (1877) . 'Taylor, H. M. Mess. Mth. 7 (1878) 22-. . Beltrami, E. Mil. I. Lomb. Ed. 13 (1880) 402-. . Catalan, E. Brux. Ac. Bll. 23 (1892) 143-. 250 3220 Trigonometric Series cases. Barbette, E. Mathesis 15 (1895) 135-. 30+.... Clausen, T. Crelle J. 8 (1832) 298-. . 2 s- tan ^- . Catalan, E. C. Liege S. Sc/Mm. 15 (1888) ^o. 1, 227-. . Ssin(n!;r7r). Kopcke, H. A. Hamb. Mth 1 . Gs. Mt. 1 (1889) 106-. class. Eogel, F. Prag Sb. (1893) (Mth.-Nt.) No. 33, 25 pp. coefficients from particular values. Bronwin, B. Ph. Mg. 34 (1849) 260-, (vi Adds.) 374-. convergence. Lobatschewsky, N. (vi Jdds.) Kazan Mm. Un. (1834) cah. 2, 226 pp. . Holmgren, H. Liouv. J. Mth. 16 (1851) 186-. . Poincare, H. Bll. As. 1 (1884) 319-. , method of increasing. Cliarlier, C. V. L. Bll. As. 3 (1886) 378- ; Stockh. Ofv. (1886) 157-. , --- . Poincare, H. Bll. As. 3 (1886) 521-. of cosines of multiple angles. Fuss, N. [1809] St Pet. Ac. Sc. Mm. 5 (1815) 115-. deduced from algebraic expansions of $lj-^~ and ev-l. Studnicka, F. J. Casopis 20 (1891) 173- ; Fschr. Mth. (1891) 264. determination of constants. Hunyady, J. (xn) Mag. Tud. Ak. Ets. 5 (No. 14) (1871) 256-. differentiation. Lerch, M. C. E. 119 (1894) 725-; Par. EC. Norm. A. 12 (1895) 351- ; Tor. Ac. Sc. At. 35 (1900) 54-. double. Arzela, C. Bologna Ac. Sc. Mm. 4 (1894) 373-. extension of theorem. Cantor, G. [1871] Acta Mth. 2 (1883) 336-. --- . Arzela, C. Em. B. Ac. Line. Ed. 1 (1885) 637-. multiplication. Pringsheim, A. Mth. A. 26 (1886) 157-. . Bourlet, C. Bll. Sc. Mth. 13 (1889) 55-. with positive coefficients. Lerch, M. Prag Sb. (1898) (Mth.-Nt.) No. 23, 19 pp. series for in terms of sines or tangents of . Rodrigues, 0. Liouv. J. Mth. 8 (1843) 225-. of sines and cosines. Le Cointe, L. A. N. A. Mth. 3 (1844) 518-. ---- of multiple angles. Dirksen, E. H. As. Nr. 1 (1823) 405-, 409- ; Berl. Ab. (1827) 85-; (1829) 169- ; Crelle J. 4 (1829) 170-. theorem. Cantor, G. Crelle J. 72 (1870) 130-. . Appett, P. Arch. Mth. Ps. 64 (1879) 95-. theorems of Cantor and Du Bois-Eeymond, extension to spherical functions. Neumann, C. G. Leip. Mth. Ps. B. 33 (1881) 1-; 35 (1883) 18-; Mth. A. 22 (1883) 406-. theory of convergence applicable to other series. Boussinesq, J. Liouv. J. Mth. 7 (1881) 147-. , fundamental formula. Bonnet, O. Bll. Sc. Mth. As. 3 (1879) 480-. in two variables. Ascoli, G. Em. E. Ac. Line. Mm. 8 (1880) 263-. Infinite Series 3220 Trigonometric Series, continued. for y ( = tan" 1 [m tan x]) , in sines of multiples of x. Saporetti, A. Bologna Ac. Sc. Mm. 8 (1887) 57-. Triple series. Eetali, V. G. Mt. 9 (1871) 266-. Unequally convergent series. Maggi, G. A. Mil. I. Lomb. Ed. 26 (1893) 368-. Utility of series in finding fluent. Young, (Dr) T. Nicholson J. 22 (1809) 213-. Various series. Sarrus, F. Gergonne A. Mth. 10 (1819-20) 217-. . Tchebiche/, P. Liouv. J. Mth. 16 (1851) 337-. . Lebesgue, V. A. N. A. Mth. 18 (1859) 433-, 460-. . Torelli, G. G. Mt. 10 (1872) 129-. . Genocchi, A. Tor. Ac. Sc. At. 10 (1875) 985-. r i" ( , 1 . } . Lorgna, A. M. Verona S. It. r=i \aK T * o) Mm. 2 (1784) 210-. : 2 + Abel,N.H. Crelle J. 1 (1826) 311-. ]n_2 + 3 n ... and l n -3 n + 5 B .... Clausen, T. Crelle J. 7 (1831) 112-. I,e x y. Malmsten, C. J. Ups. N. Acta S. Sc. 12 (1844) 293-. S (M O + MJ +...) = Sw + 2Mj + . . . , conditional proof. Dienger, J. Grunert Arch. 13 (1849) 281-. S p n xP. Lindman, C. F. Grunert Arch. 27 P ~(1856) 291-. S, ^rv. Lebesgue, V. A. N. A. Mth. 18 (a + na) K (1859) 82-. 1 + T + ( H^ 2 + I S ) + ete -> calculation by =0 ^ ' ^ ' I e"e-' ze -z^ Schlomilch, 0. Schlomilch Z.4(1859)433-. - + - + 3 + etc., a, b, c, etc. being in a.b b.c c.d arithmetical progression. Clausen, P. A. Mth. Ts. 3 (1861) 69-. J.+J-+ etc. Frost, P. [1865] QJ. Mth. 7 (1866) 370-. Besso, D. G. Mt. 10 (1872) 160-. 1 + _ L- +... + -_ and elliptic functions. ra + 1 n + 2 2n Catalan, E. C. Brux. Ac. Bll. 34 (1872) 26-, 424-. SS [(x + i) (x + j)]- 1 and other series. Glaisher, J. W. L. [1875] Mess. Mth. 5 (1876) 21. Sw n z n . Ascoli, G. Em. E. Ac. Line. At. 3 (1876) (Pte. 2) 156-. 1 + ... + -,sum(Catalan'sformula), Ti+1 n + 2 in generalisation. Tchebitchef, P. L. As. Fr. C. E. (1876) 114-. 251 3220 Infinite Products 3220 (y 1 - etc., note. Brisse, C. N. A. Mt^ 16 '(1877) 418-. Hermite, C. (xn) Mathesis (1881) 37. . Baehr, G. F. W. (xn) Mathesis 1 (1881) 58. . Catalan, E. C. (xn) Mathesis 1 (1881) 58. f (x) = f (0) + . . . x ( x ~" n ~ 1 fW ( n p) . . . (Abel) . Halphen, G. C. R. 93 (1881) 1003- ; Par. S. Mth. Bll. 10 (1882) 67-. 2jB>/w, Buniakoff sky's formula. Petr, K. (Sasopis 16 (1887) 169- ; Fschr. Mth. (1887) 172-. "2^. Jonquiere, A. Par. S. Mth. Bll. 17 n=l " (1889) 142-. 2 p and 2 ^ . Markoff, A. C. R. 109 (1889) 934-. i-()= 1 ZY-Y. Piltz,. D. Nf. Vh. (1891) /TO-S^W (In. 2) 15-. 2n-. Lafay, A. Toul. Fac. Sc. A. 6 (1892) x-2q) n + ... +rand x n - (x - q) r, J. W. I, 6 pp. x+(x- + (x- 2q) n - . . . r, values. Glaisher L. QJ. Mth. 31 (1900) 193-. 1" (x - l)+2(a?- 2)*+ ... . Glaisher, J. W. L. QJ. Mth. 31 (1900) 241-. INFINITE PRODUCTS. Racagni, G. Mil. Mm. I. Lomb. Yen. 1 (1812- 13) 59-, 341-. Le Francois, (Prof.) . Brux. Mm. Cour. 4, 14 (1841) 71 pp. Schlomilch, 0. [1851] Crelle J. 44 (1852) 344-. Weierstrass, C. [1854] Crelle J. 51 (1856) 1-. Dini, U. [1868] A. Mt. 2 (1868-69) 28-. Rodrigues, J. M. G. Teix. J. Sc. 3 (1881) 87-. Arzela, C. Bologna Ac. Sc. Mm. 4 (1882) 419- ; Bologna Rd. (1885-86) 92-. Giudice, F. G. Mt. 28 (1890) 305-, 380. Chessin,A. S. J. H. Un. Cir. [15 (1895-96) ]38-. Anissimow, W. A. Fschr. Mth. (1897) 221. Pasquale, V. de. G. Mt. 35 (1897) 259-. Application of calculus of residues. Anon. Ferussac Bll. Sc. Mth. 12 (1829) 202-. . Cauchy, A. L. C. R. 17 (1843) 572-. ' Case. Gill, C. Mth. Misc. 2 (1839) 140-. Constant G and some products. Catalan, E. Brux. Ac. Mm. 51 (1893) (No. 2) 28 pp. Convergence. Arndt, F. Grunert Arch. 21 (1853) 78-. . Pringsheim, A. Mth. A. 33 (1889) 119-. . Bruno de Cabedo, J. G. Teix. J. Sc. 10 (1891) 138-. , conditional. Pringsheim, A. Mth. A. 44 (1894) 413-. Decomposition of functions into factors. Cauchy, A. L. C. R. 19 (1844) 1069- ; 30 (1850) 186-. numbers into infinite products. Cantor, G. Z. Mth. Ps. 14 (1869) 152-. Decomposition of numbers into infinite pro- ducts (Cantor). Gegenbauer, L. Mh. Mth. Ps. 3 (1892) 87-. Elementary theory. Genocchi, A. N. A. Mth. 8 (1869) 121-. Expansion. Rogers, L. J. [1893-94] L. Mth. S. 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Mt. 2 (1864) 73-. , , functions of functions. Fais, A. G. Mt. 13 (1875) 47-. , , some functions. Schlomilch, O. Grunert Arch. 4 (1844) 364-. , , , and definite integrals connected with them. Lindman, C. F. [1865] Stockh. Ak. Hndl. 5 (1866) No. 8, 29 pp. , , independent representation. Hoppe, B. Mth. A. 4 (1871) 85-. , , . Jensen, J. L. W. V. (xn) Ts. Mth. 3 (1879) 90-. , , , and summation symbols. Hoppe, E. Crelle J. 33 (1846) 78-. , , of logarithmic functions. Schlomilch, 0. Grunert Arch. 8 (1846) 427-. ,, quotients. Meyer, F. Mh. Mth. Ps. 1 (1890) 33-. , , tangent. Schlomilch, 0. Grunert Arch. 12 (1849) 297-. , , theory. Malmsten, C. J. Stockh. Ofv. 3 (1846) 324-. -, , . Schlomilch, 0. Grunert Arch. 7 (1846) 204- ; Leip. B. 9 (1857) 163-. , , of p _ sin a; 1 + 2y cos x + 7/ 2 ' y + coax l + 2y cos x + y 2 ' Grunert, J. A. Grunert Arch. 3 (1843) 41-. Differential coefficients, higher, of Schlomilch, 0. Grunert Arch. 8 (1846) 357-. -- , , cos" 6. Schlomilch, 0. 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Mth. 16 (1857) 237-, 398-. n F (nx) =f (x) +f ( x + - J + etc. Kinkelin, H. Grunert Arch. 22 (1853) 189-. Formulae, general. Libri, G. Crelle J. 7 (1831) 57-. Function of functions, nth differential. Arez, J. G. Teix. J. Sc. 12 (1895) 110- ; Fschr. Mth. (1895) 307-. "Functionesinexplicabiles," i.e. not expansible by Taylor's theorem. Euler, L. [1780] St Pet. Ac. Sc. Mm. 4 (1813) 88-. Functions, analytical, abrupt changes of value. Du Bois-Reymond, P. [1873] (vn) Mth. A. 7 (1874) 241-. General theorems, two, including Lagrange's and Laplace's. Johnson, A. R. Mess. Mth. 14 (1885) 76-. 3240 Lagrange's Series Indeterminate algebraic expressions. Studnicka, F. J. fiasopis 22 (1893) 321- ; Fschr. Mth. (1893-94) 464. Jacobian of functions, extension of theory. Games Teixeira, F. Mh. Mth. Ps. 1 (1890) 481-. Lagrange's Series. Cayley, A. Camb. Mth. J. 3 (1843) 283-. Menabrea, L. F. [1843] Tor. Mm. Ac. 8 (1846) 91-. Caraffa, A. Palomba Kac. 1 (1845) 43-. Chio, F. [1846] Par. Mm. Sav. Etr. 12 (1854) 340-. Dienger, J. [1847] Crelle J. 42 (1852) 287-. Cayley, A. Camb. and Dubl. Mth. J. 6 (1851) 37-. Cauchy, A. L. C. E. 34 (1852) 309-, 316-, 345-. Heine, H. E. Crelle J. 54 (1857) 388. Tchebicheff, P. St Pet. Ac. Sc. Bll. 15 (1857) 289-. Oliveira, C. B. de. [1860] Ev. Brazil. 4 (1861) Rouche,E. [1861] C. E. 52 (1861) 295-; Par. Mm. Sav. Etr. 18 (1868) 457-. (Eouche.) Bertrand,J. C. E. 52 (1861) 1301-. Laurent, [Herm. non] Henry. C. E. 60 (1865) 25-. Chio, F. Tor. At. Ac. Sc. 7 (1871-72) 647-. Zolotarev, E. I. N. A. Mth. 15 (1876) 422-. Rodrigues, J. M. G. Teix. J. Sc. 4 (1883) 121- ; 8 (1887)59-. Hamy, M. Bll. Sc. Mth. 20 (1896) 213-. analogue. Cauchy, A. L. [1824] Par. Mm. Ac. Sc. 9 (1830) 104-. approximate determination of mth term where m is very great. Nekrasov, P. A. [1900] Eec. Mth. (Moscou) 21 (1901) 439-; Fschr. Mth. (1900) 268. and approximate expressions for functions of large numbers. Nekrasov, P. A. Eec. Mth. (Moscou) 12 (1885) 49-, 315-, 432, 483-, 643-; Fschr. Mth. (1885) 212-. Burmann's series. Dienger, J. [1868] Prag Ab. 3 (1870) 25 pp. controversy. Genocchi, A. Tor. At. Ac. Sc. 8 (1872-73) 18-. convergence, Cauchy's and Lagrange's con- ditions. Menabrea, L. F. C. E. 77 (1873) 1358-. , (Menabrea). Genocchi, A. C. E. 77 (1873) 1541-. demonstration. Miller, T.H. Edinb. Mth. S. P. 4 (1886) 45-. and forms, new. Meech, L. W. Des Moines Anal. 3 (1876) 33-. demonstrations, three. Rothe, H. A. Kastner Arch. Ntl. 12 (1827) 1-. expansion derived from. Glaisher, J. W. L. [1877] Mess. Mth. 7 (1878) 118-. extension. Peirce, B. Gould As. J. 5 (1858) 164. generalisation. Moth, F. X. Baumgartner Z. 7 (1830) 64-. Other Series 3240 generalisation. Gomes Teixeira, F. [1876-79] Lisb. J. Sc. Mth. 5 (1876) 203- ; 7 (1880) 247-. . Cesdro, E. N. A. Mth. 4 (1885) 316-. . Stieltjes, T. J. Par. EC. Norm. A. 2 (1885) 93-. . Bassani, A. Yen. I. At. (1886-87) 1163-. .Laplace's. Schlfijli,L. QJ. Mth. 2 (1858) 24-. requiring proof. Schlafli, L. Bern Mt. (1848) 97-. interpretation. Menabre'a, L. F. [1846] Tor. Mm. Ac. 10 (1849) 111-. Lagrange-Cauchy formula. Zinin, N. N. Vars. S. Nt. Tr. (1897) (Ps. C.) Fasc. 2, No. 6. remainder. Ramus, C. Kiob. Dn. Yd. Selsk. Afh. 9 (1842) 365-. . Popoff, A. C. E. 53 (1861) 795-. successive differentiations connected with. Knar, I. Baumgartner Z. 3 (1827) 175-. summation by definite integrals. Parseval, M. A. [1804] Par. Mm. Sav. Etr. 1 (1806) 567-. . Gruson,J.P. Berl. Ab. (1812-13) (Mth.) 31-. and summation of series. Dienger, J. Crelle J. 34 (1847) 75-. symbolic theorems derived from. Glaisher, J. W. L. QJ. Mth. 16 (1879) 263-. Laplace's series. Mainardi, G. Em. At. N. Line. 20 (1867) 164-. . Darboux, G. C. E. 68 (1869) 324-. . Saint-Germain, A. L. de (xn) Caen Ac. Mm. (1880) 112-. , remainder. McClintock, E. Am. J. Mth. 4 (1881) 96. , summation. Dirksen, E. H. Berl. Ab. (1839) (Mth.) 49-. Maclaurin's series. Adams, Jas. Thomson A. Ph. 1 (1821) 98-. . Hermite, C. [1877] Crelle J. Mth. 84 (1878) 64-. . Genocchi, A. C. E. 86 (1878) 466-. . Callandreau, O. C. E. 86 (1878) 589-; Par. S. Mth. Bll. 15 (1887) 23-. .convergence. Timmermans, J. A. Brux. Ac. Bll. 13 (1846) 53-. , demonstration, direct. Dienger, J, Grunert Arch. 12 (1849) 93-; 13 (1849) 333-. , legitimate use. Jacobi, C. G. J. Crelle J. 12 (1834) 263-. , limits. Craufurd, A.Q.G. Camb. Mth. J. 3 (1843) 84-, 159-. for real variable. Callandreau, 0. C. E. 104 (1887) 38-. , and application to potential of a homogeneous body. Callandreau, 0. C. E. 103 (1886) 864-, 954. , symbolic form, applications. Walton, W. Camb. Mth. J. 4 (1845) 176-. , theory. Scheibner, W. Leip.B. 9 (1857) 190-. theorem, demonstration. Hayes, J. S. Des Moines Anal. 8 (1881) 149-; 9 (1882) 12-. , expansion of circular functions by. Rutherford, W. Mathematician 3 (1850) 35-. , remainder. Veneziani, C. Am. J. Mth. 7 (1885) 168-. 261 3240 Maxima and Minima 3240 MAXIMA AND MINIMA. of function y = Slee, T. Thomson A. Ph. 13 (1819) 193-. Verdam, G. J. Leijd. A. Ac. (1823-24) 100 pp. Verhulst, P. F. Leijd. A. Ac. (1823-24) 30 pp. Marianini, P. D. Mod. S. It. Mm. 25 (1855) (pte. 2) 347-. Gibbes, L. R. Camb. (M.)Mth.M. 1 (1859)335-. Kleinfeller, . Z. Mth. Ps. 13 (1868) 515-. Genese, B. W. [1871] Mess. Mth. 1 (1872) 33-. _ of algebraic functions. Bary, E, (vn) Ger- gonne A. Mth. 22 (1831) 37-. . Valat, . Bordeaux Act. Ac. Sc. (1848) 625-. , new method. Cassani, P. Ven. Aten. At. 1 (1878) 55-. polynomials. Trowbridge, D. Des Moines Anal. 2 (1875) 1-. analytic theory. Schulten, N. G. of (fil.) [1839] Helsingf. Acta 1 (1840) 171-. application to central conies and quadrics. - Berard, . Gergonne A. Mth. 3 (1812-13) 105-. of Homer's and Sudan's substitutions method. Spitzer, S. Grunert Arch. 23 (1854) 100-. symbol of discontinuity. Walton, W. - Camb. and Dubl. Mth. J. 3 (1848) 201-. of arc of curve. Euler, L. [1779] St Pet. - Ac. Sc. Mm. 2 (1810) 3-. . Steiner,J. Crelle J. 17 (1837) 83-. - case. Frigelius, A. Stockh. Ak. Hndl. 37 (1816) 1-. conditional. Cauchy, A. L. C. B. 24 (1847) 757-. conjugate. Elliott, E. B. Mess. Mth. 14 (1885) 182-. criteria. Brioschi, F. Mil. G. I. Lomb. 3 (1851) 400- ; Tortolini A. 3 (1852) 322-. of 2nd degree. Noel, J. N. Quetelet Cor. Mth. 2 (1826) 71-, 135-. determination. Guldberg, C. M. Mth. Ts. 1 (1859) 113-. by interpolation, formulae. Lehmann, J. W. H. Grunert Arch. 25 (1855) 237-. distance problem. Reiss, M. Quetelet Cor. Mth. 6 (1830) 368-. distances of point from curve. Bedetti, G. Bologna N. Cm. 9 (1849) 455-. . Brenner, . Grunert Arch. 13 (1849) 167-. duplication of results. Elliott, E. B. Mess. Mth. 9 (1880) 121-. elementary. Holmgren, H. Ts. Mt. Fys. 1 (1868) 31-, 73-. . Phragmtn, L. Ts. Mt. Fys. 2 (1869) 201-. _ of explicit function. Gaudard, J. A. Gen. Civ. 2 (1873) 476-. Format's method, application to tangents and centres of gravity. Duhamel, J. M. C. C. E. 50 (1860) 741-. of figures, on plane or sphere, or in space. Steiner, J. Crelle J. 24 (1842) 93-, 189-. function on closed surface. Stieltjes, T. J. As. Fr. C. B. (1887) (Pt. 1) 168. of radii vectores from moving point to fixed centres. Lindelof, L. L. [1866] Helsingf. Acta 8 (1867) 189-. 262 . Catalan, E. C. (xn) Mathesis 2 (1882) 5-. functions. Turazza, D. A. Sc. Lomb. Ven. 10 (1840) 68-. . Lindman, C. F. Stockh. Ofv. 10 (1853) . Bjorling, E. G. Stockh. Ofv. 10 (1853) 161-. . Vashchenko-Zakharchenko, M. E. (xn) Bee. Mth. (Moscou) 3 (1868) (Pt. I) 217-. . Haluschka, F. Wien Ak. Sb. 83 (1881) (Ab. 2) 1092-. of 1 real variable. Arzeld, C. Palermo G. Sc. Nt. 14 (1879) 55-. . variable. Ermakov, V. P. [1892] Kharkov. Mth. S. Com. 3 (1893) 155- ; Fschr. Mth. (1892) 253-. or 2 variables. Walton, W. [1882] Mess. Mth. 12 (1883) 14-. more variables. Gergonne, J. D. Gergonne A. Mth. 20 (1829-30) 317-. 2 variables. Fisher, A. M. Silliman J. 5 (1822) 82-. . B. (vi Adds.) Silliman J. 8 (1824) 133-. . Bouniakowsky, V. [1829] St Pet. Ac. Sc. Mm. 1 (1831) 463-. . D. Camb. Mth. J. 2 (1841) 138-. . Aldis, W. S. QJ. Mth. 6 (1864) 301-. . Schee/er, L. Leip. Mth. Ps. B. 38 (1886) 102-. . Posse, K. A. Bee. Mth. (Moscou) 14 (1890) 591- ; Fschr. Mth. (1890) 286. . Ermakov, V. P. [1892] Bee. Mth. (Moscou) 16 (1893) 415- ; Fschr. Mth. (1892) 254-. . Bukreev, B. Ja. Bee. Mth. (Moscou) 16 (1893) 813-. . Dantscher, V. von. Mth. A. 42 (1893) 89-. . Vivanti, G. Em. B. Ac. Line. Ed. 7 (1898) (Sem. 1) 240-. , criteria. Walton, W. [1861] QJ. Mth. 5 (1862) 20-. 3 variables. Cayley, A. Camb. and Dubl. Mth. J. 1 (1846) 74-. . Bukreev, B. Ja. [1897] Bee. Mth. (Moscou) 20 (1899) 92-; Fschr. Mth. (1897) 250. n variables. Francais, J. F. Ger- gonne A. Mth. 3 (1812-13) 132-, 197-. . Schubert, F. T. [1815] St Pet. Ac. Sc. Mm. 6 (1818) 267-. . Brashmann, N. D. (xn) Mosc. Un. Mm. 8 (1835) 131-. . Bertrand, J. Liouv. J. Mth. 8 (1843) 155-. . Bouniakowsky, V. [1857] St Pet. Ac. Sc. Mm. 9 (1859) 153-. _. Zmurko, L. [1866] Wien D. 27 (1867) (2te Ab.) 63-. . Williamson, B. QJ. Mth. 12 (1873) 48-. 3240 Maxima and Minima 3240 of functions of n variables. Walton, W. [1882] Mess. Mth. 12 (1883) 42-. . Novikov, P. M. (xn) Kharkov Mth. S. Com. (1883) 43-. . Stolz, O. Wien Ak. Sb. 99 (1891) (Ab. 2a) 495-; 100 (1891) (Ab. 2a) 1167- ; 102 (1893) (Ab. 2a) 85-. . Mayer, A. Leip. Mth. Ps. B. 44 (1892) 54-. . Pierpont, J. N. Y. Am. Mth. S. Bll. 4 (1898) 535-. . Dantscher, V. von. Mth. A. 51 (1899) 227-. , application of Hessian deter- minants. Studnicka, F. Prag Sb. (1868) 67-. , characteristics. Studnicka, F. Prag Sb. (1870) 12-. , tests for distinction. Stolz, O. Wien Sb. 58 (1868) (Ab. 2) 1063- . geometrical application. Schlomilch,O. Grunert Arch. 9 (1847) 448-. , cases. G. Camb. Mth. J. 3 (1843) 237-. methods. Alexandrov, I. Fschr. Mth. (1892) 256. , and normals to curves and surfaces. Peano, G. Palermo Cir. Mt. Kd. 2 (1888) 189-. of implicit functions, and laws of reciprocity in ordinary maximum and minimum. Mayer, A. Leip. Mth. Ps. B. 41 (1889) 308-. indeterminate. Walton, W. Camb. Mth. J. 4 (1845) 253-. interpretation of imaginary roots. Alvord, B. Camb. (M.) Mth. M. 2 (I860) 237-. maxima maximorum, and minima minimorum. Cauchy, A. L. C. E. 17 (1843) 1215-. maximum solid of given surface and minimum solid of given volume. Babinet, J. C. E. 63 (1866) 361-. method. Dirksen, E. H. Berl. B. (1838) 145- ; Berl. Ab. (1841) 105-. ascribed to Fermat. Mansion, P. (xn) Mathesis 2 (1882) 193-. neutral case. Korn, A. [1900] Munch. Ak. Sb. 30 (1901) 235-. paradox. Euler, L. [1779] St Pet. Ac. Sc. Mm. 3 (1811) 16-. problems. Canterzani, S. [1808-14] Mod. Mm. S. It. 14 (1809) 167- ; 17 (1815) 241-. . Mossbrugger, L. Grunert Arch. 2 (1842) 400-. . Bonnet, 0. N. A. Mth. 2 (1843) 417- ; 3 (1844) 64-. . Mossbrugger, L. Grunert Arch. 4 (1844) 373-. . Grunert, J. A. Grunert Arch. 31 (1858) 476-; 38(1862)475-. . Battaglini, G. Nap. Ed. 2 (1863) 56-. . Grunert, J. A. Arch. Mth. Ps. 49 . Enneper, A. Z. Mth. Ps. 25 (1880) 41-. . Langley, E. M. Nt. 37 (1888) 605. . Markov, A. A. Kharkov Mth. S. Com. 1 (1889) 250- ; Fschr. Mth. (1889) 267-. to find x so that sin a; 2 sin (0 - x) is maximum or minimum. Grunert, J. A. Grunert Arch. 26 (1856) 354-. problems, geometrical. Sturm, J. C. F. Ger- gonne A. Mth. 14 (1823-24) 108-. , . Liouville, J. Liouv. J. Mth. 7 (1842) 163-. , . Thue, A. Sk. Nf. F. (1892) 352-; Fschr. Mth. (1892) 259. , , class. Schell, W. Grunert Arch. 19 (1852) 450-. , sphere touching cone. Alvord, . Smiths. Misc. Col. 33 (1888) Art. 1, 149. (Wash. Ph. S. Bll. 6 (1884).) , . Farquhar, H. Smiths. Misc. Col. 33 (1888) Art. 1, 152. (Wash. Ph. S. Bll. 6 (1884).) in turning and planing. Tobell, J. Giving. 35 (1889) 569-. question propounded by Cebysev. Markoff, A. Acta Mth. 9 (1887) 57-. questions. Gelin, (I'abbe) E. (xn) Mathesis 2 (1882) 164-. relative, application of Euler's principle. Zimmermann, -W. A. Fschr. Mth. (1899) 344-. , . Sleszinski,J.W. Fschr. Mth. (1899) 344, 345. , Euler's and Lagrange's methods. Strauch, G. W. Grunert Arch. 4 (1844) 39-. , new point of view. Kummell, C. H. A. Mth. 4 (1888) 33-. of stationary values. Nanson, E, J. Mess. Mth. 25 (1896) 129-. Steiner's theorems. Sturm, R. Crelle J. Mth. 96 (1884) 36-. of sum of powers of distances of point from system of points, lines, or planes. Wetzig, F. Crelle J. 62 (1863) 346-. sums of function and its derivatives. Cebygev, P. [1867] St Pet. Ac. Sc. Min. (Es.) 12 (*1868) (Suppl.) No. 3, 74 pp.; Liouv. J. Mth. 13 (1868) 9-. symmetric function of several variables. Bouniakowsky, V. St Pet. Ac. Sc. Bll. 12 (1854) 353-. without Taylor's theorem. Hill, E. Mess. Mth. 5 (1876) 84-. theorem. Purkiss, H. J. Mess. Mth. 1 (1862) 180-. . Walton, W. [1869] QJ. Mth. 10 (1870) 253-. . Cayley, A. QJ. Mth. 10 (1870) 262-. , demonstration. Adams, Jas. Thomson A. Ph. 10 (1817) 430-. theory. Spitzer, S. Grunert Arch. 22 (1853) 183-. . Bjorling, E. G. Stockh. Ofv. 13 (1856) 83-. . Eichelot, F. J. As. Nr. 48 (1858) 272-. . . Brenner, . Grunert Arch. 35 (1860) 157-. . Transon, A. Par. EC. Pol. J. Cah. 42 (1867) 205-. . Unferdinger, F. Arch. Mth. Ps. 53 (1871) 15-. . Gruber, J. Arch. Mth. Ps. 60 (1877) 415-. . Mayer, A. Leip. Mth. Ps. B. 41 122-. 263 3240 Maxima and Minima; etc. 3240 Maxima. circle, point and diameter. Lanavicensis (Syl- vester, J. J.). Mess. Mth. 2 (1863) 100-, 168. elementary method. Grillet, H. N. A. Mth. 9 (1850) 70-. of factorial functions. Cayley, A. Mess. Mth. 2 (1873) 129-. geometrical question. Vachette, A. N. A. Mth. 5 (1846) 45-. maximum maximorum of function \f/(rc^ 1 ) in reference to . Nekrassoff, P. Mth. A. 31 (1888) 337-. problem. Dienger, J. Grunert Arch. 14 (1850) 221-. . Rodenberg, C. Z. Mth. Ps. 24 (1879) 63-. Gilbert, L. P. (xn) Mathesis 3 (1883) 200-. of a product. Deladere&re, A. N. A. Mth. 3 (1844) 165-. . Tychsen, C. Mth. Ts. 1 (1859) 22-. question. Serret, P. N. A. Mth. 2 (1863) 79-. questions, two. Lebesgue, F. A. N. A. Mth. 2 (1863) 433-. relative, question. Ostrogradsky , . (vm) St Pet. Ac. Sc. Bll. 8 (1841) 327-. theorem. Terquem, 0. N. A. Mth. 5 (1846) 207-. value of Zlx and theorems in connection. Oettinger, L. Arch. Mth. Ps. 42 (1864) 106-. of *^. Bacaloglo, E. Grunert Arch. 36 (1861) 12-. Minima. case. Korkine, A., <& Zolotareff, G. N. A. Mth. 12 (1873) 337-. geometrical, determination of remarkable. Cellerier, C. Bll. Sc. Mth. 15 (1891) 167-. of linear function of irrational quantity, La- grange's theorem, demonstration. Sylvester, J. J. C. B. 53 (1861) 1267- ; 54 (1862) 53-. minimum area cut off by normal to ellipse. Lemoine, E. Fschr. Mth. (1893-94) 470. of polygon circumscribing re-entering curve. Goodwin, H. Camb. and Dubl. Mth. J. 3 (1848) 181-. distance. Ginger, V. Ja. [1891] Eec. Mth. (Moscou) 16 (1893) 317-. , Format's problem. Callegari, P. Bologna N. Cm. 4 (1840) 309-. quantities. Posse, K. [1880] St Pet. Ac. Sc. Mm. (Rs.) 38 (*1881) (Suppl.) No. 3, 31 pp. straight line, problem. Roche, . N. A. Mth. 2 (1843) 488-. problems. Dilettant (Pseud.). Mth. Ts. 5 (1863) 17-. . Tychsen, C. (xn) Ts. Mth. 1 (1871) 70-. problems. Hoppe, R. Arch. Mth. Ps. 62 (1878) 215-. . Brims, H. Mth. A. 20 (1882) 455-. . Markov, A. A. Kharkov Mth. S. Com. (1884) 83-; Fschr. Mth. (1885) 369. . Millar, W. J. Edinb. Mth. S. P. 7 (1889) 27-. , elementary solution. Tychsen, C. Mth. Ts. 4 (1862) 106-. , geometrical. Gruson, J. P. Berl. Ab. (1816-17) 11-. _ ( _. Neovius, E. Mth. A. 31 (1888) 359-. , railway construction. Pollack, V. Forster Al. Bauztg. 51 (1886) 45-, 53-. , sum of distances from vertices of tetra- hedron. Lindeldf, L. L. Les Mondes 10 (1866) 691-. questions. Puissant, L. Par. EC. Pol. Cor. 1 (1804-08) 430-. . Liouville, J. Liouv. J. Mth. 1 (1856) 7-. . Posse, C. Mth. A. 26 (1886) 593-. in connection with approximate representa- tion of functions. Tchebiche/, P. [1857] St Pet. Ac. Sc. Mm. 9 (1859) 201-. of transport. Dupain, J. C. N. A. Mth. 13 (1854) 141-. theorem for straight lines in space. Hossard, P., <& Poudra, . N. A. Mth. 9 (1850) 241- . in triangle, theorem. Poudra, . N. A. Mth. 7 (1848) 407-. Mean value, theorem. Bruno de Cabedo, J. G. Teix. J. Sc. 9 (1889) 129- ; Fschr. Mth. (1889) 234. Numerical identities discovered by means of differentiation and integration. Minin, A . P. Mosc. S. Sc. Bll. 78 (No. 1) (1892) 22-; Fschr. Mth. (1892) 187. Partial differences calculus applied to analytical problems. Sarrus, F. Gergonne A. Mth. 10 (1819-20) 33-. Prices, laws. Launhardt, . Hann. Archt.- Vr. Z. 32 (1886) 293-. Quadratic irrationals for representation of function. Pincherle, S. Mil. I. Lomb. Bd. 23 (1890) 373-. Bailway fares, best scale. Launhardt, . Hann. Archt.-Vr. Z. 37 (1891) 113-. Bemainder in expansion of (1 + x)~ n , and theorem on products of squares. Young, J. R. Ph. Mg. 34 (1849) 113-. Bepeated differentiation, formulae. Chrystal, G. Edinb. Mth. S. P. 3 (1885) 94-. Bodrigues's theorem. Hudson, W. H. H. [1877] Mess. Mth. 7 (1878) 117. . Glaisher, J. W. L. [1879] Mess. Mth. 9 (1880) 55-. , two demonstrations. Walton, W. QJ. Mth. 15 (1878) 335-. Series, new, for f\a + sin 2 "0 (1 - k sin 2 <)", solution. Schon- bichler, K. Wien SB. 16 (1855) 447-. '-y A~I~. Grunert, J. A. Grunert [ I J Arch. 27 (1856) 362-. , evaluation. Gauss, C. F. *Ja + bx -f ex 2 Grunert Arch. 30 (1858) 229-. l\/a*-x 2 dx in connection with other in- tegrals. Fischer, . Grunert Arch. 38 (1862) 151-. dx ~/(^ Spitzer, S. Grunert Arch. 40 (1863) 168-. [ x m dx I , a . Hermite, C. A, Mt. 1 (1867-68) 155-. am i 187 Integrals I e" in x cos (nx - cos x) dx, &c. Unfer- dinger, F. Wien Sb. 57 (1868) (Ab. 2) 611-. I x n . cos {wtlg (a + bx) } dx, &c. Unferdinger, Wien Sb. 59 (1869) (Ab. 2) 437-. \ CO s?m+i x dx. Realis, S. N. A. Mth. 13 (1874) 83-. I tj>(x \dx=.\ (x)dx, proof. Glaisher, J. W. L. Mess. Mth. 4 (1875) 186-. I (u) n du, infinite values. Schroder, E. Z. Mth. Ps. 25 (1880) 106-. , (Schroder). Schlomilch, 0. X. Z. Mth. Ps. 25 (1880) 117-. I f(z) dz over a surface containing no singu- larity of/ (z). Rados, I. [1884] Mth. Termt. Ets. 3 (1885) 46-; Mth. Nt. B. Ung. 3 (1884- 85) 186-. lf(x)cosxdx, and the equation -^ + y=f(x). Studntika, F. J. Casopis 13 (1884) 12-; Fschr. Mth. (1884) 235. I e" x f (x) dx. Gomes-Teixeira, F. Em. E. Ac. Line. Ed. 1 (1885) 278-. e z) Bierens de Haan, D. (1873) 135-. finite differences. Libri, G. Crelle J. 12 (1834) 240-. given by Schlomilch, corrections. Unfer- dinger, F. Arch. Mth. Ps. 52 (1871) 252-. with indefinite limits. Fade, H. Par. fee. Norm. A. 6 (1889) 349-. infinite limits. La Vallee-Poussin, C. de. Brux. S. Sc. A. 16 (1892) (Pt. 1) 6-, (Pt. 2) 150-. Gilbert, P. Bll. Sc. Mth. 12 (1888) 66-. , . Hocevar, F. Mh. Mth. Ps. 4 (1893) 177-. and infinite series. Kummer, E. E. Crelle J. 17 (1837) 210-, 228-. . Malmsten, C. J. Crelle J. 38 (1849) 1-. (Kummer, 1837). Giuliani, G. G. Mt. 26 (1888) 234-. , class, calculation. Stokes, G. G. Ph. Mg. 37 (1850) 68- ; Camb. Ph. S. T. 9 (1856) 166-. Pringsheim, A. Mth. cos (qx*>), sin (qx p ). (is.) Arch. Neerl. 8 A. 37 (1890) 591-. integrable by quadrature or division of circle. Eichelot,F.J. Crelle J. 21 (1840) 293-. 276 3260 Definite Integrals (simple) Definite integrals involving exponential integral. Glaisher, J. W. L. QJ. Mth. 18 (1882) 370-. in mechanics of motion of point on surface. Rouche, E. Liouv. J. Mth. 3 (1858) 337-. obtained by symbolic methods. Bromvin, B. Camb. and Dubl. Mth. J. 3 (1848) 243-. with periodic factor. Chartier, J. Liouv. J. Mth. 18 (1853) 201-. in planetary motion, numerical expression. Lefort, F. Liouv. J. Mth. 11 (1846) 142-. of products of functions, inequalities limiting value of. Imshenetskii , V. G. (xn) Kharhov Mth. S. Com. (1882) 99-. , relation between. Andreief, C. Bordeaux S. Sc. Mm. 2 (1886) 1-. . , theorems of Cebysev and Imseneckij. Andreev,K.A. (xn) Kharkov Mth. S. Com. (1882) 110-. [ve~ x dx reducible to Arndt, F. Grunert Arch. 10 (1847) 233-. representing functional operations. Pincherle, S. Acta Mth. 10 (1887) 153-. . Viterbi, A. Em. E. Ac. Line. Ed. 6 (1897) (Sen. 1) 247-. logarithms and Euler's constant. Sonin, N. Ja. Vars. S. Nt. Tr. (Mm.) 1 (1891) No. 6, 15 pp. and series. Glaisher, J. W. L. Mess. Mth. 22 (1893) 97-. summation of series. Graindorge, J. Liouv. J. Mth. 18 (1873) 129-. sums deduced from Bernoulli function. Raabe, J. L. Crelle J. 42 (1851) 348-. in theory of heat. Russell, W. H. L. Camb. and Dubl. Mth. J. 8 (1853) 157-. . Meyer, G. F. Mth. A. 3 (1871) 157-. uniformly convergent. Selivanoff, . Par. S. Mth. Bll. 10 (1882) 147-. used for expanding functions in series. Deflers, . Par. S. Phlm. Bll. (1819) 161-. integration. Rawson, R. [1844] Manch. Ph. S. Mm. 7 (1846) 464-. as element of operation in mathematics. Viterbi, A. A. Mt. 26 (1897) 261- ; 3 (1899) 299-. , paradox. Walton, W. QJ. Mth. 6 (1864) 319-. , theorem. Glaisher, J.W.L. QJ. Mth. 10 (1870) 347-. , , general. Boole, G. Liouv. J. Mth. 13 (1848) 111- ; Camb. and Dubl. Mth. J. 4 (1849) 14-. , theorems. Glaisher, J. W. L. [1878] Mess. Mth. 8 (1879) 63-. Definition of definite integral. Fouche, M. N. A. Mth. 15 (1896) 207-. . Burali-Forti, C. N. A. Mth. 15 (1896) 495-. as limiting value of a sum. Thomae, C. J. Z. Mth. Ps. 21 (1876) 224-. Derivation of two definite integrals. Zehfuss, G. Schlomilch Z. 7 (1862) 445-. Determinant of definite integrals. Tissot, A. Liouv. J. Mth. 17 (1852) 177-. Evaluation 3260 Determinant of definite integrals. Enneper, A. Z. Mth. Ps. 11 (1866) 69-. Determinantal and generating functions of Abel. Bassani, A. Nap. Ac. At. 7 (1895) No. 9, 19 pp. Determination of limits to values. Winckler, A. Wien Ak. Sb. 90 (1885) (Ab. 2) 528-. by series. Frullani, G. Mod. S. It. Mm. 20 (1828) 663-. Differentiation. Lindman, C. F. Stockh. Ofv. 38 (1881) No. 6, 3-. . Lerch, M. Prag Ceske Ak. Fr. Jos. Ez. (Tfida 2) 2 (1893) Art. 34, 15 pp. ; Fschr. Mth. (1893-94) 479. . Hallgren, E. Goteb. Hndl. 32 (1897) 83-; Fschr. Mth. (1897) 255. Dirichlet's integral. Kronecker, L. Berl. Ak. Sb. (1885) 641-. Discontinuous integrals connected with series for (l-2px + x 2 )~l. Hamilton, (Sir) W. R. Ph. Mg. 20 (1842) 288-. Division into elements, mode. Petrovitch, M. C. E. 122 (1896) 27-. Evaluation. Ellis, R. L. Camb. Mth. J. 3 (1843) 185- ; Liouv. J. Mth. 9 (1844) 422-. Lobatschewsky, W. Erman Arch. Es. 14 (1855) 232-. Gegenbauer, L. Wien Ak. Sb. 64 (1871) (Ab. 2) 703-. Todhunter, I. E. S. P. 23 (1875) 300-. Hocevar, F. [1876] Wien Ak. Sb. 74 (1877) (Ab. 2) 155-. Appell, P. C. E. 87 (1878) 874-. Spitzer, S. Arch. Mth. Ps. 62 (1878) 221-. approximate. Dirksen, E. H. Berl. Ab. (1831) 117-. . Schlomilch, 0. Grunert Arch. 9 (1847) 215-. . Grunert, J. A. Grunert Arch. 14 (1850) 225-. . Araldi, A. [1859] (ix) Mod. Ac. Sc. Mm. 3 (1861) 65-, 137. . Andr(x}.^ (x) dx. Korkine, A. G. E. 96 (1883)326-. Theorems. Winckler, A. Wien SB. 21 (1856) 389-. . Gegenbauer, L. Wien Ak. Sb. 97 (1889) (Ab. 2a) 1053-. , connection. Walton, W. [1872] QJ. Mth. 12 (1873) 126-. of Liouville, demonstrations. Glaisher, J.W.L. Mess. Mth. 4 (1875) 170-. , generalisation. Rawson, E. Mess. Mth. 5 (1876) 30-. Theory. Schlomilch, 0. Grunert Arch. 1 (1841) 417-. . Grunert, J. A. Grunert Arch. 2 (1842) 266-; 4 (1844) 113-. (Grunert). Meyer, U. H. Grunert Arch. 5 (1844) 216-. . Russell, W. H. L. [1854] Phil. Trans. (1855) 157-. . [Bierens de] Haan, D. Amst. Vs. Ak. 4 (1856) 332- ; 16 (1864) 28- ; 1 (1866) (Ntk.) 117- ; 3 (1869) (Ntk.) 323- ; 5 (1871) (Ntk.) 65-; 7 (1873) 12-; Arch. Neerl. 2 1867) 349-. . Enneper, A. Schlomilch Z. 6 (1861) 289-. . Gomes Teixeira, F. G. Teix. J. Sc. 9 (1889) 39-; Fschr. Mth. (1889) 276. . Schiff, P. Eec. Mth. (Moscou) 17 (1895) 607- ; Fschr. Mth. (1895) 321-. , elementary. Meyer, A. Liege Mm. S. Sc. 7 (1851) 510 pp. , formulae. Cauchy, A. L. [1815] Par. EC. Pol. J. 28 cah. (1841) 147-. , . Andreiewsky,. Mth. A. 4 (1871)550-. , foundations. Haan, D. Bierens de. Amst. Arch. Wisk. Gn. 1 (1856-59) 343-. Transcendents, functional, reduction. Glaisher, J. W. L. Mess. Mth. 1 (1872) 153-. , squares of. Pendlebury, R. [1871] Mess. Mth. 1 (1872) 131-. Transformation. Bassani, A. G. Mt. 25 (1887) 223-. Nielsen, N. Kjrfb. Ov. (1896) 335-. Kapteyn, W. N. Arch. Wisk. 4 (1899) 243-. and calculation. Blatek, G. [1864] Wien Sb. 50 (1865) (Ab. 2) 60-. cases. Schaar, M. Brux. Ac. Bll. 13 (1846) (pte. 2) 30-. . Schlomilch, 0. Crelle J. 36 (1848) 271-. . Hoppe, R. Crelle J. 40 (1850) 139-. . Besge, V. A. Liouv. J. Mth. 18 (1853) 112-. Transformation 3260 cases. SchlGmilch, 0. Leip. B. 9 (1857) 181- ; Schlomilch Z. 3 (1858) 115-. . Unferdinger, F. Arch. Mth. Ps. 53 (1871) 27-. . Glaisher, J. W. L. [1876] Mess. Mth. 6 (1877) 155-. . Silva, J. A. Martins da. G. Teix. J. Sc. 3 (1881) 177-. by Cebysev's formula, generalisations. And- reev, K. A. (xn) Kharkov Mth. S. Com. (1883) 19-. and determination. Cauchy, A. L. C. E. 12 (1841) 1145-. and evaluation. Haan, D. Bierens de. Amst. Vh. 8 (1862) 702 pp. formula. Jacobi,C.G.J. Crelle J. 15 (1836) 1-. general. Kummer, E. E. Crelle J. 20 (1840) theory. Dirksen, E. H. Berl. B. (1848) 152-. in variation. Sabinin, E. T. [1890] Eec. Mth. (Moscou) 15 (1891) 99- ; Fschr. Mth. (1890) 380. of D^ =0 (l-2fccos7 + fc 2 )~i into definite in- tegrals. Dirksen, E.H. Berl. B. (1848) 120-. Transition from indefinite to definite integrals. Heine, H. E. Crelle J. 51 (1856) 382-. Use of multiform functions in finding definite integrals. Raabe, J. L. Crelle J. 37 (1848) 356-. l 1 x n dxe~ axSm fix. Fuss, N. [1810] St Pet. J o cos Ac. Sc. Mm. 11 (1830) 238-. I e~ x2 dx = ^Jw. Buzengeiger, K. H. J. Zach M. Cor. 24 (1811) 169-. i . Frullani, G. Mod. S. It. Mm. 19 (1821) 223-. * z i dzF(cos z). Frullani, G. Mod. S. It. Mm. 20 (1828) 712-. 1 x 2m dx cos ax Plana, G. [1836] Quetelet Cor. Mth. 9 (1837) 47-. (sin ax, cos bx) dx, reduction. Raabe, J. L. Crelle J. 15 (1836) 355-. (x) dx when (x) is infinite or discon- tinuous. Raabe, J. L. [1837] Crelle J. 20 (1840) 173-. log (1 - 2a cos x + a 2 ) dx. Delaunay, C. Liouv. J. Mth. 3 (1838) 355-. cos as da Catalan, E.G. Liouv. J. Mth. 5 (1840) 110-. =V, for potential of ellipsoid. Plana, G. Crelle J. 20 (1840) 271-. 280 3260 Definite Integrals (simple) 3260 yf*dqi$ "*. Binet, J. P. M. C. R. 12 a (1841) 958-. I" cost (u-x sin u)du. Liouville, J. Liouv. J. Mth. 6 (1841) 36. rlL'Ljg, Bonnet, 0. Liouv. J. Mth. 6 (1841) 238-. 1 + *" / e~& x . x m ~ l dx f" cos fix . B l+X* ' J ll /~ sin j3a; . .r*"" 1 da; Svanberg,A. F. Liouv. J. Mth. 11 (1846) 197-. Jo x*-a* and Jo a^-a 2 ' ^rwdt, F. Grunert Arch. 10 (1847) 247-. (V Sil M,r./(4, Fresnel's approxima- I] (f^ ~ lS) **' * W devel P ments ' /.. cos ,<- , ' ., T > ^rwdt. F. Grunert Arch. 10 (1847) 253-. tion mehod. U. Camb. Mth. J. 2 (1841) 81-. Arndt, F. Grunert Arch. 10 (1847) 253-. da;, value. Newman, F. W. Camb. /a and Dubl. Mth. J. 2 (1847) 75-. J^lo g ( S in0)d0. Ellis,R.L. Camb.Mth.J. J arctan/(a;)( /(x) being unique function of x . f 2 ( 1841 )^ 282 -- Raabe, J. L. Ziir. Mt. 1 (Heft 1) (1847) 61-. Stockh. Ak. f' > a - ldz DiengeTj j, N. A. Mth. 7 / o 1 + z n + z m (1848) 201-. Hndl. (1841) 65-. I (sin ax, cos bx) , reduction. Raabe, J 1 /:' J. L. Crelle J. 23 (1842) 105- ; 25 (1843) 160-. . Bertrand, J. Liouv. J. Mth. /:- Besge, V. Liouv. J. Mth. 14 and (1843) 110-. f m (l-pcostx)( Jo T^ 2j COS tX +P 2 psintxf(x)dx (1849) 31-. I log (1 - 2r cos x + r 2 ) cos nx dx. Haan, D. * Bierens de. Grunert Arch. 13 (1849) 193-. /* , *****".. .,. Schl&milch,0. Grunert r 2 + 2rx cos u + x* Arch. 12 (1849) 198-. -i 1 a- 2 1 f-Jtf- Jo fa- Malmsten, C. J. Ups. N. Acta S. Sc. 12 (1844) 155-. A logjl^) ^ Serret,J.A. Liouv. J. Mth. 9 (1844) 436. R. Camb. Mth. J. 4 (1845) 143-. _ r*log(l-a)^ Schdffer, . Crelle J. 30 (1846) 277-. /.v , value imaginary A cos B sin ' values of A and B. Jacobi, C. G. J. Crelle J. 32 (1846) 8-. 1 - k ' 2 sin2 ^ J. Mth. 11 (1846) 471-. Smaasen, W. Amst. Ts. Wis. Nt. Wet. 2 (1849) 251-. I 27r f(re'i >i )e~ n 'l' i d. Dienger, J. Grunert Arch. 15 (1850) 119-. r smx dx ^ Lindmarit Cf Ff Grunert Arch. 17 (1851) 455-. /arc tan/ (a;), /(a:) being unique function of x. Boofte, J. L. Crelle J. 41 (1851) 54-. /o(^)^'(l-|- 2 "- ^^^'^ Crelle J. 42 (1852) 283-. / -dx. Lindman, C. F. Grunert Arch. 21 (1853)113-. e'^sinaxda; Schlomilch, 0. Crelle J. 33 /"GO g-gx g j n a , Jo TTiT (1846) 325-. Schlomilch, 0. Grunert Arch. 7 (1846) 38-. / cos 6a; , -s rda; and o "- 2 -" 2 / i ~^dt J ^ ' . Liouville, J. Liouv. J. J. W. L. [1871] Mess. Mth. 1 (1872) 106-. Jo (a + bt-ctY +l f*x-idx Mth. 2 (1857) 47-. I . Walton, W. [1871] QJ. Mth. 12 Mth. 2 (1857) 49-. Liouv. J. > Mth 2(1857)279 Ac. Sc. Bll. 15 (1857) 307-. [*>sinx<", Jo ^^~ d ' r - Mndt ^' ^- ^ A - Grunert Arch. 30 (1858) 171-. / e M2re ~'e-d2 for calculation of series /-i^-i-3-w I n^Tw^^- Wr " ora - W ' Mth. /2 ?1873) 184-. fc / ^ ( x> a ) ^ x> n * n differentiation. Walton, W. QJ. Mth. 12 (1873)215-. H W*>- Lyetnlkov [Letnikow], D) + etc - [^72] QJ. A. V. [1872-73] (xn) Eec. Mth. (Moscou) 7 (1874-75) (Pt. 1) 5-; (x) Bll. Sc. Mth. As' 7 (!874) 233-. I sinx 2 da;and/ cosx z dx. Glaisher, J. W. L. [1872-75] QJ. Mth. 13 (1875) 343-. Schlomikh, 0. Schlomilch Z. 4 (1859) 433-. i (b) /" / cos(x 2 )da; and / sin (x 2 ) da;. J Jo QJ. Mth. 12 (1873) 118-. Cayley, A. . Worpitzky,(Dr). nert Arch. 35 (1860) 70-. = sin^x Schlomilch Z. f | J ) logxdx between real and positive limits, G. Mt. 12 (1874) 1-. 5 (1860) 286-. ^'log 1 ]^' Wol f erg ' J -P- Grunert Arch. pfp** 1 Glaisher, J. W. L. Mess. Mth. 37 (1861) 245-. ^ 8 (1876) 168-. /""I COS 7 * 1 X I -a dx. Stefan, J. Schlomilch Z. 7 f2 xF(x)dx J f2 J o g niometrical function. Bierens de Haan, D. Amst. Ak. C08+&ixcosiaxcospxdx. Lindman,C.F. Vs. M. 12 (1878) 325-; Arch. Neerl. 13 [1864] Stockh. Ofv. 21 (1865) 373-. '^dx. Besso,n. G.Mt. 7 (1869)210-. . x At. 14 (1878) 91-. . -. Jo ~T+? * Liouville < J ' Liouv. J. Mth. ( l (u) n du, infinite value. Schroder, E. Z.Mth. 14(1869)298-. 7 Jsinx^dx, |cosx n dx, etc., between limits. ' Ps. 25 (1880) 106-. f ^ ^ ~$ ( bx ) dx, expansion for. Elliott, Oettinger.'L. Arch. Mth. Ps. 49 (1869) 51-. * . B. Ikless. Mth. 10 (1881) 118-. 282 3260 Definite Integrals (simple) Jiirgens, E. [1881] Mth. A. 19 (1882) 435-. 1 (log *) 2 - 2 log ( yij|J ~ etc. Scott, R. F. Mess. Mth. 10 (1881) 142-. /(*) + /(-*) reduction. J. A. Martins da. G. Teix. J. Sc. 3 (1881) Ax. Thomson, J.J. QJ.Mth. t dx. Catalan, E. C. Liege S. Sc. Mm. 12 (1885) No. 2, 26-. | _A _^__ \ LdQHBTTCj E C. H. 100 (1885) 624-. I . ~ - g and related integrals. Schlafli, L. Acta Mth. 7 (1885) 187-. I e ~ x (l + xla)- a ~ b dx. Stieltjes, . BU. Sc. Mth. 9 (1885) 306-. [ dx. Lerch, M. J. Prag Sb. (1886) (Mth.-Nt.) 588- ; Fschr. Mth. (1886) 258-. c, expansion. Stieltjes, T. J. Acta Mth. 9 (1887) 167-. e~**dt, approximate calculation. Mansion, P. Brux. S. Sc. A. 12 (1888) (Pt. 1) 63-. f(x)G(x)dx. Stieltjes, T. J. N. A. Mth. 7 (1888) 161-. cos p

etc., integration. Beaupain, J. Brux. Mm. Cour. 4, 51 (1889) No. 4, 60 pp. ; 52 (1890-93) No. 4, 62 pp. / W cot (x - a) dx. Gomes Teixeira, F. G. Teix. J. Sc. 9 (1889) 113- ; Fschr. Mth. -*? dx. Gomes Teixeira, F. Prag Sb. 1 (Mth.-Nt.) 118-. dx. Martins da Silva, J. A. [1890] Lisb. J. Sc. Mth. 6 (1902) 14-. ' " log { (x - a cos 0) 2 + (y-b sin 0) 2 } ? S m0 d6. Gibson, G. A. Edinb. Mth. S. P. 10 (1892) 83-. Z , formulas deduced from. Lerch, M. fiasopis 21 (1892) 218- ; Fschr. Mth. (1892) 265-. /, Multiple Integrals 3270 b F (x) ~- . Sonin, N. St Pet. Ac. Sc. Mm. 38 (1892) No. 14, 44 pp. smPxcos^x (mx) dx, integration. Stud- o cos nicka, F. J. Casopis 23 (1894) 67- ; Fschr. Mth. (1893-94)481. e ae sm m 8de. Forsyth, A. R. QJ. Mth. 27 (1895) 216-. , a 2 S ^*dx. Oltramare, G. As. Fr. C. E. (1895) (Pt. 1) 182. 00 sin ax TT _ dx=-^, generalisation. Stprmer, C. Christiania F. (1895) No. 4, 11 pp. arc tan - arc tan - dx = log proof. Lerch, M. [] 39-; Fschr. Mth. (1899) 391. 7T 2 Casopis 29 (1900) (log 2 cos ) m n d, value. Morley, F. [1899] N. Y. Am. Mth. S. BU. 7 (1901) 390-. 2 /"* j- I e-v dt, with tables of values. Burgess, J. Edinb. E. S. T. 39 (1900) 257-. 3270 Multiple integrals. Lejeune-Dirichlet, G. Crelle J. 4 (1829) 94- ; Berl. B. (1839) 18-, 61-; C. E. 8 (1839) 156-. Cauchy, A. L. C. E. 11 (1840) 1000-, 1019. Cayley, A. Liouv. J. Mth. 10 (1845) 158-, 242-. Ellis, R. L. Camb. Mth. J. 4 (1845) 64-, 116-. Bromcin, B. Ph. Mg. 28 (1846) 373-. Roberts, W. Liouv. J. Mth. 11 (1846) 201-. Schlomilch, O. Tortolini A. 3 (1852) 327-. Winckler, A. [1869] Wien Sb. 60 (1870) (Ab. 2) 379-. Scott, R. F. [1875] Mess. Mth. 5 (1876) 22-. Zinine, N. N. Fschr. Mth. (1887) 281. Sauvage, L. Mars. Fac. Sc. A. 2 (1892) 3-. Puzyna, J. Prace Mt.-Fiz. 4 (1893) 1- ; Fschr. Mth. (1893-94) 491-. Abel's theorem irf (x) Tait, P. G. [1874] Edinb. E. S. P. 8 (1875) 440-. Application of Gauss's integration method. Lowenstern, S. St Pet. Mm. Sav. Etr. 3 (1837) 279-. Stokes's formula. Vogt, . N. A. Mth. 19 (1900) 97-. Approximate multiple integration between limits by summation. Maxwell, J. C. [1877] Camb. Ph. S. P. 3 (1880) 39-. Boole's theorem. Cayley, A. Liouv. J. Mth. 13 (1848) 245-. Cases. Boole, G. [1846] Ir. Ac. T. 21 (1848) 140-. 283 3270 Multiple Integrals Cases. Liouville, J. Liouv. J. Mth. 4 (1859) 155-. . Taylor, H. M. Mess. Mth. 5 (1876) . Sersawy, F. Arch. Mth. Ps. 64 (1879) 30-. . Staeckel, . fill. Sc. Mth. 21 (1897) 31. Change of variables. Catalan, E. C. Par. S. Phlm. PV. (1839) 123-. . Baehr, G. F. W. Arch. Mth. Ps. 41 (1864) 453-. Class. Tchebicheff, P. Liouv. J. Mth. 8 (1843) 235-. . Liouville, J. Liouv. J. Mth. 1 (1856) 82-. . Lipschitz, E. Crelle J. Mth. 101 (1887) 214-. , numerical values. Hamilton, (Sir) W. R. Ph. Mg. 14 (1857) 375-. Classes. Elliott, E. B. [1876-77] L. Mth. S. P. 8 (1877) 35-, 146-. Convertent functions. Cockle, (Sir) J, [1868] Manch. Lt. Ph. S. P. 8 (1869) 2-; 9 (1870) 86-; 10(1870)1-. Curvilinear coordinates, Stokes' s theorem. Padova, E. Mil. I. Lomb. Ed. 25 (1892) 1021-. Determination by transformation of coordinates. Cauchy, A. L. C. E. 13 (1841) 34-. Differentiation with regard to parameter on which limits also depend. Zindler, K. Wien Ak. Sb. 106 (1897) (Ab. 2a) 359-. Double integrals. Schlffinilch, 0. Grunert Arch. 11 (1848) 174-. Dienger, J. Grunert Arch. 13 (1849) 286-. Steen, A. Kiob. Dn. Vd. Selsk. Skr. 1 (1849) 333-. Besge, V. A. Liouv. J. Mth. 3 (1858) 324. Le Cordier, P. C. E. 66 (1868) 707-. Besso, D. G. Mt. 10 (1872) 79-. DM Bois-Reymond, P. [1882] Crelle J. Mth. "' ""33)273-. Goursat, E. C. E. 96 (1883) 1304-. Arzela, C. Bologna Ac. Sc. Mm. 2 (1891) 133. Pochhammer, L. Mth. A. 41 (1893) 179-. of algebraic functions, reduction. Picard, E. C. E. 126 (1898) 1116-. apparently with two values. Puzyna, J. [1883] (xn) Krk. Ak. (Mt.-Prz.) Pam. 9 (1884) and binary intervals, theory. Meray, C. Par. EC. Norm. A. 16 (1899) 193-. calculation, approximate. Minding, E. F A Crelle J. 6 (1830) 91-. , case. Callandreau, 0. C. E. 96 (1883) 1125. Cases. Roberts, W. A. Mt. 3 (1869-70) 327-. . Didon, F. Par. EC. Norm. A. 7 (1870) . Orlow, G.A. N. A. Mth. 1 (1882) 311-. Cauchy's theorem. Kluyver, J. C. Fschr. Mth. (1897) 258. Double Integrals 3270 class. Goursat, E. Acta Mth. 5 (1884) 97-. including Fourier's. Du Bois-Reymond, P. Crelle J. 69 (1868) 65-. Crofton's theorem, corollary. Hoppe, R. C. E.70 (1870) 1394- ; Arch. Mth. Ps. 55 (1873) 426-. differentiation and integration. Bierens de Haan, D. Amst. Vs. Ak. 5 (1871) (Ntk.) 53-, 65-; Arch. Neerl. 6 (1871) 329-. of explicit irrational functions, theory. Bier- mann, O. Wien Ak. Sb. 98 (1890) (Ab. 2a) 340-. formulae, integration. Fuss, N. [1826] St Pt. Ac. Sc. Mm. 11 (1830) 294-. Fourier's theorem, Boole's and other proofs. Alexander, P. Edinb. Mth. S. P. 3 (1885) 45-. geometric demonstration. Chasles, M. Liouv. J. Mth. 3 (1838) 10-. evaluation. Collignon, E. As. Fr. C. E. 3 (1874) 1193-. integration. Sylvester, J. J. Ph. Mg. 14 (1839) 298-. inversion. Appell, P. Am. J. Mth. 19 (1897) 377-, 382. . Le Roux, J. C. E. 130 (1900) 882-. of 2nd kind in theory of algebraic surfaces. Picard, E. [1897] Liouv. J. Mth. 5 (1899) 5. maxima and minima. Kobb, G. Stockh. 6fv. (1890) 385- ; Fschr. Mth. (1890) 382 ; Acta Mth. 16 (1892-93) 65-; 17 (1893) 321-. and their periods, elementary theory. Marie, M. C. E. 75 (1872) 576-, 614-, 660-. Poisson's theorem, proof. Roberts, W. Liouv. J. Mth. 11 (1846) 210-. properties. Rodrigues, (Dr) . Par. S. Phlm. Bll. (1815) 34-. of rational functions, residues. Picard, E. C. E. 124 (1897) 433-. reduction, and new invariant in theory of algebraic surfaces. Picard, E. C. E. 126 (1898) 297-. to products of simple integrals. Schlomilch, O. Z. Mth. Ps. 9 (1864) 205-. quadratures. Winckler, A. [1851] Crelle J. 45 (1853) 102-. residues. Poincare, H. C. E. 102 (1886) 202- ; Acta Mth. 9 (1887) 321-. (Poincare). Marie, M. N. A. Mth. 10 (1891) 77-. , theory. Marie, M. C. E. 75 (1872) 695-, 751-. returning to same form after transformation of variables. Jacobi, C. G. J. Crelle J. 15 (1836) 193-. theory. Harnack, A. Mth. A. 26 (1886) 566-. . Pringsheim, A. Munch. Ak. Sb. 28 (1899) 59-. , Green's and Cauchy's theorems. Prings- heim, A. Munch. Ak. Sb. 29 (1900) 39-, 268-. and invariant of two triangles. Le Roux, . Par. S. Mth. Bll. 28 (1900) 168-. transformation. Jacobi, C. G. J. [1827-32] Crelle J. 2 (1827) 234-; 8 (1832) 253-, 321- ; 10 (1833) 101-. 284 3270 Multiple Integrals 3270 transformation. Cayley, A. Mess. Mth. 4 (1875) 92-. . Berger, A. Stockh. Ofv. (1887) 153-. . Stolz, 0. Mh. Mth. Ps. 9 (1898) 316-, 371; 10 (1899) 96. formula. Scheibner, W. Leip. Mth. Ps. B. 36 (1884) 185-. , Euler's. Plana, G. Brux. Ac. Bll. 10 (1843) 303-. , general. Winckler, A. Wien D. 20 (1862) 97-. by Green's theorem. Gmeiner, J. A. Mh. Mth. Ps. 4 (1893) 277-. and triple integrals, simultaneous substitutions, theory. Unferdinger, F. Wien Sb. 63 (1871) (Ab. 2) 773-. , transformation. Piola, G. Mod. Mm. S. It. 20 (1828) 272-. , . Tait, P. G. [1873] Edinb. E. S. P. 8 (1875) 209-. 4>(ax m by n )xP (ax m by n ) l dxdy, Raabe, J. L. Crelle J. 37 (1848) 345-. ~c fb 1:1: 6 J o V(c 2 - /t 2 ) (/t 2 - 6 2 ) (c 2 - v 2 ) (6 2 - v 2 ) Huberts, W. Liouv. J. Mth. 15 (1850) 238-. Equality. Besge, V. A. Liouv. J. Mth. 3 (1858) 416. Evaluation. Ellis, R. L. Camb. Mth. J. 4 (1845) 1-. , case. Gregory, D. F. Camb. Mth. J. 2 (1841) 215-. , process. Arndt, F. Grunert Arch. 5 (1844) 443-. Expansibility of quotient of two definite in- tegrals of form idxdy...dz. Schlafti, L. Crelle J. 67 (1867) 183-. Expansion. Schlomilch, 0. Schlomilch Z. 1 (1856) 75-. Formula, demonstration. Thomson, (Sir) W. Liouv. J. Mth. 10 (1845) 137-. , (Thomson). Liouville, J. Liouv. J. Mth. 1 (1856) 445-. , Fourier's, generalised by Liouville. Popov, A. T. (xii) Kazan Un. Mm. (1848) (Bk. 4) 3-. Frullanian integrals. Elliott, E. B. L. Mth. S. P. 15 (*1883-84) 12-. Integral expressing area of circle and of spherical triangle. Schlafti, L. Liouv. J. Mth. 20 (1855) 359-. Integrals of area and volume, transformations and values. Tortolini, B. G. Arcad. 80 (1839) 13-; 82 (1840) 97-. with 3 complex variables. Picard, E. C. E. 108 (1889) 132-. of Jacobi. Zinin, N. N. Vars. S. Nt. Tr. (1897) (Ps. C.) Fasc. 1, No. 3, ii. any order, and their periods, elementary theory. Marie, M. C. E. 75 (1872) 1078-, 1247-. Integrals in probability theory, two. Czuber,E. Mh. Mth. Ps. 2 (1891) 119-. representing products and powers of simple definite integrals. Raabe, J. L. [1847] Ziir. Mt. 1 (Heft 2) (1847-48) 29-; Crelle J. 48 (1854) 137-. in theory of attractions. Cayley, A. Camb. and Dubl. Mth. J. 2 (1847) 219- ; 7 (1852) 174-. unaltered in form by transformation of the variables. Gundelfinger, S. [1880] Crelle J. Mth. 91 (1881) 215-. , value of which is affected by order of integration. Decker, G. Grunert Arch. 19 (1852) 403- ; 22 (1854) 413-. Inversion. Volterra, V. Em. E. Ac. Line. Ed. 5 (1896) (Seat. 1) 289-. , mode. Appell, P. C. E. 124 (1897) 213-. Limits. MacColl, H. L. Mth. S. P. 16 (1884- 85) 142-. Line and surface integrals. Peirce, B. 0. Am. Ac. P. 26 (1891) 20-. , surf ace and volume integrals, relation among. Tait, . Edinb. E. S. P. 16 (1890) 257. Maxima and minima, criteria. Sabinine, G. [1869] St Pet. Ac. Sc. Bll. 15 (1871) 70-. Mean values of functions. Cauchy, A. L. C. B. 18 (1844) 558-; 20 (1845) 119-, 212- ; 26 (1848) 549-, 624-, 666-; 27 (1848) 6-, 37-; 29 (1849) 341-. Minimum. Lebesgue, H. C. E. 131 (1900) 935-. Multiple Gaussian sums. Weber, H. [1871] Crelle J. 74 (1872} 14-. and single integrals. Nanson, E. J. Mess. Mth. 26 (1897) 119-. Ostrogradskii's formulae. Zinin, N. N. Eec. Mth. (Moscou) 15 (1891) 645-; Fschr. Mth. (1891) 304-. Problem. Zinine, N. N. Fschr. Mth. (1888) 308-. Eeduction. Catalan, E. C. Liouv. J. Mth. 4 (1839) 323-; 6 (1841) 81-; 8 (1843) 239-; Brux. Mm. Cour. 4, 14 (1841) 47 pp. (error in Catalan's paper, 1841). Boole, G. Camb. Mth. J. 3 (1843) 277-. . Chelini, D. G. Arcad. 94 (1843) 49-. . Liouville, J. Liouv. J. Mth. 1 (1856) 289-. . Enneper, A. Gott. Nr. (1867) 164- ; Z. Mth. Ps. 15 (1870) 121-. . Letnikov, A. V. Eec. Mth. (Moscou) 14 (1890) 303- ; Fschr. Mth. (1889) 288. . Nekrasov, P. A. [1891] Eec. Mth. (Mos- cou) 16 (1893) 81-; Fschr. Mth. (1891) 305. . La Vallee-Poussin, C. J. de. C. E. 127 (1898) 950-. , application of general differentiation. Nekrasov, P. A. Eec. Mth. (Moscou) 14 (1890) 410- ; Fschr. Mth. (1889) 288-. , cases. Raabe, J. L. Crelle J. 28 (1844) 19-. , -. Schaar, M. Brux. Ac. Bll. 15 (1848) (pte. 2) 501-. , . Genocchi, A. Tortolini A. 8 (1857) 284-. , . Schlomilch, 0. Leip. B. 9 (1857) 67-; Liouv. J. Mth. 2 (1857) 206-. 285 3270 Multiple Integrals Eeduction, cases. Catalan, E. C. Liege S. Sc. Mm. 12 (1885) No. 2, 12-, 17-. , . Maggi, O. A. Mil. I. Lomb. Bd. 19 (1886) 689-. , . Sonine, N. J. Fschr. Mth. (1889) 287-. , . Zinin, N. N. Vars. S. Nt. Tr. (1889- 90) (C. E., Ps. C.) No. 6, 9-. , . Sonin, N. Ja. Bee. Mth. (Moscou) 14 (1890) 527- ; Vars. S. Nt. Tr. (Mm.) 1 (1891) No. 4, 7 pp. , . Black, A. [1897] Camb. Ph. S. T. 16 (1898) 219-. , direct, of class. Silva, J. A. Martins da. G. Teix. J. Sc. 3 (1881) 49-. , formula. Sonin, N. Ja. Vars. S. Nt. Tr. (1889-90) (C. R., Ps. C.) No. 2, 13-; Fschr. Mth. (1889) 287. of generalised. La Vallee-Poussin, C. J. de. Liouv. J. Mth. 5 (1899) 191-. , quaternion formulae. Carvallo, E. Par. S. Mth. Bll. 18 (1890) 80-. , Zinin's treatment of problem. Nekrasov, P. A. Bee. Mth. (Moscou) 16 (1893) 823-. Space and surface integrals. Fiske, T. S. [1891] A. Mth. 6 (1891-92) 61-, 131. Surface and curvilinear integrals. Gazzaniga, T. Mil. I. Lomb. Bd. 33 (1900) 567-. integrals, condition of dependence on boun- dary alone, new interpretation. Meray, C. C. B. 128 (1899) 913-. , definition. Lebesgue, H. C. B. 131 (1900) 867-. , evaluation, and application to expansion, in series, of potential of ellipsoids. Hobson, E. W. L. Mth. S. P. 24 (1893) 80-. in problem of 3 bodies. Wetter, A. As. Nr. 74 (1869) 85-. , transformation into multiple line in- tegrals. Larmor, J. Mess. Mth. 17 (1888) 23-. , . McAulay, A. Mess. Mth. 18 (1889) 139-. Theorem. Volterra, V. Tor. Ac. Sc. At. 32 (1896) 597- or 859-. of Fourier. Glaisher, J. W. L. [1876] Mess. Mth. 6 (1877) 30-. Frullani, extensions. Leudesdorf, C. L. Mth. S. P. 9 (1877-78) 118-. Hadamard. Hurwitz, A. C. B. 128 (1899) 350-. Kronecker. Koenigs, G. Liouv. J. Mth. 2 (1896) 41-, Theorems. Jacobi, C. G. J. Crelle J. 12 (1834) 1-. , general. Ellis, R. L. Camb. and Dubl. Mth. J. 1 (1846) 1-. of Green and Cauchy. Chessin, . C. B. 128 (1899) 604-. Theory. Andrade, J. C. B. 119 (1894) 1192-. of numbers, application to integral cal- culus. Gegenbauer, L. Wien Ak. Sb. 102 (1893) (Ab. 2a) 927-. Transformation. Haedenkamp, H. Crelle J. 22 (1841) 184-. . Boole, G. Camb. Mth. J. 3 (1843) 216- ; 4 (1845) 20-. Calculus of Variations 3280 Transformation. Svanberg, A. F. Stockh. Ofv. 2 (1845) 228-; Ups. N. Acta S. Sc. 13 (1847) 1-. . Chelini, D. G. Arcad. 106 (1846) 127-. . Genocchi, A. Tortolini A. 4 (1853) 401-. . Russell, W. H. L. QJ. Mth. 4 (1861) 163-, 316-. . Holmgren, H. [1864] Stockh. Ak. Hndl. 5 (1866) (No. 6) 40 pp. . Tannenberg, W. de. Bordeaux S. Sc. PV. (1897-98) 109-. , case. Haedenkamp, H. Crelle J. 20 (1840) formula. Cayley, A. Camb. and Dubl. Mth. J. 4 (1849) 50-. . Brioschi, F. Bm. B. Ac. Line. Bd. 2 (1886) (Son. 2) 111-. theorem. Minchin, G. M. [1871] QJ. Mth. 12 (1873) 172-. theorems of Gauss and Stokes. Predella, P. Mil. I. Lomb. Bd. 20 (1887) 215-. of variables. Ostrogradsky, M. A. [1836] St Pet. Ac. Sc. Mm. 3 (1838) 401-. Triple integrals, case. Le Paige, C. M. M. H. H. Lie"ge S. Sc. Mm. 9 (1882) (No. 3) 4 pp. embracing potential of homogeneous ellipsoid as particular case. Mertens, F. Gott. Nr. (1887) 269-. , reduction to quadratures. Winckler, A. [1853] Crelle J. 50 (1855) 1-. , transformation by changing order of integration. Winckler, A. Crelle J. 45 (1853) 168-. , and determination, case. Unferdinger, F. Wien Sb. 61 (1870) (Ab. 2) 105-, 417-. -, variation. Abbott, R. B. S. P. 4 (1838) 42. / dxdy ...dz, whose limits are p 1 = a^ ... + h l z>0, p 2 >0, ...p n >0, and ... + z*0, p z >Q, ...p n >0, and + 2 a F ' G - [ 188 ] Z. Mth. Ps. 26 (1881) 231-. Isotropic functions. Cauchy, A, L. C. E. 26 (1848) 549-, 624-, 666- ; 27 (1848) 6-. Lateral curves, theory. Thieme, F. E. Arch. Mth. Ps. 58 (1876) 185-. Linear functions of complex variable. Cole, F. N. A. Mth. 5 (1889-90) 121-, iv. Logarithmic potential. Tatarinov, I. V. Mosc. S. Sc. Bll. 96 (No. 1) (1899) 7 Maximum and minimum. Scheibner, W. Leip. Mth. Ps. B. 40 (1888) 4-. Operations, cases. Pincherle, S. Bologna Ac. Sc. Mm. 7 (1886) 393-. Orders of infinity of functions. Pincherle, S. Bologna Ac. Sc. Mm. 5 (1883) 739-. Oscillations of harmonic functions of two real variables and of functions of one complex variable. ScJiottky, F. Crelle J. Mth. 117 (1897) 225-. Periodic function of one variable with more than two periods impossible, theorem of Jacobi. Casorati, F. Mil. I. Lomb. Ed. 15 (1882) 623-. functions. Berger, A. Bll. Sc. Mth. As. 6 (1874) 72-. Periods of integral of any order. Marie, M. C. E. 46 (1858) 738-. Plane ?i-line, metric geometry. Morley, F. N. Y. Am. Mth. S. T. 1 (1900) 97-. 291 T2 3600 Complex Variables Plane n-line, metric theorems. Loud, F. H. N. Y. Am. Mth. S. T. 1 (1900) 323-. Point involutions in plane of complex variable. Vries, J. de. Mh. Mth. Ps. 3, (1892) 285-. Principles. Eiquier, . Par. EC. Norm. A. 8 (1891) 59-, 141- ; 9 (1892) 281-. , fundamental. Cellerier, C. Bll. Sc. Mth. 14 (1890) 142-. , generalisation. Scheffers, G. Leip. Mth. Ps. B. 45 (1893) 828-; 46 (1894) 120-. of Weierstrass. Pincherle, S. G. Mt. 18 (1880) 178-, 317-. Problem. Puzyna,J. Fschr. Mth. (1888) 411. Properties. Trzaska, W. Par. T. Nauk Sc. Pam. 1 (*1871) 109- ; Bll. Sc. Mth. As. 6 (*1874) 152-. Eational functions. Beke, E. Mth. A. 47 (1896) 441-. Real functions of complex variables. Murphy, E. Ph. Mg. 2 (1833) 287-. , given over any aggregate of points, analytic representation. Phragmen, E. Palermo Cir. Mt. Rd. 14 (1900) 256-. part of any function of complex variable. Bougaieff, . Bll. Sc. Mth. 11 (1887) 61-. Representation. Fuchs, L. [1872] Crelle J. 75 (1873) 177- ; 76 (1873) 175-. , approximate forms for. Pincherle, S. Bologna Ac. Sc. Mm. 10 (1889) 77-. , conformal. Wagner, H. A. Hamb. Mth. Gs. Mt. 1 (1889) 64-. , . , given by quadratic fractional relation. Holzmuller, F. G. Mth. A. 18 (1881) 289-. by definite integrals, cases. Somigliana, C. Mil. I. Lomb. Rd. 21 (1888) 431-. , theory of images. Harris, E. A. A. Mth. 4 (1888) 65-, 128. Roots considered as functions of a variable parameter. Puiseux, V. C. R. 30 (1850) 171. Series representing arbitrary functions in separate regions of convergence. Pringsheim, A. Mth. A. 22 (1883) 109-. , terms of which are functions of complex variable. La Valtee Poussin, C. de. G. Teix. J. Sc. 11 (1892) 77-. Simplifications. Pringsheim, A. Mth. A. 47 (1896) 12 1-. Singular functions, representation, analytic. Tonelli, A. Rm. R. Ac. Line. Rd. 1 (1885) 124-. lines of analytic functions. Painleve, P. Toul. Fac., Sc. A. 2 (1888) B, 130 pp. points of a certain function. Dell' Agnola, C. A. Ven. I. At. (1898-99) (Ft. 2) 525-, 669-. Singularities of analytic functions, class. Pringsheim, A. Mth. A. 50 (1898) 442-. (two), comparison. Pincherle, S. Rm. R. Ac. Line. Rd. 3 (1887) (Sem. 2) 310-. , especially functions defined by differential equations. Painleve, P. C. R. 131 (1900) 489-. , single-valued and general. Chessin, A. S. A. Mth. 11 (1896-97) 52-. function depending on two given functions. Pincherle, S. Rm. R. Ac. Line. Rd. 8 (1899) (Sem. 1) 228-. Uniform Functions 3610 Theorem. Lerch, M. Prag Sb. (1885) (Mth.- Nt.) 351- ; Fschr. Mth. (1886) 347. , Abel's, demonstration of non-existence of any other. Konigsberger, L. [1885] Munch. Ak. Sb. 15 (1886) 462-. , Cauchy's, on roots, two deductions from. Mittag-Leffier, G. Stockh. Ofv. 31 (1874) No. 7, 23-. , fundamental. Ascoli, G. [1870] A. Mt. 4 (1870-71) 31-. , new. Jensen, J. L. W. V. Acta Mth. 22 (1899) 359-. Theorems, two. San-Martino, A. N. A. Sc. Nt. 4 (1845) 353-. Total variation in describing contour. Cauchy, A. L. C. R. 40 (1855) 651-, 713-, 804-. Transcendental functions, class. Picard, E. C. R. 123 (1896) 1035-. , genus zero and genus one. Laguerre, E. C. R. 95 (1882) 828-. , integrals. Koenigsberger, L. Gott. Nr. (1884) 116- ; Crelle J. Mth. 98 (1885) 97-. , , reduction. Koenigsberger, L. Am. J. Mth. 11 (1889) 221-. , new, class. Picard, E. Acta Mth. 18 (1894) 133- ; 23 (1900) 333-. , roots being all transcendental numbers. Gegenbauer, L. Wien Ak. Sb. 108 (1899) (Ab. 2a) 423-. Transformation, orthomorphic, of circle into itself. Cayley, A. Edinb. Mth. S. P. 8 (1890) 91-. Transformations, algebraic. Pagani, G. M. Brux. Ac. Bll. 7 (1840) (pte. 2) 50-. y=oo Weierstrass' s function 2 b"cos(a v Trx). Tauber, A. Mh. Mth. Ps. 8 V (1897) 330-. Wronskians, fundamental property. Demoulin, A. Mathesis 17 (1897) 62-. (x, y)+ty(x,y)=F(x + iy) , conditions. Dien- ger, J. Grunert Arch. 10 (1847) 422-. Tan~' ( + irj) , reduction to form x + iy. Unfer- dinger, F. Arch. Mth. Ps. 49 (1869) 478-. , value. Unferdinger, F. Z. Mth. Ps. 14 (1869) 77-. (x + yi) + (x- yi) , Abel's expression for. Glaisher, J. W. L. [1872] Mess. Mth. 2 (1873) 12-. II (x + iy + r), properties. Schlomilch, 0. Leip. B. 24 (1872) 26-; Z. Mth._Ps. 17 (1872) 248-. (x + y V^l) + (x - y \/^T) , value. Oltra- mare, G. As. Fr. C. R. (1885) (Ft. 2) 127-. 3610 Uniform functions of one variable. Bouquet, J., <& Briot, . Par. EC. Pol. J. cah. 36 (1856) 85-, 133-, 199-. Bertrand, J. C. R. 48 (1859) 427-. Imshenetskil, V. G. (xii) Kharkov Mth. S. Com. (1880) 173-. Farkas, J. C. R. 96 (1883) 1646-. Vivanti, G. G. Mt. 25 (1887) 54-, 232-, 312. Additions to work of Weierstrass and Mittag- Leffler. Casorati,F. A. Mt. 10 (1880-82)261-. 292 3610 Uniform Functions of One Variable 3610 Analytic functions. Range, C. Acta Mth. 6 (1885) 229-. . Stackel, P. Crelle J. Mth. 112 (1893) 262-. , arithmetical properties. Stackel, P. Mth. A. 46 (1895) 513-. , expansion. Painleve', P. C. R. 128 (1899) 1277-. ; extension of Taylor's series. Mittag- Leffler, G. Nt. 62 (1900) 419. , representation. Herz, N. [1879] Z. Mth. Ps. 25 (1880) 125-. , . Painleve, P. C. R. 126 (1898) 200-. , . Mittag-Leffler, G. C. R. 128 (1899) 1212-. 1 series. Gerbaldi, F. Rv. Mth. 6 (1896- 99) 24-. , theorems. Mittag-Leffler, G. Bll. Sc. Mth. As. 3 (1879) (Ft. 1) 269-. , theory, conception of derived function. Vivanti, G. Palermo Cir. Mt. Rd. 13 (1899) 263-. , values at periphery of convergence circle. Puzyna, J. Krk. Ak. (Mt.-Prz.) Rz. 6 (1893) 311- ; Crc. Ac. Sc. Bll. (1893) 200-. point (x . y). Appell, P. Acta Mth. 1 (1882) 109-. ( ) , nature of algebraic relation between two uniform functions. Goursat, E. C. R- 97 (1883) 1048-. Approximation, method. Jablonski, E. C. R. 116 (1893) 19-. Arbitrary functions, representation. Runge, C. Acta Mth. 7 (1885) 387-. Arithmetical means. Gutzmer, A. G. Teix. J. Sc. 8 (1887) 147- ; N. A. Mth. 8 (1889) 101-. Case. Stieltjes, . C. R. 101 (1885) 153-. (Stieltjes). Hermite, C. C. R. 101 (1885) 112-. Cauchy's formula / ^ dx = 2iirF (a), applica- tion. Schonholzer, J. Bern Mt. (1874) (Ab.) 255-. integral. Morera, G. Mil. I. Lomb. Rd. 22 (1889) 191-. theorem. Kronecker, L. Berl. Ak. Sb. (1885) 785-. , demonstration. Goursat, E. Acta Mth. 4 (1884) 197-. , . Bruno de Cabedo, J. G. Teix. J. Sc. 9 (1889) 7-. , extension. Ravut, L. N. A. Mth. 16 (1897) 365-. on integral of function, demonstration. Falk, M. Ups. S. Sc. N. Acta 12 (1885) No. 3, 18 pp. value of complex function at any point of area, deduction from. Grandi, A. G. Mt. 7 (1869) 374-. lf(x) dx round a closed curve. Mittag- Leffler, G. Gott. Nr. (1875) 65-. Characteristic conditions. Volterra, V. A. Mt. 11 (1882-83) 1-. Class, determination. Sparre, de. C. R. 102 (1886) 740-. Coefficients and roots of an integral transcen- dental function, relations. Pincherle, S. Mil. I. Lomb. Rd. 11 (1878) 391-. Construction. Ovidio, E. d\ Tor. Ac. Sc. At. 2.8 (1893) 812-. Continuation. Picard, E. C.R. 128 (1899)193-. of analytic functions. Borel, E. C. R. 128 (1899) 283-. , analytic, of functions representing interior of conic on half plane. Lindemann, F. Munch. Ak. Sb. 26 (1897) 401-. _, _, generalisation. Fabry, E. C. R. 128 (1899) 78-. , , . Borel, E. C. R. 130 (1900) 1115-. of monogenic functions. Mittag-Leffler, G. Stockh. Ofv. (1898) 247-, 263-, 375-; Fschr. Mth. (1898) 358-. . . Viterbi, A. Palermo Cir. Mt. Rd. 12 (1898) 95-. single-valued function represented by double infinite series. Kluyver, J. C. Amst. Ak. Vs. 8 (1900) 40-; Amst. Ak. P. 2 (1900) 24-. Curves represented by imaginary equation. David, . Toul. Ac. Sc. Mm. 6 (Sent. 2) (1884) 239-. Differential equations, class. Picard, E. Bll. Sc. Mth. As. 4 (1880) (Ft. I) 416-. Equality of two products, theorem. Cauchy, A. L. C. R. 16 (1843) 886-. Expansion in any domain. Painleve, P. C. R. 126 (1898) 318-. power series. Pringsheim, A. Munch. Ak. Sb. 25 (1896) 75-. Expressions representing distinct functions in different parts of plane. Lerch, J. Bll. Sc. Mth. 10 (1886) 45-. Factorial in complex quantities, and some definite integrals. Raabe, J. L. Sch. Gs. N. D. 8 (1847) 19 pp. , m t -=-&!, limiting value when k increases indefinitely. Raabe, J. L. Crelle J. 48 (1854) 130-. Formula. Hermite, C. Am. J. Mth. 6(1884)60-. Function corresponding to given continuum. Osgood, W. F. N. Y. Am. Mth. S. Bll. 5 (1899) 14-. existing only in interior of circle. Stieltjes, T. J. Bll. Sc. Mth. 11 (1887) 46-. with isolated singularities, expansion in series. Bucca, F. Palermo Cir. Mt. Rd. 11 (1897) 90-. natural boundary. Osgood, W. F. N. Y. Am. Mth. S. Bll. 4 (1898) 417- ; 5 (1899) 17-. a certain singular point. Arcais, F. d\ Yen. I. At. (1897-98) 1048-. two singular points expressed in a series, converging for all values of the variables except along the arc of a circle or a straight line. Ermakov, V. P. Rec. Mth. (Moscou) 16 (1893) 518- ; Fschr. Mth. (1892) 354-. Functions of complex variable, special form. Seidel, L. [1869] Crelle J. 73 (1871) 297-. connected by algebraic relation. Picard, E. Bll. Sc. Mth. As. 4 (1880) (Ft. 1) 416- ; 7 (1883) (Ft. 1) 107-. 293 3610 Uniform Functions of One Variable 3610 Functions connected by algebraic relation, theorem. Picard, E. ActaMth. 11 (1887-88) with "coupures" (sections). Picard, E. C. E. 94 (1882) 1405-. ( ). Appell, P. C. E. 96 (1883) 1018-. line of essential singularities, decomposi- tion into prime factors. Picard, E. C. E. 92 (1881) 690-. , (Picard). Dautheville, S. Mntp. Ac. Mm. 11 (1892) 45-. in neighbourhood of a singular point. Picard, E. C. E. 89 (1879) 745-. presenting lacunae. Goursat, E. C. E. 94 (1882) 715-. represented by series convergent within certain limits. Ascoli, G. Mil. I. Lomb. Ed. 19 (1886) 173- . Holomorphic functions. Cesdro, E. C. E. 99 (1884) 26-; G. Mt. 22 (1884) 191-. . Hermite, C. Liouv. J. Mth. 1 (1885) 9-. . Bassi, A. G. Mt. 36 (1898) 100- ; 37 (1899) 326-. in an elliptic field. Pascal, E. Em. E. Ac. Line. Ed. 5 (1896) (Sem. 1) 319-. . Cazzaniga, T. Tor. Ac. Sc. At. 33 (1897) 574- or 808-. , expansion in series in area bounded by circular arcs. Appell, P. Acta Mth. 1 (1882) 145-. , genus zero or unity, roots of derivative. Bassi, A. Mil. I. Lomb. Ed. 28 (1895) 979-. , quotient of two, and recurring series. Presle, de. Par. S. Mth. Bll. 19 (1891) 114-. , roots of derivative. Bassi, A. Mil. I. Lomb. Ed. 28 (1895) 1119-. and meromorphic functions in rational and in elliptic field. Cazzaniga, T. Tor. Ac. Sc. At. 33 (1897). 743- or 983-. Integrability of function of one variable, new form of condition for. Stolz, 0. Wien Ak. Sb. 107 (1898) (Ab. 2a) 736-. Integrals between imaginary limits. Cauchy, A. L. Ferussac Bll. Sc. Mth. 3 (1825) 214-. . Dienger, J. Crelle J. 37 (1848) 363-; 39 (1850) 62-. , Cauchy 's theorem. Lerch, M. Prag Sb. (1887) (Mth.-Nt.) 673-. Integration, application of free path. Christen , T. Arch. Mth. Ps. 16 (1898) 1-. Irrational function of an imaginary variable. Cayley, A. [1876] L. Mth. S. P. 8 (1877) Isogonal transformation, statics of polynomials. Harris, E. A. A. Mth. 6 (1891-92) 77-. Laguerre's rank of uniform function with infinite number of zeros. Puzyna, J. von. Mh. Mth. Ps. 3 (1892) 1-.- Laurent's theorem, demonstration. Scheeffer, L. Acta Mth. 4 (1884) 375-. , for monogenic function with n singular points. Mittag-Leffler, G. Stockh. Ofv. 40 (1883) No. 9, 5-; Acta Mth. 4 (1884) Laurent's theorem, extension. Hermite, C. Crelle J. Mth. 116 (1896) 85-. Linear dependence. Bocher, M. [1900] N. Y. Am. Mth. S. Bll. 7 (1901) 120-, 234. Maximum of/(x), (= So^a^Y Puzyna, J. Prace Mt.-Fiz. 6 (1895) 1-; Fschr. Mth. (1895) 444-. Meromorphic functions. Borel, E. C. E. 120 (1895) 303-. Monodromic functions having a characteristic equation. Pincherle, S. Mil. I. Lomb. Ed. 12 (1879) 536-. , important class. Pincherle, S. G. Mt. 18 (1880) 92-. and monogenic functions. Cauchy, A. L. C. E. 43 (1856) 13-. Monogenic functions. Craig, T. [1893] N. Y. Mth. S. Bll. 3 (1894) 78-. , representation, analytic. Mittag-Leffler, G. Acta Mth. 4 (1884) 1-. , , . Volterra, V. Tor. Ac. Sc. At. 34 (1898) 348- or 492-. , , . Mittag-Leffler, G. [1898-1900] Camb. Ph. S. T. 18 (1900) 1-. , singular points of which make an aggregate of first kind, analytic representa- tion. Mittag-Leffler, G. Stockh. Ofv. 39 (1882) No. 2, 11-, No. 4, 21-. and holomorphic functions, inversion, cyclic. Eodrigues, (le capit.) J. M. As. Fr. C. E. (1893) (Pt. 1) 174; Ev. Mt. 5 (1895) 52-. monotypical functions. Cauchy, A. L. C. E. 32 (1851) 484-. Number of periods. Milhll, K. von der. Basel Vh. 9 (1893) 78-. . Astor, A. N. A. Mth. 15 (1896) 227-. Partial fractions found by complex functions. Hankel, H. Z. Mth. Ps. 9 (1864) 425-. Power series, non-continuable. Vivanti, G. Ev. Mt. 3 (1893) 111-, 127. , relation between real and imaginary parts. Tauber, A. Mh. Mth. Ps. 2 (1891) 79-. Products of variable factors. Andre, D. Par. S. Phlm. Bll. 2 (1890) 151-. Property. Poincare, H. C. E. 92 (1881) 1335-. Proposition of Hermite. Eaffy, L. Par. EC. Norm. A. 2 (1885) 99-. Eadius of convergence. Fuchs, L. Crelle J. Mth. 106 (1890) 1-. Eational functions with arbitrary limiting point, representation, analytic. Mittag-Leffler, G. Stockh. Ofv. 34 (1877) No. 1, 33-. , class, representation, analytic. Mittag- Leffler, G. Stockh. Ofv. 33 (1876) No. 6, 3-. with finite number of limiting points, representation, analytic. Mittag-Leffler, G. Stockh. Ofv. 34 (1877) No. 2, 31-. , representation, analytic. Mittag-Leffler, G. Stockh. Ofv. 34 (1877) No. 1, 17-. , by quotient of two converging power series. Mittag-Leffler, G. Stockh. Ofv. 34 (1877) No. 3, 5-. 294 3610 Uniform Functions of One Variable 3610 Eational integral function of a complex variable. Stieltjes, T. J. (jun.) Arch. Neerl. 18 (1883) 1-. Eepresentation, approximate, by rational frac- tions. Fade, H. C. R. Ill (1890) 674-. by infinite products and series of partial fractions. Frenzel,C. Z. Mth. Ps. 24 (1879) 316-. Residues. Tortolini, B. G. Arcad. 63 (1834- 35) 86- ; 67 (1836) 179-. . Gregory, D. F. Camb. Mth. J. 1 (1839) 145-. . Oltramare, G. C. R. 12 (1841) 953-. (Oltramare). Cauchy, A. L. C. R. 13 (1841) 296-. . Cauchy, A. L. C. R. 41 (1855) 41-. . Oltramare, G. Gen. I. Nt. Mm. 3 (1855) 13pp. , application. Amigues, E. N. A. Mth. 12 (1893) 142-. , in analysis. Cauchy, A. L. C. R. 32 (1851) 207-. , to integrals. Cauchy, A. L. C. R. 23 (1846) 321-. , mathematical physics. Cauchy, A. L. Par. Mm. Ac. Sc. 7 (1827) 463-. , resolution of transcendental func- tions. Cauchy, A. L. C. R. 32 (1851) 267-, 354-. , applications. Heine, H. E. [1879] Crelle J. Mth. 89 (1880) 19-. and calculus of limits, relations. Cauchy, A. L. [1831-37] Ferussac Bll. Sc. Mth. 16 (1831) 116- ; Mod. Mm. S. It. 22 (1839) 91-. , Cauchy's formula, demonstration. Mansion, . Brux. S. Sc. A. 24 (1900) (Pt. 1) 88-. , formulae, generalisation. Amanzio, D. G. Mt. 15 (1877) 257-. , work on. Kummell, C. H. Des Moines Anal. 6 (1879) 1-, 41-, 173-. , extension to space. Carvallo, E. Par. S. Mth. Bll. 24 (1896) 180- . , formulae. Cauchy, A. L. C. R. 12 (1841) 871-. , and their application. Roselli, E. G. Arcad. 114 (1848) 1-, 121- ; 115 (1848) , functions. Falk, M. Ups. S. Sc. N. Acta Vol. extra ord. (1877) (No. 7) 32 pp. of functions, theory, application. Dint, U. Rm. R. Ac. Line. Mm. 2 (1895) 495-; A. Mt. 1 (1898) 39-. , principles (Cauchy). Eadicke, G. Crelle J. 25 (1843) 216-. " , and uses (Cauchy). Piola, G. Opusc. Mt. Fis. 2 (1834) 237-. , propositions, fundamental. Cauchy, A. L. C. R. 19 (1844) 1337-. , theory. Cauchy, A. L. C. R. 44 (1857) 406-. Series, two, representing same function in one part of plane. Kapteyn, W. N. Arch. Wisk. 3 (1898) 225-. Singular points of function given by series, and impossibility of analytic continuation. Fabry, E. Par. EC. Norm. A. 13 (1896) 367-. Singularities of analytic function, operations adding or removing. Pincherle, S. A. Mt. 4 (1900) 219-. , arrangement of infinite number. Velt- mann, W. Z. Mth. Ps. 27 (1882) 176-. on convergence circle of analytic functions. Pincherle, S. Bologna Rd. 2 (1898) 77-. , essential. Vivanti, G. Palermo Cir. Mt. Rd. 3 (1889) 223-. of F(x)= 2 az 2 . Fredholm,!. Stockh. Ofv. (1890) 131- ; Fschr. Mth. (1890) 410. Symmetrical expressions formed from values of modulus of function. Lewicki, W. Prace Mt.-Fiz. 6 (1895) 5- ; Fschr. Mth. (1895) 445. Synectic functions. Pringsheim, A. Munch. Ak. Sb. 26 (1897) 167-. Theorem. Lerch, M. Prag Sb. (1885) (Mth.- Nt.) 400-. of Bertini, demonstration. Liiroth, J. Mth. A. 42 (1893) 457- ; 44 (1893) 539-. , fundamental. Morera, G. Mil. I. Lomb. Rd. 19 (1886) 304-. of Mittag-LefHer, extension. Phragmen, E. C. R. 128 (1899) 1434-. Puiseux, on different paths leading to same value. Cauchy, A. L. C. R. 42 (1856) 663-. . Uniform function near singular point can approach infinitely near to any value. Proof. Holder, O. Mth. A. 20 (1882) 138-. of Weierstrass in an elliptic field. Cazzaniga, T. Mil. I. Lomb. Rd. 31 (1898) 1065-. Theory. Weierstrass, C. T. W. Berl. Ak. Ab. (1876) (Mth.) 11-. . Mittag-Leffler, G. C. R. 94 (1882) 414-, 511-, 713-, 781-, 938-, 1040-, 1105-, 1163- ; 95 (1882) 335-. . Goursat, E. C. R. 96 (1883) 565-. . Stfickel, P. Crelle J. Mth. 106 (1890) 189-. . Desaint, L. Par. EC. Norm. A. 14 (1897) 311-. . HaUgren, E. Goteb. Hndl. 32 (1897) 53- ; Fschr. Mth. (1897) 355. Transcendental function discovered by Fred- holm. Mittag-Leffler, G. C. R. 110 (1890) 627-. functions, class. Poincarf, H. C. R. 103 (1886) 862- ; Liouv. J. Mth. 6 (1890) 313-. , . Craig, T. Am. J. Mth. 16 (1894) 207-. , determination of genus. Laguerre, E. C. R. 94 (1882) 635-. , resolution into partial fractions. Bukreje/, B. [1886] Fschr. Mth. (1887) 381-. Transformation of imaginary trigonometric binomials into exponentials. Matzka, W. Prag Sb. (1864) (pt. 2) 116-. Uniform substitutions, convergence. Lemeray, E. M. C. R. 123 (1896) 793-; 124 (1897) 1220-. Zeros, distribution. Cesdro, E. N. A. Mth. 6 (1887) 36-. S e-Jpsinpx. Lerch, M. Mh. Mth. Ps. 8 P ~(1897) 377-. 295 3620 Multiform Functions of One Variable; Riemann Surfaces 3620 3620 Multiform functions of one variable ; Riemann surfaces. (See also 4060, 6420, 8075.) Algebraic configurations admitting involution. Fuchs, L. Berl. Ak. Sb. (1886) 797-. uniform automorphic transformations. Hurwitz, A. Gott. Nr. (1887) 85-. functions, application of a theorem in, to partial differential equations. Fuchs, L. Berl. Ak. Sb. (1887) 159-. , branches. Ascoli, G. Mil. I. Lomb. Ed. 18 (1885) 279-. , contours round singular points. David, . Toul. Ac. Sc. Mm. 8 (1886) 295-. , , theory. Rados, G. Mth. Termt. Ets. 3 (1885) 185-. , . Klein, F. D. Mth. Vr. Jbr. 2 (1893) 61-. Analytic point describing cycle , relations between residues. Appell, P. C. E. 95 (1882) 714-. , theorems. Appell, P. C. E. 95 (1882) 624-. polydromic functions. Volterra, V. Em. E. Ac. Line. Ed. 4 (1888) (Sem. 2) 355-. Arbitrary function, class, determination. Thome, L. W. Crelle J. Mth. 108 (1891) 335-. Cauchy's theorem. Tonelli, A. Em. E. Ac. Line. Ed. 1 (1885) 785-. Class. Picard, E. C. E. 88 (1879) 852-. Coinciding points in ditropic (2, 2) corre- spondence. Broden, T. Stockh. Ofv. (1893) 45-. Continuity. Neumann, C. Leip. Mth. Ps. B. 40 (1888) 120- . or discontinuity, simple method of in- vestigating. Neumann, C. G. Leip. Mth. Ps. B. 35 (1883) 85-. Contours traced round given points. David, . Toul. Ac. Sc. Mm. 9 (1887) 179-. Cross-cuts. Laguerre, E. C. E. 99 (1884) 1065-. Curves, points of which correspond to several values of the parameters. Thomae,J. Leip. Mth. Ps. B. 41 (1889) 365-. Doubly periodic functions. Pick, G. Wien Ak. Sb. 92 (1886) (Ab. 2) 893-. with-lacunae. Niccoletti, 0. G. Mt. 32 (1894) 364-. Function theory and differential equations. Kimigsberger, L. D. Nf. Vh. (1894) (Th. 2, Halfte 1) 5. Functions belonging to a group of linear substitutions. Wilczynski, E. J. Am. J. Mth. 21 (1899) 85-. of complex variable, investigation. Riemann, B. Crelle J. 54 (1857) 101-. , especially integrals of differential equations, representation near singular points. Ham- burger, M. [1876] Crelle J. Mth. 83 (1877) 185-. of quintuply connected surface. Thomae, J. Z. Mth. Ps. 11 (1866) 427-. General equation of 4th degree. Boer, F. de. Amst. Ak. Vs. M. 20 (1884) 413- ; Fschr. Mth. (1884) 79. Integral with double circuit. Pochliammer, L. Mth. A. 35 (1890) 470- . Integrals of complete differentials with two terms, analytic theorems. Riemann, B. Crelle J. 54 (1857) 105-. differential equations of 1st order. Vivanti, G. A. Mt. 19 (1891-92) 29-. solutions of differential equations, 2nd order, with three singular points. K$pinski, S. Krk. Ak. (Mt.-Prz.) Ez. 17 (1900) 112- ; Ore. Ac. Sc. Bll. (1899) 67-. Laplace's transformation and applications. Pincherle, S. Bologna Ac. Sc. Mm. 8 (1887) 125-. Linearly related functions, theory. Heun, K. Acta Mth. 11 (1887-88) 97-; 12 (1889) 103-. Multiformity, Diihring's conception, applica- tions. Wessely, K. Arch. Mth. Ps. 16 (1898) 225-. Periodic functions, generalisation. Floquet, G. Nancy S. Sc. Bll. (1897) 260-. Periodicity. Benthem, A. N. Arch. Wisk. 3 (*1877) 186-. Polynomials, geometry. Lucas, F. Par. EC. Pol. J. Cah. 46 (1879) 1-. Eelations of reality in the normal curve of 0's, belonging to any genus. Klein, F. Mth. A. 42 (1893) 1-. Eepresentation. Desaint, L. C. E. 130 (1900) 1296-. , conformal, by rational functions. Neovius, E. R. Helsingf. Acta 17 (1891) 1-. Eiemann integrals, theory. Baker, H.F. Mth. A. 45 (1894) 118-. , (Baker). Hensel, K. Mth. A. 45 (1894) 598-; 46 (1895) 160. EIEMANN SUEFACES. Berlin*, E. Em. E. Ac. Line. Ed. 3 (1894) (Sem. 1) 106-. algebraic configurations on, moduli. Fricke, R. D. Mth. Vr. Jbr. 3 (1894) 93-. analysis situs of. Durege, H. Wien Ak. Sb. 69 (1874) (Ab. 2) 115-. , theorems. Thomae, J. Z. Mth. Ps. 12 (1867) 361-. analytic functions on. Vitali, G. Palermo Cir. Mt. Ed. 14 (1900) 202-. with branch points given. Hurwitz, A. Mth. A. 39 (1891) 1-. variable. Hoyer, P. Mth. A. 47 (1896) 47-. canonical cross-cuts. Liiroth, J. Erlang. Ps. Md. S. Sb. 15 (1883) 24-. form and dissection. Clifford, W. K. L. Mth. S. P. 8 (1877) 292-. case. Burnside, W. L. Mth. S. P. 22 (1891) 410-. class non-transformable into simply connected surfaces. Puchta, A. [18801 Wien Ak. Sb. 82 (1881) (Ab. 2) 260-. connection of Bernoulli's functions. Jonquiire, A. Stockh. Ak. Hndl. Bh. 16 (Afd. 1) (1891) No. 6, 28 pp. 296 3620 Riemann Surfaces in connection with curve \ 3 /* + ^v + v s \ = 0. Haskell, M. W. Am. J. Mth. 13 (1891) 1-. construction for inverse of rational functions by conformal representation. Bmiton, C. L. A. Mth. 12 (1898-99) 1-. of monogenic analytic functions on. Ott, K. Mh. Mth. Ps. 4 (1893) 367-. "coupures," and cross-cuts, and concepts of integration. Casorati, F. A. Mt. 15 (1887- 88)223-; 16(1888-89) 1-. cross-cuts, transformation. Wellstein, J. Mth. A. 52 (1899) 433-. deficiency. Baker, H. F. Mth. A. 45 (1894) 133-. function constructed on. Wilczynski, E. J. N. Y. Am. Mth. S. Bll. 5 (1899) 388-. for Galois's resolvent of certain modular equa- tions. Dyck, W. Mth. A. 18 (1881) 507-. given, algebraic functions appertaining to. Thomae, C. J. Mth. A. 18 (1881) 443-. , integration of V 2 w=0 on. Ascoli, G. [1887] Stockh. Ak. Hndl. Bh. 13 (Afd. 1) (1888) No. 2, 83 pp. , = (Ascoli). Enestrom, G. Stockh. Ofv. (1887) 99-; Fschr. Mth. (1887) 29. integration of V 2 w + fc 2 = on. Sommerfeld,A. Gott. Nr. (1895) 267-. linearly bounded. Schonflies, A. Gott. Nr. (1892) 257-. models for. Durege, H. Prag Sb. (1864) (pt. 2) 112-. new kind. Klein, F. (vm) Erlang. Sb. Ps. Md. S. 6 (1874) 102-, 133- ; (x) Mth. A. 7 (1874) 558-; 10 (1876) 398-. order of branch points. Hensel, K. Berl. Ak. Sb. (1895) 933-. prime form on, Eitter's. Fricke, R. Gott. Nr. (1900) 314-. forms on. Klein, F. C. B. 108 (1889) 134-. . Wellstein, J. Gott. Nr. (1900) 380-. regular, group and irrationality. Dyck, W. Mth. A. 17 (1880) 473-. , of 3rd species, and corresponding " normal curve" of 4th order. Dyck, W. Mth. A. 17 (1880) 510-. representation of function in, with variable multipliers of periodicity. Suchar, P. I. [Bucarest S. Sc. Bll. 7 (1898)] 143-. sections. Lilroth, J. Mth. A. 4 (1871) 181-. symmetrical, and periodicity moduli of corre- sponding Abelian normal integrals. Weichold, G. Z. Mth. Ps. 28 (1883) 321-. theory. Clebsch, E. F. A. [1872] Mth. A. 6 (1873) 216-. three-leaved. Baur, L. Crelle J. Mth. 119 (1898) 171-. and four-leaved. Hensel, K. Berl. Ak. Sb. (1895) 1103-. treatment of unicursal curves by. Osgood, W. F. N. Y. Am. Mth. S. Bll. 2 (1896) 168-. use in applied mathematics. Car slaw, H. S. B. A. Ep. (1900) 644-. Eiemann theorem, demonstration. Prym, F. E. Crelle J. Mth. 83 (1877) 251-. Expansions in Series 3630 Eiemann theorem, demonstration. Croullebois, M. C. E. 93 (1881) 719-. theory. Klein, C. F. [1882] Mth. A. 21 (1883) 141-. (Klein). Noether, H. Z. Mth. Ps. 27 (1882) (H.-lt. Ab.) 201-. , extension. Volterra, V. Em. E. Ac. Line. Ed. 3 (1887) (Sem. 2) 281- ; 4 (1888) (Sem. 1) 107-, 196-, (Sem. 2) 485. , note. Lippich, F. Wien Ak. Sb. 69 (1874) (Ab. 2) 91-. Eoch theorem. Landsberg, G. [1897] Mth. A. 50 (1898) 333-. . Fields, J. C. [1900] N. Y. Am. Mth. S. Bll. 7 (1901) 244, 254-. , extension to linear systems of forms. Eitter, E. Gott. Nr. (1894) 328-. Eolle's theorem, extension. Veneziani, C. J. H. Un. Cir. [4] (1885) 78-. Surface of two sheets, theory of functions in. Prym, F. E. [1866] Ziir. N. D. Sch. Gs. 22 (1867) 47 pp. Surfaces, relationship, in Eiemann's sense. Lippich, F. [1873] Mth. A. 7 (1874) 212-. Symmetric and periodic functions derived from algebraic function. Petrovitch, M. Bll. Sc. Mth. 20 (1896) 108-. Theta functions, representation of quotient of two. Thomae, J. Mth. A. 6 (1873) 603-. Transformation of functions, theory. Weber, H. Crelle J. 76 (1873) 345-. ^t where Rl and * 2 belong to one Eiemann's surface. KriegvonHochfelden, F. (Frhr.) Wien Ak. Sb. 94 (1887) (Ab. 2) 221-. 3630 Expansions in series of func- tions, other than powers of the variable. Gomes Teixeira, F. Bll. Sc. Mth. 12 (1888) 272-. Algebraic expressions, expansion in a class of. AM, (Dr) M. Wien Ak. Sb. 52 (1866) (Ab. 2) 453-. functions, cases. Deruyts, J. Brux. Ac. Mm. 49 (1890-93) (No. 5) 20 pp. Analytic functions, expansion in given func- tions. Frobenius, G. [1870] Crelle J. 73 (1871) 1-. , integral rational functions, case. Hilbert, D. Gott. Nr. (1897) 63-. , systems, and series formed with them. Pincherle, S. A. Mt. 12 (*1883-84) 11-, 107-. Arbitrary functions, expansion in given func- tions. Lorenz, L. V. (xn) Ts. Mth. 6 (1876) 129-. powers of trigonometric functions. Sincov, D. M. Kazan S. Ps.-Mth. Bll. 4 (1895) 199- ; Fschr. Mth. (1895) 471. Bernoulli's functions, expansion in. Eogel, F. Prag Sb. (1896) (Mth.-Nt.) No. 31, 48pp. with fractional indices. Sincov, D. Kazan S. Nt. (Ps.-Mth.) P. 8 (1890) 291- ; Fschr. Mth. (1890) 443-. 297 3630 Expansions in Series Bernoulli's functions, generalised, applications. ImSeneckij, V. G. [1885] St Pet. Ac. Sc. Mm. (Rs.) 52 (1886) (Suppl.) No. 2, 62 pp. ; Fschr. Mth. (1886) 373-. polynomials and their applications. Sonine, N. J. Fschr. Mth. (1888) 424. Cases. Deruyts, J. Brux. Mm. Cour. 4, 47 (1886) No. 6, 18 pp. Convergence - subtrahends for trigonometric functions expressed in infinite sums. Boliannan, E. D. A. Mth. 1 (1884-85) 49-. Deduction of C, S, from c+is= s 7i -V 1 ^>*. x=0 *1 C x Pdnek, A. Casopis 17 (1888) 227-; Fschr. Mth. (1888) 263. Determination of coefficients in (n, 2) + ... where f(n) = A l (n, \} + A z (n, 2) + ... and f (n, k) = a 1 cf> (n, 1) + 0^ (n, 2) + .... Minin, A. P. Eec. Mth. (Moscou) 13 (1886) 535-; Fschr. Mth. (1887) 173-. Effective validity, cases. Pincherle, S. Em. E. Ac. Line. Ed. 5 (1896) (Sem. 1) 27-. Expansion in continued fractions. Hurwitz, A. Acta Mth. 11 (1887-88) 187-. -- denominators of convergents of continued fraction. Rouche, E. Par. EC. Pol. J. 37" eah. (1858) 1-. -- Euler's functions. Rogel, F. Prag Sb. (1896) (Mth.-Nt.) No. 42, 9pp. -- factorials. Schldmilch, 0. Leip. B. 15 (1863) 58-. --- of the variable. Graves, C. Ir. Ac. P. 3 (1847) 455-. harmonic functions. Steklov, V. A. Kharkov Mth. S. Com. 5 (1896?) 60-; Fschr. Mth. (1896) 203- ; C. E. 128 (1899) 279-. -- infinite series. Morera, G. Tor. Ac. Sc. At. 21 (1885) 892-. -- inverse factorial series. Kluyver, J. C. N. Arch. Wisk. 4 (1899) 74-; Fschr. Mth. (1899) 372. -- other functions. Hugoniot, . C. E. 95 (1882) 907-. ---- (Hugoniot). DM Bois-Reymond, P. C. E. 96 (1883) 61-. --- , form of residue. Lagrange, C. C. E. 98 (1884) 1422-. , any number of variables. Lagrange, G. Brux. Mm. Cour. 4, 48 (1886) No. 2, 16 pp. -- polynomials. Pincherle, S. C. E. 107 (1888) 986-. . Lean, L. Par. S. Mth. Bll. 27 (1899) 194-. -- powers of n algebraic functions from n equations. Arzela, C. G. Mt. 11 (1873) 368-. ---- a polynomial. Jacobi, C. G. J. [1847] Crelle J. 53 (1857) 103-. ------ e Laguerre, E. [1878-79] C. E. 86 (1878) 383-; Crelle J. Mth. 88 (1880) Multivariable Functions 3640 Expansion in trigonometric functions of im- aginary arcs. Clapeyron, E., & "Lame, G. Crelle J. 6 (1830) 45-. trigonometric functions sin^a;, cos*>x, x p when p is irrational. Yakovlev, N. Y. (xn) Kazan Un. Mm. (1857) (Bk. 4) 84-. Formula of Darboux. Mansion, P. Brux. S. Sc. A. 13 (1889) (Pt. 2) 108-. Hermite. Pincherle, S. Em. E. Ac. Line. Ed. 1 (1885) 267-. Functions of fractional powers of sine and cosine. Stein, J. P. Gergonne A. Mth. 15 (1824-25) 150-. one real variable, expansion in Jacobian functions. Dini, U. A. Mt. 10 (1880-82) 145-. Generating functions, fundamental theorem. Lerch, M. Prag Ceske Ak. Fr. Jos. Ez. (Trida 2) 1 (1892) Art. 33, 7pp.; Fschr. Mth. Holomorphic function within a given contour, expansion in polynomials. Renaux, . C. E. 129 (1899) 473-, 626. Integral function, expansion in denominators of certain regular continued fractions, pro- perty of. Gegenbauer, L. Wien Ak. Sb. 98 (1890) (Ab. 2a) 867-, Lagrange's series, generalisation. Amigues, E. C. E. 116 (1893) 368-, 429. -- , series analogous to. Amigues, E. C. E. 99 (1884) 1149-. -- , --- . Bougalev, N. C. E. 131 (1900) 793-. Series with imaginaries. Denzler, W. Zur. Vjschr. 1 (1856) 333-. of rational functions. Borel, E. C. E. 130 (1900) 1061-. Uniform functions, expansion in polynomials. Painleve; P. C. E. 102 (1886) 672-; 129 (1899) 27-. Wronski's law, elementary demonstration. Lagrange, C. Brux. Mm. Cour. 4, 47 (1886) No. 2, 8 pp. (x - z) m , expansion in powers of (z 2 1). Laguerre, E. C. E. 86 (1878) 956-. /(#), expansion in series 35-. . Humbert, G. Par. S. Mth. Bll. 8 (1880) 124-. Rogel, F. Prag Sb. (1892) (Mth.-Nt.) 3-. *, expansion in powers of sin x. Gomes Teixeira, F. N. A. Mth. 15 (1896) 270-. 3640 Functions of several variables. Trzaska, W. Par. T. Nauk Sc. Pam. 2 (*1872) 27-; Bll. Sc. Mth. As. 6 (*1874) 156. Mittag-Leffler, G. Stockh. Ofv. 34 (1877) No. 10, 3-, 17-. (Mittag-Leffler.) Weierstrass, C. T. W. Berl. Ak. Mb. (1880) 707-. ( -- .) Hermite, C. [1881] Crelle J. Mth. 92 (1882) 145-. Combescure, E. C. E. 96 (1883) 235-, 483-. Cousin, P. Acta Mth. 19 (1895) 1-. Lebesgue, H. C. E. 128 (1899) 811-. 298 3640 Multivariable Functions Algebraic functions of 2 variables. Hensel, K. D. Mth. Vr. Jbr. 7 (1899) (Heft 1) 58-. , new theory. Hensel, K. D. Mth. Vr. Jbr. 8 (1900) (Heft 1) 221-. several variables. Nother, M. Gott. Nr. (1869) 298-. Analytic continuation. Goursat, E. C. E. 128 (1899) 591-. functions in n dimensions, theorems. Scheffen, G. C. K. 116 (1893) 1242-. of several variables. Osgood, W. F. Mth. A. 52 (1899) 462-; 53 (1900) 461-. , expansion. Painleve, P. C. B. 129 (1899) 92-. Conjugate functions in n variables. Volterra, V. Km. E. Ac. Line. Ed. 5 (1889) (Sem. 1) 599-. Differential quotients. Czuber, E. Wien Ak. Sb. 101 (1892) (Ab. 2a) 1417-. Differentials. Goursat, E. Par. EC. Norm. A. 2 (1885) 255-. , successive. Darboux, G. C. E. 93 (1881) 1123- ; 94 (1882) 575-; Bll. Sc. Mth. As. 5 (1881) 376-, 395-. Expansion, conditions of reality. Schulten, N. G. of (fil.\. [1843] Helsingf. Acta 2 (1847) 347-. in series, form when curtailed. Eiqnier, . C. E. 126 (1898) 1558-. Functions of two real variables, Schonflies's theorem. Osgood, W. F. Gott. Nr. (1900) 94-. , . Bernstein, F. Gott. Nr. (1900) 98-. variables. Poincare, H. Acta Mth. 2 (1883) 97-. , behaviour near a zero point. Brill, A. Munch. Ak. Sb. 21 (1892) 207-. , boundary conditions for value. Ascoli, G. Mil. I. Lomb. Mm. 16 (1891) 197-. , class. Appell, P. Acta Mth. 2 (1883) 71-. , divisibility by another function, case. Berry, A. L. Mth. S. P. 30 (1899) 271-. , graphical representation. Klejber, I. A. Es. Ps.-C. S. J. 19 (Ps.) (1887) 18-. , inversion. Fuchs, L. Berl. Ak. Sb. (1887) 99-. , limiting values, two theorems. Du Bois-Reymond, P. [1876] Mth. A. 11 (1877) 145-. n real variables. Giuliani, G. G. Mt. 29 (1891) 234-. variables arising from inversion of integrals. Fuchs, L. Berl. Ak. Sb. (1883)507-. Gauss's formula for interlaced curves and ex- tension to higher configurations. Dyck, W. Munch. Ak. Sb. 25 (1896) 447-. Hyperspaces, complex variables in. Volterra, V. Em. E. Ac. Line. Rd. 5 (1889) (Sem. 1) 291-; 6 (1890) (Sem. 2) 241-. , functions of, and their differential para- meters. Volterra, V. Em. E. Ac. Line. Ed. 5 (1889) (Sem. 1) 630-. Integral function of n elements, number of values. Netto, E. Crelle J. Mth. 85 (1878) 327-. Algebraic Functions 4000 Integrals and expansions in series, cases. Bucca,F. Palermo Cir.Mt. Ed. 12 (1898)260-. Kronecker's characteristic. Dyck,W. Munch. Ak. Sb. 25 (1896) 261-. Limiting values. Stolz, 0. Mth. A. 26 (1886) 83-; Innsb. Nt. Md. B. 17 (1887) xxvi-. Periodic functions of two variables. Appell, P. Liouv. J. Mth. 7 (1891) 157-. Potential theory. Dyck, W. Munch. Ak. Sb. 25 (1896) 261-, 447-; 28 (1899) 203-. Eational functions of n variables, classes. Kronecker, L. Berl. Ak. Sb. (1886) 251-. , conjugate values. Netto, E. Crelle J. Mth. 100 (1887) 436-. Eelation between p integral functions of p - 1 variables. Pen-in, R. C. E. 106 (1888) 1789-. Series integral with reference to several variables. Dautheville, S. Par. EC. Norm. A. 2 (1885) (Suppl.) 59 pp. Theorem, demonstration. Trzaska, W. Par. T. Nauk Sc. Pam. 2 (*1872) 39-; Bll. Sc. Mth. As. 6 (*1874) 156. Theory. Baker, H. F. Camb. Ph. S. T. 18 (1900) 408-. , formula. Pincherle, S. Stockh. Ofv. (1886) 51-. , , and integration of total differentials. Jaggi, E. N. A. Mth. 16 (1897) 297-. Transcendental functions, property of class. Volterra, V. Em. E. Ac. Line. Ed. 2 (1886) (Sem. 2) 21 1-. Undeveloped functions, theory. Schwarz ,H.A. Berl. Ak. Sb. (1897) 948-. Uniform analytic functions of n variables, theory. Biermann, O. Wien Ak. Sb. 89 (1884) (Ab. 2) 266-. functions defined by inversion of total differentials. Painleve, P. C. E. 122 (1896) 660-, 769-. expressible as quotient of two power series, theorem. Hurwitz, A. [1882] Crelle J. Mth. 95 (1883) 201-. of 2 variables, class. Picard, E. C. E. 95 (1882) 594-. connected by an algebraic equa- tion. Giinther, P. Crelle J. Mth. 109 (1892) 199-. , and a group of linear substitu- tions. Picard, E. C. E. 94 (1882) 579-, 837-. , poles. Autonne, L. Palermo Cir. Mt. Ed. 10 (1896) 196-. several independent variables, poles. Autonne, L. C. E. 124 (1897) 139- ; Acta Mth. 21 (1897) 249-. Zeros. Dyck, W. Munch. Ak. Sb. 28 (1899) 203-. Algebraic functions and their integrals. 4000 General. Rydberg, J. R. Lund. Un. Acta 15 (1878-79) (Mth.) (No. 2) 18 pp. Algebraic functions. Cockle, Jos. Camb. and Dubl. Mth. J. 2 (1847) 267-; 3 (1848) 179-. 299 4000 Algebraic Functions and their Integrals. General 4000 Algebraic functions. Puiseux, V. Liouv. J. Mth. 15 (1850) 365-; 16 (1851) 228-. (Puiseux). Canchy, A. L. C. B. 32 (1851) 276-. . Hermite, C. C. B. 32 (1851) 458-. , application in geometry. Brill, A., & Nother, M. [1873] (vn) Mth. A. 7 (1874) 269-. , class. Schldmilch, 0. Grunert Arch. 10 (1847) 67-. , continuity and discontinuity, conditions. Gascheau, . Toul. Ac. Sc. Mm. 2 (1880) (Pt. 2) 122-. and curves, property. Picard, E. C. B. 91 (1880) 214-, 724-. , fundamental theorem. Voss, A. Mth. A. 27 (1886) 527-. , number of arbitrary constants. Roch, G. [1864] Crelle J. 64 (1865) 372-. , property. Darboux, G. C. E. 94 (1882) 575-. , rationalisation. Polewski, S. Arch. Mth. Ps. 69 (1883) 149-. , representation, approximate, by means of rational functions. Bortolotti, E. Em. E. Ac. Line. Ed. 8 (1899) (Son. 1) 57-. , , invariant. Noether, M. Mth. A. 17 (1880) 263-. , Eiemann's theorem, demonstration. Herz, N. [1881] Arch. Mth. Ps. 68 (1882) 14-. , singular points, application of Newton's polygon to theory. Baker, H. F. Camb. Ph. S. T. 15 (1894) 403-. , theorem. Noether, M. Crelle J. Mth. 92 (1882) 301-. , theory, development. Brill, A., < Noether, M. D. Mth. Vr. Jbr. 3 (1894) 107-. , values, connexion between. Runge, C. Crelle J. Mth. 97 (1884) 337-. integrals, theorem. PtaSickij,I.L. Kharkov Mth. S. Com. 1 (1889) 74-; Fschr. Mth. (1888) 296-. Arbitrary function, determination. Cellerier, C. Liouv. J. Mth. 8 (1843) 245-. functions, elimination. Hermite, C. B. A. Ep. 42 (1872) 233-. Class of functions connected with =(*-a a ) (*-,)...(*-,), theory. Wellstein, J. Mth. A. 52 (1899) 440-. Coincidence of function with algebraic functions in infinite number of places. Schering, E. [1879] Gott. Ab. 27 (1881) (Mth.) 62 pp. Continued fractions, new applications. Markov, A. St Pet. Ac. Sc. Mm. 3 (1896) No. 5, 50pp. Differentials, integrals of which are algebraic functions, form. Kapteijn, W. Amst. Ak. Vs. M. 17 (1882) 93-. Expressions, integrals of which are algebraic. Liouville, J. [1832-37] Par. EC. Pol. J. 22 cah. (1833) 124-, 149- ; Crelle J. 10 (1833) 347-; C. E. 5 (1837) 330-. Formula of Jacobi, demonstration. Brioschi, F. Tortolini A. 1 (1858) 117-. Fractional functions, decomposition and trans- formation. Rangoni, L. [1827] (vm) Mod. Mm. Ac. Sc. 1 (1833) (pte. 2) 254-. Functions in general, fundamental theories established without reduction to the canon- ical form. Tikhomandritskii, M. A. (xn) Kharkov Mth. S. Com. (1883) 79-; Fschr. Mth. (1885) 423. . with multiple periods. Casorati, F. C. E. 57 (1863) 1018-. Goniometric functions, relation to certain alge- braic expressions. Studnicka, F. J. Casopis 19 (1890) 249-; Fschr. Mth. (1891) 444. Identities, generalisation. Oltramare, G. Gen. I. Nt. Mm. 16 (1886) No. 4, 109 pp. Integrals of algebraic differentials. Ramus, C. Crelle J. 24 (1842) 69-. functions. Ostrogradsky, M. A. [1842] St Pet. Ac. Sc. Bll. 1 (1848) 113-. . Pellet, A. E. C. E. 90 (1880) 676-. , reduction to integrals of rational functions. Formenti, C. Mil. I. Lomb. Ed. 14 (1881) 667-; 15 (1882) 165-. . , theorems. Poisson, S. D. Crelle J. 12 (1834) 89-. , theory. Jttrgensen, C. Kiob. Ov. (1839) 6-; Crelle J. 23 (1842) 126- ; N. Mg. Ntvd. 3 (1843 ?) 85-. and transcendental formulae. Tortolini, B. G. Arcad. 56 (1832) 81-. . I^A X . Guic!iard,C. Par. EC. Norm. ; JKM A. 5 (1888) 193-. Integration in finite explicit functions, deter- mination of algebraic part. Piuma, C. M. (vm) Tortolini A. 4 (1861) 154-. form of cubics and quartics sym- metrical in x and -. Toropov, K. A. Kharkov Mth. S. Com. (1885) 3-; Fschr. Mth. (1885) 258. , Jacobi-Mayer method, improvement. Lie, S. Christiania F. 15 (1873) 282-. Inverse derived functions. Buquoy, G. von. Gilbert A. 67 (1821) 202-. Inversion of system of functions. Lipschitz, R. Gott. Nr. (1870) 439-. Isoperimetric equations and integrability. Brioschi, F. [1853] Mod. Mm. S. It. 25 (1855) 205-. Periodic function, calculation of coefficients from given mean values. Krttger, A. As. Nr. 82 (1873) 333-. Periods of integrals. Marie, M. C. E. 36 (1853) 437-. Powers of functions, expansions, cases. Andre, D. [1877] Par. S. Mth. Bll. 6 (1878) 120-. Product of several integrable and finite functions. Ascoli, G. Mil. I. Lomb. Ed. 12 (1879) 372-. Eesolution of a class of functions. Jiirgensen, C. [1838] Kiob. Dn. Vd. Selsk. Afh. 8 (1841) 1-; Crelle J. 19 (1839) 84-. Sums depending on positive values of any function. Cebysev, P. St Pe"t. Ac. Sc. Mm. 1 (1895) No. 7, 20pp.; Fschr. Mth. (1895) 441-. 300 UNIVERSITY B&NTA BARBARA COLLEGE UBRAKT 4010 Theorem, fundamental. Noether, M. Mth. A. 6 (1873) 351- ; 30 (1887) 410- ; 34 (1889) 450- ; 40 (1892) 140-. , (Noether). Baker, H. J. Mth. A. 42 (1893) 601-. , ( ). Scott, C. A. Mth. A. 52 (1899) 593-. , , demonstration. BoJiannan, R. D. Am. J. Mth. 19 (1897) 192. Transcendental functions, classification. Liou- ville, J. [1835] Liouv. J. Mth. 2 (1837) 56-; 3 (1838) 523-. Undeveloped functions, new proof of funda- mental theorem. Grave, D. Fschr. Mth. (1899) 379. 4010 Algebraic functions of one variable. Betti, E. Pisa A. Un. Tosc. 7 (Sc. Cosm.) (1862) 101-. Hensel, K. D. Mth. Vr. Jbr. 1 (1892) 56-. Algebraic configuration of nth rank in domain of n + 1 magnitudes. Biermann, O. Wien Ak. Sb. 95 (1887) (Ab. 2) 802-. differentials, integration. Ptasickij, I. L. Kharkov Mth. S. Com. 1 (1889) 61-; Fschr. Mth. (1888) 296-. functions of one and two variables. Nother, M. Gott. Nr. (1871) 267-. integrals. Cremona, L. [1869] Bologna Ac. Sc. Mm. 10 (1870) 3-. logarithmic integrals of algebraic functions, extension of Abel's theorem. Konigsberger, L. Mth. A. 17 (1880) 560-. part of integral of rational functions, determination. Gomes Teixeira, F. Bm. B. Ac. Line. Bd. 1 (1885) 187-. systems, reduction to canonical form. Hensel, K. Crelle J. Mth. 117 (1897) 129-. Application of theory of linear differential equations. Thomt, L. W. Crelle J. Mth. 104 (1889) 1-; 112 (1893) 165-. Canonical systems. FiscJier,K. Crelle J. Mth. 117 (1897) 1-. Circular arc polygons. Schdnflies, A. Mth. A. 42 (1893) 377-. and triangles. Schonjlies, A . Mth. A. 44 (1894) 105-. quadrilaterals and principle of symmetry. Fricke, R. Mth. A. 44 (1894) 565-. Definition by cubic equation. Betti, E. G. Mt. 3 (1865) 143-. . Floquet, G. Par. S. Mth. Bll. 23 (1895) 76-. Factorisation of integral function of one variable into factors without multiple roots. Engel, F. Leip. Mth. Ps. B. 49 (1897) 603-. Function (x) = ^| . Tanner, H. W. L. [1879] Mess. Mth. 9 (1880) 109-. (x - 1). Humbert, G. Par. S. Mth. Bll. 9 (1881) 56-. f(x) = (x + 1) (x -l)(x-2). Bigler, U. Arch. Mth. Ps. 14 (1896) 337-. Algebraic Functions of One Variable 4010 Functions analogous to Legendre's polynomials. Hugoniot, . C. B. 95 (1882) 983-. of form/(.r) = cx + d > symbolic powers and roots. Johnson, W. W. Mess. Mth. 9 (1880) ax + b powers. Tanner, H. W. L. cx + d Mess. Mth. 9 (1880) 139-. Fundamental system for given class domain,, determination. Mertens, F, Wien Ak. Sb 102 (1893) (Ab. 2a) 497-. systems for. Baker, H. F. L. Mth. S. P. 26 (1895) 107-. Geometrical study. Tognoli, O. G. Mt. 22, (1884) 308- ; 23 (1885) 247-, 345-. Hyperelliptic integrals. Chessin, A. S. [1897] N. Y. Am. Mth. S. Bll. 4 (1898) 93-. Implicit function defined by algebraic equa- tion, values. Marie, M. C. B. 47 (1858> 145-. Integral algebraic functions, representation. Hensel, K. Crelle J. Mth. Ill (1893). 139-. rational functions with integral coefficients,, irreducibility. Hilbert, D. Crelle J. Mth. 110 (1892) 104-. x 2m dx , value. Popov, A. T. (xn) -LJ. 1 Kazan Un. Mm. 6 (1871) 236-. , evaluation. Worpitzky, J. Z. Mth. Ps. 19 (1874) 90-. Integrals of algebraic expressions with several variables. Gronwall, H. Stockh. Ofv. (1896) 67-. value, theory. Bertrand, J. Par. S. Phlm. PV. (1843) 126. factorial functions, inversion. Picard, E. Am. J. Mth. 16 (1894) 111-. form I 7^=^ ' reduction. Stuart, G. H. QJ. Mth. 18 (1882) 245-. Integration of differentials containing square root of polynomial of 3rd or 4th degree. Tchebicheff, P. [1854] Liouv. J. Mth. 2 (1857) 1-. 4th degree or cube root of polynomial of 3rd degree. Aleksyeev, N. N. (xn) Bee. Mth. (Moscou) [1] (1866) 187-. in finite series, cases. Pezzi, F. Verona S. It. Mm. 4 (1788) 577-; 6 (1792) 256-. , graphical treatment, and applications. Massau, J. Bv. Un. Mines 16 (1884) 243-, 579-; 17 (1885) 1-; 20 (1886) 1-, 475-; 21 (1887) 85-, 308-, 572-; 22 (1887) 1-. of an irrational formula. Maurice, F. Bb. Brit. 17 (1801) 103-. irrational functions. Casorati, F. Tortolini A. 7 (1856) 223-. , reduction formulae. Azzarelli, M. Bm. N. Line. Mm. 1 (1887) 247-. by substitution, case. Luchterhandt, A. R~ Crelle J. 17 (1837) 248-. 301 4010 Functions of One Variable Integration by use of imaginaries. Martin, H. N. A. Mth. 2 (1863) 57-. and Rumovski, S. [1807] St Pet. Ac. Sc. Mm. 2 (1807-08) 84-. x m dx Nicolai, F. B. G. As. Nr. 5 (1827) 371-. . dx. Tchebicheff, Jx* + ax 3 + P. [1860] St Pet. Ac. Sc. Bll. 3 (1861) ...+px dx Grelle, F. Z. Mth. Ps. 16 (1871) 254-. Isodynamic lines of a polynomial, electric determination. Lucas, F. C. E. 106 (1888) 587-. Limiting values of integrals. Markov, A. A. St Pet. Ac. Sc. Mm. 10 (1900) No. 9, 34 pp. ; Fschr. Mth. (1900) 313-. Linear functions. Pellet, A. E. C. E. 90 (1880) 1111-. of one variable. Christoffel, E. B. Crelle J. 55 (1858) 281-. Normal basis and absolute fundamental system, connexion between. Baur, L. Mth. A. 49 (1897) 73-. Periodicity, multiple, in functions of one variable. Casorati, F. Mil. I. Lomb. Ed. 16 (1883) 815-. Polynomials, statics. Lucas, F, Par. S. Mth. Bll. 17 (1889) 17-. Eational fractional functions, representation by. Biermann, 0. Wien Ak. Sb. 89 (1884) (Ab. 2) 84-. integral functions, type. Laguerre, . C. E. 98 (1884) 79-. operations. Noether, M. Mth. A. 23 (1884) 311-. Eeciprocity theorem. Study, E. Leip. Mth. Ps. B. 42 (1890) 153-. Simple transcendental functions, elementary proof of periodicity. StudniCka, F. J. Casopis 22 (1893) 209- ; Fschr. Mth. (1893- 94) 737. Theorem of Abel. Stickelberger, L. Crelle J. Mth. 82 (1877) 45-. , fundamental. Bertini, E. Mth. A. 34 (1889) 447-. , . Hensel, K. Crelle J. Mth. 115 (1895) 254-. of Hadamard, application. Borel, E. Bll. Sc. Mth. 18 (1894) 22-. -- Netto. Bertini, E. Mth. A. 35 (1890) -- Noether. Stickelberger, L. Mth. A. 30 (1887) 401-. Weierstrass. Noether, M. Crelle J. Mth. 97 (1884) 224-. Algebraic Functions 4020 Theory. Dedekind, B., & Weber, H. [1880] Crelle J. Mth. 92 (1882) 181-. . Kneser, A. Mth. A. 29 (1887) 171-. . Hensel, K. Crelle J. Mth. 109 (1892) 1-; Acta Mth. 18 (1894) 247-. . Baur, L. Crelle J. Mth. 116 (1896) 167-. , arithmetical. Landsberg, G. Gott. Nr. (1895) 407-. Transcendental functions, arising from re- peated integrations. Kummer, E. E. Crelle J. 21 (1840) 74-, 193-, 328. Trinomial integrals. Besge, V. A . Liouv. J. Mth. 4 (1859) 194. Values assumed by a rational integral function between assigned limits of the variable. Laguerre, . C. E. 98 (1884) 136-. of x oo , etc. Nievengloski, G. H. N. A. Mth. 7 (1848) 425-. (l-x m )(i-x m -i)...n-x m -* +l ) . i - '- - '- - '- is an integral (l-x)(l-x*)...(l-x) function of x when m and /j. are integers, and m >>. Perott, J. Par. S. Mth. Bll. 10 (1882) 87-. 4020 Algebraic functions of several variables. Convergence and double limits. Osgood, W. F. N. Y. Am. Mth. S. Bll. 3 (1897) 59-. Formula of Leaute, extension. Karagiannides, . N. A. Mth. 17 (1898) 539-. Theorem of Kronecker on functional determi- nant of 3 equations. Marenghi, C. Ven. I. At. (1897-98) 1688-. Theory. Vessiot, E, Toul. Fac. Sc. A. 10 (1896) D, 14 pp. (f + " = lf within. Most, It. Z. Mth. Ps. 14 (1869) 422-. 2 variables. Landsberg, G. Berl. Ak. Sb. (1900) 296-. change of independent variables. Gergonne, J. D. Gergonne A. Mth. 1 (1810-11) 251-. class. Picard, E. C. E. 90 (1880) 1119-. first differentials being given _, implicitly. Woisard, J. L. Par. Mm. Sav. Etr. 3 (1832) polynomials of two variables and approximate calculation of double integrals. Appell, P. Toul. Fac. Sc. A. 4 (1890) H, 20 pp. representability by double trigonometric series. Ascoli, G. Em. E. Ac. Line. Mm. 4 (1879) 253- ; Mil. I. Lomb. Ed. 15 (1882) 543-. resolution into partial fractions. Hoyer, P. Mth. A. 47 (1896) 113-. and theorem of Noether. Brill, A. Mth. A. 39 (1891) 129-. theory. Picard, E. Liouv. J. Mth. 5 (1889) 135-. . Kobb, G. Liouv. J. Mth. 8 (1892) 385-; Par. S. Mth. Bll. 21 (1893) 76-. . Grave, D. Fschr. Mth. (1899) 378-. ,new. Hensel, K. Acta Mth. 23 (1900) 339-. transformation. Euler, L. [1779] St Pet. Ac. Sc. Mm. 3 (1811) 43-. 302 4030 Logarithmic, Circular, Exponential Functions 3 or more variables. approximation. Didon, F. C. E. 70 (1870) 749-. functions analogous to sine and cosine. Glaisher, J. W. L. [1878] QJ. Mth. 16 (1879) 15-. integrability. Brunn, H. [1848J St Pe"t. Ac. Sc. Bll. 7 (1849) 126-. integration. Minich, S. E. At. Sc. It. (1844) 62-. , formulae. Minich, S. E. Yen. Mm. I. 6 (1856) 473-. rational functions, representation, analytic. Mittag-Leffler, G. Stockh. Ofv. 34 (1877) A T o. 10, 3-, 17-. theory. AW, M. [1875] Wien Ak. Sb. 72 (1876) (Ab. 2) 289-. . Capelli, A. Nap. Ed. 29 (1890) 297-. 4030 Logarithmic, circular, exponential functions. CIECULAE FUNCTIONS. Eaabe, J. L. Crelle J. 16 (1837) 219-. Calculation of tables. Olivier, L. Crelle J. 2 (1827) 252-. Circular arcs, reduction to imaginary logarithms. Calandrelli, G. [1825] Mod. Mm. S. It. 20 (1828) 45-. functions, analogues. Appell, P. C. E. 84 (1877) 1378-. , analysis. Eees, E. van. Quetelet Cor. Mth. 6 (1830) 277-. , analytic derivation. Tralles, J. G. Berl. Ab. (1812-13) (Mth.) 161-. , theory. Chessin, A. S. Am. J. Mth. 19 (1897) 217-. , approximate formulas and application. Fein, A. (ix) Forster Al. Bauztg. 38 (1873) 34-. , approximation by means of algebraic functions. Laguerre, E. C. E. 90 (1880) 304-. , astronomical formulae. Le Verrier, U. J. J. (x) Par. Obs. A. 1 (1855) 73-; 2 (1856) 301 + [167] pp.; 3 (1857) 199-, [1]-; 4 (1858) 1-, [1]-; 5 (1859) 1-; 6 (1861) 435 pp.; 10 (1874) 1-, [1]-; 11 (1876) 560 + [136] pp.; 12 (1876) 76 + [178] + A. 80 + [A. 286] pp. ; 13 (1876) 1-, [1]- ; 14 (1877) Ft. 1, A. 1-, [A. 1]-, Ft. 2, 1-, [I]-- , differential, integration in finite form. Du Bourguet, . Gergonne (1813-14) 72-. , , . Gruson, J. P. Berl. Ab. (1820-21) 44-. , differentiation, successive. Colombier, P. A. G. N. A. Mth. 9 (1850) 29-; 1 (1862) , formula. Catalan, . As. Fr. C. E. (1889) (Pt. 1) 232. 4030 Circular functions, fundamental principles. Sarrus, F. Gergonne A. Mth. 11 (1820-2li 323-. , properties, new proof. Fleck, (I'abbe) J. M. (xn) Metz Ac. Mm. 54 (1874) 337-. , integration. Hermite, C. [1872] L. Mth. S. P. 4 (1871-73) 164-. , inverse, differential coefficients. Wiiatley, F. Mth. Gz. 1 (1900) 83-. , 2nd order, integrals. Meyer, V. H. Grunert Arch. 17 (1851) 426-. , principal values. Stolz, O. Innsb. Nt. Md. B. 17 (1887) 67-. , resolution into partial fractions. Oliveira Eamos, J. C. d\ G. Teix. J. Sc. 7 (1886) 7-; Fschr. Mth. (1885) 231-. , de Stainville's theorem. Gergonne, J. D. Gergonne A. Mth. 13 (1822-23) 270-. , summation. Jarrett, T. B. A. Ep. (1847) (pt. 2) 5. and hyperbolic functions. Imshenetskii, V. G. (xn) Kazan Un. Mm. (1862) (Ps.- Mth.) 19 pp. . Giudice, F. G. Mt. 27 (1889) 124-. , analogy. Mossotti, O. F. Tortolini A. 2 (1851) 474- ; Pisa A. Un. Tosc. 6 (1861) , logarithmic tables. Forti, A. (vi Adds.) N. Cim. 17 (1863) 243-. , relations. Eealis, S. Mathesis 6 (1886) 7-. Concepts cos x and sin x, extension. Beyssell, . Grunert Arch. 31 (1858) 299-. Cosines, sums of series of like powers. Eadicke, A. Mathesis 4 (1884) 161-. Cotangent and cosecant, fractional series for, algebraic proof. Glaisher, J. W. L. [1880] QJ. Mth. 17 (1881) 211-. Definite integrals containing circular functions. Haan, D. Bierens de. Haarl. Ntk. Vh. Mtsch. 17 (1862) 199 pp. Expansion. Oliveira, C. B. Ev. Brazil. 3 (I860) 214-. of arc in terms of tangent. Schlomilch, 0. Grunert Arch. 16 (1851) 230-. circular and exponential functions. Ampere, A. M. Gergonne A. Mth. 15 (1824-25) 369-. . Demoulin, A. Brux. Ac. Bll. 19 (1890) 541-. cosines and sines of multiple arcs. Maiden, H. Ph. Mg. 20 (1842) 113-. in powers of arc. Schellbach, C. H. Crelle J. 16 (1837) 363-. A. Mth. 4 terms of arc, and of arc in terms of tangent. Gergonne, J. D. Gergonne A. Mth. 3 (1812-13) 344-. exponential in powers of sine and cosine of arc. David, . [1882] Toul. Ac. Sc. Mm. 5 (1883) (San. I.) 148-. fractional powers of cosines. Crelle, A. L. Gergonne A. Mth. 13 (1822-23) 213- ; Crelle J. 7 (1831) 253-, 314-. 303 4030 Circular Functions, Expansion 4030 of fractions containing sines and cosines of multiple arcs. Trembley, J. Berl. Mm. Ac. (1802) 31-. in indefinite products. Gergonne, J. D. Ger- gonne A. Mth. 1 (1810-11) 116-. of infinite products for trigonometric functions. Schlomilch, 0. Schlomilch Z. 3 (1858) 389-. logarithmic, exponential and trigonometric functions. Bangma, O. S. Amst. Mengel- werk 2 (1816) 92-. powers of trigonometric ratios in series of of 6 a in powers of sin 6 . Scherk, H. F. Crelle J. 11 (1834) 101-. (l + 2l'cos

')- n in series of cosines of multiples. Jacobi, C. G.J. Crelle J. 15 (1836) 205-. cos n (0 + A0) in powers of cos 0. Cauchy, A. L. C. E. 15 (1842) 411-. sin-ia-. Raabe, J. L. Crelle J. 25 (1843) 169-. . Schlomilch, O. Schlomilch Z. 1 (1856) 181-. by Maclaurin's theorem. Chevilliet, . ascending powers of variable. Jeffery, H. M. N. A. Mth. 13 (1874) 209- [1860] QJ. Mth. 5 (1862) 91-. sines of multiple angles. Krafft, W. L. [1802] St Pet. Ac. Sc. N. Acta 14 (1805) 290-. in products. Fade, H. N. A. Mth. 17 (1898) 312-. trigonometric functions. Young, J. JR. Ph. Mg. 5 (1834) 198-. as infinite products. Dienger, J, Arch. Mth. Ps. 43 (1865) 474-. in infinite series. Burg, A. "Wien Jb. Pol. I. 19 (1837) 147-. products of binomial factors. Rodriguez, O. Liouv. J. Mth. 8 (1843) 217-. in trigonometric series. Walton, W. [1871] QJ. Mth. 12 (1873) 146-. of sinn0 and cosnfl. Eider, L. [1780] St Pet. Ac. Sc. Mm. 5 (1815) 57-. in terms of sin 6 and cos 6, demonstra- tion. Kausler, C. F. [1810] St Pet. Ac. Sc. Mm. 4 (1813) 256-. (Euler). Fuss, N. [1813] St Pet. Ac. Sc. Mm. 6 (1818) 100-. , cosn0 and sin" 6, cos" 0. Crelle, A. L. Crelle J. 5 (1830) 197-. and cos nd in ascending powers of sin and cos 0. Gregory, D. F. Camb. Mth. J. 2 (1841) 129-. terms of powers of sin 6 and cos0, by de Moivre's theorem. Sylvester, J. J. QJ. Mth. 4 (1861) 159-. series of sines and cosines, of multiples of 0. Poisson, S. D. Par. EC. Pol. Cor. 2 (1809-13) 212- ; Ferussac Bll. Sc. Mth. 4 (1825) 140-, 344-. . Kummer, E. E. Crelle J. 14 (1835) 110-. in terms of multiples of 0. Simony, 0. Arch. Mth. Ps. 55 (1873) 64-. (a + bcosx) n , general term. Degen, C. F. (vi Adds.) Kiob. Ov. (1819-20) 3-. cos" 6 in cosines of multiples of 0. Plana, G. Gergonne A. Mth. 11 (1820-21) 84-. in sines and cosines of multiples of 6. Michel, P. Gergonne A. Mth. 13 (1822-23) cosines of multiples of d. Olivier, L. Crelle J. 1 (1826) 16-. tan d in powers of d, law. Bouvier, L. C. Gergonne A. Mth. 14 (1823-24) 347-. (cosx), etc. Ohm, M. Ferassac Bll. Sc. Mth. 5 (1826) 250-. 0" in terms of sin 6. Scholtz, E. J. Crelle J. 3 (1828) 70-. cos nx/cos n x and sin nxjcos" x in powers of tan*. Arndt,F. Grunert Arch. 4 (1844)441-. sine and cosine in powers of arc. Schlomilch, O. Grunert Arch. 5 (1844) 326-. cos n ax. Dienger, J. Grunert Arch. 11 (1848) 331-. cos m x in series of cosines for all values of n. Neivman, F. W. Camb. and Dubl. Mth. J. 3 (1848) 61-. ; in series of even multiples of . l-/JiCOS Wolfers, J. P. Grunert Arch. 21 (1853) 190-. sine and cosine. Grunert, J. A. Grunert Arch. 29 (1857) 452-. sin rax and cosnx in powers of sin a; and cos a;. Baehr, G. F. W. Amst. Vs. Ak. 10 (1860) 86-. . Yvon-Villarceau, A. J. F. C. R. 82 (1876) 1469-. . Catalan, E. C. N. A. Mth. 2 (1883) 529-. sin a; and cos a;. Almqvist, P. W. Ts. Mt. Fys. 3 (1870) 121-. in infinite series. Meusburger, K. Mh. Mth. Ps. 6 (1895) 256-. in series of exponential functions with real exponents. Popov, A. T. (xn) Kazan Un. Mm. 11 (1875) 661-. cos n0 in powers of cosines, etc. Walton, W. [1872] QJ. Mth. 12 (1873) 168-. sine and cosine of arc. Sang, E. [1877] Edinb. E. S. P. 9 (1878) 400-. . David, . Par. S. Mth. Bll. 11 (1883) 72-. sec a; and tan a:. Andre, D. C. B. 88 (1879) 965-. cot x. Le Paige, C. M. M. H. H. C. E. 88 (1879) 1075-. (1 + b cos + &' cos ' + ...)". Oppolzer, T. von. As. Nr. 105 (1883) 367-. - ~ . Catalan, E. C. Liege S. Sc. Mm. 15 (1888) No. 1, 82-. sin n and cos nip in powers of cos . Ocagne, M. d'. G. Teix. J. Sc. 9 (1889) 161-. tan -1 x, in convergent series. Saalschiltz, . Konigsb. Schr. 40 (1899) [28]-. Function of 3 angles, first differentials of which are also to be regarded as angles. Schlafli, L. [1852] Crelle J. 48 (1854) 292-. Functions of small angles, method of finding from five-figure tables. Pfeil, L. von. Arch. Mth. Ps. 58 (1876) 147-. 304 4030 Logarithmic, Circular, Exponential Functions Gregory's series. Chartres, B. Nt. 42 (1890) 341. Hyperbolic base, analytical value. Hill, T. Am. As. P. 19 (1870) 21-. logarithms of first n-i numbers, formula for. Tilly, J. M. de. Brux. Ac. Bll. 35 (1873) 30-. Logarithmic series, and series for tan" 1 6, with alternating signs, sum. Unferdinger, F. Wien Sb. 55 (1867) (Ab. 2) 75-, 332. Parabolic trigonometric functions. Egidi, G. Em. N. Line. At. 47 (1894) 16-. trigonometry. Giinther, . D. Nf. Tbl. (*1881) [20]-. Periodic functions, real, derived from equation i^V = F(x). Weierstrass, C. Berl. Mb. (1866) 97-, 185. -- , simplest. Schellbach, C. H. Crelle J. 48 (1854) 207-. Periodicity of sin a; and cos a;. Studnicka, . Palermo Cir. Mt. Ed. 13 (1899) 378-. Powers of sines, series, summation. Ritten- house, D. [1792] Am. Ph. S. T. 3 (1793) 155-. --- , , . Nulty, 0. [1816] Am. Ph. S. T. 1 (1818) 395-. Problems. Adams, Jas. Thomson A. Ph. 13 (1819) 188-. Product series for cos x and summation for secx. Weyr, E. Bll. Sc. Mth. 12 (1888) 25-. Eepresentation of sine and cosine as infinite products. Hoppe, R. Grunert Arch. 27 (1856) 170-. Secant and tangent coefficients, formulae for determination. Schlomilch, 0. Grunert Arch. 16 (1851) 411-. Series, value. Glaisher, J. W. L. [1878] Mess. Mth. 8 (1879) 127-. Side of regular n gon, series for finding. Oppert, J. As. Fr. C. E. (1896) (Ft. 1) 133. Sines of fourth order F Kapteyn, J. C., Kapteyn, W. Amst. Ak. Vh. 24 (1886) 98 pp. Tangent and cotangent series, new deduction of Euler's. Studnicka, F. J. Prag Sb (1887) (Mth.-Nt.) 103-. Transcendental functions, method of treatment differing from Euler's, and theorem. Pezzi, F. Verona S. It. Mm. 5 (1790) 417-. Transformation involving de Moivre's theorem. Gamier, J. G. Brux. Ac. Bll. 6 (1839) 151-. ' Trigonometric and astronomical infinite series. Grunert, J. A. Grunert Arch. 18 (1852) 420-. ----- . Ligowski, W. Arch. Mth. Ps. 56 (1874) 328-. developments, and integration of trigono- metric functions. Richardson, J. M . Franklin I. J. 32 (1856) 409-. 4030 Trigonometric functions, integration of general classes. Philbrick, P. H. Des Moines Anal 9 (1882) 177- ; 10 (1883) 9-. , new researches. Lyetnikov, A. V. [1882] (XH) Eec. Mth. (Moscou) 10 (1882-83) (Ft. 1) 227-. , periodicity. Mantel, W. N. Arch. Wisk. 3 (1898) 279-; Fschr. Mth. (1898) 376. , properties deduced from addition theorems. Muirhead, R. F. Edinb. Mth. S. P. 14 (1896) 127-. , two reciprocal relations, and application to 4 dimensions. Hoppe, R. Arch. Mth. Ps. 67 (1882) 412- . and hyperbolic functions, multiplication. Trzaska, W. Par. T. Nauk So. Pam. 10 (*1878) Art. 8, 7 pp. series. Dobinski, G. [1876] Arch. Mth. Ps. 63 (1879) 380-. , Abel's transformation. Lerch, M- Prag Ceske Ak. Fr. Jos. Ez. (Trida 2) 5 (1896) Art. 24, 5 pp. ; Fschr. Mth. (1896) 203. Trigonometry, Dr Stewart's theorems, analytical demonstration. Ellis, R. L. Camb. Mth. J. 2 (1841) 271-. (Z + 2Z'cos0 + 2Z"cos')- n in series of cosines of multiples. Jacobi, C. G. J. Crelle J. 15 (1836) 205-. . Bjorling, E. G. Stockh. Ofv. 6 (1849) 171- ; Grunert Arch. 21 (1852) 26-. . Dienger, J. Grunert Arch. 12 (1849) 409-. sin oo and cos oo . Glaisher, J. W. L. [1870] Mess. Mth. 5 (1871) 232-. Usmx+ Fcos x + W. Hermite, C. Crelle J. 76 (1873) 303-. arc sin (x + iy) . Cayley, A. QJ. Mth. 15 (1878) . i? 1 -- J / (sin x, cos x) dx. Hermite, C. [1879?] G. Teiz. J. Sc. 2 (1880) 65-. cos e + i sin 6, and its application to some equations. Airy, (Sir) G. B. [1887] Camb. Ph. S. P. 6 (1889) 104-. So;*" 1 cos ka and similar series deduced from i* ^^a' Studnicka, F. J. Casopis 19 n 1 - xe ia 890) 113- ; Fschr. Mth. (1890) 261. sm (xir) ^ continued pro d uc t for. McClintock, . N. Y. Mth. S. Bll. 2 (1893) 147-. Sa: n arc tan K n ~ l = &7T/4, integral solutions. Stormer, C. C. E. 122 (1896) 175-, 225-. sin 6, expression as an infinite product. Dixon, A. C. Mth. Gz. 1 (1900) 130-. LOGAEITHMIC AND EXPONENTIAL FUNCTIONS. Compound interest, approximate calculation of r. Lemannier, H. N. A. Mth. 3 (1864) 337-. l ' 305 Logarithmic, Circular, Exponential Functions 4030 Compound interest and current accounts. Starkov, A. N. Es. S. Nt. Mm. (Mth.) 3 (*1881). Discontinuous logarithmic functions. Bugaev, N. V. [1887-88] Eec. Mth. (Moscou) 13 (1886) 757-; 14 (1890) 1- ; Fschr. Mth. (1888) 186. 4030 Equations of form S F (z) dz Laguerre, E. C. B. 93 (1881) 1000-. Expansion. Bangma, 0. S. Amst. Mengel- werk 2 (1816) 92-. . Cerquero, J. S. Quetelet Cor. Mth. 10 (1838) 383-. in convergent series. Oliveira, C. B. Bv. Brazil. 3 (1860) 205-. series. Gergonne, J. D. Gergonne A. Mth. 5 (1814-15) 363-. Exponential equations, reduction to algebraic form. Leyomnark, G. A. Stockh. Ak. Hndl. 29 (1808) 284-; 31 (1810) 101-. , solution. Arnott, G. A. W. Tilloch Ph. Mg. 49 (1817) 321-. expressions, imaginary. Roy, A. Mathe- matician 3 (1850) 13-. forms, indeterminate. Franklin, F. Am. J. Mth. 1 (1878) 368-. functions. Hermite, C. C. E. 77 (1873) 18-, 74-, 226-, 285-. . Laguerre, E. [1879] Par. S. Mth. Bll. 8 (1880) 11-. __. Gegenbauer, L. Wien Ak. Sb. 95 (1887) (Ab. 2) 846- ; Mh. Mth. Ps. 5 (1894) 303-. , application, case. Hilgard, J. E. Am. As. P. 20 (1871) 61-. , case. Pagani, G. M. Brux. Ac. Bll. 13 (1846) (pte. 2) 347-. , expansion in continued fractions. Pad, H. Par. EC. Norm. A. 16 (1899) 395-. , Lindemann's theorem. Vahlen, T. Mth. A. 53 (1900) 457-. , periodicity. Studnicka, F. J. Casopis 23 (1894) 1-; Fschr. Mth. (1893-94) 737. , representation by infinite product. Lipschitz, E. C. E. 99 (1884) 701-. , . Bassani, A. G. Teix. J. Sc. 11 (1892) 93-. , table, 12 place. Newman, F. W. [1876] Camb. Ph. S. T. 13 (1883) 145-. functions, tables. Glaisher, J. W. L. [1877] Camb. Ph. S. T. 13 (1883) 243-. , theorejns. Glaisher, J. W. L. Mess. Mth. 5 (1876) 164-. , theory. Graves, J. T. B. A. Ep. (1834) 523-. , , arithmetical propositions derived from. Lipschitz, . Liouv. J. Mth. 2 (1886) 219-. and imaginary quantities, theory. Wool- house, W. S. B. Mth. Misc. 1 (1838) 336-. logarithmic quantities. Toblini, G. Poligrafo 1 (1830) 21-. theorems, method. Strong, T. Silli- man J. 48 (1845) 36-. other functions, treatment by equiangular spiral. Schellbach, C. H. Crelle J. 17 (1837) 321-. Exponential series. Barfuss, F. W. Grunert Arch. 5 (1844) 155-. , error in theorem. Fontana, G. Verona S. It. Mm. 2 (1784) 123-. , inequalities. Tweedie, C. Edinb. Mth. S. P. 17 (1899) 33-. , connected with. Gibson, G. A. Edinb. Mth. S. P. 18 (1900) 84-. and other expansions. Dupuis, N. F. Cn. E. S. P. & T. 7 (1890) (Sect. 3) 16-, 21-. , sine, etc., series with alternating signs, sum. Unferdinger, F. Wien Sb. 56 (1867) (Ab. 2) 257-. symbolic operations. Scott, R. F. Mess. Mth. 10 (1881) 142-. operators, theorems. Glaisher, J. W. L. QJ. Mth. 15 (1878) 266-. theorem. Schaeberle, J. M. A. Mth. 3 (1887) 156. transformation. Laisant, C. A. As. Fr. C. E. (1879) 206-. and trigonometric functions, differentiation. Martin, Art. Des Moines Anal. 6 (1879) 23-. Exponentials, expansion in a series of. Popoff, A. [1874] Crelle J. Mth. 80 (1875) 204. , imaginary. Tarry, G. N. A. Mth. 14 (1895) 269-. , , expansion. Davies, T. S. Mathe- matician 3 (1850) 66-. and logarithms, differentiation. Schlomilch, 0. Grunert Arch. 11 (1848) 386-. , expansion. Ocagne, M. d'. N. A. Mth. 1 (1882) 241-. as limiting values of ( 1 + - j , CUja - 1) n (n = oo ) . Wulf, T. Mh. Mth. Ps. 8 (1897) 43-. , theory, application of geometry. Schulten, N. G. af. (vm) Helsingf . Acta 5 (1858) 437-. , successive, of Euler. Grillet, J. H. Liouv. J. Mth. 10 (1845) 233-. , theory. Laurent, H. N. A. Mth. 13 (1874) 5-. Hyperbolic areas, and natural logarithms. Grunert, J. A. Grunert Arch. 25 (1855) 82-. and cyclic sine and cosine, functions repre- senting generalisation. Hellwig, C. Grunert Arch. 35 (1860) 186-. functions. Fuss, N. [1810] St Pe"t. Ac. Sc. Mm. 11 (1830) 220-. . Hottel, J. N. A. Mth. 3 (1864) 416-. . Azzarelli, M. Em. At. N. Line. 24 (1871) 112-. . Laisant, C. A. Bordeaux S. Sc. Mm. 10 (1875) 233-. . Gallatly, W. Mth. Gz. No. 5 (1895) 40-. . Haskell, M. W. N. Y. Am. Mth. S. Bll. 1 (1895) 155-. in connection with the hyperbola. Crawford, L. Edinb. Mth. S. P. 13 (1895) 153-. , functions of. Nicodemi, R. G. Mt. 15 (1877) 193-. 306 4030 Logarithmic, Circular, Exponential Functions Hyperbolic functions, tables. Jordann, . Crelle J. 16 (1837) 196. , , new. Forti, A. [1881] Z. Mth. Ps. 27 (1882) (H.-lt. Ab.) 1-. in teaching. Barbarin, P. Ens. Mth. 2 (1900) 443-. , theorems. Cantor, M. Grunert Arch. 19 (1852) 88-. , theory. Barsotti, G. Lucca At. Ac. 15 (1855) 441-. , , and application. Grunert, J. A. Grunert Arch. 38 (1862) 48-. , , 'to resistance coefficients. Gronau, J. F. W. Danzig Schr. 1 (1865) (Heft 2) 79 pp. , , geometrical. Thomson, W. L. Edinb. Mth. S. P. 17 (1899) 2-. logarithms. Formula I dx (- Lx)^, &c. Ferroni, P. [1794] Siena At. Ac. 8 (1800) Indices in higher arithmetic and application to logarithmic tables. Agardh, J. M. Stockh. Ac. Hndl. (1838) 253-. Infinite products, cases. Stern, M. A. Crelle J. 27 (1844) 279-. Integral in dynamics, Euler. Dostor, G. J. N. A. Mth. 11 (1852) 45-. transcendentals (resolvable into factors of form e p W ( 1 ) , P(x) being a polynomial). Poincare, H. C. R. 95 (1882) 23-. Integrals, class, expressible by logarithms alone. Piuma, C. M. A. Mt. 7 (1875-76) 18-. of irrational algebraic functions expressible by logarithms alone. Piuma, C. M. (vm) Tortolini A. 4 (1861) 1-. . Aleksyeev, N. N. (xn) Rec. Mth. (Moscou) [1] (1866) 173. Irrationals \/2 and \/3 cannot be Napierian logarithms of rational numbers. Schulten, N. G. af (fil.) [1844] Helsingf. Acta 2 (1847) 681-. Logarithm, common, method of finding. Bitten- house, D. [1795] Am. Ph. S. T. 4 (1799) , expansion, case. Schlomilch, 0. Leip. Mth. Ps. B. 39 (1887) 53-. , expression for integral power. Walton, W. Camb. Mth. J. 2 (1841) 102-. , isolated, to many decimal places. Lefort, F. N. A. Mth. 6 (1867) 308-. of n ! , when n is large. Lerch, M. Casopis 24 (1895) 129- ; Fschr. Mth. (1895) 472. , Napierian, of a number and its integral powers are incommensurable with number, etc. Pollak, (Dr) . [1849] Haidinger B. 6 (1850) 95-. Logarithmic and algebraic functions giving logarithmic integrals. Hill, C. J. D. Lund Acta Un. 4 (1867) (Mth.) No. 1, 28 pp. 4030 Logarithmic calculation, errors from using pro- portional parts. Vincent, A.J.H. Gergonne A. Mth. 16 (1825-26) 19-. , circular and exponential functions, dif- ferentials. Gruson, J. P. Berl. Ab. (1820- 21) 49-. and circular functions. Lambert, C. J. N. A. Mth. 12 (1853) 302-. , development of properties from definite integral. Entleutner, A. F. [1878] Arch. Mth. Ps. 63 (1879) 225-. , formulas relating to. Terquem, 0. N. A. Mth. 5 (1846) 79-, 151-, 221-, 349-, 411- ; 7 (1848) 269-. series. Adams, Jas. Thomson A. Ph. 4 (1822) 261-. exponential series, summation. Chessin, A. Am. J. Mth. 16 (1894) 164-, 290. formulas derived from a differential equa- tion. Durege, . Schlomilch Z. 3 (1858) 241-. , rectification. Graves, J. T. [1828] Phil. Trans. (1829) 171-. functions, integration by. Dolbnia, I. P. Rec. Mth. (Moscou) 18 (1896) 150-. integral and allied functions, calculation. Bretschneider, C. A. Schlomilch Z. 6 (1861) 126-. , constant, numerical determination. Arndt, F. Grunert Arch. 11 (1848) 315-. , , value. Lindman, C. F. Grunert Arch. 29 (1857) 238-. , , . Oettinger, L. [1861] Crelle J. 60 (1862) 375-. , rth integral. Glaisher, J. W. L. [1882] Mess. Mth. 12 (1883) 120-. , theory. Bretschneider, C. A. Crelle J. 17 (1837) 257- , . Schlomilch, 0. Grunert Arch. 9 (1847) 5-, 307-. , . Beez, R. Grunert Arch. 19 (1852) 419-. integrals, 2nd order. Newman, F. W. Camb. and Dubl. Mth. J. 2 (1847) 77-, 172-. Bierens de Haan, D. Amst. ~/' Vs. Ak. 1 (1866) (Ntk.) 117-. and other series, connection. Grunert, J. A. Grunert Arch. 8 (1846) 272-. problems. Thiele, T. N. Mth. Ts. 3 (1861) 145- ; 4 (1862) 17-, 93-, 145-. progressions. Barbette, E. Mathesis 18 (1898) 135-. series. Servois, J. F. Gergonne A. Mth. 2 (1811-12) 178-. . Dantscher, V. von. Mh. Mth. Ps. 6 (1895) 390-. , interpolation. Meikle, J. Assur. Mg. 6 (*1857) 200-. of Wronski. Dickstein, S. Prace Mt.- Fiz. 4 (1893) 88-; Fschr. Mth. (1893-94) 422. sines, construction of canon. Sang, E. Edinb. R. S. P. 12 (1884) 601-. 307 u2 4030 Logarithmic Tables Hessel, J. F. C. 4030 Logarithmic systems (three). Grunert Arch. 14 (1850) 97-. , classification. Stringham, I. [1891] Am. J. Mth. 14 (1892) 187-. Logarithmic tables. Bresson, . (vn) Eouen Bll. S. Em. (1841) 162-. Shortrede, R. Beng. J. As. S. 11 (1842) 40-. Galbraith, W. Edinb. N. Ph. J. 39 (1845) 169-. Riese, (Prof.) von. Eheinl. Westphal. Sb. " sviii-. rens de. Amst. Vs. Ak. 14 (1862) As. S. M. Not. 33 (1873) 14 (1857) xxviii-. aan,D. Biere 15-. Tennant, J. F. 563-. August's. Lehmann, J. W. H. As. Nr. 43 (1856) 225-. Bremiker's, errata. Fabritius, W. As. Nr. 85 (1875) 319-. Callet's, errata. Secretan, . C. B. 44 (1857) 1276-. compact, use. Templeton, R. Ph. Mg. 30 (1865) 221-. construction. Knight, T. Phil. Trans. (1817) 217-. . Serret, J. A. N. A. Mth. 11 (1852) 2 2-. , formulae. Lavernede, T. Gard Not. Tr. Ac. (1807) 179-. early. Bierens de Haan, D. Ph. Mg. 45 (1873) 371-. errata. Stanley, A. D. Silliman J. 5 (1848) 398-. of 1849, errata. Shortrede, R. As. S. M. Not. 27 (1867) 272-. errors, common to. Babbage, C. [1827] As. S. Mm. 3 (1829) 65. exercise on. Tail, P. G. Edinb. Mth. S. P. 5 (1887) 101-. and factor tables. Drach, S. M. [1873] (rx) Mess. Mth. 3 (1874) 6. 5 figures. Bouche, A. [1858] M.-et-L. Mm. S. Ac. 5 (1859) 23-. 7 figures, smallest number of entries necessary in. Steggall, J. E. A. Edinb. Mth. S. P. 10 (1892) 35-. 10 figures. Carlini, F. Wien. Jb. Pol. I. 10 (1827) 77-. 12 figures, from 1 to 120,000, by Thomson. Glaisher, J. W. L. As. S. M. Not. 34 (1874) 447-; 35(1875)135. 20 figures, Callet's, Hermann's corrections. Stampfer, S. (vra) Wien Sb. 1 (1848) 27 figures, Thoman's. Bertrand, J. J. Sav. (1870) 750-. Kohler's. Lehmann, J. W. H. As. Nr. 44 (1856) 209-. proof that large errors arise from use in certain cases. Voll, W. Oken Isis (1826) 677-. and trigonometric tables, of Bureau du Cadastre. Pr&ny, R. de. Par. Mm. de PI. 5 (1802) (H.) and trigonometric tables, of Bureau du Cadastre. Delambre, J. B. J. Par. Mm. de PI. 5 (1804) (H.) 56-. , . Lefort, F. C. B. 46 (1858) 994-; (vi Adds.) Par. A. Obs. 4 (1858) [123]-. , . Sang, E. [1874] Edinb. B. S. P. 8 (1875) 421-. , (Sang) . Lefort, P.A.F. Edinb. B. S. P. 8 (1875) 563-. Vega's. Mollweide, C. Zach M. Cor. 21 (1810) 284-. , errata. Bremiker, (Prof.) . As. Nr. 85 (1875) 373-. to 200,000. Sang, E. [1871] Edinb. B. S. T. 26 (1872) 521-. Logarithmic theorem. Catlin, M. Mth. Misc. 1 (1838) 113-. transcendents. Glaisher, J. W. L. Mess. Mth. 1 (1872) 138-. Logarithms. Laurent, H. Ens. Mth. 2 (1900) 450-. of algebraic numbers, arithmetical property. Stormer, C. Par. S. Mth. Bll. 28 (1900) 146-. and antilogarithms. Gram, J. P. Ts. Mth. 4 (1886) 1-; Fschr. Mth. (1886) 1122. , calculation. Weddle, T. Mathe- matician 1 (1847) 17-. , to 12 or 16 places, improved bi- modular method, with brief tables. Ellis, A. J. B. S. P. 31 (1881) 381-. , approximate formulae for. Buquoy, G. von. Oken Isis 21 (1828) 1203-. , arithmetical solutions. Guilmin,A. N. A. Mth. 5 (1846) 429-. , base, expression. Baltzer, R. Grunert Arch. 18 (1852) 405-. , calculation. Manning, T. Phil. Trans. (1806) 327-. , . Transon, (Prof.) A. N. A. Mth. 10 (1851) 71-. , . Tagert, . Grunert Arch. 27 (1856) 132-. , . Oliveira, C. B. Bv. Brazil. 3 (1860) 181-. , . Wace, H. B. A. Ep. 43 (1873) (Sect.) 24-; Mess. Mth. 3 (1874) 66-. , by algebraic fractions. Murphy, R. Ph. Mg. 19 (1841) 479-. , , best methods. Franchini, P. Lucca At. Ac. 6 (1830) 349-. , of characteristics. Holten, C. Mth. Ts. 5 (1863) 65-. , , elementary. Paugger, F. Arch. Mth. Ps. 42 (1864) 197-. , , , of common. Sturm, J. B. Grunert Arch. 24 (1855) 228-. , , formula. Du Bourguet, . Gergonne A. Mth. 2 (1811-12) 69-, 286-. , , formulae. Lavernede, T. Gergonne A. Mth. 1 (1810-11) 18-, 78-. , to many figures. Burnier, F. [1871] Laus. Bll. S. Vd. 11 (1873) 147-. , , easy method. Stille, W. A. Des Moines Anal. 10 (1883) 143-. 308 4030 Logarithmic, Circular, Exponential Functions 4030 Logarithms, calculation, potential radix as means. Ellis, A. J. E. S. P. 31 (1881) 398-; 32 (1881) 377-. , , series for. Allman, W. [1796] Ir. Ac. T. 6 (1797) 391-. , , (Halley's). Murray, E. [1801] Ir. Ac. T. 9 (1803) 3-. , , . Wallace, (Prof.) W. [1808] Edinb. K. S. T. 6 (1812) 269-. , , . Thomson, (Prof.) J. [1838] Edinb. E. S. T. 14 (1840) 217-. , canon. Kulik, J. P. As. Nr. 4 (1826) 47-. , criticism of Thiele. H., E. L. Mth. Ts. 4 (1862) 113-, 161-. , definition. Haskell, M. W. N. Y. Mth. S. Bll. 2 (1893) 164-. , , geometrical (polygonal spirals). Bar- barin, P. As. Fr. C. E. (1895) (Ft. 2) 257. of different orders of numbers. Grillet, J. H. Liouv. J. Mth. 10 (1845) 233-. and exponentials, definitions. Stringham, I. N. Y. Mth. S. Bll. 2 (1893) 168-. , 4 figure Gaussian, in new arrangement. Wittstein, T. As. Nr. 51 (1859) 125-. , hyperbolic and Napierian. Wackerbarth, A. D. As. S. M. Not. 31 (1871) 263-. , . Dubois, E. Les Mondes 27 (1872) 651-. , imaginary, Graves's theory. Young, J. E. Ph. Mg. 25 (1844) 401-. of imaginary quantities. Cayley, A. [1867] L. Mth. S. P. 2 (1869) 50-. , impossible. Gregory, D. F. Camb. Mth. J. 1 (1839) 226-. with large number of places. Perott, J. Bll. Sc. Mth. 11 (1887) 51-. of large numbers. Le Barbier, . Gergonne A. Mth. 20 (1829-30) 366-. Napier and Briggs. Blanken, H. van. Arch. Wisk. Gn. 1 (1856-59) 66-. , natural, hyperbolic and Napierian, dis- cussion of terms. D g. Ts. Mt. Fys. 4 (1871) 28-. , , of numbers 1-25, to 81 figures. Vietz, . As. Nr. 4 (1826) 53-. , , systematic treatment. Matzka, W. Prag Sb. (1878) 206-. of negative quantities. Vincent, A. J. H. Gergonne A. Mth. 15 (1824-25) 1-; 16 (1825- 26) 92-. (Vincent). Stein, J. P. Gergonne A. Mth. 15 (1824-25) 230-. , new kind. Hill, C. J. D. Crelle J. 70 (1869) 282-. of numbers, calculation. Huygens, . C. E. 66 (1868) 565-. , , Huygens' method. Bertrand, J. N. A. Mth. 7 (1868) 229-. , , . Thoman, F. C. E. 66 (1868) 662-. order N, application of logarithmic and Archimedes' spirals. Cabreira, A. Lisb. J. Sc. Mth. 5 (1898) 226-. , origin, geometrical, and trigonometry of parabola. Booth, J. B. A. Ep. (1856) 68-. Logarithms and powers, principal. Bjbrling, E. G. Stockh. Ofv. 9 (1852) 100- ; Grunert Arch. 21 (1853) 1-. , proportional, table. Shortrede, E. Beng. J. As. S. 10 (1841) 713-. , representation, geometrical. Fischer, E. G. [1804] Berl. Ab. (1804-11) tilth.} 1-. of sines and tangents of small angles. Burnier, F. Laus. Bll. S. Vd. 8 (1864-65) sum and difference of two numbers, calculation from logarithms of the num- bers. Seeling, P. Arch. Mth. Ps. 45 (1866) 451-. or difference, calculation, tables. Matthiessen, E. A. As. Nr. 2 (1824) 103-, 121-. , theory. Bouvier, L. C. Gergonne A. Mth. 14 (1823-24) 275-. , . Stein, J. P. Gergonne A. Mth. 15 (1824-25) 105-. , . Pagani, G. M. Brux. Ac. Bll. 6 (1839) 259-. , . Matzka, W. Grunert Arch. 15 (1850) 121-. , . Mourgues, . N. A. Mth. 11 (1852) 109-. , . Cayley, A. Ph. Mg. 11 (1856) 275-; 12 (1856) 354-. , . Berdelle, C. As. Fr. C. E. (1889) (Pt. 2) 38-. , , elementary. Jamet, V. Mathesis 8 (1888) 89-. and trigonometric functions, expansion in series. Jack, J. Edinb. Mth. S. P. 13 (1895) 132-. of trigonometric quantities, formulae for finding from one another. Wallace, (Prof.) W. [1823] Edinb. E. S. T. 10 (1826) 148-.| ratios, tables. Elphinstone, H. W. QJ. Mth. 2 (1858) 222-. , trigonometric, tables. Escher, P. Grunert Arch. 23 (1854) 264-. of unity. Graves, J. T. Ph. Mg. 8 (1836) 281-. M, Briggs' modulus, value to 205 places. Shanks, W. [1854-71] E. S. P. 6 (1854) 397-; 20 (1872) 27-. M, fractions for value. Templeton, E. Ph. Mg. 30 (1865) 434-. Napierian logarithms of commensurable num- bers or of algebraic irrationals. Lindemann, C.L.F. C. E. 95 (1882) 72-. , and exponential functions, analytical . theory. Meray, C. Toul. Fac. Sc. A. 4 (1890) Q, 35 pp. Natural logarithm of number, method of finding without logarithmic tables. Spiegler, G. (xn) Mag. Ak. Ets. (1858) 299-. Notation for printing complicated exponents. Glaisher, J. W. L. [1872] Mess. Mth. 2 (1873) 107-. Number whose logarithm is equal to itself, form. Farren, E. J. Assur. Mg. 3 (*1853) 323-. Numbers whose logarithms are rational. Arndt, F. Grunert Arch. 6 (1845) 57-. 309 Logarithmic, Circular, Exponential Functions 4030 Potential expression ((!))*. Kinkelin, H. Grunert Arch. 26 (1856) 304- . Eeciprocal series of sines and cosines of arcs in arithmetical progression. Lorgna, A. M. Turin Mm. Ac. 3 (1786-87) 215-. Series formed from series for log(l + :r) etc. Hellwig, C. Grunert Arch. 21 (1853) 43-. - for roots and logarithms. Bouvier, L. C. Gergonne A. Mth. 16 (1825-26) 294-. l_^-)-.i J..., note. Lionnet, E. N. A. Mth. 18 (1879) 509-. Stirling's approximation to n ! Allardice, E. E. Edinb. Mth. S. P. 6 (1888) 22-. formula, proof by logarithmic inequalities. Mansion, P. Mathesis 20 (1900) 265-. theorem, elementary proof. Franklin, F. J. H. Un. Cir. [3] (1884) 57. Transcendental functions arising from repeated integration of rational functions. Jonquiere, A. Stockh. Ak. Hndl. Bh. 15 (Afd. 1) (1890) No. 1, 50pp. , class, integration. Winckler, A. [1874] Wien Ak. Sb. 70 (1875) (Ab. 2) 17-. integrals of form / e~ x ^fxdx. Thomson, (Sir) W. QJ. Mth. 1 (1857) 316-. Wronski, formula, 4030 I" I + constant, \F(x)dx = l{W where 1 dFx 1 d?Fx ~\ ^ x=Fx+ _.__ + __. ^- + ...J. Rodrigues, J. M. G. Teix. J. Sc. 3 (1881) 55-. \e nx x m X 1 X^X^dx, where X^^ are poly- nomials in x. Melanderhjelm, D. [1798] St Pet. Ac. Sc. N. Acta 12 (1801) 114-. to* ' Bidme ' G ' Turin Mm ' Ac ' ( 1805 - 8 ) (pte. 2) 19-. -. Caluso, (Vabbe) T. V. de. Mod. Mm. S. It. 12 (1805) 268-. 1^. Bessel,F. W. Zach M. Cor. 22 (1810) 395-; Konigsb. Arch. 1 (1812) 1-, 239-. . Buzengeiger, K. H. J. Zach M. Cor. 28 (1813) 373-. . Casinelli, A. (s. L.). Bologna N. Cm. 3 (1839) 97-. x x =A, solution. Leyonmark, G. A. Stockh. Ak. Hndl. 31 (1810) 161-. x xn . Carlini, F. Mil. Mm. I. Lomb. Yen. 1 (1812-13) 167-. e 2 **, trigonometric expression for. Busse, F. G. von. Oken Isis (1825) 394-, 528-. log ( - x) and Bouvier. Busse, F. G. von. Oken Isis (1825) 1048-. 0*. Libri, G. Crelle J. 6 (1830) 67-. ir=?logi. Schellbach, C. H. Crelle J. 9 (1832) 404-. le x ydx, integration where y is algebraic. Liouville, J. Crelle J. 13 (1835) 93-. log (1 + x) m . Casinelli, A. (s. L.). Bologna N. Cm. 4 (1840) 147-. ^ , limiting value for x=. Schulten, N. G. af (fil.) Helsingf. Acta 1 (1840) logo; and a x , differentials. Grunert, J. A. Grunert Arch. 1 (1841) 204-. log a; dx, higher integrals. Arndt, F. Grunert Arch. 4 (1844) 436-. e v ~ 19 , use in transformations. Goodwin, H. Camb. Mth. J. 4 (1845) 177-. expansion. Meyer, U. H. Grunert Arch. 9 (1847) 101-. e 31 , expansion in series. Brassinne, E. N. A. Mth. 7 (1848) 30-. a x >x, condition. Hessel, J. F. C. Grunert Arch. 14 (1850) 93-. xv and log (x). Bjorling, E. G. Stockh. Ak. Hndl. (1852) 121-. Pr^^' f^ dx 2 , /C=, / f****- te J r, , Roberts, W. Liouv. J. Mth. 17 (1852) 117-. x a and a x , differentials, calculation. Schlomilch, O. N. A. Mth. 12 (1853) 31-. log x and tan ~ l x, differentials, and expansions in series. SouMet, . N. A. Mth. 12 (1853) 438-. Iog e 2, Iog e 3, Iog e 5, log, 10 to 205 places (Napierian). Shanks, W. [1854-71] E. S. P. 6 (1854) 397-; 20 (1872) 27-. " ' = JL [go H . e -a _ e -an . e a^ Zehfuss, G. Grunert Arch. 30 (1858) 465-. e+w. Dienger, J. Grunert Arch. 33 (1859) 481. {(*)}, log {(*)}, theory. Almqvist, P. W. Stockh. Ofv. 20 (1863) 203-. y = e* x , representation in form y = S[A m e mx ]. Spitzer, S. Arch. Mth. Ps. 42 (1864) 104-. e x , limits. Steen, A. (xn) Ts. Mth. 3 (1867) 29-. f f (x\ I e* J ~Jr , integration. Tissot, A . Par. EC. Pol. J V x J. 43 e cah. (1870) 201-. y = x n e* x *, representation in form y = S [A m e*]. Spitzer, S. Arch. Mth. Ps. 52 (1871) 364-. y = x n e alfi , representation in form y = S [A m e" 1 *]. Spitzer, S. Arch. Mth. Ps. 52 (1871) 368-. log M = log . log e, value to 24 places, also log Mir. Gray, P. Mess. Mth. 5 (1876) 172-. x n e* x *, transformation. Spitzer, S. Arch. Mth. Ps. 58 (1876) 431-. e', period. Yvon-Villarceau, A. J. F. C. E. 83 (1876) 594-. 310 4040 Elliptic and Single Theta Functions 4040 log(l + x), expansion. Fuhrmann, W. Arch. Mth. Ps. 62 (1878) 220. r* _z* I z n e 2 zx dz. Laguerre,E. [1878] Par. S. Mth. Bll. 7 (1879) 12-. log 2, &c., calculation (Gauss). Cayley, A. Mess. Mth. 8 (1879) 125-. I fL^: Z . Laguerre, E. Par. S. Mth. Bll. 7 (1879) 72-. I \ogsinxdx, determination. Ligowski, W. [1879] Arch. Mth. Ps. 65 (1880) 110-. F and (-!)*. Laisant, C. A. Par. S. Mth. Bll. 8 (1880) 109-. e*, log e (lx), (l + x) m , series for. HOI, M. J. M. Camb. Ph. S. P. 5 (1886) 415-. e az :(e z -\), resolution into partial fractions. Stolz, O. Innsb. Nt. Md. B. 15 (1886) 84-. COSP x sin" x dx, integration. Laisant, C. A. As. Fr. C. E. (1889) (Pt. 2) 225-. e x , regular combined fractions for. Fade, H. C. E. 112 (1891) 712-. log x and e*, independent definitions. McClin- tock, E. Am. J. Mth. 14 (1892) 72-. I cos yx dxl(a 2 + b 2 x y ). Oltramare, G. As. Fr. C. E. (1895) (Pt. 2) 167-. , expression by logarithms. Dolbnia, I. P. Bll. Sc. Mth. 19 (1895) 76- ; Eec. Mth. (Moscou) 18 (1896) 108-. e^), series for. Petrovitch, M. Palermo Cir. Mt. Ed. 14 (1900) 22-. 4040 General properties of elliptic functions and single theta func- tions ; addition-theorem. (See also 8050, 8060.) Addition. of arguments in elliptic functions. Hermite, C. G. Teix. J. Sc. 11 (1892) 65-. periodic functions, 2nd order. Fontene, G. C. E. 122 (1896) 172-. elliptic functions. Cayley, A. Crelle J. 41 (1851) 57-. . Darboux, G. Par. EC. Norm. A. 4 (1867) 81-. . Catalan, E. C. C. E. 78 (1874) 1479-. . Frobenius, G., (& Stickelberger, L. [1879] Crelle J. Mth. 88 (1880) 146-. . Cayley, A. Mess. Mth. 14 (1885) 56-. , formula. Gilbert, L. P. (xii) Brux. S. Sc. A. 4 (1880) (Pt. 2) 87-. , formulae. Bjorling, E. G. [1866] Arch. Mth. Ps. 47 (1867) 399-. of elliptic functions, formulae of Hermite. Novarese, E. [1882] Tor. Ac. Sc. At. 17 (1881) 607-. , Jacobi. Broch, 0. J. C. E. 59 (1864) 999-. , 1st kind. Catalan, E. C. Brux. Ac. Bll. 27 (1869) 145-. , . Catalan, E. Brux. Ac. Mm. 45 (1884) (No. 3) 20 pp. , , and Euler's integral. laggi, E. N. A. Mth. 19 (1900) 443-. integrals, geometrical application. Hoppe, E. Arch. Mth. Ps. 64 (1879) 274-. equation. Thomson, J. J. [1879] Mess. Mth. 9 (1880) 52-. for elliptic integrals, 1st kind, geometric construction. Hermite, C. Bll. Sc. Mth. As. 2 (1871) 21-. , 2nd kind, proof by ^-series. GlaisJier, J. W. L. [1882] Mess. Mth. 12 (1883) 43-. , 3rd kind. Glaisher, J. W. L. [1880-81] Mess. Mth. 10 (1881) 124-. , spherical triangle proof. Johnson, W. W. QJ. Mth. 18 (1882) 365-. equations for elliptic and theta functions of sum of n arguments. Wilkinson, M. M. U. L. Mth. S. P. 13 (1881-82) 110-. ADDITION-THEOBEM. Story, W. E. Am. J. Mth. 7 (1885) 364-. Mukhopadhyay, A. QJ. Mth. 21 (1886) 212-. Giinther, P. Crelle J. Mth. 109 (1892) 213-. Teixeira, J. P. G. Teix. J. Sc. 11 (1892) 136-; Fschr. Mth. (1893-94) 767. for elliptic integrals. Olsson, 0. Ts. Mth. 5 (1887) 33-; Fschr. Mth. (1887) 455. , construction, geometric. Morera, G. [1880] Tor. Ac. Sc. At. 15 (1879) 649-. , demonstration. Gwyther, E. F. Manch. Lt. Ph. S. P. 22 (1883) 81-. , derivation from equation + -rr = 0. Graefe, F. [1880] Crelle J. Mth. 90 (1881) 83-. , Euler's. Eichelot, F. J. Crelle J. 44 (1852) 277-. , , Degen's attempt to generalise. Bjerknes, C. A. Bb. Mth. (1888) 1-. , 1st kind, demonstration, Sturm's. SehriJter, H. Z. Mth. Ps. 17 (1872) 508-. , , demonstrations. Bigler, U. Arch. Mth. Ps. 7 (1889) 401-. . , , derivation, method of Sturm. Much, . Z. Mth. Ps. 26 (1881) 333-. , , 2nd and 3rd kinds. Dolbnia, I. Kazan S. Nt. (Ps.-Mth.) P. 7 (1889) 364-; N. A. Mth. 8 (1889) 204-. , 2nd kind, demonstrations. Walker, J. J. L. Mth. S. P. 13 (1881-82) 172-. , and 3rd kinds. Forsyth, A. B. Mess. Mth. 15 (1886) 49-. , theory. Schellbach, C. H. Crelle J. 54 (1857) 59-. 311 4040 Elliptic and Single Theta Functions extension. Winckler, A. Wien SB. 44 (1861) (Ab. 2) 477-. and inversion problem. Brill, A. Mth. A. 17 (1880) 87-. of Jacobi and of Weierstrass. Study, E. Am. J. Mth. 16 (1894) 156-. proof. Griffiths, J. L. Mth. S. P. 13 (1881- 82) 177-. for theta functions. Caspary, F. C. B. 104 (1887) 1255-. transcendental functions. Pokrovskij, P. M. Eec. Mth. (Moscou) 18 (1896) 121- ; Fschr. Mth. (1896) 360. ' , integration of differential equations leading to. Steen, A. Kjob. Skr. 8 (1870) 1-; Resume, 15-. Hu) and for cotangent. Schottky, F. Crelle J. Mth. 110 (1892) 324-. g? (u), elementary deduction. Anissimow, . [1893] Fschr. Mth. (1893-94) 767. . Stdckel,P. Mth. A. 47 (1896) 604. Arithmetico-geometric mean. Borchardt, C.W. [1858] Crelle J. 58 (1861) 127-. . Frisby, E. Des Moines Anal. 6 (1879) 10-. . Lohnstein, T. Z. Mth. Ps. 33 (1888) 129-, 318. of 4 elements (Borchardt). Schering, K. Crelle J. Mth. 85 (1878) 115-. Class dxlU x ^+ ... where U x =(a,b,c,d,e)(x, 1). Russell, R. QJ. Mth. 20 (1885) 179-. dxlU x *+... where U x =(a, b, c, d, e) (x, - I) 4 . Russell, R. QJ. Mth. 20 (1885) 265-. Closed polygons in- and circum-scribed to conies. Oekinghaus, E. Arch. Mth. Ps. 6 (1888) 186-. Complete elliptic integrals. Smith, H. J. S. Km. B. Ac. Line. T. 1 (1877) 42-. , functions derivable from, and con- nected with Legendre's functions. Kleibner, J. Mess. Mth. 22 (1893) 1-. , 1st kind, formula relating to. Besso, D. Bm. B. Ac. Line. Ed. 4 (1895) (Sem. 1) 229-. , , and Legendre's coefficients, connection. Stuart, G. H. Mess. Mth. 13 (1884) 161-. , , quotient, extension of theorem of Biemann. Torelli, G. Nap. Bd. 29 (1890) 238-. , and 2nd kinds, approxima- tions. Lodge, A. Mess. Mth. 28 (1899) 17-. , , continued fraction for ratio. Baraniecki, M. A. Par. T. Nauk S~c. Pam. 7 (*1875) Art. 5, 8 pp. , , representation in simple form. Radicke, A. Z. Mth. Ps. 21 (1876) 442-. , 2nd kind, expression by sines of multiples of modular angle. Glaisher, J. W. L. Ph. Mg. 19 (1885) 504-. , , reduction to complete elliptic integrals of 1st kind for same modulus. Meissel, E. Arch. Mth. Ps. 56 (1874) 337-. 4040 Conjugate functions of Cartesians and other quartics. Greenhill, A. G. [1881] Camb. Ph. S. P. 4 (1883) 77-. Definite theta-function integrals, cases. Rogers, L. J. [1894] L. Mth. S. P. 26 (1895) 145-. Differential equation in elliptic functions, geo- metrical illustration of solution. Hart, H. Mess. Mth. 4 (1875) 125-. , Euler's. Halphen, G. H. Palermo Cir. Mt. Bd. 2 (1888) 40-. , 2nd order, for & (u). Mittag-Leffler, G. C. K. 117 (18931 92-. , remarkable, integration. Picart, A. C. B. 66 (1868) 1192-. satisfied by I2q + 2q*2q 9 + .... Jacobi, C. G. J. Liouv. J. Mth. 14 (1849) 181-. equations, class. Catalan, . As. Fr. C. B. (1886) (Pt. 1) 85. and elliptic series. Liouville, J. Liouv. J. Mth. 14 (1849| 225-. expression, reduction. Manning, H. P. Am. J. Mth. 13 (1891) 185-. dt-\lt -a.t-p.t-y.t-8, reduction to standard form. Griffiths, J. L. Mth. S. P. 14 (1882-83) 196-. Doubly periodic functions. Liouville, J. Crelle J. Mth. 88 (1880) 277-. arising from x 3 + y 3 -3axy = l. Dixon, A. (7. QJ. Mth. 24 (1890) 167-. considered as limits of algebraic functions. Biehler, C. [1879] Crelle J. Mth. 88 (1880) 185-. with essential singular points. Picard, E. C. B. 89 (1879) 852-. expansion in Fourier's series. Charlier, C. V. L. Stockh. Ak. Hndl. Bh. 12 (Afd. 1) (1887) No. 9, 24 pp.; Fschr. Mth. (1887) 448. trigonometric series. Krause, M. D. Nf. Tbl. (1889) 188-. expression by definite integrals and arithmetical invariants. Poincare, H. As. Fr. C. B. (1881) 109-. Fourier's expansion for, new method. Mohr- mann, G. Arch. Mth. Ps. 12 (1894) 1-. Hermite's memoir. Cauchy, A. L. C. B. 32 (1851) 442-. of 2nd kind. Mittag-Leffler, G. C. B. 90 (1880) 177-. . Gegenbauer, L. [1882] Wien Ak. Sb. 86 (1883) (Ab. 2) 969-. . Goursat, E. C. B. 98 (1884) 35-. . Teixeira, J. P. G. Teix. J. Sc. 9 (1889) 103- ; Fschr. Mth. (1889) 471. , application. Hermite, . Par. EC. Norm. A. 2 (1885) 303-. , differential equations satisfied by. Krause, M. Leip. Mth. Ps. B. 42 (1890) 55-, 268-, 430- ; 43 (1891) 32-, 289-, 597-. and 3rd kinds. Krause, M. Leip. Mth. Ps. B. 41 (1889) 347-. 3rd kind. Appell, P. Par. EC. Norm. A. 1 (1884) 135- ; 2 (1885) 9-; 3 (1886) 9-; C. B. 101 (1885) 1478-. 312 4040 Doubly Periodic Functions of 3rd kind, differential relations in reference to. Krause, M. Leip. Mth. Ps. B. 41 (1889) 110-. , resolution into simple elements. Appell, P. C. E. 97 (1883) 1419-. monodromic and monogetiic. Meray, C. C. B. 40 (1855) 787-. periods. Dantscher von Kollesberg, V. Innsb. Nt. Md. B. 15 (1886) 79-. resolution into simple elements. Hermits, . Toul. Fac. Sc. A. 2 (1888) C, 12 pp. . Weyr, E. fill. Sc. Mth. 12 (1888) 246-. and singly periodic functions, analogy between formula. Glaisher, J. W. L. B. A. Ep. (1881) 548-. theorem. Jordan, C. C. E. 70 (1870) 1128-. , Liouville's. Rink, H. J. Z. Mth. Ps. 28 (1883) 48-. , proof. Schumann, A. Z. Mth. Ps. 27 (1882) 125-. uniform, differential relations. Jahnke, E. Crelle J. Mth. 112 (1893) 265-. Weierstrass's. Forsyth, A. R. Mess. Mth. 15 (1886) 165- ; QJ. Mth. 22 (1887) 1-. Doubly periodic series, examples. Appell, P. N. A. Mth. 15 (1896) 126-. Elliptic differential equation. Battaglini, G. [1879] Em. E. Ac. Line. Mm. 5 (1880) 50-. , application of binary quadratic forms to. Battaglini, G. [1885] Nap. Ac. At. 2 (1888) No. 4, 11 pp. , multiplier, Despeyrous's. Du Bois- Reymond, P. [1882] Mth. A. 21 (1883) 255-. , of general. Winckler, A. Wien Ak. Sb. 95 (1887) (Ab. 2) 209-. differentials, general condition for finite integrability. Bugaev, N. V. St Pet. Ac. Sc. Mm. (Rs.) 69 (1892) (Suppl.) No. 8, 16 pp. ; Fschr. Mth. (1892) 422-. double series. Scheibner, W. Leip. Mth. Ps. B. 42 (1890) 130-. ELLIPTIC FUNCTIONS. Abel, N. H. Crelle J. 2 (1827) 101- ; 3 (1828) 160-. (Abel.) JacoU, C. G. J. Crelle J. 3 (1828) 88. Jacobi, C. G. J. Crelle J. 3 (1828) 192-, 303-, 403- ; 4 (1829) 185-. Legendre, A. M. As. Nr. 6 (1828) 205-. Abel, N. H. As. Nr. 7 (1829) 33-; Crelle J. 4 (1829) 85-, 194- ; 5 1829) 336-; 6 (1830) 73-. Jacobi, C. G. J. Crelle J. 4 (1829) 371- ; 6 (1830) 397-; 7 (1831) 41-. Gudermann, C. Crelle J. 16 (1837) 78-. Verhulst, P. F. Brux. Ac. Bll. 7 (1840) 322-. Bronwln, B. Camb. Mth. J. 3 (1843) 123-. Liouville, J. [1844-51] Liouv. J. Mth. 20 (1855) 203-. Bronwin, B. Ph. Mg. 26 (1845) 75-. Elliptic Functions 4040 (Bronwin.) Cayley, A. Ph. Mg. 26 (1845) Bronwin, B. Ph. Mg. 27 (1845) 42-. Hermite, C. Nancy Mm. S. Sc. (1845) 201-. Jacobi, C. G. J. Liouv. J. Mth. 11 (1846) 97-. Roberts, W. Liouv. J. Mth. 12 (1847) 449-. Despeyrous, C. Dijon Mm. Ac. (1849) (pte. 2) 41-. Cayley, A. Camb. and Dubl. Mth. J. 5 (1850) 201-. Malmsten, C. J. Crelle J. 39 (1850) 116-. Meyer, M. H. Grunert Arch. 16 (1851) 365-. Peche, F. Haidinger Ab. 4 (1851) (Ab. 3) 19-. Sturm, J. C. F. C. E. 42 (1856) 988-. Rothig, O. [1857] Crelle J. 56 (1859) 197-. Strebor [Roberts, W.] N. A. Mth. 19 (1860) 185-. Mathet, G. Liouv. J. Mth. 6 (1861) 329-. Mathieu, E. [1862] Par. EC. Pol. J. Cah. 42 (1867) 265-. Catalan, E. C. Em. At. N. Line. 20 (1867) 171-. Klein, F. D. Nf. B. (*1877) 104. Konigsberger, L. Epm. Mth. 1 (1877) 191-, 340-. Gruss, G. Prag Sb. (1878) 246-. Farkas, J. C. E. 90 (1880) 1482-. Glaisher, J. W. L. [1880] QJ. Mth. 17 (1881) 50-. Ferrers, N. M. Mess. Mth. 10 (1881) 102-. Glaisher, J. W. L. [1881-83] Mess. Mth. 11 (1882) 81-, 120- ; 12 (1883) 174- . Sparre, (lecomte)de. Brux. S. Sc. A. 10 (1886) (Pt. 2) 129- ; 11 (1887) (Ft. 2) 200- ; 12 (1888) (Pt. 2) 1-. Teixeira, J. P. G. Teix. J. Sc. 9 (1889) 132-; Fschr. Mth. (1889) 471-. Appell, P. C. E. 110 (1890) 32-. Grave, P. Kazan Un. Mm. 1897 (Pt. 3) 61-, (Pt. 4) 127- ; Fschr. Mth. (1897) 388. Holder, 0. Konigsb. Schr. 38 (1897) [53]-. Pierpont, J. N. Y. Am. Mth. S. Bll. 5 (1899) 490-. Abelians, and transcendentals of higher orders, geometric origin and representation. Yvon- Villarceau, A. J. F. Liouv. J. Mth. 4 (1878) 305-. actually multiplied together. Kiepert, L. [1872] Crelle J. 76 (1873) 21-. algebraic relations. David, . Toul. Ac. Sc. Mm. 6 (Sem. 2) (1884) 81-. and alternants, identities connected with. Muir, T. [1900] Edinb. E. S. T. 40 (1905) 187-. anharmonic properties of cubics. Ktodnicki, G. Wiad. Mt. 4 (1900) 232-. applications. Clebsch, A. [1855] Crelle J. 53 (1857) 292-. . Hermite, C. C. E. 85 (1877) 689-, 728-, 821-, 870-, 984-, 1085-, 1185- ; 86 (1878) 271-, 422-, 622-, 777-, 850- ; 89 (1879) 1001-, 1092- ; 90 (1880) 106-, 201-, 478-, 643-, 761- ; 93 (1881) 920-, 1098- ; 94 (1882) 186-, 372-, 477-, 594-, 753-. 313 4040 Elliptic and Single Theta Functions Picard, E. C. E. 89 (1879) 4040 applications. 74-. . Stouff, X. N. A. Mth. 15 (1896) 262-. . Sparre, (le comte) de. Par. S. Mth. Bll. 28 (1900) 52-. of Abel's theorem. Vries, J. de. Amst. Ak. Vs. 4 (1896) 96-. to celestial mechanics. Liouville, J. Liouv. J. Mth. 1 (1836) 445-. discontinuous functions. Bugaev, N. V. (xn) Bee. Mth. (Moscou) 11 (1883-84) 201-, 415-, 515- ; 12(1885) 3-. of formulae of spherical trigonometry. Boklen, 0. Grunert Arch. 40 (1863) 27-. to functions differing very little from zero. Zolotarev, E. St Pet. Ac. Sc. Mm. (Es.) 30 (1877) (Sappl.) No. 5, 71 pp. , geometric. Pact, P. G. Mt. 12 (1874) 93-, 97-. , . Ahlborn, H. Hamb. Mth. Gs. Mt. 1 (1889) 63. to maxima and minima. Zolotarev, E. I. [1877] St Pet. Ac. Sc. Bll. 24 (1878) 305-. mechanics. Padova, E. Yen. I. At. (1889-90) 547-. motion of body. Padova, E. Ven. I. At. (1889-90) 437-. problem in geometry. Boyd, J. H. A. Mth. 6 (1891-92) 93-, 163-. problems of mean values. Mukhopadhyay, A. [1888] Beng. As. S. J. 58 (Pt. 2) (1890) 199-, 213-. corresponding to function cos x + i sin x. Le- monnier, H. Par. EC. Norm. A. 4 (1875) 371- ; 6 (1877) 79-. deduction of g-series from g-products. Glaisher, J. W. L. Mess. Mth. 16 (1887) 133-. and definite integrals, and infinite series. Lerch, M. Prag Ceske Ak. Fr. Jos. Ez. (Tfida 2) 1 (1892) Art. 8, 11-, Art. 25, 6 pp.; 2 (1893) Art. 23, 42 pp.; 4 (1895) Art. 1, 55 pp.; Fschr. Mth. (1892) 445; (1893-94) 791- ; (1895) 501-. definition. Varicak, V. N. A. Mth. 14 (1895) 14-. of 2nd degree. Folk, M. Ups. S. Sc. N. Acta 16 (1893) No. 1, 30 pp. derivation. Verdam, G. J. Amst. I. (1842) 57-. differential equation in. Cayley, A. Mess. Mth. 4 (1875) 69-, 110- ; 6 (1877) 29. involving. Palmstrflm, A. Berg. Ms. Aarb. (1894-95) No. 14, 16 pp. , new. Brioschi, F. Mil. I. Lomb. Ed. 10 (1877) 417-. equations in. Genocchi, A. [1864] Tor. Mm, Ac. 23 (1866) 223-. for periods. Brioschi, F. A. Mt. 14 (1886-87) 238-. satisfied by K, E, etc. Kleiber,J. Mess. Mth. 18 (1889) 167-. relations. Martins da Silva, J. A. G. Teix. J. Sc. 6 (1885) 169-. differentiation with regard to modulus. Hermite, C. [1879] As. Nr. 96 (1880) 321-. of theory of binary forms. Faa de Bruno, formulae arising from. Wilkin- F. Am. J. Mth. 5 (1882) 1-. 8OW) 3/ . M . fj. L . Mth . S . P . 13 (1881-82) to theory of equations of higher degree. 212- Konig, G. (xn) Mag. Tud. Ak. Etk. (Mth.) ! O r function of modulus. Bierens [1] (No. 8) (1871) 34 pp. de Haatli D . Amst . Ak . V h. 18 (1879) 33 pp.; numbers. Joubert, P. C. E. 50 Arch. Neerl. 15 (1880) 225-. (1860) 774-, 832-, 907-, 1040-, 1095-, periods and invariants. Frobenius, 1145-. . Nazimov, P. S. Mosc. Un. Mm. (Ps.-Mth.) 5 (1885) 67-; Par. EC. Norm. A. 5 (1888) 23-, 147-. perturbations. Gylden, J. A. H. Liouv. J. Mth. 2 (1876) 411-. twisted quartics. Mathews, G. B. L. Mth. S. P. 19 (1889) 507-. trajectory. Gossot . Ev. Mar. et Col. 105 (1890) 225-, 417-. and binary quartics, connection. Study, E. Am. J. Mth. 17 (1895) 216-. calculation, approximate. Verhulst, P. F. Brux. Ac. Bll. 6 (1839) (pte. 2) 424-. cases. Schlomilch, 0. Schlomilch Z. 2 (1857) 49-; Liouv. J. Mth. 2 (1857) 43-. of St Venant's problem integrable by. Noccioli, G. N. Cim. 33 (1893) 208-. and circular functions, continued fractions for. Malmsten, C. J. Camb. and Dubl. Mth. J. 4 (1849) 286. class of integrals related to. Eamus, C. Kiob. Ov. (1835-36) 4-. and complex variable. Herrmann, 0. Z. Mth. Ps. 28 (1883) 193-, 257-. construction for amplitudes. Chasles, M. C. E. 19 (1844) 1239-. G. t & St'ickelberger, L. [1881] Crelle J. Mth. 92 (1882) 311-. and equation of 4th order. Pellet, A. E. Par. S. Mth. Bll. 14 (1886) 90-. especially elliptic arc, relations of curve of elasticity. Bender, C. Arch. Mth. Ps. 60 (1877) 117-. and Eulerian integrals. Lerch, M. Prag Sb. (1897) (Mth.-Nt:) No. 28, 11 pp. evaluation of definite integrals by. Roberts, W. Liouv. J. Mth. 11 (1846) 157-. existence domain, determination. Lerch, M. Mh. Mth. Ps. 11 (1900) 107-. expansion. Nicolls, W. [1883] Ir. Ac. T. 28 (1880-86) 267-. . Jack, J. Edinb. Mth. S. P. 14 (1896) 35-. as continued fractions. Rogers, L. J. L. Mth. S. P. 19 (1889) 550-. of imaginary period when modulus is small enough. Schlafli, L. A. Mt. 3 (1869-70) 243-. in increasing powers of variable. Hermite, C. [1875] Crelle J. Mth. 81 (1876) 220-. power series. Krause, M. Leip. Mth. Ps. B. 46 (1894) 30-. 314 4040 expansion of Elliptic Functions 2K 2K . + t 1K A r am x . cos* 8 am x . sin*' x [* *e 2K : I A e am x . Jo * dx in sines and cosines of multiple angles. Meyer, G. 0. Crelle -T. 37 (1848) 273-, 281-. expansions, new. Gylden, H. Stockh. Ofv. (1886) 197- ; Fschr. Mth. (1886) 387-. in trigonometric series. Appell, P. Par. S. Mth. Bll. 13 (1885) 13-. (Appell). Poincare, H. Par. S. Mth. Bll. 13 (1885) 19-. . Presle, de. Par. S. Mth. Bll. 14 (1886) 131-. . Krause, , <& Mohrmann, . Mth. A. 32 (1888) 331-. . Lerch, M. Acta Mth. 12 (1889) 51-. expressing definite integrals. Bronwin, B. Mathematician 2 (1847) 297-. expression for, by ratio of two series. Appell, P. Am. J. Mth. 14 (1892) 9-. formula. Glaisher, J. W. L. Mess. Mth. 7 (1878) 144-. . Cayley, A. Mess. Mth. 8 (1879) 126-. , Cayley 's. Forsyth, A, R. Mess. Mth. 14 (1885) 23-. formulas. Jacobi, C. G. J. Crelle J. 30 (1846) 269-. . Dillner, G. [1863] Stockh. Ak. Hndl. 5 (1866) No. 3, 19 pp. . Syloiv, L. [1864] Christiania F. 7 (1865) 68-. . Brioschi, F. Em. E. Ac. Line. T. 2 (1878) 115-. . Glaisher, J. W. L. [1879-82] B. A. Ep. (1879) 269; QJ. Mth. 16 (1879) 382-; 19 (1883) 22-. . Craig, T. Am. J. Mth. 5 (1882) 62-. . Wilkinson, M. M. U. L. Mth. S. P. 13 (1881-82) 106-. . Martins da Silva, J, A. G. Teix. J. Sc. 5 (*1883) 75-. . Cayley, A. Mess. Mth. 14 (1885) 21-. . Stieltjes, T. J. (jun.). Amst. Ak. Vs. M. 2 (1886) 101-. . Albeggiani, M. L. Palermo Cir. Mt. Ed. 1 (1887) 350-, 398. . Glaisher, J. W. L. Mess. Mth. 16 (1887) 67-. . Brioschi, F. Tor. Ac. Sc. At. 26 (1891) 586-. . Goursat, E. Par. S. Mth. Bll. 23 (1895) 18-. , application to geometric problem. Eichelot, F. J. Crelle J. 38 (1849) 353-. , class derivable from (/-series in. Glaisher, J. W. L. Mess. Mth. 14 (1885) 102-. , connection. Cayley, A. Mess. Mth. 9 (1880) 23-. , and deduction of fundamental theorem. Richelot, F. J. [1854] Crelle J. 50 (1855) 41-. , deduction of trigonometric formulae from. Glaisher, J. W. L. B. A. Ep. (1880) 477-. 4040 formulae, derivation from confocal conies. Griffiths, J. [1882] L. Mth. S. P. 14 (1882- 83) 46-. , , method. Glaisher, J. W. L. [1881] Camb. Ph. S. P. 4 (1883) 186-. , geometric interpretation. Cayley, A. (xn) J. H. Un. Cir. [1] (1882) 238. , new. Brun, F. de. Stockh. Ofv. (1897) 309- ; (1898) 523-. , systems. Glaisher, J. W. L. [1880-81] Mess. Mth. 10 (1881) 104- ; 11 (1882) 78. functions connected with. Cauchy, A. L. C. E. 17 (1843) 640-. and geodesic lines. Braunmuhl,A.von. Mth. A. 26 (1886) 151-. geometric considerations. Kiipper, C. Crelle J. 63 (1864) 40-. construction. Essen, E. Grunert Arch. 22 (1854) 241-. interpretation. Janssen van Raaij, W. H. L. Haarl. Ms. Teyl. Arch. 6 (1900) 151-. investigations. Tortolini, B. Tortolini A. 3 (1860) 179- ; (vm) 4 (1861) 204-. origin and fundamental formulae. Despeyrous, C. Toul. Mm. Ac. 5 (1873) 211-. properties relative to. Serret, J. A. Liouv. J. Mth. 8 (1843) 495-. . . Roberts, W. Liouv. J. Mth. 10 hyperbolic and cyclic functions, numerical calculation of arguments. Runge, C. Acta Mth. 15 (1891) 221-. functions, relations. Griffiths, J. L. Mth. S. P. 6 (1874-75) 98-. , . Sincov, D. Kazan S. Ps.-Mth. Bll. 9 (1900) 27-. hyperelliptic functions, approximation method. Merrifidd, C. W. Phil. Trans. (1860) 223-; (1862) 417-. , reports on progress. Russell, W. H. L. B. A. Ep. 39 (1869) 334-; 40 (1870) 102- ; 42 (1872) 334-; 43 (1873) 307-. , representation, geometric. Serret, J. A. Par. Mm. Sav. Etr. 11 (1851) 103-. identities. Glaisher, J. W. L. [1874-76] Ph. Mg. 47 (1874) 437-; 50 (1875) 539-; Phil. Trans. 165 (1876) 489-; L. Mth. S. P. 7 (1875-76) 61-; Mess. Mth. 6 (1877) 102-. . Cayley, A. QJ. Mth. 16 (1879) 281-. . Glaisher, J. W. L. As. Fr. C. E. 9 1880) 223-. , case. Rogers, L. J. [1893] L. Mth. S. P. 25 (1894) 44-. imaginary period. Johnson, W. W. Am. J. Mth. 6 (1884) 246-. integrals, calculation. Steen, A. (xn) Ts. Mth. 3 (1867) 153-. , with regard to modulus. Salvert, F. de. C. E. 124 (1897) 1008-, 1186 ; Brux. S. Sc. A. 21 (1897) (Pt. 1) 119-. related to. Lerch, M. Acta Mth. 22 (1899) 365-. , relations. Herz, N. [1881] Arch. Mth. Ps. 67 (1882) 343-. and integrals, and Jacobian functions, theory. Bolzani, J. F. (xn) Kazan Un. Mm. (1856) (Bks. 3 & 4) 170 pp. 315 4040 Elliptic and Single Theta Functions 4040 integration by. Bouquet, J., & Briot, . C. B. 61 (1855) 1229-. . Dolbnia, I. Kazan S. Ps.-Mth. Bll. 1 (1891) 46-; Fschr. Mth. (1891) 463-. of equation connected with. MacMahon, P. A. B. A. Bp. (1882) 450-. interpolation problem. Appell, P. Bll. Sc. Mth. 10 (1886) 109-. . Schottky, F. Crelle J. Mth. 107 (1891) 189-. inverse. Cayley, A. Camb. Mth. J. 4 (1845) 257-. , integrals expressed by. Greenhill, A. G. Camb. Ph. S. P. 3 (1880) 361-. investigations. Klein, F. Mth. A. 26 (1886) 455-. involved in definite integrals. Glaisher, J. W. L. B. S. P. 29 (1879) 331-. system of integrals. Glaisher, J. W. L. QJ. Mth. 19 (1883) 145-. Jacobian. Stringham, I. A. Mth. 8 (1893-94) 105-. of 1st kind. Serret, J. A. Liouv. J. Mth. 8 (1843) 145-. . Iaggi,E. N. A. Mth. 17 (1898) 367-. , fundamental properties, geometric explanation. Lemeray, E. M. N. A. Mth. 19 (1900) 255-, 289-. , geometric demonstration of two theorems. SchlSfii, L. Grunert Arch. 12 (1849) 188-. , study. Leaute, H. Par. EC. Pol. J. Cah. 46 (1879) 65-. - , Jacobi's 1st and 2nd theorems, supplements to. Essen, E. Grunert Arch. 21 (1853) 241-, 418-. i representation by arcs of curves. Allegret, \A.] C. B. 70 (1870) 1032-. , gauche biquadratics. Leaute, H. C. B. 83 (1876) 527-. , , geometric. Roberts, W. Liouv. J. Mth. 8 (1843) 263-. , , . Serret, J. A. Camb. and Dubl. Mth. J. 1 (1846) 187-. , , . Davidov, A. Y. (xii) Bee. Mth. (Moscou) 2 (1867) (Pt. 1) 129-. and 2nd kinds. Lobatto, R. Amst. N. Vh. 12 (1846) 119-. . Bellavitis, G. Padova Bv. Period. 1 (1851-53) 149-. . Kupper, G. Crelle J. 55 (1858) 89,-. , as functions of amplitude. Liouville, J. [1833] Par. EC. Pol. J. 23 cah. (1834) 37-. , modulus. Liouville, J. Liouv. J. Mth. 5 (1840) 34-, 441-. 2nd kind. Frobenius, G. [1881] Crelle J. Mth. 93 (1882) 53-. and 3rd kinds, expansion in trigonometric series. Krause, M. Mth. A. 30 (1887) 425-, 516- ; 33 (1889) 108- ; 35 (1890) 577-. , results in. Griffiths, J. B. S. P. 38 (1885) 206-, 316-. 3rd kind. Salvert, (le vicomte) de. . Brux. S. Sc. A. 21 (1897) (Pt. 2) 131-. i calculation, method. Somoff, J. Crelle J. 47 (1854) 269-. of 3rd kind, reduction to functions of two arguments. Verhulst, P. F. Quetelet Cor. Mth. 11 (1839) 225-. , representation, geometric. Tortolini, B. G. Arcad. 100 (1844) 257-. , , . Drobisch, M. W. Grunert Arch. 24 (1855) 320-. logarithm, behaviour. Schottky, F. Crelle J. Mth. 108 (1891) 342-. method, short. Meech, L. W. Des Moines Anal. 4 (1877) 129-, 161- ; 5 (1878) 9-. , (Meech). Kummell, C. H. Des Moines Anal. 5 (1878) 17-, 97-. modulus, formulas relative to. Hermite, C. Liouv. J. Mth. 9 (1864) 313-. as function of quotient of two periods. Hermite, C. A. Mt. 3 (1869-70) 81-. numerators and denominators satisfying partial differential equation. Jacobi, C. G. J. Crelle J. 36 (1848) 81-. of 2nd order. Bricard, R. Par. S. Mth. Bll. 25 (1897) 212- ; 26 (1898) 91-. , properties. Berger, A. Ups. S. Sc. N. Acta 14 (1891) No. 12, 50 pp. 3rd order, differential equation. Pfannen- stiel, E. Ups. S. Sc. N. Acta 15 (1895) No. 3, 18 pp. nth order. Rados, G. Mth. Termt. Ets. 6 (1888) 353-. and orthogonal transformations. Wilkinson, M. M. U. L. Mth. S. P. 14 (1882-83) 203-. periodicity. Benthem, A. N. Arch. Wisk. 3 (* 1877) 186-. , double. Franklin, F. Am. J. Mth. 11 (1889) 283-. periods, existence of two. M6ray, C. Par. EC. Norm. A. 1 (1884) 177 r . and their powers, expansion. Andre, D. C. B. 83 (1876) 135-. , in series. Andre, D. Par. EC. Norm. A. 6 (1877) 265-. property, demonstration. Walton, W. [1870] QJ. Mth. 11 (1871) 177-. and g-series. Glaisher, J. W. L. Mess. Mth. 13 (1884) 126-. quadriquadric curve. Cayley, A. Mth. A. 25 (1885) 152-. of nth rank. Klein, F. Leip. Mth. Ps. B. 36 (1884) 61-. rational combinations. Meyer, C. O. [1885] Crelle J. 56 (1859) 314-. and reciprocal factorials. Cauchy,A.L. C.B. 17 (1843) 825-. recurring relation. Jonquieres, E. de. C. B. 101 (1885) 415-. reduction of some definite integrals to, and applications. Tortolini, B. G. Arcad. 116 (1848) 137-, 265-. integrals to. Dienger, J. Grunert Arch. 11 (1848) 94-; 13 (1849) 424-. . Tortolini, B. Tortolini A. 3 (1860) 183-. , cases. Sparre, (le comte) de. Brux. S. Sc. A. 8 (1884) (Pt. 2) 97-; 9 (1885) (Pt. 2) 205- ; 21 (1897) (Pt. 2) 45-. representation. Liouville, J. C. B. 21 (1845) 1255-. 316 4040 Elliptic Functions 4040 representation, conformal, of curvature lines of ellipsoid by. Neovius, E. R. Helsingf. Acta 15 (1888) 363-. by cubic curve. Juel, C. Sk. Nf. F. (1892) 344-; Fschr. Mth. (1892) 455. of curves of 3rd degree by. Picquet, . Par. EC. Pol. J. 54 (1884) 31-. by doubly infinite products. Fiske, T. S. [1891] N. Y. Mth. S. Bll. 1 (1892) 61-. elastic curve. Greenhill, A. G. Mess. Mth. 5 (1876) 180- ; 6 (1877) 182-. , geometric. Strnad, A. Arch. Mth. Ps. 61 (1877) 321-. , , of equation, and applications. Trudi, N. Nap. Ac. Sc. Mm. 1 (1852-54) 63-. , , equations relative to. Serret, J. A. Liouv. J. Mth. 10 (1845) 351-. , of three. Eoberts, W. Liouv. J. Mth. 9 (1844) 155-. , , (Roberts). Serret, J. A. Liouv. J. Mth. 9 (1844) 160. , graphic, and binodal quartic. Cayley, A. Camb. Ph. S. T. 14 (1889) 484-. by power series. Jacobi, C. G. J. Crelle J. 54 (1857) 82-. of any quantity by sin am (u + w, k). Richelot, F. J. Crelle J. 45 (1853) 225-. as quotient of convergent series. Kneser, A. Mth. A. 32 (1888) 309-. by rational functions of exponential function. Schwarz, H. A. Berl. Ak. Sb. (1894) 1187-. Eiemann's lectures. Stahl, H. Z. Mth. Ps. 45 (1900) 216-. roots of unity occurring in. Molien, T. Leip. Mth. Ps. B. 37 (1885) 25-. series connected with. Gutzmer, A, G. Teix. J. Sc. 8 (1887) 33-. , new. Faa de Bruno, F. C. E. 95 (1882) 22-. solution of x 3 + i/3-l = 0. Cayley, A. [1881] Camb. Ph. S. P. 4 (1883) 106-. = (Cayley). Greenhill, A. G. [1882] Camb. Ph. S. P. 4 (1883) 223-. special. Thomae, C. J. Z. Mth. Ps. 27 (1882) 181-. spherical triangle in. Johnson, W. W. QJ. Mth. 19 (1883) 185-. and spherical trigonometry, connexion. Glaisher, J. W. L. QJ. Mth. 17 (1881) 353-. and orthogonal substitutions. Study, E. Leip. Mth. Ps. Ab. 20 (1893) 85-. summation formula. Alexeief, . Bll. Sc. Mth. As. 1 (1877) 44. theorem. Jacobi, C. G. J. [1827] As. Nr. 6 (1828) 133-. (Jacobi). Plana, G. [1828] Tor. Mm. Ac. 33 (1829) 333-. ( ). Poisson, S. D. Par. Mm. Ac. Sc. 10 (1831) 73-. . Richelot, F. J. Crelle J. 32 (1846) 219. . Cayley, A. L. Mth. S. P. 10 (1878-79) 43-. . Hermite, C. C. E. 90 (1880) 1096-. . Glaisher, J. W. L. B. A. Ep. (1882) 456-. . Phragmen, E. Stockh. Ofv. (1884) No. 9, 199- ; Acta Mth. 7 (1885) 33-. theorem. Falk, M. Acta Mth. 7 (1885) 197-. . Lerch, M. G. Teix. J. Sc. 8 (1887) 3-. . Moore, E. H. Palermo Cir. Mt. Ed. 4 (1890) 186-. .fundamental. Pringstieim, A. Mth. A. 27 (1886) 151-. of Landen. Cayley, A. [1881] L. Mth. S. P. 13 (1882) 47-. theorems, 2. Scheibner, W. Leip.B. 11 (1859) 159-. . Falk, M. Stockh. Ak. Hndl. Bh. 14 (Afd. 1) (1889) No. 1, 30 pp. , geometric. Roberts, W. Ir. Ac. P. 3 (1847) 77-. and trigonometric theorems. Glaisher, J. W. L. [1880] Mess. Mth. 10 (1881) 92-. theory. Jacobi, C. G. J. [1827-32] Par. EC. Norm. A. 6 (1869) 127-. . Ivory, J. Phil. Trans. (1831) 349-. . Jacobi, C. G. J. Crelle J. 8 (1832) 413-. . Bromcin, B. Ph. Mg. 22 (1843) 258-. (Bronwin). Cayley, A. Ph. Mg. 22 (1843) 358-. (Cayley). Bronwin, B. Ph. Mg. 23 (1843) 89-. . Jacobi, C. G. J. Crelle J. 26 (1843) 93-. . Serret, J. A. Par. S. Phlm. PV. (1845) 81-. . Eisenstein, G. Crelle J. 30 (1846) 185- ; 32 (1846) 59-; 35 (1847) 137-, 185-. . Cayley, A. Camb. and Dubl. Mth. J. 2 (1847) 256- ; 3 (1848) 50-. . Malmsten, C. J. Stockh. Ak. Hndl. (1847) 71-. . Hermite, C. Camb. and Dubl. Mth. J. 3 (1848) 54-; C. E. 29 (1849) 594. . Krusemarck, R. [1852] Crelle J. 46 (1853) 189-. . Richelot, F. J. C. E. 49 (1859) 641-. . Betti, E. Tortolini A. 3 (1860) 65-, 298- ; 4 (1861) 26-, 57-, 297-. . Plana, G. [1860] Tor. Mm. Ac. 20 (1863) 189-, 422-. . Hermite, C. Liouv. J. Mth. 7 (1862) 25- ; C. E. 57 (1863) 613-, 993-. . Rubini, R. G. Mt. 1 (1863) 33-, 118-, 140-, 291-. . Thiele, T. N. (xn) Ts. Mth. 3 (1873) 65-. . Weyr, E. Prag Sb. (1876) 172-. . Frobenius, G., & Stickelberger, L. Crelle J. Mth. 83 (1877) 175-. . Kronecker, L. Berl. Ak. Mb. (1881) 1165-. . Stolz, O. Innsb. Nt. Md. B. 12 (1882) xiv-. . Kronecker, L. Berl. Ak. Sb. (1883) 497-, 525-. . Brioschi, F. A. Mt. 12 (*1883-84) 49-. . Mansion, P. Brux. Ac. Bll. 8 (1884) 180-. . Schroeter, H. Acta Mth. 5 (1884) 205-. . Kronecker, L. Berl. Ak. Sb. (1885) 761- ; (1886) 701-. . Lipschitz, R. Par. EC. Norm. A. 2 (1885) 315-. . Mertens, F. Wien Ak. Sb. 91 (1885) (Ab. 2) 974-. . Weber, H. 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Ps. 48 (1868) 121-. , elementary. Laurent, H. N. A. Mth. 16 (1877) 78-, 211-, 361-, 385-, 433-, 481- ; 17 1878 119-, 247-, 385-, 537-; 18 (1879) 126-, 145-. , equations in, and geometric theorems connected therewith. Fergola, E. Bm. S. It. Mm. 4 (1882) No. 6, 10 pp. , formula, fundamental, direct verification. Jacobi, C. G. J. Crelle J. 36 (1848) , formulae, three. Martins da Silva, J. A. Bll. Sc. Mth. 10 (1886) 78-. , Gauss. Pepin, T. Km. N. Line. Mm. 9 (1893) (Pt. 2) 1-; 10 (1894) 125-. , . Gilnther, P. Liouv. J. Mth. 3 (1897) 95-. of Legendre. Caldarera, F. Palermo G. Sc. Nt. 1 (1866) 38-. and theory of numbers. Kronecker, L. Liouv. J. Mth. 3 (1858) 265-. theory, outline. Heine, H. E. Crelle J. 39 (1850) 122-. , partial fractions in. Gilnther, P. Crelle J. Mth. 113 (1894) 262-. , point in. Lipschitz, E. 0. S. C. K. 97 (1883) 14H-. , (Lipschitz). Hermite, C. C. B. 97 (1883) 1414-. , . Martins da Silva, J. A. Brux. Ac. Bll. 10 (1885) 79-. . . Koenigs, G. Toul. Fac. Sc. 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S. It. Mm. 8 (1892) No. 3, 18 pp. in finite form. Bugaev, N. V. [1891] Bee. Mth. (Moscou) 16 (1893 259-; Fschr. Mth. (1892) 421-. of line-element by. Selander, E. Helsingf. Ofv. 34 (1892) 203-. formula y - sin (A + B + C +...), deduction from. Griffiths, J. L. Mth. S. P. 15 (*1883-84) 78-. = , (Griffiths). Cayley, A. L. Mth. S. P. 15 (*1883-84) 81. formulas. Cayley, A. Mth. A. 12 (1877) 369-. . Torelli, G. A. Mt. 15 (1887-88) 67-. as functions of modulus. Torelli, G. G. Mt. 12 (1874) 168-. general forms. Scheibner, W. Leip. Mth. Ps. B. 43 (1891) 575-. geometrical, analytical and dynamical signi- ficance. Oekinghaus, E. Arch. Mth. Ps. I (1884) 337-; 4 (1886) 279-. and hyperelliptic integrals. Neumann, C. [1870] Mth. A. 3) 1871) 611-. . Oekinghaus, E. Arch. Mth. Ps. II (1892) 132-. with imaginary modulus. Hermite, . [1890] Hamb. Mth. Gs. Mt. 3 (1900) 22-. parameters. Booth, J. Ph. Mg. 7 (1854) 213-. in indeterminate form, two. Darboux, G. Bordeaux S. Sc. 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Mth. 5 (1871) 245-. , illustration by certain link-work. Emch, A. A. Mth. 1 (1900) 81-. 31$ 4040 Elliptic Integrals 4040 of 1st kind, infinitely many normal forms. Klein, C. F. Munch. Ak. Sb. 10 (1880) 533-. , interpretation, geometric. Mansion, . Brux. S. Sc. A. 13 (1889) (Pt. 1) 47-. , inversion. Kluyver, J. C. N. Arch. Wisk. 4 (1899) 109- ; Fschr. Mth. (1899) 400. , new transcendental transforming into a circular function. laggi, E. N. A. Mth. 19 (1900) 537-. , normal forms of 3rd and 5th degree. Bianchi, L. Mth. A. 17 (1880) 234-. , representation , geometric . Morgan, A . Edinb. Mth. S. P. 15 (1897) 2-. , transformation of integrals of irrational differentials to normal form. Steen, A. Kjob. Skr. 8 (1870) 183-. and 2nd kinds, differential equations satisfied by moduli. Fuchs, L. Crelle J. Mth. 112 (1893) 156-. , series for calculating, etc. Gudermann, C. Crelle J. 16 (1837) 366-; 17 (1837) 382-. , 2nd and 3rd kinds, solution of by. Simony, 0. Arch. Mth. Ps. 55 (1873) 193-. 2nd kind. Thomae, C. J. Crelle J. Mth. 93 (1882) 69-; 94 (1883) 241- ; Z. Mth. Ps. 27 (1882) 179-. , functions inverse to. Glaisher, J. W. L. Camb. Ph. S. P. 5 (1886) 372-. , representation, mechanical. Darwin, G. H. Mess. Mth. 4 (1875) 113-. and 3rd kinds, expression by functions. Pokorny, M. Z. Mth. Ps. 10 (1865) 317-. 3rd kind. Newman, F. W. Camb. and Dubl. Mth. J. 8 (1853) 190-. . Hermite, C. C. E. 94 (1882) 901-. . Kummell, C. H. A. Mth. 2 (1885- 86) 73-, 104-. . Weyr, E. [1891] Prag Ceske Ak. Fr. Jos. Ez. (Tfida 2) 1 (1892) Art. 6, 9 pp. . Mansion, . Brux. S. Sc. A. 21 (1897) (Pt. 1) 15-. , formula. Cayley, A. L. Mth. S. P. 13 (1882) 175-. , formulae, complementary moduli. Salvert, F. de. C. E. 120 (1895) 1208-. , reducible to 1st kind. Saalschiitz , L. Z. Mth. Ps. 34 (1889) 199-. , reduction. Gudermann, C. Crelle J. 14 (1835) 169-, 185-. linear transformation. Burnside, W. Mess. Mth. 21 (1892) 170-. of 1st and 2nd order, relation between. Childe, (Rev.) G. 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(Rs.) 16 (*1870) 1-. Integrations over surface of ellipsoid. Tallqvist, . H. Helsingf. Acta 18 (1891) 329-. Invariants of $ (u) when -~ = \/ - 7. Kluyver, J. C. N. Arch. Wisk. 3 \1898) 36-. Inversion of algebraic integral. Dillner, G. Stockh. Ofv. (1885) No. 4, 3- ; Fschr. Mth. (1885) 378. Jacobian equation. Sterba, J. Crelle J. Mth. , 122 (1900) 198-. functions with 1 argument, theory. Brioschi, F. G. Mt. 2 (1864) 8-, 33-, 129-. , expansion in powers of variable. Brioschi, F. Tortolini A. 1 (1858) 41-. Kleiber's functions K and G. Glaisher, J. W. L. Mess. Mth. 22 (1893) 71-. Kronecker's double series (, 77, u, v, w). Lerch, M. Mh. Mth. Ps. 5 (1894) 367-. Lambert's series. La Vallee Poussin, C. J. de. Brux. S. Sc. A. 20 (1896) (Pt. 1) 56-. Lame's equation. Brioschi, F. C. B. 85 (1877) 1160- ; 86 (1878) 313-. -- . Hermite, C. [1877] A. Mt. 9 (1878-79) -- . Lindemann, C. L. F. B. A. Bp. (1883) Lemniscate functions, expansion coefficients. Hurwitz, A. Gott. Nr. (1897) 273-; Mth. A. 51 (1899) 196-. Linear differential equations. Stenberg, E. A. Helsingf. Acta 16 (1888) 333-. with doubly periodic coefficients. Malmborg, M. Ups. irsk. (1897) (Mt. Nt.) 31pp. homogeneous differential equations, with doubly periodic coefficients. Stenberg, E. A. Helsingf. Acta 16 (1888) 493-, 555-; 19 (1893) No. 11, 7 pp. Periodic equation. Rodrigues, J. M. G. Teix. J. Sc. 6 (1885) 191-. function of one variable with more than two periods impossible, theorem of Jacobi. Caso- rati, F. Mil. I. Lomb. Ed. 15 (1882) 623-. functions. Laurent, H. N. A. Mth. 6 (1867) 267-. , generalisation. Picard, E. C. B. 89 ' (1879) 140-. of 2nd kind. Rausenberger, 0. Mth. A. 20 (1882) 550-. , theory, fundamental theorem. Prings- heim, A. [1900] Munch. Ak. Sb. 30 (1901) 541-. , uniform, of not more than two periods. Gade, K. (xn) Ts. Mth. 5 (1881) 181-. Problem of Abel, complement to. Bougaief, . C. B. 113 (1891) 1025-. proposed by Hermite, solution. Le Vavasseur, . Toul. Fac. Sc. A. 8 (1894) G, 3 pp. " illipti Par. . Greenhill, A. G. [1888] L. Mth. S. P. 20 (1889) 213-. . Floquet, G. C. B. 121 (1895) 805-, Pseudo-elliptic integrals. Goursat, E. S. Mth. Bll. 15 (1887) 106-. . Halphen, G. H. C. B. 106 (1888) 1263-. . Burnside,W. Mess. Mth. 22 (1893) 83-. , Abel's. Dolbnia, I. P. Kazan S. Nt. (Ps.-Mth.) P. 8 (1890) 252-; Liouv. J. Mth. 6 (1890) 293-. depending on cube root of trinomial. - ,, . Dolbnia, J. Bll. Sc. Mth. 17 (1893) 125-. - _. Markoff, A. Mth. A. 47 (1896) 598- J_ _ radicals of 4th an v d 6th ; de g r ees. "rV-r 1 ? ^ a / 1 1 n ^ Hermite, C. [1879] Crelle Dolbnia, J. Bll. Sc. Mth. 17 (1893) 288-. J. Mth. 89 (1880) 9-. and their dynamica i applications. , new ivppliCcition. (ji/luCH, J A., Jd. ~ r*V> Q T> o^ C. B. 93 (1881) 537- S) 195- "o^'iao^oa 11 ' Craw f rd ' L ' Q J " Mth ' -, evaluation. Gilnther, S. Par. S. t (loyoj yo . , , factors. 29 (1898) 196-. , special case. Stenberg, E. A. Mth. 10 (1887) 339-. , . Jamet, V. C. B. Ill (1890) 638-. , theorems. Brioschi, F. C. B. 92 (1881) 325-. Lame-Hermite differential equation. Brioschi, F. St Pet. Ac. Sc. Bll. 35 (1894) 449-. Crawford, L. QJ. Mth. Acta Legendre's "Fonctions elliptiques," revision of chaps, xxiv and xxvi. Greenhill, A. G. QJ. Mth. 19 (1883) 225-. Lemniscate function, representation, geometric. Krimphoff, W. Crelle J. Mth. 110 (1892) 73-. sn u, multiplication. Schwering, K. Crelle J. Mth. 107 (1891) 196-. Mth. Bll. 10 (1882) 88-. ^-formula leading to expression for E. Cayley, A. Mess. Mth. 6 (1877) 63-. for sine-amplitude, method of obtaining. Glaisher, J. W. L. [1879] L. Mth. S. P. 11 (1879-80) 45-. g-series. Heine, H. E. Crelle J. 32 (1846) 210- ; 34 (1847) 285-. . Catalan, E. Par. S. Mth. Bll. 19 (1891) 145-. , formula due to Jacobi. Catalan, E. C. Liege S. Sc. Mm. 15 (1888) No. 1, 32-. , formulae. Catalan, E. C. Liege S. Sc. Mm. 15 (1888) No. 1, 197-, 241-. , relation of Kronecker's. Igel, B. Mh. Mth. Ps. 8 (1897) 354-. for v/2K and 2Glir, coefficients. Glaisher, Ith. J. W. L. QJ. Mth. 21 (1886) 60-. 322 4040 Elliptic and Single Theta Functions g-series for kp sn u, kp en u, p dn u, process for squaring. Glaisher, J. W. L. Mess. Mth. 16 (1887) 145-. Quadratic forms, Kronecker's theorems. Her- mite, C. Liouv. J. Mth. 9 (1864) 145-. Quadrature of the ellipsoid. Carda, K. Mh. Mth. Ps. 7 (1896) 129-. Quadri-quadric porism. Matheics, G. B. QJ. Mth. 30 (1899) 16-. Eational functions of elliptic form. Pokrovskij, P. M. [1900] Rec. Mth. (Moscou) 21 (1901) 387-; Fschr. Mth. (1900) 445. Rectangular transformation, Wilkinson's. Cayley, A. L. Mth. S. P. 14 (1883) 222-. Reduction of j- , when U is of fourth order. Cayley, A. Camb. and Dubl. Mth. J. 1 (1846) 70-. Relation of Jacobi, arithmetical deduction. Lipschitz, E. Acta Mth. 7 (1885) 95-. Legendre. Kronecker, L. Berl. Ak. Sb. (1891) 323-, 343-, 447-, 905-. Rotation of body, theory. Marcolongo, It. Palermo Cir. Mt. Rd. 7 (1893) 26-. Segner's numbers. Catalan, E. C. Li6ge S. Sc. Mm. 15 (1888) No. 1, 113-. Series representing the twelve elliptic and the four zeta functions. Glaisher, J. W. L. Mess. Mth. 18 (1889) 1-. Sigma function. Tognoli, 0. G. Mt. 25 (1887) 367-. , general function satisfying equation for. Delisle, A. Mth. A. 30 (1887) 91-. , Weierstrass's fundamental theorem. Caspary, F. Crelle J. Mth. 96 (1884) 182-. functions. Loria, G. G. Mt. 31 (1893) 179-. of several arguments, new method. Jahnke, E. Z. Mth. Ps. 37 (1892) 178-. , elliptic, as special case of hyperelliptic (r-functions. Bolza, 0. N. Y. Am. Mth. S. T. 1 (1900) 53-. , even. Pascal, E. A. Mt. 23 (1895) 181-. , , Brioschi's recursion formula for expansion. Bolza, O. Am. J. Mth. 21 (1899) 175-. , expression of orthogonal system by, and application to rotation. Jahnke, E. Liouv. J. Mth. 5 (1899) 155-. , general, formation. Krazer, A. Mth. A. 33 (1889) 591-. functions and theta functions, Weierstrass's, properties. Lerch, M. Prag Sb. (1887) (Mth.-Nt.) 426-. series, Weierstrass's, numerical coeffi- cients in. Haussner, B. Gott. Nr. (1894) 299-. a (u), Weierstrass's, and allied functions. Kluyver, J. C. N. Arch. Wisk. 3 (1898) 80-; Fschr. Mth. (1897) 392. Sine of amplitude, inverse and square, ex- pansion in powers of the variable. Hermite, C. As. Fr. C. R. (1875) 131-. Subsidiary elliptic function pm (u, k). Griffiths, J. L. Mth. S. P. 15 (*1883-84) 219-. 4040 Symmetric functions of I, G, E in terms of Q. Glaisher, J. W. L. Mess. Mth. 21 (1892) 65-. Theorem of Dirichlet. Cayley, A. Camb. and Dubl. Mth. J. 9 (1854) 163-. Poncelet on \/a + bx + ex 2 . Sylvester, J. J. Ph. Mg. 20 (1860) 307-. THETA FUNCTIONS. attached to plane algebraic curve without singularities. Pick, G. Wien Ak. Sb. 94 (1887) (Ab. 2) 739-. breaking up into products of such functions, after transformation. Wiltheiss, E. Mth. A. 26 (1886) 127-. of class 3. Thomae, J. Leip. Mth. Ps. B. 39 (1887) 100-. connexion with elliptic integrals. Scheibner, W. Leip. Mth. Ps. B. 41 (1889) 86-, 245-. constant factor in expression of (x) as an infinite product. St Germain, A. de. [1873] Par. S. Mth. Bll. 2 (1874) 62-. differential equation and expansion. Wiltheiss, E. Mth. A. 29 (1887) 272-. elliptic constant (0). Thomae, J. Z. Mth. Ps. 11 (1866) 247-. and eta functions, expansion by products. i. Bll. 15 216-. Presle, de. Par. S. Mth. (1887) evaluation, and integrals of 2nd kind. Eoch, G. Z. Mth. Ps. 11 (1866) 53-. expression as definite integrals. Glaisher, J. W.L. [1877] Camb. Ph. S. P. 3 (1880) 61-. , series summation leading to. Glaisher, J. W. L. B. A. Rp. (1876) (Sect.) 15-; Camb. Ph. S. P. 6 (1889) 129-. of direction-cosines of two orthogonal systems of coordinates by. Kotter, F. Berl. Ak. Sb. (1895) 807- ; Crelle J. Mth. 116 (1896) 213-. elliptic integrals of 2nd and 3rd kinds by. Eoch, G. Z. Mth. Ps. 10 (1865) 317-. Euler's angles by. Caspary, F. Bll. Sc. Mth. 13 (1889) 89-. (x) as definite integral. Glaisher, J. W. L. Mess. Mth. 5 (1876) 173; 17 (1888) 152-. formula. Craig, T. Am. J. Mth. 5 (1882) 350-. , fundamental, new demonstration. Kapteyn, W. Bll. Sc. Mth. 15 (1891) 125-. of Hermite. Humbert, G. [1880] Par. S. Mth. Bll. 9 (1881) 42-. , Jacobi's. Gutzmer, A. Crelle J. Mth. 110 (1892) 177-. , , geometrical proof. Baker, H. F. Mth. A. 43 (1893) 593-. formulae, general. Krazer, A. Mth. A. 52 (1899) 369-. , Jacobi's. Kronecker, L. Crelle J. Mth. 102 (1888) 260-. with fractional characteristics. Krause, M. Mth. A. 26 (1886) 569-. integration by. Appell, P. C. R. 90 (1880) 1207-. Jacobi's. Frenet, F. Bordeaux Mm. S. Sc. 8 (1870) 177-. ^ , theory. Herstowski, F. Mth. A 11 (1877) 1-. 323 x2 4040 Elliptic and Single Theta Functions logarithmic derivatives of higher order. Ber- tolani, G. G. Mt. 34 (1896) 135-. products of 3 and 4. Scheibner, W. Crelle J. Mth. 102 (1888) 255-. representation of invariants. Bolza, O. Gott. Nr. (1887) 418- ; Mth. A. 30 (1887) 478-. motion of top. Marcolongo, R. G. Teix. J. Sc. 14 (1900) 169-. quotient of two. Thomae, J. Mth. A. 6 (1873) 603-. resolution into factors. Humbert, G. C. B. 126 (1898) 394-. resultants and discriminants. Schlesinger, 0. Mth. A. 33 (1889) 411-. and sigma functions, logarithmic derivatives of higher order than 2nd. Bertolani, G. G. Mt. 33 (1895) 139-. , and orthogonal systems. Caspary, F. Liouv. J. Mth. 6 (1890) 376-. theorem, fundamental. Craig, T. [1891] J. H. Un. Cir. [11 (1891-92)] 42. , , Jacobi's. Caspary, F. C. B. 104 (1887) 1094-. theorems. Lerch, M. Mh. Mth. Ps. 9 (1898) 177-. theory. Enneper,A. Z. Mth. Ps. 12 (1867) 79-. .exposition. Caspary, F. C.B. Ill (1890) 225-. , illustration. Cayley, A. Mess. Mth. 7 (1878) 27-. triple. Cayley, A. Mess. Mth. 7 (1878) 48-. vanishing of. Riemann, [G. F.~] B. [1865] Crelle J. 65 (1866) 161-. even. Frobenius, G. Gott. Nr. (1888) 67-. of one variable, algebraic identities used to obtain relations for. Caspary, F. Mth. A. 28 (1887) 493-. , expression of orthogonal systems by. Caspary, F. C. B. 107 (1888) 859-, 934. with fractional characteristics. Voss, R. Arch. Mth. Ps. 4 (1886) 385-. zero value. Cayley, A. Crelle J. Mth. 100 (1887) 87-. for zero values, relations of derivatives. Pascal, E. A. Mt. 24 (1896) 23-. and zeta functions, values for certain values of argument. Glaisher, J. W. L. E. S. P. 29 (1879) 351-. Theta products, relations. (Jordan, P. [1865] Crelle J. 66 (1866) 185-. , , linear. Krazer, A. Acta Mth. 17 (1893) 281-. series, analogy, and some special Heine's hypergeometric series. Lerch, M. [1893] Prag Cesk6 Ak. Fr. Jos. Ez. (Tfida 2) 3 (1894) Art. 5, 10 pp.; Fschr. Mth. (1893-94) 794. , and connexion with double integrals. Lipps, G. F, Leip. Mth. Ps. B. 44 (1892) 340-, 473-. , constant factors. Frobenius, G. Crelle J. Mth. 98 (1885) 244-. , , in casep = 3. Klein, F. L. Mth. S. P. 20 (1889) 235-. series, convergence. Krazer, A. Mth. A. 49 (1897) 400-. 4040 Transcendental functions. Bidone, G. Tor. Mm. Ac. 23 (1818) 295-; 24 (1820) 255-. . Jacobi, C. G. J. As. Nr. 6 (1828) 33-; Crelle J. 3 (1828) 376- ; 15 (1836) 199-. . Clausen, T. As. Nr. 19 (1842) 181-. , calculation. Petersen, J. P. C. (xn) Ts. Mth. 6 (1876) 1-. , comparison, theory. Ellis, R. L. B. A. Ep. (1846) 34-. and their Fourier developments. Kry- gowski, Z. Prace Mt.-Fiz. 5 (1894) 70-. having property /(* + )=/(*),/(* + <>')=/(*), 8(z), S(z) denoting a doubly periodic function. Picard, E. C. E. 86 (1878) 657-. , representation by arcs of curves. Allegret, [A]. Par. EC. Norm. A. 2 (1873) 149-. space-coordinates, class. Staude, 0. Acta Mth. 10 (1887) 183-. Transformation formula y = sin (L + A + B + C + ...), some consequences. Griffiths, J. L. Mth. S. P. 17 (1887) 11-. Values of integrals in terms of fc 2 . Glaisher, J. W. L. Mess. Mth. 22 (1893) 145-. Weierstrass's equation of three terms, generali- sation. Morley, F. N. Y. Am. Mth. S. Bll. 2 (1896) 21-. formula. Picard, E. Bll. Sc. Mth. 24 (1900) 30-. formulae applied to binary quart* cubic. Baker, H. F. QJ. Mth. 24 (1890) 1-. formulae applied to binary quartic and ternary al(x) functions. Cayley, A. Liouv. J. Mth. 7 (1862) 137-. , expansion in powers of variable. Andre, D. Liouv. J. Mth. 5 (1879) 31-. Zeta-function. Glaisher, J. W. L. [1885-86] Camb. Ph. S. P. 5 (1886) 221- ; Mess. Mth. 15 (1886) 92-. , complex, map; a condenser problem. Michell, J. H. Mess. Mth. 23 (1894) 72-. -- , process of squaring. Glaisher, J. W. L. Mess. Mth. 16 (1887) 150-. I fl X Richelot, A. Crelle J. 32 (1846) 213-. ' reduction to elli P tic integrals. lcnf^** f**W)BinosW 10g UJtf' Jo l-fc'sinV etC ' Roberts, W. Liouv. J. Mth. 11 (1846) 343-. y- 1 2.^. , reduction to elliptic integral of tri- J \X gonometric form. Plana, G. Crelle J. 36 (1848) 1-. 2?L + ^=0. Cayley, A. L. Mth. S. P. 8 vz vr (1877) 184-. = , integration formulae. Cayley, A. [1878] Mess. Mth. 8 (1879) 60-. = , Eichelot's integral. Cayley, A. Mess. Mth. 23 (1894) 42-. , reduction to elliptic integral. (l- Cayley, A. Mess. Mth. 11 (1882) 142-. 324 4050 Elliptic Functions, Division, etc. 4050 p dn u, squaring. Ferrers. N. M. Mess. Mth. 16 = G *( U - ) ( u - b )( u ~ c ) ( u ~ tion, new method. Kapteyn, W. Par. EC. Norm. A. 9 (1892) 35-. , expression by logarithms. Dol- the bnia, I. A. Bll. Sc. Mth. 19 (1895) 76-; Eec. Mth. (Moscou) 18 (1896) 108-. 6 [wW(z) - GJ], resolution into factors. Lacour, E. Par. EC. Norm. A. 13 (1896) 415-. 4050 Multiplication, division, transformation of elliptic functions; modular functions. (See also 4440.) Algebra, transcendental, researches. Franchini, P. Mod. S. It. Mm. 16 (1813) 223-; 17 (1815) 262-. Bisection. Igel,B. Crelle J. Mth. 119 (1898) 50-. Class-invariants. Mathews, G. B. [1890] L. Mth. S. P. 21 (1891) 234-. numbers of binary quadratic forms, relations. Hurwitz, A. Mth. A. 25 (1885) 157-. Congruence groups of 6th order. Fricke, E. Mth. A. 29 (1887) 97-. Determinants formed from elliptic functions. Lerch, M. Prag Ceske Ak. Fr. Jos. Hz. (Tfida 2) 2 (1893) Art. 5, 5 pp. ; Fschr. Mth. (1893-94) 790. DIVISION. Abelian relations for. Engel, F. Leip. Mth. Ps. B. 36 (1884) 32-. algebraic equations determining. Kronecker, L. Berl. Ak. Mb. (1875) 498-. of argument. Emmanuel, D. [Bucarest S. Sc. Bll. 6 (1897)] 20-. differential equation satisfied by roots of equations for. Migotti, A. Wien Ak. Sb. 94 (1887) (Ab. 2) 748-. of lemniscate. Liouville, J. C. E. 17 (1843) 635-. . Kohl, E. Wien Ak. Sb. 98 (1890) (Ab. 2o) 364-. . Mathews, G. B. L. Mth. S. P. 27 (1896) 367-. . Eipa, P. G. Mt. 38 (1900) 7-. , equation, irreducibility, etc. Eisenstein, G. Crelle J. 39 (1850) 160-, 224-. , , separation into four factors. Schwering, K. Crelle J. Mth. 110 (1892) 42-; 112 (1893) 37-. , equations, resolution. Schwering, K. Crelle J. Mth. Ill (1893) 170-. periods. Brioschi, F. Mil. I. Lomb. Ed. 26 (1893) 727-; 27 (1894) 186-. , algebraic equations connected with. Palmstrjm, A. Berg. Ms. Aarb. (1897) No. 4, 17pp. , resolvent of algebraic equation deter- mining. Kiepert, L. Gott. Nr. (1885) 257-. of periods by 7, forms of equation determining. Palmstrflm, A. Arch. Mth. Ntvd. 19 (1897) No. 5, 17 pp. , Galois's resolvent. Halphen, G. H. C. E. 108 (1889) 476-. , . Brioschi, F. C. E. 109 (1889) 520-. by 9. Burnside, W. Mess. Mth. 22 (1893) 89-. theta-function. Klein, C. F. Mth. A. 17 (1880) 565-. Doubly periodic functions. Cayley, A. Liouv. J. Mth. 10 (1845) 385- ; 19 (1854) 193-. , rational functions analogous to. Vasil'ev, A. V. (xn) Kazan Un. Mm. [16] (1880) (Pt. 2) 121-. Elliptic formulae of Jacobi, demonstration. Somoff, J. [1851] St Pet. Ac. Sc. Bll. 10 (1852) 65-. functions. Klein, F. D. Nf. B. (*1877) 104. , expansion, method. Schlomilch, O. Leip. B. (1853) 25-. , in series. Broch, 0. J. [1863] Sk. Nf. F. 9 (1865) 146-. .theory. Krause,M. Mth. A. 12 (1877) 1-. , . Eausenberger, 0. Crelle J. Mth. 93 (1882) 328- ; 94 (1883) 251-. , . Weierstrass, C. T. W. Berl. Ak. Sb. (1882) 443- ; (1883) 193-, 265-, 1271-. snnu. Cayley, A. Crelle J. 37 (1848) 58. of IK. Forsyth, A. E. Mess. Mth. 12 (1883) 134-. . Glaisher, J. W. L. Mess. Mth. 20 (1891) 191-. , etc. Burnside, W. Mess. Mth. 12 (1883) 154-. integral, 1st kind, expansion, cases. Catalan, E. C. As. Fr. C. E. 6 (1877) 214-. integrals, 1st and 2nd kinds, expansion in recurring series. Farkas, J. C. E. 92 (1881) 181-. , , series. Hermite, C. A. Mt. 2 (1868-69) 97-. , theory. Newman, F. W. B. S. P. 9 (1857-59) 704-. normal curves and connected modular func- tions. Klein, F. Leip. Mth. Ps. Ab. 13 (1887) 337-. Equation between 2 transcendental functions, demonstration. Enneper, A. QJ. Mth. 1 (1857) 272-. Equations, 5th order, equations of icosahedron in solution of. Klein, C. F. Mil. I. Lomb. Ed. 10 (1877) 253-. Formulae for sn 9w. Sudo, 0. Tok. Coll. Sc. J. 7 (1895) 283-. sn 10u, en lOw, dn 10w in terms of sn u. Sakai, E. Tok. Coll. Sc. J. 7 (1895) 285-. Functions with several periods not interchange- able. Rausenberger, 0. Mth. A. 20 (1882) Gauss's pentagramma mirificum. Schlesinger, L. Mth. Termt. Ets. 17 (1899) 526- ; Mth. Nt. B. Ung. 17 (1901) 20-. 325 4050 Modular Equations and Functions 4050 Hermite's ^-function, linear transformation. Konigsberger, . Mth. A. 3 (1871) 1-. Integral occurring in Cauchy's transformation of elliptic integral. Lerch, M. D. Nf. Vh. (1894) (Th. 2, Halfte 1) 8-. Jacobian functions. Frobenius, G. Crelle J. Mth. 97 (1884) 16-, 188-. , generalisation of formula. Hoppe, R. Arch. Mth. Ps. 70 (1884) 400-. Kleinian functions of 3rd family, generalisation. Cassel, G. Stockh. Ofv. (1890) 237- ; Fschr. Mth. (1890) 410. Landen's substitution. Koldcek, F. Casopis 18 (1889) 21- ; Fschr. Mth. (1889) 495. Linear equation connecting complete function of 1st kind with modulus. Tannery, J. C. B. 86 (1878) 811-. . Goursat, E. Par. S. Mth. Bll. 10 (1882) 44-. Modular correspondences of 7th kind. Fricke, E. Gott. Nr. (1892) 272-. Modular equations, Schroter, H. E. Crelle J. 58 (1861) 378-. Jordan, C. C. E. 66 (1868) 308-. Smith, H. J. S. Bm. B. Ac. Line. Mm. 1 (1877) 136-. Klein, C. F. Mil. I. Lomb. Bd. 12 (1879) 21-. Pick, G. Wien Ak. Sb. 91 (1885) (Ab. 2) 138-. Russell, R. [1890] L. Mth. S. P. 21 (1891) 351-. Brioschi, F. Bm. B. Ac. Line. Bd. 2 (1893) (Sem. 2) 185- ; 5 (1896) {Sem. 2) 333-. Pierpont, J. N. Y. Am. Mth. S. Bll. 3 (1897) 279-. and application to theory of numbers. Krause, M. Mth. A. 12 (1877) 419-. class. Brioschi, F. C. B. 112 (1891) 28-. and curves, singularities. Smith, H. J. S. L. Mth. S. P. 9 (1877-78) 242-. of 8th degree, new form. Brioschi, F. Am. J. Mth. 13 (1891) 381-. , property of reduced form. Brioschi, F. Bm. E. Ac. Line. Bd. 1 (1885) 514-, 583-. , reduction. Hermite, C. Tortolini A. 2 (1859) 59-. 12th degree, resolvent of llth degree. Klein, C. F. Bm. B. Ac. Line. T. 3 (1879) 177-. discriminants. Krause, M. Mth. A. 8 (1875) 539- ; 9 (1876) 554-. equations analogous to. Brioschi, F. C. B. 47 (1858) 337-. . Joubert, P. C. B. 47 (1858) 341-. form in transformation of 3rd order. Cayley, A. Mess. Mth. 12 (1883) 173-. Galois' s group. Gierster, J. Leip. Mth. Ps. B. 37 (1885) 291- ; Mth. A. 26 (1886) 309-. , sub-groups in case of a transformation of prime degree. Gierster, J. Mth. A. 18 (1881) 319-. irrational. Fiedler, E. W. Ziir. Vjschr 30 (1885) 129-. reduction. Betti, E. Tortolini A. 4 (1853) 81-. of degree. Sylow, L. [1870] Christiania F. 13 (1871) 387-. resolution, cases. Morera, G. Mil. I. Lomb. Bd. 18 (1885) 629-. theory. Schroter, H. E. Liouv. J. Mth. 3 (1858) 258-. . Hermite, C. C. B. 48 (1859) 940-, 1079-, 1095- ; 49 (1859) 16-, 110-, 141-. , algebraic proofs. Igel, B. Mh. Mth. Ps. 3 (1892) 349-. K \ - K'\'. Russell, R. [1887] L. Mth. S. P. 19 (1889) 90-. MODULAB FUNCTIONS. Klein, C. F. [1879] Munch. Ak. Sb. 10 (1880) 89-. Rausenberger, 0. Mth. A. 20 (1882) 45-. Wirtinger, W. Mh. Mth. Ps. 1 (1890) 429-. Cayley, A. Mth. A. 47 (1896) 1-. Weber, H. Mth. A. 47 (1896) 6-. Lewicki, W. Prace Mt.-Fiz. 8 (1897) 5-. demonstration of Hermite's transformation tables. Schlqfli,L. Crelle J. 72 (1870) 360-. functions akin to. Stouff, X. Toul. Fac. Sc. A. 5 (1891) C, 16 pp. and Gauss's theory of arithmetico-geometric mean. Schlesinger, L. Berl. Ak. Sb. (1898) 346-. integrals. Gudermann, C. Crelle, J. 18 (1838) 1-, 142-, 220-, 303- ; 19 (1839) 45-, 119-, 244- ; 20 (1840) 62-, 103- ; 21 (1840) 240- ; 23 (1842) 301- ; 25 (1843) 281- ; 41 (1851) 93-; [Latin version with preface and annotations by J. F. Stader] Tortolini A. 2 (1851) 557- ; 3 (1852) 442- ; 5 (1854)81-. invariants, differential equations. Scheibner, W. Leip. Mth. Ps. B. 41 (1889) 331-. of lowest degrees. Klein, F. Leip. Mth. Ps. B. 37 (1885) 70-. and multiplicator equations of 1st degree, foundations of theory. Hurwitz, A. Mth. A. 18 (1881) 528-. representation by infinite products. Mangoldt, H. von. Gott. Nr. (1885) 313-. , and extension to their functions. Mangoldt, H. von. Gott. Nr. (1886) 1-. special subgroups. Fricke, R. Mth. A. 31 (1888) 227-. theory. Dienger, J. Grunert Arch. 11 (1848) 395- ; 13 (1849) 1- ; 21 (1853) 344- ; 22 (1854) 362-. . Dedekind,R. Crelle J. Mth. 83 (1877) 265-. X1. Kupper, K. Prag Sb. (1896) (Mth.-Nt.) No. 43, 11 pp. -- , order 3, genus arbitrary. Kilpper, C. J. Prag Ab. 1 (1886) No. 1, 26 pp. , C n p with special systems g^ } , g^ . Kilpper, K. Prag Ab. 3 (1890) No. 4, 18 pp. -- , CJj with 13 given double points, system. KUpper, C. Prag Sb. (1898) (Mth.-Nt.) No. 1, 7 pp. differential equations, integration. Jacobi, C. G. J. Crelle J. 32 (1846) 220-. --- , . Weierstrass, C. Berl. Mb. (1862) 127-. differentials, integration by logarithms. Pick, G. A. Wien Ak. Sb. 85 (1882) (Ab. 2) 643-. equations, integration. Brioschi, F. Crelle J. 55 (1858) 56-. , invariant. Brun, F. de. Stockh. Ofv. (1897) 173- ; Fschr. Mth. (1897) 400-. >- formulae. Brioschi, F. C. B. 102 (1886) 239-, 297-. modular functions. Burkhardt,H. Mth. A. 36 (1890) 371- ; 38 (1891) 161- ; 41 (1893) 3 13-. multiplicator equation. BurkJiardt, H. Gott. Nr. (1889) 553-. 334 4060 Abelian Integrals Hyperelliptic surfaces, general theory. Hum- bert, G. Liouv. J. Mth. 9 (1893) 29-, 361-. system of general form, explicit expression of algebraic integral. Salvert, F. de. C. E. 116 (1893) 243-. systems of barriers. Thompson, H. D. Am. J. Mth. 15 (1893) 91-. Integral equations, new, of Jacobi's system of differential equations. Richelot, F. J. Crelle J. 25 (1843) 97-. Integrals, class, involving roots of polynomials. Minding, E. F. A. Crelle J. 10 (1833) 195-, 292. of solutions of 2nd order of differential equations, with 3 singular points. Kejainski, S. Krk. Ak. (Mt.-Prz.) Ez. 17 (1900) 112- ; Crc. Ac. Sc. Bll. (1899) 67-. Inversion problem. Jacobi, C. G. J. Crelle J. 32 (1846) 176-. , Jacobi's. Goursat, E. Bll. Sc. Mth. 19 (1895) 24-. , , generalisation. Lindemann, C. L. F. [1878] Freiburg B. 7 (1880) 273-. , __, solution. Stahl, H. Crelle J. Mth. 89 (1880) 170- ; 111 (1893) 98-. Kleinian prime forms and automorphic func- tions. Ritter, E, Mth. A. 44 (1894) 261-. Minding' s integral \dx(\.-a?)~, reduction to elliptic integral of 1st kind. Richelot, F. J. Crelle J. 9 (1832} 407-. Multiply-tangential curves to algebraic curves. Weiss, W. Mh. Mth. Ps. 7 (1896) 370-. Number of periods of uniform function. Astor, A. N. A. Mth. 15 (1896) 227-. functions. MUhll, K. von der. Basel Vh. 9 (1893) 78-. Parameters, representation of algebraic curves by. Enriques, F. Palermo Cir. Mt. Ed. 10 (1896) 30-. Quadruple theta functions, relations satisfied by. Craig, T. Am. J. Mth. 6 (1884) 205-. Eepresentation of function in Eiemann surface, with variable multipliers of periodicity. Suchar, P. I. [Bucarest S. Sc. Bll. 7 (1898)] 143-. Eiemann's integrals of 1st species, canonical form. Christoffel, E. B. A. Mt. 9 (1878- 79) 240-. Theorem in geometry of n dimensions. Segre, C. Mth. A. 30 (1887) 308. Theorems. Abel,N.H. N. A. Mth. 4 (1845) 536-. Theta functions, Jacobian, partial values, and Gauss's posthumous papers. Goring, W. Mth. A. 7 (1874) 310-. with ^-series, characteristics formed of thirds of integers. Braunmilhl, A. von. Erlang. Ps. Md. S. Sb. 18 (1886) 37-; Munch. Ak. Ab. 16 (1888) 325-; Mth. A. 32 (1888) 513-. , nths of integers. Braunmiihl, A. von. Mth. A. 37 (1890) 61-. , theorems. Bock, W. Hamb. Mth. Gs. Mt. 2 (1890) (Festschr., Tl. 2) 74-. Transcendental functions. Abel, N. H. (n Adds.) Norske Vd. Skr. 2 (1824-27) (2d Heft) Periodic Functions 4070 Transcendental f unctionswith algebraic differen- tials, summation. Jilrgensen, C. Kiob. Dn. Vd. Selsk. Afh. 8 (1841) 17-; Crelle J, 19 (1839) 113-. Transcendents, of which hyperelliptic functions are special case. Laguerre, E. C. E. 84 (1877) 643-. Univocal correspondences on elliptic curves (application of Abelian integrals). Patrassi, P. G. Mt. 34 (1896) 195-. hyperelliptic curves. Kantor, S. Pa- lermo Cir. Mt. Ed. 9 (1895) 65-. p ~, integration. Abel, N. H. Crelle J. 1 (1826) 185-. , Abel's theorems, demonstration. Dolb- nia, I. P. Kazan S. Nt. (Ps.- Mth.) P. 6 (1888) 307- ; Fschr. Mth. (1888) 486. U/l + cur 3 . dx. Minding, E. F. A. Crelle J. 9 (1832) 295-. , integration, E, F, \(/, being rational in- tegral functions of same variable. Mainardi, G. Yen. Mm. I. 2 (1845) 401-. Al(x), Al 1 (x), Al% (x), expansion in powers of modulus. Andre, D. Liouv. J. Mth. 5 (1879) 131-. here fx> y=0 and {x> =0> 3 (x, y) and theorem of Jacobi. Forsyth, A. R. QJ. Mth. 18 (1882) 313-. fG(x) dx ^ Guichardt Cm c B> 104 ( 1887 ) J Kj R (x) 1494-. 4070 Periodic functions of several variables ; general theta func- tions. Abelian functions. Poincare, H. C. E. 124 (1897) 1407 -. , addition. Salvert, (le vicomte) de. Brux. S. Sc. A. 17 (1893) (Pt. 1) 70-; 18 (1894) (Pt. 2) 41-. -- , generalisation. Picard, E. C. E. 98 (1884) 665-. -- , integration by. PoincarS, H. C. E. 92 (1881) 913-. -- , 1st order, modular equations for trans- formation of 3rd degree. Konigsberger, . [1868] Mth. A. 1 (1869) 161-. -- with 4 periods. Roberts, M. [1870] A. Mt. 4 (1870-71) 95-. -- and quadratic forms of n differentials. Lipschitz,R. [1871] Crelle J. 74 (1872) 150-. , singular. Humbert, G. C. E. 126 (1898) 508-. , transformation. Humbert, G. C. E. 126 (1898) 814-. -- of 2 variables. Pokrovsky, P. Bll. Sc. Mth. 20 (1896) 86-. ----- , addition theorem. Pokrowsky, P. [1895] D. Mth. Vr. Jbr. 4 (1897) 137-. 335 4070 General Periodic and Theta Functions 4070 Abelian functions of 2 variables, complex multiplication. Wiltheiss, E. E. Mth. A. 21 (1883) 385-. . , differential equations for periods. Brioschi, F. Em. B. Ac. Line. Ed. 4 (1888) (Sem. 2) 341-, 429-. , expression in series. Brioschi, F. Em. E. Ac. Line. Ed. 2 (1886) (Sem. 1) 199-, 215-. , modular equations in trans- formation of 3rd order. Brioschi, . Em. E. Ac. Line. Ed. 1 (1885) 769-. , new moduli for. Brioschi, F. Em. E. Ac. Line. Ed. 2 (1886) (Sem. 1) 159-. , transformation. Wiltheiss, . Crelle J. Mth. 96 (1884) 17-. 4 variables. Schottky, F. Crelle J. Mth. 102 (1888) 304-. any number of variables. Klein, F. Gott. Nr. (1887) 515-. a- functions. Pascal, E. A. Mt. 17 (1889-90) 81-, 197- ; 18 (1890) 1-. , term in expansion. Schroder, J. [1891] Hamb. Mth. Gs. Mt. 3 (1900) 7-. 6 functions of 3 variables, partial differential equations. Wiltheiss, E. Gott. Nr. (1889) 381- ; Mth. A. 38 (1891) 1-. integrals, reduction. Poincare, H. C. E. 102 (1886) 915-. Algebraic surfaces, class. Humbert, G. [1889] Palermo Cir. Mt. Ed. 4 (1890) 54-. Characteristics, |>-rowed, theorem. Moore, E. H. N. Y. Am. Mth. S. Bll. 1 (1895) 252-. Convergence of Jacobi's 6 series with Eiemann's moduli. Christoffel, E. B. Ziir. Vjschr. 41 (1896) (Festschr., Th. 2) 3-. Differential equations, integrals of which are Abelian functions of 1st kind, two systems. Caspary,F. C. E. 112 (1891) 1305-. of theta functions, formation. Caspary, F. C. E. 112 (1891) 1120-. Doubly periodic functions, uniform, behaviour at essential singularities. Appell, P. C. E. 94 (1882) 936-. Elliptic Abelian functions of 4 variables. Schott* ky, F. Crelle J. Mth. 108 (1891) 147-, 193-. Function, single- valued, of n variables cannot have more than 2n periods. Riemann, [G. F.-] B. Crelle J. 71 (1870) 197-. Functions of 3rd kind. Tichomandrickij, M. A. Kharkov Mth. S. Com. 5 (1896?) 182- ; Fschr. Mth. (1896) 360. with 4 pairs of periods. Humbert, G. C. E. 130 (1900) 483-. 2 real periods, class. Staude, 0. Mth. A. 29 (1887) 468-. of 2 variables analogous to modular func- tions. Picard, E. Acta Mth. 2 (1883) 114-. Weierstrass's elliptic func- tions. Pokrovskij, P. M. Bee. Mth. (Moscou) 18 (1896) 598-; Fschr. Mth. (1896) 360. with 3 or 4 pairs of periods. Appell, P. C. B. 90 (1880) 174-. several pairs of periods. Appell, P. C. E. 110 (1890) 181-. and 4 periods, inverses of Abelian integrals of 1st class. Rosenhain, G. [1846] Par. Mm. Sav. Etr. 11 (1851) 361-. Functions of n variables admitting 2n systems of periods, Eiemann's theorem. Picard, E., & Poincare, H. C. E. 97 (1883) 1284-. Hyperabelian functions and surfaces represent- ablebythem. Humbert, G. C. B. 129 (1899) 667. Inversion problem. Elliot, Z. C. E. 90 (1880) 1466-. , Jacobi's. Noether, M. Erlang. Ps. Md. S. Sb. 16 (1884) 88-. Jacobian functions of fcth order of 2 variables. Witting, A. Mth. A. 29 (1887) 157-. 3 variables. Frobenius, G. Crelle J. Mth. 105 (1889) 35-. several variables. Weierstrass, C. T. W. Bll. Sc. Mth. As. 6 (1882) (Pt. 1) 136-. Lame's differential equation, generalisation. GrSnwall, H. Stockh. Ofv. (1898) 357-. Linear equations integrable with, the help of X m (x, y). Appell, P. Par. EC. Norm. A. 5 (1888) 211-. Periodic functions, 2w-ply. Wirtinger, W. Mh. Mth. Ps. 6 (1895) 69- ; 7 (1896) 1-. , , application to total differential equa- tions. Gronwall, H. Stockh. Ofv. (1896) 295-; Fschr. Mth. (1896) 285-. , , of r variables. Weierstrass, C. T. W. Crelle J. Mth. 89 (1880) 1-. , special 4-ply. Roch, G. Z. Mth. Ps. 11 (1866) 463-. of 2 variables. Appell, P. C. B. Ill (1890) 636- ; Liouv. J. Mth. 7 (1891) 157-. , n variables. Puiseux, V. C. E. 43 (1856) 321-. , , demonstration of a leading theorem. Weierstrass, C. T. W. Berl. Ak. Mb. (1876) 680-. , , most general uniform and 2n- ply. Weierstrass, C. Berl. Mb. (1869) 853-. Periodicity, general, theory. Rausenberger, 0. Mth. A. 18 (1881) 379-. Periods of double integrals. Picard, E. C. B. 102 (1886) 349-. . Poincare, H. C. B. 125 (1897) 995-. of algebraic functions. Picard, E. C. B. 125 (1897) 1068-. , calculation. Picard, E. C. B. 102 (1886) 410-. and two-dimensional cycles on algebraic surfaces. Picard, E. C. B. 126 (1898) 1457-. uniform 2n-ply periodic functions. Hur- witz,A. [1882] Crelle J. Mth. 94 (1883) 1-. Potential, properties ; and Abelian functions. Poincare, H. Acta Mth. 22 (1899) 89-. Quadruply periodic expressions. Picard, E. Par. S. Mth. Bll. 17 (1889) 131-. depending on 2 variables. Picard, E. C. E. 108 (1889) 557-. . Appell, P. C. E. 108 (1889) 607-. functions of 2nd kind, systems of diffe- rential equations satisfied by. Krause, M. C. E. 126 (1898) 1086-, 1489-, 1618- ; 127 (1898) 91- ; A. Mt. 1 (1898) 265-. 336 4070 Multiperiodic Functions. General Theta Functions 4070 Quadruply periodic functions of 2nd and 3rd kinds. Krause, M. Liouv. J. Mth. 3 (1887) 87-. satisfying systems of differential equations. Krause, M. N. Y. Am. Mth. S. T. 1 (1900) 287-. of 2 variables. Picard, E. C. B. 124 (1897) 1490-. , of 3rd kind. Appell, P. Par. EC. Norm. A. 7 (1890) 143-. series. Picard, E. C. B. 108 (1889) 659-. Eosenhain's fundamental formulas, simple de- monstration. Caspary, F. Mth. A. 30 (1887) 571-. Sigma functions of 2 variables, expansion in series. Brioschi, F. Gott. Nr. (1890) 236-. Theta characteristics. Noether, M. Erlang. Ps. Md. S. Sb. 11 (1879) 198-. , demonstration of theorem of Biemann's. Stahl, H. Crelle J. Mth. 88 (1880) 273-. constants, 2 variables, certain relations among. Hutchinson, J. I. N. Y. Am. Mth. S. T. 1 (1900) 391-. formula, general, derivation. Prym, F. E. Acta Mth. 3 (1883) 216-. , Biemann's, demonstration. Prym, F. E. [1882] Acta Mth. 3 (1883) 201-. , , derivation. Prym, F. E. Crelle J. Mth. 93 (1882) 124-. , , generalisation. Krazer, C. A. J., & Prym, F. E. Acta Mth. 3 (1883) 240-. function, infinite variety of forms. Weber, H. Crelle J. 74 (1872) 57-. functions. Enneper, A. Gott. Nr. (1883) 175-, 261-. , characteristics. Jordan, C. C. E. 88 (1879) 1020-, 1068- ; Par. EC. Pol. J. Cah. 46 (1879) 35-. , of which are formed from thirds of integers. Krazer, C. A. J. Mth. A. 22 (1883) 416-. , double. Cayley, A. [1877-78] Crelle J. Mth. 83 (1877) 210-, 220- ; 85 (1878) 214- ; 87 (1879) 74-. ,, addition. Cayley, A. [1879] Crelle J. Mth. 88 (1880) 74-. Gott. J. Jena. Z. 20 (1887) 581-. , , transformation. Cayley, A. QJ. Mth. 21 (1886) 142-. , equation between. Enneper, A. Nr. (1878) 550-. , equations for 6 (0, 0, ... 0). Fuchs, L. Crelle J. 73 (1871) 324-. Theta functions with multiple arguments. Eoch, G. Crelle J. 66 (1866) 177-. , multiple, theorem. Cayley, A. Mth. A. 16 (1880) 115-. , multiplication of four, formula. Smith, H. J. S. L. Mth. S. P. 1 (1866) No. 8, 14 pp. , particularly those of 2 variables. Forsyth, A. E. [1881] Phil. Trans. 173 (1883) 783-. , quadruple. Craig, T. Am. J. Mth. 6 (1884) 14-, 183-. , single and double. Cayley, A. [1879] Phil. Trans. 171 (1880) 897-. , theorems, generalisation of two. Elliot, Z. C. E. 90 (1880) 352-. , transformation. Frobenius, G. [1879] Crelle J. Mth. 89 (1880) 40-. , theory, especially for 3 variables. Weber, H. A. Mt. 9 (1878-79) 126-. , triple. Cayley, A. [1878] Crelle J. Mth. 87 (1879) 134-, 165-, 190-. , vanishing. Neumann, C. G. Leip. Mth. Ps. B. 35 (1883) 99-. of 2 variables. Caspary, F. [1881] Crelle J. Mth. 94 (1883) 74-. . Darboux, G. C. E. 92 (1881) 685-, 1493-. . Schleiermacher, L. Erlang. Ps. Md. S. Sb. 18 (1886) 22-. . and corresponding Kummer'a surface. Schleiermacher, L. Mth. A. 50 (1898) 183-. , derivatives in terms of squares of theta functions. Jahnke, E. C. B. 126 (1898) 1083-. . , differential relations for. Jahnke, E. Leip. Mth. Ps. B. 52 (1900) 140-. , connected with. Krause, M. Mth. A. 26 (1886) 1- ; A. Mt. 15 (1887- 88) 173-. , expressions for elements of orthogonal system. Jahnke, E. C. B. 126 (1898) 1013-. , Fourier's series in domain of. Krause, M. Mth. A. 27 (1886) 419-. and generalised Kummer's surface. Wirtinger, W. Mh. Mth. Ps. 1 (1890) 113-. , orthogonal system formed from. Jahnke, E. Berl. Ak. Sb. (1896) 1023- ; Crelle J. Mth. 119 (1898) 234-. , systems for derivatives. Jahnke, E. C. E. 125 (1897) 486-. , reduction to theta functions of one variable. Appell, P. C. E. 94 (1882) 421-. , form of arguments, and determination '_ ^ relations. Staude, O. Mth. of (0, . . . ) as function of class moduli. Fuchs, L. Crelle J. 73 (1871) 305-. , function analogous to. Appell, P. Mars. Fac. Sc. A. 1 (1892) 47-. , general. Noether, M. [1879] Erlang. Ps. Md. S. Sb. 12 (1880) 1-. , . Wirtinger, W. Wien Az. 32 (1895) 58-. on a general Eiemann's surface. Klein, F. C. E. 108 (1889) 277-. A. 24 (1884) 281-. , . Igel, B. Mh. Mth. Ps. 2 (1891) 157-. , between the 16. Schottky, F. Crelle J. Mth. 105 (1889) 233-. , transformation. Krause, M. Mth. A. 25 (1885) 323-. , , application to arithmetico- geometric mean of 4 elements. Hettner, G. Crelle J. Mth. 112 (1893) 89-. VOL. I. 337 4400 Other Special Functions. General Theta functions of 3 variables. Borchardt, C. W. [1878] Crelle J. Mth. 87 (1879) 169-. ' . Krause, M. C. K. 131 (1900) 1188-. , algebraic investigations. Schottky, F. Crelle J. Mth. 105 (1889) 269-. 4 variables. Noether, M. Erlang. Ps. Md. S. Sb. 10 (1878) 87-; Mth. A. 14 (1879) 248-. n variables. Noether, M. Mth. A. 16 (1880) 270-. . Poincare", H. Par. S. Mth. Bll. 11 (1883) 129-. . Frobenius, G. Crelle J. Mth. 96 (1884) 100-. , addition theorem. Frobenius, G. Crelle J. Mth. 89 (1880) 185-. , . Caspary, F. Crelle J. Mth. 97 (1884) 165-. , definition. Schottky, F. Crelle J. Mth. 107 (1891) 117-. , formation of orthogonal systems by. Caspary, F. C. R. 104 (1887) 490-. , reduction to theta functions of fewer variables. Appell, P. Par. S. Mth. Bll. 10 (1882) 59-. , theory. Caspary, F. Crelle J. Mth. 96 (1884) 324-. , transformation, principal. Frobenius, G. Crelle J. Mth. 95 (1883) 264-. , vanishing. Tichomandritzky, M. A. [1900] N. Y. Am. Mth. S. Bll. 7 (1901) 64-. p variables. Dalwigk, F. von. Ac. Nt. C. N. Acta 57 (1892) 221-. , addition theorem. Stahl, H. Crelle J. Mth. 88 (1880) 117-. series, convergence. Thomae, C. J. Z. Mth. Ps. 25 (1880) 43-. , ra-tuple, infinite, theory. Meissel, E. Crelle J. 48 (1854) 324-. Uniform functions with several non-inter- changeable periods. Rausenberger, 0. Mth. A. 20 (1882) 187- ; 21 (1883) 59- ; 25 (1885) d log 0(0, 0, ... 0), determination by class- moduli. Thomae, J. [1865] Crelle J. 66 (1866) 92-. (0, 0, ... 0), determination by class-moduli of algebraic functions. Thomae, J. [1869] Crelle J. 71 (1870) 201-. (x, y,z) = ,'S e w*+*w*+ zeros . Rivereau, n U'~albe) . Mars. Fac. Sc. A. 2 (1892) 59-. p (Mi + M 2 +...2n)> expression by means of a Pfaffian. Fontene, G. Par. EC. Norm. A. 13 (1896) 469-. Other special functions. 4400 General. Bernoullians, Baabe's. Dickson, J. D. H. [1888] L. Mth. S. P. 20 (1889) 14-. Bernoulli's function. Hermite, C. [18741 Crelle J. Mth. 79 (1875) 339-. 4400 Bernoulli's function. Glaisher, J. W. L. QJ. Mth. 29 (1898) 1-. , Baabe's monograph. Saalschiitz, L. Z. Mth. Ps. 42 (1897) 1-. , and its use in semi-convergent series. Schlomilch, O. Schlomilch Z. 1 (1856) 193-. functions. Imschenetsky , V. G. G. Mt. 9 (1871) 87-. . SchrSder, . D. Nf. Vh. (1890) (Th. 2) 5-. , application to series. Berger,A. Stockh. Ofv. (1891) 523- ; Fschr. Mth. (1891) 274. with arbitrary indices. Sincov, D. Kazan S. Ps.-Mth. Bll. 1 (1891) 234-; Fschr. Mth. (1892) 404-. , class of functions related to. Beaupain, J. Brux. Mm. Cour. 4, 59 (1901) (Pt. 1) 33 pp. , connection with Biemann's surface. Jonquiere, A. [1890] Stockh. Ak. Hndl. Bh. 16 (Afd. 1) (1891) No. 6, 28 pp. , definitions. Renfer, H. Bern Mt. (1900) 1-. , expansion in. Rogel, F. Arch. Mth. Ps. 17 (1900) 129- . , generalisation. Imshenetsky, V. G. St Pet. Ac. Sc. Mm. 31 (1883) No. 11, 58 pp. and numbers. Berger, A. Stockh. Ak. Handl. Bh. 13 (Afd. 1) (1888) No. 9, 43 pp. ; Stockh. Ofv. (1888) 433-; Acta Mth. 14 (1890-91) 249-. , resolution into partial fractions. Worpitzky, J. Z. Mth. Ps. 29 (1884) 45-. polynomial. Plainer, G. Mil. I. Lomb. Ed. 25 (1892) 1179-. polynomials. Sonin, , & Hermite, . Crelle J. Mth. 116 (1896) 133-. , approximate values. Appell, P. N. A. Mth. 6 (1887) 547-. and numbers. Pietrocola, C. G. Mt. 34 (1896) 48-. Functional operations represented by definite integrals. Pincherle, S. Acta Mth. 10 (1887) 153-. Functions allied to trigonometric functions. Knar, J. Grunert Arch. 27 (1856) 365-. analogous to algebraic functions, property. Picard, E. C. B. 89 (1879) 1106-. functional determinants. Rosanes, J. [1872] Crelle J. 75 (1873) ^166-. with lacunae. Goursat, E. Bll. Sc. Mth. 11 (1887) 109-. . Poincare; H. Am. J. Mth. 14 (1892) 201-. . Cayley,. QJ. Mth. 26 (1893) 279-. . Krygowski, Z. Par. S. Mth. Bll. 25 (1897) 240- ; PraceMt.-Fiz. 9 (1898) 213- ; Fschr. Mth. (1898) 352. . Pscheborsky, A. P. Fschr. Mth. (1897) 357. , case. Goursat, E. Bll. Sc. Mth. 17 (1893) 247-. , Goursat's theorem. Pszeborsky, A. P. Fschr. Mth. (1897) 356-. lacunary circle. Gomes Teixeira, F. Bll. Sc. Mth. 17 (1893) 29-. 338 4400 Other Special Functions. General Eulerian Functions 4410 Functions representable by Heine's series. Thomae, J. [1868] Crelle J. 70 (1869) 258-. of real variables possessing distributive property. 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Crelle J. 11 (1834) 193-. Prym's function, generalisation. Nielsen, N. N. Ts. Mth. 11 (B) (1900) 73-; Fschr. Mth. (1900) 440-. Pseudo- and hyper-Bernoullian functions and their applications. Lang champs, G. de. Brux. Mm. Cour. 4, 52 (1890-93) No. 1, 41 pp. n fn\ Eecurring function c n S (-!)"( ) c . o vy Lerch, M. Casopis 22 (1893) 31-; Fschr. Mth. (1893-94) 409-. Eepeated exponentials. Grave, D. N. A. Mth. 17 (1898) 80-. Eiemann's f(s) function, generalisation. Mellin, H. Helsingf. Acta 24 (1899) No. 10, 50pp. for odd values of argument. Lerch, M. G. Teix. J. Sc. 14 (1900) 65-. Theorem of Stieltjes, and functions denned by continued fractions. Koch, H. von. Par. S. Mth. Bll. 23 (1895) 33-. Transcendental function, new. Soldner, J. Zach M. Cor. 23 (1811) 182-. , , in relation with functions T and Z. Blaserna, P. Em. E. Ac. Line. Mm. 1 (1894) 499-. functions. Broch, 0. J. Crelle J. 20 (1840) 178- ; N. Mg. Ntvd. 2 (1840) 297-. with algebraic differentials. Hermite, C. C. E. 18 (1844) 1133-. Transcendental functions, arithmetical pro- perty. Hurwitz, A. Mth. A. 32 (1888) 583-. -, case. Holder, O. Gott. Nr. (1886) 514-. , classification ; and general^, theory of integration. Drach, J. Par. EC. Norm. A. 15 (1898) 243-. , comparison. Boole, G. Phil. Trans. (1857) 745-. , theorem of Sturm. Liouville, J. Liouv. J. Mth. 1 (1836) 269-. , Z' (a) and Ga, and calculation of their values for several values of a. Lindman, C. F. [1863] Stockh. Ak. Hndl. 5 (1866) No. 2, 18 pp. quantities. Liouville, J. Liouv. J. Mth. 16 (1851) 133-. Transcendentally transcendental functions. Moore, E. H. Mth. A. 48 (1897) 49-. Trigonometric null-representations, special class. Rogel, F. Arch. Mth. Ps. 15 (1897) 431-. (x'~v p ~ 1 F(x)(R(xf))*vdx. Broch, O.J. Crelle J. 23 (1842) 145-, 201-. a a ' , development. Eisenstein, G. Crelle J. 28 (1844) 49-. and corresponding invert functions. Woepcke, F. Crelle J. 42 (1851) 83-. z*. Le Cointe, J. L. A. (vm) A. Mt. 5 (1863) 106-. 2e zknix l(w + k)*, k = Q to oo . Lerch, M. Acta Mth. 11 (1887-88) 19-. (f n (x). Gegenbauer, L. Wien Ak. Sb. 97 (1889) (Ab. 2a) 259-. , addition theorem. Gegenbauer, L. Wien Ak. Sb. 102 (1893) (Ab. 2a) 942-. y defined by 2a rs x r y a = Q. Melander, K. Fschr. Mth. (1890) 409. S ( . h ( - w) m - cot ) , new form of \sinrz . Loria, G. Prag Sb. (1897) (Mth.-Nt.) No. 18, 6 pp. tables. Bellavitis, G. [1873] Ven. I. Mm. 18 (1874) 125-. theorem, demonstration. Hoppe, E. Arch. Mth. Ps. 41 (1864) 65-. theorems. Catalan, E. Brux. Ac. Bll. 22 (1891) 459-. theory. Schaar, M. Brux. Mm. Cour. 4, 22 (1846-47) 16 pp. . Stern, M. A. Gott. Stud. (1847) 283-. . Pochhammer, L. Mth A. 35 (1890) 495-. , explanation. PreobraZenskij , P. V. Mosc. S. Sc. Bll. 65 (No. 2) (1890) 55-. , formula, fundamental. Tortolini, B. Tortolini A. 5 (1854) 292-. , formulae. Serret, J. A. Liouv. J. Mth. 8 (1843) 489-. and other uniform functions. Bourguet, L. Acta Mth. 2 (1883) 261-. with unlimited argument. Hankel, H. Z. Mth. Ps. 9 (1864) 1-. Euler's constant. Lerch, M. G. Teix. J. Sc. 13 (1897) 129-. and Binet's function. Catalan, E. C. C. B. 77 (1873) 198- ; Liouv. J. Mth. 1 (1875) 209-. , expressions for. Lerch, M. Prag Sb. (1897) (Mth.-Nt.) No. 42, 5 pp. E, calculation. Shanks, W. [1867-71] E. S. P. 15 (1867) 429-; 16 (1868) 154, 299-; 18 (1870) 49 ; 20 (1872) 29-. sum formula, representation of functions by. Lipschitz, E. Crelle J. 56 (1859) 11-. Factorials, Stirling's theorem 1 . 2 . 3 ... n=j2nTr . n n . e~ n , proof. Glaisher, J. W. L. [1877] QJ. Mth. 15 (1878) 57-. Formula of Eaabe, on I log F(x + u) dx. Lerch, M. G. Mt. 26 (1888) 39-. Gamma functions. Enneper, A. Schlomilch Z. 7 (1862) 189-. Mellin, H. Stockh. Ofv. 40 (1883) No. 5, 3-. Brunei, G. Bordeaux S. Sc. Mm. 3 (1887) 1-. Mansion, P. Brux. S. Sc. A. 17 (1893) (Pt. 1) 4-. Baker, H. F. Mess. Mth. 25 (1896) 125-. 340 4410 Gamma Functions 4410 Lerch, M. Prag Cesk6 Ak. Fr. Jos. Ez. (TKda 2) 5 (1896) Art. 14, 37 pp. ; Fschr. Mth. (1896) 333. analogues to. Alekseevskij, V. P. Kharkov Mth. S. Com. 1 (1889) 169- ; Fschr. Mth. (1890) 439-; Leip. Mth. Ps. B. 46 (1894) 268-. and analytical factorials. Hoppe, E. [1845] Crelle J. 40 (1850) 152-. with arbitrary parameter. Bigler, . Crelle J. Mth. 102 (1888) 237-. and beta functions, relations between. Spitzer, S. D. Nf. Tbl. (*1876) (Beil.) 63. , series connected with. Lerch, M. G. Teix. J. Sc. 10 (1891) 103-. of a complex. Sharpe, H. J. Mess. Mth. 13 (1884) 111-. deduced from S(ic + n)-*. Lerch, M. Prag Ceske Ak. Fr. Jos. Ez. (Tfida 2) 8 (1899) Art. 35, 9 pp. ; Fschr. Mth. (1899) 392. and definite integrals. Matthiessen, L. Z. Mth. Ps. 12 (1867) 302-. derivatives. Jeffery, H. M. [1862] QJ. Mth. 6 (1863) 82-. double. Barnes, E. W. [1900] Phil. Trans. (A) 196 (1901) 265-. , genesis. Barnes, E. W. L. Mth. S. P. 31 (1900) 358-. evaluation of infinite factorials by. Weyr, E. Casopis 22 (1893) 161- ; Fschr. Mth. (1893- 94) 750. expansion embracing series of Stirling and Kummer. Landsberg, G. Brux. Mm. Cour. 8, 55 (1896-98) No. 4, 28 pp. and integrals depending on it. Gilbert, L. P. [1873] Brux. Ac. Mm. 41 (Pt. 1) (1875) (No. 3) 60 pp. expression of definite integrals by. Graf, J. H. Bern Mt. (1884) (Heft 2, Ab.) 46-. expressions in terms of complete elliptic integrals. Glaislier, J. W. L. Mess. Mth. 24 (1895) 27-. and factorials. Ohm, M. Crelle J. 36 (1848) 277-. figurate numbers, binomial coefficients in relation to. Matthiessen, . D. Nf. Tbl. (*1868) 128-. formula. Lerch, M. Mh. Mth. Ps. 8 (1897) 187-. _ J" <(> (x + a) a*' 1 do = (- If I* (x)dx*. Liouville, J. Crelle J. 12 (1834) 273-. formulae. Genocchi, A. Brux. Ac. Bll. 21 (1854) 84-. . Lindhagen, A. Stockh. Ofv. 40 (1883) No. 2, 27-. function related to. Schlomilch, O. X. Z. Mth. Ps. 25 (1880) 335-. Gauss's expression for. Arndt, F. Grunert Arch. 10 (1847) 250-. T(M). Zehfuss, G. 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Nr. 129 (1892) 281- ; 130(1892) 363-. expansion of function with 2 variables in. Cantor, G. [1871] Mth. A. 5 (1872) 133-. -- any functions by. Neumann, C. Crelle J. 67 (1867) 310-. in series by. Konig, J. Mth. A. 17 (1880) 85. expression of function in terms of. Sheppard, W. E. QJ. Mth. 23 (1889) 223-. ' in semi-convergent series for large values of variable. Callandreau, O. Bll. Sc. Mth. 14 (1890) 110-. formula, demonstration. Giuliani, G. G. Mt. 22 (1884) 201-. involving. Graf. J. H. Sch. Nf. Gs. Vh. (1896) 59-. function related to. Lommel, E. C. J. [1875] Mth. A. 9 (1876) 425-. integration of equation x m +* 2))t+i =t y = by. Lommel, E. Mth. A. 2 (1870) 624-. formulas. Lerch, M. Prag Ceske Ak. Fr. Jos. Rz. (Tfida 2) 5 (1896) Art. 23, 16 pp. ; Fschr. Mth. (1896) 233-. integration formulas connected with. Gegen- bauer, L. Mh. Mth. Ps. 5 (1894) 53-. 1st kind. Wagner, C. Bern. Mt. (1894) 204-. . Gegenbauer, L. Mh. Mth. Ps. 10 (1899) 189-. , addition theorem, generalisation. Wendt, C. Mh. Mth. Ps. 11 (1900) 125-. and a continued fraction. Graf, I. H. A. Mt. 23 (1895) 45-. , roots, new properties. Gegenbauer, L. Liege S. Sc. Mm. 2 (1900) No. 3, 7pp. and 2nd kinds. Nicolas, J. Par. EC. Norm. A. 11 (1882) (Suppl.), 90 pp. , descending series for. McMahon, J. A. Mth. 8 (1893-94) 57-. 2nd kind. Gegenbauer, L. Wien Sb. 65 (1872) (Ab. 2) 33-; 66 (1872) (Ab. 2) 220-. . Bocher, M. A. Mth. 6 (1891-92) 85-. . Julius, V. A. Arch. Neerl. 28 (1895) 221-. . Crelier, L. Bern. Mt. (1897) 61-. . Otti, H. Bern Mt. (1898) 1-. Neumann's researches. Schlafli, L. [1870] Mth. A. 3 (1871) 134-. 2nd order. Coates, C. V. QJ. Mth. 20 (1885) 250- ; 21 (1886) 183-. Oth order, roots. Bocher, M. N. Y. Am. Mth. S. Bll. 5 (1899) 385-. and P-functions, roots. Van Vleck, E. B. Am. J. Mth. 19 (1897) 75-. product J m (x)J n (x). Orr, W. McF. [1899] Camb. Ph. S. P. 10 (1900) 93-. products and squares. Neumann, C. Mth. A. 2 (1870) 192. properties. Hobson, E. W. L. Mth. S. P. 28 (1897) 370-. derived from continued fractions. Crelier, L. A. Mt. 24 (1896) 131-. with purely imaginary index. Bocher, M. A. Mth. 6 (1891-92) 137-. and related functions, roots. McMahon, J. A. Mth. 9 (1894-95) 23-. representation of arbitrary functions by. Weber, H. [1872] Mth. A. 6 (1873) 146-. by definite integrals. Gubler, E. Ziir. Vjschr. 33 (1888) 130-. of definite integrals by. Wagner, C. Bern Mt. (1895) 115- ; (1896) 53-. roots. Bocher, M. N. Y. Am. Mth. S. Bll. 3 (1897) 205-. series, expression for a rational polynomial in. McMahon, J. Am. As. P. (1900) 42-. solution of some differential equations by. Pearson, K. Mess. Mth. 9 (1880) 127-. tables. Brit. Ass. Comm. B. A. Ep. (1889) 28-; (1893) 227-; (1896) 98-. theory. Lommel, E. C. J. [1870-79] Mth. A. 3 (1871) 475-; 4 (1871) 103- ; 14 (1879) 510-; 16 (1880) 183-. _. Weber, H. Mth. A. 37 (1890) 404-. . Meissel, E. As. Nr. 128 (1891) 435-. , deduction of formulas. Ball, L. de. As. Nr. 128 (1891) 1-. zero. Eudzki, M. P. Prace Mt.-Fiz. 3 69-; Fschr. Mth. (1892) 107-. 343 4420 Bessel's Functions, etc. transformation of origin of reference. Walker, G. T. Mess. Mth. 25 (1896) 76-. zeros. Hurwitz, A. Mth. A. 33 (1889) 246-. . Macdonald, H. M. L. Mth. S. P. 29 (1898) 575-; 30 (1899) 165-. . Schafheitlin, P. Crelle J. Mth. 122 (1900) 299-. _ Bessel's and hypergeometric series, differential equations for. Schafheitlin, P. Crelle J. Mth. 114 (1895) 31-. Catalan's theorem on roots of generalised. Gegenbauer, L. Wien Ak. Sb. 94 (1887) (Ab. 2) 801-. Circular opening, intensity relationships of diffraction phenomenon. Steiner, L. Mth. Termt. Ets. 12 (1894) 44-; Mth. Nt. B. Ung. 11 (1894) 362-. Cylindrical functions. Ermakof, V. Mth. A. 5 (1872) 639-. Giinther, S. Arch. Mth. Ps. 56 (1874) 292-. Beltrami, E. Tor. Ac. Sc. At. 16 (1880) 201-. Sonine, N. J. Mth. A. 16 (1880) 1- ; 30 (1887) 582-. analogues to. Giuliani, G. G. Mt. 25 (1887) 198-. application of theorem of Jacobi's to. Schlafti, L. [1871] A. Mt. 5 (1871-73) 199-. definite integrals with. Hankel, H. [1869] Mth. A. 8 (1875) 453-. expansions in terms of. Nielsen, N. N. Ts. Mth. 9 (B) (1898) 73-; 10 (B) (1899) 73-; Fschr. Mth. (1898) 361- ; (1899) 374-. history. Maggi, G. A. Em. E. Ac. Line. T. 4 (1880) 259-. 1st kind, theorems. Marcolongo, R. Nap. Ed. 28 (1889) 96-. and 2nd kinds. Hankel, H. [1868] Mth. A. 1 (1869) 467-. 2nd kind, for small arguments. Morton, W. B. [1900] Nt. 63 (1900-01) 29. product of two. Nielsen, N. Mth. A. 52 (1899) 228-. representation of a function of two variables by. Mehler, F. G. (1871] Mth. A. 5 (1872) 135-. _ Differential equation of functions of elliptic cylinder. Lindemann, C. L. F. Mth. A. 22 (1883) 117-. equations for series F (p, a ; x) , F (p, a, r ; x) . Pochhammer, L. Mth. A. 41 (1893) 197-. Discontinuous integral. Gubler, E. Mth. A. 48 (1897) 37-. Expansion of potential in morphometry of earth's surface. Kovesligethy, R. Mth. Termt. Ets. 13 (1895) 167- ; Mth. Nt. B. Ung. 13 (1897) 365-. Expansions in series of X n functions of func- tions of one variable, uniformity. Dim, U. [1873] A. Mt. 6 (1875) 216-. JF-series, generalised, reduction of differential equation. Pochhammer, L. Crelle J. Mth. 112 (1893) 58-. Hypergeometric Functions 4420 F-series, 3rd order, differential equations. Pochhammer, L. Mth. A. 46 (1895) 584-. , 4th order, differential equations. Poch- hammer, L. Mth. A. 50 (1898) 285-. Fluctuating functions, numerical calculation of roots. Rayleigh, J. W. Strutt (Lord). (xi) L. Mth. S. P. 5 (1873-74) 119-. Fourier-Bessel functions. Heine, E. Crelle J. 69 (1868) 128-. . Kapteyn, W. Par. EC. Norm. A. 10 (1893) 91-. , expansion of function in squares and products of. Neumann, C. Leip. B. 21 (1869) 221-. Functions arising in continued fraction for e * 1 , addition theorem. Gegenbauer, L. Wien Ak. Sb. 85 (1882) (Ab. 2) 491-. from expansion of (1 -2ax + a 2 a 2 )~^~. Escary, . C. E. 86 (1878) 114-, 1451-. "differing as little as possible from zero." Tchebitchef, P. L. Liouv. J. Mth. 19 (1874) 319-. Gaussian function F(a, 1, y, x), expansion as continued fraction. Tiiome, W. L. Crelle J. 66 (1866) 322-. F(a, /3, y, x), differential equation satisfied by. Mathieu, E. L. Liouv. J. Mth. 8 (1882) 357-. quotient j^f fr~^ r > expansion as continued fraction. Thome, W. L. Crelle J. 67 (1867) 299-. Gauss's polynomials X n . Catalan, E. Brux. Ac. Mm. 49 (1890-93) (No. 1) 34 pp. series. Thomae, J. Gott. Nr. (1884) 493-. F (a, |8, y, x), functions representable by, theory. Riemann, B. Gott. Ab. 7 (Mth.) (1856-57) 3-. Geometrical series, 2nd order. Hochheim, A. Magdeb. Nt. Vr. Jbr. u. Ab. (1886) 137- ; (1887) 25-. Harmonics, tesseral, functions of more than 2 variables analogous to. Hill, M. J. M. Camb. Ph. S. T. 13 (1883) 273-. Hypergeometric differential equation. Besso, D. Em. E. Ac. Line. T. 7 (1883) 230. equations. Besso, D. Em. E. Ac. Line. Ed. 3 (1894) (Sem. 2) 393-. equation. Humbert, G. Par. S. Mth. Bll. 8 (1880) 112-. , derivation. Pochhammer, L. Crelle J. 73 (1871) 85-. , special cases. Boyd, J. H. A. Mth. 7 (1892-93) 145-. expansions, notation suitable for. Radau, R. C. E. 98 (1884) 39-. HYPEEGEOMETEIC FUNCTIONS. Pincherle, S. G. Mt. 32 (1893) 209-, 375. algebraic equations soluble by. Lachtin, L. K. [1892] Eec. Mth. (Moscou) 16 (1893) 597-; 17 (1895) 1-; Fschr. Mth. (1893-94) 753-. transformation. Papperitz, E. [1885-86] Mth. A. 27 (1886) 315-. 344 4420 Hypergeometric Functions 4420 analogues to. Goursat, E. Par. EC. Norm. A. 3 (1886) 107- . arising from Gauss's equation. Halphen, G.H. G. E. 92 (1881) 856-. and Bessel's functions, roots. Porter, M. B. Am. J. Mth. 20 (1898) 193-. with branchings (2, 3, 7), (2, 4, 7), arithmetical theory. Fricke, E. Mth. A. 41 (1893) 443-. case. Markov, A. A. [1892] St Pet. Ac. Sc. Mm. 41 (1893) No. 2, 37 pp. last element of which has special value. Hey- mann, W. Z. Mth. Ps. 44 (1899) 280-. expression as definite integrals. Mellin, H. Helsingf. Acta 20 (1895) No. 1, 39 pp.; Fschr. Mth. (1895) 481. extension of Eiemann's problem to functions of 2 variables. Picard, E. [1880] Par. EC. Norm. A. 10 (1881) 305-. . Goursat, E. C. E. 95 (1882) 903-, 1044- ; 96 (1883) 185-. of form pp'p" p.p + l.p'p' + l p".p" + l + l.q'q" + 1.2.q'.q' + l q".q" + l' Thomae, C. J. Crelle J. Mth. 87 (1879) 26-. and gamma functions, application of Cauchy's theorem. Mellin, H. Helsingf. Acta 21 (1896) No. 1, 115pp. generalisation. Pincherle, S. Em. E. Ac. Line. Ed. 4 (1888) (Sem. 1) 694-, 792-. history. Papperitz, E. Dresden Isis Sb. (1889) (Ab.) 61-. 2nd kind and higher order. Beaupain, J. Brux. Mm. Cour. 4, 54 (1896) No. 1, 46 pp., No. 8, 46 pp. higher order. Goursat, E. Par. EC. Norm. A. 12 (1883) 261-, 395-. . LUnikov, A. V. Eec. Mth. (Moscou) 14 (1890) 216- ; Fschr. Mth. (1890) 443. . Beaupain, J. Brux. Mm. Cour. 4, 56 (1897-98) No. 2, 52 pp. nth order. Pochhammer, L. [1869] Crelle J. 71 (1870) 316-. . Fuchs, L. Crelle J. 72 (1870) 255-. simultaneous linear differential equations in- tegrable by. Heymann, W. Z. Mth. Ps. 32 (1887) 176-. with singularity. Eitter, E. Mth. A. 48 (1897) theorem, fundamental. Baraniecki, M. A. Par. T. Nauk Sc. Pam. 8 (*1876) Art. 2, 19 pp. ; Bll. Sc. Mth. As. 1 (*1877) (Pt. 2) 2 variables. Goursat, E. C. E. 95 (1882) 717-. . Appell, P. Liouv. J. Mth. 10 (1884) 407-. , formulas. Appell, P. Liouv. J. Mth. 8 (1882) 173-. several variables. Lauricella, G. Palermo Cir. Mt. Ed. 7 (1893) 111-. Hypergeometric series. Hypergeometric integral, general, fundamental values. Eadicke, A. Z. Mth. Ps. 22 (1877) 87-. integrals, nth order. Pochhammer, L. [1870] Crelle J. 73 (1871) 135-. quantities, calculation. Multedo, . Ge- nova Mm. I. Ligure 1 (1806) 1-; Geneva Mm. Ac. 2 (1809) 230-. Gauss, C. F. Gott. Cm. 2 (1811-13) 46 pp. Rummer, E. E. Crelle J. 15 (1836) 39-, 127-. Cayley, A. Ph. Mg. 16 (1858) 356-. Dronke, A. Schlomilch Z. 8 (1863) 401-. Schlrifli, L. [1870] Mth. A. 3 (1871) 286-. Gegenbauer, L. Arch. Mth. Ps. 55 (1873) 284-. Meissel, E. [1874] Arch. Mth. Ps. 57 (1875) Markoff, A. C. E. 114 (1892) 54-; Mth. A. 40 (1892) 313-. Graf, I. H. G. Mt. 36 (1898) 233-. case. Cayley, A. QJ. Mth. 16 (1879) 268-. class of equations of 6th order soluble by. Besso, D. [1882] Em. E. Ac. Line. Mm. 14 (1883) 30-. in connexion with observation polygons. Pear- son, K. Ph. Mg. 47 (1899) 236-. definite integrals connected with. Eajewski, J. Krk. Ak. (Mt.-Prz.) Ez. 20 (1890) 272-; Crc. Ac. Sc. Bll. (1890) 61-. and definite integrals, theorem. Catalan, E. C. Liege S. Sc. Mm. 15 (1888) No. 1, 46-. definite integrals satisfying differential equa- tion. Torelli, G. Em. S. It. Mm. 7 (1890) No. 4, 18 pp. differential and difference equations integrable by. Heymann, W. Crelle J. Mth. 122 (1900) 164-. equation. Jacobi, C. G. J. Crelle J. 56 (1859) 149-. . Spitzer, S. Crelle J. 57 (1860) 78-. (Spitzer). Borchardt, C. W. (vi Adds.) Crelle J. 57 (1860) 81. . Markov, A. A. Kharkov Mth. S. Com. (1886) 51-, 95-; Mth. A. 28 (1887) 586-; 29 (1887) 247-. . Pochhammer, L. Mth. A. 38 (1891) 586-. , 2nd solution. Johnson, W. W. Mess. Mth. 17 (1888) 35-. equations integrable by. Heymann, W. Z. Mth. Ps. 29 (1884) 144-. satisfied by. Pochhammer, L. Mth. A. 33 (1889) 353-. divergent (or semiconvergent). Orr, W. McF. [1898] Camb. Ph. S. T. 17 (1899) 171-, 283-. elementary treatment. Thomae, C. J. Z. Mth. Ps. 26 (1881) 314- ; 27 (1882) 41-. extension of theorem. Jackson, (Eev.) F. H. L. Mth. S. P. 28 (1897) 475-. finite, discriminant. Hilbert, D. Crelle J. Mth. 103 (1888) 337-. form of square. Clausen, T. Crelle J. 3 (1828) 89-. more general integral representation. Schaf- heitlin, P. Crelle J. Mth. 103 (1888) 89-. Heine's, analogy of series. Lerch, M. [1893] Prag Ceske Ak. Fr. Jos. Ez. (Trida 2) 3 (1894) Art. 5, 10 pp.; Fschr. Mth. (1893-94) 794. integral function equal to product of two. Markov, A. A. Kharkov Mth. S. Com. 3 (1893) 252-; Fschr. Mth. (1892) 408-. 345 4420 Hypergeometric Functions, etc. Legendre's Functions 4420 order, especially 1 + * 2 x + .... Me, J. 1869 Mth. A.' 2 1870 427 and Jacobi's polynomials. Appell, P. C. E. 89 (1879) 31-. 3rd order, special case. Saalschiltz, L. Z. Mth. Ps. 36 (1891) 278-, 321-. higher Thomae, J. [1869] Mth. A. 2 (1870) 427- , relations between. Thomae, J. Leip. Mth. Ps. B. 43 (1891) 459-. orders. Mellin, H. Helsingf. Acta 23 (1897) No. 7, 10pp. with 5 parameters, differential equation. Markov, A. A. St P6t. Ac. Sc. Mm. 5 (1897) No. 5, 23 pp.; Fschr. Mth. (1897) 288-. product of two, theorems. Orr, W. McF. [1898] Camb. Ph. S. T. 17 (1899) 1-. representation by definite integral. Schafheit- lin, P. Mth. A. 30 (1887) 157-. roots. Gegenbauer, L. Mh. Mth. Ps. 2 (1891) 125-. . Porter, M. B. N. Y. Am. Mth. S. Bll. 3 (1897) 274-; 6 (1900) 280-. with two singular points, more general, diffe- rential equation. Pochhammer, L. Crelle J. Mth. 102 (1888) 76-. special case, identities. Blissard, J. [1864] QJ. Mth. 7 (1866) 155-, 223-. theory. Gegenbauer, L. Wien Ak. Sb. 100 (1891) (^6. 2a) 225-. transformation into continued fraction. Muir, T. L. Mth. S. P. 7 (1875-76) 112-. 2 variables. Appell, P. C. B. 90 (1880) 977-. Picard, E. C. B. 104 (1887) 896-. , contiguous functions relative to. Levavasseur, . C. B. 115 (1892) 1255-. , and linear partial differential equations. AppcU, P. C. B. 90 (1880) 296-, 731-. , uniform functions arising from. Goursat, E. C. B. 104 (1887) 893-. zeros. Hurwitz, A. Gott. Nr. (1890) 557-. . Klein, F. Mth. A. 37 (1890) 573-. F (a, /3, 7, x), cases where it is algebraic function of its fourth element. Schwarz, H. A. [1871-72] Sch. Gs. Vh. 54 (1871) 74-; Crelle J. 75 (1873) 292-. F 1 (a, /3, p, 7; x, y), partial differential equations satisfied by. Le Vavasseur, R. Toul. Fac. Sc. A. 7 (1893) F, 205 pp. for x\ Jackson, (Rev.) F. H. Edinb. Mth. S. P. 15 (1897) 90-. Hypergeometric theorems Q=F(-2m-l, -n, -2n, 2) (-2m, -n, -2n,2). Bretschneider, C. A. Grunert Arch. 13 (1849) 223. transcendentals, representation by uniform functions. Papperitz, E. Mth. A. 34 (1889) Hyperspherical functions, and development of any function in series of. Letnikov, A. V. Bee. Mth. (Moscou) 12 (1885) 205-. Icosahedra, regular. Klein, C. F. Erlang. Ps. Md. S. Sb. 9 (1877) 16-, 70-, 179- ; Mth. A. 12 (1877) 503-. Integral I P dx and allied forms between arbi- trary limits. Hargreaves, R. [1897] L. Mth. S. P. 29 (1898) 115-. Kleiber's functions K t and G 4 . Glaisher, J. W. L. Mess. Mth. 22 (1893) 71-. Rummer's equation. Goursat, $. Helsingf. Acta 15 (1888) 45-. , integration. Lindeldf, E. Helsingf. Acta 19 (1893) No. 1, 31 pp. Lame's, Laplace's and Bessel's functions, con- nexion between. Raentzschel, E. Z. Mth. Ps. 31 (1886) 25-. Laplace's and Bessel's functions, relation between. Rayleigh, J. W. Strutt (Lord). L. Mth. S. P. 9 (1877-78) 61-. coefficients, expansion of integral functions in series, of. Cayley, A. Camb. and Dubl. Mth. J. 3 (1848) 120-. , F (/j. w) in series of. Goodwin, H. Camb. Mth. J. 2 (1841) 223-. 6g coefficients. Radau, R. Bll. As. 13 (1896) 5-. equation. Tychsen, C. (xn) Ts. Mth. 6 (1876) 165-. functions, equation, solution. Graves, G. Ir. Ac. P. 6 (1853-54 162-, 186-, 221-. , , . Hamilton, (Sir) W. R. Ir. Ac. P. 6 (1853-54) 181-. Legendre's coefficients. Todhunter, I. B. S. P. 27 (1878) 381-. . Webb, R. R. [1879] Mess. Mth. 9 (1880) 125-. and complete elliptic integral, connexion between. Stuart, G. H. Mess. Mth. 13 (1884) 161-. , product of any two as series of Legendre's coefficients. Adams, J. C. [1877] E. S. P. 27 (1878) 63-. , theorems. Zanotti Bianco, 0. Tor. Ac. Sc. At. 22 (1886-87) 225-; 23 (1887-88) 5. , value for n great. Sharpe, H. J. QJ. Mth. 24 (1890) 383-. equation, solution in a particular case. Forsyth, A. R. Mess. Mth. 16 (1887) 168-. LEGENDBE'S FUNCTIONS. RoucM, E. C. B. 47 (1858) 917-. Laurent, H. Liouv. J. Mth. 1 (1875) 373-. (Laurent.) Heine, H. E. Liouv. J. Mth. 2 (1876) 155-. Catalan, E. C. As. Fr. C. B. 5 (1876) 68-; Brux. Mm. Cour. 8, 31 (1881) (No. 3) 64 pp. ; Brux. Ac. Sc. Mm. 43 (1882) (No. 9) 10 pp.; 44 (1882) (No. 1) 103 pp. ; 46 (1886) (No. 5) 28 pp.; 47 (1889) (No. 1) 12 pp., (No. 4) 12 pp., (No. 5) 24 pp. Laguerre, E. C. E. 91 (1880) 849-. Deruyts, J. Liege S. Sc. Mm. 11 (1885) No. 8, 22 pp. analogues to. Didon, F. Par. EC. Norm. A. 5 (1868) 229-. 346 4420 Legendrian and Hypergeometric Functions 4420 analogues to. Deruyts, J. Liege S. Sc. Mm. 14 (1888) No. 2, 15 pp. calculation for high values of n. Callandreau, O. Bll. Sc. Mth. 15 (1891) 121-. expansion of function in series of functions similar to. Andreev, K. A. St Pet. Ac. Sc. Mm. (Rs.) 51 (1885) 1-; F.schr. Mth. (1885) 500. functions connected with, and derivable from complete elliptic integrals. Kleiber, J. Mess. Mth. 22 (1893) 1-. similar to. CebySev, P. [1869] St Pet. Ac. Sc. Mm. (Es.) 16 (*1870) 131-. . Posse, K. A. Kharkov Mth. S. Com. (1885) 155- ; Fschr. Mth. (1886) 430. generalisation. Hansted, B. [1881] G. Teix. J. Sc. 4 (1883) 53-. arising from development of .am [l-2euc + a 2 ] 2 Escary, . Liouv. J. Mth. 5 (1879) 47-. for n infinite. Giuliani, G. G. Mt. 22 (1884) 236-. and Lamp's functions. Schmit, N. C. Liege Mm. S. Sc. 13 (1858) 289-. new form for. Catalan, E. Par. S. Mth. Bll. 16 (1888) 128-. polynomials analogous to. Didon, F. Par. EC. Norm. A. 7 (1870) 247-. properties, general. Biervliet, A. van. (xn) Brux. S. Sc. A. 7 (1883) (Ft. 1) 68-, (Pt. 2) 402-. , new. Catalan, E. C. Lie"ge S. Sc. Mm. 15 (1888) No. 1, 122-. of series involving. Plarr, G. C. E. 44 (1857) 984-. transformation into definite integral. Silva, J. A. Martins da. G. Teix. J. Sc. 3 (1881) Legendre's polynomials. Hermits, C. G. Teix. J. Sc. 6 (1885) 81-; Palermo Cir. Mt. Ed. 4 (1890) 146-. Stieltjes, T. J. Toul. Fac. Sc. A. 4 (1890) G, 17pp. Hermite, C. Crelle J. Mth. 107 (1891) 80-. Morera, G. Palermo Cir. Mt.Ed. 11 (1897) 176-. asymptotic expression of. Lerch, M. [1891] Prag Ceske Ak. Fr. Jos. Ez. (Tfida 2) 1 (1892) Art. 8, 27-. class of associated polynomials analogous to. Deruyts, J. Liege S. Sc. Mm. 14 (1888) No. 3, 14 pp. generalisation. Clariana Eicart, L. [1892] Barcel. Ac. Bl. 1 (1892-1900) 73-. of formula. Gegenbauer, L. Palermo Cir. Mt. Ed. 12 (1898) 21-. and Hermite' s and Polignac's. Catalan, E. C. Brux. Ac. Mm. 49 (1890-93) (No. 4) 48 pp. limit. Gomes Teixeira, F. Prag Sb. (1886) (Mth.-Nt.) 19-. theorem. Stieltjes, . C. E. 100 (1885) 439-. theory. Hilbert, D. Acta Mth. 18 (1894) 155-. Legendrian coefficient, formulas for rth integral. Glaisher, J. W. L. [1882] Mess. Mth. 12 (1883) 120-. Polygons, area, theorem. Tardy, P. Tortolini A. 3 (1852) 116-. Poly-harmonic functions, i.e. solutions of / d* d 2 \ V^ 2 + d?) ' in simply connected plane space. Almansi, E. Palermo Cir. Mt. Ed. 13 (1899) 225-. Polynomials A^ n of 2 variables analogous to Jacobi's. Appell, P. [1880] Arch. Mth. Ps. 66 (1881) 238-. Pm,n f Didon, generating function. Orlow, G. A. N. A. Mth. 20 (1881) 481-. Eeduction of a certain integral. Zinin, N. N. Eec. Mth. (Moscou) 14 (1890) 549-. Eiemann's P-function, mathematical repre- sentation. Radicke, A. Epm. Mth. 1 (1877) 50-. functions of 2nd order with four branch- points. Heun, K. Mth. A. 33 (1889) 161-. Eodrigues's theorem (formula for / d \i (E) = (*-!) Hudson, W. H. H. [1877] Mess. Mth. 7 (1878) 117. , demonstration. Hogg, R. W. Mess. Mth. 14 (1885) 20. Series of Gauss. Tano, F. G. Mt. 9 (1871) 60-. , generalised form, cases. Glaisher, J. W. L. L. Mth. S. P. 9 (1877-78) 197-. of Heine. Tano, F. G. Mt. 9 (1871) 60-. , elements, threefold symmetry in. Rogers, L. J. L. Mth. S. P. 24 (1893) 171-. , higher. Thomae, J. A. Mt. 4 (1870-71) 105-. , transformation. Rogers, L. J. Mess. Mth. 23 (1894) 28-. (x-l) x(x-T)(x-2) (z-l)(z-2) x(x-l)(x-2)(x-3) z (z - 1) (z - 2) (z - 3) summation. Grunert, J. A. Crelle J. 2 (1827) 358-. -+..., summation. Schlomilch, 0. Grunert Arch. 3 (1843) 442-. F(v, x, k, a) with relation to Euler's formula (also expressed as sum of definite integrals). Lipschitz,R. Crelle J. 54 (1857) 313-. 2 n A n X n . Ascoli, G. A. Mt. 7 (1876) 258-. Solution of Steinthal, A. E. QJ. Mth. 18 (1882) 330-. Spherical harmonics. Betti, E. [1867] A. Mt. (1867-68) 81-. Sylvester, J. J. Ph. Mg. 2 (1876) 291-, 400. Beltrami, E. Mil. I. Lomb. Ed. 20 (1887) 469-. 347 4420 Spherical Harmonics 4420 Frischauf, J. Crelle J. Mth. 107 (1891) 87-. Caspary, F. Crelle J. Mth. 107 (1891) 137- ; Par. S. Mth. Bll. 19 (1891) 11-. Besso, D. G. Teix. J. Sc. 12 (1895) 65-; Fschr. Mth. (1893-94) 818. Gegenbauer, L. Mh. Mth. Ps. 7 (1896) 35-. Rogel, F. Prag Sb. (1898) (Mth.-Nt.) No. 30, 3 pp. Tallqvist, H. Helsingf. Acta 26 (1900) No. 4, 87pp. application. NiemiJller, F. Z. Mth. Ps. 24 (1879) 57-. to Mittag-Leffler's theorem, and to deter- mination of functions with lacunae. Pincherle, S. Bologna Ed. (1883) 100-. of principle of maximum and minimum exponents to determination. Mazing, A. [1891] Kec. Mth. (Moscou) 16 (1893) 173-. to Segner's numbers. Caspary, F. Brux. Mm. Cour. 4, 52 (1890-93) No. 8, 15 pp. of theorem in differentiation. Hobson, E. W. [1892] L. Mth. S. P. 24 (1893) 55-. and associated functions. Basset, A. B. Mess. Mth. 13 (1884) 147-. cylindrical functions. Olbricht, R. Ac. Nt. C. N. Acta 52 (1888) 1-. definite integrals connected with. Frost, P. QJ. Mth. 19 (1883) 242-. as determinants. Heun, K. Gott. Nr. (1881) 104-. . Kronecker, L. Gott. Nr. (1881) 271-. determination of Green's function. Dougall, J. Edinb. Mth. S. P. 18 (1900) 33-. differentiation. Gallop, E. G. [1896] L. Mth. S. P. 28 (1897) 183-. expression for. Dougall, J. Edinb. Mth. S. P. 8 (1890) 81-. extension. Pincherle, S. Bologna Ac. Sc. Mm. 1 (1890) 337-. formula. Fujisawa,R. Mess. Mth. 21 (1892) 40-. functions analogous to. Giuliani, G. G. Mt. 25 (1887) 203-. general form. Dedekind, R. Ziir. Vjschr. 4 (1859) 346-; 5 (1860) 66-. generalisation. Pincherle, S. Bologna Ed. (1891-92) 31-. identities in connexion with. Vivanti, G. Z. Mth. Ps. 36 (1891) 1-. any index, expression and expansion. Lyetnikov, A. V. (xn) Eec. Mth. (Moscou) 10 (1882-83) (Pt. 1) 383-. 2nd kind, roots. Hermite, C. Toul. Fac. Sc. A. 4 (1890f I, 10 pp. , . Stieltjes, T. J. Toul. Fac. Sc. A. 4 (1890) J, 10 pp. , . Hermite, C. Prag Ab. 4 (1892) No. 2, 10 pp. logarithms of 1st five orders for every 5 degrees. Schmidt, A. Terr. Mag. 1 (1896) 73-. higherorder. Giuliani,G. G.Mt. 26 (1888)155-. property. Clebsch, A. [1861] Crelle J. 60 (1862) 343-. representation by an integral. Walter, A. Steierm. Mt. (1887) 25-. roots of equations occurring in. Markoff, A. Mth. A. 27 (1886) 143-, 177-. series. Dint, U. [1873] A. Mt. 6 (1875) 112-, series, convergence. Frischauf, J. Steierm. Mt. (1886) 3-. , and their differentials, tables for approxi- mate evaluation. Schmidt, A. Terr. Mag. 4 (1899) 59-. theorems. Pizzetti, . Palermo Cir. Mt. Ed. 13 (1899) 89-. , generalisation. Gegenbauer, L. Palermo Cir. Mt. Ed. 13 (1899) 92-. theory. Schendel, L. Crelle J. Mth. 80 (1875) 86-; 82 (1877) 158-. . Hermite, C. C. E. 86 (1878) 1515-. . Frischauf, J. Steierm. Mt. (1886) 19-. , extension. Frombeck, H. Wien Ak. Sb. 70 (1874) (Ab. 2) 61-. transformation of coordinates. Schmidt, A. Z. Mth. Ps. 44 (1899) 327-. one variable, coefficients. Bauer, G. Crelle J. 56 (1859) 101-. , and coefficients of series of spherical functions. Winckler, A. [1860] Wien D. 21 (1863) 37-. P(jji), regarded as a function of n, zeros. n Macdonald, H. M. [1899] L. Mth. S. P. 31 (1900) 264-. Theory of potential, expansion in series in. Johamon, A. M. Ups. Arsk. (1893) (Mt. Nt.) Zonal harmonics, table. Perry, J. [1890] L. Ps. S. P. 11 (1892) 221- ; Ph. Mg. 32 (1891) 512-. -- , , corrections to Perry's. Farr, C. C. Ph. Mg. 49 (1900) 572. [1 - a cos <(> + a' 2 ] A d, integration. Moth, F. X. As. Nr. 4 (1826) 539-. (1 - 2a cos + a 2 ) ~ ^, expansion in powers of a, coefficients. Plana, G. Zach Cor. 14 (1826) 125-. (1 -2xz + z 2 )"^, expansion. Ivory, J. Liouv. J. Mth. 2 (1837) 105-. , . Liouville,J. Liouv. J. Mth. 2 (1837)135-. , in powers of z. Schaar, M. Brux. Ac. Bll. 15 (1848) (pte. 2) 506-. F( n ', application to phenomena which are functions of latitude and longitude. Neumann, F. E. As. Nr. 15 (1838) 313-. (1 - 2at + a 2 ) ~ i, expansion by definite integrals. Dirksen, E. H. Berl. B. (1843) 111-. [a 2 - 2aa' (cos w cos + sin a> sin cos (6 - 0')) + a' 2 ] ~ *, expansion. Jacobi, C. G. J. Crelle J. 26 (1843) 81-. sin transformation. Liouville, J. Liouv. J. Mth. 11 (1846) 458-. l + *. v + X ( X + V v + &c> ad inf _ ScMffer, . y y(y + i) Crelle J. 37 (1848) 127-. Y n , series, convergence. Bonnet, 0. C. E. 28 (1849) 349-, 378-. and X n , expansion in series of. Bonnet, O. Liouv. J. Mth. 17 (1852) 265-. 348 4420 Legendrian and Bessel Functions Polymorphic Functions 4430 J (k) and J T (k) (or 1 and 1^) from k = to A=15-5, table. Meissel, E. Berl. Ak. Ab. (1888) (Anh. Mth.) 23 pp. I functions, expansions. Meissel, E. As. Nr. 127 (1891) 359-. ij', theory. MmseZ, . As. Nr. 128 (1891) 145-. J n -(x), new formulae. Kapteyn, W. Bll. Sc. Mth. 16 (1892) 41-. ^ w +i( x ) = ( integral), place of roots. Rudski, M. P. Liege S. Sc. Mm. 18 (1895) No. 3, 29 pp. O n (x\ and S n (x). Crelier, L. C. E. 125 (1897) 421-, 860-. (1 - '2aH + a 2 ) * , expansion in powers of a. Hansen, P. A. Leip. Ab. Mth. Ps. 1 (1852) 123-. ^{r-2 + r '2 _ 2rr' (cost/cost/' + sinf/sin U'cosJ) } , expansion of negative odd powers. Hansen, P. A. [1854] Leip. Ab. Mth. Ps. 2 (1855) 283-. Schlomilch Z. 3 (1858) 244-. (1 - 2qx + q 2 ) ~, and {1 - 2q [cos cos 0' + sin sin 0' cos (\f/ - 1 expansion. 4M#, (Dr) Jkf. Wien Ak. Sb. 51 (1865) (Ab. 2) 429-. ^f ,' ^ J)i theory. Thomae, J. [1867] Z a Mth. Ps. 14 (1869) 48-. E n . Hermite, C. A. Mt. 3 (1869-70) 83. Y m and X m . Gegenbauer, L. Wien Sb. 65 "(1872) (Ab. 2) 373-; 66 (1872) (Ab. 2) 55-; 68 (1873) (Ab. 2) 357-. , integral expressions for. Gegenbauer, L. Wien Sb. 66 (1872) (Ab. 2) 374-. X~ r+1 , expansion in terms of. Gegenbauer, L. Wien Sb. 66 (1872) (Ab. 2) 415-. tJl-2-riU+i) 1 , expansion of negative odd powers. Backlund, J. 0. [1877] St Pe"t. Ac. Sc. Bll. 24 (1878) 509-. G v n (x). Gegenbauer, L. Wien Ak. Sb. 75 (1877) \Ab. 2) 891-. Y n , final value for infinitely increasing values of n. Escary, . As. Fr. C. E. 8 (1879) 273-. , expansions in series whose terms are these functions. St Germain, A. de. C. B. 88 (1879) 1186-, 1313-. P(cos7), n infinite. Heine, H. E. [1880] Crelle J. Mth. 90 (1881) 329-. P n (0037), development. Pleskot, A. Prag Sb. (1893) (Mth.-Nt.) No. 17, 7 pp.; Fschr. Mth. (1893-94) 820. J -I ^ l> J_! tC ' '' J_i J / P i sin mOdn. Forsyth, A. R. QJ. Mth. 17 (1881) 37-. C v n (x) where (1 - 2zx + a; 2 )-" ="2" c" n (x) z n . Gegenbauer, L. Wien Ak. D. 48 (1884) (Ab. 2) 293-. , theorems. Gegenbauer, L. Wien Ak. D. 57 (1890) 425-. , on roots. [Coefficient of z 2 in expansion of (l-2zx + z z )~ v .] Gegenbauer, L. Amst. Ak. Vs. 8 (1900) 250- ; Amst. Ak. P. 2 (1900) 196-. Y m (x), addition theorem. Gegenbauer, L. Wien Ak. Sb. 92 (1886) (Ab. 2) 1340-. T(x). Gegenbauer, L. Wien Ak. Sb. 95 "(1887) (Ab. 2) 274-. X n =0, roots. Stieltjes, T. J. Acta Mth. 9 (1887) 385-. Gubler, E. Mth. A. 49 (1897) 583- Y and y lt table. Smith, B. A. Mess. Mth. 26 (1897) 98-. G (x) , G 1 (x) , and J n (x *Ji) , numerical com- putation. Aldis, W. S. E. S. P. 66 (1900) 32-. 4430 Polymorphic functions. Other functions which may be denned by definite integrals. (See also 4860.) Abelian generating functions. Minine, A. Les Mondes 2 (1882) 126-. . Pareto, V. Crelle J. Mth. 110 (1892) 290-. Associated functions and potential of spherical segment. Beltrami, E. N. Cim. 14 (*1883) 139-. polynomials ( | U^ *V V , idx = 0\ . Didon, F. Par. EC. Norm. A. 6 (1869) 111-. Conical functions, adjunctive, properties of integrals. Leonhardt, G. Mth. A. 19 (1882) 578-. Definite integrals to express the roots of u m -pu n -q = 0. Nekrasov, P. A. Eec. Mth. (Moscou) 13 (1886) 739-; Fschr. Mth. (1888) 304. , functions involved under. Gomes de Souza, J. E. S. P. 8 (1856-57) 146-. Double gamma function. Barnes, E. W. [1900] Phil. Trans. (A) 196 (1901) 265-. functions, genesis. Barnes, E. W. L. Mth. S. P. 31 (1900) 358-. Formula of Anger for 1 2>r cos (ha. - k sin a) da = 2ir ij and analogous formulae. Cauchy, A. L. C. B. 39 (1854) 129-. Functions analogous to Laplace's. Routh, E. J. L. Mth. S. P. 11 (1879-80) 92-. Legendre's. Aleksyeev, N. N. (xn) Eec. Mth. (Moscou) 5 (1870) (Pt. 1) 125-. , arbitrary, expressed by double integrals. Pioch, A. Brux. Mm. Cour. 4, 15 (1841-42) 74pp. 349 4440 Funtions, arbitrary, expressed by double in- tegrals. Meyer, A. [1850] Crelle J. 43 (1852) 60-. , class, represented by definite integrals. Goursat, E. [1882] Acta Mth. 2 (1883) 1-. , expansion, by definite integrals. Lorenz, L. Mth. Ts. 2 (1860) 161-. , in form of definite integrals. Pochhammer, L. Crelle J. Mth. 104 (1889) 152-. Gamma and related functions. Scheeffer, L. Crelle J. Mth. 97 (1884) 230-. G (r, v) -integrals, calculation. Brit. Ass. Comm. (Pearson, K.). B. A. Up. (1896) 70-. , tables. Brit. Ass. Comm. (Pearson, K.). B. A. Kp. (1899) 65-. Integral and sum between same limits, dif- ference. Imschenetsky, V. G. G. Mt. 9 (1871) 87-. Laplace's transformation and applications. Pincherle, S. Bologna Ac. Sc. Mm. 8 (1887) 125-. Toroidal functions. Hicks, W. M. Phil. Trans. 172 (1882) 609-. Automorphic Functions 4440 Transcendental jendental function I -.- = 1. , C. F. [1874] Ups! S. Sc. N. = 1. Lind- Acta 9 (1875) (No. 5) 48 pp. sidered by functions considered by Legendre, two. Lerch, M. Prag Sb. (1893) (Hth.-Nt.) No. 25, 5pp. Transcendents arising from integration of ho- mogeneous functions. Winckler, A. [1864] Wien Sb. 50 (1865) (Ab. 2) 531-. P (u ; v a , v 2 , . . . v p ; c z , c 2 , . . . c p ; s) . Lerch, M. Prag Sb. (1893) (Mth.-Nt.) No. 24, 7 pp. 4440 Automorphic functions. (See also 1210, 4050.) Burnside, W. [1891-92] L. Mth. S. P. 23 (1892) 49-, 281-. Stackel, P. Crelle J. Mth. 112 (1893) 287-. Abelian functions, generalisation. Picard, E. C. E. 98 (1884) 665-. Algebraic surf aces, class. Humbert, G. Palermo Cir. Mt. Ed. 4 (1890) 54-. Analogue to exponential function. Alekseevskij , V. P. Kharkov Mth. S. Com. 4 (1895) 253- ; Fschr. Mth.' (1895) 466. Automorphic elementary forms. Fricke, R. Gott. Nr. (1900) 303-. forms of arbitrary class. Ritter, E. Gott. Nr. (1893) 120-. function groups, application of ideals to. Fricke, R. Gott. Nr. (1894) 106-. functions of two variables. Picard, E. Acta Mth. 1 (1882) 297-; C. E. 96 (1883) 320- ; C. E. 97 (1883) 1045-. n variables, theory. Wirtinger, W. Wien Ak. Sb. 108 (1899) (Ab. 2o) 1239-. prime forms, representation in theta-series. Burkhardt, H. Mth. A. 42 (1893) 185-. Connexion with algebraic functions. Whittaker, E. T. [1898] Phil. Trans. (A) 192 (1899) 1-. Cyclic linear transformation. Ermakov, V. P. Eec. Mth. (Moscou) 14 (1890) 427-. Differential equations of 3rd order satisfied by automorphic functions. Hurwitz, A. Mth. A. 33 (1889) 345-. Equations in 2 variables, automorphic for a system of rational uniform transformations. Schwarz, C. H. A. Crelle J. Mth. 87 (1879) 139-. , . Hettner, G. Gott. Nr. (1880) 386-. Fuchsian functions. Poincare, H. C. E. 92 (1881) 333-, 395-, 957, 1198-, 1274-, 1484- ; 93 (1881) 301-, 581- ; 94 (1882) 163-, 1038-, 1166- ; 95 (1882) 626- ; 96 (1883) 1485- ; Acta Mth. 1 (1882) 193-. Mittag-Leffler, G. B. A. Ep. (1894) 577. application, and properties. Poincare, H. C. E. 92 (1881) 859-. of two complex variables. Kgpinski, S. [1895] Krk. Ak. (Mt.-Prz.) Ez. 10 (1896) 211- ; Mth. A. 47 (1896) 573-. and equation V-u = e u . Poincare, H. C. E. 126 (1898) 627-. existence, theorem. Bagnera, F. Ev. Mth. 6 (1896-99) 31-. of first family, representation by infinite pro- ducts. Biermann, 0. Mh. Mth. Ps. 1 (1890) 49- ; 3 (1892) 143-. hyperfuchsian functions. Picard, E. C. E. 98 (1884) 563-. , cases. Picard, E. C. E. 101 (1885) 1127-. in hypergeometric series of two variables. Picard, E. C. E. 99 (1884) 852- ; Par. EC. Norm. A. 2 (1885) 357-; Par. S. Mth. Bll. 15 (1887) 148-. . Levavasseur, . C. E. 115 (1892) 1006-. and ternary quadratic forms. Picard, E. Acta Mth. 5 (1884) 121-. and indefinite ternary quadratic forms. Poin- care, H. C. E. 102 (1886) 735-. and quadratic forms. Picard, E. Am. J. Mth. 11 (1889) 187-. . Stouff, X. Toul. Fac. Sc. A. 8 (1894) D, 20 pp. theory. Poincare, J. H. (xn) Caen Ac. Mm. (1882) 3-. . Biermann, 0. Wien Ak. Sb. 92 (1886) (Ab. 2) 1137-. . Schlesinger, L. Mth. Termt. Ets. 6 (1888) 36- ; Mth. Nt. B. Ung. 6 (1889) 337- ; Crelle J. Mth. 105 (1889) 181- ; C. E. 114 (1892) 1100-, 1409-. , Poincare's. Weber, H. Gott. Nr. (1886) 359-. _, _ (Weber). Cassel, G. Stockh. Ak. Hndl. Bh. 16 (Afd. 1) (1891) No. 2, 16 pp. ; Fschr. Mth. (1891) 438. theta-fuchsian functions. Ovidio, E. d\ Tor. Ac. Sc. At. 29 (1894) 741-. transformation. Poincare, H. C. E. 102 (1886) . Stouff , X. Par. EC. Norm. A. 5 (1888) 219-. 350 zeta-fuchsian functions. Poincare, H. Acta Mth. 5 (1884) 209-. 4440 Automorphic Functions Lame's and other Functions 4450 Schwarzian triangles. Kapteijn, W. N. Arch. Wisk. 1 (1895) 185- ; Fschr. Mth. (1893-94) 698 ; Bll. Sc. Mth. 18 (1894) 36-. with simple nodal point. Schilling, F. [1896] D. Mth. Vr. Jbr. 5 (1901) (Heft 1) 73-. s-function with branching (2, 3, 7). Fricke, E. Gott. Nr. (1892) 279-. , Schwarz's, with complex exponents, fundamental domain. Schilling, F. Gott. Nr. (1894) 261-. , , , geometrical theory. Schilling, F. Mth. A. 46 (1895) 62-, 529-. , , geometrical theory. Schilling, F. Mth. A. 44 (1894) 161-. s-f unctions, related. Papperitz, E. Mth. A. 25 (1885) 212- ; 26 (1886) 97-. Substitutions of 2nd degree. Laurent, H. N. A. Mth. 16 (1897) 389-. Symmetric /^-functions with singularity. Schil- ling, F. Mth. A. 51 (1899) 481-. Ternary automorphic transformations, infinite, theorem. Fano, G. Bm. B. Ac. Line. Bd. 8 (1899) (Sem. 1) 562-. Theory. Dixon, A. G. L. Mth. S. P. 31 (1900) 297-. . Lindemann, F. [1899-1900] Munch. Ak. Sb. 29 (1900) 423- ; 30 (1901) 493-. , and theory of groups. Fricke, E. Gott. Nr. (1892) 453- ; Mth. A. 42 (1893) 564-. , numbers, relations between. Fricke, E. D. Mth. Vr. Jbr. 6 (1899) (Heft 1) 94-. Transformation. Fricke, E. Mth. A. 44 (1894) 97-. of 5th and 7th orders of special automorphic functions. Fricke, E. Acta Mth. 17 (1893) 345-. Uniform automorphic forms of zero class. Eitter, E. Gott. Nr. (1892) 283- ; Mth. A. 41 (1893) 1-. functions. Klein, C. F. Mth. A. 19 (1882) 565-; 20 (1882) 49-. . Schottky,F. Mth. A. 20 (1882) 299-. functions reproduced by linear substitutions. Poincare, H. Mth. A. 19 (1882) 553-; 20 (1882) 52-. Fuchsian groups. Stouff, X. Toul. Fac. Sc. A. 4 (1890) P, 25 pp. and quadratic forms. Stouff, X. Toul. Fac. Sc. A. 3 (1889) B, 28 pp. reducible to modular groups. Bianchi, L. Em. B. Ac. Line. Bd. 6 (1890) (Sem. 1) 375-. , theory. Poincare, H. Acta Mth. 1 (1882) 1-. limiting circle. Nekrasov, P. A. [1889-91] Bee. Mth. (Moscou) 14 (1890) 537-; Fschr. Mth. (1890) 428; Mth. A. 38 (1891) 82-. . Anisimov, V. A. [1891-92] Bee. Mth. (Moscou) 16 (1893) 234- ; Fschr. Mth. (1892) 387 ; Mth. A. 40 (1892) 145-. Fuchs's new rule for finding the radius of the discriminating circle, explanations to supply insufficiency. Nekrasov, P. A. Bee. Mth. (Moscou) 16 (1893) 219- ; Fschr. Mth. (1892) 387-. Functions unaltered by certain linear transfor- mation of argument, class. Schottky, F. Crelle J. Mth. 101 (1887) 227-. linear substitutions. Fuchs, L. Gott. Nr. (1882) 81-. Group of 360 collineations in plane. Fricke, E. D. Nf. Vh. (1896) (Th. 2, Hdlfte 1) 5. 504 operations. Fricke, E. Mth. A. 52 (1899) 321-. Groups appearing in theory of analytic func- tions. Medolaghi, P. C. B. 126 (1898) 1188-. Invariant of congruent figures. Andrade, J. N. A. Mth. 8 (1889) 150-. expression for Poincar6's substitutions. Brun, F. de. Stockh. Ofv. (1890) 265-; Fschr. Mth. (1890) 410. Invariants of Schwarzian polygon of genus zero. Pick, G. Mth. A. 42 (1893) 489-. Kleinian functions of third family, generalisa- tion. Cassel, G. Stockh. Ofv. (1890) 237- ; Fschr. Mth. (1890) 410. Modular equation of 8th degree. Brioschi, F. [1878] Bm. B. Ac. Line. T. 3 (1879) 45-. , Jacobi's. Brioschi, F. Mth. A. 15 (1879) 241-. equations, class. Brioschi, F. A. Mt. 9 (1878-79) 167-. , depression. Klein, C. F. [1878] Mth. A. 14 (1879) 417-. .theory. Hitrwitz,A. Gott. Nr. (1883)350-. Octahedron and equation of 4th order. Puchta, A. Wien Ak. D. 41 (1879) (Ab. 2) 57-. function. Cayley,A. QJ. Mth. 16 (1879)280-. Polyhedral functions and Schwarzian deriva- tive. Cayley, A. [1880] Camb. Ph. S. T. 13 (1883) 5-. Quadratic quaternary forms and applications to Fuchsian groups. Stouff, X. Toul. Fac. Sc. A. 6 (1892) G, 19 pp. corresponding hyperabelian groups. Picard, E. C. B. 98 (1884) 904-. Bepresentation by infinite products. Stahl, H. Mth. A. 33 (1889) 291-, 604. 4450 Other functions which may be defined by linear differential equations. Lame's functions. Analytical functions of one variable depending on two real arbitrary constants. Picard,, . C. B. 114 (1892) 1310-. Biharmonic functions. Arcais, F. d'. Ven. I. At. (1898-99) (Pt. 2) 479-. Complete functions, of 1st kind. Tannery, J. C. B. 86 (1878) 950-. f*dx ( x dx \ Dilogarithms. (Given by / / j ^J Mantel, W. N. Arch. Wisk. 3 (1898) 292- ; Fschr. Mth. (1898) 377. Function analogous to modular functions. Poincare, H. C. B. 93 (1881) 138-. Functions arising from inversion of integrals of linear differential equations. Fuchs, L. Bll. Sc. Mth. As. 4 (1880) 278-, 328-. 351 4450 Lame's and other Functions 4450 Functions defined by differential equation of 1st order. Vivanti, G. A. Mt. 16 (1888-89) 117-. equations. Bouquet, J. , & Briot, . C. B. 39 (1854) 368-. (Bouquet & Briot). Cauchy, A.L. C. B. 40 (1855) 557-. . Poincari, H. Par. EC. Pol. J. Cah. 45 (1878) 13-. . Fine, H. B. Am. J. Mth. 11 (1889) 317-. of 1st order. Petrovitch, M. Par. S. Mth. Bll. 24 (1896) 58-. two simultaneous differential equa- tions. Guldberg, A. Mh. Mth. Ps. 2 (1891) 147-. a system of linear homogeneous dif- ferential equations of one or several variables, properties. Sauvage, L. Par. EC. Norm. A. 11 (1882) 33-. , expansion in integrals of linear differential equations of 2nd order. Pochhammer, L. [1871] Crelle J. 74 (1872) 315-. of parabolic cylinder, differential equation. Haentzschel, E. Z. Mth. Ps. 33 (1888) 22-. with recurring derivatives. Glaisher, J. W. L. [1872] L. Mth. S. P. 4 (1871-73) 113-. satisfying ^ = y. Schapira, . D. Nf. Tbl. (*1879) 171-. General co-functions, theory and application. Solutions of ^ = y. Sapiro, G. [Schapira, #.] N. Es. S. Nt. Mm. (Mth.) 3 (*1881); D. Nf. Tbl. (1881) [1]-. Integrals of solutions of differential equations of 2nd order, with 3 singular points. Kejpinski, S. Krk. Ak. (Mt.-Prz.) Ez. 17 (1900) 112- ; Ore. Ac. Sc. Bll. (1899) 67-. Lame-Hermite differential equation. Brioschi, F. St Pet. Ac. Sc. Bll. 35 (1894) 449-. Lame's equation. Brioschi, F. C. B. 85 (1877) 1160- ; 86 (1878) 313-. . Hermite, C. [1877] A. Mt. 9 (1878- 79) 21-. . Lindemann, C. L. F. B. A. Ep. (1883) 351. . Greenhill, A. G. [1888] L. Mth. S. P. 20 (1889) 213-. . Floquet, G. C. B. 121 (1895) 805-, 920. . Markoff, A. Mth. A. 47 (1896) 598-. , integration. Hermite, C. [1879] Crelle J. Mth. 89 (1880) 9-. , new application. Gyldfn, J. A. H. C. E. 93 (1881) 537-. , solution. Crawford, L. QJ. Mth. 27 (1895) 93-. , , factors. Crawford, L. QJ. Mth. 29 (1898) 196-. , special case. Stenberg, E. A. Acta Mth. 10 (1887) 339-. , . Jamet, V. C. E. Ill (1890) 638-. , theorems. Brioschi, F. C. B. 92 (1881) 325-. LAME'S FUNCTIONS. Heine, H. E. Crelle J. 56 (1859) 79-, 87-. Klein, C. F. Mth. A. 18 (1881) 237-. Schllifli, L. Em. E. Ac. Line. Mm. 4 (1887) 37-. Jaerisch, P. Hamb. Mth. Gs. Mt. 1 (1889) 212-. Moser, C. Arch. Sc. Ps. Nt. 8 (1899) 377-. and Hermite's function, zero values. Markov, A. A. Kharkov Mth. S. Com. 5 (1896?) 74-; Fschr. Mth. (1896) 334. of 1st kind, existence and number. Heine, [H.] E. Berl. Mb. (1864) 13-. , any order. Heine, H. E. [1862] Crelle J. 62 (1863) 110-. 2nd kind. Bigler, U. Arch. Mth. Ps. 12 (1894) 113-, 225-. Klein's theorem. Crawford, L. QJ. Mth. 30 (1899) 307-. and Laplace's and Bessel's functions, connexion between. Haentzschel, E. Z. Mth. Ps. 31 (1886) 25-. of 2nd order. Waelsch, E. Wien Ak. Sb. 105 (1896) (Ab. 2a) 741-. different orders. Heine, H. E. Crelle J. 60 (1862) 252-. and other functions. Stieltjes, T. J. Acta Mth. 6 (1885) 321-. theory. Heun, K. Mth. A. 33 (1889) 180-. . Klein, F. Gott. Nr. (1890) 85-. in " Theory of Heat." Escary, . C. B. 87 (1878) 646-; 88 (1879; 1027- ; 90 (1880) 1341-. with given zeros. Johanson, A. M. Stockh. Ofv. (1900) 675-. Lamp's numbers, generalisation. Berger, A. Ups. S. Sc. N. Acta 15 (1895) No. 10, 33 pp. products. Sa/ord, F. H. Am. J. Mth. 21 (1899) 1-. Polyharmonic functions. Volterra, V. Ven. I. At. (1898) 233-. -, i.e. solutions of = o t in simply connected plane space. Almansi, E. Palermo Cir. Mt. Ed. 13 (1899) 225-. Polynomials arising from (1 - 2xz + z z )~^' Amanzio, D. Nap. Ac. At. 7 (1895) No. 10, 35pp. , Stieltjes'. Van Vleck, E. B. N. Y. Am. Mth. S. Bll. 4 (1898) 426-. Potential functions in n dimensional space. Biermann, 0. Lotos 34 (1885) 93-. Eiemann's P-function, mathematical represen- tation. Radicke,A. Epm. Mth. 1 (1877) 50-. Bing functions. Gegenbauer, L. Wien Ak. Sb. 100 (1891) (Ab. 2a) 745-. Series and differential equations. Bervi, N. V. [1894] Eec. Mth. (Moscou) 17 (1895) 838- ; Fschr. Mth. (1895) 275. Specification of n functions in plane, problem of Eiemann. Schlesinger, L. C. E. 126 (1898) 723-. Successive approximations, singular examples. Picard, E. C. B. 126 (1898) 497-. Transcendental functions defined by differential equations of 2nd order. Painlevi, P. C. B. 116 (1893) 566-. (l-2az + a 2 )-*, expansion. Radau, R. Par. Obs. A. (Mm.) 18 (1885) D, 20 pp. 352 4470 Integral Functions 4460 Functions which may he de- fined by functional equations. (See also 6030.) Algebraic operations of higher order; e.g. a aetc< Schubert, H. C. H. Hamb. Mth. Gs. Mt. 1 (1889) 62. Formation of a function F (x) having property F[(x)]=F(x). Appell.P. C. E. 88(1879) 807-. Function given by / (z) + a0 (z) = expanded in infinite series ; applications. Rodrigues,J.M. Fschr. Mth. (1884) 219-. f (z) = ---- , definition by a system of functional properties. Moore, E. H. A. Mth. 9 (1894-95) 43-. Functional equations, solutions. Pick, G. Mth. A. 24 (1884) 590-. relation (u) . (v) = (u + v) . Wilson, E. B. A. Mth. 1 (1900) 47-. Functions satisfying equation relations. Magnus, L. J. Crelle J. 5 (1829) 365-. Lottner, C. L. E. [1850] Crelle J. 46 (1853) 367-. such that F ( sin |. Appell, P. C. E. 88 (1879) 1022-. Generalisation, calculus. Oltramare, G. As. Fr. C. E. (1887) (Pt. 2) 285-. Integral of function when only characteristic functional equation is given. Pexider, J. Casopis 29 (1900) 33-; Fschr. Mth. (1900) 306. Iterative functions. Schroder, E. [1869] Mth. A. 3 (1871) 296-. . Lemeray, E. M. As. Fr. C. E. (1895) (Pt. 2) 149-. Transcendental function, Soldner's. Buzen- geiger, K. H. J. Zach M. Cor. 26 (1812) 285-. 4470 Integral functions. Galois, E. Liouv. J. Mth. 11 (1846) 408-. Picard, E. C. E. 88 (1879) 1024- ; 89 (1879) 662-; Par. EC. Norm. A. 9 (1880) 145- ; Par. S. Mth. Bll., 11 (1883) 136-. Guichard, . Par. EC. Norm. A. 1 (1884) 427-. Vivanti, G. G. Mt. 26 (1888) 303-. Laurent, H. Par. EC. Pol. J. 60 (1890) 107-. Desaint, . C. E. 120 (1895) 548-. Hadamard, . C. E. 122 (1896) 1257-. Laurent, H. N. A. Mth. 15 (1896) 23-. Peter sen, J. Acta Mth. 23 (1900) 85-. Wigert, S. Stockh. Ofv. (1900) 1001-. Divisibility. Koch, H. von. Stockh. Ofv. (1893) 449-. Integral Functions 4470 Divisibility of function of n 2 elements by their determinant. Mertens, F. Wien Ak. Sb. 81 (1880) (Ab. 2) 260-. properties of functions of higher derivatives. Meyer, F. Mth. A. 36 (1890) 435-, Divisor, common, of two functions of one variable. Netto, E. Crelle J. Mth. 106 (1890) 81-. Divisors. Mertens, F. Wien Ak. Sb. 97 (1889) (Ab. 2a) 618-. Extension of property of polynomials to inte- gral functions. Borel, E. C. E. 123 (1896) 556-. Factorisaton. Konig, G. [1882] (xn) Mth. Termt. Ets. 2 (1884) 45-. and related elimination problems. Konig, J. [G.] Mth. A. 15 (1879) 161-. Factors. Selivano/, D. Par. S. Mth. Bll. 13 (1885) 119-. Function of 2 integral arguments. Muir, T. Edinb. Mth. S. P. 3 (1885) 19-. satisfying certain conditions, degree. Tchebitchef, P. L. Par. S. Mth. Bll. 3 (1875) 103. Functions assuming given values at given points. Cazzaniga, P. A. Mt. 10 (1880- 82) 279-. and continued fractions. Christoffel, E. P. [1876] A. Mt. 8 (1877) 1-. of form G(x). Hadamard, . C. E. 114 (1892) 1053-. of m systems each of n variables. Mertens, F. Mh. Mth. Ps. 4 (1893) 193-, 297-. and types of increase. Borel, E. C. E. 126 (1898) 321-. Hilbert's theorem, extension to infinite poly- nomials. Capelli, A. Nap. Ed. 35 (1896) 321-. Homogeneous functions. Andre, D. Par. S. Phlm. Bll. 1 (1899) 124. Irreducibility of functions of integral functions with integral coefficients. Netto, E. Giessen Oberh. Gs. B. 31 (1896) 113- ; Mth. A. 48 (1897) 80-. Irreducible polynomials. Kahan, W. F. Fschr. Mth. (1899) 100. Polynomials satisfying relation of 3rd order. Weill, . N. A. Mth. 4 (1885) 184-. Properties. Hadamard, J. Liouv. J. Mth. 9 (1893) 171-. . Desaint, . C. E. 124 (1897) 746-. Eational function, transformation. Laisant, . Par. S. Mth. Bll. 20 (1892) 6-. functions, theorems. Grunert, J. A. Grunert Arch. 31 (1858) 27-. Eedueible functions of n variables. Meyer, F. Mth. A. 30 (1887) 30-. Eeduction. Pellet, A. E. Par. S. Mth. Bll. 19 (1891) 48-. to canonical form of functions of n variables. Mirimanov, D. S. Eec. Mth. (Moscou) 19 (1897) 629 (bis)- ; Fschr. Mth. (1897) 88. Eesultants of n forms with n variables, deter- mining properties. Mertens, F. Wien Ak. Sb. 93 (1886) (Ab. 2) 527-. System of functions A + Bi{/', theorem. Nother, M. [1872] Mth. A. 6 (1873) 351-. 353 4800 Differential Equations. General 4800 Theorem of Picard, demonstration. Borel, E. C. B. 122 (1896) 1045-. , (Borel). Picard, E. C. B. 122 (1896) 1048. Theorems. Vivanti, G. G. Mt. 22 (1884) 243-, 378. Theory. Schou, E. C. B. 125 (1897) 763-. Transcendental functions. Vivanti, G. G. Mt. 23 (1885) 96-; Mil. I. Lomb. Ed. 32 (1899) 569-. Zeros. Borel, E. Acta Mth. 20 (1897) 357-. , position. Witting, A. Z. Mth. Ps. 30 (1885) 274-. ^ (a, b) and x ( a &)> elementary properties. Lerch, M. Casopis 24 (1895) 25-, 118-, 228-; Fschr. Mth. (1895) 217-. Differential equations. 4800 General. (See also 8080 Connexes.) Algebraic equations. Casorati, F. A. Mt. 9 (1878-79) 41-, 106-. , number of independent transcendentals belonging to. Koenigsberger, L. Mth. A. 30 (1887) 299-. Analogues. Bourlet, C. Par. EC. Norm. A. 16 (1899) 333-. Analytical remarks on Ouroussof ' s work. Mind- ing, E. F. A. [1865] St P3 (*) y" + 02 (*) y' + 0i (*) y = o. svan- berg, A. F. Ups. N. Acta S. Sc. 1 (1851) 1-. criteria. Minich, S. R. Yen. At. 2 (1843) 377-. . Challis, J. Ph. Mg. 24 (1844) 94-. . Volpicelli, P. Em. At. 4 (1850-51) 622. criterion, Euler's, reduction of conditions to single equation. Winckler, A. [1883] Wien Ak. Sb. 88 (1884) (Ab. 2) 820-. factor, means of determining. Maximovitch, W. C. E. 97 (1883) 1544-. immediate, of differential expression, any order, conditions. Pujet, A. C. E. 82 (1876) 740-. and integration. Combescure, E. [1871] A. Mt. 5 (1871-73) 20-. of ordinary equations. Peano, G. Mth. A. 37 (1890) 182-. --- . Me, G. Mth. A. 43 (1893) 553-. --- . Arzela, C. Bologna Ac. Sc. Mm. 5 (1895-96) 257-. Integrals, algebraic, of Rummer's equation. Goursat, E. C. E. 103 (1886) 993- ; Liouv. J. Mth. 3 (1887) 255-. of 1st class, representation by fundamental system. Hensel.K. Crelle J. Mth. 117 (1897) 29-. 356 4810 Existence Theorems Integrals, continuous and discontinuous. Cauchy, A. L. C. B. 29 (1849) 548-. of equation, 1st order, degree. Autonne, . C. B. 118 (1894) 1184-. , linear independent, of 1st class, number. Christoffel, E. B. A. Mt. 10 (1880-82) 81-. of ordinary equations, initial conditions de- termining. Niccoletti, 0. Tor. Ac. Sc. At. 33 (1897) 514- or 746-. . , system, unique determination by Cauchy 's initial conditions. Painlev$, . Par. S. Mth. Bll. 28 (1900) 191-. partial equations, domains of convergence for. Koenigsberger, L. Crelle J. Mth. 112 (1893) 181-. , rational, of Rummer's equation. Goursat, E. Mth. A. 24 (1884) 445-. of system of total differential equations, convergence of expansions. Meray, , & Eiquier, . Par. EC. Norm. A. 6 (1889) 355-; 7 (1890) 23-. ^ + py = 0, conditions for finiteness. dx' 2 Zukovskij, N. E. Bee. Mth. (Moscou) 16 (1893) 582-. Integration, complete, of given system of ordinary equations, possibility. Lipschitz, R. [1868-76] A. Mt. 2 (1868-69) 288-; Bll. Sc. Mth. As. 10 (1876) 149-. Irreducibility. Konigsberger, L. Crelle J. Mth. 92 (1882) 291-. Linear partial equation. Bendixson, I. Par. S. Mth. Bll. 24 (1896) 220-. Ordinary equations. Picard, E. Par. S. Mth. Bll. 19 (1891) 61-. . Peano, G. N. A. Mth. 11 (1892) 79-. . Arzela, C. Bologna Ac. Sc. Mm. 6 (1896-97) 131-. . Lindelof, E. Helsingf. Acta 21 (1896) No. 7, 13 pp. Orthoic systems. Eiquier, . C. B. 125 (1897) 1159. Partial equations. Mie, G. Z. Mth. Ps. 37 (1892) 151-, 193-. containing any number of functions and variables. Darboux, G. C. E. 80 (1875) 101-. . Genocchi, A. C. E. 80 (1875) 315-. , general demonstration. Me"ray,C. Liouv. J. Mth. 6 (1880) 235-. , any order. Darboux, G. C. B. 80 (1875) 317-. Any system. Meray, C. C. E. 80 (1875) 389-. . Eiquier, . C. B. 114 (1892) 731- ; Par. EC. Norm. A. 10 (1893) 65-, 123-, 167-. System of ordinary equations, order. Jacobi, C. G. J. Crelle J. 64 (1865) 297-. partial equations. Goursat, E. Par. S. Mth. Bll. 26 (1898) 129-. Systems, cases. Eiquier, . C. E. 125 (1897) 933-. , most general. Eiquier, C. Par. EC. Norm. A. 14 (1897) 99-. Solution and Reduction 4820 Systems of partial equations. Koenigsberger, L. Mth. A. 42 (1893) 485-. . Stackel, P. Crelle J. Mth. 119 (1898) 339-. , priority claim. Delassus, E. Par. EC. Norm. A. 14 (1897) 243-. Au=ke u . Picard, E. C. E. 116 (1893) 1015-. 4820 Methods of solution and re- duction of ordinary differential equations. Adjoint equation, and systems of equations. Borel, E. Par. EC. Norm. A. 9 (1892) 63-. Algebraic equations. Kimigsberger, L. D. Nf . Tbl. (1889) 197-. , 1st order. Vivanti, G. Palermo Cir. Mt. Ed. 4 (1890) 9-. , and degree. Voss, A. Mth. A. 23 (1884) 157-. , roots of which are integrals of Eiccati's equation. Autonne, L. Liouv. J. Mth. 6 (1900) 157-. homogeneous equations. Appell, P. C. E. 104 (1887) 1776-. Application of transformation groups. Page, J. M. A. Mth. 9 (1894-95) 59-. Applications of calculus of symbols. Russell, W. H. L. Phil. Trans. (1861) 69- ; (1862) 253-; (1863) 517-. Approximate methods. Durand,W.F. A. Mth. 12 (1898-99) 110-. Behaviour near singular points. Dulac, H. C. B. 129 (1899) 276-. Change of dependent and independent variables, applications. Niven, C. QJ. Mth. 10 (1870) 312-. independent variable. Thomas, L. N. A. Mth. 9 (1850) 365-. . Spitzer, S. C. B. 49 (1859) 48-, 270-. . Glashan, J. C. Am. J. Mth. 3 (1880) 190-. . variables, formulae. Fais, A. Bologna Ed. (1877) 60-. Clairaut's equation. Mansion, P. [1876] Mess. Mth. 6 (1877) 90-. , analogues of. Goursat, E. Par. S. Mth. Bll. 23 (1895) 88-. , , of higher order. Raffy, L. Par. S. Mth. Bll. 25 (1897) 71-. generalised, and equations of 3 variables. Gasparis, A. de. Nap. Ed. 6 (1847) 149-. Cremonian quadratic substitutions, application to equations, 1st order. Autonne, L. C. E. 105 (1887) 929- ; 106 (1888) 262-. Depression of equations. Cunningham, (Lt.- Col.) A. Mess. Mth. 17 (1888) 118- ; 18 (1889) 122-. Determination, explicit, of equations, 2nd order, with fixed critical points. Painleve, P. C. E. 126 (1898) 1329-. Differential resolvents. Spottiswoode, W. QJ. Mth. 6 (1864) 262-. and partial differential resolvents. Raw- son, R. Manch. Lt. Ph. S. P. 21 (1882) 59-. 357 4820 Ordinary Differential Equations, Solution and Reduction 4820 Equations. Fuss, N. [1818] St Pet. Ac. Sc. Mm. 9 (1824) 115-. Besge, V. A. Liouv. J. Mth. 4 (1859) 72; 18 (1873) 139-. Frahm, W. [1873] Mth. A. 8 (1875) 35-. Le Paige, C. M. M. H. H. (xn) Brux. S. Sc. A. 1 (1877) (Pt. 2) 51-. Tanner, H. W. I.. [1877] Mess. Mth. 7 (1878) 107-. Lindman, C. F. Stockh. Ofv. 37 (1880) No. 1, 3-. Harley,E. QJ. Mth. 17 (1881) 352- ; 18(1882) 41-. Poincare, H. C. E. 98 (1884) 793-. Sharpe, H. J. Mess. Mth. 13 (1884) 66-. Jamet, V. C. E. 104 (1887) 844-. Guldberg, A. C. E. 117 (1893) 614-. Chini, M. Tor. Ac. Sc. At. 34 (1898) 56-. adjoint to equation, nth order. Grilnfeld, E. Crelle J. Mth. 122 (1900) 43-, 88. with algebraic integral. Brioschi, F. Mil. I. Lomb. Ed. 9 (1876) 786- ; 10 (1877) 48-. arbitrary parameter. Picard, E. C. E. 118 (1894) 760- . class. Liouville, J. Liouv. J. Mth. 6 (1841) 448. . Boole, G. (vi Adds.) Ph. Mg. 30 (1847) 96-. class. Eosanes, J. Z. Mth. Ps. 16 (1871) 263-. . Jablonski, E. As. Fr. C. E. (1877) 188-. . Haag, . Par. S. Mth. Bll. 8 (1880) 80-. . Catalan, E. Brux. Ac. Bll. 12 (1886) . Vessiot, E. Par. EC. Norm. A. 10 (1893) 53- ; C. E. 116 (1893) 959-, 1112-. . Lindhagen, A. N. Ts. Mth. 10 (B) (1899) 35- ; Fschr. Mth. (1899) 297. reducible to linear equations. Appell, . C. E. 107 (1888) 776-. with complete differentials, two systems. Stodoikiewicz, A. J. Krk. Ak. (Mt.-Prz.) Pam. 12 (1886) (93)-; Fschr. Mth. (1886) 307. connected with orthogonal trajectories. Euler, L. [1781] St Pet. Ac. Sc. Mm. 11 (1830) 125-. of 1st degree. Trembley, J. Turin Mm. Ac. (1790-91) 1-. 2nd degree. Fuss, N. [1819] St Pet. Ac. Sc. Mm. 11 (1830) 258-. . Enneper, A. Schlomilch Z. 8 (1863) 58-. , general integral of which is uniform. Painleve, P. C. E. 117 (1893) 211-. form y m = x m (Ax-y" + Bxy' + Cy). Spitzer, S. Arch. Mth. Ps. 58 (1876) 100-. with fractional indices. Besge, V. A. Liouv. J. Mth. 9 (1844) 294. general integral of which is linear fractional function of arbitrary constants. Wallenberg, G. Crelle J. Mth. 121 (1900) 210-. geometrical interpretation. Fouret, G. C. E. 102 (1886) 415-. having fundamental system of integrals. Guld- berg, A. C. E. 116 (1893) 964-. . Lie, S. C. E. 116 (1893) 1233-. integrated by differentiation. Eaffy, L. Par. S. Mth. Bll. 23 (1895) 50-. invariant under transformation xx' = l, yx'=y'. . Elliott, E. B. Mess. Mth. 19 (1890) 7-. not containing the independent variable. Zehfuss, G. (xi) Z. Mth. Ps. 3 (1858) 247-. occurring in Laplace's kinetic theory of tides. Ling, G. H. A. Mth. 10 (1895-96) 95-. of multiplier. Genocchi, A. Mil. I. Lomb. Ed. 10 (1877) 553-. 1st order. Liouville, R. C. E. 103 (1886) 476- ; 106 (1888) 1648- ; Par. EC. Pol. J. 62 (1893) 181-. . Vessiot, . C. E. 116 (1893) 427-. . Korkine, A. C. E. 122 (1896) 1183- ; 123 (1896) 38-, 139; Mth. A. 48 (1897) 317-. . Petrovitch, M. C. E. 122 (1896) 1261-. . Painlere, P. C. E. 122 (1896) 1319- ; 123 (1896) 88-. . Cahen, A. C. E. 127 (1898) 1196-. . Horn, J. Mth. A. 51 (1899) 360-. , class, and invariant formations at- taching to them. Liouville, E. C. E. 105 (1887) 460-, 496. ,2 variables. Ermakov, V. P. Fschr. Mth. (1887) 316-. and 2nd orders, 2 variables. Weiler, A. Grunert Arch. 29 (1857) 1-. 2nd order. Besge, V. A. Liouv. J. Mth. 9 (1844) 336. . Hazzidakis, J. N. [1880] Crelle J. Mth. 90 (1881) 74-. . Guldberg, A. N. Ts. Mth. 7 (B) (1896) 1-; Fschr. Mth. (1896) 250. , with fixed critical points. Painleve, P. C. E. 126 (1898) 1185-, 1697-. 3rd order. Goursat, E. C. E. 98 (1884) 419-. . Klein, F. Mth. A. 23 (1884) 587-. . Kummer, E. E. Crelle J. Mth. 100 (1887) 1-. 4th order. Pochhammer, L. Crelle J. Mth. 104 (1889) 116-. with periodic coefficients, series attached to theory of. Liapouno/, A. C. E. 123 (1896) 1248-. possessing fundamental integrals. Lie, S. Leip. Mth. Ps. B. 45 (1893) 341-. relating to breaking of railway bridges. Stokes, G. G. Camb. Ph. S. T. 8 (1849) 707-. Euler's factor. Lorenz, L. V. (xn) Ts. Mth. 3 (1873) 174- ; 4 (1874) 33-. . Heun, K. Mth. A. 31 (1888) 363-. . , application. Peterson, J. Mth. Ts. 4 (1862) 139-. and Poisson's equations. L6vy, L. N. A. Mth. 8 (1889) 545-. Exact differential. Autonne, L. N. A. Mth. 15 (1896) 232-. 358 4820 Ordinary Differential Equations, Solution and Reduction 4820 General equations reducible to quadratures. Maximovitch, W. C. E. 101 (1885) 809-. Greatest and least exponents in theory of dif- ferential equations, complete particular in- tegrals. Bugaev, N. V. [1891] Eec. Mth. (Moscou) 16 (1893) 39- ; Fschr. Mth. (1891) 313. Homogeneous equations. Lindelof, E. Par. S. Mth. Bll. 24 (1896) 35-. Integrability conditions for linear equations. Eusjan, C. K. Fschr. Mth. (1899) 314-. , formula, generalisation. Eestellini, L. G. Mt. 36 (1898) 262-. Integrable case of + (T + iv'kx) y' + w"ky = 0. Thomae, C. J. Z. Mth. Ps. 21 (1876) 100-. cases of special equation. Heymann, W. Crelle J. Mth. 113 (1894) 84-. class of differential and finite difference equations. Sylvester, J. J. [1881] (xn) J. H. Un. Cir. [1] (1882) 178. forms of linear equations, 2nd order. Liou- ville, E. C. E. 100 (1885) 235-. total differential equations, theory. Guld- berg, A. Crelle J. Mth. 122 (1900) 34-. Integrals. algebraic, of Lame's equation. Brioschi, F. Em. E. Ac. Line. Ed. 1 (1892) (Sem. 2) 327-. particular. Bugaev, N. V. [1892-97] Eec. Mth. (Moscou) 17 (1895) 399-; 19 (1897) 662-; Fschr. Mth. (1893-94) 538. cases. Herz, N. [1881] Arch. Mth. Ps. 67 (1882) 312-. of equation, 1st order, at singular point, values. Fuchs, L. Berl. Ak. Sb. (1886) 279-. equations, 1st order, with limited number of values. Painleve, P. C. E. 114 (1892) 107-, 280-. -- with periodic coefficients, form. Anisimov, V. A. Eec. Mth. (Moscou) 20 (1899) 411- ; Fschr. Mth. (1898) 286-. ----- , . Anisimov, V. [1900] Eec. Mth. (Moscou) 21 (1901) 62-. ----- , . Hermite, C. [1900] Eec. Mth. (Moscou) 21 (1901) 66-. expansions in series. Picard, E. C. E. 128 (1899) 1363-. --- , cases. Liouville, E. C. E. Ill (1890) 597-. first, of systems. Painleve, P. C. E. 124 (1897) 136-. as functions of their initial values. Niccoletti, 0. Em. E. Ac. Line. Ed. 4 (1895) (Sem. 2) 316-. ---- . Holmgren, E. Stockh. Ak. Hndl. Bh. 25 (1900) (Afd. 1) No. 4, 19 pp. general, of Bessel's equation, formulse for nu- merical calculation. Niembller, F. Z. Mth. Ps. 25 (1880) 65-. , equation of 1st order, form. Fuchs, L. Berl. Ak. Sb. (1884) 1171-. , certain systems of simultaneous dif- ferential equations. Seiliger, D. Kazan Un. Mm. (1897, Pt. 1) (Non-o/. Sect.) 69-. particular, with fractional indices. Bugaev, N. V. [1891] St Pet. Ac. Sc. Mm. (Es.) 67 (1892) (Suppl.) No. 4, 12 pp. ; Fschr. Mth. (1891) 313. and general, connexion between. Bochow, Z. Mth. Ps. 33 (1888) 101-. of special equation, form. Haentzschel, E. Crelle J. Mth. 112 (1893) 148-. system, calculation by Cauchy-Lipschitz method. Painleve, . Par. S. Mth. Bll. 27 (1899) 149- of total differential equations. Sauvage, . Liouv. J. Mth. 10 (1884) 387-. systems. Forsyth, A. E. [1899] Camb. Ph. S. T. 18 (1900) 35-. total differentials. Poincare, H. C. E. 99 (1884) 1145-. . Picard, E. C. E. 100 (1885) 843-. and class of algebraic surfaces . Picard , E. C. E. 99 (1884) 1147-. uniform, of equations of 1st order and genus zero. Petrovitch, . C. E. 118 (1894) 1190-. of : ~ z* + C. Wood, De V. Franklin I. J. 47 (1864) 28-. f(y, x, -^J = 0, representation in space. Autonne, L. C. E. 105 (1887) 850-. Integrating equation. Bierens de Haan, D. Amst. Ak. Vs. M. 14 (1879) 162-. factor. Buchwaldt, F. Mth. Ts. 5 (1863) 33-, 82-. , determination. Maksimovich, V. P. Kazan S. Nt. (Ps.-Mth.) P. 2 (1884) 19-. -of elliptic equation. Sturm, E. Mth. A. 21 (1883) 446-. equations, 1st order. Winckler, A. Wien Ak. Sb. 99 (1891) (Ab. 2a) 457-, 875-. expressions, 1st order. Collet, . C. E. 68 (1869) 799-. , , for any number of variables. Collet, . Par. EC. Norm. A. 7 (1870) 59-. homogeneous equations. Le Francois, (Prof.) . Brux. Ac. Bll. 13 (1846) 159-. system of equations, new. Jacobi, C. G. J. Crelle J. 27 (1844) 199- ; 29 (1845) 213-, 333-. , theory. Dg. Ts. Mt. Fys. 2 (1869) 127-. ,3 variables. De Morgan, A. QJ. Mth. 2 (1858) 323-. factors of one or several differential forms. Chini, M. Mil. I. Lomb. Ed. 33 (1900) 601-. system of linear homogeneous equa- tions. Escherich, G. von. Wien Ak. Sb. 101 (1892) (Ab. 2a) 1222-. Integration. Brassinne, E. Toul. Mm. Ac. 5 (1839) 139-. Bertrand, J. Liouv. J. Mth. 17 (1852) 175-. HiU, C. J. D. Sk. Nf . F. 8 (1860) 525-. Steen, A. Ts. Mt. Fys. 3 (1870) 44-. Spitzer, S. Z. Mth. Ps. 52 (1871) 1-, 16 ; 53 (1871) 1-. 359 4820 Ordinary Differential Equations, Integration 4820 Russell, W. H. L. Mess. Mth. 10 (1881) 38-. Heymann, W. Z. Mth. Ps. 27 (1882) 1- ; 29 (1884) 257-. La Vallee-Poussin, C. de. Brux. S. Sc. A. 17 (1893) (Ft. 1) 8-. Guldberg, A. C. B. 121 (1895) 49-; Crelle J. Mth. 118 (1897) 158-. algebraic. Poincare, H. C. E. 112 (1891) 761-. , of equations, 1st order and degree. Caz- zaniga, P. G. Mt. 18 (1880) 72-. , , . Poincare, H. Palermo Cir. Mt. Ed. 11 (1897) 193-. , linear equations, 3rd order. Boulanger,A. C. B. 124 (1897) 1011-. application of Leibnitz's formula. Tychsen, C. Mth. Ts. 6 (1864) 177-. approximate, of equations, 1st order. Se- verini, C. Mil. I. Lomb. Ed. 31 (1898) 950-. , of 1 variable. Heun, K. Z. Mth. Ps. 45 (1900) 23-. , 2 variables. Kramp, G. Ger- gonne A. Mth. 10 (1819-20) 1-. , linear equations, cases. Sparre, (le couite) de. Brux. S. Sc. A. 23 (1899) (Pt. 2) 176-. , , 2nd order, cases. Sparre, M. de. Em. E. Ac. Line. Ed. 7 (1898) (Sem. 2) 111-, of Brassinne's equation. Longchamps, G. de. As. Fr. C. E. (1890) (Pt. 2) 146-. cases. Favre-Rollin, A. M. Liouv. J. Mth. 1 (1836) 339-. --. Ellis, R. L. Camb. Mth. J. 2 (1841) 169-, 193-. . Bronwin, B. Camb. Mth. J. 3 (1843) 29-, 175-. . Huet, . N. A. Mth. 5 (1846) 165-. . Turquan, . N. A. Mth. 5 (1846) 396-. . Spitzer, S. [1849] Haidinger B. 6 (1850) 86-. . Dupain, J. C. N. A. Mth. 10 (1851) 418-. . Bjorling, E. G. Stockh. Ak. Hndl. (1854) 149- ; Liouv. J. Mth. 3 (1858) 417-. . Spitzer, S. Schlomilch Z. 3 (1858) 106-. . Lorenz, L. Mth. Ts. 3 (1861) 102-. . Stem, A. Mth. Ts. 5 (1863) 113- ; (xn) Ts. Mth. 2 (1866) 138- ; 3 (1867) 56-. . Allegret, [A.] C. E. 66 (1868) 1144-. . Letter, G. Mittag. Ts. Mt. Fys. 3 (1870) 28-. . Turquan, L. V. [1874] Par. S. Mth. Bll. 3 (1875) 40-. . Lyetnikov, A. V. (xn) Eec. Mth. (Moscou) 8 (1876) (Pt. 1) 498-. . Halphen, G. H. C. E. 88 (1879) 562-. . Spitzer, S. Arch. Mth. Ps. 64 (1879) 393-. . Toropov, K. A. Kharkov Mth. S. Com. (1884) 199- ; Fschr. Mth. (1885) 309-. . Arneberg, A. Ts. Mth. 3 (1885) 168- ; Fschr. Mth. (1885) 328. of class of equations. Cauchy, A. L. Par. S. Phlm. Bll. (1818) 17-. (Cauchy). Poisson, S. D. Par. S. Phlm. Bll. (1818) 19-. of class of equations. Woisard, J. L. Ger- gonne A. Mth. 15 (1824-25) 121-. . Liouville, J. Liouv. J. Mth. 3 (1838) 31-. , 2nd order. Laguerre, . C. E. 67 (1868) 1130- ; 69 (1869) 1125-. , , in explicit finite quantities. Liouville, J. Liouv. J. Mth. 4 (1839) 423-. linear equations. Spitzer, S. [1863] Arch. Mth. Ps. 43 (1865) 478-. and classification of equations between x and y admitting of a group of transformations. Lie, S. Arch. Mth. Ntvd. 8 (1883) 187-, 249-, 371- ; 9 (1884) 431-. of complete differentials. Morera, G. Mth. A. 27 (1886) 403-. definite, of a system of equations. Cauchy, A. L. C. E. 43 (1856) 497-. of differential equations or partial differential equations. Cauchy, A. L. Par. S. Phlm. Bll. (1822) 49-; C. E. 11 (1840) 1-. by differentials to any index. Serret, J. A. Liouv. J. Mth. 9 (1844) 193-. of equation in analysis of functions of 2 variables. Eider, L. [1777] St Pet. Ac. Sc. Mm. 11 (1830) 131-. equations, coefficients of which are algebraic functions of the independent variable. Spitzer, S. Wien SB. 29 (1858) 53-. with complete differentials. Stodotkiewicz , A. J. Prace Mt.-Fiz. 1 (1888) 78-; Fschr. Mth. (1888) 355. in Euler's integral calculus, cases. Grunert, J. A. Grunert Arch. 32 (1859) 239-. with fractional index. Liouville, J. [1836] Par. EC. Pol. J. 25 cah. (1837) 58-. , 1st order. Bohlmann, G. Crelle J. Mth. 113 (1894) 207- ; 115 (1895) 89-. , . Severini, C. Mil. I. Lomb. Ed. 31 (1898) 657-. , and degree, 3 variables. Volpicelli, P. Em. At. 1 (1847-48) 108-. , , 2nd degree, several variables. Raabe, J. L. Baumgartner Z. 8 (1830) 425- ; 9 (1831) 36-. , , by equation of 2nd order, case. Dienger, J. Arch. Mth. Ps. 46 (1866) 317-. , , where independent variable does not appear. Raschke, W. Acta Mth. 14 (1890-91) 31-. , ,2 variables. Minding, E. F. A. [1845-61] St Pet. Ac. Sc. Bll. 4 (1845) 378-; St Pet. Ac. Sc. Mm. 5 (1863) No. 1, , 2nd order. Cayley, A. Ph. Mg. 42 (1871) 197-. , , application of principle of last multiplier. Laguerre, . Bll. Sc. Mth. As. 2 (1871) 246-. , , cases. Brioschi, F. Tortolini A. 2 (1851) 497-. , , . Wolfers, J. P. Grunert Arch. 28 (1857) 271-. , , . Malmsten, C. J. Ts. Mt. Fys. 2 (1869) 71-. , , . Laguerre, E. [1870] (x) Par. S. Phlm. Bll. 7 (1871) 158-. 360 4820 Ordinary Differential Equations, Integration 4820 of equations, 2nd order, cases. Steen, A. [1882] Kjob. Dn. Vd. Selsk. Skr. 1 (1880-85) 331-. , , _. Gylden, H. Stockh. Ak. Hndl. Bh. 12 (Afd. 1) (1887) No. B, 15 pp. , , Challis's method, and application to calculus of variations. Ehrhorn, . Mth. Ps. 66 (1881) 113-. , and degree, cases. Lobatto, R. Grunert Arch. 30 (1858) 292-. , , geometrical significance of primitive. Tychsen, C. Mth. Ts. 5 (1863) 118-. , th order, 1st degree. Malmsten, C. J. Stockh. Ak. Hndl. (1842) 145-. ,2 variables. Kramp, C. Gergonne A. Mth. 11 (1820-21) 97-, 345-. , . Stodoikiewicz,A.J. (xn) Krk. Ak. (Mt.-Prz.) Bz. & Sp. 10 (1883) 225-. under finite form. Genocchi, A. [1864-72] Tor. Mm. Ac. 23 (1866) 299-; Tor. Lav. Sc. Fis. Mt. (1869) 88-; Tor. At. Ac. Sc. 7 (1871-72) 682-. formula. Pezzi, F. Turin Mm. Ac. (1790-91) 79-. of functions, case. Dienger, J. Grunert Arch. 12 (1849) 203-. geometric method, case. Crofton, M. W. Mess. Mth. 3 (1866) 106. , of L(xdy-ydx)-Mdy + Ndx=0, L, M, N being linear functions of x and y. Fouret, G. C. B. 78 (1874) 1837-. of homogeneous equations. Cauchy, A. L. C. B. 13 (1841) 564-. . Minding, E. F. A. Liouv. J. Mth. 4 (1859) 273-. method, Brisson's, for equations with constant coefficients. Mansion, P. N. A. Mth. 11 (1872) 118-. , general. Schmidten, H. G. [1826] Kiob. Dn. Vd. Selsk. Afh. 3 (1828) 1-; Crelle J. 1 (1826) 137-. methods. Aniaimov, V. A. Vars. S. Nt. Tr. (1897) (Ps. C.) Fasc. 1, Mm. 1, 35 pp.; Acta Mth. 51 (1899) 181-. problem, solution. Lagrange, C. Brux. Obs. A. (As.) 1 (1896) 40 pp. " lofo by separation of symbols of operation. Francois, J. F. Gergonne A. Mth. 3 (1812-13) 244-. series. Cayley, A. Mess. Mth. 5 (1871) 77-. . Sabinine, G. Bll. Sc. Mth. As. 2 (1878) (Ft. 1) 284-. . Poincare, H. C. B. 94 (1882) 577-. of -^ - - - -r =y- Bach, . Par. EC. Norm. A. 3 (1874) 47-. of simultaneous equations. Boole, G. B. S. P. 12 (1862-63) 13-. , case. Steen, A. (xn) Ts. Mth. 1 (1865) 35-. , 1st order. Bouquet, . Bll. Sc. Mth. As. 3 (1872) 265-. symmetrical equations. Imchenetsky, B. St P (x, y) dy = Q, case. Harkema, C. N. A. Mth. 13 (1874) 545-. (xdy - ydx) (a + bx + cy) - dy (a' + b'x + c'y) + dx (a" + b"x + c"y) = 0. Allegret, A. C. E. 83 (1876) 1171. - y g - | (g) 2 = 6/(x),/being a polynomial of 2nd degree. Laguerre, E. [18771 Par - S. Mth. Bll. 6 (1878) 121-. Alexeeff, N. C. E. 87 (1878) 641-. _/^ |t A = 0. Hansen, P. C. V. [1881] Kjob. Ov. (1881-82) 156- (& Resume, 7-). Ldx + Mdy + N (xdy - ydx) = 0. Hocevar, F. Wien Ak. Sb. 85 (1882) (Ab. 2) 848-. -r-^ + p = P + fiip + /3 2 p 2 .... Lindstedt, A. As. Nr. 103 (1882) 257-; 104 (1883) 145-. = (Lindstedt). Gylden, J. A. H. As. Nr. 103 (1882) 321-. (A~&* + B~ 3 y* + Clx'y + Dp + E^y + F 3 \ f ~ ' (xdy -ydx)) Heymann, W. Z. Mth. Ps. 28 (1883) 54-.

when it is an exact differential coefficient. Foldberg, P. T. N. Ts. Mth. 1 (B) (1890) 23-; Fschr. Mth. (1890) 281-. y"=Ay* + By 2 + Cy + D + (Ey + F) y'. Mittag- Leffler, G.- Acta Mth. 18 (1894) 233-. Newton-Cotes method of quadratures. Jung, V. Prag Ceske Ak. Fr. Jos. Ez. (Tfida 2) 8 (1899) Art. 17, 12 pp. ; Fschr. Mth. (1899) 279. Non-linear equations, cases. Liouville, R. C. B. 103 (1886) 457-. , class. Liouville, R. C. E. 103 (1886) 520-, 571. , 2nd order, with doubly periodic co- efficients. Gylden, H. C. E. 122 (1896) 160-, 585-. Partial equations occurring in theory of conjugate functions. Volterra, V. Palermo Cir. Mt. Ed. 3 (1889) 260-. Paths described by ships to perform certain evolutions. Bernotti, R. [1897] Ev. Mar. et Col. 137 (1898) 341-. Eeciprocal equations. Orloff, . Brux. Ac. Bll. 33 (1872) 113-. Eeduction to canonical form. Pennacchietti, . G. Mt. 29 (1891) 239-. ---- . Riquier, C. Par. EC. Norm. A. 14 (1897) 259-. of complete system to single system of ordinary equations. Mayer, A. Leip. Mth. Ps. B. 43 (1891) 448-. ---------- . Schur, F. Leip. Mth. Ps. B. 44 (1892) 177-. order. Konigsberger, L. Mth. A. 26 (1886) 110-. --- by known integrals. Koenigsberger, L. Mth. A. 31 (1888) 302-. to new variables. Baehr, G. F. W. Amst. Vs. Ak. 3 (1855) 233-. Separation of variables in Mdx + Ndy = 0. Harkema, C. N. A. Mth. 16 (1877) 215-. 115 Solution. Roberts, S. QJ. Mth. 3 (1860) 218-. asymptotic, cases. Picard, E. C. E. (1892) 1030-. method, Petzval's. Spottiswoode, W. QJ. Mth. 5 (1862) 154-. Boole's. Rawson, R. Mess. Mth. 5 (1876) 138-. cases. Bronwin, B. Ph. Mg. 32 (1848) 256-. (Bronwin). Boole, G. Ph. Mg. 32 (1848) 413-. . Williamson, B. Ph. Mg. 11 (1856) 364-. . Jackson, F. H. Edinb. Mth. S. P. 12 (1894) 109-. of class of difference or differential equations. Sylvester, J. J. Am. J. Mth. 4 (1881) 260-. common to two equations, orders m + n, m, Libri's theorem. Liouville, J. C. E. 8 (1839) 790-. of differential resolvent. Russell, W. H. L. Manch. Lt. Ph. S. Mm. 2 (1865) equation, 2nd order, properties. Routh, E. J. L. Mth. S. P. 16 (1884-85) 245-. of sources. Hammond, J. Am. J. Mth. 5 (1882) 218-. equations in series. Glaisher, J. W. L. [1878] Mess. Mth. 8 (1879) 20-. homogeneous equation. Forsyth, A. R. Mess. Mth. 18 (1889) 136-. and integration. West, E. Liouv. J. Mth. 7 (1881) 5-; 8 (1882) 19-, 125- ; 9 (1883) 301-. numerical. Runge, C. Mth. A. 46 (1895) 167- . periodic, of linear equations. Bendixson, I. Stockh. Ofv. (1896) 193-. regular, of a system. Sauvage, L. Par. EC. Norm. A. 3 (1886) 391- ; 5 (1888) 9-; 6 (1889) 157-. singular. Fine, H. B. Am. J. Mth. 12 (1890) 295-. . Page, J. M. Am. J. Mth. 18 (1896) 95-. 362 4820 Ordinary Differential Equations, Solution and Reduction 4820 singular, of algebraic equations, 1st order. Hamburger, M. Crelle J. Mth. 112 (1893) 205-. ' , equations, 1st order. Darboux, G. Bll. Sc. Mth. As. 4 (1873) 158-. , , . Kapteyn, W. Bll. Sc. Mth. 12 (1888) 135-. , , . Predella, L. G. Mt. 33 (1895) 31-, 183-. , , . Maddism, I. QJ. Mth. 28 (1896) 311-. , , . Toussaint, H. Em. N. Line. Mm. 13 (1897) 283-. , , . Petrovitch, M. Mth. A. 50 (1898) 103-. , , higher order. Hamburger, M. Crelle J. Mth. 121 (1900) 265-. , homogeneous equations. Johnson, W. W. Mess. Mth. 16 (1887) 186-. , integrable equations, 1st order. Work- man, W. P. QJ. Mth. 22 (1887) 175-, 308-. , simultaneous equations. Goursat, E. Am. J. Mth. 11 (1889) 329-. , and theory of congruencies. Dixon, A. C. [1894] Phil. Trans. (A) 186 (1896) 523-. , a system. Biermann, O. Wien Ak. Sb. 90 (1885) (Ab. 2) 897-. , of equations with n dependent variables. Hamburger, M. Crelle J. Mth. 122 (1900) 322-. symbolic and by definite integrals of -~=x m y. Fields, J. G. Am. J. Mth. 8 (1886) 367-. dy of ~=f(x, y) without Cauchy-Lipschitz con- dition. Osgood, W. F. Mh. Mth. Ps. 9 331-. Substitution of variables as method of finding integrating factor. Ermakov, V. P. (xn) Kharkov Mth. S. Com. (1881) 3-. Successive approximations. Picard, E. C. E. 115 (1892) 543-. . Bendlxson, I. Stockh. Ofv. (1893) 599-. . Lindelof, E. Liouv. J. Mth. 10 (1893) . Picard, E. Liouv. J. Mth. 9 (1893) 217-. . Bugaev, N. F. [1896] Eec. Mth. (Moscou) 19 (1897) 1-; Fschr. Mth. (1896) 70, 72. . Picard, E. Am. J. Mth. 20 (1898) 87-. . Davidoglou, A. C. B. 130 (1900) 692-, 1241-. , application to equations, 1st order. Lindelof, E. C. B. 118 (1894) 454-. , , . Picard, E. C. B. 118 (1894) 457-. , , . Horn, J. Mth. A. 51 (1899) 346-. divergent approximations. Picard, E. C. B. 118 (1894) 899-. Symbolic decomposition. Cockle, Jos. Ph. Mg. 24 (1862) 288-. System of equations, determination of variables satisfying. Cauchy, A. L. C. B. 23 (1846) 485-, 529-. Systems of equations. Combescure, E. [1874] Crelle J. Mth. 80 (1875) 33-. --- . Halphen, G. H. C. E. 92 (1881) 1404-. --- . Guldberg, A. C. B. 117 (1893) 215-. --- , C ase. Halphen, G. H. C. B. 92 (1881) 1101-. --- , . Brioschi, F. C. B. 92 (1881) 1389-. --- , 1st order, having fundamental sys- tems of integrals. Vessiot, E. Toul. Fac. Sc. A. 8 (1894) B, 33 pp. --- , 2nd order, in involution. Goursat, E. Par. EC. Pol. J. 3 (1897) 75-. , integration of which is reducible to that of total differentials. Riquier, . C. E. 127 (1898) 809-, 1194-. Theorem of mean, extension to equations, 1st order. Petrovitch, M. C. B. 128 (1899) 981-. Poincare on systems of 1st order. Bendixson, . C. B, 118 (1894) 971-. Total algebraic differentials. Picard's equation Pdx + Qdy = when F (x, y , z) = 0. Noether, M. Mth. A. 29 (1887) 339-. Transformation, case. Hargreave, (7. J. Ph. Mg. 30 (1847) 8-. of equations, 1st order, case. Painleve, P. C. E. 110 (1890) 840-. --- with 2 variables. Zeuthen, H. G. N. Ts. Mth. 1 (B) (1890) 1- ; Fschr. Mth. (1890) 316. and integration of differential systems. Delassus, E. Par. EC. Norm. A. 14 (1897) 195-. : --- simultaneous equations, several variables. Brassinne, E. Toul. Mm. Ac. 5 (1839) 129- ; Liouv. J. Mth. 10 (1845) 194-. of linear equations, 2nd order. Spottiswoode, W. Tortolini A. 3 (1852) 26-. --- , , with linear coefficients. Bolza, O. Am. J. Mth. 15 (1893) 264-. -- ^2 + ^ + *& + 03 = - Heymann, W. Z. Mth. Ps. 27 (1882) 374-. Transformations. Bianchi, G. Tortolini A. 3 (1852) 347-. Variation of arbitrary constants. Liouville, J. Liouv. J. Mth. 3 (1838) 342-. -- constants in solutions. Brunacci, V. Turin Mm. Ac. (1804-05) (pte. 2) 3-. Sturm, J. C. F. (gr-l)V L'l. 1 (1833) 247-. y" + f (*) y' + F (y) y'- = o. Liouville, J. Liouv. J. Mth. 7 (1842) 134-. ^| + / (x) sin y + F (x) cos y + (x) = 0. Besge, V. A. Liouv. J. Mth. 11 (1846) 445. 363 4830 Partial Differential Equations, First Order 4830 = +y. Vieille, J. N. A. Mth. 5 (1846) 509-. d x { (x - x 3 ) y x } - xy = 0. Liouville, J. Liouv. J. Mth. 19 (1854) 151-. -~=f(x,y). Liouville, J. Liouv. J. 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Bll. 26 (1898) 77-. of equations of dynamics. Painleve, P. C. E. 123 (1896) 636-. and application to problem of three bodies. Painleve, P. C. E. 123 (1896) 871-. Solution. cases. C., G. Camb. Mth. J. 1 (1839) 162-. complete, and transformation. Mayer, A. Gott. Nr. (1872) 405-. of Euler's equations of motion by elliptic functions. Greenhill, A. G. QJ. Mth. 14 (1877) 265-. Lagrange's rule, demonstration, and history of proofs. Chrystal, G. [1891] Edinb. E. S. T. 36 (1892) 551-. Lie's. Dixon, A. C. Camb. Ph. S. P. 9(1898) 279-. method, general, for equations with several dependent variables. Tanner, H. W. L. L. Mth. S. P. 11 (1879-80) 72-. of reciprocation. M'Cowan, J. Edinb. Mth. S. P. 11 (1893) 2-. methods of Lagrange and Monge. Watson, H. W. QJ. Mth. 6 (1863) 146-. particular, of Fix, y, z, p, q)=0. Madsen, (Kapt.)V.H.O. (xn) Ts. Mth. 2 (1866) 130-. of problems of mechanics by successive approxi- mations. Picard, E. Tor. Ac. Sc. At. 34 (1898) 6-. singular. Carmichael, JR. Ph. Mg. 15 (1858) 522-. . Weber, H. Crelle J. 66 (1866) 193-. . Yaroshenko, S. P. (xn) Eec. Mth. (Moscou) 5 (1870) (Pt. 1) 383-. 367 4830 Partial Differential Equations, First and Second Orders Mil. I. Lomb. Ed. 9 4840 singular. Casorati, F. (1876) 522-. . Darboux, G. Par. Mm. Sav. Etr. 27 (1883) No. 2, 243 pp. Holler, J. Stockh. Ofv. (1888) 463-. Egorow, D. T. Fschr. Mth. (1898) 305-. of coincidence curve. Bjorling, C. F. E. Stockh. Ak. Hndl. Bh. 12 (Afd. 1) (1887) Vo. 7, 16 pp. and envelopes. Cockle, (Sir) Jas. Manch. Lt. Ph. S. P. 24 (1885) 10-, 23-. , of equations representing lines at infinity. Mansion, P. B. A. Ep. (1875) (Sect.) 19-. , -- in 2 variables. Landre, C. L. As. Fr. C. E. (1879) 241-. of system of simultaneous non-linear equa- tions. Nanson, E. J. E. S. P. 23 (1875) 510; 24 (1876) 337-. L - QJ. Mth. 8 (1867) 252-. Substitution of complex values in equations with 2n variables. Imshenetsky,V.G. Liege S. Sc. Mm. 7 (1878) (No. 2) 6 pp. System of equations. Combescure, E. C. E. 74 (1872) 977-. --- . Volterra, V. Tor. Ac. Sc. At. 30 (1895) 445-. -- equivalent to those in problem of n bodies but admitting one more integral. Ebert, W. C. E. 131 (1900) 251-. Systems, complete, of homogeneous linear equations. Schur, F. Crelle J. Mth. 108 (1891) 313-. of equations. Du Bois-Reymond, P. [1867] Crelle J. 68 (1868) 180-. --- . Backlund, A. V. [1876] Mth. A. 9 (1877) 412-. --- analogous to equations of an involution. Beudon, J. Liouv. J. Mth. 5(1899) 351-. 1st order. Beudon, J. Par. EC. Norm. A. 15 (1898) 229-. --- deprived of integrals. Meray, C. C. E. 106 (1888) 648-. --- with one unknown function. Delassus, E. Par. EC. Norm. A. 14 (1897) 109-. several dependent variables. Tanner, H. W. L. L. Mth. S. P. 10 (1878- 79) 55-. -- involution. Weber, E. von. Mth. A. 49 (1897) 543-. Theorem of Jacobi's. Bonnet, 0. C. E. 45 (1857) 581-. -- Lie. Bendixson, I. Stockh. Ofv. (1892) 301-. Theory. Yanishevskil, E. P. (xn) Kazan Un. Mm. (1863) (Ps.-Mth., No. 2) 100 pp. Darboux. G. C. E. 70 (1870) 746-. Mansion, P. [1873] Brux. Mm. Cour. 8, 25 (1875) No. 1, xvi + 282 pp. Kowalevsky, S. von. [1874] Crelle J. Mth. 80 (1875) 1-. Backlund, A. V. Mth. A. 17 (1880) 285-. Czuber,E. WienAk. Sb.l03(1894)(^6.2a)295-. Tannenberg, W. de. C. E. 120 (1895) 674-. Saltykow, N. C. E. 129 (1899) 195-. analytic. Delassus, E. Liege S. Sc. Mm. 20 (1898) No. 3, 84 pp. and classification. Lie, S. Gott. Nr. (1872) 473-. direct application of first principles. Larmor, J. E. S. P. 43 (1888) 176-. of equations with 3 variables. Du Bois- Reymond, P. Crelle J. 64 (1865) 271-. general. Liouville, J. Liouv. J. Mth. 1 (1856) 345-. . Lie, M. S. Christiania F. (1874) 198- ; (1875) 1- ; Mth. A. 9 (1876) 245- ; 11 (1877) 464-. Lie's. Biermann, 0. Leip. Mth. Ps. B. 48 (1896) 665-. of linear equations. Krankenhagen, (Dr) . Gott. Nr. (1880) 197-. Transformation of equations of dynamics. Radau, R. Par. EC. Norm. A. 5 (1868) 311-. . Painleve, P. C. E. 115 (1892) 714-, 874 ; 123 (1896) 392-. motion of a particle. Cerruti, V. Em. E. Ac. Line. T. 3 (1879) 196-. three bodies problem. Brioschi, F. C. E. 66 (1868) 710-. , method of Jacobi, extension. Meray, C. Par. EC. Norm. A. 7 (1890) 217-. , orthogonal, applicable to equations of dy- namics. Radau, R. C. E. 67 (1868) 316-. Transformations of equations of three bodies problem. Siacci, F. Tor. At. Ac. Sc. 6 (1870-71) 440-. principal function. Cauchy, A. L. C. E. 13 (1841) 40-, 97-, 397-, 455-, 487-, 1127- ; 14 (1842) 1-. Variation of arbitrary constants in problems of dynamics. Brioschi, F. Tortolini A. 4 (1853) 298-. definite integral and theory of equations. Marcolongo, R. Nap. Ed. 27 (1888) 500-. 4840 Methods of solution and re- duction of partial differential equations of the second and higher orders. Abel's theorem. Cayley, A. [1880-81] Phil. Trans. 172 (1881) 751- ; Camb. Ph. S. P. 4 (1883) 119-. , equation connected with. Cayley, A. B. A. Ep. (1881) 534-. of partial differential equations. Delassus, E. Par. EC. Norm. A. 14 (1897) 21-. Analogues of equations of 1st order. Beudon, J. C. E. 126 (1898) 324-, 388-. Arbitrary functions, calculation. Beudon, J. C. E. 128 (1899) 1215-. , elimination. Garbieri, G. Yen. I. At. (1886-87) 831-. , , new method. Minich, S. R. C. E. 84 (1877) 1496- ; 87 (1878) 161-. Changes of variables. Beudon, J. Par. S. Mth. Bll. 28 (1900) 107-. 368 4840 Partial Differential Equations. Second and Higher Orders 4840 Change of variables, equations, 2nd order. Bounsinesq, J. C. E. 74 (1872) 730-. Characteristics. Beudon, J. Par. S. Mth. Bll. 25 (1897) 108-. of equations, 2nd order, theory. Bdcklund, A . V. Mth. A. 13 (1878) 411-. in 3 variables. Weber, E. von. Mth. A. 47 (1896) 229-. , and Huygens' principle. Coulon, J. C. E. 130 (1900) 1064-. of linear equations, 2nd order, with constant coefficients. Coulon, J. Bordeaux S. Sc. PV. (1899-1900) 24-. , theory. Goursat, E. C. E. 126 (1898) 1332-. Equations. Falk, M. (xn) Ts. Mth. 5 (1881) 97-. Moutard, T. Par. EC. Pol. J. 64 (1894) 55-. Nekrasov, P. A. Eec. Mth. (Moscou) 18 (1896) 468-; Fschr. Mth. (1896) 271, 272. Duport, H. Liouv. J. Mth. 6 (1900) 41-. and bilinear systems. Weber, E. von. Miinch. Ak. Sb. 29 (1900) 231-. cases. Glaisher, J. W. L. B. A. Ep. (1882) 454-. . Picard, E. C. B., 116 (1893) 454-. . Jamet, V. Par. EC. Norm. A. 13 (1896) 95-. . Goursat, E. Par. S. Mth. Bll. 25 (1897) 36-. . Schi/, P. A. Fschr. Mth. (1899) 332. , exceptional. Cockle, (Sir) J. L. Mth. S. P. 10 (1878-79) 105-. class. Fredholm, I. C. E. 129 (1899) 32-. . Lauricella, G. A. Mt. 4 (1900) 281-. , invariants of which are equal. Guichard, C. Par. EC. Norm. A. 7 (1890) 19-. , , 3 variables. Poisson, S. D. Par. EC. Pol. Cor. 2 (1809-13) 410-. in conduction of heat. Liouville, J. Liouv. J. Mth. 1 (1836) 33-. with constant coefficients and non-analytic functions. Borel, E. C. E. 121 (1895) 933-. and definite integrals. Sarrus, F. Gergonne A. Mth. 12 (1821-22) 254-. with equal characteristics. Kiirschak, J. Mth. A. 37 (1890) 317-. to a family of envelopes. Walton, W. Camb. Mth. J. 4 (1845) 152-. of form - = zf (xy) , classes, 2 special. Medolaghi, P. G. Mt. 35 (1897) 267-. group. Bohlin, K. Stockh. Ofv. (1887) 333-. integrable without use of arbitrary functions under the sign / . Pistoye, de. C. E. 78 (1874) 1102-. of light vibrations. Brill, A. [1868] Mth. A. 1 (1869) 225-. linear. Appell, P. C. E. 90 (1880) 296-, 731-. . Petot, A. C. E. Ill (1890) 522-. . Delassus, E. Par. EC. Norm. A. 13 (1896) 339-. . Le Eoux, J. Liouv. J. Mth. 4 (1898) 359-. linear. Vesnot, E. C. E. 128 (1899) 544-. , cases. Darboux, G. G. E. 95 (1882) 69-. , . Holmgren, E. Ups. Arsk. (1897) (Mt. Nt.) 33 pp. ; Fschr. Mth. (1898) 314-. , , with characteristics real. Delassus, E. C. E. 119 (1894) 40-; Par. EC. Norm. A. 12 (1895) (Suppl.) 53-. , class. Picard, E. C. E. 107 (1888) 476-. , . Koch, H. von C. E. 121 (1895) 517-. , . Drach, J. Bll. Sc. Mth. 21 (1897) 140-. , with analytic integrals. Picard, E. C. E. 121 (1895) 12-. t with constant coefficients and last term variable. Cauchy, A. L. [1822] Par. EC. Pol. J. 19 e cah. (1823) 510-. , 3rd degree. Trembley, J. [1798] St Pet. Ac. Sc. N. Acta 13 (1802) 101-. , and their germ integrals. Earnshaw, S. [1874] Camb. Ph. S. P. 2 (1876), 360-. , Laplace's method. Goursat, E. C. E. 122 (1896) 169-. , system. Horn, I. Acta Mth. 12 (1889) 113-. , , characteristic surface corresponding to. Cauchy, A. L. C. E. 13 (1841) 184-. , with 2 independent variables. Darboux, G. C. E. 105 (1887) 199-. 2nd order. Darboux, G. Par. EC. Norm. A. 7 (1870) 163-. Mainardi, G. Em. At. N. Line. 26 (1873) 79-. Poenisch, E. Santiago de Chile Un. A. 82 (1892) 675-. Stouff, X. C. E. 118 (1894) 1320. Child, M. Tor. Ac. Sc. At. 30 (1895) 494- ; 31 (1895) 400- or 568-. Goursat, E. C. E. 120 (1895) 712-. Koch, H. von. Stockh. Ofv. (1895) 721-. Dini, U. Em. E. Ac. Line. Ed. 5 (1896) (Sem. 2) 381-, 421- ; 6 (1897) (Sem. 1) 5-, 45-; Em. E. Ac. Line. Mm. 3 (1899) 33-. Foray th, A. E. [1897] Camb. Ph. S. T. 16 (1898) 191-. Dixon, A.C. B. A. Ep. (1899) 646. Goursat, E. Toul. Fac. Sc. A. 1 (1899) 31-, 439-. admitting an infinite group of point trans- formations. Medolaghi, P. A. Mt. 1 (1898) 229-. bilinear, geometrical interpretation. Egoroic, D. T. Fschr. Mth. (1898) 306. case. Tonelli, A. Em. E. Ac. Line. Ed. 4 (1888) (Sem. 2) 384-. J Weingarten, J. C. E. 116 (1893) 493-. characteristics, coalescence of 2 systems. Weber, E. von. C. E. 124 (1897) 1215-. __ t _-. Goursat, E. C. E. 124 (1897) 1294-. with characteristics real. Coulon, J. C. E. 128 (1899) 1386-. class. Stephan, E. Par. EC. Norm. A. 3 (1866) . 'Bdcklund, A. V. Erlang. Ps. Md. S. Sb. 8 (1876) 112-. . Picard, E. Palermo Cir. Mt. Ed. 5 (1891) 308-. VOL. I. 369 4840 Partial Differential Equations. Second and Higher Orders 4840 class. Kiirschdk,J. Mth. Termt. Ets. 15 (1897) 225- ; Mth. Nt. B. Ung. 14 (1898) 285-. , including Laplace's equation. Hearn, G. W. Phil. Trans. (1846) 133-. , reduction. Oster, B. Arch. Mth. Ps. 17 (1900) 321-. form f(r, s, t) = 0. Guldberg, A. N. Ts. Mth. 10 (B) (1899) 82- ; Fschr. Mth. (1899) 324. having general first integral. Tanner, H. W. L. [1877] Mess. Mth. 7 (1878) 89-. highest differential coefficient appearing in linear form. Liouville, B. C. B. 98 (1884) 216-. integrable by ordinary differential equations. Weber, E. van. Munch. Ak. Sb. 26 (1897) 425-. linear. Engel, F. Leip. Mth. Ps. B. 34 (1882) 39-. . Liouville, R. .C..B..98 (1884) 569-, 704. . Bianchi, L. Km. B. Ac. Line. Rd. 5 (1889) (Sem. 2) 35-. -^-. Du Bois-Reymond, P. Crelle J, Mth. 104 (1889) 241-. . Krukovskij, A. L [1894] Eec. Mth. (Moscou) 17 (1895) 759- ; Fschr. Mth. (1895) 393-. . Petot, A. C. E. 119 (1894) 510-. . Picard, E. Bll. Sc. Mth. 23 (1899) 150-. , cases. Levy, L. C. E. 100 (1885) 98-; Par. EC. Pol. J. 56 (1886) 63-. , . Le Roux, J. C. E. 123 (1896) 1052-. , . Guichard, G. C. B. 131 (1900) 100-. , class. Picard, E. Acta Mth. 12 (1889) 323-. , with constant coefficients. Coulon, J. C. E. 130 (1900) 765-. , general. Paoli, P. [1802] Mod. S. It. Mm. 10 (1803) 249-. , theorem. Picard, E. C. E. 107 (1888) 939-. , invariants belonging to. Burgatti, P. Em. E. Ac. Line. Ed. 5 (1896) (Sem. 2) 433-. , one parameter, case. Horn, J. Mth. A. 52 (1899) 271-. , system. Liouville, R. C. B. 104 (1887) 1496-. , with 2 independent variables. Le Roux, J. Par. EC. Norm. A. 12 (1895) 227-. , , theory. Pfannenstiel, E. [1882] Ups. S. Sc. N. Acta 11 (1883) (No. 9) 86pp. , variables. Cotton, E. C. B. 123 (1896) 936-; 124 (1897) 744-. methods of Charpit and Lagrange. Mainardi, G. Mil. G. I. Lomb. 9 (1856). 94-. point invariants. Medolaghi,P. Em. E. Ac. Line. Ed. 6 (1897) (Sem. 2) 247-, 313-. simultaneous, with 3 variables. Weber, E. . van. Munch. Ak. Sb. 25 (1896) 101-. soluble by differentiation. Tanner, H. W. L. [1876] Mess. Mth. 6 (1877) 32-. system. Niccoletti, 0. Em. B. Ac. Line. Ed. 4 (1895) (Sem. 1) 197-. with 2 independent variables. Konig, J. Mth. A. 24 (1884) 465-. , , theory. Levy, H. C. E. 75 (1872) 1094-. -r 3 independent variables. Vivanti, G. Mth. A. 48 (1897) 474-, 608. and 3rd order, linear, with 2 independent vari- ables. Picart, A. N. A. Mth. 2 (1883) 34-. higher orders. Falk, M. Ups. Arsk. (1875) 40pp. Higher orders. 3rd order. Guldberg, A. Christiania Skr. (Mth.-Nt. Kl.) (1900) No. 5, 43 pp. , admitting an intermediate integral. Guldberg, A. C. B. 130 (1900) 1452-. , case. Tonelli, A. Em. E. Ac. Line. Ed. 4 (1888) (Sem. 2) 458-. , linear, with 3 variables. Fries, J. de. N. Arch. Wisk. 10 (1884) 66-. , of orthogonal systems of surfaces. Ltvy, M. C. E. 77 (1873) 1435-. and higher orders, linear. Moon, R. Ph. Mg. 21 (1886) 63-. 4th order. Somigliana, C. A. Mt. 18 (1890) 59-. . Fredholm,!. Stockh. Ofv. (1899) 935-. higher order. Gutzmer, A. Liouv. J. Mth. 6 (1890) 405-. , extension of Biemann's method to. De- lassus, . C. B. 117 (1893) 510-. , with intermediate first integrals. Back- lund, A. V. [1876] Mth. A. 9 (1877) 199- ; 13 (1878) 69-. any order. Davidov, A. Y. [1865] (xn) Bee. Mth. (Moscou) [1] (1866) 351-. , and their reduction to linear system of 1st order. Cauchy, A. L. C. E. 15 (1842) 131-. , any number of variables. Backlund, A. V. Lund Un. Acta 12 (1875-76) (Mth., No. 2) 4 pp. , theory, general. Lie, S. Leip. Mth. Ps. B. 47 (1895) 53-. ?ith order, class. Tonelli, A. Em. E. Ac. Line. Ed. 5 (1889) (Sem. 1) 178-. , formulae relative to. Tonelli, A. Bm. E. Ac. Line. Ed. 5 (1889) (Sem. 1) 508-. of potential. Picard, E. C. E. 90 (1880) 601-. satisfied by concomitants of quantics. For- syth, A.R. L. Mth. S. P. 19 (1889) 24-. certain definite integrals. Torelli, G. Nap. Bd. 29 (1890) 123-. similar to those of hydrodynamics, general properties. Craig, T. [1881] Am. Ac. Mm. 10 (Pt. 2) (1882) 362-. simultaneous. Folkierski, W. Par. T. Nauk Sc. Pam. 3 (*1873) 1-; Bll. Sc. Mth. As. 6 (*1874) 156-. . Dixon, A. C. [1899] Phil. Trans. (A) 195 (1901) 151-. .linear. Painleve,. C. E. 104(1887)1497-. systems, in involution. Goursat, E. C. E. 122 , , cases. Appell, P., & Picard, . C. B. (1896) 1258-. 92 (1881) 692-. with 2 independent variables. Moutard, T. , theory. Campbell, J. E. [1898] L. Mth. C. R. 70 (1870) 834-. S. P. 31 (1900) 235-. 370 4840 Partial Differential Equations. Second and Higher Orders 4840 system. Goursat, E. C. K. 104 (1887) 1361-. . Picard, E.. C. B. 112 (1891) 685-. systems. Picard, E. C. E. 114 (1892) 805-. _. Beudon, J. C. E. 120 (1895) 304-. , characteristics of which depend on finite number of parameters. Beudon, J. Par. EC. Norm. A. 13 (1896) (Suppl.) 51 pp. reducible to ordinary differential equations. Beudon, J. C. E. 127 (1898) 1003-. , symmetric. Somigliana, C. A. Mt. 22 (1894) 143-. in theory of asymmetric algebra. Forsyth, A. R. Camb. Ph. S. T. 16 (1898) 291-. and theory of surfaces, relation between. -Stouff, X. Par. EC. Norm. A. 13 (1896) 9-. Existence of irreducible equations. Koenigs- berger, L. Berl. Ak. Sb. (1894) 989-. Extension of Cauchy's theorem to most general systems. Delassus, E. C. E. 122 (1896) 772- ; Par. EC. Norm. A. 13 (1896) 421-. General equation, geometrical significance. Rolling, . Mth. Ts. 5 (1863) 79-. Generation. Minich, S. R. Yen. At. 4 (1845) 377-. Integrability of Airy's equations. Moon, R. E. S. P. 15 (1867) 486-. Integrable equations of 2nd order, two. Goursat, E. C. E. 129 (1899) 31-. types of equations of 2nd order. Goursat, E. C. E. 127 (1898) 854-. Integrals. Poisson, S. D. Par. S. Phlm. Bll. 3 (1804) 227- ; Par. EC. Pol. J. 19 e call. (1823) 215-. Cauchy, A. L. C. E. 16 (1843) 572-, analytic, of linear equations with 2 variables. Le Roux, J. C. E. 125 (1897) 1015-. Delasmu, . C. E. 118 (1894) 968-. , |^ 2 = ^ . Le Roux, J. Bll. Sc. Mth. 19 (1895) 127-. complete, of some classes of equations, any order. Pennacchietti,'G. Mil. I. Lomb. Ed. 17 (1884) 279-. , and general solution of equations. Ser- sau-y,V. Wien Ak. D. 53 (1887) (Ab. 2) 1-. , of homogeneous equations which do not contain explicitly the unknown function. Gubkin, I. 'S. Mosc. S. Sc. Bll. 73 (No. 1) (1891) 9-; Fschr. Mth. (1891) 396-. of equations, 2nd order. Serdobinskij , V. E. Eec. Mth. (Moscou) 13 (1886) 511- ; Fschr. Mth. (1887) 354. -- , -- , cases. Picard, E. C. E. Ill (1890) 487-.. first, of equations of 2nd degree in 4 or 5 variables. Trembley, J. [1805] St Pet. Ac. Sc. Mm. 4 (1811) 120-. , ----- and of 3rd in 3 variables. Trembley, J. [1803] St Pet. Ac. Sc. N. Acta 15 (1806) 257-. first, of equations, 2nd order. Boole fnon st Pa - Ac - So - B11 - " - . -- , cases. Tanner, H. W. L. Mess. Mth. 5 (1876) 133-. forms of arbitrary functions. Challis, J. Ph. Mg. 6 (1829) 296-. general, character. Forsyth, A. R. [18971 L. Mth. S. P. 29 (1898) 5-. , of equations with constant coefficients Tortolini, B. [1854] Mod. S. It. Mm. 25 (1855) 310-. , -- of form Starkov, A. N. Es. S. Nt. Mm. (Mth.) 2 (*1879). for given initial conditions. Goursat E C E. 125 (1897) 640-. intermediate, of equations, 2nd order. Goursat, E. Acta Mth. 19 (1895) 285-; C. E. 127 (1898) 603-. of linear equations. Cauchy, A. L. C. E. 16 (1843) 469-. number of arbitrary functions. Gomes Teixeira, F. Bordeaux S. Sc. Mm. 2 (1878) 315-. of partial differential equations and difference equations, arbitrary functions. Oltramare, G. As. Fr. C. E. (1895) (Pt. 2) 171-. problem on determination. Goursat, E. C. E. 121 (1895) 671-. singularities. Horn, J. Mth. A, 33 (1889) 310-. of j + = 0. Matthiessen, L. Z. Mth. Ps. 16 (1871) 228-. Integrating factor, . theory. Lie, M. S. Chris- . tiania F. (1874) 242-. of s + Qpq + Rp + Sq + T-O, determina- . tion. Johnsen, S. JV. (xn) Ts. Mth. 2 (1878) 129-. Integration. Cauchy, A. L. [1823] ' Par. Mm. Ac. Sc. 9 (1830) 97-. Pagani, G. M. Quetelet Cor. Mth. 6 (1830) 286-. Bronwin, B. Camb. Mth. J. 2 (1841) 209-. Weiler, A. Grunert Arch, 33 (1859) 171-. Bour, E. Par. EC. Pol. J. cah. 39 (1862) 149-. Graindorge, J. (x) Liege S. Sc. Mm. 5 (1873) No. 3, xii + 189 pp. . Peterson, K. M. [1876-79] (xn) Eec. Mth. (Moscou) 8 (1876) (Pt. 1) 291- ; 9 (1878-81) (Pt. 1) 137- ; 10 (1882-83) (Pt. 1) 169-. Nazimov, P. S. Mosc. Un. Mm. (Ps.-Mth.) 1 (*1880) 1-, 257-. Sersawy, V. Wien Ak. D. 49 (1885) (Ab. 2) Antomari, X. Par. S. Mth. Bll. 19 (1891) 154-. application of contact-transformations. Brill, J. N. A. Mth. 10 (1891) 362-. ' 371 AA 4840 Partial Differential Equations, Integration application of differentio-differential relations. SapoSnikov, N. A. Mosc. S. Sc. Bll. 73 (No. 2) (1891) 20-; Fschr. Mth. (1891) 398-, case. Zaremba, S. Par. EC. Norm. A. 7 (1890) 135-. cases. Christoffel, E. B. Gott. Nr. (1871) 435-. . Steen, A. (xn) Ts. Mth. 6 (1876) 69-. of class of equations. Duhamel, J. M. C. C. E. 3 (1836) 638-. linear equations by definite integrals. Lie, M. S. [1881] Arch. Mth. 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[1881] Mth. A. 19 (1882) 435-. Casorati, F. Em. E. Ac. Line. T. 6 (1882) 121-. Pfannenstiel, E. Goteb. Hndl. 17 (1882) 1-. Kapteijn, W. Amst. Ak. Vs. M. 18 (1883) 95-; Arch. Neerl. 18 (1883) 105-. Floquet, G. Par. EC. Norm. A. 1 (1884) 405-. Thome, L. W. Crelle J. Mth. 96 (1884) 185- ; 107 (1891) 79. Brioschi, F. A. Mt. 13 (1885) 1-. Griinfeld, E. Crelle J. Mth. 98 (1885) 333-. Lie, S. Christiania F. (1885) No. 21, 4 pp. Starkov, A. P. Kazan S. Nt. (Ps.-Mth.) P. 3 (1885) 79- ; Fschr. Mth. (1885) 283-. Cayley, A. QJ. Mth. 21 (1886) 321- ; Crelle J. Mth. 100 (1887) 286-. Escherich, G. von. Wien Ak. D. 51 (1886) 1-. Thome, L. W. Crelle J. Mth. 99 (1886) 88- ; 101 (1887) 203-. Cayley, A. Crelle J. Mth. 101 (1887) 209-. Painleve, P. C. E. 105 (1887) 58-. Volterra, V. Em. E. Ac. Line. Ed. 3 (1887) (Sem. 1) 393-; Em. S. It. Mm. 6 (1887) No. 8, 104 pp. ; Palermo Cir. Mt. Ed. 2 (1888) 69-. Fuehs, L. Berl. Ak. Sb. (1888) 1115-, 1273- ; (1889) 713- ; (1890) 21-. Anisimov, V. A. Mosc. Un. Mm. (Ps.-Mth.) 8 Bigiavi, C. Em. E. Ac. Line. Ed. 5 (1889) (Sem. 1) 651- ; 6 (1890) (Sem. 1) 86-. Cels, J. C. E. Ill (1890) 98-; Par. EC. Norm. A. 8 (1891) 341-. Vessiot, E. C. E. 112 (1891) 778-. Markoff, A. C. E. 113 (1891) 685-, 790-, 1024-. Painleve, . C. E. 113 (1891) 739-. Rudskij, M. N. Es. S. Nt. Mm. (Mth.) 13 (1891) 1-; Fschr. Mth. (1891) 340. Cels, J. C. E. 115 (1892) 1057- ; 116 (1893) 176-. Starkov, A. N. Es. S. Nt. Mm. (Mth.) 14 (1892) 191- ; Fschr. Mth. (1892) 291. 375 4850 Theory of Ordinary Linear Differential Equations 4850 Case, and its application to trinomial equations. Floridia, G. G. Mt. 31 (1893) 354-; 32 (1894) 86-; 33 (1895) 13-, 145-, 242-, 327-, 379-. Metzler, G. F. A. Mth. 9 (1894-95) 171- ; 11 Heymann, W. Mth. A. 26 (1886) 534-. having hypergeometric series for integral. Goursat, E. Par. EC. Norm. A. 10 (1881) (1896-97) 1-. (Suppl.) 142 pp. Anisimov, F. A. [1898] Eec. Mth. (Moscou) 20 _ ( 3rd or ; der . Starkov, A. N. Es. S. Nt. Mm. (1899) 316-. Igel, B. Mh. Mth. Ps. 9 (1898) 124-. Grilnfeld, E. Mh. Mth. Ps. 9 (1898) 372. B6cher, M. A. Mth. 12 (1898-99) 45-. Dini, U. A. Mt. 2 (1899) 297-; 3 (1899) 125-. Horn, J. Mth. A. 53 (1900) 177-. Adjoint and associated equations, nth order. Grilnfeld, E. Crelle J. Mth. 121 (1900) 218-. equations. Pick, G. Wien Ak. Sb. 101 (1892) (Ab. 2a) 893-. . Igel, B. Mh. Mth. Ps. 5 (1894) 163- ; 10 (1899) 229-. . Gilnther, P. Crelle J. Mth. 117 (1897) 168. Algebraically integrable equations. Frobenius, G. Crelle J. Mth. 80 (1875) 183-. . Fuchs, L. Berl. Ak. Sb. (1890) in relation to Fuchsian functions. Viterbi, A. G. Mt. 35 (1897) 150- ; 36 (1898) 55-. homogeneous equations. Boulanger, A. Par. EC. Pol. J. 4 (1898) 1-. Analogies with algebraic equations. Mainardi G. Em. At. N. Line. 26 (1873) 386-. Application of determinant formula. Anisimov, V. A. Vars. S. Nt. Tr. (1897) (Ps. C.) Fasc. 3, Mm. 2, 16 pp.; Fschr. Mth. (1898) 263 ; Mth. A. 51 (1899) 388-. infinite determinants. Koch, H. von. Acta Mth. 15 (1891) 53-; 16 (1892-93) 217-. to homogeneous equations. Koch, H. von. Stockh. Ofv. (1890) 225-, 499-; Fschr. Mth. (1890) 316. integral calculus. Brassinne, E. Toul. Mm. Ac. 4 (1860) 510-. theory of substitutions. Jordan, C. Par. S. Mth. Bll. 2 (1874) 100-. Arbitrary constants, determination. Trudi, N. [1864] (xi) Nap. Ac. At. 2 (1865) No. S, 25pp. Associated equations. Fuchs, L. Berl. Ak. Sb. (1899) 182-. Binary forms, variables of which are funda- mental integrals of equation, 2nd order. Vernier, P. Par. S. Mth. Bll. 22 (1894) 133-. Binomial biordinal. Cockle, (Sir) J. L. Mth. S. P. 11 (1879-80) 123-. equations. Steen, A. (xn) Ts. Mth. 4 (1874) 104-. . Craig, T. [1886] J. H. Un. Cir. [61 (1886-87) 28-. with algebraic coefficients, class. Appell, P. C. E. 94 (1882) 202- ; Par. EC. Norm. A. 12 (1883) 9-. , nth order, case. Pochliammer, L. Mth. A. 38 (1891) 247-. 376 (1885) 13-; Fschr. Mth. (1885) , . Markov, A. A. [1892] Eec. Mth. (Moscou) 16 (1893) 352- ; Fschr. Mth. (1892) 313. satisfied by squares of X%y" + X l y' + X y = 0. Spitzer, S. Schlomilch Z. 4 (1859) 73-. y= I e w ( m+x ) (u - x) A ~! (u - /3) *-i log [(m + x) (u -"a) (u - /S)l du. Spitzer, S. Schlomilch Z. 3 (1858) 178-. Cases. Cockle, Jas., & Harley, R. Manch. Ph. S. P. 3 (1862-63) 16-. . Michaelis, J. P. Lux. S. Sc. Mm. 8 (1865) 160-. . Hermite, C. [1874] Crelle J. Mth. 79 (1875) 324-. . Harley, E. B. A. Ep. (1878) 466-. . Steen, A. Acta Mth. 3 (1883) 277-. . Goursat, E. C. E. 98 (1884) 1248-. . Brioschi, F. Mth. A. 26 (1886) 106-. . Gutzmer, A. [1891-95] G. Teix. J. Sc. 10 (1891)3-; 13 (1897)3-. . Jackson, F. H. Edinb. Mth. S. P. 14 (1896) 104-. . Markov, A. A. St Pet. Ac. Sc. Mm. 3 (1896) No. 10, 17 pp. . Fano, G. Em. E. Ac. Line. Ed. 8 (1899) (Sem. 1) 285-. Class. Picard, E. C. E. 90 (1880) 128-. . Appell, P. C. E. 91 (1880) 972-; 96 (1883) 1018-. . Halphen, G. H. C. E. 92 (1881) 779-. . Craig, T. Am. J. Mth. 7 (1885) 279-. . Cels,J. C.E. Ill (1890) 879-; 112(1891) 985-. . Fuchs, L. Berl. Ak. Sb. (1896) 753-. . Horn, J. Mth. A. 50 (1898) 525-. discussed by Mittag-Leffler. Brioschi, F. A. Mt. 11 (1882-83) 81-. , special. Hamburger, . Crelle J. Mth. 103 (1888) 238-. Common divisors and multiples of linear ex- pressions. Heffter, L. D. Mth. Vr. Jbr. 4 (1897) 131-. Complete cubic resolvent, explicit form. Harley, (Rev.) R. B. A. Ep. (1886) 439-. Construction of equation from given element of integral function. Heun, K. Am. J. Mth. 11 (1889) 215-. , particular integrals of which are products of those of 2 given equations. Spitzer, S. Arch. Mth. Ps. 42 (1864) 62-. , squares of those of X 2 y" + X,y' + X tt y = 0. Spitzer, S. Arch. Mth. Ps. 42 (1864) 64. f fm+x satisfied by y = e A J * ^ "*. Spitzer, S. Arch. Mth. Ps. 42 (1864) 345-. equations of higher order, cases. Spitzer, S. Arch. Mth. Ps. 65 (1880) 321-. 4850 Theory of Ordinary Linear Differential Equations 4850 Criticoids. Cockle, (Sir) J. [1869-71] Ph. Mg. 39 (1870) 201- ; 41 (1871) 360-. . Raw son, E. B. A. Ep. (1887) 604-. , inverse problem. Cockle, (Sir) Jas. Ph. Mg. 12 (1881) 189-. Decomposition, theory. Cayley, A. QJ. Mth. 21 (1886) 331-. Determinantal relation in theory. Koenigsberger, L. Crelle J. Mth. 105 (1889) 170-. . Igel, B. Mh. Mth. Ps. 4 (1893) 381-. Differential resolvents. Harley, (Rev.) R. Manch. Lt. Ph. S. Mm. & P. 5 (1892) 79-. , results connected with. Harley, (Rev.) R. B. A. Ep. (1896) 714-. Distributive operations, non-homogeneous equa- tions. Pincherle, S. Em. E. Ac. Line. Ed. 5 (1896) (Sem. 1) 301-. Divergent series in theory. Horn, J. D. Mth. Vr. Jbr. 8 (1900) (Heft 1) 219-. Elementary divisors of determinants, appli- cation to homogeneous equations. Sauvage, L. Mars. Fac. Sc. A. 6 (1897) Fasc. 5, 9 pp. Elimination. Favre-Rollin, A. M. Liouv. J. Mth. 1 (1836) 88-. Equation with constant coefficients, equal roots, treatment. Routh, E. J. QJ. Mth.- 19 (1883) 262-. having second member, application of Hermite's method. Darboux, G. Bll. Sc. Mth. As. 3 (1879) 325-. for hypergeometric series of nth order. Pochhammer, L. Crelle J. Mth. 108 (1891) 50-. including Lame's as special case. Darboux, G. C. E. 94 (1882) 1645-. Equations. admitting integrals (4) connected by quadratic relation. Goursat, E. Par. S. Mth. Bll. 25 (1897) 95-. , logarithmic differentials of which are doubly periodic functions. Fuchs, L. Liouv. J. Mth. 4 (1878) 125-. algebraic analogy. Mainardi, G. Mm. At. N. Line. 26 (1873) 384-. with algebraic integrals. Jordan, C. [1877] Crelle J. Mth. 84 (1878) 89-. . Poincarf, H. C. E. 92 (1881) 698-. algebraic, 1st order and degree, singular points. Bjorling, C. F. E. Arch. Mth. Ps. 4 (1886) 358-. automorphic by change of function and variable. Appell, P. ActaMth. 15 (1891) 281- ; C. E. 112 (1891) 34-. belonging to same Eiemann's class. Marotte, F. C. E. 125 (1897) 84-. type as their adjoint equations. Fano, G. Tor. Ac. Sc. At. 34 (1898) 260- or 415-. . Fuchs, R. Crelle J. Mth. 121 (1900) 205-. with coefficients algebraic. Painleve, P. C. E. 106 (1888) 535-. . Guichard, C. Acta Mth. 12 (1889) 57-. with coefficients algebraic. Thomt, L. W. Crelle J. Mth. 122 (1900) 1-. functions of independent variable. Appell, P. C. E. 92 (1881) 61-. , many-valued. Thome, L. W. Crelle J. Mth. 115 (1895) 33-, 119- ; 119 (1898) 131- ; 121 (1900) 1-. constant. Lorenz, L. Mth. Ts. 4 (1862) 38-. . Picquet, H. Par. EC. Pol. J. Call. 53 (1883) 79-. . Meray, C. Bll. Sc. Mth. 12 (1888) 196-. . Auric, . N. A. Mth. 13 (1894) 47-. . Franklin, F. Am. J. Mth. 19 (1897) 92-. . Gonzalez, M. Arg. S. Ci. A. 47 (1899) 178-. , d'Alembert's method. Stodoikiewicz^ A. J. Krk. Ak. (Mt.-Prz.) Ez. 6 (1893) 105- ; Crc. Ac. Sc. Bll. (1892) 294-. , particular case. Ocagne, M. d'. N. A. Mth. 3 (1884) 138-. doubly periodic. Picard, E. C. E. 90 (1880) 293-. . Mittag-Leffler, G. C. E. 90 (1880) 299-. . Appell, P. C. E. 92 (1881) 1005-. . Picard, E. Crelle J. Mth. 90 (1881) 281-. . Floquet, G. C. E. 98 (1884) 38-, 82-; Par. EC. Norm. A. 1 (1884) 181-. . Bigiavi, C. Em. E. Ac. Line. Ed. 6 (1890) {Sem. 1) 339-. . Malmborg, M. Ups. Arsk. (1897) (Mt. Nt.) 31 pp. periodic. Floquet, G. Par. EC. Norm. A. 12 (1883) 47-. , and their elementary integrals. Cauchy, A. L. Par. Mm. Ac. Sc. 22 (1850) 587-. rational. Schafheitlin, P. Crelle J. Mth. 106 (1890) 285-; 111 (1893) 44-. . Koch, H. von. C. E. 116 (1893) 91-. . , determination of groups. Poincare, H. C. E. 96 (1883) 691-, 1302-. , relations due to definite integrals. Mellin, H. Helsingf. Acta 21 (1896) No. 6, 57 pp. , solution of which depends on quad- rature of algebraic irrational. Dillner, G. C. E. 91 (1880) 616-, 687-, 721- ; 92 (1881) 117, 235-, 289-. real, singular points. Bocher, M. N. Y. Am. Mth. S. Bll. 5 (1899) 275-. variable. Brassinne, E. Toul. Mm. Ac. 6 (1839-41) 50-. . Fuchs, L. Crelle J. 66 (1866) 121- ; 68 (1868) 354-. connected with algebraic surface. Klein, F. [1894] D. Mth. Vr. Jbr. 4 (1897) 91-. cubic equation. Cockle, J. Ph. Mg. 24 (1862) 37-. equivalent to Lagrange's adjoint equations. Brioschi, F. A. Mt. 24 (1896) 339-. 377 integrable, class, new. Halphen, G. H. C. B. 101 (1885) 1238-. by elliptic functions, class. Brioschi, F. Rm. B. Ac. Line. T. 4 (1880) 241-. ---- , , generation. Brioschi, F. A. Mt. 10 (1880-82) 74-. -- hyperbolic sine. Imseneckij, V. G. [1891] Bee. Mth. (Moscou) 16 (1893) 177- ; Fschr. Mth. (1891) 311-. -- special functions. Schou, E. S. Kjab. Ov. (1897) 433-. systems. Mayer, A. Mth. A. 5 (1872) 448-. integrals having fixed branch-points. Fuchs, L. Berl. Ak. Sb. (1884) 699-. one singularity at finite distance. Giinther, P. Crelle J. Mth. 105 (1889) 1-; 119 (1898) 330-. regular at a singularity. Bendixson, I. Stockh. Ofv. (1892) 279-. satisfying relations of form 4850 Theory of Ordinary Linear Differential Equations 4850 exact, symbolic treatment. Johnson, W. W. Am. J. Mth. 10 (1888) 94-. of form (mx 2 + nx +p) y" + (qx + r) y' + sy = 0. Spitzer, S. Schlomilch Z. 4 (1859) 250-. *() =x m (Axz' + Bz). Spitzer, S. Grunert Arch. 38 (1862) 458-. xyW - myP-U = ay . Spitzer, S. Grunert Arch. 40 (1863) 232-. (a, + b*x) y" + (a, + b t x) y' + (a + b x) y = 0. Spitzer, S. Arch. Mth. Ps. 58 (1876) 361-. and group-theory. Marotte, F. Toul. Fac. Sc. A. 12 (1898) H, 92 pp. homogeneous. Catalan, E. C. Brux. Ac. Bll. 43 (1877) 335-. . Stenberg, E. A. Stockh. Ofv. (1884) No. 5, 143-. . Lie, S. Leip. Mth. Ps. B. 43 (1891) 253-. . Bendixson, I. [1892] Stockh. Ofv. (1892) 91-; Stockh. Ak. Hndl. Bh. 18 (Afd. 1) (1893) No. 1, 29 pp. . Gutzmer, A. Crelle J. Mth. 115 (1895) 79-. , with algebraic relations between funda- mental integrals. Wirtinger, W. Wien Ak. Sb. 98 (1890) (Ab. 2a) 66-. , . Wallenberg, G. Crelle J. Mth. Ill (1893) 89-. , . Fano, G. Mth. A. 53 (1900) 493-. , class, special. Guldberg, A. Christiania F. (1894) No. 9, 12 pp. ; Fschr. Mth. (1893- 94) 545-. , with coefficients doubly periodic. Stenberg, E. A. Helsingf. Acta 16 (1888) 493-, 555-; 19 (1893) No. 11, 7 pp. , common particular integrals, conditions for. Griinfeld, E. Z. Mth. Ps. 30 (1885) 210-. , general integral of which is uniform. Floquet, G. C. B. 121 (1895) 676-. , group-theory. Beke, E. Mth. A. 49 (1897) 573-. , with homogeneous relations between integrals. Fuchs, L. Acta Mth. 1 (1882) 321-. , properties. Stenberg, E. A. Acta Mth. 8 (1886) 119-. , regularity of system. Sauvage, L. Mars. Fac. Sc. A. 4 (1895) Fasc. 3, 14 pp. , relations of two, to each other. Thome, . D. Nf. Tbl. (*1873) 50-. , resolvents. Beke, M. Mth. Termt. Ets. 16 (1898) 407-. , special. Hurwitz, A. Mth. A. 26 (1886) , symmetric functions in. Beke, E. Mth. A. 45 (1894) 295-. , with symmetric integral determinant. Hazzidakis, J. N. Crelle J. Mth. Ill (1893) 315-. , systems, property. Hazzidakis, J. N. [1880] Crelle J. Mth. 90 (1881) 80-. , M+N^=0, proof that (Mx + Ny)~i is an integrating factor. Stokes, G. G. Camb. Mth. J. 4 (1845) 241-. Appell, P. C. B. 93 (1881) 699-. irreducible, property. Ratner, E. Mth. A. 32 (1888) 566-. Laplace's method. Goursat, E. Am. J. Mth. 18 (1896) 347-. .linearoid. Wilczynski, E. J. Am. J. Mth. 21 (1899) 354-. non-homogeneous. Thome, L. W. Crelle J. Mth. 107 (1891) 51-. , class. Floquet, G. Par. EC. Norm. A. 4 (1887) 111-. , and their functional operations. Pincherle, S. Bologna Bd. (1895-96) 77-. any order, with fixed critical points. Painleve, P. G. B. 130 (1900) 1112-. -- 2 variables. Libri , G. C. B. 8 (1839) 732-. -- , --- (Libri). Liouville, J. C. B. 8 (1839) 792-. 1st order and degree, with 2 variables, singular points. Fouret, G. Par. S. Mth. Bll. 19 (1891) 128-. , fundamental invariants of system. Griinfeld, E. Z. Mth. Ps. 39 (1894) 237-. , general integral of which is function with finite number of branches. Cohen, A. Toul. Fac. Sc. A. 1 (1899) 239-, 469-. , reduction of solution to continued fraction, case. Laguerre, (le capit.) E. C. B. 98 (1884) 209-. 2nd order. Sturm, J. C. F. Liouv. J. Mth. 1 (1836) 106-. Svanberg, A. F. Stockh. Ak. Hndl. (1837) 207-. Cockle, Jas. Mess. Mth. 1 (1862) 118-, 164-, 241-. Heine, E. Berl. Mb. (1864) 13-. Gegenbauer, L. Arch. Mth. Ps. 55 (1873) 252-. Moutard, T. C. B. 80 (1875) 729-. Pepin, T. C. B. 82 (1876) 1323-. Tait, P. G. [1876] Edinb. B. S. P. 9 (1878) 93-. Pepin, T. A. Mt. 9 (1878-79) 1-. Brioschi, F. As. Fr. C. B. 8 (1879) 278-. 378 4850 Theory of Ordinary Linear Differential Equations (Ord. 2) 4850 Laguerre, E. [1879] Par. S. Mth. Bll. 8 (1880) 35-. Tannery, J. Par. EC. Norm. A. 8 (1879) 169-. Gylden, J. A. H. C. B. 90 (1880) 208-, 344-. Hermite, C. [1880] A. Mt. 10 (1880-82) 101-. Humbert, G. Par. EC. Pol. J. Cah. 48 (1880) 207-. Picard, E. C. E. 90 (1880) 1479-. mttag-Leffler, G. C. E. 91 (1880) 978-. Brioschi, F. C. E. 93 (1881) 941-. Gylden, H. Stockh. Ak. Hndl. Bh. 14 (Afd. 1) (1889) A T o. 6, 23 pp. Petrovitch, M. Par. S. Mth. Bll. 25 (1897) 221- Thome, L. W. Crelle J. Mth. 117 (1897) 185-. Liapounqff, A. C. B. 128 (1899) 910-. with algebraic integrals. Fuchs, L. Crelle J. Mth. 81 (1876) 97-; 85 (1878) 1-. . Jordan, C. C. E. 82 (1876) 605-. . Ermakov, F. P. [1894] Bee. Mth. (Moscou) 17 (1895) 500-. allied to Hermite's equation. Vleck, E. B. van. Am. J. Mth. 21 (1899) 126-. with arbitrary parameter. Picard, E. C. E. 118 (1894) 379-. arithmetical developments in theory. Fricke, E. D. Mth. Vr. Jbr. 2 (1893) 59-. cases. Molins,H. Toul. Mm. Ac. 2 (1858) 19-. . Hoppe, R. Z. Mth. Ps. 9 (1864) 56-. . Craig, T. Am. J. Mth. 8 (1886) 180-. . Lohnstein, . Z. Mth. Ps. 38 (1893) 27-. . Igel, B. Mh. Mth. Ps. 5 (1894) 85-. , special. PocMiammer, L. Mth. A. 41 (1893) 174-. class. Brioschi, F. A. Mt. 9 (1878-79) 11-; 10 (1880-82) 4-. . Fuchs, L. Crelle J. Mth. 100 (1887) 189-. , and equation of 5th degree. Besso, D. Em. E. Ac. Line. Ed. 2 (1886) (Sem. 1) 593-. integrable by elliptic functions. Krause, M. Berl. Ak. Sb. (1900) 258-. with coefficients binomial. Zmurko, L. D. Nf. Tbl. (*1881) [28]-. doubly periodic, case. Elliot, Z. Acta Mth. 2 (1883) 233-. functions. Krause, M. Leip. Mth. Ps. B. 44 (1892) 15-, 238-. linear, case. Pochhammer, L. Mth. A. 36 (1890) 84- ; 38 (1891) 225-. non-analytic, singular points. Bocher, M. N. Y. Am. Mth. S. T. 1 (1900) 40-, 507. periodic. Liapunov, A. M. Kharkov Mth. S. Com. 5 (1896?) 190- ; Fschr. Mth. (1897) 289. transcendent. Cassel, G. Stockh. Ofv. (1890) 381-. variable. Tychsen, C. (xn) Ts. Mth. 1 (1871) 28-. in connexion with plane curves of 4th degree. Gordon, P. Mth. A. 46 (1895) 606-. with fixed critical points. Painleve, P. C. E. 117 (1893) 686-; 127 (1898) 541-, 945-; 129 (1899) 750-, 949-. fundamental invariants. Vogt, H. Par. EC. Norm. A. 6 (1889) (SupplJ 71 pp. Hermite's, application of doubly periodic functions of Weierstrass. Stenberg, E. A. Helsingf. Acta 16 (1888) 205- ; Fschr. Mth. (1886) 296. homogeneous. Foldberg, P. Ts. Mth. 4 (1886) 81-; Fschr. Mth. (1887) 323. . Chini, M. Tor. Ac. Sc. At. 33 (1897) 505- or 737-. , functions satisfying. Floquet, G. Nancy S. Sc. Bll. (1895) 78-. . , integrable by doubly periodic functions of 2nd kind. Naetsch, E. Leip. Mth. Ps. B. 45 (1893) 538-. , properties, and algebraic equations. Besso, D. [1882] Em. E.Ac. Line. Mm. 14(1883)14-. , system of solutions of theorem. Fuchs, L. Bll. Sc. Mth. As. 4 (1880) 334-. of hyperbolic type, for which Laplace's series is finite in one sense only. Niccoletti, 0. Em. E. Ac. Line. Ed. 6 (1897) (Sem. 1) 334-. , both senses. Niccoletti, O. Em. E. Ac. Line. Ed. 6 (1897) (Sem. 1) 307-. integrability by algebraic functions, cases. Hansen,P.C.V. (xn) Ts. Mth. 6 (1876) 144-. integrable, class. Grave, D. A. Bee. Mth. (Moscou) 14 (1890) 197- ; Fschr. Mth. (1889) 325-. ' integral forms. Liouville, E. Par. EC. Pol. J. 56 (1886) 7-. for which inverse function has finite multi- formity. Schlesinger, L. Crelle J. Mth. 110 (1892) 265-. non-homogeneous, properties. Besso, D. [1882] Em. E. Ac. Line. Mm. 14 (1883) 40-. polynomials satisfying. Humbert, G. Par. S. Phlm. Bll. 4 (1880) 238-. , roots. Bocher, M. N. Y. Am. Mth. S. Bll. 4 (1898) 256-. primary forms occurring in. Schlesinger, L. Crelle J. Mth. 110 (1892) 130-. property. Brioschi, F. A. Mt. 10 (1880-82) 1-. reduction to canonical form. Klein, F. Mth. A. 38 (1891) 144-. refutation of Pepin's theorem. Fuchs, L. C. E. 82 (1876) 1494- ; 83 (1876) 46-. relation corresponding to. Winckler, A. [1877] Wien Ak. Sb. 76 (1878) (Ab. 2) 173-. with relation between particular integrals. Winckler, A. Wien Ak. Sb. 92 (1886) (Ab. 2) 7-. satisfied by function of independent variable of finite number of terms. Molitis, H. Toul. Mm. Ac. 2 (1858) 357-. series relative to. Liapounoff, A. C. E. 131 (1900) 1185-. theorem of Hermite. Fuchs, L. C. E. 85 (1877) 947-. theorems, generalisation. Casorati, F. A. Mt. 10 (1880-82) 224-. and higher orders. Libri, G. [1833] Par. Mm. Ac. Sc. 14 (1838) 94-. 3rd and higher orders. 3rd order. Cockle, (Sir) Jas. [1864-76] QJ. Mth. 7 (1866) 316- ; 8 (1867) 373-; 14 (1877) 340-. . Laquerre, E. C. E. 88 (1879) 116-. . Co'mbescure, E. C. E. 88 (1879) 275-. 379 4850 Theory of Ordinary Linear Differential Equations (Ord. 3 n) 4850 3rd order. Goursat, E. C. E. 98 (1884) 609-. . Halphen, . Mth. A. 24 (1884) 461-. . Painleve; P. C. E. 104 (1887) 1829-. , case. Malmsten, C. J. Stockh. Ofv. 5 (1848) 189-. , class integrable by hypergeometric series. Besso, D. Em. E. Ac. Line. Mm. 19 (1884) 251-. , with coefficients doubly periodic functions. Krause, M. Leip. Mth. Ps. B. 45 (1893) 35-. , with fixed critical points. Painleve, P. C. E. 130 (1900) 879-. t satisfied by automorphic functions. Hunvitz, A. Mth. A. 33 (1889) 345-. and 4th orders. Campbell, D. F. QJ. Mth. 31 (1900) 161-. 4th order. Halphen, G. H. C. E. 97 (1883) 247-. , class. Goursat, E. Par. S. Mth. Bll. 11 (1883) 144-. , , and equation of 5th degree. Besso, D. Em. E. Ac. Line. Ed. 1 (1885) 183-, 233-, 884. , integrable by hypergeometric series. Besso, D. Em. E. Ac. Line. Mm. 19 (1884) 245-. , determination of group of rationality. Marotte, F. C. E. 126 (1898) 715-. , integrable when particular integral is known. Enriques, F. Mil. I. Lomb. Ed. 29 (1896) 257-. , invariants. Halphen, G. H. Acta Mth. 3 (1883) 325-. , reduction, case. Goursat, E. C. E. 100 (1885) 233-. 5th order, integral curves of which are contained in algebraic configurations. Fano, G. Mil. I. Lomb. Ed. 32 (1899) 843-. and 6th orders, integral curves of which are contained in a quadric. Fano, G. Tor. Ac. Sc. At. 34 (1898) 285- or 388-. ?ith order, and their n adjoint equations. Griinfeld, E. Crelle J. Mth. 117 (1897) 273-. , , connexion between funda- mental determinants. Griinfeld, E. Crelle J. Mth. 115 (1895) 328-. , classes. Stodotkiewicz, A. J. Krk. Ak. (Mt.-Prz.) Ez. 6 (1893) 145- ; Crc. Ac. Sc. Bll. (1892) 373-. , non-essential singular points. Griinfeld, E. Wien Az. 26 (1890) 65-. , possible forms. Starkov, A. P. Kazan S. Nt. (Ps.-Mth.) P. 3 (1885) 263-. ordinary differential or with finite differences. Poincare, H. Am. J. Mth. 7 (1885) 203-. with parameter variable. Appell, P. As. Fr. C. E. 8 (1879) 253-. . Horn, J. Mth. A. 52 (1899) 340-. particular integrals all of same form. Stem, A. Kjob. Ov. (1866) 77-; Resume, 10-; QJ. Mth. 8 (1867) 228-. in particular that satisfied by series 1 a/30 a.a + l./3.j8 + 1.0.0 + l 2 Forsyth, A. R. QJ. Mth. 19 (1883) 292-. problem. Halphen, G. H. Liouv. J. Mth. 1 (1885) 11-. reducible. Heffter, L. Crelle J. Mth. 122 (1900) 39-. , class. Bigiavi, C. A. Mt. 19 (1891-92) 97-. reduction. Pochhammer, L. Crelle J. Mth. 110 (1892) 188-. to integrable forms. Halphen, M. Par. Mm. Sav. Etr. 28 (1884) No. 1, 301 pp. simultaneous. Mainardi, G. Mil. At. I. Lomb. 1 (1858) 95-. . Pieard, E. C. E. 90 (1880) 976-, 1065-, 1118-. . Tilly, J. de. Mathesis 5 (1885) 121-. , with coefficients variable, class. Griln- icald, A. K. Prag Sb. (1869) 63-. differential or difference equations with con- stant coefficients. Sylvester, J. J. Am. J. Mth. 4 (1881) 321-. or single, with absolute term. Franz, J. D. Nf. Vh. (1896) (Th. 2, Hdlfte 1) 45. singular point, fundamental equation. Gilnther, P. Crelle J. Mth. 106 (1890) 330-. . , , roots. Hamburger, M. [1877] Crelle J. Mth. 84 (1878) 264-. points. Pochhammer, L. [1870] Crelle J. 73 (1871) 69-. . Bendixson, I. Stockh. Ofv. (1895) 81-. solutions of which satisfy quadratic relation. Halphen, G. H. C. E. 101 (1885) 664-. with substitution-groups independent of para- meters. Fuehs, L. Berl. Ak. Sb. (1892) 157- ; (1893) 975- ; (1894) 1117-. uniform integrals. Hirsch, A. Konigsb. Schr. 33 (1892) [16]-. only. Stenberg, E. A. Helsingf. Acta 16 (1888) 333-. 1 independent variable. Appell, P. C. E. 90 (1880) 1477-. 2 variables, property. Ramus, C. Kiob. Dn. Vd. Selsk. Afh. 9 (1842) 379- ; Crelle J. 23 (1842) 372-. , theorem. Ramus, C. Crelle J. 24 (1842) 260-. Edinb. Equivalence of systems. Chrystal, G. E. S. T. 38 (1897) 163-. Euler's method demonstrated by the integrat- ing equation. Bier ens de Haan, D. Amst. Vs. Ak. 6 (1872) 122- ; Arch. Neerl. 7 (1872) 273-. , extension. Inueneckij, V. G. St Pet. Ac. Sc. Mm. (Rs.) 42 (*1882) 1-. Extension of Galois's ideas to theory. Pieard, E. C. E. 121 (1895) 789-; Mth. A. 47 (1896) 155-. Factorisation of operator. Pincherle, S. Bo- logna Ed. 4 (1900) 101-. Fuchsian theory, extension to system of partial differential equations. Horn, I. Acta Mth. 14 (1890-91) 337-. of homogeneous equations. Zajaczkowski, W. Krk. Ak. (Mt.-Prz.) Pam. 13 (1887) (1)- ; Fschr. Mth. (1887) 298-. Fundamental equation. Casorati, F. Mil. I. Lomb. Ed. 13 (1880) 176-. 4850 Theory of Ordinary Linear Differential Equations 4850 Fundamental equation of homogeneous equa- tions. Hamburger, M. Crelle J. Mth. 115 (1895) 343-. , theory. Schlesinger, L. Crelle J. Mth. 114 (1895) 143-. equations. Gilnther, P. Crelle J. Mth. 107 (1891) 298-. of homogeneous equations. Beke, E. [1898] Mth. Nt. B. Ung. 15 (1899) 273-. property, demonstrations. Mansion, P. Brux. Ac. Bll. 38 (1874) 578- ; Mess. Mth. 4 (1875) 177- ; 7 (1878) 188- ; Arch. Mth. Ps. 58 (1876) 99-. General equation. Oliver, J. E, A. Mth. 3 (1887) 109-. , 2nd order. Cockle, (Sir) Jas. L. Mth. S. P. 19 (1889) 257-. Geometrical considerations. Fano, G. Rm. E. Ac. Line. Rd. 4 (1895) (Sem. 1) 18-. Groups. Picard,E. Bll. Sc. Mth. 9 (1885)202-. Gylden-Linstedt equation generalised. Picart, L. Bordeaux S. Sc. PV. (1896-97) 103- ; (1897-98) 145-. Hamburger sub-groups of canonical funda- mental system. Schlesinger, L. Crelle J. Mth. 114 V (1895) 159-, 309-. d?u Integrability conditions for ^ + x m u 0. Florov, P. S. (xn) Kharkov Mth. S. Com. (1883) 129- ; Fschr. Mth. (1884) 288. Integrals. Heffter, L. Crelle J. Mth. 109 (1892) 222-. algebraic. Goursat, E. Bll. Sc. Mth. As'. 6 (1882) 120- ; C.R. 100 (1885) 1329-. , of equations with coefficients rational. Autonne, L. Par. EC. Pol. J. 54 (1884) 1-. , , 2nd order. Vernier, P. C. R. 118 (1894) 1317-. , , , method of obtaining. Pepin, T. Rm. N. Line. At. 34 (1881) 243-. , form. Konigsberger, L. [1882] Mth. A. 21 (1883) 454-. relations between. Koenigsberger, L. Mth. A. 31 (1888) 220-. asymptotic representation. Kneser, A. Mth. A. 49 (1897) 383-. . Horn, J. Mth. A. 49 (1897) 453-. behaviour near singular point. Horn,J. Crelle J. Mth. 120 (1899) 1-; 122 (1900) 73-. cases. Graefe, F. [1880] Crelle J. Mth. 91 (1881) 262-. complete integral from particular integrals with 2nd term zero. Minich, S. R. [1851] Yen. Mm. I. 5 (1855) 43-. distinction in sub-groups. Casorati, F. C. R. 92 (1881) 175-, 238-. of equations with coefficients constant. Mai- nardi, G. Rm. At. N. Line. 26 (1873) 388-. , introduction of algebraic func- tions of principal variable. Hincks, E. Ir. Ac. T. 13 (1818) 115-. periodic. Anisimov, V. A. Vars. S. Nt. Tr. (Mm.) 3 (1896) 23 pp. ; Fschr. Mth. (1896) 258; Vars. S. Nt. Tr. (1897) (Ps. C.) Fasc. 2, No. 6. of equations with coefficients variable, form. Hamburger, M. Crelle J. 76 (1873) 113-. , properties. Tannery, J. Par. EC. Norm. A. 4 (1875) 113-. , 2nd order, bilinear relations between con- stants. Kepiriski, S. Krk. Ak. (Mt.-Prz.) Rz. 7 (1895) 384- ; Fschr. Mth. (1895) 445-. , , form. Steen, A. Kiob. Ov. (1874) 1-. , , infinite values. Gromeka, I. S. Kazan S. Nt. (Ps.-Mth.) P. 6 (1888) 14-; Fschr. Mth. (1887) 322. , higher order. Schellbach, C. H. Crelle J. 16 (1837) 352-. evaluation and reduction of principal function. Cauchy, A. L. C. R. 9 (1839) 637-. exchange of argument and parameter. Fro- benius, G. [1873] Crelle J. Mth. 78 (1874) 93-. expansion in series. Fuchs, L. [1870] A. Mt. 4 (1870-71) 36-. . Fabre, A. As. Fr. C. R. (1894) (Pt. 2) 158-. form. Jiirgens, E. Crelle J. Mth. 80 (1875) 150-. . Johanson, A. M. Stockh. Ofv. (1889) 583- ; (1890) 283- ; Fschr. Mth. (1890) 311-. forms. Picard, E. C. R. 94 (1882) 418-. fundamental, and their derivatives, algebraic relations for homogeneous equation, 2nd order. Koenigsberger, L. Mth. A. 31 (1888) 75-. , , , 3rd order. Koenigs- berger, L. Crelle J. Mth. 103 (1888) 274-. , derivatives of same function. Craig, T. Am. J. Mth. 8 (1886) 85-. , of homogeneous equation, 2nd order, rela- tions. Konigsberger, L. Mth. A. 22 (1883) 269-. general, of equation, 3rd order. Starkov, A. P. Kazan S. Nt. (Ps.-Mth.) P. 3 (1885) 86-; Fschr. Mth. (1885) 323, 324. and particular, connexion. Konigsberger, L. Crelle J. Mth. 91 (1881) 265-. of homogeneous equation connected by homo- geneous relation y^ - y<,y s y 4 ... y p+l = 0. Wallenberg, G. Crelle J. Mth. 114 (1895) 181-. , expansion near singular point. Bucca, F. Palermo Cir. Mt. Rd. 14 (1900) 115-. equations, 2nd order, real roots. Gegen- bauer, L. Wien Ak. Sb. 109 (1900) (Ab. 2a) 61-. , 6th order, homogeneous quadratic relations between. Bosenkranz, M. Z. Mth. Ps. 35 (1890) 82-, 129-. , recurring formulae for. Heffter, L. Crelle J. Mth. 106 (1890) 269-. , (Heffter). Fuchs, L. Crelle J. Mth. 106 (1890) 283-. and invariants of homogeneous equation, analytic representation. Mittag-Leffler, G. Acta Mth. 15 (1891) 1-. irregular. Poincare, H. C. R. 101 (1885) 939-, 990- ; Acta Mth. 8 (1886) 295- ; 10 (1887) 310-. 381 4850 Theory of Ordinary Linear Differential Equations Eec. Mth. 4850 irregular. Nekrasov, P. A. [1894] (Moscou) 17 (1895) 498-. . Horn, J. G. E. 126 (1898) 205-. , of equations, 2nd order. Horn, J. Acta Mth. 23 (1900) 171-. logarithmic, of non-homogeneous equations. Xonigsberger, L. Crelle J. Mth. 90 (1881) 267- ; 94 (1883) 291-. , , form. Koehler, C. Z. Mth. Ps. 33 (1888) 231- ; 34 (1889) 36-. , regular equations. Heun, K. Am. J. Mth. 10 (1888) 205-. of non-homogeneous equations. Koenigsberger, L. Crelle J. Mth. 99 (1886) 10-. between each pair of singular points, relations. ' Fuchs, L. Crelle J. 76 (1873) 177-. , with constants of group substitutions. Fuchs, L. Berl. Ak. Sb. (1892) 1113-. particular. Florov, P. S. Kharkov Mth. S. Com. (1886) 31- ; Fschr. Mth. (1886) 303-. , common to two equations. Escherich, G. von. [1882] Wien Ak. D. 46 (1883) (Ab. 2) 61- ; 47 (1883) (Ab. 2) 1-. , of homogeneous equation, product. Besso, D. [1882] Em. E. Ac. Line. Mm. 14 (1883) 3-. , non-homogeneous equations. Pincherle, S.' Em. E. Ac. Line. Ed. 6 (1890) (Sem. 1) 199-. , reduced equation expressed by particular integrals of given equation. Starkov, A. P. Kazan S. Nt. (Ps.-Mth.) P. 3 (1885) 253-; Fschr. Mth. (1885) 284. nth particular, of y n + Py( n ~ l ) + Qy( n ~^ + ... Sy'+Ty = 0. Malmsten, C. J. Crelle J. 39 (1850) 91-. properties. Schlesinger, L. Orv.-Termt. Ets. (Termt. Szak) (1892) 341-. rational. Sincov, D. Kazan S. Ps.-Mth. Bll. 7 (1898) 1-, 137- ; Kazan Un. Mm. (1898) (Pt. 4) 179-, (Pts. 5 & 6), (Ptf. 7 & 8) 137-, (Pt. ,9) ; Fschr. Mth. (1898) 273- ; N. A. Mth. 17 (1898) 411-. algebraic, of equations with algebraic rational coefficients. Nekrasov, P. A. Kharkov Mth. Com. 4 (1895) 225-; Fschr. Mth. (1893-94) 534, 536. . fractional, method of finding. IniSeneckij, V. G. St Pet. Ac. Sc. Mm. (Rs.) 55 .(1887) . (Suppl.) No. 9, 55 pp. ; Fschr. Mth. (1-887) 296-; St Pet, Ac. Sc. Mm. (Rs.) 58 (1888) 1- ; Fschr. Mth. (1888) 317-. , -7- (Imseneckij). Nekrasov, P-. A. [1893} ; Eec. Mth. (Moscou) 17 (1895) 341-, 391- ; Fschr. Mth. (1893-94) 534-. - , ( ). Posse, K. A. [1893] Eec. . Mth. (Moscou) 17 (1895) 386-. , . Gilnther, N. Eec. Mth. . (Moscou). 17 (1895) 680- ; Fschr. Mth. (1893-94) 534, 537. , (Imseneckij). Nekrasov, P. A. Eec. Mth. (Moscou) 18 (1896) 337-; Fschr. Mth. (1896) 258. , method of finding. Ermakoff, W. P. Fschr. Mth. (1893-94) 534, 536-. --, (Ermakov). Nekrasov, P. A. Eec. Mth. (Moscou) 18 (1896) 275- ; Fschr. Mth. (1896) 258. ' rational particular, investigation by integrating factor. Andreev, K. A. Kharkov Mth. S. Com. 4 (1895) 177- ; Fschr. Mth. (1893-94) 534, 535-. real, of equations, 3rd order, zeros. Davidoglou, . C. E. 130 (1900) 399-. regular. Frobenius, G. Crelle J. Mth. 80 (1875) 317-. . Koch, H. von. Acta Mth. 18 (1894) 337-. representation. Fuchs, L. [1872] Crelle J. 75 (1873) 177- ; 76 (1873) 175-. in series of binomial equations. Johnson, W. W. Am. J. Mth. 11 (1889) 37-. of solutions of equations, 2nd order, with 3 singular points. K$pinski, S. Krk. Ak. (Mt.-Prz.) Ez. 17 (1900) 112- ; Crc. Ac. Sc. Bll. (1899) 67-. homogeneous equations, 2nd order. K&ihski, S. Krk. Ak. (Mt.-Prz.) Ez. 5 (1893) 264- ; Crc. Ac. Sc. Bll. (1893) 20-. supplementary. Heymann, W. Crelle J. Mth. 98 (1885) 231-. of system of equations, expansion near singular point. Bendixson, I. Stockh. Ofv. (1894) 141-. , points. Horn, J. Crelle J. Mth. 116 (1896) 265-; 117 (1897) 104-, 254-. , 1st order, with constant co- efficients when characteristic equation has equal roots. Baehr, G. F. W. As. Fr. C. E. 11 (1882) 92-. uniform. Koch, -H, von. C. E. 116 (1893) 365-. , of homogeneous equation with doubly periodic coefficients. Sjoblom, E. Stockh. Ofv. (1884) No. 5, 155- ; Fschr. Mth. (1884) 282. , > equations with doubly periodic co- efficients, general form.- Stenberg, E. A. Acta Mth. 15 (1891) 259-. zeros, real. Kneser, A. Mth. A. 42 (1893) 409-. of -r^ + gx m y = Q. Plana, G. Tor. Mm. Ac. ' 26 (1821) 519-, 557-. _j + il' + j = 0. Lipschitz, R. Crelle J. 56 (1859) 189-. If (x) cos x dx, and the equation ~ + y = f (x) . Studnicka, F. J. Casopis 13 (1884) 12-; Fschr. Mth. (1884) 235. conditions for finiteness. Zu- Eec. Mth. (Moscou) 16 (1893) kovskij, N. E. 582-. Integrating factor. Florov, A. P. Kharkov ' Mth. S. Com. (1887) 47- ; Fschr. Mth. (1887) 298. of equations, 2nd order. Imshenetsky , V. G. Liege S. Sc. Mm. 9 (1882) (No. 10) 7 pp. factors. Halphen, G. H. C. E. 97 (1883) 1408-, 1541- ; 98 (1884) 134-. 382 Integration. Gergonne A. Mth. 11 4850 Theory of Ordinary Linear Differential Equations 4850 of class of equations, 3rd order. Zorawski, K. Krk. Ak. (Mt.-Prz.) Bz. 14 (1899) 141- ; Ore. Ac. Sc. Bll. (1897) 303-. complete equations. Studnicka, F. Prag Sb. (1870) 76-. , 1st order, with variable coefficients. Valles,F. Gergonne A. Mth. 17 (1826-27) 72-. convergence of series occurring in. Kneser, A. Mth. A. 47 (1896) 408-. by doubly periodic functions. Goursat, E. Par. S. Mth. Bll. 12 (1884) 96-. of equations with any coefficients. Andre, D. C. E. 84 (1877) 1018-. coefficients constant. Boole, G. Camb. Mth. J. 2 (1841) 114- . . Conchy, A. L. C. E. 13 (1841) 109- Sehmidten, H. G. (1820-21) 269-. Bees, R. van. Quetelet Cor. Mth. .2 (1826) 332-. Conti, C. A. Sc. Lomb. Yen. 5 (1835) 134-. Tortolini, B. G. Arcad. 67 (1836) 179-. Lobatto, R. Amst. N. Vh. 6 (1837) 83-. Cauchij, A. L. C. E. 8 (1839) 827-, 845-, 889-, 931-. Bromvin, B. Mathematician 2 (1847) 204- ; Camb. and Dubl. Mth. J. 3 (1848) 35-. Malmsten, C. J. Camb. and Dubl. Mth. J. 4 (1849) 286-. Mainardi, G. Tortolini A. 1 (1850) 50-, 251-. Bromvin, B. Ph. Mg..2 (1851) 477-; 3 (1852) 187-. Russell, IF. H. L. Camb. and Dubl. Mth. J. 9 (1854) 104-. Petzval, J. Grunert Arch. 28 (1857) 300-. Rouche, E. Mntp. Ac. Sc. Mm. 4 (1858-60) 195-. Wetter, A. Crelle J. 55 (1858) 274- ; Grunert Arch. 35 (1860) 440-. Bruno, G. [1860] (vi Adds.) Tor. Ac. Sc. Mm. 21 (1864) 29-. Lommel, E. Grunert Arch. 40 (1863) 101-. Halphen, G. C. B. 58 (1864) 471-. Waszmuth, A. Arch. Mth. Ps. 45 (1866) 70-. Seydler, A. Casopis 1 (*1872) 195- ; Fschr. Mth. (*1872) 152-. Starkov, A. N. Es. S. Nt. Mm. (31th.) 1 (*1878). Florov, P. S. Kharkov. Mth. S. Com. (1884) 143- ; Fschr. Mth. (1885) 327-. Vessiot, E. Par. EC. Norm. A. 9 (1892) 197-. Stodotkiewicz, A. J. Prace Mt.-Fiz. 6 (1895) 129- ; Fschr. Mth. (1895) 398. algebraic. Poincare, H. C. E. 97 (1883) 984-, 1189-. . Painlei-e, P. C. E. 119 (1894) 37-. , of class of equations. Goursat, E. C. E. 96 (1883) 323-. t special equations. Hansen, P. C. V. (xn) Ts. Mth. 4 (1880) 148-. application of co-functions. Schapira, H. D. Nf. Tbl. (1884) 57 by auxiliary equations. ImSeneckij, V. G. [1890] St Pet. Ac. Sc. Mm. (Rs.) 64 (1891) (Suppl.) No. 8, 47 pp. ; Fschr. Mth. (1891) 3 10-. in case analogous to that of algebraic equation with equal roots. Minich, S. R. At. Sc. It. (1842) 415-. cases. Dienger, J. N. A. Mth. 6 (1847) 124-. . Spitzer, S. Wien SB. 26 (1857) 449-. . Seydler, A. J. Bll. Sc. Mth. As. 6 (1874) 94-. . Spitzer, . D. Nf. Tbl. (*1875) 49-. . Winckler, A. Wien Ak. Sb. 71 (1875) (Ab. 2) 5-. . Spitzer, S. Crelle J. Mth. 88 (1880) 343-. . Steen, A. Bll. Sc. Mth. As. 5 (1881) (Pt. 1) 30-. of class of equations. Mittag-Leffler, G. Stockh. . Ofv. 36 (1879) No. 3, 17-. (1842) 266-. 3 (1854) 52-. (1867) 507-." Dam, . Liouv. J. Mth. 7 Combescure, E. Gould As. J. Tychsen, C. Z. Mth. Ps. 12 Collet, J. Par. EC. Norm. A. 4 (1887) 129-. . Oltramare, G. As. Fr. C. E. (1887) (Pt. 2) 75-. . Vaschy, . Par. EC. Pol. J. 63 (1893) 39-. and complete. Mainardi, G. Mil. G. I. Lomb. 7 (1855) 19-. , and last term function of x. Herapath, J. Ph. Mg. 2 (1827) 419- ; 3 (1828) 19-. , 2 variables. Minich, S. R. Padova N. Sag. 4 (1838) 402-. , . Tortolini, B. A. Mt. 6 (1864) 249-. containing x to first power. Spitzer, S. Wien SB. 26 (1857) 479-. doubly periodic and integrals non- uniform functions of variable. Mittag- Leffler, G. Helsingf. Acta 12 (1883) 59-. periodic. Boltzmann, L. Wien Sb. 58 (1868) (Ab. 2) 54-. rational. Autonne, L. Par EC. Pol. J. cah. 51 (1882) 93-. variable. Yur'ev, S. A. (xn) Eec. Mth. (Moscou) [1] (1866) 451-. _ equal roots. Lamia, K. Baum- gartner Z. 4 (1828) 35-. ' , 1st order, 2 variables. Bouvier, L. C. Gergonne A. Mth. 15 (1824-25) 41-. _, , ^. and 2nd orders. Tychsen, C. (xn) Ts. Mth. 4 (1868) 163-. , 2nd order. Weiler, A. Grunert Arch. 32 1859) 184-. ; . Schlomilch, 0. Schlomilch Z. 5 (1860) 323-. , . De Jong, J. Amst. Ak. Vs. M. 8 (1874) 28- ; Arch. Neerl. 9 (1874) 411-. 1, _ _. Winckler, A. Wien Ak. Sb. 75 (1877) (Ab. 2) 577-. _ __, . StodoVtiewicz, A. J. Krk. Ak. (Mt.-Prz.) Ez. 15 (1887) 36-; Fschr. Mth. (1887) 323. _ _, . Tilly, J. M. de. Brux. Mm. Cour. 8, 40 (1887) No. 3, 98 pp. 383 4850 Theory of Ordinary Linear Differential Equations 4850 of equations, 2nd order, with coefficients linear. Zmurko, W. [L.] (XH) Krk. Ak. (Mt.-Prz.) Pam. 8 (1883) 74-. , , . Stodotkieu-icz, A. J. [1883] (xn) Krk. Ak. (Mt.-Prz.) Pam. 9 (1884) 113- ; Fschr. Mth. (*1884) 286-. , , functions of the in- dependent variables. Winckler, A. Wien Sb. 67 (1873) (Ab. 2) 143-. , , rational, in finite form. Pepin, . [1862] (vm) A. Mt. 5 (1863) 185-. , , by development in continued fractions with Abelian integrals. Heun, K. Mth. A. 30 (1885) 553-. , , on which evection depends. Gylden, J. A. H. C. K. 93 (1881) 127-. , , forms, theory. Brioschi, F. Mth. A. 11 (1877) 401-. , , , . Forsyth, A. E. QJ. Mth. 23 (1889) 45-. , , by Gauss's series. Thoniae, C. J. Z. Mth. Ps. 19 (1874) 273-. , ,2 variables. Buttel, P. Grunert Arch. 23 (1854) 410-. , , or more variables. Weiler, A. [1854] Crelle J. 51 (1856) 105-. , 3rd order. Besso, D. Mod. Ac. Sc. Mm. 7 (1890) 245-. , higher orders. Malmsten, C. J. Stockh. Ofv. 5 (1848) 135-. , nth order. Stodotkiewicz, A. J. Crc. Ac. Sc. Bll. (1893) 241-. , , case. Hossenfelder, E. Mth. A. 4 (1871) 195-. , , . Alekseevskij, V. P. Kharkov Mth. S. Com. (1884) 222-; Fschr. Mth (1885) 328. , , with constant coefficients and equal roots. M., F. (vi Adds.) Gergonne A. Mth. 3 (1812-13) 46-. , particular integrals of which can be ex- pressed in terms of one another. Libri, G. [1830] Crelle J. 10 (1833) 167-. ,2 variables. Canterzani, S. [1798] Mod. Mm. S. It. 8 (1799) 307-. error attributed by Herapath to Lagrange. Alpha Beta (a. /3.). (vi Adds.) Ph. Mg. 3 . F. E. S. (vi Adds.) Ph. Mg. 3 (1828) 97-. . Herapath, J. Ph. Mg. 3 (1828) 21CK by finite explicit functions. Steen, A, Kjob. Dn. Vd. Selsk. Skr. 10 (1875) 519-. in finite form. Andre, D. C. E. 88 (1879)230-; Liouv. J. Mth. 6 (1880) 27- ; 7 (1881) 283-. . Stoddtkieicicz, A. J. [1892] Krk. Ak. (Mt.-Prz.) Rz. 6 (1893) 100- ; Crc. Ac. Sc. Bll. (1892) 296-; Prace Mt.-Fiz. 4 (1893) 31. general, of equations of any degree and with any number of variables. Euler, L. [1779] St Pet. Ac. Sc. Mm. 4 (1813) 43-. of general equations of order above 2nd by finite number of quadratures, impossibility. Starkov, A. P. Kazan S. Nt. (Ps.-Mth.) P. 3 (1885) 256- ; Fschr. Mth. (1885) 309. general formula for equation with 2nd member and constant coefficients. Lyetnikov, A. V. [1879] (xn) Eec. Mth. (Moscou) 9 (1878-81) (Pt. 1) 550-. graphic, of equations, 1st order. Czuber, E. Z. Mth. Ps. 44 (1899) 41-. of Hermite's equations of 3rd and 4th orders, integrals having poles of 1st order. Mittag- Leffier, G. Helsingf. Acta 12 (1883) 409- ; A. Mt. 11 (1882-83) 65-. homogeneous equation in annular space containing no singularities. Mittag-Leffler, G. Stockh. Ofv. (1889) 415-. equations, classes. Mittag-LeJfter, G. Stockh. Ofv. 37 (1880) No. 6, 59-. , 2nd order, case. Besso, D. Mod. Ac. Sc. Mm. 7 (1890) 239-. by quadratures. Schlesinger, L. Crelle J. Mth. 116 (1896) 97-. Kummer's equation. Lindelnf,E. Helsingf. Acta 19 (1893) No. 1, 31 pp. mechanical, of equations, 2nd order, with variable coefficients. Thomson, (Sir) W. R. S. P. 24 (1876) 269-. , general equation, any order, with variable coefficients. Thomson, (Sir) W. R. S. P. 24 (1876) 271-. method. Woodward, E. S. N. Y. Am. Mth. S. Bll. 6 (1900) 366, 369-. , d'Alembert's. Brassinne, E. Toul. Mm. Ac. 1 (1844) 91-. , . application to equations with constant coefficients. Stodrtkieivicz , A. J. Krk. Ak. (Mt.-Prz.) Rz. 6 (1893) 105- ; Crc. Ac. Sc. Bll. (1892) 294-. , Brisson's first, for linear and finite difference equations. Mansion, P. [1869] Brux. Mm. Cour. 8, 22 (1872) (No. 1) 32 pp. for equations with one independent variable. Caque, F. Liouv. J. Mth. 9 (1864) 185-. , general, for special class of equations. Steen, A. Kjob. Ov. (1866) 13-. , Laplace's, for linear and finite difference equations. Darboux, G. Bll. Sc. Mth. As. 3 (1879) 209-. for simultaneous equations, 1st order. Cauchy, A. L. C. R. 43 (1856) 127-. of non-homogeneous equations. Heymann, W. Z. Mth. Ps. 30 (1885) 27-, 79- ; 32 (1887) 22-. when polynomial has equal roots. Plana, G. [1826] Tor. Mm. Ac. 31 (1827) 377-. of Riccati's equation y" + 2zy' + z 2 y = by ex- pansion of ye ~ ^ z in infinite series. Clariana Eicart, L. Fschr. Mth. (1900) 352. by series. Frobenius, G. Crelle J. 76 (1873) 214-. . Peano, G. Tor. Ac. Sc. At. 22 (1886- 87) 437- ; Mth. A. 32 (1888) 450-. of simultaneous equations. Prouhet, E. N. A. Mth. 5 (1866) 323-. (2), d'Alembert's. Kryger, C. (xii) Ts. Mth. 5 (1875) 38-. ( ) in 71 variables. Eaabe, J. L. Ziir. Mschr. 1 (1856) 130-. by sines of higher orders. Yron-Villarceau, A. J. F. C. R. 90 (1880) 721-, 767-; 91 (1880)13- ; Par. Ing. Civ. Mm. (1880) (v. 2) 18-. 384 4850 Theory of Ordinary Linear Differential Equations 4850 of system of equations. Sturm, J. C. F. Ferussac Bll. Sc. Mth. 12 (1829) 313-. , 1st order. Grilnfeld, E. Wien Ak. D. 54 (1888) (Ab. 2) 93-; Wien Az. 25 (1889) 37-. partial differential equations, 1st order. Zajaczkoicski, W. Arch. Mth. Ps. 56 (1874) 163-. simultaneous partial differential equations. Schiff, P. [1891] St Pet. Ac. Sc. Bll. 34 (1892) 407-. systems of simultaneous homogeneous equa- tions. Spdta, C. G. Mt. 38 (1900) 217-. _ d ^y + M _liL +1 ) N \ y = Q. Paoli, P. [1827] dx 2 \ x 2 J Mod. S. It. Mm. 20 (1828) 183-. x 2 (a + bx n ) dry' + x(c + ex n ) dy'dx Sauer, C. G. Crelle J. 2 (1827) 93-. (mx* + nx+p)y" - (qx + r) y' + sy=0. Liouville, J. Par. EC. Pol. J. 21 cah. (1832) 163-. _ l = (a + px)y. Scherk, H. F. Crelle J. 10 (1833) 92-, 279. = (Scherk). Jacobi, C. G. J. Crelle J. 10 (1833) 279. d Spitzer, S. Arch. Mth. Ps. 42 (1864) 102-. - xOT -p^ = J/> m being a negative integer. Spitzer, S. Arch. Mth. Ps. 42 (1864) 328-. - (a + bx + ca; 2 ) (b + 2cx) y" + A (a + bx + ex 2 ) y' + B(b + 2cx)y = 0. Spitzer, S. Arch. Mth. Ps. 42 (1864) 330-. - (b + 2cx)y" + A (a+bx + cx*)y' + B(b + 2cx) (a + bx + cx^y^Q. Spitzer, S. Arch. Mth. Ps. 42 (1864) 331. - (b + 2cx) y" + A (a + bx + ex 2 ) y' + B(b + 2cx)y = 0. Spitzer, S. Arch. Mth. Ps. 42 (1864) 332. - (m + x) (n + x) y" +( m -n)y'- X 2 (i + xfy = 0. Spitzer, S. Arch. Mth. Ps. 42 (1864) 375-. Spitzer, S. Z. Mth. Ps. 10 (1865) 155-. xy( n ) + ay(' l -n = bxy. Spitzer, S. Z. Mth. Ps. 10 (1865) 221-. ~ = ay + \f/(x). Eogner, J. Arch. Mth. Ps. 45 (1866) 315-. Malmsten, C. J. x ~ Camb. and Dubl. Mth. J. 5 (1850) 180-. xyW-y = Q. Spitzer, S. Grunert Arch. 26 (1856) 57-. y( n > = Ax m y' + Bx m ~ l y. Spitzer, S. Grunert Arch. 28 (1857) 254-. yW = Ax m y" + Bx- 1 y' + Cx m ~' 2 y. Spitzer, S. Grunert Arch. 29 (1857) 403- ; 30 (1858) 76-; 33 (1859) 118- ; 38 (1862) 77-. xy"-y = Q. Spitzer, S. Schlomilch Z. 2 (1857) 165-. ( 2 + b 2 x) y" + (etj + fcj a;) y' + (a + b x)y = Q. Spitzer, S. Wien SB. 25 (1857) 31-. x 2 (ojj + b z x) y" + x ( fll + 6, x) y' Spitzer, S. Schlomilch Z. 3 "(1858) 53-. x 3 (a, + b,ar) y'" + x 2 (a 2 + b 2 x) y" Spitzer, S. Schlomilch Z. 3 (1858) 55-. am jfm = x ' m ~i~m' W r eiler,A. Grunert Arch. 31 (1858) 44-. x^ n yW = Axy' + By. Spitzer, S. Grunert Arch. 33 (1859) 413-. Grunert Arch/ 38 (1862) 462-. (n + b n x ) l J n + (n-l + b n-l x ) 2/"" 1 + + (a 1 +b 1 x) y'+(a + b x) y = 0. Spitzer, S. Schlomilch Z. 7 (1862) 264-. xy( r >-y( r ~' i } + mx' 2 y = 0. Spitzer, S. Grunert Arch. 40 (1863) 21-. Spitzer, S. Schlomilch Z. 8 (1863) 66-. sy" + (r + qx) y' + (p + nx + mx z ) y = 0. Spitzer, S. Schlomilch Z. 8 (1863) 123-. VOL. i. 385 Arch. Mth. Ps. 46 (1866) 25-. Spitzer, S. Arch. Mth. Ps. 47 (1867) 110-.' - (a 2 + b 2 x + c^)^+(a 1 + b i x) ( ^ + a y = (). Holmgren, H. Stockh. Ak. Hndl. 7 (1867- 68) (No. 9) 58 pp. kov, A. V. [1876] Eec. Mth. (Moscou) 14 (1890) 205- ; Fschr. Mth. (1889) 333-. j e = ax m y. Molins, H. Toul. Ac. Sc.Mrn. 8 (1876) 167-. - -j-j H + F=0. Worms de Eomilly, P. Liouv. J. Mth. 4 (1878) 177-. - y" + Py'+Qy = 0. Alexeeff, N. As. Fr. C. E. 8 (1879) 190-. - ~^ + 2cotx . ~-y 0. Graindorge, J. (xii) Kharkov Mth. S. Com. (1880) 46-. - x' 2 y'" + Axy" + By' + Cx^y = 0. Aleksyeevskil, V. P. (xn) Kharkov Mth. S. Com. 115- ; Fschr. Mth. (1884) 288. - xy" + my' + nyx = 0. Stodoikiewicz , A. J. [1883] (xn) Krk. Ak. (Mt.-Prz.) Ez. & Sp. 11 (1884) 155-, XLIX-; Fschr. Mth. (1884) 287.' ij, V. P. Kharkov Mth. S. Com. (1884) 41-; Fschr. Mth. (1884) 289-. of (c. 2 x 2 + b z x + a 2) y" + (biX- application of Zmurko's improvement of Laplace's method. Rajewski, J. Krk. Ak. (Mt-Prz.) Pam. 9 (1884) 120- ; Fschr. Mth. (1884) 287-. 2 (a. + b^x) ^ 8 = 0. Nekrasov, P. A. [1889] Bee. Mth. (Moscou) 14 (1890) 344- ; Fschr. Mth. (1889) 334. ^ = X (ay 2 + by + c) + X 1 , where X and X 1 are functions of x and a, b, c, are constants. Stoddikieivicz, A. J. Prace Mt.-Fiz. 3 (1892) 52- ; Fschr. Mth. (1892) 300. 4850 Theory of Ordinary Linear Differential Equations 4850 Laplace's equation and formula of Abel. Cailler, C. Bll. Sc. Mth. 23 (1899) 26-. Multiform functions belonging to group of linear substitutions. Wilczynski, E. J. Am. J. Mth. 21 (1899) 85-. Operations in general, and equations of infinite order. Bourlet, C. Par. EC. Norm. A. 14 (1897) 133-. Origin of some equations. Brioschi, F. [1881] Em. B. Ac. Line. T. 6 (1882) 42-. Oscillation theorems of Sturm and Klein. Bocher, M. N. Y. Am. Mth. S. Bll. 4 (1898) 295-, 365- ; 5 (1899) 22-. Pfaffian equation, 2nd order, linear. Stodoi- kiewicz, A. J. Krk. Ak. (Mt.-Prz.) Bz. 13 (1886) 160- ; Fschr. Mth. (1885) 315. Picard's equations, reduction and integration. Naetsch, E. Leip. Mth. Ps. B. 48 (1896) 1-. 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[6J (1886-87) 29-. application of calculus of finite differences. Russell, W. H. L. QJ. Mth. 1 (1857) 23-. cases. David, C. A. [1862] (xn) -Lille S. Mm. 9 (1863) 409-. conditions that two equations should have p common solutions. Lemonnier, H. C. E. 95 (1882) 476-. conjugate, in differential and difference equa- tions. Pincherle, S. Bm. B. Ac. Line. Ed. 4 (1895) (Son. 2) 228-. by continued fractions. Laguerre, E. [1878- 85] C. B. 87 (1878) 923- ; Par. S. Mth. Bll. 8 (1880) 21-; Liouv. J. Mth. 1 (1885) 135-. dependence on certain parameters. Fuchs, L. Berl. Ak. Sb. (1895) 905-. depending on multiple types. Routh, E. J. L. Mth. S. P. 14 (1882-83) 144-. Invariants. Cayley, A. QJ. Mth. 21 (1886) 257-. . Brioschi, F. Acta Mth. 14 (1890-91) 233-. . Mittag-Leffler, G. Crelle J. Mth. 114 (1895) 306-. associated with equations. Platts, C. QJ. Mth. 25 (1891) 300-. of general equation, and relation to theory of continuous groups. 13outon, C. L. Am. J. Mth. 21 (1899) 25-. Irreducibility. Koenigsberger, L. Gott. Nr. (1883) 390- ; Crelle J. Mth. 96 (1884) 123- ; Berl. Ak. Sb. (1899) 672- ; Mth. A. 53 (1900) 49-. , concept. Frobenius, G. Crelle J. 76 (1873) 236-. of homogeneous equations. Beke, E. Mth. A. 45 (1894) 278-. Iterated operators, equations arising from. Gutzmer, A. Prag Sb. (1892) (Mth.-Nt.) 54-. Summer's equation. Forsyth, A. R. QJ. Mth. 19 (1883) 125-. , analogue of. Goursat, E. C. B. 99 (1884) 777-, 858-. of hypergeometric series. Goursat, E. Helsingf. Acta 15 (1888) 45-. , 3rd order, particular case. Cayley, A. Mess. Mth. 20 (1891) 75-. Lagrange's determinantal equation of small oscillations, occurrence of equal roots in. Purser, F. B. A. Bp. (1878) 463-. Lame-Hermite equation. Brioschi, F. St Pet. Ac. Sc. Bll. 35 (1894) 449-. equations for two variables, generalised. Krause, M. Leip. Mth. Ps. B. 50 (1898) (Mth.) 192-, 231-. Lamp's equation. Pick, G. A. Wien Ak. Sb. 96 (1888) (Ab. 2) 872-. , generalisation. Palmstrfmi, A. Berg. Ms. Aarb. (1897) No. 11, 9 pp. , Hermite's case. Klein, F. Mth. A. 40 (1892) 125-. , second solutions. Crawford, L. Edinb. Mth. S. P. 16 (1898) 57-. - ^| - [n (n + l)$u + B}y = Q. Palmstrfin, A. Christiania Skr. (Mth.-Nt. Kl.) (1894) No. 6, 12 pp. 386 4850 Theory of Ordinary Linear Differential Equations 4850 of equation in perturbations. Motoda, T. As. J. 9 (1890) 124-. equations with constant coefficients. Gregory, D. F. (vi Adds.) Camb. Mth. J. 1 (1839 22- 278. , 2nd order, with linear coefficients. Graf, J. H. Mth. A. 45 (1894) 235-. , , product of two. Besso, D. Em. E. Ac. Line. Mm. 19 (1884) 219-. , 3rd and 4th orders, relations between. Campbell, D. F. N. Y. Am. Mth. S. fill. 5 (1899) 327, 332-. expression of arbitrary constant in terms of initial conditions. Eouth, E. J. QJ. Mth. 19 (1883) 262-. history. Enestrom, G. Bb. Mth. (1897) 43-. integral rational, of equation, 2nd order. Berger, A. Ups. S. Sc. N. Acta 15 (1895) No. 4, 28 pp. method for large class of equations. Graves, C. Ir. Ac. P. 6 (1853-54) 34-. periodic. Bendixson, I. Stockh. Ofv. (1896) 193-. regular, of system of equations. Sauvage, . C. E. 108 (1889) 174-. representation near singular point. Griinfeld, E. Z. Mth. Ps. 36 (1891) 21-. second special, of equation, 2nd order. Hoppe, E. Arch. Mth. Ps. 64 (1879) 379-. in series. Hammond, J. L. Mth. S. P. 6 (1874- 75) 67-. , arithmetical nature of coefficients. Pincherle, S. Crelle J. Mth. 103 (1888) 84-; Palermo Cir. Mt. Ed. 2 (1888) 153-. by symbols. Graves, C. Ir. Ac. P. 3 (1847) 536. synthetic, and deformation. Cockle, (Sir) Jas. QJ. Mth. 23 (1889) 1-. of system of equations. Gilnther, S. Arch. Mth. Ps. 57 (1875) 240-. Sturm's "theorem of comparison." Bocher, M. N. Y. Am. Mth. S. Bll. 6 (1900) 96, 100. Successive approximations method. Picard, E. Par. S. Mth. Bll. 22 (1894) 52-. . Zaremba, S. Liouv. J. Mth. 3 (1897) 311-. System of (2) equations of which the one gives integrating factors of the other. Eicci, G. G. Mt. 15 (1877) 135-. simultaneous equations. Meray, C. C. E. 81 (1875) 1203-. Systems of equations identical with their ad- joint. Goursat, E. C. E. 106 (1888) 187-. , 1st order. Koch, H. von. C. E. 116 (1893) 179-. , , admitting only regular solu- tions. Griinfeld, E. Wien. Az. 26 (1890) 66-. with 1 independent variable. Horn, J. Mth. A. 39 (1891) 391- ; 40 (1892) 527-. formed from equations with 1 independent variable. Darboux, G. C. E. 90 (1880) 524-, 596-. of homogeneous equations. Sauvage, L. Toul. Fac. Sc. A. 8 (1894); 9 (1895) 100 + 76pp. Systems of simultaneous equations, class. Ibach, L. F. N. A. Mth. 3 (1884) 172-. , (Ibach). Tardy, P. N. A. Mth. 3 (1884) 257-. Theorem. Russell, W. H. L. B. A. Ep. (1879) 263. . Darboux, G. C. E. 94 (1882) 1456-. of Fuchs, point in Jordan's proof. Gerbaldi, F. Ev. Mt. 1 (1891) 125-. Malmsten. Joachimsthal, F. Crelle J. 40 (1850) 48-. Theorems. Besso, D. Em. E. Ac. Line. Mm. 10 (1881) 252-. . Bocher, M. N. Y. Am. Mth. S. Bll. 6 (1900) 279-. 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Sc. 2 (1880) 183- ; (xn) Ts. Mth. 5 (1881) 5-. , formula of Halphen. Torelli, G. Nap. Ed. 29 (1890) 233-. , group, determination. Harotte, F. C. E. 124 (1897) 608-. of ordinals, new. Cockle, (Sir) Jas. Ph. Mg. 13 (1882) 44-. f , rational. Goursat, E. Par. EC. Norm. A. 2 (1885) 37-. Transformed equation, Euler's. Pincherle, S. Crelle J. Mth. 119 (1898) 347-. , , of homogeneous equations of Fuchs' s class. Schlesinger, L. Crelle J. Mth. 117 (1897) 148-. (D + a) n y = X. De Morgan, A. Camb. Mth. J. 4 (1845) 60-, 96. ^M.= y Besge,V.A. Liouv. J. Mth. 11 (1846) 96. y" + -y' + ny = 0. Lebesgue, V. A. Liouv. J. Mth. 11 (1846) 338-. 387 4860 Integration by Definite Integrals 4860 . y = 0, and other equations. Bras- sinne, E. Toul. Mm. Ac. 2 (1846) 326-; 3 (1847) 204- ; 1 (1863) 483-. z"- 1 K + b n x) y (n] + xn ~ 2 K-i + V-i*) 2/ ( "-" + - + (a 1 + b 1 x)y' + b y = 0. Malmsten, C. J. Crelle J. 39 (1850) 99-. y" x + y-* + Ax m y = Q. Malmsten, C. J. 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De Morgan, A. [1854] (rx) Camb. Ph. S. P. 1 (1866) 136-. . Bertrand, J. C. E. 45 (1857) 617-. of equations, 1st degree. Autonne, L. Par. EC. Pol. J. 61 (1891) 35-; 62 (1892) 47-. , general. Peano, G. Tor. Ac. Sc. At. 33 (1897) 9-. , paradox in. Tanner, H. W. L. Mess. Mth. 10 (1881) 158-. Transcendental integrals of equations with coefficients of 2nd degree. Winckler, A. Wien Ak. Sb. 84 (1882) (Ab. 2) Transcendents. Konigsberger, L. Acta Mth. 3 (1883) 1-. Transformation. Russell, W. H. L. Camb. and Dubl. Mth. J. 9 (1854) 81-. . Zorawski, K. Krk. Ak. (Mt.-Prz.) Ez. 6 (1893) 67-; Crc. Ac. Sc. Bll. (1892) 292-. =/ (u, z), proof of theorem of Cauchy. Ascoli, G. [1871] A. Mt. 5 (1871-73) 14-. * (y') + yt (y') + X(V') = O- Heymann, W. [1878] Z. Mth. Ps. 24 (1879) 252-. -& . Sonin, N. Ja. [1894] St Pet. Ac. Sc. Bll. 2 (1895) 93-; 3 (1895) 339-. dx 4880 General theory of ordinary equations, not linear, of order higher than first. Fais, A. G. G. B. Abridged form of equations Mt. 12 (1874) 320-. Addition theorem. Mittag-Lejfler, A. Ep. (1894) 561. Algebra and group theory of equations. Loewy, A. Mh. Mth. Ps. 8 (1897) 225-. Class of equations. Picard, E. C. B. 104 (1887) 41-. including those of all geodesies. Liou- ville, R. C. E. 105 (1887) 1062-. Construction of equations of form 1 cPz ;dxTy = X ^> admitting an explicit general integral. Mmt- tard, T. Par. EC. Pol. J. Cah. 45 (1878) Equation of algebraic curve. Chrystal, G. Edinb. Mth. S. P. 3 (1885) 95-. Mm. 2 379-. , 2nd order. Enneper, A. Z. Mth. Ps. 15 (1870) 56-. , nth order, 2 variables, conditions for deri- vation from equation of (n - l)th order. Raabe, J. L. Crelle J. 31 (1846) 181-. Equations, general integral of which is uni- form. Painleve, P. Par. S. Mth. Bll. 28 (1900) 201-. , , class. Picard, E. C. E. 110 (1890) 877-. , 2 general, of 2nd order, 7ith degree, derived from the known linear equation. Rawson, R. Mess. Mth. 2 (1873) 185-. integrable by meromorphic doubly periodic functions. Petrovitch, M. Acta Mth. 22 (1899) 379-. , integral of which has finite number of determining values. Painleve, P. C. E. 116 (1893) 88-, 173-. in lunar theory, two. Cayley, A. [1871] As. S. M. Not. 32 (1872) 31-, 201-. , 2nd order. Vessiot, E. Toul. Fac. Sc. A. 9 (1895) F, 26 pp. , , class. Wallenberg, G. Crelle J. Mth. 120 (1899) 113-. , , with fixed critical point and uniform correspondence of surfaces. Painleve", P. C. E. 117 (1893) 611-. , and 3rd orders, normal form. Pick, G. Mth. A. 38 (1891) 139-. , 3rd order, case. Brioschi, F. E. S. P. 27 (1878) 126-. , , Schwarzian derivative. Klein, C. F. Leip. Mth. Ps. B. 35 (1883) 1-. possessing fundamental first integrals. Guldberg, A. Christiania F. (1894) No. 1, 41 pp. ; Fschr. Mth. (1893-94) 544-. Homogeneous equations, 2nd order. Steen, A. (xn) Ts. Mth. 1 (1883) 11-. , . Appell, P. Toul. Fac. Sc. A. 3 (1889) K, 12 pp. , . Wall Wallenberg, G. Crelle J. Mth. 119 (1898) 87-. Integrability. Bach, , & Stoffel, . Liouv. J. Mth. 7 (1862) 49-. , conditions in quadratures of equation of 2nd order. Yur'ev, S. A. [1869] (xn) Eec. Mth. (Moscou) 4 (186970) (Pt. 1) 246-. Integrals, asymptotic representation for large real values of argument. Kneser, A. Crelle J. Mth. 116 (1896) 178-; 117 (1897) 72-; 120 (1899) 267-. common to 2 systems. Capelli, A. Nap. Ed. 39 (1900) 100-. several systems. Pennacchietti, G. Catania Ac. Gioen. At. 3 (1891) 1-. , form near critical points. Picard, E. C. E. 87 (1878) 430-, 743-. , singular. Zajaczkowski, W. [1873] (xn) Krk. Ak. (Mt.-Prz.) Pam. 1 (1874) 45-. Integrating factor of equations of form A + By' + Cy' m + Dy' m + l + Ey' m ~ l y" = 0. Andreevskii, M. A. (xn) Eec. Mth. (Moscou) 4 (1869-70) (Pt. 1) 143-. 392 4880 Higher Orders Integration. Malmsten, C. J. Stockh. Ak. Hndl. 3 (1859-60) 94 pp. ; Liouv. J. Mth. 7 (1862) 257-. . Schapira, . D. Nf. Vh. (1893) (Th. 2, Hfilfte 1) 15-. , application of differential quotients with general index. Most, E. Z. Mth. Ps. 16 (1871) 190-. of equation of conies. Prevost, G. Bor- deaux S. Sc. PV. (1897-98) 47-. equations, 2nd order. Svanberg, A. F. Stockh. Ak. Hndl. (1833) 1-; Ups. N. Acta S. Sc. 1 (1851) 261-. , . Guldberg, A. Christiania F. (1895) No. 6, 48 pp. ; Fschr. Mth. (1896) 249-. , , cases. Winckler, A. [1874] Wien Ak. Sb. 70 (1875) (Ab. 2) 149-. , , . Pajak, S. Prace Mt.- Fiz. 11 (1900) 32- ; Fschr. Mth. (1900) 350-. , , class. Strong, T. Silliman J. 42 (1842) 273-. , , by factors. Andreiewsky, . C. B. 68 (1869) 716-. , , with 2 variables. Tychsen, C. Mth. Ts. 6 (1864) 1-. of high order. Cardinali, F. [1806] Mod. Mm. S. It. 13 (1807) 381-. -*g + ()Vag + -.*.* S being functions of x, y. Tychsen, C. Mth. Ts. 5 (1863) 173-. - |^ + (1 - ft) p = jS + etc. Backlund, J. 0. As. Nr. 103 (1882) 323-. Irreducible functions, properties. Konigsberger, L. Crelle J. Mth. 95 (1883) 171-. Last multiplier. Winckler, A. [1879] Wien Ak. Sb. 80 (1880) (Ab. 2) 948-. Polynomials satisfying differential equation, 3rd order. Appell, P. As. Fr. C. E. 8 (1879) 253-. Eepresentation of functions, between certain limits, by series of values of function satis- fying equation of 2nd order. Picart, A. N. A. Mth. 2 (1883) 109-. Singularities depending on integral constants. Fransen, A. E. Stockh. Ofv. (1897) 31 7-. , essential. Painleve, P. C. E. 116 (1893) 362-. , . Picard, E. C. E. 116 (1893) 365; Acta Mth. 17 (1893) 297-. Solutions, particular, of equations, 2nd order, in 2 variables. Weiler, A. Grunert Arch. 32 (1859) 286-. , rational fractional. Andreev, K. A. Kharkov Mth. S. Com. 4 (1895) 150- ; Fschr. Mth. (1893-94) 537. , . Markov, A. A. Kharkov Mth. S. Com. 4 (1895) 175-. , , Bugaev's method compared with others. Im&eneckij, V. G. Kharkov Mth. S. Com. 4 (1895) 60-; Fschr. Mth. (1893- 94) 537. Kharkov Mth. S. Com. 4 (1895) 146-'; Fschr. Mth. (1893-94) 537. Differential Forms 5200 Solutions, singular. Girault, C. Caen Ac. Mm. (1855) 211-. , . Cockle, (Sir) Jas. QJ. Mth. 12 (1873) 305-. , . Hollman, P. J. N. Arch. Wisk. 8 (*1882) 23-. , . Mitchell, H. Edinb. Mth. S. P. 16 (1898) 64-. , , general theory. Zajaczkoicski, W. (xn) Krk. Ak. (Mt.-Prz.) Pam. 3 (1877) 1-. , , geometric meaning. Fine, H. B. Am. As. P. (1887) 64. Systems with fixed critical points. Painleve, P. C. E. 130 (1900) 767-. Theory, history, 1878-93. Craig, T. N. Y. Mth. S. Bll. 2 (1893) 119-. Transformation. Cockle, (Sir) J. [1881] Mess. Mth. 11 (1882) 109-. of equation of even order to isoperimetric form. Imchenetsky, V. [1886] St Pet. Ac. Sc. Bll. 31 (1887) 283-. >S+-f(al) +*=> *""" * Mess. Mth. 5 (1871) 90-. ^frrJ\ = - 4 T rV Tait, P. G. Edinb. dr \pdrj k Mth. S. P. 4 (1886) 23-. y" = Ay 3 + By 2 + Cy + D + (Ey + F) y'. Fransen, A. E. Stockh. Ofv. (1895) 223-; Fschr. Mth. (1895) 374-. Differential forms and differential invariants. 5200 General. Differential functions. Guerin, V. C. E. 60 (1865) 519-. operators, some families. Perrin, R. C. E. 106 (1888) 1131-. polynomials, theorem on class. Chini, M. Yen. I. At. (1893-94) 872-. Exponential differences and differentials, Wronski's theory. Krauze, W. Prace Mt.- Fiz. 5 (1894) 160- ; Fschr. Mth. (1893-94) 459-. ---- , -- . Dickstein, S. Prace Mt.- Fiz. 5 (1894) 169- ; Fschr. Mth. (1893-94) 459-. Integrability conditions. Kronecker, L. Crelle J. 59 (1861) 311-. Integration of algebraic differential expressions in finite form. Petersen,J. Gott. Nr. (1878) 68-. differential forms, five. Martin, Art. Des Moines Anal. 4 (1877) 175-. ___ formulae, two. Malfatti, G. F. Turin Mm. Ac. 4 (1788-89) 53-. -- exact differentials. Combescure, E. Mntp. Ac. Mm. 11 (1892) 59-. method, Bertrand's, application to total differential equation in many variables. Stodotkiewicz, A. J. (xn) Krk. Ak. (Mt.- Prz.) Pam. 8 (1883) 137-. Invariant differential operators. Capelli, A. Nap. Ed. 26 (1887) 110-. 393 5210 Linear Differential Forms; Pfaffians 5210 Linear congruences. Giddberg, A. 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Grunert Arch. 23 (1854) 453- ; 30 (1858) forms; Algebraic integral of two differential equations. Roberts, R. A. [1890] L. Mth. S. P. 22 (1891) 28-. integrals of differential equations, 1st order and degree. Poincare, H. Palermo Cir. Mt. Rd. 5 (1891) 161-. Binomial differentials. Combescure, E. Bll. So. Mth. 11 (1887) 245-. . Kapteyn, W. Bll. Sc. Mth. 12 (1888) 44-. Common multiples of linear differential ex- pressions. Heffter, L. Crelle J. Mth. 116 (1896) 157-. Condition that total differential equation Pdx + Qdy + Rdz + etc. = admit of single primitive. Van Velzer, C. A. [1879] (xn) J. H. Un. Cir. [1] (1882) 35. be integrable. Lovett, E. 0. A. Mth. 1 (1900) 175-. Differential expressions and Pfaff's problem. Cartan, E. Par. EC. Norm. A. 16 (1899) 239-. , problem. Torelli, G. A. Mt. 13 (1885) 23-. forms. Combescure, E. C. R. 70 (1870) 1164-. parameters, 1st order. Banal, E. G. Mt. 30 (1892) 235-. problem, theory. Lie, S. Christiania F. 14 (1872) 132-. systems - with uniform general integral. Painleve, P. C. R. 131 (1900) 497-, 534. Equation xdx + x^d.^ + x 2 dx 2 + ... + x n dx n = integrable by a primitive equation. Zajacz- kowski, W. Par. T. Nauk Sc. Pam. 6 (*1875) Art. 4, 14 pp. Equations of form dz = Z (Udx + Vdy) , where U, V, z, are functions of x and y, and Z a function of z. Hansen, C. (xn) Ts. Mth. 4 (1868) 188-. Exact homogeneous differentials. Combescure, E. Bll. Sc. Mth. 11 (1887) 243-. Integrability conditions. Laurent, H. N. A. Mth. 6 (1887) 274-, 305-. of certain differential expressions. Sincov, D. Kazan S. Ps.-Mth. Bll. 9 (1900) 38-. (A + Cz) dx+(B + Dz) dy + Kdz = 0. Sonln, N. Y. [1870] (xn) Rec. Mth. (Moscou) 6 (1872-73) (Pt. 1) 278-. dv=Mdx + Ndy +Pdz. PerevoScikov , D. M. St Pet. Ac. Sc. Mm. (Es.) 22 (*1873) 201-. X^ + X.-,dx. 2 +... + X n dx n = 0. Stodotkie- wicz, A. J.' Prace Mt.-Fiz. 7 (1896) 6-; Fschr. Mth. (1896) 271. Linear differential forms. Pincherle, S. Rm. R. Ac. Line. Rd. 1 (1892) (Sem. 1) 273-. , theory. Bortolotti, E. A. Mt. 23 (1895) 309-. equations with total differentials. Bianchi, L. Rm. R. Ac. Line. Rd. 5 (1889) (Sem. 1) 312-. MacMahon's differential equation. Cayley, A. QJ. Mth. 19 (1883) 182-. Normals to surface. Bertrand, J. C. R. 17 (1843) 1277-. Operation O having reference to Cayley 's func- tions A (x). Cayley, A. QJ. Mth. 13 (1875) 369-. (Cayley's), reduction to polar operations. Capelli, A. Mth. A. 29 (1887) 331-. Pfaffian equation, infinitesimal automorphic transformations. Vivanti, G. Palermo Cir. Mt. Rd. 12 (1898) 1-. , . Weber, E. von. Palermo Cir. Mt. Rd. 12 (1898) 133-. , transformations. Enqel, F. Leip. Mth. Ps. B. 51 (1899) (Mth.) 296-. , integration, Clebsch's 2nd method. Brill, J. L. Mth. S. P. 31 (1900) 315-. , , general. Sauer, C. G. Oken Isis (1825) 1294-. , shortest integral curves. Lilientlml, E. von. Mth. A. 52 (1899) 417-. 2Xdx = Q. Eussjan, C. Fschr. Mth. (1896) 269-. equations. Lovett, E. 0. C. R. 129 (1899) 274-. and line complexes. Weber, E. von. [1900] Munch. Ak. Sb. 30 (1901) 393-. of special type, systems. Brill, J. Mess. Mth. 26 (1897) 183-. , system. Eusjan, C. K. Fschr. Mth. (1899) 315-. , , normal number of complete in- tegrals. Eusjan, C. K. Fschr. Mth. (1900) 367. 394 5210 Linear Differential Forms; Pfaffians 5210 Pfaffian equations, systems. Weber, E. von. Munch. Ak. Sb. 25 (1896) 423-. , . Brill, J. QJ. Mth. 30 (1899) 221-. , . Kowalewski, G. Leip. Mth. Ps. B. 51 (1899) (Mth.) 265-. , , invariant theory. Engel, F. Leip. Mth. Ps. B. 41 (1889) 157- ; 42 (1890) 192-. , , . Weber, E. von. Leip. Mth. Ps. B. 50 (1898) (31th.) 207-. expressions and equations. Lie, S. Leip. Mth. Ps. B. 48 (1896) 405-. , multilinear differential covariants. Brill, J. L. Mth. S. P. 30 (1899) 263-. having a vacant minor, development. Muir, T. [1900] Edinb. R. S. T. 40 (1905) 49-. system, reducibility to given number of terms. Weber, E. von. [1900] Munch. Ak. Sb. 30 (1901) 273-, 434. systems and line geometry. Weber, E. von. Leip. Mth. Ps. B. 52 (1900) 179-. theorem, extension. Hamburger, M. Crelle J. Mth. 110 (1892) 158-. Pfaffians, functions allied to. Tanner, H. W. L. Q.I. Mth. 16 (1879) 34-. , property. Baker, H. F. L. Mth. S. P. 29 (1898) 141-. , theorem. Tanner, H. W. L. [1878] Mess. Mth. 8 (1879) 56-. , . Zajaczkoicski, W. (xn) Krk. Ak. (Mt.- Prz.)Ez.&Sp. 7(1880)67-; (xi) Mess. Mth. 10 (1881) 36-. Pfaff s integration method for equation in In variables. Jacobi, C. G. J. Crelle J. 2 (1827) 347-. . Binet, J. P. M. C. E. 14 (1842) 1049- ; 15 (1842) 74-. -, proof of theorem relative to. Zajacz- kowski, L. Arch. Mth. Ps. 47 (1867) 106-. problem. Frobenius, G. [1876] Crelle J. Mth. 82 (1877) 230-. . Hamburger, M. [1876] Arch. Mth. Ps. 60 (1877) 185-. . Darboux,G. Bll. Sc. Mth. As. 6 (1882) 14_, 49_ ; c. R. 94 (1882) 835-. . Morera, G. [1883] Tor. Ac. Sc. At. 18 (1882) 521-. . Stodotkiewicz, A. J. [1892] Prace Mt.- Fiz. 4 (1893) 56-; C. E. 115 (1892) 592-; 119 (1894) 489-. . Engel, F. Leip. Mth. Ps. B. 48 (1896) 413-. . Zantschewsky, J. Par. EC. Norm. A. 13 (1896) 267-. , generalisation. Tanner, H. W. L. L. Mth. S. P. 11 (1879-80) 131-. , integration method. Lie, S. Christiania F. 15 (1873) 321-. , Pfaff 's solution. Mayer, A. C. G. Mth. A. 17 (1880) 523-. , theorem of Jacobi. Cayley, A. Crelle J. 57 (1860) 273-. , theory. Lie, M. S. Arch. Mth. Ntvd. 2 (1877) 338-. transformation. Rusjan, C. Prace Mt.-Fiz. 8 (1897) 61-; 9 (1898) 61-; Fschr. Mth. (1897) 300- ; (1898) 303. Solutions, singular, of total differential equa- tions. Guldberg,A. Christiania Skr. (Mth.- Nt. Kl.) (1899) No. 4, 26 pp. Theorems. Le Pont, H. G. Teix. J. Sc. 8 (1887) 37-, 65-, 175-. Theory of system of equations. Stodolkievitz , A. J. C. E. 120 (1895) 825-. Total differential (ternary) equations. Cockle, (Sir) Jas. [1876] Manch. Lt. Ph. S. P. 16 (1877) 66-. __ ( ) . Rawson, R. Manch. Lt. Ph. S. P. 16 (1877) 114-. -- ( ) . Cockle, (Sir) Jas. Manch. Lt. Ph. S. P. 20 (1881) 119-. -- equations. Guldberg, A. Fschr. Mth. (1900) 368. --- , application of 2-periodic functions to. Gronwall, H. Stockh. Ofv. (1896) 295-; Fschr. Mth. (1896) 285-. --- denning arithmetico-geometric mean of 4 elements. Borchardt, C. W. Par. S. Mth. Bll. 7 (1879) 124-. , reduction from 3 to 2 variables. Minich, S. R. [1879] Em. S. It. Mm. 4 (1882) No. 1, 47 pp. --- , singularities. Gronwall, H. Stockh. Ak. Hndl. Bh. 24 (Afd. 1) (1899) No. 9, 22pp. --- , system. Stodoikieicicz , A. J. Krk. Ak. (Mt.-Prz.) Ez. 2 (1892) 299-; Crc. Ac. Sc. Bll. (1891) 283. , , integration. Bendixson, I. Stockh. Ofv. (1892) 271-. --- , systems. Gronwall, H. Stockh. Ofv. (1895) 729-; Fschr. Mth. (1895) 399. -- , . Hadamard, . Bordeaux S. Sc. PV. (1894-95) 17-. --- unconditionally integrable. Guldberg, A. Mh. Mth. Ps. 7 (1896) 332-. -- in 3 or more variables, primitive equation. Paoli, P. [1814] Mod. S. It. Mm. 17 (1815) 104-. differentials of 1st kind. Noether, H. Erlang. Ps. Md. S. Sb. 18 (1886) 11-. , integrals, quartic surfaces admitting. Berry, A. Camb. Ph. S. T. 18 (1900) 333-. and partial differential equations. Natani, L. [1860] Crelle J. 58 (1861) 301-. Transformation of linear differential expres- sion. Tanner, H. W. L. QJ. Mth. 16 (1879) 45-. X-*dx+Y-$dy + Z-%dz = Q. MacMahm, P. A. [1882] QJ. Mth. 19 (1883) 158-. X l dx 1 + X^dx^ + . . . + X n dx n = 0, integrability conditions. Stodoikiewicz , A. J. Prace Mt.- Fiz. 6 (1895) 20-; Fschr. Mth. (1895) 378. /zO, 2b i dx i = Q. Duport, . Liouv. J. th. 3 (1897) 17-. a.- Mt X, dx, + X,dx, . . . , canonical forms. Rmissiane, C. Mth. A. 50 (1898) 247-. 395 5220 5230 5220 Differential forms of the second and higher orders. (See also 8450.) Clairautian functions and equations. Cunning- ham, A. [1876] E. S. P. 25 (1877) 43-. Differential criticoid. Cockle, (Sir) Jas. Ph. Mg. 50 (1875) 440-. forms of infinitely high order. Zorawski, K. Krk. Ak. (Mt.-Prz.) Ez. 6 (1893) 419- ; Fschr. Mth. (1893-94) 456-. parameter of 2nd order, properties. Haton de la Goupilliere, . C. E. 101 (1885) 18-. parameters. Somigliana, C. Mil. I. Lomb. Ed. 22 (1889) 275-. of functions. Cornbescure, E. Par. EC. Norm. A. 7 (1878) 409-. , especially V 2 , definition. Bous- sinesq, J. C. E. 95 (1882) 479-. , theory. Beltrami, E. Bologna Ac. Sc. Mm. 8 (1868) 549-; Bologna Ed. (1869) 55-. quadratic form, second parameter. Bel- jankin, I. [1900] Kazan S. Ps.-Mth. Bll. 10 (1901) 181-. forms, classification. Eicci, G. Em. E. Ac. Line. Ed. 4 (1888) (Sem. 1) 203-. , parameters and invariants. Ricci, G. A. Mt. 14 (1886-87) 1-. , simultaneous transformation. Knob- lauch, J. Crelle J. Mth. 115 (1895) 185-. , theory. Eicci, G. A. Mt. 12 (* 1883- 84) 135-. , . Tannenberg, W. de. C. E. 119 (1894) 321-. quadratics, invariants. Maschke, H. N. Y. Am. Mth. S. T. 1 (1900) 197-, 508. Differentials of functions of several indepen- dent variables. Goursat, E. C. E. 101 (1885) 309-. Homogeneous expressions of higher order in several variables. Andreevskil, M. A. [1869] (xn) Eec. Mth. (Moscou) 4 (1869-70) (Pt. 1) Invariable expressions. Padova, E. Em. E. Ac. Line. Mm. 4 (1887) 4-. Quadratic forms of n differentials. Lipschitz, E. Crelle J. 71 (1870) 274-. , and Abel's transcendents. Lipschitz, E. [1871] Crelle J. 74 (1872) and linear differential forms, invariants. Hessenberg, G. Acta Mth. 23 (1900) 121-. Ternary quadratic differential forms, canonical types. Koenigs, G. C. E. 100 (1885) 789. Total differential equations. Guldberg, A. C. E. 126 (1898) 1335- ; 127 (1898) 1199-. ,2nd order. Guldberg, A. Christiania Skr. (Mth.-Nt. Kl.) (1898) No. 11, 24 pp. , , systems. Pascal, E. C. E. 130 (1900) 645-. , 3rd order, theory. Pascal, E. Mil. I. Lomb. Ed. 33 (1900) 675-. , any order. Pascal, E. Mil. I. Lomb. Ed. 33 (1900) 287-. 5230 Transformation of differen- tial forms, including- tangential (or contact) transformations. Adjoint group, theory. Ahrens, W. Leip. Mth. Ps. B. 49 (1897) 358-. operation. Pincherle, S. Bologna Ed. 2 (1898) 130-. Algebraic configurations with continuous in- tegrable group of projective automorphic transformations. Fano, G. Ven. I. At. (1895-96) 1069-. non-integrable group of projective automorphic transformations. Fano,G. Tor. Ac. Sc. Mm. 46 (1896) 187-. equations admitting infinitesimal trans- formations. Lie, M. S. Christiania F. (1881) No. 15, 6pp. Associated differential forms. Halphen, G. Liouv. J. Mth. 6 (1890) 211-. Change of variable. Mertens, F. Krk. Ak. (Mt.-Prz.) Ez. 20 (1890) 267-; Fschr. Mth. (1890) 282. Continuous projective groups, simply infinite, in ordinary space. Pittarelli, G. A. Mt. 22 (1894) 261-. Differential parameters. Pennacchietti, G. Ca- tania Ac. Gioen. At. 9 (1896) Mem. 1, 11 pp. quotients of higher order. Sonin, N. Ja. St Pet. Ac. Sc. Bll. 1 (1894) 321- ; Fschr. Mth. (1893-94) 456. Equations admitting a group of transforma- tions. Lie, M. S. Arch. Mth. Ntvd. 7 (1882) 443-; 8 (1883) 187-, 249-, 371-. , and group theory. Lie, S. Leip. Mth. Ps. B. 47 (1895) 261-. of problems of dynamics admitting an in- finitesimal transformation. Stnckel, P. C. E. 119 (1894) 508-, 723-. . (Claim of priority.) Staude, 0. C. E. 119 (1894) 903. . (Staude's claim of priority.) Stdckel, P. C. E. 119 (1894) 1189. Infinitesimal adjoints of plane curve. Ocagne, M. d\ N. A. Mth. 19 (1900) 219-. deformations. Padova, E. Em. E. Ac. Line. Ed. 5 (1889) (Sem. 1) 174-. Integrability factor, theory. Lie, M. S. Chris- tiania F. (1874) 242-. Lie, theorem of. Tannenberg, W. de. Bor- deaux S. Sc. PV. (1897-98) 49-. Lie's equations, double generalisation. Vessiot, E. C. E. 125 (1897) 1019-. Linear homogeneous group. Maurer, L. Leip. Mth. Ps. B. 46 (1894) 215-. Orbits of point on surface admitting infini- tesimal transformations. Staude, 0. Leip. Mth. Ps. B. 44 (1892) 429-; 45 (1893) 511-. Orthogonal substitutions generated by repeti- tion of infinitesimal substitution. Taber, H. L. Mth. S. P. 26 (1895) 364-. Partial differential equations, 2nd order, ad- mitting an infinite group of point trans- formations. Medolaghi, P. A. Mt. 1 (1898) 229-. 396 5230 Transformation of Differential Forms 5230 Eeciprocants, theory. Perrin, E. C. B. 102 (1886) 351-. Eeversion of partial differential expressions with two independent and two dependent variables. Elliott, E. B. [1890] L. Mth. S. P. 22 (1891) 79-. Total differentials of first kind. Noether, M. Erlang. Ps. Md. S. Sb. 18 (1886) 11-. TEANSFOEMATION. Sylvester, J. J. E. S. P. 7 (1854-55) 219-. Cockle, (Sir) Jas. Ph. Mg. 13 (1882) 357-. Stackel, P. Crelle J. Mth. Ill (1893) 290-. automorphic, of V V 2 = 0. Levi-Civita, T. Ven. I. At. (1897-98)" 1399-. infinitesimal, of sum of squares. Beez, . Z. Mth. Ps. 43 (1898) 65-, 121-, 277-. of Biicklund. Clairin, J. Bll. Sc. Mth. 24 (1900) 284-. for pseudospherical surfaces. Bianchi, L. Em. E. Ac. Line. Ed. 1 (1892) (Sem. 2) 3-. triply orthogonal pseudospherical systems. Bianchi, L. Em. E. Ac. Line. Ed. 1 (1892) (Sem. 2) 156-. biordinal of Schwarz's. Cockle, (Sir) Jas. Mess. Mth. 11 (1882) 49-. continuous, discrete groups. Ahrens, W. Mth. A. 50 (1898) 518-. Cremonian, continuous groups. Enriques, F., & Fano, G. A. Mt. 26 (1897) 59-. of curves and surfaces. Bficklund, A. V. Lund Acta Un. 10 (1873) (Mth.) No. 2, 12 pp. differential operator. HatJiaway, A. S. [1883] (xn) J. H. Un. Cir. [3] (1884) 35-. differentials of 1st order and 2nd degree by generalised elliptic coordinates. Henrici, 0. [1865] Crelle J. 65 (1866) 1-. dualistic differential. Elliott, E. B. B. A. Ep. (1891) 568 ; L. Mth. S. P. 23 (1892) 188-. of equations, 1st order. Zorawski, K. Krk. Ak. (Mt.-Prz.) Bz. 6 (1893) 67-; Crc. Ac. Sc. Bll. (1892) 292-. group, finite continuous, functions represent- ing. Sehur, F. Mth. A. 41 (1893) 509-. groups, continuous. Maurer, L. Mth. A. 39 (1891) 409-. , , equations defining. Engel, F. Mth. A. 27 (1886) 1-. , , theory. Campbell, J. E. L. Mth. S. P. 28 (1897) 381-. of 4-dimensional manifoldness. Kowalew- ski, G. Leip. Mth. Ps. B. 50 (1898) (Mth.) 60-. , elementary notions. Combebiac, . N. A. Mth. 18 (1899) 347-. , finite continuous, composition. Killing, W. Mth. A. 31 (1888) 252-; 33 (1889) 1-; 34 (1889) 57- ; 36 (1890) 161-. , , greatest sub-groups. Killing, W. Mth. A. 36 (1890) 239-. , , theory. Schur, F. Mth. A. 35 (1890) 161- ; 38 (1891) 263-. relating to certain differential equations. Picard, E. C. E. 106 (1888) 118-. of homogeneous expressions. Christoffel, E. B. Berl. Mb. (1869) 1-; Crelle J. 70 (1869) 46-, infinitesimal. Saltykoic, N. Liouv. J. Mth. 3 (1897) 429-. . Lovett, E. O. B. A. Ep. (1899) 648-. , of concentric conies. Lovett, E. 0. N. Y. Am. Mth. S. Bll. 4 (1898) 520-. , especially contact transformations of plane. Mayer, A. Leip. Mth. Ps. B. 45 (1893) 697-. , of plane. Zorawski, K. Leip. Mth. Ps. B. 52 (1900) 77-. projective. Lovett, E. O. N. Y. Am. Mth. S. Bll. 4 (1898) 515-. , monomial group defined by. Torelli, G. Palermo Cir. Mt. Ed. 8 (1894) 41-. , , finite equations. Torelli, G. Nap. Ed. 33 (1894) 91-; 34 (1895) 56-. , reduction to normal form. Engel, F. Leip. Mth. Ps. B. 46 (1894) 73-. , of typical Cremonian groups. Fano, G. Em. E. Ac. Line. Ed. 7 (1898) (Sem. 1) 332-. /3C/Y /St/V /5t/\ 2 n leaving equation ^J + (_ J + (^_J =0 unaltered. Campbell, J. E. Mess. Mth. 28 (1899) 97-. lengths of arcs on surfaces unaltered. Campbell, J. E. L. Mth. S. P. 29 (1898) 249-. Lie's, sub-groups of a group. Maillet, E. C. E. 130 (1900) 1449-. linear homogeneous. Engel, F. Leip. Mth. Ps. B. 49 (1897) 249-. of linear partial differential operators by ex- tended linear continuous groups. Elliott, E. B. L. Mth. S. P. 29 (1898) 439-. ordinals, new. Cockle, (Sir) Jas. Ph. Mg. 13 (1882) 44-. point, infinitesimal elements of 2nd order in. Hadamard, . Bordeaux S. Sc. PV. (1895- 96) 11-. of quadratic differential expressions. Voss, A. E. Mth. A. 16 (1880) 129-. form. Levy,M. C. E. 86 (1878) 463-. tangential. Fouret, G. C. E. 85 (1877) 134-. . Engel, F. Mth. A. 23 (1884) 1-. . Lovett, E. 0. [1898-1900] N. Y. Am. Mth. S. Bll. 4 (1898) 402- ; 7 (1901) 113, 115-. . Levi-Civita, T. Ven. I. At. (1899-1900) (Ft. 2) 671-. , analytic theory. Lie, S. Christiania F. 15 (1873) 237-. , of circles in plane, synthetic determination of all. Scheffers, G. Leip. Mth. Ps. B. 51 (1899) (Mth.) 145-. , class. Lovett, E. 0. C. E. 127 (1898) 480-. , , determination. Scheffers, G. Acta Mth. 14 (1890-91) 111-. , of developable. Vivanti, G. Palermo Cir. Mt. Ed. 5 (1891) 205-. , surfaces. Lovett, E. 0. Palermo Cir. Mt. Ed. 13 (1899) 210-. , and differential equations. Lie, S. Leip. Mth. Ps. B. 50 (1898) (Mth.) 113-. , generalised, theory. Liebmann, H. Leip. Mth. Ps. B. 51 (1899) (Mth.) 354-. , group of linear, groups of finite order con- tained in. Autonne, . C. E. 102 (1886) 313-. , infinitesimal, of mechanics. Lie, S. Leip. Mth. Ps. B. 41 (1889) 145-. 397 5240 Differential Invariants tangential, infinitesimal, of optics. Lie, S. Leip. Mth. Ps. B. 48 (1896) 131-. , invariant theory. Lie, S. Christiania F. 14 (1872) 133- ; Mth. A. 8 (1875) 215-. , leaving ratio of measures of curvature un- altered. Vivanti,G. Z.Mth.Ps. 37 (1892) 1-. , Lie's. Mayer, A. C. G. Gott. Nr. (1874) 317-. , theory, and theory of perturbations. Lovett, E. O. Am. As. P. (1897) 58-. , and line geometry. Lie, S. Leip. Mth. Ps. B. 49 (1897) 687-. , between line and sphere, infinite continuous group. Lovett, E. O. G. E. 129 (1899) 405-, 1296. , problem. Vivanti, G. Palermo Cir. Mt. Ed. 5 (1891) 269-. , properties in connexion with curvature. Mehmke, R. Z. Mth. Ps. 38 (1893) 7-. , of surfaces, theory. Bdcklund, A. V. Mth. A. 19 (1882) 387-. , theory. Lie, S. Leip. Mth. Ps. Ab. 14 (1888) 535-. , . LUienthal, E. von. Mth. A. 50 (1898) 303-. theory, general. Lie, S. Leip. Mth. Ps. B. 47 (1895) 494-; 48 (1896) 390-. of 4S r8 a r8 -r . -T , reciprocity law. Beltrami, E. [1866] r G. Mt. 5 (1867) 24-. Uniform substitutions. Podetti, F. G. Mt. 35 (1897) 264-. Universal invariants, application to point and contact groups in space of two dimensions. Eabut, . Par. EC. Pol. J. 4 (1898) 137-. 5240 Differential invariants. (See also 1230.) Combescure, E. [1870] Mntp. Mm. Ac. Sect. Sc. 7 (1867-71) 447-. Lie, S. Mth. A. 24 (1884) 537-. Sylvester, J. J. C. B. 102 (1886) 31-. Appell, P. C. R. 105 (1887) 55-. Papperitz, . D. Nf. Tbl. (1887) 229. Goursat, E. C. E. 107 (1888) 898-. Appell, P. Liouv. J. Mth. 5 (1889) 361-. Rivereau, P. Liouv. J. Mth. 8 (1892) 233-. Levi-Civita, T. Yen. I. At. (1893-94) 1447-, Bouton, C. L. N. Y. Am. Mth. S. Bll. 4 (1898) 313-. Wallenberg, G. Crelle J. Mth. 121 (1900) 200-. Application to linear homogeneous equations. Wallenberg, G. Crelle J. Mth. 113 (1894) 1-. Binomial biordinals. Cockle, (Sir) Jas. L. Mth. S. P. 12 (1880-81) 63-; Ph. Mg. 24 (1887) 249-. Canonical expansions in series, coefficients being invariants of continuous group. Tresse, A. C. E. 114 (1892) 1256-. Class. Malet, J. C. [1881] Phil. Trans. 173 (1883) 751-. , relative to linear equations. Poincare, H. C. E. 94 (1882) 1402-. Differential Invariants 5240 Configurations of three dimensions. Cotton, E. Toul. Fac. Sc. A. 1 (1899) 385-. Conjugate annihilators of homogeneous and isobaric differential expressions. Rogers, L. J. Mess. Mth. 18 (1889) 153-. Connexion between elements of orthogonal 9 and 16 systems. Jahnke, E. Crelle J. Mth. 118 (1897) 224-. Continuous groups. Lie, S. Christiania F. (1884) No. 8, 4 pp., No. 9, 4 pp. ; (1885) A T o. 15, 4 pp. transformation groups, extension. Zorawski, K. Krk. Ak. (Slt.-Prz.) Rz. 4 (1893) 34-; Fschr. Mth. (1893-94) 635. Coresolvents. Cockle, Jas. [1862] QJ. Mth. 6 (1863) 9-, 151-, 226-. Covariant belonging to quadratic invariant. Ricci, G. Em. E. Ac. Line. Ed. 3 (1887) (Sem. 1) 15-. Covariants. Cockle, (Sir) Jas. [1863-64] Ph. Mg. 27 (1864) 225-; 28 (1864) 205-. of plane curves. Gwyther, R. F. [1893] Phil. Trans. (A) 184 (1894) 1171-. Critical functions. Cockle, Jas. QJ. Mth. 4 (1861) 97-, 265-. . Warren, J. W. [1863-64] (vi) QJ. Mth. 6 (1864) 231-, (vni) 372-. Deformation invariants. Zorawski, K. Acta Mth. 16 (1892-93) 1-. Differential equation of most general substitu- tion of 1 variable. MacMahon, P. A. Ph. Mg. 23 (1887) 542-. operators, certain linear, interchange of variables in. Elliott, E. B. [1889] Phil. Trans. (A) 181 (1891) 19-. used to form system of invariants, etc. Elliott, . L. Mth. S. P. 26 (1895) 185-. parameters, simple or simultaneous. Morin, P. C. E. 66 (1868) 601-. resolvents. Cockle, (Sir) Jas. [1864] Manch. Lt. Ph. S. P. 4 (1865) 38-. . Spottiswoode, W. QJ. Mth. 6 (1864) 262-. of algebraic equations, higher degree. Lachtin, L. K. [1896-98] Eec. Mth. (Moscou) 19 (1897) 21 1-, 393- ; 20 (1899) 260- ; Fschr. Mth. (1897) 88- ; (1898) 74. and partial differential resolvents. Raw- son, R. Manch. Lt. Ph. S. P. 21 (1882) 59-. Functional invariants, class. Forsyth, A. R. [1888] Phil. Trans. (A) 180 (1890) 71-. Function- and invariant-theory of binomial configurations. Welhtein, J. Mth. A. 52 (1899) 70-. Fundamental invariants. Vogt, H. Par. EC. Norm. A. 6 (1889) (Suppl.) 71 pp. General quartine or incriticoid of 4th degree. Barley, (Rev.) R. Ph. Mg. 26 (1888) 456-. Group of motions and their invariants. Lie, S. Leip. Mth. Ps. B. 45 (1893) 370-. theory and its applications. Zorawski, K. Krk. Ak. (Mt.-Prz.) Ez.6 (1893) 289-; Crc. Ac. Sc. Bll. (1893) 145-. Homographic invariants and quotient deriva- tives. Forsyth, A.R. Mess. Mth. 17 (1888) 154-. Indefinite quadratic invariants. Bianchi, L. Em. E. Ac. Line. Mm. 5 (1888) 539-. 398 5240 Differential Invariants Infinitesimal transformations of plane curve with corresponding differential invariants. Arvay, W.,(x) in sines and cosines of multiples of x. Bessel, F. W. As. Nr. 16 (1839) 229-. Expansions of functions employed in physics, general reason for use. Boussinesq, J. C. E. 92 (1881) 513-. implicit functions in trigonometric series, and their transformation to isotropic means. Cauchy, A. L. C. E. 38 (1854) 910-. in series of multiple sines and cosines. Cayley, A. Camb. Mth. J. 3 (1843) 162-. particular solutions of equations of elasticity. Lauricella, G. Tor. Ac. Sc. At. 34 (1898) 11-. , trigonometric. Lerch, . Toul. Fac. Sc. A. 3 (1889) C, 11 pp. , , of periodic solutions of V 2 F = 0. Appell, P. C. B. 102 (1886) 1439- ; Liouv. J. Mth. 3 (1887) 5-. Fourier's double integral theorem. Glaisher, J. W. L. Mess. Mth. 2 (1873) 20-. , Boole's and other proofs. Alex- ander, P. Edinb. Mth. S. P. 3 (1885) 45-. , failing cases. Alexander, P. Edinb. Mth. S. P. 3 (1885) 12-. integrals, more general theorem. Griln- ivald, F. K. Z. Mth. Ps. 9 (1864) 131-. expansion for doubly periodic functions, new method. Mohrmann, G. Arch. Mth. Ps. 12 (1894) 1-. expansions of functions in trigonometric series. Tliomson, (Sir) W. Camb. Mth. J. 2 (1841) 258-. transcendent functions. Krygowski, Z. Prace Mt.-Fiz. 5 (1894) 70-. formula and its analogues. Pellet, A. C. E. 106 (1888) 1062-. applied to differentiation. Liouville, J. Crelle J. 13 (1835) 219-. generalised. Liouville, J. Liouv. J. Mth. 1 (1836) 102-. integrals. Du Bois-Reymond, P. Mth. A. 4 (1871) 362-. problem. LeRoy,E. C. R. 120 (1895) 179-. theorem. Gregory, D. F. Camb. Mth. J. 3 (1843) 288-. . La Vallee-Poussin, C. de. Brux. S. Sc. A. 15 (1891) (Ft. 1) 39-. , general extension. Preston, T. Ph. Mg. 43 (1897) 281-. , generality of new theorem analogous to. Stoney, G. J. Ph. Mg. 43 (1897) 139-. , proof. Bonnet, 0. N. A. Mth. 3 (1844) , . Walton, W. [1865] QJ. Mth. 8 (1867) 136-. , , Cauchy's. Schouten, G. N. Arch. Wisk. 18 (1891) 146-. and sin oo and cos oo . Moon, R. Ph. Mg. 26 (1845) 483-. theorems. Hobson, E. W. Mess. Mth. 11 (1882) 11-. , proof. Schldmilch, 0. Crelle J. 36 (1848) 268-. Functions expressible by series of periodic terms. Ivory, J. Ph. Mg. 9 (1836) 161-. Harmonic analyser, Henrici and Coradi's. Grabowski, L. Wiad. Mt. 4 (1900) 236-. components, computation (for yearly and daily periods). Strachey, (Lt.-Gen.) R. B. S. P. 42 (1887) 61-. of periodic motion. Clifford, W. K. [1873] (vn) Mess. Mth. 3 (1874) 153-. curves, graphic analysis. Wedmore, E. B. Elect. 35 (1895) 512-. decomposition of functions, and allied ex- pansions. Johnson, A. R. L. Mth. S. P. 19 (1889) 75-. expansion of arbitrary function. Zaremba, S. Liouv. J. Mth. 6 (1900) 47-. series. Schldmilch, O. Z. Mth. Ps. 14 (1869) 250-. . Cesdro, E. (xn) Mathesis 1 (1881) 51-, 80, 143-. Integral formulas representing arbitrary func- tions, range of application. Du Bois-Rey- mond, P. [1874] Crelle J. Mth. 79 (1875) 38-. Ae~ ar Neumann's potential function S . Bel- trami, E. Em. E. Ac. Line. Ed. 4 (1895) (Sem. 2) 177-. Periodic curves, analysis. Wedmore, E. B. I. Elect. E. J. 25 (1897) 224-. function cos x - % cos 3x + % cos 5x etc. Wil- braham, H. Camb. and Dubl. Mth. J. 3 (1848) 198-. A + A l cos + B 1 sin ... + A n cos n + B n sin n, transformation. Stieltjes, T. J. (jun.) N. Arch. Wisk. 9 (*1882) 111-. series, summation. Raabe, J. L. Crelle J. 15 (1836) 355-. , value at certain limits. Newman, F. W. Camb. and Dubl. Mth. J. 3 (1848) 108-. Periodicities, investigation of hidden, with application to supposed 26 day period. Schuster, A. Terr. Mag. 3 (1898) 13-. , lunar and solar, of earthquakes. Schuster, A. Nt. 56 (1897) 321. in series of observations, method of ascer- taining. Nene.V.N. B. S. P. 36 (1884) 361-. Phases of function, calculation. Cabreira, A. [1900] Lisb. J. Sc. Mth. 6 (1902) 55-. Series of sines and cosines of multiple angles, numerical values of coefficients. Lubbock, J. W. Ph. Mg. 33 (1848) 106-. Summation of periodic functions. Gylden, J. A. H. (x) Stockh. Ak. Hndl. 11 (1872) No. 1, 15 pp. ; Par. EC. Norm. A. 8 (1879) 203-. Trigonometric series, convergence for arbitrary function between given limits. Lejeune- Dirichlet, G. Crelle J. 4 (1829) 157- ; 17 (1837) 35-. , representation of arbitrary functions by. Lejeune-Dirichlet, G. Epm. Ps. 1 (1837) 152-. , . Harnack, A. Mth. A. 17 (1880) 123-. Wave motion, supposed proof of theorem in. Stoney, G. J. Ph. Mg. 43 (1897) 368- ; 44 (1897) 98-, 206-. , . Preston, T. Ph. Mg. 43 (1897) 458-. 402 Harmonic Analysis; Series other than Fourier's 5620 5620 Harmonic analysis; series other than Fourier's. Spheri- cal and ellipsoidal harmonics. (See also 3220.) Cylindrical functions, development of zero in series of. Nielsen, N. Mth. A. 52 (1899) 582-. , integrals, and Fourier series. Hankel, H. [1869] Mth. A. 8 (1875) 471-. Ellipsoidal harmonics. Niven, W. D. [1890] Phil. Trans. (A) 182 (1892) 231-. Expansion of given function in series according to harmonic functions. Steklov,V.A. Khar- kov Mth. S. Com. 5 (1896?) 60-; Fschr. Mth. (1896) 203- ; C. B. 128 (1899) 279-. {JS 2 - 2Rr [cos u cos u' cos (x - x') + sin u sin u' cos (y - y')] + r 2 }" 1 . Stieltjes, T. J. Liouv. J. Mth. 5 (1889) 55-. Fourier-Bessel functions, representation of arbitrary function by. Harnack, A. Leip. Mth. Ps. B. 39 (1887) 191-. Functional series, expansion of function in. Johnson, A. R. Mess. Mth. 15 (1886) 81-. Functions X, Y, etc., possessing property that = 0, and expansible in form aX + pY . , construction. Routh, E. J. L. Mth. S. P. 10 (1878-79) 146-. Fundamental functions and expansion in terms of them. Renaux, . C. B. 129 (1899) 545-. Harmonic analysis. Perry, J., & Hunt, H. F. L. Ps. S. P. 14 (1896) 1-; Ph. Mg. 40 (1895) 506-. , application to dynamical theory of tides. Hough, S. S. Phil. Trans. (A) 189 (1897) 201-. expression of daily variation of solar radia- tion, etc. Hargreaves, R. Camb. Ph. S. P. 9 (1898) 532-. functions for elliptic cone. Hobson, E. W. L. Mth. S. P. 23 (1892) 231-. , expansion of arbitrary function in series of. Zaremba, S. C. B. 128 (1899) 1088-. of Poincare. Stekloff, W. Toul. Fac. Sc. A. 2 (1900) 273-. , theorem. Arone, G. D. d\ C. B. 114 (1892) 1055-. of 3 variables, analogous to 0-Fuchsian functions. Stouff, X. Toul. Fac. Sc. A. 4 (1890) A, 15 pp. , new theorems. Sanctis, L. de. A. Mt. 4 (1900) 161-. several real variables. Arone, G. D. d\ C. B. 118 (1894) 342-. periodic infinite series, summation. Raabe, J. L. Crelle J. 23 (1842) 105- ; 25 (1843) 160-. series with infinitely small terms. Fontana, G. Verona S. It. Mm. 2 (1784) 123-. Harmonics of a ring. Niven, W. D. L. Mth. S. P. 24 (1893) 272-. Lame's equation. Escary, . C. B. 91 (1880) 40-, 102-. Lame's and spherical functions, expansion by. Lindemann, C. L. F. [1881] Mth. A. 19 (1882) 323-. Laplace's coefficients. Glaisher, J. W. L. L. Mth. S. P. 6 (1874-75) 126-. . Hall, A. A. Mth. 3 (1887) 1-. , equation of. Moon, R. Ph. Mg. 40 (1870) 434-. , expansion, determination of numerical factors in. Tarleton, F. A. Ir. Ac. P. 1 (1889-91) 16-. and functions. Heath, D. D. OJ. Mth. 7 (1866) 23-. , proof that F (p, u) may be expanded into a series of. Procul. [Pseudon.] (vi Adds.) Ph. Mg. 25 (1844) 337-. , transformation. Plarr, G. Edinb. B. S. T. 36 (1892) 19-. values of integral I Q n Q n 'dfj., Q n> Q n > J o being Laplace's coefficients of orders n, n', with application to theory of radiation. Rayleigh, (Lord). Phil. Trans. 160 (1870) 579-. , F (0, !/<) developable in only one series of. Pratt, J. H. Ph. Mg. 8 (1836) 474-. b ( ^, use of continued fractions in calcu- lation. Callandreau, O. Par. EC. Pol. J. cah. 45 (1878) 91-. dynamical equations, general integration. Hough, S. S. [1897] Phil. Trans. (A) 191 (1898) 139-. functions. Cayley, A. Liouv. J. Mth. 13 (1848) 275-. . Schlafli, L. A. Mt. 1 (1867-68) 243-. , Dirichlet's proof of expansion possibility. Poincare, H. C. B. 118 (1894) 497-. , expansion of arbitrary function in. Echols, W. H. A. Mth. 12 (1898-99) 164-. , fundamental principle, geometrical proof. Pratt, J. H. Ph. Mg. 24 (1862) 504-. of higher order, development of function of several variables according to. Mehler, F. G. Crelle J. 66 (1866) 161-. , property. Smith, Arch. Camb. Mth. J. 2 (1841) 192. , series expressing. Darboux, G. Liouv. J. Mth. 19 (1874) 1-. integrals, definite integrals allied to. Un- ferdinger, F. Wien Sb. 55 (1867) (Ab. 2) 90-. Legendre's functions, expansion of potential function l/B*-i in. Routh, E. J. L. Mth. S. P. 26 (1895) 481-. Linear, cylindric, spherical, and allied func- tions, expansion of functions in terms of. Alexander, P. Edinb. E. S. T. 33 (1888) 313-. Periodic series, expansion of function. Encke, J. F. Berl. As. Jb. (1860) 313-. . , sums, critical values. Stokes, G. G. [1847] Camb. Ph. S. T. 8 (1849) 533-. Potentials of ellipsoid and ellipse, expansion in series. Niven, W. D. Phil. Trans. 170 (1880) 379-. 403 cc 2 5620 Harmonic Analysis Series for function of elliptic coordinates. Escary, . As. Fr. C. B. (1887) (Pt. 2) 63-. of roots of transcendental equation, repre- sentation of arbitrary function by. Fujisau-a, E. Tok. Coll. Sc. J. 2 (1889) 1-. Solar radiation, distribution on surface of earth, and dependence on astronomical ele- ments. Hargreaves, R. [1896] Camb. Ph. S. T. 16 (1898) 58-. Spherical analysis. Eastwood, G. Camb. (M.) Mth. M. 2 (I860) 163-, 186-, 271-, 301- ; 3 (1861) 78-. function P"(cos0), Dirichlet's integral ex- pression for, and analogous form for cylin- drical function J (x). Mehler, F. G. [1871] Mth. A. 5 (1872) 141-. functions, differentials. Most, R. [1868] Crelle J. 70 (1869) 163-. , geometric significance. Heine, E. Crelle J. 68 (1868) 386-. , series, extension of theorem of Cantor's to. Neumann, C. G. Leip. Mth. Ps. B. 33 (1881) 1-; 35 (1883) 18-; Mth. A. 22 (1883) 406-. , proceeding by. Thome, W, L. Crelle J. 66 (1866) 337-. , , with application to cylindrical functions. Bauer, G. Miinch. Ak. Sb. 5 (1875) 247-. , theory. Bruns, H. [1880] Crelle J. Mth. 90 (1881) 322-. , , Sidler's. Schlafli, L. Ziir. Vjschr. 8 (1863) 205-, 324-. harmonic analysis, definite integrals. Niven, W. D. Phil. Trans. 170 (1880) 379-. , theorem. Darwin, G. H. Mess. Mth. 6 (1877) 165-. of nth order, general canonical form. Thomson, (Sir) W. B. A. Bp. 41 (1871) (Sect.) 25-. harmonics. Tail, P. G. [1871] Edinb. B. S. P. 7 (1872) 589-. . Niven, W. D. [1877-78] Mess. Mth. 7 (1878) 1-, 131-. . Callandreau, O. C. E. 99 (1884) 23-. . Beltrami, E. C. B. 110 (1890) 934-. . Dyson, F. W. QJ. Mth. 26 (1893) 30-. , canonical form. Clifford, W. K. B. A. Bp. 41 (1871) (Sect.) 10. , cases. Burnside, W. Mess. Mth. 14 (1885) 122-. of complex degree, with application to physical problems. Hobson, E. W. [1887] Camb. Ph. S. T. 14 (1889) 211-. 2nd kind, expression in finite form. Bryan, G. H. Camb. Ph. S. P. 6 (1889) 293-. or Laplace's functions. Tallqvist, H. Helsingf. Acta 26 (1900) No. 4, 87 pp. representing magnetic potential of earth. Schuster, A. B. A. Bp. (1898) 752-. , systems. Hobson, E. W. L. Mth. S. P. 22 (1891) 431-. , theory. Neumann, C. Leip. Mth. Ps. Ab. 13 (1887) 401-. Equations of Physics 5630 Spherical harmonics, theory. Beltrami, E. C. B. 116 (1893) 181- ; Mil. I. Lomb. Bd. 29 (1896) 793-. of unrestricted degree, order, and argu- ment, type. Hobson, E. W. [1896] Phil. Trans. (A) 187 (1897) 443-. and hyper-spherical functions, theory. Neumann, C. Z. Mth. Ps. 12 (1867) 97-. Tesseral harmonics, functions connected with. Leahy, A. H. B. S. P. 56 (1894) 45-. Toroidal functions. Basset, A. B. Am. J. Mth. 15 (1893) 287-. Zonal harmonics, expansion of elliptic inte- grals by. Hargreaves, R. Mess. Mth. 26 (1897) 89-. 5630 Generalities on the differen- tial equations of mathematical physics. Branched (verzweigte) potentials in space. Sommerfeld, A. L. Mth. S. P. 28 (1897) 395-. Conjugate potential functions, generalisation. Appell, P. Mars. Fac. Sc. A. 2 (1892) 53-. Differential alternating functions in astronomy. Cauchy, A. L. C. B. 11 (1840) 297-, 377-, 432-. Equation in mathematical physics, integration. Carvallo, E. Par. S. Mth. Bll. 22 (1894) 234-. of motion of heat in thin prism, complete integration. Pagani, G. J/. Quetelet Cor. Mth. 3 (1827) 237-. in theory of propagation of electricity. Picard, E. Par. S. Mth. Bll. 22 (1894) 2-. Equations in conduction of heat. Sturm, J. C. F. Liouv. J. Mth. 1 (1836) 373-. of dynamics. Boole, G. Phil. Trans. (1863) 485-. . Ball, (Sir) R. Aust. As. Bp. (1895) 215-. , fundamental. Janaud, . Arch. Mth. Ps. 67 (1882) 160-. mathematical physics. Poincare, H. Palermo Cir. Mt. Bd. 8 (1894) 57-. . Steklov, V. A . Bee. Mth. (Moscou) 19 (1897) 469 (bis)-. mechanics. Grosso, R. del. G. Mt. 4 (1866) 243-. motion, property. Poisson, S. D. Par. S. Phlm. Bll. (1816) 109-. -physics, integration. Mathieu,E. Liouv. J. Mth. 17 (1872) 249-. for problems of oscillation. Abraham, M. Mth. A. 52 (1899) 81-. sound transmission through tube, Pois- son's solution. Moon, R. [1866] B. S. P. 15 (1867) 306-. of thermodynamics, integrating factors. Fliegner, A. Ziir. Vjschr. 40 (1895) 278-. vibrator affected by its own reflected vibrations, and analogous problems. Waals, J. D. van der (jun.). Amst. Ak. Vs. 8 (1900) 638-; Amst. Ak. P. 2 (1900) 534-. 404 5630 Differential Equations of Physics. General 5630 Equilibrium of fluids, studies on Green's work. Biittner, F. Leip. Jablon. Preisschr. 36 (1900) 98 pp. Existence proofs for potential in plane and in space. Harnack, A. Leip. Mth. Ps. B. 38 (1886) 144-. Function Z (x, y, z), applications to mathe- matical physics. Appell, P. Acta Mth. 8 (1886) 265-. Functions of Laplace and Bessel, expression of velocity potential by. McMahon, J. N. Y. Am. Mth. S. Bll. 2 (1896) 173-. satisfying equation VF=0. Appell, P. C. E. 96 (1883) 368-. of 3 variables satisfying equation V 2 .F=0. Appell, P. Acta Mth. 4 (1884) 313-. General equations of dynamical problems, particular integrals, use when general inte- gration is impossible. Barnard, J. G. Am. As. P. 25 (1876) 49-. Green's function. Eicci, G. Ven. I. At. (1884-85) 1439-. Huygens' principle, demonstration. Tedone, 0. Km. E. Ac. Line. Ed. 5 (1896) (Sem. 1) 357-, 483. Integral of Laplace's equation in finite terms. Earnshaic, S. B. A. Kp. (1880) 486-. i (l-a?)y'ds ; in theory of heat. Ivory, J. ?h. Mg. 1 (1827) 324-. . Poisson, S. D. Ph. Mg. 2 (1827) 11-. Integrals of differential Aquations, calculation by Cauchy-Lipschitz method. Painleve, P. C. E. 128 (1899) 1505-. partial differential equations, form. Poisson, S. D. Par. S. Phlm. Bll. (1817) 180-. Kirchhoff's diffusion equations, reduction. Farkas, G. Mth. Termt. E Mth. Nt. B. Ung. 16 (1899) 97-. Laplace's double equation, integration. Al- mansi, E. Em. E. Ac. Line. Ed. 9 (1900) (Sem. 1) 298-. equation. Boole, G. B. A. Ep. (1845) (pt. 2) 2 ; Camb. and Dubl. Mth. J. 1 (1846) 10-. . Carmichael, E. Ir. Ac. P. 6 (1853) 216-. . Donkin, W. F. [1856] Phil. Trans. (1857) 43-. , and analogues. Carmichael, E. Camb. and Dubl. Mth. J. 7 (1852) 126- ; Ph. Mg. 6 (1853) 273-. , quaternion integration. PreobraZenskij, V. V. Kazan S. Nt. (Ps.-Mth.) P. 2 (1884) 46-; Fschr. Mth. (1884) 308-. , most general solution of given degree. Hobson, E. W. L. Mth. S. P. 26 (1895) 492-. in toroidal coordinates, solution by method of imaginary inversion. Safford, F. H. A. Mth. 12 (1898-99) 27-. , transformations leading to new solu- tions. Forsyth, A. E. L. Mth. S. P. 29 (1898) 165-. extended equation V 2 V 2 = 0, integration. Lauricella, G. Ven. I. At. (1898) 236-. Linear equations occurring in mathematical physics. Macfarlane, A. Am. As. P. (1897) 54. partial differential equations. Jarisch, P. Hamb. Mth. Gs. Mt. 2 (1890) (Festschr., Tl. 2) 110-. , especially general equation of motion of elastic fluids, integration. Pois- son, S. D. Par. Mm. Ac. Sc. 3 (1818) 121-. relative to motion of heat in solids, integration. Pagani, G. M. Brux. Ac. Sc. Mm. 8 (1834) 32pp. Maxwell's theory of electricity, mathematically considered. Planck, M. D. Mth. Vr. Jbr. 7 (1899) (Heft 1) 77-. Molecular motion in gases, integration of equa- tions. Boltzmann, L. [1876] Wien Ak. Sb. 74 (1877) (Ab. 2) 503-. Operator V 2 , properties. Haton de la Goupil- liere, . Brux. S. Sc. A. 9 (1885) (Pt. 2) 151-. Partial differential equations of mathematical physics. Poincare, H. Am. J. Mth. 12 (1890) 211-. , solutions, multiform. Carslaw, H. S. [1898] L. Mth. S. P. 30 (1899) 121-. , , simple. Mathieu, E. L. Liouv. J. Mth. 5 (1879) 5-. , 4th order, VVw = 0. Mathieu, E. Liouv. J. Mth. 14 (1869) 378-. , systems, integration. Cauchy, A. L. C. E. 13 (1841) 1-. and vibrating surfaces. Biot, J. B. Par. Mm. de PI. 4 (1802) 21-. of small vibrations of elastic bodies, integration. Ostrogradsky, M. A. [1832] Ostrogradsky, M. A. [1829] St Pet. Ac. Sc. Mm. 1 (1831) 455-. Polynomials, conjugate, class. Deruyts, J. Brux. Mm. Cour. 4, 48 (1886) No. 3, 20 pp. Potential function, 3rd derivative. Pad, P. Genova Un. At. [11] (1892) 593-. theory. Dyck, W. Munch. Ak. Sb. 25 (1896) 261-, 447-; 28 (1899) 203-. Schwarz's integrals of functions given by V 2 F i+1 + F i = 0. Andrade, . Par. S. Mth. Bll. 22 (1894) 50-. '(4) Solution of - + etc. = 47r/>, theorems. Thomson, (Sir) W. Camb. and Dubl. Mth. J. 3 (1848) 84-. Stationary temperatures. Lauricella, G. Pa- lermo Cir. Mt. Ed. 11 (1897) 189-. Successive divergent approximations, example. Picard, E. Par. S. Mth. Bll. 28 (1900) 137-. Transformation, application of Dirichlet's method. Meyer, F. Mth. A. 26 (1886) 509-. Uniform functions F satisfying V 2 F=0. Mit- tag-Leffler, G. As. Fr. C. E. (1888) (Pt. 1) 153. 405 5630 Equations of Physics Variation of arbitrary constants in mechanics. Poisson, S. D. Par. Mm. Ac. Sc. 1 (1816) -- - ---- . Lubbock, J. W. Ph. Mg. 11 (1837) 492- ; 12 (1838) 254-. Vector conditions of integrability. Tait, P. G. [1877] Edinb. B. S. P. 9 (1878) 527-. -v^ - a 2 V*=\f/(t, x, y, z), integration. Liou- ville, J. Liouv. J. Mth. 4 (1839) 1-. "V" equal to positive unity (Lagrange). e. 2) Fiona, G. Brux. Ac. Bll. 8 (1841) (pte. . Thomson, (Sir) W. Liouv. J. Mth. 12 (1847) 493-. V%=0, solution by method of quaternions. Boole, G. Ir. Ac. P. 6 (1853-54) 375-. = , integration. Dint, U. A. Mt. 5 (1871- 73) 305-. = , . Ascoli, G. Mil. I. Lomb. Ed. 19 (1886) 285-. = , in non-Euclidean space. Bianchi, L. Em. E. Ac. Line. Ed. 9 (1900) (Sem. 2) , 333 ~- -pg = a 2 -r-g, integration. Mainardi, G. Tor- tolini A. 5 (1854) 161-, 273-, 459-. = , applications in acoustics and optics. Klinkerfues, W. Gott. Nr. (1868) 469-. = in theory of sound. Amagat, [E. H."] J. de Ps. 9 (1880) 56-. - Crelle J> 72 V 2 v = 0, reduction to ordinary differential equa- tions. Wangerin, F. H. A. Berl. Ak. Mb. (1878) 152-. = , V 2 v=-K*v, V z v = c^v, new solutions. Forsyth, A. E. Mess. Mth. 27 (1898) 99-, 190. = , solutions. Bromwich, T. J. I' A. L. Mth. S. P. 30 (1899) 281-. ^S + ( + ^2 + -)% = 0. integration. Boussinesq, J. C. E. 94 (1882) 514-. V 2 /=0 with n variables. Giuliani, G. G. Mt. 25 (1887) 109-. V 2 z = 0. Du Bois-Reymond, P. Crelle J. Mth. 103 (1888)' 204-. V 2 <=0 in orthogonal coordinates. Petrini, H. Fschr. Mth. (1890) 982-. V 2 M+fc%=0. B6cher, M. Am. J. Mth. 15 (1893) 78-. = in space of n dimensions. Marcolongo, . A. Mt. 24 (1896) 301-. , G. Palermo Cir. __ R. du dv 8 dx + d~ + dz = Morera, Mt. Ed. 7 (1893) 288-. V 2 w=fce tt on closed Eiemann surface. Picard, E. Liouv. J. Mth. 9 (1893) 273-. V 2 * + F/(x7/2)<|=0. Petrini, H. Stockh.Ak. Hndl. Bh. 19 (Afd. 1) (1894) No. 4, 36 pp. V 2 u=F(u, x, y), integration. Picard, E. C. E. 124 (1897) 1488-. Integration by Series 5640 V 4 u = 0, integration. Almansi, E. Tor. Ac. Sc. At. 34 (1898) 68- or 92-. = . Goursat, . Par. S. Mth. Bll. 26 (1898) 236-. = , integration. Almansi, E. Em. E. Ac. Line. Ed. 8 (1899) (Sem. 1) 104-, 618. = , . Boggio, T. Ven. I. At. (1899- 1900) (Pt. 2) 497-. V-u = e u and Fuchsian functions. Poincare, H. Liouv. J. Mth. 4 (1898) 137-. V 2 0= -j-y for space of n dimensions. Tedone, 0. A. Mt. 1 (1898) 1-. V 2 + +/=0. Zaremba, S. Par. S. Mth. Bll. 26 (1898) 70-. = , and harmonic functions. Zaremba, S. Par. EC. Norm. A. 16 (1899) 427-; Prace Mt.-Fiz. 11 (1900) 99-; Fschr. Mth. (1900) 375. V^^O, integration. Almansi, E. A. Mt. 2 (1899) 1-. V B %, integration. Brajcev, I. R. Eec. Mth. (Moscou) 21 (1901) 490- ; Fschr. Mth. (1900) 376. V' 2 u=k 2 u, problems relating to. Picard, . Par. S. Mth. Bll. 28 (1900) 186-. 5640 Integration of the differen- tial equations of mathematical physics by series. Convergence of series in Kepler's problem (Carlini). Jacobi, C. G. J. As. Nr. 30 (1850) 197. serving as integrals of differential equations of 1st order. Nazimov, P. S. Kazan S. Nt. (Ps.-Mth.) P. 8 (1890) 145- ; Fschr. Mth. (1890) 325. Divergent series in differential equations. Horn , J. D. Mth. Vr. Jbr. 8 (1900) (Heft 1) 219-. Expansions occurring in theory of conduction of heat. Poincare, H. C. E. 118 (1894) 383-. Formula for calculation of a certain potential. Lerch, . Liouv. J. Mth. 5 (1899) 427-. Functions satisfying the equation V 2 w = 0. Ascoli, G. Mil. I. Lomb. Ed. 18 (1885) 474_, 546_, 599-, 617-. Gauss's series in experimental physics. Tarn- men, H. G. Carl Epm. 18 (1882) 129-. Heating of a cylinder. Laar, J. J. van. Haarl. Ms. Teyl. Arch. 6 (1900) 65-. Integrals of differential equations near essen- tial singularities (viz . ' ' normal ' ' series in Poincare's sense). Rajewski, J. Krk. Ak. (Mt.-Prz.) Pam. 17 (1890) 166- ; Crc. Ac. Sc. Bll. (1890) 62-. , particular, of V*F=F, and their expansion in spherical harmonics. Neumann, C. Leip. Mth. Ps. B. 38 (1886) 75-. Integration of equations of motion. Sokoloff, J. C. E. 55 (1862) 99-. sound with resistance terms. Boussinesq, J. C. E. 118 (1894) 162-, 223-, 271-. 406 5650 Solution by Definite Integrals Integration by means of simple solutions. Somigliana, C. Mil. I. Lomb. Ed. 24 (1891) 1005-. of V 2 /=& 2 / for elliptic and parabolic co- ordinates. Hartenstein, J. H. Arch. Mth. Ps. 14 (1896) 170-. Integrations relating to elastic equilibrium. Mathieu, E. L. Par. EC. Pol. J. Cah. 48 - (1880) 163-. Laplace's equation, form, special. Niven, W. D. [1880] Mess. Mth. 10 (1881) 114-. , solution, particular. Jacobi, C. G. J. Crelle J. 36 (1848) 113-. Maxwell's equations of electro-magnetic field, solution. Fitzgerald, G. F. B. A. Ep. (1890) 755-. Propagation of electricity. Poincare, H. C. E. 117 (1893) 1027-. , partial differential equation in. Picard, E. C. E. 118 (1894) 16-. Solutions of (F 2 + fc 2 )^=0 in elliptic coordi- nates, and physical applications. (F 2 written for V 2 .) Maclaurin, R. C. [1898] Camb. Ph. S. T. 17 (1899) 41-. V 4 k = 0, solutions in series. Marcolongo, E. Nap. Ed. 28 (1889) 149-. + -= ~ = 6 (t), electric interpretation. Eeignier, C., & Bary, P. Cherb. S. Sc. Nt. Mm. 26 (1889) 65-. Dirichlet's Problem 5660 Integration of V*u = Q (2 variables). Levi- Civita, T. Tor. Ac. Sc. At. 33 (1897) 692- or 932-. 5650 Integration of the differen- tial equations of mathematical physics by definite integrals. 5660 Dirichlet's problem and analogous problems affected by boundary conditions. Analytic theory of heat, problem in. Stekloff", W. C. E. 126 (1898) 1022-. Boundary condition and discontinuities, func- tions determined from. Burnside, W. L. Mth. S. P. 22 (1891) 346-. conditions, determination of integrals of partial differential equations by. Picard, E. Par. S. Mth. Bll. 22 (1894) 103-. , given, determination of function in a circle from. Veltmann, W. Z. Mth. Ps. 26 (1881) 1-. , , solutions of certain differential equations satisfying. Brill, J. Camb. Ph. S. P. 6 (1889) 344-. value problem for continuous plane curve with no cusp. Noble, C. A. Gott. Nr. (1896) 191-. Contour values, determination of integrals of differential equations by. Picard, E. C. E. 130 (1900) 447-. -- , ---- linear equations, 2nd order, by. Picard, E. Liouv. J. Mth. 6 (1900) 129-. -- , ---- partial differential equations by. Picard, E. C. E. 109 (1889) 499-; Par. EC. Pol. J. 60 (1890) 89-; Liouv. J. Duhamel, J. M. C. Liouv. J. Mth. 19 (1854) Mth - 2 ( 1896 ) 295 ~ 337-. Definite integrals suggested by problems in theory of electricity. Thomson, (Sir) W. Camb. and Dubl. Mth. J. 2 (1847) 109-. Double definite integrals, employment in find- ing the form of waves. Cauchy, A. L. C. E. 13 (1841) 188-. Finite integrals of partial differential equations occurring in physics. Earnshaw, S. Ph. , 2nd order, by. Picard, E. C. E. 128 (1899) 1487-. Cooling of isotropic body, two theorems of ' Andradez, . N. A. Mth. 8 mg Mg. 4 (1877) 213-. Fresnel's integrals, geometric significance. Umov, N. N. Es. S. Nt. Mm. (Mth.) 6 (1885) 57-; Fschr. Ps. (1885) (Ab. 2) 15-. Integration of V 2 u = on given Eiemann surface. Ascoli, G. [1887] Stpckh. Ak. Hndl. Bh. 13 (Afd, 1) (1888) No. 2, 83pp. = (Ascoli). Enestrom, G. Stockh. Ofv. (1887) 99-; Fschr. Mth. (1887) 29. V 4 u = (2 variables). Almansi, E. Tor. Ac. Sc. At. 31 (1895) 527- or 881-. = in a plane circular field. Lauri- cella, G. Tor. Ac. Sc. At. 31 (1895) 610- or 1010-. = (Lauricella and Almansi). Vol- terra, V. Tor. Ac. Sc. At. 31 (1895) 618- or 1018-. V 2 u + fc 2 M = on Eiemann surfaces. Sommerfeld, A. Gott. Nr. (1895) 267-. Poincare'. (1889) 435-. Dirichlet's principle. Weierstrass, K. [1870] Wiad. Mt. 2 (1898) 1-. -- . Phragmen, E. Stockh. Ofv. (1891) 745-. -- . Arzela, C. Bologna Ed. 1 (1897) 71-. -- . Volterra, V. Palermo Cir. Mt. Ed. 11 (1897) 83-. -- . Hilbert, D. D. Mth. Vr. Jbr. 8 (1900) (Heft 1) 184-. , application to function. Graf, J. H. Bern Mt. (1891) 95-. in connection with V 2 u = ke u . Schulz, W. N. Y. Am. Mth. S. Bll. 5 (1899) 219, 221-. t demonstration, new. Phragmen, E. Stockh. Ofv. (1892) 511-. -- , , preliminary assumptions. Heine, E. Gott. Nr. (1871) 375-. _ , , Eiemann's. Weber, H. [1869] Crelle J. 71 (1870) 29-. , generalisation. Eiemann, . C. E. 106 (1888) 123-. -- j substitute for. Dahvigk, F. von. Gott. Nr. (1894) 160-. 407 5660 Dirichlet's Problem and its Analogues 5560 Dirichlet's principle, substitute for, in certain cases. Neumann, C. Leip. Mth. Ps. B. 47 (1895) 185-. problem. Riemann, J. Par. EC. Norm. A. 5 (1888) 327-. . Grave, D. A. As. Fr. C. R. (1895) (Pt. 2) 111-. . Zaremba, S. Par. EC. Norm. A. 14 (1897) 251-. . Fredholm, I. Stockh. Ofv. (1900) 39-. __. ,. . Integration of = under given con- ditions as to boundary and discontinuity. Schwarz, H. A. Berl. Mb. (1870) 767-. = . Prym, F. E. Crelle J. 73 (1871) 340-. V 2 u = within a circular area. Ascoli, G. Mil. I. Lomb. Rd. 17 (1884) 627-. = for some simple areas. Ascoli, G. Mil. I. Lomb. Rd. 18 (1885) 252-. -- for 2 dimensions, special case. Kluyver, --- = in any Riemann area. Ascoli, J. C. Acta Mth. 21 (1897) 265-. in domains, with discontinuous boun- daries. Schwarz, H. A. Ziir. Vjschr. 15 (1870) 272-. and its extension to general linear equations, 2nd order. Paraf, A. Toul. Fac. Sc. A. 6 (1892) H, 75 pp. , generalisation, and linear partial dif- ferential equations, 2nd order. Picard, E. C. R. 130 (1900) 1088-. and harmonic functions belonging to closed surface. Le Roy, . C. R. 123 (1896) 986-. , questions relating to. Liapounoff, A. C. R. 125 (1897) 808- ; Liouv. J. Mth. 4 (1898) 241-. -- , solutions by combination of Neumann's and Schwarz 's methods. Kohn, A. Mth. A. 53 (1900) 593-. Distribution of electricity, problem of. Poin- care, H. C. R. 104 (1887) 44-. , , and Neumann's problem. Stekloff, W. C. R. 125 (1897) 1026-. Equation of vibrations of a membrane. Poin- care, H. C. R. 118 (1894) 447-. Equations of conduction of heat, integration. Le Roy, E. Par. EC. Norm. A. 14 (1897) 379-; 15 (1898) 9-. Fundamental functions and Dirichlet'sproblem. Stekloff, W. C. R. 131 (1900) 870-. -- , existence. Stekloff, W. C. R. 128 (1899) 808-. -- .theory. Stekloff, W. C.R. 128 (1899) 984-. problems of mathematical physics. Stekloff, W. C. R. 128 (1899) 588-. ----- , general methods for solution. Stekloff, W. Toul. Fac. Sc. A. 2 (1900) 207-. Green's function, behaviour at limit of its region. Neumann, C. Leip. Mth. Ps. B. 40 (1888) 163-. -- for system of non-intersecting spheres. Burnside, W. L. Mth. S. P. 25 (1894) 94-. -- , theory. Zaremba, . Par. S. Mth. Bll. 24 (1896) 19-. functions in 1 dimension. Burkhardt, H. Par. S. Mth. Bll. 22 (1894) 71-. second function. Arcais, F. d\ Ven. I. At. (1897-98) 1664-. Integrals of Laplace's equation. Capelli, A. G. Mt. 23 (1885) 123-. Integration of 3- -r dt/ 2 Neumann, C. Crelle J. 59 (1861) 335-. Mth. A. 1 (1869) 1-. Weber ' G. Mil. I. Lomb. Rd. 18 (1885) 718-. = . Ohnesorge, 0. Arch. Mth. Ps. 2 (1885) 53-. V 4 u = for plane regions. Venske, 0. Gott. Nr. (1891) 27-. V*u=fu. Lindeberg, J. W. C. R. 130 (1900) 1539-. V 2 w=e" on a closed surface. Picard, E. Bll. Sc. Mth. 24 (1900) 196-. Inverse square, actions following law of. Vaschy, . As. Fr. C. R. (1894) (Pt. 2) 141-. Method of arithmetical mean. Neumann, C. Leip. Mth. Ps. Ab. 13 (1887) 705- ; 14 (1888) 563-. , Neumann's. Tauber, A. Mh. Mth. Ps. 5 (1894) 137-. , . Stekloff, W. C. R. 131 (1900) 987-, 1182-. successive approximations, Picard's. Zaremba, S. C. R. 124 (1897) 554-. Neumann's and Gauss's problems. Stekloff, W. C. R. 130 (1900) 480-. method and Dirichlet's problem. Poiucare, H. [1895] Acta Mth. 20 (1897) 59-. . Stekloff, W. C. R. 130 (1900) 396-, 1599-. . Korn, A. C. R. 130 (1900) 557, 1238- ; 131 (1900)26-. (Korn). Stekloff, W. C. R. 130 (1900) 826-. of solving Dirichlet's problem. La Vallee- Poussin, C. de. Brux. S. Sc. A. 18 (1894) (Pt. 1) 11-. Partial differential equation in theory of mem- branes. Picard, E. C. R. 117 (1893) 502-. equations, 2nd order, with imaginary characteristics. Picard, E. C. R. 122 (1896) 417-. Poisson's integral. Bncher, M. N. Y. Am. Mth. S. Bll. 4 (1898) 424-. Potential, existence, proof. Kirchhoff, G. Acta Mth. 14 (1890-91) 179-. function in plane space bounded by two circles with given values at boundaries. Kluyver, J. C. N. Arch. Wisk. 3 (1898) 230- ; Fschr. Mth. (1897) 363-. functions expressed in curvilinear coordi- nates. Painleve, P. [1889] Lille Tr. Mm. 1 (1889-91) Mem. 1, 28 pp. theory, solution of fundamental problem in. Schiitz, J. R. Gott. Nr. (1895) 1-. Problem of hydrodynamics. Lejeune-Dirichlet, G. [1857] Gott. Ab. 8 (1858-59) (Mth.) 3-. , Dirichlet's. Dedekind, R. [1860] Crelle J. 58 (1861) 217-. 408 6000 Functional and Difference Equations 6020 Problem of hydrodynamics, Dirichlet's. Brios- chi, F. [1860] Crelle J. 59 (1861) 63-. Eesidual integral in -=a 2 V 2 u. Hadamard, . Par. S. Mth. Bll. 28 (1900) 69-. Solution of Laplace's equation, finite and con- tinuous with normal deviation given at all points of bounding surface. Stekloic, W. A. Fschr. Mth. (1897) 364-. Steady flow of electricity on rectangular plate. Mathieu, E. As. Fr. C. E. (1886) (Pt. 2) 1-. Stokes's theorem for hydrokinetic circulation, proof. Mukhopadhyay , A. Beng. As. S. J. 59 (Pt. 2) (1891) 59-. Theorem of Picard. Kobb, G. C. E. Ill (1890) 726-. Poincare. Zaremba, S. C. E. 127 (1898) 215-. Thermal equilibrium of closed radiating surface. Picard, E. C. E. 130 (1900) 1499-. Values on closed surface, determination of integrals of partial non-linear equations by. Le Roy, E. C. E. 124 (1897) 1508-. Vaschy's theorem on any vectorial distribution. Larose, . Par. S. Mth. Bll. 24 (1896) 177- % -- + ( -^ = Q. Liouville, J. Liouv. J. Mth. dx* dy- 8 (1843) 265-. V 2 u = e u . Picard, E. Liouv. J. Mth. 4 (1898) 313-. Difference equations and functional equations. 6000 General. Analytic factorials, reduction in series. Novi, G. G. Mt. 2 (1864) 1-, 40-. Difference equation, 2nd order. Cayley, A. QJ. Mth. 14 (1877) 23-. Differential coefficients, calculation from tables. Sheppard, W. F. Nt. 60 (1899) 390-. and finite differences. Makeham, W. M. I. Act. J. 16 (* 1872) 98-. equations, numerical solution. Ldska, W. Prag Sb. (1897) (Mth.-Nt.) No. 35, 10 pp. Euler' s formula, deduction. Zbikovskij, A. [1870] St Pet. Ac. Sc. Mm. (Rs.) 18 (*1871) Finite differences. Lobatto, R. Amst. N. Vh. 6 (1837) 1-. increments, generalisation of formula for. Stieltjes, T. J. N. A. Mth. 7 (1888) 26-. Generic equations, class. Blissard, J. QJ. Mth. 5 (1862) 325-. Interpolation formula. Roe, E. D. [1900] N. Y. Am. Mth. S. Bll. 7 (1901) 202, 203-. Lagrange's calculus of functions, superiority to infinitesimal calculus. Magistrini, G. B. [1832] Bologna Mm. Ac. Sc. 1 (1850) 93-. Linear difference forms, algebra. Pincherle, S. Bologna Ac. Sc. Mm. 5 (1895-96) 87-. Numbers, A m O n class, properties. Blissard, J. QJ. Mth. 8 (1867) 85-; 9 (1868) 82-, 154-. Numbers, < = [> - (f ) (p - 1)-+ ... Ocagne, M. d'. G. Teix. J. Sc. 8 (1887) 171- ; 10 (1891) 185-. Progressions. Jadanza, N. G. Mt. 7 (1869) 117-. Eecurring systems of 3rd order, especially periodic systems. Bortolotti, E. Palermo Cir. Mt. Ed. 5 (1891) 129-. Sum of functions of quantities in arithmetical progression. Richmond, H. W. Mess. Mth. 21 (1892) 29-. Summation formula of Euler. Enestrom, G. Stockh. Ofv. 36 (1879) No. 10, 3-. -- Stirling. Sonin, N. Ja. [1889] Vars. S. Nt. Tr. (1889-90) (C. R., Ps. C.) No. 2, 1-. formulas of Euler and Stirling. Sonin, N. Ja. [1889] Vars. S. Nt. Tr. (1889-90) (C. R., Ps. C.) No. 1, 2-. -- in finite differences. Glaisher, J.W. L. QJ. Mth. 29 (1898) 303-. 6020 Solution of equations of finite differences. Murphy, R. [1835] Camb. Ph. S. T. 6 (1838) 91-. Ellis, R. L. Camb. Mth. J. 4 (1845) 182-. Siverly, W. Des Moines Anal. 1 (1874) 8-. Pincherle, S. Em. E. Ac. Line. Ed. 3 (1894) (Sem. 1) 12-, 99-. Case. Catalan, E. C. Liouv. J. Mth. 3 (1838) 508- ; 4 (1839) 95-. . Walton, W. [1867] QJ. Mth. 9 (1868) 108-. . Sylvester, J. J. Ph. Mg. 8 (1879) 120-. Class of equations of integrable form. Syl- vester, J. J. C. E. 54 (1862) 129-, 170-. Difference and differential equations, con- nexion between. Mellin, H. Acta Mth. 9 (1887) 137-. ---- , of integrals. Tartolini, B. G. Arcad. 97 (1843) 45-. ---- , integrable class. Sylvester, J. J. [1881] (xn) J. H. Un. Cir. [1] (1882) 178. --- , law of reciprocity. Spitzer, S. Grunert Arch. 33 (1859) 415-. Equation connected with ^ (e~*). Vivanti, G. G. Teix. J. Sc. 11 (1892) 167-. , 2nd order. Sylvester, J. J. Ph. Mg. 37 (1869) 225-. __, __ , general. Muir, T. Ph. Mg. 17 (1884) 115-. of nth order has n constants in its solution. Enestram, G. Stockh. Ofv. (1886) 247-; Fschr. Mth. (1886) 303-. Equations for approximating to differential equation. Coriolis, G. 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Integration of linear equations, any order. Libri, G. Crelle J. 12 (1834) 234-. -- with partial finite differences. Eytelwein, J. A. fieri. Ab. (1824) 53-. variable coefficients. Binet, J. P. M. Par. Ac. Sc. Mm. 19 (1845) 639-. method, Brisson's first, for linear and finite difference equations. Mansion, P. [1869] Brux. Mm. Cour. 8, 22 (1872) (No. 1) 32 pp. for equations, coefficients of which are algebraic functions of independent variable. Spitzer, S. Grunert Arch. 32 (1859) 334-. -- , Laplace's,ior linear and finite difference equations. Darboux, G. Bll. Sc. Mth. As. 3 (1879) 209- . of partial finite and infinite difference equa- tions. Paoli, P. Mod. S. It. Mm. 8 (1794) 575-. -- f(x + n) = (x)f(x). Spitzer, S. Grunert Arch. 38 (1862) 456-. + (a + bn) A$(n) + c (n) = 0. Thomae, J. Z. Mth. Ps. 16 (1871) 146-, 428-. -- 4fe) = ty ( -l) + ty (n -2) + 1. Tagiuri, A. G. Mt. 31 (1893) 95-. Inverse calculus of finite differences. Bjorling, E. G. Ups. N. Acta S. 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Fac. Sc. A. 11 (1897) E, 110 pp. linear equations. Lemeray, . C. E. 125 (1897) 1160- ; 126 (1898) 949- ; As. Fr. C. E. (1900) (Pt. 2) 208-. relation of 2 variables x, y. Roselli, E. G. Arcad. 115 (1848) 129-. Functions, calculus. Russell, W. H. L. Phil. Trans. (1862) 265-. determined by functional equation in 2 variables. Giudice, F. Palermo Cir. Mt. Ed. 2 (1888) 28-, 94-; 3 (1889) 273-; 4 (1890) 67-. -------- . Peano, . Palermo Cir. Mt. Ed. 2 (1888) 94, 187-. 411 6030 Functional Equations Functions determined by functional equation in 2 variables. Vivanti, . Palermo Cir. Mt. Ed. 4 (1890) 66-, 69. satisfying certain functional equations. Wirtinger, W. Wien Ak. Sb. 99 (1891) (Ab. 2a) 918-. Generating functions. Rangoni, L. Mod. Mm. S. It. 19 (1821) 241-, 659-. . Ostrogradsky, M. A. St Pet. Ac. Sc. Bll. 1 (1836) 73-. and repeated differentiation. Jilrgensen, U. (vi Adds.) Kiob. Ov. (1830-31) 2-. Integrals. Koenigs, G. C. B. 99 (1884) 1016- ; Par. EC. Norm. A. 1 (1884) (Suppl.) 41 pp. Iteration and general algorithms. Lemeray, . Par. S. Mth. Bll. 26 (1898) 10-. , problem. Lean, L. Par. S. Mth. Bll. 26 (1898) 5-. , , application of doubly periodic functions to solution. Lemeray, . Par. S. Mth. Bll. 27 (1899) 282-. Iterations and reversion series. Zorawski, K. Krk. Ak. (Mt.-Prz.) Bz. 9 (1895) 240-; Ore. Ac. Sc. Bll. (1894) 242. Iterative and other functional equations, con- ditions of holomorphism for integrals. Koenigs, G. C. B. 101 (1885) 1137-. Lagrange-Foncenex lever problem. Genocchi, A. Tor. At. Ac. Sc. 4 (1868-69) 323-; Brux. Ac. Bll. 36 (1873) 181-. Linear difference forms equivalent to their adjoints. Bortolotti, E. Bm. B. Ac. Line. Ed. 7 (1898) (Sem. 1) 257-, (Sew. 2) 46-. Meaning of equation U^=V 2 , where U and V are products of linear functions. Kirkman, T. P. Camb. and Dubl. Mth. J. 5 (1850) 97-. Eeduction of functional equations to finite difference equations. Herschel, (Sir) J. F. W. [1820] Camb. Ph. S. T. 1 (1822) 77-. Bepeated functions. Tralles, J. G. Berl. Ab. (1814-15) (Mth.) 21 6-. Beversion of series as connected with \pa.- l ^ax = x. Homer, W. G. Thomson A. Ph. 11 (1818) 108-. Solution with conditions. Jensen, J. L, W. V. N. Ts. Mth. 8 (B) (1897) 25-; Fschr. Mth. (1897) 346. of a functional equation and application to parallelogram of forces and to curves of equilibration. Wallace, W. (Prof.) [1839] Edinb. B.- S. T. 14 (1840) 625-. used by Poisson for parallelogram of forces. Perkins, G. R. Silliman J. 42 (1842) 69-. Symbolic equation, and its consequences. Pincherle, S. Bm. B. Ac. Line. Ed. 8 (1899) (Sem. 1) 152-. equations. Pincherle, S. Bologna Ac. Sc. Mm. 5 (1895-96) 663-. Theory of functional equations. Bugaev, N. V. [1878] (xn) Bee. Mth. (Moscou) 9 (1878-81) (Ft. 1) 109-. Transcendental functions, class. Winckler, A. Wien SB. 44 (Ab. 2) (1861) 477-. Use of functional equations in elementary geometry. Sigma, (vi Adds.) Tilloch Ph. Mg. 65 (1825) 101-. Geometry 6400 \f/ n x=x. Homer, W. G. Thomson A. Ph. 10 (1817) 341-. = . Herapath, J. Thomson A. Ph. 8 (1824) 420-. f{z,f(x, y)} symmetrical in x, y, z, determina- tion. Abel, N. H. Crelle J. 1 (1826) 11-. x (*?}. Ramus, C. Crelle J. 9 (1832) 359-. Sylvester, J. J. QJ. Mth. 3 (1860) 186-. , coefficients consta ologna Ac. Sc. Mm. 9 "Lh v (x + a v ) =f(x), coefficients constant. B (1888) 45-. Pincherle, S. = , rational in x. Pincherle, S. Bo- logna Ac. Sc. Mm. 9 (1888) 181-. GEOMETRY. 6390 General. Algorithms, geometry. Seguin, (aine) . Moigno An. Cosmos (1862) 451-. , Equation of the beautiful. Lagout, E. Les Mondes 31 (1873) 480- . Geometrical investigation, modern tendency. Segre, C. Ev. Mt. 1 (1891) 46-, 154-. -- , -- (Segre). Peano, G. Ev. Mt. 1 (1891) 66-, 156-. Geometry, linear and metric. Klein, F. [1871] Mth. A. 5 (1872) 257-. Beductio ad absurdum, origin and avoidance. Hennessy, J. P. B. A. Ep. (1857) (pt. 2) 3. Foundations. 6400 General. Construction, geometrical, as proof of existence in Greek mathematics. Zeuthen, H. G. N. Ts. Mth. 3 (A) (1892) 105- ; Fschr. Mth. (1892) 48- ; Mth. A. 47 (1896) 222-. Extension. Riccobelli, F. Brescia Cm. (1829) 104-. Functions, conception of, in teaching ele- mentary geometry. Besso, D. G. Mt. 7 (1869) 131-. Geometrical ideas, origins. Laquiere, E. M. As. Fr. C. B. (1881) 162-. intuition and analysis, connection. Schlofii, L. Mil. I. Lomb. Bd. 5 (1872) 290-. -- , example. Monro, C. J. QJ. Mth. 2 (1858) 225-. Method in geometry. Tannery, J. Ens. Mth. 1 (1899) 333-. Morphology, analytic. Peirce, B. Am. As. P. (1855) 67-. Perimeters and areas of circles, Legendre's demonstration. Breton [de Champ], P. N. A. Mth. 4 (1845) 415-. Batio, principles of antecedental calculus. Glenie, J. [1794] Edinb. B. S. T. 4 (1798) 65-. 412 6410 Principles of Geometry ; Hyperspace 6410 6410 Principles of geometry ; non- Euclidean geometries; hyper - Absolute conic, Cayley's theory. Scott, C. A. N. Y. Am. Mth. S. Bll. 3 (1897) 235-. geometry. Bolyai de Bolya, J. Bordeaux Mm. S. Sc. 5 (1867) 206-. (Bolyai). Schmidt, F. D. Mth. Vr. Jbr. 4 (1897) 107-. ( ). Macaulay, F. S. Mth. Gz. 1 (1900) 25-, 49-. (_). sttickel, P. Mth. Termt. Ets. 18 (1900) 241- ; Mth. Nt. B. Ung. 17 (1901) '-. Time, A. Arch. Mth. Ntvd. 10 (1886) 304-. , dualism. Thue, A. Ts. Mth. 3 (1885) 129- ; Fschr. Mth. (1885) 507-. , parallel construction. Simon, M. Crelle J. Mth. 107 (1891) 84-. Abstract geometry. Cayley, A. [1869] Phil. Trans. 160 (1870) 51-. Analytical non-Euclidean geometry. Haus dor/, F. Leip. Mth. Ps. B. 51 (1899) (Mth.) 161-. . Barbarin, P. Brux. Mm. Cour. 8, 60 (1900-01) No. 1, 167 pp. Angles in a triangle, Legendre's principle. Knight, T. Nicholson J. 27 (1810) 285-. , second theorem, demonstra- tion. Tilly, de. Mathesis 19 (1899) 5-. , theorems on sum. Dehn, M. Mth. A. 53 (1900) 404-. , sum. Tilly, de. Mathesis 19 (1899) 265-. . , , Thibaut's demonstration. Germar, F. H. Grunert Arch. 15 (1850) 361-. Angular geometry. Heatly, S. G. T. Beng. J. As. S. 11 (1842) 230-, 782-. Anharmonics. Lovett, E. 0. L. Mth. S. P. 29 (1898) 566-. Aninvolutive metrics. Fontene, G. Par. S. Mth. Bll. 26 (1898) 176-. Area of spherical triangle, generalisation. Stouff, X. C. E. 122 (1896) 303-. and volume, concept. Hadamard, . Bordeaux S. Sc. PV. (1896-97) 25-. , (Euclid's assumption). Thue, A. Arch. Mth. Ntvd. 10 (1886) 181- ; Fschr. Mth. (1885) 507. Asymptotes. Laurent, H. N. A. Mth. 7 (1868) 413-. Bipolar geometry, principles. Leonardt, G. Z. Mth. Ps. 27 (1882) 346-. Calculus of distances, areas, etc., and its rela- tions to other forms of space. Cotterill, T. Camb. and Dubl. Mth. J. 7 (1852) 275-. Circular points at infinity. Mannheim, A. As. Fr. C. B. 2 (1873) 82-. Coincidence axiom and Euclid i. 4. Bogoro- dickij, V. Kazan Un. Mm. (1892) (Pt. 2) 129-. Commensurables, theory. Sang, E. Edinb. B. S. T. 23 (1864) 721-. Compasses, Euclid's use of. Zeuthen, H. G. (xn) Ts. Mth. 6 (1882) 75-. Content, theory. Ball, (Sir) E. S. 118871 Ir. Ac. T. 29 (1887-92) 123-. Continuity, geometrical, history. Taylor, C. [1880] Camb. Ph. S. P. 4 (1883) 14-. , postulate. Veronese, G. Bm. E. Ac. Line. Ed. 6 (1897) (Sem. 2) 161-. of space. Strong, W. M. N. Y. Am. Mth. S. Bll. 4 (1898) 443-. Continuum, rectilinear, and axiom V of Archimedes. Veronese, G. Em. E. Ac. Line. Mm. 6 (1889) 603-. Definitions. Olivier, L. Crelle J. 1 (1826) 241-. and axioms in Barrow's Euclid. Saint, W. Nicholson J. 23 (1809) 377-. propositions, fundamental, with reference to teaching. Newcomb, S. Nt. 21 (1880) 293-. Differential equations of geometry. Stringham, I. B. A. Ep. (1899) 646-. 3-dimensional space of constant curvature, problems of mechanics. Francesco, D, de. [1900] Nap. Ac. At. 10 (1901) No. 4, 38 pp., No. 9, 33 pp. , elementary remarks. DroUsch, M. W. Leip. Mth. Ps. B. 28 (1876) 268-. , geometry. Calinon, A. Nancy S. Sc. Bll. (1891) 35-. , . Newcomb, S. Crelle J. Mth. 83 (1877) 293-. 4:-Diinensional Space. Eodivell, G. F. Nt. 8 (1873) 8-. Durege, . D. Nf. Tbl. (*1880) 175-. Billson, C. J. Leic. S. T. 1 (1889) (Pt. 2) Preble, W. P. Science 19 (1892) 304-. De la Rive, . C. E. 120 (1895) 983-. Analogue to quaternions. Wijthoff, W. A. Amst. Ak. Vs. 6 (1898) 520-. Angles, calculation. Hoppe, E. Arch. Mth. Ps. 67 (1882) 269- ; 69 (1883) 278-. between 2 planes. Schoute, P. H. N. Arch. Wisk. 3 (1898) 111-. , tetratops. Hoppe, E. Arch. Mth. Ps. 69 (1883) 287-, 297-, 439-. Complex of first order. Ascione, E. Em. E. Ac. Line. Ed. 6 (1897) (Sem. 1) 162-. Concept. Wake, C. S. Science 19 (1892) 331-. Crystals, types of regular system. Schoute, P. H. As. Fr. C. E. (1895) (Pt. 2) 278-. Figure of six points. Richmond, H. W. OJ. Mth. 31 (1900) 125-. Figures. Stringham, W. I. [1880] (xn) J. H. Un. Cir. [1] (1882) 33. . Hoppner, J. D. Edinb. Mth. S. P. 14 (1896) 121. , regular, linearly bounded. Hoppe, E. Arch. Mth. Ps. 67 (1882) 29-. , , , internal angles. Hoppe, E. Arch. Mth. Ps. 68 (1882) 110-. Geometry, descriptive. Veronese, G. Yen. I. At. 8 (1881-82) 987-. , and non-Euclidean geometry. Bruckner, M. Zwick. Vr. Nt. Jbr. (1893) 1-. 413 6410 Principles of Geometry ; Hyperspace XII) 6410 Geometry, proofs. Forchhammer, G. Ts. Mth. 5 (1881) 157-. , theorem. Schlegel, S. F. V. As. Fr. C. E. (1887) (Ft. 2) 264-. History. Haft, F. Arg. S. Ci. A. 30 (1890) 337-. Intersection of straight lines and planes. Segre, C. Palermo Cir. Mt. Bd. 2 (1888) 45-. Kinematics, 4-dimensional, applied to 3 di- mensions. Oss, S. L. van. N. Arch. Wisk. 3 (1898) 165- ; Fschr. Mth. (1897) 492. Plane, place in. Hoppe, E. Arch. Mth. Ps. 68 (1882) 378-. Prismoid. Schoute, P. H. Amst. Ak. Vh. (Sect. 1) 5 (1897) No. 2, 20 pp. Projection of general locus into space of 3 dimensions. Craig, T. Am. J. Mth. 2 (1879) 252-. Projections, 3-dimensional, of 4-dimensional regular bodies or cells. Schlegel . D. Nf . Tbl. (1884) 68-. Quadratic spaces, special cluster. Schoute, P. H. D. Nf. Vh. (1900) (Th. 2, Hdlfte 1) Realisation. Hall, T. P. Science 19 (1892) 272-. . Sanford, E. C. Science 19 (1892) 332. Eotation of rigid system. Stringham, I. Am. As. P. (1884) 55-. Sections of regular hypersolids. Stott, A. N. [1900] Amst. Ak. Vh. (Sect. 1) 7 (1901) No. 3, 21 pp. Solids. Durege, H. Wien Ak. Sb. 83 (1881) (Ab. 2) 1110-. Systems of "spheres." Le Vavasseur, . Toul. Ac. Sc. fill. 2 (1899) 71-. Volume of pentahedron. Liers, E. Arch. Mth. Ps. 12 (1894) 344-. 6-dimensional space, homogeneous coordi- nates, etc. Aschieri, F. G. Mt. 12 (1874) 368-. n-Dimensional Space. Betti, E. A. Mt. 4 (1870-71) 140-. Lunde, J. L. Arch. Mth. Ntvd. 12 (1888) 364-. Badoureau, A. Rv. Sc. 46 (1890) 586-. Mansion, P. Brux. S. Sc. A. 17 (1893) (Pt. 1) 12-. ' Chandler, C. H. Wise. Ac. T. 11 (1898) 239-. Analogue to maximum tetrahedron of given surface. Borchardt, C. W. Berl. Ab. (1866) (Mth.) 121-. Angles. Hoppe, E. Arch. Mth. Ps. 66 (1881) 448; D. Nf. B. (*1883) 55-. . Cassani, P. Rm. R. Ac. Line. Rd. 1 (1885) 133- ; Ven. I. At. (1894-95) 388-. between 2 spaces. Castelnuovo, G. Ven. I. At. (1884-85) 1331-. Complex variables. Volterra, V. Rm. R. Ac. Line. Rd. 5 (1889) (Sem. 1) 158-. Concept. Cassani, P. [1869] (rx) Ven. Aten. At. 6 (1870) 63-. Coordinates, hypotheses permitting introduc- tion. Enriques, F. Palermo Cir. Mt. Rd. 12 (1898) 222-. Curvature. Folie, F. Leip. As. Gs. Vjschr. 35 (1900) 332-. Curve, number of independent equations repre- senting. Veronese, G. Mth. A. 18 (1881) 448. Curves, class. Lasker, E. Nt. 52 (1895) 596. Definition, and hypothesis of continuity in our space. Loria, G. G. Mt. 25 (1887) 97-. Distance relations. Study, E. Z. Mth. Ps. 27 (1882) 140-. Figures, regular, linearly bounded. Hoppe, E. Arch. Mth. Ps. 68 (1882) 151-. Geometry. Cayley, A. Camb. Mth. J. 4 (1845) 119-. . Halphen, G. H. [1873] Par. S. Mth. Bll. 2 (1874) 34-. . Jordan, C. Par. S. Mth. Bll. 3 (1875) 103-. . Veronese, G. Rm. R. Ac. Line. T. 5 (1881) 303-, 333-. . Schlegel, S. F. V. Par. S. Mth. Bll. 10 (1882) 172-. . Cassani, P. G. Mt. 23 (1885) 1-. . Dickstein, S. Prace Mt.-Fiz. 1 (1888) 129- ; Fschr. Mth. (1888) 656. . Veronese, G. Palermo Cir. Mt. Rd. 6 (1892) 42-. . Caisani, P. Ven. I. At. (1893-94) 820-. . Vasiljev, A. V. Kazan S. Ps.-Mth. Bll. 4 (1895) (Pro*.) 16-. . Dupuis, N. F. Cn. R. S. P. & T. 3 (1897) (Sect. 3) 3-. . Barbarin, P. As. Fr. C. R. (1898) (Pt. 2) , history and state. Schlegel, V. Lpldina. 22 (1886) 92-, 108-, 133-, 149-, 160- ; Ens. Mth. 2 (1900) 77-. Lines, property. Lovett, E. 0. Am. J. Mth. 22 (1900) 226-. Manifoldness. Mlodzeevskij, B. K. Mosc. Un. Mm. (Ps.-Mth.) 8 (1889) xn-t- 155 pp. ; Fschr. Mth. (1891) 863-. Maximum, geometrical question, extension. Geiser, C. F. Mil. I. Lomb. Rd. 1 (1868) 778-. Means of arriving at conception of hyperspace. Farjon, F. N. A. Mth. 14 (1895) 101-. Measures. Cesdro, E. G. Mt. 24 (1886) 49-. Metric functions, fundamental, in spaces of n dimensions and constant curvature. Ovidio, E. d'. Rm. R. Ac. Line. 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Gott. Nr. (1872) 157-. of figures by correlative figures of ordinary geometry. Spottiswoode, W. C. E. 81 (1875) 875-, 961-. on plane, of ordinary space by quadric. Aschieri, E. Mil. I. Lomb. Ed. 18 (1885) 494-. , regular solids. Schlegel, V. Pa- lermo Cir. Mt. Ed. 5 (1891) 1-. Six-hundred cell (hyper-polyhedron) and its self-corresponding displacements. Oss, S. L. van. [1899] Amst. Ak. Vh. (Sect. 1) 7 (1901) No. 1, 18 pp.; Nt. 61 (1899-1900) 304. Sphere, potential. Giuliani, G. N. Cim. 21 (1887) 260-. Spheres, ?i-fold linear systems. Del Re, A. Palermo Cir. Mt. Ed. 2 (1888) 124-. Theory corresponding to curvature of ordinary space. Lie, S. Gott. Nr. (1871) 191-, 535-. Volume calculation, three theorems. Hoppe, E. Arch. Mth. Ps. 69 (1883) 385-. Direction. Blakslee, T. M. Des Moines Anal. 9 (1882) 15-, 51-. , arguments against disuse of term. Warren, S. E. [1898] N. Y. Am. Mth. S. fill. 5 (1899) 88, 94-. , geometry of. Laguerre, E. Par. S. Mth. Bll. 8 (1880) 196-. Distance, analytical theory. Beltrami, E. Mil. I. Lomb. Ed. 5 (1872) 294-. relations, theory. Dillner, G. Ups. Vet. S. Arsk. 2 (1861) 1-. Divisibility of consecutive integer products, geometric analogy. Pincherle, S. Bologna Ed. (1892-93) 17-. magnitude. MacWhorter, A. Silliman J. 6 (1848) 329-. Equivalence of figures, proof of fundamental axiom. Veronese, G. Ven. I. At. (1894-95) 421-. in geometry, concept founded on number. Calinon, A. Nancy S. Sc. Bll. (1900) 85-. , definition. Sbrana, S. Ev. Mt. 4 (1894) 147-. , group of, and its kinematic bases. An- drade, J. C. E. 126 (1898) 1775-. Euclid, Bk. xn. Walsh, J. Tilloch Ph. Mg. 64 (1824) 181-. Euclidean geometry, defence. Lagrange, C. Brux. Ac. Bll. (1899) 506-. without Euclid's postulate. Delbceuf, J. Liege S. Sc. Mm. 19 (1897) No. 3, 116 pp. and non-Euclidean geometry. Andrade, J. Ens. Mth. 2 (1900) 298-. Euclid's 3rd postulate, substitute for. Muir- head, R. F. Glasg. Ph. S. P. 12 (1880) 112-. 5th postulate and principle of homogeneity. Flauti, V. Nap. Ed. 1 (1842) 61 (161)-. Finite equality of two plane figures. Rethy, M. Mth. Tennt. Ets. 12(1894)279-; Mth. Nt. B. Ung. 12 (1895) 72-. Finitely equal surfaces. Rethy, M. Mth. Termt. Ets. 8 (1890) 176- ; 11 (1893) 103- ; Mth. Nt. B. Ung. 8 (1891) 170- ; 11 (1894) ' _ (Eethy). Dobriner, H. Mth. A. 42 (1893) 275-. Functional notation, application to demonstra- tion of fundamental propositions. M., F. Bb. Un. 12 (1819) 85-. , geometrical theorems. Anon. (vi 534) Gergonne A. Mth. 10 (1819-20) 161. Fundamental figures in Euclidean space, measurement. Doehlemann, K. Z. Mth. Ps. 41 (1896) 265-. of 1st rank, geometric definition of pro- jectivity. Klein, C. F. Mth. A. 17 (1880) 52-. Fundamental Principles. Nieuport, C. F. F. de. [1819] Brux. Ac. Sc. Mm. 1 (1820) 435-. Price, B. Ashmol. S. P. 2 (1843-52) 177-. Wedgwood, H. B. A. Ep. (1845) (pt. 2) 2-. Matthes, C. J. Amst. Ts. Wis. Nt. Wet. 3 (1850) 35-. Gardiner, M. 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B. 42 (1890) 284-, 355-; 44 (1892) 106- ; C. E. 114 (1892) 461-. Mansion, P. Brux. S. Sc. A. 14 (1890) (Pt. 2) 35-. Killing, W. Crelle J. Mth. 109 (1892) 121-. Wiener, H. D. Mth. Vr. Jbr. 1 (1892) 45-; 3 (1894) 70-. Schmidt, H. Crelle J. Mth. 112 (1893) 112-, 319-. Peano, G. Ev. Mt. 4 (1894) 51-. Bonnel, J. Lyon Ac. Sc. Mm. 3 (1895) 241-, 351- ; 4 (1896) 129-, 373-, 433-; 5 (1898) 401-. Burkhardt, H. Gott. Nr. (1895) 114-. Cassani, P. Yen. I. At. (1898) 42-. Hilbert, D. Par. EC. Norm. A. 17 (1900) 103-. General geometry. Tarry, G. As. Fr. C. E. (1889) (Pt. 2) 60-; (1890) (Pt. 2) 152- ; (1891) (Pt. 2) 90-; (1892) (Pt. 2) 132- ; (1893) (Pt. 2) 217- ; (1894) (Pt. 2) 184-. or metageometry, Schering's researches. Mansion, P. Brux. S. Sc. A. 16 (1892) (Pt. 1) 51-. , and its three subdivisions. Man- sion, P. [1895] Brux. Ac. Bll. 29 (1895) 495-; Wiad. Mt. 1 (1897) 1-, 68-. Geometrical reasoning, facts in. Hennessy, J. P. Ph. Mg. 5 (1853) 127-. Geometry, linear, superficial and solid. Heatly, S. G. T. I. Ev. 1 (1837) 755-; 2 (1838) 38-, 82-, 151-, 205-. , new elements, and proofs of laws of opera- tion. Smith, S. Am. I. T. (1850) 207-. , , founded on principle of analogues. Valat, . Bordeaux Act. Ac. Sc. 27 (1865) 61-; 28 (1866) 35-. as purely logical system. Fieri, M. [1900] N. Y. Am. Mth. S. Bll. 7 (1901) 158, 171-, 234. , system derived from ideas of direction and position. Wedgwood, H. (vi Adds.) Ph. Mg. 10 (1856) 459-. without axioms (Perronet Thompson). Newman, F. W. W. Eng. J. 1 (1836) 68-. Gravity in Gaussian space. Schering, E. Gott. Nr. (1870) 311-. multiply extended Gaussian and Eie- mann .spaces. Schering, E. Gott. Nr. (1873) 149-. Higher geometry, laws. Boklen, 0. Grunert Arch. 33 (1859) 111-. Imaginaries in pure geometry. Scott, C. A. N. Y._Am. Mth. S. Bll. 6 (1900) 163-. , \/ - 1 in geometry. Shadow. (Pseudon.) Ph. Mg. 29 (1846) 171-. Imaginary elements in geometry. McMahon, J. Am. As. P. (1888) 62. forms in geometry. Mouchot, A. C. E. 103 (1886) 1110-. , realisation and use. Marie, M. N. A. Mth. 9 (1890) 60-, 161-, 375-, 435-, 508- ; 10 (1891) 172-, 276-, 329-, 373-, 417-, 459-. Imaginary forms in geometry, realisation and use (Marie). Boguslavsky, A. J. Fschr. Mth. (1891) 726-. points in space, geometrical representation. Molenbroek, P. [1890-91] N. Arch. Wisk. 18 (1891) 132- ; Arch. Mth. Ps. 10 (1891) 261- ; N. A. Mth. 10 (1891) 434-. Indirect proof, admissibility. Sylvester, J. J. Ph. Mg. 4 (1852) 366-. , (Sylvester). Adamson, Jas. Ph. Mg. 5 (1853) 297-, 332-, 405-. Infinity, concept, in elementary geometry. Ripert, L. Ens. Mth. 2 (1900) 370-. , , (Eipert). Appell, P. Ens. Mth. 2 (1900) 205-. Length of arc, concept, as depending on con- tinuity. Ascoli, G. Mil. I. Lomb. Ed. 17 (1884) 564-. , concept, and the straight line. Bettazzi, R. A. Mt. 20 (1892-93) 19-. of curve, concept. DM Bois-Reymond, P. Acta Mth. 6 (1885) 167-. and space, concept. Wilkie, D. Queb. T. Lt. S. 2 (1831) 64-. LobaSevskij's imaginary or pan-geometry. Lo- batschewsky, N. (vi Adds.) Kazan Mm. Un. (1835) cafe. 1, 3-; (1836) cah. 1,3-; Crelle J. 17 (1837) 295-; (xn) Kazan Un. Mm. (1835, Bk. 3) 3-; (1836, Bk. 2) 3-; (1836, Bk. 3) 3-; (1837, Bk. 1) 3-; (1838, Bk. 1) 3-; (1838, Bk. 3) 3-; Liege S. Sc. Mm. 2 (1900) No. 5, 101 pp. ; (xn) Kazan Un. Mm. (1855, Bk. 1) 1-; (iv) Erman Arch. Es. 17 (1858) 397-. . Cayley, A. Ph. Mg. 29 (1865) 231-. . Bonnel, J. Lyon Ac. Mm. (Sc.) 29 (1888) 291-. . Kagan, W. Fschr. Mth. (1896) 373. , additions. Varisco, D. G. Mt. 28 (1890) 181-, 240. , construction of tangents to circles, and doubly infinite manifoldness of spheres. Simon, M. D. Mth. Vr. Jbr. 3 (1894) 80-. , equation of horicycles and hori- spheres. Mansion, P. Brux. S. Sc. A. 23 (1899) (Pt. 1) 32-. , intermediate planes (Zwischen- ebene) and limit surfaces (Grenzfliiche). Simon, M. [1898] D. Mth. Vr. Jbr. 7 (1899) (Heft 1) 67-. , limit of a triangle. Dauge, F. Mathesis 18 (1898) 267-. , parallels, construction. Engel, F. Kazan S. Ps.-Mth. Bll. 7 (1898) 118-. , , theory, and nature of space. Levett, R. [1877] Binn. Ph. S. P. 1 (1879) (Pt. 2) 18-. and real linear transformations. Schwarz, A. Wien Ak. Sb. 99 (1891) (Ab. 2a) 153-. , series. Efimov, M. N. A. Mth. 19 (1900) 28-. , space of five dimensions, properties. Mansion, P. Brux. S. Sc. A. 23 (1899) (Pt. 1) 5. 416 6410 Lobac'evskij's imaginary or pan-geometry, . Mt. 15 (1877) 336-. Non-Euclidean Geometry 6410 theorem. Zolt, A. de. G. Magnitude and direction, science of. Goodwin, H. [1845] Camb. Ph. S. T. 8 (1849) 278-. Manifoldness of two dimensions and of finite extent, simplest continuous. Frankland, F. W. N. Z. I. T. 9 (1876) 272-. -- ------ , (Frankland). Monro, C. J. Nt. 15 (1877) 547 ; 22 (1880) 218. ---- ---- - , -- (Monro) . Frank- land, F. W. Nt. 22 (1880) 170-. -------- , -- (Frankland). Skey, W. [1880] N. Z. I. T. 13 (*1881) 100-. -------- , -- (Skey). Frank- land, F. W. [1884] N. Z. I. T. 18 (1886) 58-. -------- , -- (Frankland). Hogben, G. [1886] N. Z. I. T. 19 (1887) 510-. Maxima and minima, geometrical. Hurwitz, A. Sch. Nf. Gs. Vh. (1896) 57-. Mechanical evaluation of plane surfaces. Cue- noud, S., & Piccard, . [1860] Laus. Ell. S. Vd. 7 (1864) 93-. principle, problems solved by. Breton [de Champ], P. Liouv. J. Mth. 3 (1838) 488-. Metaphysics of geometry. Dandelin, G. P. [1842] Brux. Mm. Ac. Sc. 17 (1844) 44 pp. Minima, Fermat's problem, generalisation. Loria, G. Mathesis 19 (1899) 131-. NON-EUCLIDEAN GEOMETBY. Beltrami, E. G. Mt. 6 (1868) 284-. Klein, [C.] F. Mth. A. 4 (1871) 573-; 6 (1873) 112- ; 7 (1874) 531-. (Klein.) Cayley, A. Mth. A. 5 (1872) 630-. Ball, E. S. B. A. Ep. (1880) 476-. Chrystal, G. Edinb. B. S. P. 10 (1880) 638-. Reyes y Prosper, V. Mth. A. 29 (1887) 154-. (Beltrami, 1868.) Commines de Marsilly, L. J. A. de. As. Fr. C. E. (1888) (Ft. 2) 121-. Mansion, P. Brux. S. Sc. A. 13 (1889) (Pt. 1) 57-. Cayley, A. [1890] Camb. Ph. S. T. 15 (1894) 37-. Klein, [C.] F. Mth. A. 37 (1890) 544-. Lindemann, . Konigsb. Schr. 32 (1891) (Sb.) 20-. Miller, G. A. Science 20 (1892) 370-. Poincare, H. Nt. 45 (1892) 404-. Gerard, . N. A. Mth. 12 (1893) 74-. McClintock, E. N. Y. Mth. S. Bll. 2 (1893) 21-, 51. La Vallee-Poussin, G. J. de. Brux. S. Sc. A. 19 (1895) (Pt. 2) 17-. Neovius, E. R. Helsingf. Ofv. 37 (1895) 222-. Dauge, . Mathesis 16 (1896) 7-. Isely, L. Neuch. S. Sc. Bll. 24 (1896) 137-. Mansion, P. Brux. S. Sc. A. 21 (1897) (Pt. 1) 117- ; Mathesis 17 (1897) 112-, 134-, 158- ; 18 (1898) 33-. Engel, F. Leip. Mth. Ps. B. 50 (1898) (Mth.) 181. Villareal, F. Arg. S. Ci. A. 46 (1898) 23-, 67-, 129-. Bianchi, L. A. Mt. 2 (1899) 95-. Duport, H. Par. S. Mth. Bll. 27 (1899) 138-. Frolov, (le gen.) M. As. Fr. C. E. (1899) (Pt. 2) 70-; Ens. Mth. 2 (1900) 179-, 293-. Barbarin, P. Ens. Mth. 2 (1900) 306. application to quadratic forms. Poincare, H. As. Fr. C. E. (1881) 132-. question in kinematics. Ball, R. S. B. A. Ep. (1881) 535-. calculation in non-Euclidean forms of space. Killing, W. [1879] Crelle J. Mth. 89 (1880) 265-. case. Johnson, W. W. N. Y. Mth. S. Bll. 2 (1893) 158-. circles in. Battaglini, G. [1873] (rx) Em. E. Ac. Line. At. 1 (1875) 53-. circular affinity. Battaglini, G. Nap. Ed. 15 (1876) 219-. conies, properties. Story, W. E. Am. J. Mth. 5 (1882) 358-; (xn) J. H. Un. Cir. 2 (1883) 71-. constructions, singularities. Bouniakoivsky , V. St Pet. Ac. Sc. Mm. 18 (1872) (No. 7) 16 pp. cubics. Stecker, H. F. [1898-1900] N. Y. Am. Mth. S. Bll. 5 (1899) 3, 9-; Am. J. Mth. 22 (1900) 31-. Foncenex's lever theory. Genocchi, A. [1877] Tor. Ac. Sc. Mm. 29 (1878) 365-. formula. Janssen van Raay, W. H. L. N. Arch. Wisk. 3 (1898) 77-. formulae, elementary. Francesco, D. de. G. Mt. 37 (1899) 98-. history. McClintock, E. N. Y. Mth. S. Bll. 2 (1893) 144-. . Halsted, G. B. Am. As. P. (1899) 53-. . Monck, W. H. S. Pop. As. 8 (1900) 333-. and hyperspace, bibliography. Halsted, G. B. Am. J. Mth. 1 (1878) 261-, 384-; 2 (1879) 65-. length, definition. Alexejewsky, W. P. Fschr. Mth. (1898) 411. maxima problems. Bang, A. S. Ts. Mth. 5 (1887) 136- ; Fschr. Mth. (1887) 515. measurements, rational, in plane. Busche, E. Hamb. Mth. Gs. Mt. 3 (1900) 37-. motion. Neicson, H. B. N. Y. Am. Mth. S. Bll. 4 (1898) 2, 5-. passage in Gauss. Mansion, P. D. Mth. Vr. Jbr. 7 (1899) (Heft 1) 156-. 9-point theorem of Feuerbach, generalisation. Frattini, G. G. Mt. 16 (1878) 298-. 5 points, relation between distances. Mansion, P. Brux. S. Sc. A. 15 (1891) (Pt. 1) 8- ; 20 (1896) (Pt. 1) 62-. or 6 points, relation between distances in Euclidean and non-Euclidean geometry. Mansion, P. Brux. S. Sc. A. 19 (1895) (Pt. 2) 189-. Stokes's theorem, extension. Ricci, G. Ven. I. At. (1896-97) 1536-. 417 . DD 6410 Principles of Geometry 6410 straight lines. Petersen, J. Kjtfb. Ov. (1900) 305-. and teaching. Andrade, J. Ens. Mth. 2 (1900) 114-. theorems. Simon, M. Mth. A. 48 (1897) 607. volume of a body, expression for. Mansion, P. Brux. S. Sc. A. 21 (1897) (Pt. 1) 118-. Numbers and ratios in geometry. Maiziere, A. Lille Mm. S. (1833) 1-. Paradoxes connected with mathematics of in- finity. DomenechyEstapd, J. [1894] Barcel. Ac. Mm. 1 (1892-1900) 315- . Plane angle, definition. Bonnel, J. Lyon Ac. Mm. (Sc.) 31 (1892) 39-. , concept, fundamental. Gerling, C. L. Crelle J. 20 (1840) 332-. , definition, and axiom raised to theorem. Valeriani, V. G. Mt. 7 (1869) 376. , general theory. Crelle, A. L. Berl. Ab. (1834) 23-. in infinite dimension space, concept. Pincherle, S. Bologna Ed. 2 (1898) 71-. polygons and angles as system of absolute magnitudes. Stolz, 0. Mh. Mth. Ps. 5 (1894) 233-; 7 (1896) 296. triangle, extension of elementary theorems to general space. Mehmke, B. Arch. Mth. Ps. 70 (1884) 210-. Point pairs, and circles in space, treated by Grassmann's method. Mttller, E. Mh. Mth. Ps. 7 (1896) 77-. , power of, with regard to five circles or six spheres. Lucas, E. Em. B. Ac. Line. Mm. 2 (1878) 449-. , straight line and circle, concepts. R., F. Bv. Ligure 2 (1843) 132-. Problems, methods of solution. Egidi, G. Em. N. Line. At. 40 (1887) 9-. , --- not based on proportions. Airy, (Sir) G. B. [1857] Camb. Ph. S. T. 10 (1864) 166-. Products of numbers, properties. Teixeira, J. P. G. Teix. J. Sc. 14 (1900) 179-. Projections, areas of, applications of Tinseau's theorem. Cabreira, A. [1897] Lisb. J. Sc. Mth. 5 (1898) 93-. Project! ve relations of space of various dimen- sions. Veronese, G. Mth. A. 19 (1882) 161-. Proportion, equimultiples in ancient theory. Trudi, N. Nap. G. Mt. 1 (1863) 337-. , Euclid's theory. Hitt, M. J. M. [1897] Camb. Ph. S. T. 16 (1898) 227-. , -- . Gibson, G. A. Edinb. Mth. S. P. 16 (1898) 99-. , , filling a gap in. Nordmark, Z. Ups. N. Acta S. Sc. 6 (1799) 174-. , reduced to formulae. Peano, G. Mathesis 10 (1890) 73-. , geometrical. Marrat, W. Tilloch Ph. Mg. 33 (1809) 426-. and ratio, Euclid's theory; and nature of quantity. Powell, B. [1836] Ashmol. S. T. 1 (1838) No. 5, 64 pp. , theory. Chelini, D. G. Arcad. 73 (1837) 166-. Proportion, theory. Gibson, G. A. Edinb. Mth. S. P. 18 (1900) (App.) 26 pp. , , geometrical. Hoppe, R. Arch. Mth. Ps. 62 (1878) 153-. Pseudospherical functions and their applications to pseudospherical geometry. Jakovkin, A. F. Kazan S. Ps.-Mth. Bll. 9 (1900) (Prot.) 45-. geometry. Cayley, A. E. S. P. 37 (1884) 82-. , theorem. Beltrami, E. G. Mt. 10 (1872) 53. Quantities, geometrical and hyper-geometrical. Polignac, A. (Prince de). G. E. 52 (1861) 24-. , hyper-geometrical. Polignac, A. (Prince de). C. E. 56 (1863) 381-. , positive and negative, in geometry. Pourtales, L. A. de. Neuch. Bll. 2 (1846-47) 127-. , principles applicable to geometry. Valat, . Bordeaux Act. Ac. Sc. (1849) 535-. Eadical axis, existence demonstrated without Euclid's postulate. Reyes Prosper, V. Fschr. Mth. (1895) 571. Eatio, Archimedes' principle. Terquem, 0. N. A. Mth. 3 (1844) 586-. Eatios, arithmetical and geometrical. Bianchi, G. Palomba Eac. 3 (1847) 155-. , duplicate. Hain, E. Arch. Mth. Ps. 60 (1877) 404-. Eectangle, geometrical existence. Collignon, E. As. Fr. C. E. (1899) (Pt. 2) 87-. Eiemann planes and Euclidean spheres, identity. Lechalas, G. Brux. S. Sc. A. 20 (1896) (Pt. 2) 167-. , non-identity. Mansion, P. Brux. S. Sc. A. 20 (1896) (Pt. 2) 178-. Eiemann's geometry, fundamental principle. Mansion, P. Mathesis 14 (1894) 180- ; B. A. Ep. (1894) 579-. , interpretation of a theorem. Dauge, F. Mathesis 18 (1898) 5-. and metageometry, principles. Mansion, P. Mathesis 15 (1895) 63-. Segment, postulate of, deduced from that of angle. Sbrana, S. Bv. Mt. 3 (1893) 179-. Signs of distances. Fontene, G. Ens. Mth. 1 (1899) 272-. , rule of. Laguerre, . N. A. Mth. 9 (1870) 175-. , use. Fontene, G. Ens. Mth. 1 (1899) 123-. + and - . Haillecourt, A. N. A. Mth. 7 (1848) 83-. . De Morgan, A. Camb. and Dubl. Mth. J. 6 (1851) 156- ; 7 (1852) 242-. Space and body, knowledge. Wedgwood, H. [1850] Camb. Ph. S. T. 9 (1856) 157-. , concept. Hoppe, . D. Nf. Tbl. (*1875) 142-. , , and application to natural science. Harnack, A. [1878] Dresden Isis Sb. (1879) 178-. , , empirical and ideal. Kapke, . Hamb. Mth. Gs. Mt. 1 (1889) 274-. , , an extension of concept of length. Graberg, F. Ziir. Vjschr. 31 (1886) 339-. 418 6410 Principles of Geometry. Parallels 6410 Space, concept, and laws of geometry. P., E. Kosmos (Lw.) 11 (1886) 530-. , , (E. P.). Eaciborski, A. Kosmos (Lw.) 12 (1887) 27-. , concepts, fundamental. Graberg, F. Ziir. Vjschr. 37 (1892) 274-. , correspondence between non-Euclidean, and Euclidean. Porchiesi, A. Bologna Ac. Sc. Mm. 5 (1883) 421-. and its dimensions. Whitmell, C, T. Card. Nt. S. T. 25 (Ft. 1) (1893) 6-. , elementary geometry theorems. Seipp, H. Arch. Mth. Ps. 12 (1894) 16-. , elliptic. Ball, (Sir) E. S. Nt. 33 (1886) 86-. , form. Hadamard, . Bordeaux S. Sc. PV. (1897-98) 83-. , functional, geometry. Pincherle, S. Bo- logna Ed. 1 (1897) 85-. , geometry. Randolph, E. V. Nost. Eng. Mg. 29 (1883) 137-. magnitude, successive developments. Gra- berg, F. Ziir. Vjschr. 32 (1887) 191-. measurement, growth of ideas concerning. Graberg, F. Ziir. Vjschr. 37 (1892) 49-. , properties. Poncelet, J. V. [ascribed to Gergonne in vol. II. C. S. P.] Gergonne A. Mth. 17 (1826-27) 265-. , (Poncelet). Gergonne, J. D. Gergonne A. Mth. 17 (1826-27) 272-. , recent researches. Becker, J. C. Z. Mth. Ps. 17 (1872) 314-. of spheres, Euclidean. Aschieri, F. Mil. I. Lomb. Ed. 19 (1886) 355-, 416-, 449-. , spherical, and figures therein. Porter, M. B. Texas Ac. Sc. T. 1 (No. 2) (1893) 45-. Spaces of constant curvature, fundamental theory. Beltrami, E. [1868] A. Mt. 2 (1868 -69) 232-; Par. EC. Norm. A. 6 (1869) 347-. , (Beltrami). Schlajli, L. [1871] A. Mt. 5 (1871-73) 178-. , (Schlafli). Beltrami, E. A. Mt. 5 (1871-73) 194-. , metrics of. Giudice, F. Tor. Ac. Sc. At. 35 (1900) 119- or 193-. , negative. Backlund, A. V. Lund. Un. Acta 19 (1882-83) (Mth. No. 1) 48 pp. , positive. Killing, W. [1877] Crelle J. Mth. 86 (1879) 72-. , parallel and perpendicular. Nicoli, F. Mod. Ac. Sc. Mm. 10 (1894) 257-; 11 (1895) 239-. , various, comparability. Lechalas, G. [1900] N. Y. Am. Mth. S. Bll. 7 (1901) 158, 179-. Spheres of second species. Mannoury, G. N. Arch. Wisk. 4 (1899) 83-, 129. Spherical geometry, fundamental theorems. Sikttel, V. Kazan S. Ps.-Mth. Bll. 2 (1893) 94-; 4 (1895) 18-; Arch. Mth. Ps. 15 (1897) 159-, 403- ; 17 (1900) 337-. Statical proofs in geometry. Brassinne, E. Toul. Mm. Ac. 2 (1858) 144-. Straight line and curve, fundamental ideas. Lamarle, E. Brux. Ac. Sc. Mm. 30 (1857) 43pp. Straight line, Euclid's definition. Sikstel, V. A. Kazan S. Nt. (Ps.-Mth.) P. 8 (1890) 288. joining two given points, construction. Burnside, W. [1897] L. Mth. S. P. 29 (1898) 125-. and plane, definitions. Bonnel, J. Lyon Ac. Mm. (Sc.) 30 (1889-90) 147-. , generations. Cassani, P. Ven. Aten. At. 2 (1879) 46-. , plane and sphere. Calinon, A. Nancy S. Sc. Bll. (1887) 1-. as shortest distance between two points. Hilbert, D. Mth. A. 46 (1895) 91-. lines and planes, unreal. Pasch, M. Mth. A. 32 (1888) 159-. Superposition, geometry independent of. Pagnini, L. A. Firenze Opusc. Sc. 14 (1812) 3-. Surfaces of constant curvature, two dimensional geometry. Calinon, A. Nancy S. Sc. Bll. (1895) 1-. Symbolical geometry. Hamilton, (Sir) W. E. Camb. and Dubl. Mth. J. 1 (1846) 45-, 137-, 256-; 2 (1847) 47-, 130-, 204- ; 3 (1848) 68-, 220- ; 4 (1849) 84-, 105-. Symbols, algebraic, application to geometry. Gregory, D. F. Camb. Mth. J. 2 (1841) 1-. Theory of Parallels. Jeu/rain, A. Par. Tr. S. Amat. 2 (1808) 78-. Gergonne, J. D. Gergonne A. Mth. 3 (1812- 13) 353-. L. (yi Adds.) Gilbert A. 52 (1816) 451-. Hoffmann, J. J. I. Gilbert A. 54 (1816) 314-. Blum, H. C. Gilbert A. 57 (1817) 222-. Ivory, J. Tilloch Ph. Mg. 59 (1822) 161-. Meikle, H. Tilloch Ph. Mg. 60 (1822) 423-. Walsh, J. Tilloch Ph. Mg. 63 (1824) 100-, 271-. Stein, J. P. Gergonne A. Mth. 15 (1824-25) 77- ; 16 (1825-26) 45-. Servois, J. F. Gergonne A. Mth. 16 (1825-26) 233-. Colburn, W. Bost. J. Ph. 3 (1826) 81-. (Colburn.) Strong, T. Bost. J. Ph. 3 (1826) 371 [271]-. (Strong.) Colburn, W. Bost. J. Ph. 3 (1826) 490-. Knar, J. Baumgartner Z. 3 (1827) 414- ; 4 (1828) 427-. Lampredi, U. G. Arcad. 40 (1828) 19-. E. . . . r, J. F. Mitau Quatember 2 (1830) (Bd. 3) 35-. Doppler, C. Wien Jb. Pol. I. 17 (1832) 167-. Legendre, A. M. Par. Mm. Ac. Sc. 12 (1833) 367-. Crelle, A. L. Crelle J. 11 (1834) 198. Le Monnier, J. F. B. Cherb. Mm. S. Ac. (1838) 320-. Amiot, B. Rouen Ac. Tr. (1839) 144-. Powell, B. [1842] Ashmol. S. T. 2 (1838-52) 47 pp. Schulten, N. G. af (fil.) [1842] Helsingf. Acta 3 (1852) 351-, 783-, 791-. Bouniakowsky, V. [1843-51] St Pet. Ac. Sc. Mm. 6 (1850) 87-, 207- ; St Pet. Ac. Sc. Bll. 9 (1851) 49-. 419 DD 2 6410 Principles of Geometry. Parallels 6410 Meikle, H. Edinb. N. Ph. J. 36 (1844) 310- ; 43 (1847) 255-. Minarelli, C. N. A. Mth. 8 (1849) 312-. Lionnet, E. N. A. Mth. 9 (1850) 37. Anon, (vi 840) Nap. At. Ac. 6 (1851) 1-. Cabot, . N. A. Mth. 14 (1855) 230-. Hennessy, J. P. Ph. Mg. 12 (1856) 283-, 371-. Osorio, R. G. Coimbra I. 4 (1856) 86-. Schulten, N. G. af (fil.) Helsingf. Acta 4 (1856) 25-. Stevelly, J. B. A. Ep. (1856) (Ft. 2) 8-; Ph. Mg. 12 (1856) 220-. Vincent, A. J. H. C. K. 42 (1856) 1107-. Day, A. Ph. Mg. 13 (1857) 156-. Bouche, A. M.-et-L. Mm. S. Ac. 4 (1858) 162-. Buniakovskij, V. Ja. St Pet. Ac. Sc. Mm. (Es.) 2 (*1862) 19-; St Pe"t. Ac. Sc. Bll. 5 1863) 387-. Thompson, T. P. E. S. P. 12 (1862-63) 268-. Lobatcheu-sky, N. I. Bordeaux Mm. S. Sc. 4 (cah. 1) (1866) 83-. Grunert, J. A. Arch. Mth. Ps. 47 (1867) 307-. Baltzer, E. Leip. B. 22 (1870) 95-. Dillner, G. Ts. Mt. Fys. 3 (1870) 1-. Bunyakovskii, V. Y. [1872] (xn) Eec. Mth. (Moscou) 6 (1872-73) (Pt. 1) 77-. Johnson, W. W. Des Moines Anal. 3 (1876) 103-. Budden, E. Nt. 35 (1887) 92-. (Budden's demonstration.) Henrici, 0. Nt. 35 (1887) 100. Duclout, J. A-rg. S. Ci. A. 30 (1890) 308-, 351-, 467-; 31 (1891) 5-, 111-, 161-, 213-, 257- ; 32 (1891) 72-, 132-. Application of algebraic functions to proof of properties of parallels. Ivory, J. (vi Adds.) Tilloch Ph. Mg. 63 (1824) 161-, 246-. law of logic. Matzka, W. Grunert Arch. 8 (1846) 320-. Bolyai, letter of 1823. Szily, K. Mth. Termt. Ets. 5 (1887) 187- ; Mth. Nt. B. Ung. 5 (1886-87) 189-. Catalan's postulate. Breton [de Champ], P. N. A. Mth. 7 (1848) 93-, 401-. Definition of parallels. Bonnel, J. [1887] Lyon Ac. Mm. (Sc.) 29 (1888) 41-. , importance. Hennessy, J. P. Ph. Mg. 12 (1856) 452-. Dodgson's theory. Tucker, E. [1888] Nt. 39 (1889) 175. Euclid's axiom. Olivier, L. Crelle J. 1 (1826) 151-. . Flye Sainte-Marie, C. [1870] (ix) Par. S. Phlm. Bll. 7 (1871) 9-. . Lionnet, E. C. E. 70 (1870) 31-. . Stolz, O. [1885] Innsb. Nt. Md. B. 15 (1886) 25-. . Simon, M. D. Mth. Vr. Jbr. 1 (1892) 39, 41. . Mansion, P. Mathesis 16 (1896) 109- ; 18 (1898) 188-. . Frolov,(legen.)M. Mathesis 16 (1896) 225- ; As. Fr. C. E. (1898) (Pt. 2) 104-. . Brassai, S. Mag. Tud. Ak. Ets. 9 (1898) 415-. Euclid's axiom. Sincov, D. M. Kazan S. Ps.- Mth. Bll. 9 (1900) (Prot.) 44. . Tikhomandritzky, M. Ens. Mth. 2 (1900) 385-. . Mansion, P. Ens. Mth. 2 (1900) 457. , consequences. Sikstel, W. Fschr. Mth. (1896) 373-. , demonstration. Limrick, P. As. Ee- searches 7 (1801) 449-. , . Anon, (vi 548) Gergonne A. Mth. 14 (1823-24) 269-. , . Bouvier, L. C. Gergonne A. Mth. 17 (1826-27) 152-. , . Wilkie, D. [1832] Queb. T. Lt. S. 3 (1837) 72-. , . Twining, A. C. Silliman J. 1 (1846) 89-. , . Horlych, H. T. Grunert Arch. 18 (1852) 455-. , . Lamarle, E. Brux. Ac. Bll. 23 (1856) (pte. 2) 408-, 637-. , . Kulik, J. P. Prag Sb. (1860) 20-. , . Murray, B. A. Ph. Mg. 33 (1867) 264-. , . Twining, A. C. Am. As. P. 18 (1869) 39- ; Am. J. Sc. 4 (1872) 333-. , . Rossi Re, V. de. Em. N. Line. At. 31 (1878) 461-. , , Bertrand's. Luroth, J. Z. Mth. Ps. 21 (1876) 294-. , , . Becker, J. C. [1876] Z. Mth. Ps. 22 (1877) 60-. , , impossibility of, by plane construc- tion. Hcritel, J. Bordeaux Mm. S. Sc. 8 (1870) xi-. , , possibility of, by stereometry. Gunther, S. G. Mt. 14 (1876) 97-. and the foundations of geometry. Coin- mines de Marsilly, (le gen.) de. As. Fr. C. E. (1889) (Pt. 2) 88-. Lacroix's. Berolle, . Les Mondes 11 (1866) 55- ; 12 (1866) 513-. , principles of mechanics and geometry in relation to. Genocchi, A. [1869] Mod. S. It. Mm. 2 (1876) 153-. , Saccheri's researches. Mansion, P. Brux. S. Sc. A. 14 (1890) (Pt. 2) 46-. Geometrical theory. LUdicke,M.A.F. Gilbert A. 72 (1822) 423-. Lambert's theory. Stackel, P. Bb. Mth. (1899) 107-. Legendre's proof of properties of lines. Young, G. P. On. J. 1 (1856) 519-. LobaSevskij ' s views prior to 1826. Wassiljef, A . D. Mth. Vr. Jbr. 4 (1897) 88-. Metternich's theory. Wachter, . Lindenau Z. 2 (1816) 64-. Non-Euclidean geometry and theory of parallels, opinions of Dutch mathematicians. Janssen van Eaay, W. H. L. [1900] Kazan S. Ps.- Mth. Bll. 10 (1901) 1-. Plane geometry. Sexe, S. A. N. Mg. Ntvcl. 19 (1873) 282-. Point and line calculus. Graefe, F. Arch. Mth. Ps. 15 (1897) 34-, vn-. Problems, celebrated. Franchini, P. Lucca At. Ac. 8 (1835) 91-. 420 6420 Topology of Space and Hyperspace 6420 Twist and stretch, centres of finite. Genese, R. W. B. A. Ep. (1888) 579-. Volume in any space, concept. Loria, G. G. Mt. 26 (1888) 96-. 6420 Topology of space and hyper- 3620, 8075.) Simony, 0. Mth. A. 19 (1882) 110- ; 24 (1884) 253-. (Listing.) Tait, P. G. [1883] Ph. Mg. 17 (1884) 30-. Dyck, Jr. Leip. Mth. Ps. B. 37 (1885) 314- ; 38 (1886) 53-; 39 (1887) 40-; Mth. A. 32 (1888) 457- ; 37 (1890) 273-. Brunn, H. Z. Mth. Ps. 37 (1892) 106-. Poincare, H. C. B. 115 (1892) 633-; Par. '. Connected surfaces, class. Emch, A. [1894] Kan. Un. Q. 3 (1895) 153-. , simply. Schonflies, A. Gott. Nr. (1899) 282-. Connection of spaces. Tonelli, A. Km. E. Ac. Line. Ed. 2 (1875) 594- ; 6 (1890) (Sem. 1) 139-. , cyclomatic laws. Mannoury, G. N. Arch. Wisk. 3 (1898) 126-. surfaces. Klein, C. F. Mth. A. 7 (1874) 549- ; 9 (1876) 476-. . Bortolotti, E. Bologna Ed. (1889-90) 132-. . Tonelli, A. Em. E. Ac. Line. Ed. 2 (1893) (Sem. 1) 15-; 4 (1895) (Sem. 1) 300-. , theorem. Bortolotti, E. Em. E. Ac. Line. Ed. 5 (1889) (Sem. 2) 229-. Contours traced on surfaces. Brunei, G. Bor- deaux S. Sc. PV. (1896-97) 128-. Pol. J. 1 (1895) 121 pp. ; Palermo Cir. Mt. Curves, algebraic. ' Hulburt, L. S. ' N. Y. Mth. S. Bll. 1 (1892) 197-. Dissection of surfaces, Eiemann's theorem. Koehler, C. Crelle J. Mth. 109 (1892) 118-. Division, homogeneous, of space. Kelvin, (Lord). E. S. P. 55 (1894) 1-. , , . Vaes, F. J. N. Arch. Wisk. 4 (1899) 107- ; Fschr. Mth. (1899) 439-. , regular, of space. Schoenflies, A. Gott. Nr. (1888) 223-. Ed. 13 (1899) 285-. Dorsten, E. H. van. N. Arch. Wisk. 20 (1893) 113-. Boguslavskij, A. I. Kharkov Mth. S. Com. 4 (1895) 86-. Aspects of n points in plane to observer in plane. Perrin, R. Par. S. Mth. Bull. 10 (1882) 103-. Betti's numbers. Poincare, H. C. E. 128 (1899) 629-. Branched circuits. Wassmuth, A. Mh. Mth. Ps. 7 (1896) 49-. Cayley's color-diagrams, representation of rota- tion groups. Maschke, H. Am. J. Mth. 18 (1896) 156-. Cell-structure, natural, ideal geometrical form. Bryant, (Mrs) . L. Mth. S. P. 16 (1884-85) 311-. Close, partitions. Cayley, A. Ph. Mg. 21 (1861) 424-. Closed contours, double points. Hadamard, . Bordeaux S. Sc. PV. (1899-1900) 4-. curves. Schonflies, A. Gott. Nr. (1896) 79-. , , , non-Euclidean, into regular poly- hedra. Bianchi, L. Em. E. Ac. Line. Ed. 2 (1893) (Sem. 2) 65-. , , and plane. Fedorow, E. von. Munch. Ak. Ab. 20 (1900) 465-. , , by 14-sided figures and cubes. Schoute, P. H. Amst. Ak. Vs. 3 (1895) 15-. of space, with minimum partitional area. Thomson, (Sir) W. (Lord Kelvin). ActaMth. 11 (1887-88) 121-. Divisions of space. Moore, E. H. (jun.), and Little, C.N. Am. J. Mth. 8 (1886) 127-. , congruent. Schlegel, V. Arch. Mth. Ps. 10 (1891) 154-. , general theorems. Tait, P. G. B. A. Double surfaces. Coar, H. I. Science 22 Ep. (1876) (Sect.) 29. , plane, intersect an even number of times, applications. Tait, P. G. [1876] Edinb. E. S. P. 9 (1878) 237-. surfaces, number of maximum and minimum normals from point. Reech, F. Par. EC. Pol. J. 37 e eah. (1858) 169-. Combinations of resistances. Macmahon, (Major) P. A. Elect. 28 (1892) 601-. Configuration of 2" circles. Godt, W. Mth. A. 47 (1896) 564-. Configurations on any surfaces. Brunei, G. Bordeaux S. Sc. Mm. 2 (1891) xxii-; (1897-98) 224-. closed surfaces. Brunei, G. Bordeaux S. Sc. PV. (1896-97) 4-. closed surfaces having one summit and one face. Brunei, G. Bordeaux S. Sc. PV. (1894-95) 13-. , reciprocal configurations. Brunei, G. Bordeaux S. Sc. PV. (1894-95) 63-. Congruency of plane surfaces. Dobriner, H. Mth. A. 42 (1893) 285-. (1893) 328-. Euclidean movements, general transformation of group. Page, J. M. A. Mth. 12 (1898-99) 87-. Examples from thread manufacture. Hartig, E. Dresden Tsis Sb. (1895) 34. Folding rows and sheets of postage stamps. Schoute, P. H. Amst. Ak. Vs. M. 9 (1892) 226- ; Fschr. Mth. (1892) 507-. Four colours problem (of maps). Kempe, A. B. Am. J. Mth. 2 (1879) 193-. ( ). Story, W. E. Am. J. Mth. 2 (1879) 201-. 2 (1879) 2 ~~ ( " Kempe, A. B. Nt. 21 (1880) Lucas, E. Ev. Sc. 6 (1883) Heawood, P. J. QJ. Mth. 24 (1890) 332-; 29 (1898) 270-. Generation of lines and surfaces. Conti, C. [1825] Padova N. Sag. 3 (1831) 25-. Hypothesis of Le Bel and Van't Hoff . Brown, A. C. Edinb. Mth. S. P. 2 (1884) 28-. 421 6420 Topology of Space and Hyperspace 6420 Interlacing curves. Brunei, G. Bordeaux S. Sc. Mm. 5 (1895) xxxi-. , Gauss's formula, and extension to higher configurations. Dyck, W. Miinch. Ak. Sb. 25 (1896) 447-. surfaces, case. Brown, C. Edinb. E. S. P. 13 (1886) 382-. Intersecting spaces, problem. Fieri, M. Mil. I. Lomb. Ed. 26 (1893) 534- ; 27 (1894) 258- ; 28 (1895) 441-. Knot circles, examples of reading. Kirkman, (Rev.) T. P. Edinb. B.,S. P. 13 (1886) 693-. of cravat. Lucas, E. (xn) Mathesis 3 (1883) 54-. , insoluble, equation of curve, with solution in 4th dimension. Hoppe, R. Arch. Mth. Ps. 64 (1879) 224. , linear section. Kirkman, (Rev.) T. P. Edinb. E. S. P. 13 (1886) 514-. , solution in 4th dimension. Hoppe, R. Arch. Mth. Ps. 65 (1880) 423-. Knots. Tait, P. G. [1877] Edinb. E. S. P. 9 (1878) 306- ; Edinb. E. S. T. 28 (1879) 145-. . Little, C. N. Conn. Ac. T. 7 (1885-88) 27-. . Tait, P. G. Edinb. E. S. T. 32 (1887) 327-, 493-. . Brunn,H. Munch. Ak. Sb. 22 (1893) 77-. , algebraic. Meyer, F. Edinb. E. S. P. 13 (1886) 931-. , alternate, of order eleven. Little, C. N. Edinb. E. S. T. 36 (1892) 253-. , amphicheiral forms. Tait, P. G. [1877] Edinb. E. S. P. 9 (1878) 391-. , classification according to connections. Simony, 0. D. Nf. Vh. (1894) (Th. 2, Hfilfte 1) 7-. , , topological apparatus. Simony, O. D. Nf. Tbl. (1887) 110- . and connections in certain closed surfaces. Dingeldey, F. Wien Ak. Sb. 98 (1890) (Ab. 2a) 79-. , cruciform surfaces, figure obtained by join- ing ends in pairs. Simony, O. [1881] Wien Ak. Sb. 84 (1882) (Ab. 2) 237-. , double point of plane curve, resolution in 3-dimensional space, and related mechanical problem. Schlegel, S. F. V. Z. Mth. Ps. 28 (1883) 105-. on endless cords. Kempe, A. B. [1886] Edinb. E. S. P. 14 (1888) 36-. of fewer 'than ten crossings, enumeration, description and construction. Kirkman, (Rev.) T. P. Edinb. E. S. T. 32 (1887) 281-. Listing and Tait. Kirkman, (Rev.) T. P. Edinb. E. S. P. 13 (1886) 363-. , measurement of beknottedness. Tait, P. G. [1877-79] Edinb. E. S. P. 9 (1878) 289- ; 10 (1880) 48-. , and new theorems in higher arithmetic. Simony, O. [1887] D. Nf. Tbl. (1887) 229-; Wien Ak. Sb. 96 (1888) (Ab. 2) 191-. , non-alternate, with census for order ten. Little, C. N. Edinb. E. S. T. 39 (1900) 771-. , , of orders eight and nine. Little, C. N. Edinb. E. S. T. 35 (1890) 663-. , properties. Tait, P. G. [1877] Edinb. E. S. P. 9 (1878) 403. Knots, sevenfold knottiness. Tait, P. G. [1877] Edinb. E. S. P. 9 (1878) 363-. , theorems, demonstration. Kirkman, (Rev.) T. P. Edinb. E. S. P. 13 (1886) 359-. , , general. Koller, L. Wien Ak. Sb. 89 (1884) (Ab. 2)250-. , unifilar, of ten crossings. Kirkman, T. P. Edinb. E. S. T. 32 (1887) 483-. Konigsberg bridges, Euler's problem. Conpy, E. N. A. Mth. 10 (1851) 106-. , . Saalschiitz, L. [1875] Konigsb. Schr. 16 (1876) (Sb.) 23-. Labyrinths, problem. Wiener, C. [1872] Mth. A. 6 (1873) 29-. , . Tarry, G. As. Fr. C. E. (1886) (Pt. 2) 49-; N. A. Mth. 14 (1895) 187-. Listing's theorem. Cayley, A. Mess. Mth. 2 (1873) 81-. Looped line, possibility of traversing without repetition or interruption. Hierholzer, C. [1871] Mth. A. 6 (1873) 30-. Manifoldness, ^-dimensional, representation. EberJiard, V. Mth. A. 36 (1890) 121-. ' ' Nachbar ' ' configurations and metacyclic groups. Heffter, L. Mth. A. 50 (1898) 261-. territories, problem. Heffter, L. Mth. A. 38 (1891) 477-. Network, construction. Brunei, G. Bordeaux S. Sc. PV. (1895-96) 62-. , paths on. Brunei, . Bordeaux S. Sc. Mm. 5 (1895) xxxix. Networks. Lucas, E. As. Fr. C. E. (1887) (Pt. 1) 173-. . Brunei, G. Bordeaux S. Sc. Mm. 5 (1895) 165-. on closed surface. White, H. S. N. Y. Am. Mth. S. Bll. 4 (1898) 376-. , problems. Brunei, G. Bordeaux S. Sc. Mm. 4 (1894) ix-. , properties. Bugaev, N. V. (xn) Eec. Mth. (Moscou) 2 (1867) (Pt. 2) 87-. ' , regular. Brunei, G. Bordeaux S. Sc. PV. (1894-95) 3-. on surfaces of deficiency higher than 1. White, H. S. [1896] N. Y. Am. Mth. S. Bll. 3. (1897) 116-. (White). Davis, E. W. N. Y. Am. Mth. S. Bll. 4 (1898) 529-. , theorem. Thue, A. Ts. Mth. 3 (1885) 102-. One-sided surface, example. Hoppe, R. Arch. Mth. Ps. 57 (1875) 328-. , "marrowbone." Schoute, P. H. Edinb. E. S. P. 19 (1893) 208-. of Mobius. Maschke, H. N. Y. Am. Mth. S. T. 1 (1900) 39. surfaces. Brunei, G. Bordeaux S. Sc. Mm. 2 (1891) xxxvii. , class. Feldblum, M. Wiad. Mt. 1 (1897) 101- ; Fschr. Mth. (1897) 579. , discovery. Stdckel, P. Mth. A. 52 (1899) 598-. , and singular points of plane curves. De- launay, N. Par. S. Mth. Bll. 26 (1898) 43-. and spaces. Brunei, G. Bordeaux S. Sc. PV. (1895-96) 26-. Packing of spheres. Hoppe, R. Arch. Mth. Ps. 56 (1874) 302-. 422 6420 Topology Packing of spheres. Giinther, S. Arch. Mth. Ps. 57 (1875) 209-. . Gwyther, R. F. Manch. Lt. Ph. S. P. 25 (1886) 35-. . Reynolds, 0. Manch. Lt. Ph. S. P. 25 (1886) 95-. . Aldis, W. S. Nt. 39 (1889) 581-. . Greenhill, A. G. Nt. 40 (1889) 10-. . Liceing, G. D. Nt. 40 (1889) 55. Point groups, regular, theorems. Rohn, K. Mth. A. 53 (1900) 440-. pairs, intervals between, theorems. Padoa, A. Bv. Mth. 6 (1896-99) 35-. systems, regular plane, of unlimited extent. Sohncke,L.A. (vin) Crelle J. 77 (1874)47-. Points in closed line, position relations. Vailati, G. Ev. Mt. 5 (1895) 75-. on line, position relations. Vailati, G. Ev. Mt. 5 (1895) 183-. in space, regular distributions. Bravais, A. [1848] C. E. 27 (1848) 601- ; Par. S. Phlm. PV. (1848) 69-; Par. EC. Pol. J. 33' cah. (1850) 1-. Polyhedra, classification. Davis, E. W. J. H. Un. Cir. [6] (1886-87) 122. .convex. Berger,A. Stockh.Ofv. (1898) 497-. , Euler's equation, generalisation for 7i-fold space. Stringham, W. I. [1880] (xn) J. H. Un. Cir. [1] (1882) 35. , relation on (p + l)-ply connected surface. Kapteyn, W. Amst. Ak. Vs. 4 (1896) 199- ; Fschr. Mth. (1896) 380. , theorem and Cauchy's. Grunert, J. A. Crelle J. 2 (1827) 367. , Descartes's. Halsted, G. B. A. Mth. 1 (1884-85) 138-. , regular, elementary properties. Schoute, P. H. Fschr. Mth. (1891) 699-. , , in 4-dimensional space. Schoute, P. H. Amst. Ak. Vh. (Sect. 1) 2 (1894) No. 2, 12 pp. ; No. 4, 17 pp. ; No. 1, 26 pp. , , , bluntness. Heyl, P. R. N. Y. Am. Mth. S. Bll. 4 (1898) 292, 293-. , and semiregular, in n-dimensional space. Cesdro, E. Lisb. Ac. Sc. Mm. 6 (Pt. 2) (1887) No. 5, 75 pp. Polyhedron, Kirkman's, problem connected with. Brunei, . Bordeaux S. Sc. Mm. 5 (1895) xxix-. with trihedral summits, theorem. Tait, P. G. Edinb. E. S. T. 29 (1880) 657-. Quincunx, application in science and weaving. Koturnitzky, P. [1893] Fschr. Mth. (1893 -94) 876-. Eegion of feth rank, division by n regions of (*-l)th rank. Pilgrim, L. [1878] Z. Mth. Ps. 24 (1879) 188-. Eegions and aspects relative to points in a plane. Laisant, C. A. Par. S. Mth. Bll. 10 (1882) 52-. curves which fill them, analytical repre- sentation. Cesdro, E. Bll. Sc. Mth. 21 (1897) 257-. , number of, into which plane and space are divisible. Laisant, C. A. As. Fr. C. B. (1881) 71-. .theory. Cahen,E. N. A. Mth. 16 (1897) 533-. Analytical Geometry 6430 Eelations, elementary. Mobius, A. F. Leip. B. 15 (1863) 18-. Eepresentetion of points in n-dimensional space, groups which r arbitrary points can repre- sent. Schlegel, V. Arch. Mth. Ps. 10 (1891) 283-. space by triplets of points on a line. Hirst, T.A. L. Mth. S. P. 2 (1869) 48. ruled space by a system of conies in a plane. Aschieri, F. Mil. I. Lomb. Ed. 12 (1879) 265-. Eing formed closed knotless ribands, surfaces obtained from. Simony, O. [1880] Wien Ak. Sb. 82 (1881) (Ab. 2) 691-. , hollow flexible circular, results of dividing by closed cuts. Simony, O. [1882-83] Wien Ak. Sb. 85 (1882) (Ab. 2) 907- ; 87 (1883) (Ab. 2) 556-; 88 (1884) (Ab. 2) 939-. Straight line, definition. Brunei, G. Bordeaux S. Sc. Mm. 3 (1893) xxx-. System of two tetrads of circles, and other systems of two tetrads. Cayley, . Camb. Ph. S. P. 8 (1895) 54-. Tait's diagram. Polignac, de. Par. S. Mth. Bll. 27 (1899) 142-. Theorem. Klein, F. Gott. Nr. (1872) 290-. , and application to theory of numbers. Busche, E. [1898] Hamb. Mth. Gs. Mt. 3 (1900) 333-. Three-dimensional spaces. Dyck, W. B. A. Ep. (1884) 648-. Trigraphic fields of space. Russell, J. W. B. A. Ep. (1894) 578-. Twists, class, figures formed by cutting. Schuster, J. L. Wien Ak. Sb. 97 (1889) (Ab. 2a) 217-. , model of half-twist surface. Brown, C. Edinb. E. S. P. 13 (1886) 513-. 6430 Methods of analytical geo- metry. (See also 0840.) Hachette, J. N. P., & Monge, . Par. EC. Pol. J. ll e cah. (1802) 143-, 170-. Fuss, N. [1808] St Pet. Ac. Sc. Mm. 4 (1813) 221-. Monge, G. Par. EC. Pol. J. 8 (1809) 68-. Gergonne, J. D. Gard Not. Tr. Ac. (1810) 168-. Binet, J. P. M. Par. EC. Pol. J. 16 e cah. (1813) 280-. Hachette, J. N. P. Par. EC. Pol. Cor. 3 (1814-16) 132-. Gergonne, J. D. Gergonne A. Mth. 8 (1817-18) 156-. Hube, K. (vi Adds.) Krk. Eoczn. Uniwers. 9 (1824) 76-. Lloyd, H. Dubl. Ph. J. 1 (1825) 307-. Pliicker, J. Crelle J. 6 (1830) 107-. Conti, C. Padova N. Sag. 3 (1831) 90-. Wilder, C. Silliman J. 20 (1831) 285-. Bellavitis, G. Poligrafo 13 (1833) 53-. Brunn, H. St Pet. Ac. Sc. Bll. 2 (1837) 275-; 6 (1840) 305-. Chelini, D. G. Arcad. 75 (1838) 80-, 279-; 76 (1838) 3-, 257-. D. Camb. Mth. J. 1 (1839) 74-. 423 6430 Methods of Analytical Geometry 6430 Greatheed, S. S. (vi Adds.) Camb. Mth. J. 1 (1839) 168-. Boole, G. Camb. Mth. J. 2 (1841) 179-. Doppler, C. Bohm. Gs. Ab. 2 (1841-42) 533-. Padula, F. Nap. Ed. 2 (1843) 425- ; 3 (1844) 241-, 321-, 401- ; 4 (1845) 15-. Rutherford, W. Ph. Mg. 22 (1843) 353-. Bianchi, G. Palomba Eac. 1 (1845) 191-. Cauchy, A.L. C. E. 21 (1845) 305-. Dobson, T. Mathematician 1 (1845) 166-. Finlay, R. Mathematician 1 (1845) 258-, 294-; 2 (1847) 1-, 84-, 143-, 237- ; 3 (1850) 20-, 113-, 259-. Luca, F. de. Nap. Ms. 6 (1845) 369-. Briot, C. Lyon Mm. Ac. 2 (1847) 429-. Ellis, A.J. B. A. Ep. (1855), (Pt. 2) 5-. Brassai, 8. (xn) Mag. Ak. Ets. (1857) 165-, 561- ; (1858) 565-. Grunert, J. A. Grunert Arch. 29 (1857) 235-. Padula, F. II Giamb. Vico 3 (1857) 133- ; Nap. At. Ac. 1 (1863) No. 4, 14 pp. Araldi, A. [1859] (ix) Mod. Ac. Sc. Mm. 3 (1861) 113-. Bellavitis, G. Yen. Mm. I. 8 (1859) 241-. Fiedler, W. Schlomilch Z. 7 (1862) 53-. Clayeux, . C. E. 56 (1863) 788-. Cassani,P. (ix) Yen. Aten. At. 10 (1873) 114-. Despeyrous, C. Toul. Ac. Sc. Mm. 7 (1875) 259-. Frombeck, H. [1876] Wien Ak. Sb. 74 (1877) (Ab. 2) 399-. Lucas, E. (xn) Mathesis 1 (1881) 65-. Malcor, E. Ev. Mar. et Col. 81 (1884) 653- ; 82 (1884) 149-, 483- ; 83 (1884) 376-, 880-. MukJwpadhyay, A. [1887] Beng. As. S. J. 56 (Pt. 2) (1888) 288-. Story, W. E. Am. J. Mth. 9 (1887) 38-. Tilly, J. de. Brux. Mm. Cour. 8, 47 (1892-93) No. 5, 80 pp. Aragdn,A. Mex.S. "Alzate" Mm. 8 (1894)173-. Brill, J. Mess. Mth. 25 (1896) 49-. d'Alembert-Carnot geometrical paradox and its resolution. Turquan, L. V. N. A. Mth. 7 (1868) 437-. . Sylvester, J. J. Mess. Mth. 14 (1885) 92-. Algebra, application to geometry, and vice-versa. Schulten, N. G. af (fil.) [1853] Helsingf. Acta 4 (1856) 29-. of plane and of space. Boguslavskij, A. I. [1890-91] Eec. Mth. (Moscou) 14 (1890) 600- ; 15 (1891) 683-; 16 (1893) 113- ; Fschr. Mth. (1891) 734-. Algebraic principle, single, deduction of geo- metric theorems. Zeuthen, H. G. [1879-80] L. Mth. S. P. 10 (1878-79) 196- ; 11 (1879- 80) 156-. Analytical and geometrical methods, advantages of their separation. Woodhouse, R. Phil. Trans. (1802) 85-. operations, generalisation by help of geometry. Neovius, E. [1876] Helsingf. Acta 11 (1880) 111-. Angle between two lines given by their equations. Heal, W. E. Des Moines Anal. 5 (1878) 148. Area of polygon in terms of coordinates of vertices. Stainville, de. Gergonne A. Mth. 1 (1810) 190-. Area of polygon in terms of coordinates of vertices. Terquem, O. N. A. Mth. 9 (1850) 65. . Prouhet.E. N. A. Mth. 15 (1856) 373-. . Veltmann, W. Z. Mth. Ps. 32 (1887) 339-. triangle. Lowry, W. L. [1874] Mess. Mth. 4 (1875) 129. and tetrahedron volume, in Cartesian, tetrahedral and quadriplanar coordinates. Grunert, J. A. Arch. Mth. Ps. 53 (1871) 317-. Areas and centres of polygons. Padula, F. Nap. At. I. Inc. 1 (1864) 37-. of surfaces and volumes of solids, methods of approximation. Bugaev, N. V. [1897] Eec. Mth. (Moscou) 20 (1899) 451- ; Fschr. Mth. (1898) 252. Axes, Conditions for system of three, being mutually rectangular. Dauge, F. Mathesis 14 (1894) 85- ; 15 (1895) 250-. , 3, problem. Puttkammer, J. N. van. Leijd. A. Ac. (1822-23) 25 pp. , rectangular, nine angles between two systems of 3 (Plana). Reiss, M. Brux. Ac. Bll. 9 (1842) 6-. Barycentric calculus of Mobius. Allardice, R. E. Edinb. Mth. S. P. 10 (1892) 2-. Binary forms, application to analytical geo- metry. Laguerre, E. Liouv. J. Mth. 1 (1875) 99-. Cartesian geometry. Battaglini, G. G. Mt. 29 (1891) 3-, 93-, 195-, 298-. , higher, foundations. Folie, F. [1869-79] Brux. Ac. Sc. Mm. 39 (1872) 142 pp. ; Brux. Ac. Bll. 36 (1873) 620- ; Bll. Sc. Mth. As. 3 (1879) 278-. , historical development and bases. Corzan-Avendano, G. [1864] (xn) Mag. Ak. Ets. (Mth. Term.) 6 (1865) 3-. Circle and concentric rectangular hyperbola, connection. Godefroy, A. N. N. Arch. Wisk. 18 (1891) 119-. Circular relation (Kreisverwandtschaft) in geo- metric terms. Mobius, A. F. Leip. Ab. Mth. Ps. 2 (1855) 529-. of Mobius. Cayley, A. QJ. Mth. 2 (1858) 162. . Enneper, A. Z. Mth. Ps. 13 (1868) 261-. . Lucas, E. N. A. Mth. 15 (1876) 205-. (Lucas). Cassani, P. G. Mt. 14 (1876) 347-. . Cayley, A. L. Mth. S. P. 8 (1877) 220-. , equation. Grunert, J. A. Arch. Mth. Ps. 44 (1865) 376-. , . Zahradmk, K. Arch. Mth. Ps. 56 (1874) 15-. Complex elements. Tognoli, 0. G. Mt. 31 (1893) 137-. quantities, application to plane geometry. Brill, J. Mess. Mth. 16 (1887) 8-. Condition that straight line should be normal to a conic in trilinear coordinates. Pinker- ton, R. H. Edinb. Mth. S. P. 9 (1891) 2-. 424 6430 Analytical Geometry. Coordinates 6430 Coordinates. Enharmonic. Hamilton, (Sir) W. E. NH. Bv. 7 (1860) (P.) 242-, 325-, 506-. areal. Hart, H. Mess. Mth. 11 (1882) 104-. , method connected with orthogonal projec- tion. Slwrpe, J. W. [1879] Mess. Mth. 9 (1880) 10-. axial. Esson, W. QJ. Mth. 10 (1870) 113-. . Ocagne, M. d'. N. A. Mth. 3 (1884) 545-. (d'Ocagne). Cesdro, E. N. A. Mth. 4 (1885) 256-. barycentric. Cesdro, E. Mathesis 10 (1890) 177-. biangular. Walton, W. [1866] QJ. Mth. 9 (1868) 47-. . Ford, P. [1873] (rx) QJ. Mth. 13 (1875) . Genese, R. W. [1881] QJ. Mth. 18 (1882) 150-. . Biggin, T. QJ. Mth. 25 (1891) 237-. . Bertelsen, B. P. N. Ts. Mth. 8 (A) (1897) , and application to lines and conies. Azzarelli, M. 443-. Em. N. Line. At. 28 (1875) bilinear tangential. Loud, F, H. Des Moines Anal. 10 (1883) 50-. bipolar. Barbarin, P. N. A. Mth. 1 (1882) 15-. . Fries, J. de. Amst. Ak. Vs. 4 (1896) 219- ; Fschr. Mth. (1896) 491. , and geodesy. Grunert, J. A. Grunert Arch. 32 (1859) 444-. bipunctual. Franklin, F. Am. J. Mth. 1 (1878) 148-. 3 circles and 4 spheres, coordinate system, formulae. Lucas, E. [1876] A. Mt. 8 (1877) 187- ; Par. S. Mth. Bll. 5 (1877) 136-. circular. Stammer, W. Crelle J. 44 (1852) 295-. , applications. Franklin, F. Am. J. Mth. 12 (1890) 161-. curvilinear. Combescure, E. Par. EC. Norm. A. 4 (1867) 93-. cyclic. Ocagne, M. d'. Mathesis 7 (1887) 148-. cylindrical. Brenner, . Grunert Arch. 13 (1849) 244-. homogeneous. Terquem, 0. N. A. Mth. 8 (1849) 113-. . Trudi, N. G. Mt. 1 (1863) 11-, 47-, 148- ; Nap. At. I. Inc. 1 (1864) 77-. . Kohn, G. D. Mth. Vr. Jbr. 4 (1897) 162-. , in imaginary geometry, and application to systems of forces. Cox, H. QJ. Mth. 18 (1882) 178-. , metrical relations. Ovidio, E. d'. G. Mt. 11 (1873) 197-. , perpendicularity and parallelism in. Giu- dice, F. Tor. Ac. Sc. At. 34 (1898) 153- or 277-. , plane. Heger, R. Z. Mth. Ps. 15 (1870) 117-, 389-. homogeneous, points, planes, and lines in. Ovidio, E. d'. G. Mt. 8 (1870) 241-. , in space. Heger, R. Z. Mth. Ps. 16 (1871) 1-. linear, introduction into geometry of plane. Koehler, C. Z. Mth. Ps. 32 (1887) 152-. multipolar. Fries, J. de. Haarl. Ms. Teyl. Arch. 5 (1898) 99-. oblique. Baltzer, R. [1850] Crelle J. 46 (1853) 145-. . Silva, D. A. da. Lisb. Ac. Sc. Mm. 5 (Pt. 1) (*1875) No. 2, 20 pp. , 3, angle between 2 lines or 2 planes re- ferred to. Whewell, W. [1823] Camb. Ph. S. T. 2 (1827) 197-. , at equal angles, and point, line, and plane in space. Schmidt, G. [1850] Haidinger Ab. 4 (1851) (Ab. 4) 13-. , straight line and plane. Franqais, J. F. Par. EC. Pol. Cor. 1 (1804-08) 337-. , . Haedenkamp, H. Grunert Arch. 3 (1843) 67-. orthogonal, new system. Thomson, (Sir) W. Liouv. J. Mth. 12 (1847) 256-. parallel. Ocagne, M. d'. N. A. Mth. 3 (1884) 410-, 456-, 516- ; 6 (1887) 493- ; 8 (1889) 568- ; 9 (1890) 445-, 471- ; Brux. S. Sc. A. 14 (1890) (Pt. 1) 47- ; Par. S. Mth. Bll. 18 (1890) 108-. pentahedral. Whitworth, W. A. Mess. Mth. 3 (1866) 146-. peripolar. Neumann, C. G. [1877-80] Leip. Mth. Ps. B. 29 (1877) 134- ; Leip. Mth. Ps. Ab. 12 (1883) 363-. in piano, systems. Hill, T. Am. As. P. (1857) 42-; (1858) 1-. point, and tangential, correlations between systems. Ocagne, M. d'. N. A. Mth. 11 (1892) 70-. of points, lines and planes. Casorati, F. Mil. I. Lomb. Ed. 10 (1877) 11-, 40-. polar. Genese, R. W. Mess. Mth. 5 (1876) 14-. . Loria, G. Ens. Mth. 1 (1899) 357-. , equation of normals and tangents in. Jefdbek, V. Casopis 18 (1889) 245- ; Fschr. Mth. (1889) 694. , formula from figure. Unferdinger, F. Arch. Mth. Ps. 48 (1868) 106-. linear, in piano. Weinmeister, J. P. Z. Mth. Ps. 21 (1876) 301-. , straight line and parabola. Davies, T. S. Ph. Mg. 21 (1842) 190-. polygonal and polyhedric, etc. Sautreaux, F. Brux. Mm. Cour. 8, 36 (1884) No. 3, 14 pp. principle, origin and development. Gilnther, A.W.S. (xn) Niirnb.Nt.Gs.Ab. 6 (1877)1-. protective. Fiedler, W. Ziir. Vjschr. 15 (1870) 152-. quadrilinear. Whitworth, W. A. Mess. Mth. 1 (1862) 193- ; 3 (1866) 71-, 173-. quadriplanar. Salmon, G. QJ. Mth. 4 (1861) 231-, 271. , sphere in. Whitworth, W. A. Mess. Mth. 1 (1862) 162-. and tangential, principal radii of curvature of surface in. Jeffery, H. M. [1872] QJ. Mth. 12 (1873) 86-. 425 6430 Analytical Geometry. Coordinates 6430 quadriplanar tetrahedral volume, and triangular area in space. Fazzari, G. G. Mt. 29 (1891) 247-. reciprocal to the Cartesian. Schlegel, S. F. V. Z. Mth. Ps. 23 (1878) 195-. --- polar. Schlegel, V. As. Fr. C. E. (1885) (Ft. 2) 156-. rectangular. Bartlett, W. P. G. Camb. (M.) Mth. M. 1 (1859) 180-. axes, nine angles between two systems. Reiss, M. Quetelet Cor. Mth. 11 (1839) 119- . -- , ---- -- (Eeiss). Plana, G. Brux. Ac. Bll. 8 (pte. 2) (1841) 68-. , ------ (Pl Ac. Bll. 9 (1842) 6-. pte. 2) Plana) . Reiss, M. Brux. -- , variable system of three. Painvin, L. N. A. Mth. 10 (1871) 414-. , relation of cosines. Bar-re de Saint-Ve- nant, . Liouv. J. Mth. 9 (1844) 270-, 310-. , --- . Schldfli,L. Bern Mt. (1848) 27-. relative signs. De Morgan, A. Ph. Mg. 9 (1836) 249-. in space. Zeuthen, H. G. [1868] Sk. Nf. F. 10 (1869) 148-. spherical. Greatheed, S. S. [ascribed to Gre- gory, D. F., in vol. III. C.S.P.] Camb. Mth. J. 1 (1839) 193- ; 2 (1841) 37-. . Cayley, A. Camb. and Dubl. Mth. J. 1 (1846) 22-. system for circle in a plane and for sphere in space. Berg, F. J. van den. N. Arch. Wisk. 1 (1895) 11- ; Fschr. Mth. (1893-94) 1106. , new. PlUcker, J. Crelle J. 5 (1829) 1-. , . Chasles, M. Quetelet Cor. Mth. 6 (1830) 81-. , . Frankenbach, F. W. Eiga Cor.-Bl. 17 (1868) 117-. , : theorems on conies. Bobillier, . Gergonne A. Mth. 18 (1827-28) 320-. , : --- (Bobillier) . Gergonne, J. D. Gergonne A. Mth. 19 (1828-29) 32-, 220-. , two parallel axes. Schwering, K. Z. Mth. Ps. 21 (1876) 278-. systems. Fiedler, W. Schlomilch Z. 4 (1859) 91. . Chelini, D. Bologna Mm. Ac. Sc. 3 (1863) 3-. . Arcais, F. d\ [1877] G. Mt. 16 (1878) 18-. . Perrin, E. As. Fr. C. E. (1897) (Pt. 2) 90-. , simplest, to fix a point. Lemoine, E. Par. S. Mth. Bll. 16 (1888) 162-. tangential. Booth, J. E. S. P. 9 (1857-59) 175-. . South, E. J. QJ. Mth. 8 (1867) 111-. . Taylor, C. [1880] Mess. Mth. 10 (1881) 13-. . Ocagne, M. d\ N. A. Mth. 3 (1884) 410-, 456-, 516-, 545- ; 4 (1885) 110-. . Aubert, P. Edinb. Mth. S. P. 12 (1894) 55-. , Brianchon's theorem, etc. Grunert, J. A. Arch. Mth. Ps. 53 (1871) 443-. , system. Balitrand, . N. A. Mth 12 (1893) 256-. tetrahedral. Fiedler, W. Schlomilch Z. 8 (1863) 45-. . Neuberg, J. [1869] Brux. Mm. Cour. 8, 21 (1870) (No. 2) 52 pp. , general equation of second degree referred to. Allen, A. J. C. L. Mth. S. P. 13 (1882) 165-. and quadriplanar. Ferrers, N. M. QJ. Mth. 5 (1862) 172-. . Grunert, J. A. Arch. Mth. Ps. 53 (1871) 193-. , system. Schlegel, V. Par. S. Mth. Bll. 23 (1895) 216-. and triangles, constant relations. Toplitz, J. Z. Mth. Ps. 14 (1869) 253-. triangular. Torry, A. F. Mess. Mth. 1 (1862) 220-. transformation. Livet, . Par. EC. Pol. J. 13 e cah. (1806) 270-. . Francais, J. F. Par. EC. Pol. J. 14 cah. (1808) 182-. . Hachette, J. N. P. Par. EC. Pol. Cor. 2 (1809-13) 6-, 247-. . Guinard, J. B. (vi Adds.) Gand A. Ac. (1821-22) 41 pp. . Hube, K. (vi Adds.) Krk. Eoczn. Uniwers. 13 (1829) 91-. . Lowenstern, S. As. Nr. 17 (1840) 369-. . Steczkowski, J. K. Grunert Arch. 22 (1854) 356. . Dostor, G. J. Grunert Arch. 26 (1856) 121-. . Roberts, S. QJ. Mth. 2 (1858) 172-. . Lebesgue, V. A. N. A. Mth. 2 (1863) 392-. . Walton, W. [1864] QJ. Mth. 7 (1866) 102-. . Klein, F. [1869] Mth. A. 2 (1870) 366-. . Cayley, A. [1878] Camb. Ph. S. P. 3 (1880) 178-. . Fiedler, W. Zur. Vjschr. 24 (1879) 145-. . Reye, T. Crelle J. Mth. 94 (1883) 312-. . Ziegel, R. Arch. Mth. Ps. 17 (1900) 263-. , Euler's formulae. Jacobi, C. G. /. Crelle J. 2 (1827) 188-. , . Lobatto, R. Amst. Ts. Ws. Nt. Wet. 1 (1848) 254-. , . Hallsten, K. Helsingf. Ofv. 20 (1878) 24-. , , demonstration, and deduction of Monge's. Grunert, J. A. N. A. Mth. 5 (1846) 414-. , formula containing symbol denoting given rotation. Graves, C. Ir. Ac. P. 5 (1853) 140-. , Monge's formulae. Encke, J. F. Berl. As. Jb. (1832) 305-. , and signs of angles and distances in piano. Folie, F. Brux. Ac. Bll. 41 (1876) 86-. in space. Gergonne, J. D. Gergonne A. Mth. 7 (1816-17) 54-. . Grunert, J. A. Crelle J. 8 (1832) 153-. . Tortolini, B. G. Arcad. 59 (1833) 52-. 426 6430 Analytical Geometry. Coordinates 6430 transformation in space. Wolf, E. Bern Mt. (1847) 24- ; Grunert Arch. 13 (1849) 274-. . Hesse, L. O. Crelle J. 63 (1864) 247-. . Bardelli, G. Mil. I. Lomb. Kd. 2 (1869) 248-. . Le Paige, G. M. M. H. H. As. Fr. C. E. (1876) 384-. . Borletti, F. Mil. I. Lomb. Ed. 15 (1882) 252-. triangular, and analytical geometry of plane in space. Loria, G. G. Mt. 24 (1886) 164-. , system. Sondat, P. N. A. Mth. 12 (1893) 360-, 503-. triaxial, in equations of first and second degree. Veltmann, W. Arch. Mth. Ps. 64 (1879) 113-. , theorem. Dedekind, R. Crelle J. 50 (1855) 272-. trigonic. Walton, W. QJ. Mth. 9 (1868) 340-. . Levi, S. G. Mt. 14 (1876) 353-. . Genese, R. W. L. Mth. S. P. 12 (1880-81) 157-. trilinear. Slesser, G. M. QJ. Mth. 4 (1861) 134-. . Esson, W. Mess. Mth. 1 (1862) 113-. . Grunert, J. A. Grunert Arch. 38 (1862) 289-. . Faure, H, N. A. Mth. 2 (1863) 289-. . C., J. B. Cn. J. 9 (1864) 249- ; 10 (1865) 334-. . Clifford, W. K. [1864] QJ. Mth. 7 (1866) 54- ; 8 (1867) 16-, 119-. . Walker, E. (vi) Mess. Mth. 2 (1864) 40-. . Steen, A. (xn) Ts. Mth. 3 (1867) 129- ; 4 (1868) 6-; 5 (1869) 33-; 6 (1870) 52-, 113-, 171-. . Pendlebury, R. Mess. Mth. 4 (1868) 138. . London, J. [1870] Cn. J. 13 (1873) 62-. . Grave, P. Kazan S. Ps.-Mth. Bll. 3 (1893) 82-; Fschr. Mth. (1893-94) 1063. , angles and distances in. Vogt, . N. A. Mth. 11 (1892) 148-. , and applications to straight lines and conies. Foglini, G. Em. N. Line. At. 30 (1877) 159-. , points. Imshenetskil, V. G. (xn) Kazan Un. Mm. (1862) (Ps.- Mth.) 25 pp. areal, results involving. Anglin, A. H. Edinb. Mth. S. P. 11 (1893) 8-. , circular points, use of coordinates. Faure, H. N. A. Mth. 3 (1884) 140-. , contact and curvature of curves, and theorems of conies. Grunert, J. A. Arch. Mth. Ps. 53 (1871) 443-. , equations for straight line. Whitworth, W. A. Mess. Mth. 1 (1862) 42-, 92-. , of focoids. Davis, R. F. Mth. Gz. 1 (1900) 336-. , inscribed and circumscribed circles of triangle in. Grunert, J. A. Arch. Mth. Ps. 52 (1871) 331-. , principles of geometry in. Tillol, . Toul. Mm. Ac. 5 (1867) 18-. and quadriplanar. Greer, H. R. QJ. Mth. 6 (1863) 237-. , straight line and plane in. Esson, W. QJ. Mth. 5 (1862) 373-. trilinear, spherical ellipse. Jeffery, H. M. [1866] QJ. Mth. 8 (1867) 283-. and tetrahedral. Giudice, F. Palermo Cir. Mt. Ed. 12 (1898) 278-. , fundamental equations. Zeuthen, H. G. (xii) Ts. Mth. 5 (1869) 129-. , theorem. Mullins, W. E. Mess. Mth. 1 (1862) 111-. , transversals, seventy-two propositions. Martin, (Rev.) H. [1865] Edinb. E. S. T. 24 (1867) 37-; Mess. Mth. 4 (1868) 40-, 65-. , triangle, area. Walker, W. F. [1864] QJ. Mth. 7 (1866) 122-. , , fundamental, property. Turnbull, J. C. QJ. Mth. 5 (1862) 236. trimetric point, first formulae. Hain, E. [1881] Arch. Mth. Ps. 67 (1882) 425-. tripolar. Lucas, E. Mathesis 9 (1889) 129-, 173-. Curves as envelopes of a straight line. Dyrion, L. N. A. Mth. 7 (1868) 176-. , symmetrical equation. Holditch, H. QJ. Mth. 2 (1858) 38. Demonstration of elementary geometry theorems. Reuschle, [C. G.] Crelle J. 24 (1842) 171-. 3-dimensional analytical geometry. Greatheed, S. S. [ascribed to Gregory in vol. III. C.S.P.] Camb. Mth. J. 1 (1839) 37-. , formulae. Bartels, . [1825] St Pet. Mm. Sav. Etr. 1 (1831) 77-. , oblique axes. Weddle, T. Mathe matician 3 (1850) 169-, 233-; (Suppl.) 5-. Direction, analytical designation. Wessel, C. Arch. Mth. Ntvd. 18 (1896) No. 1, 69 pp. cosine of straight line, expression in rational fractions. Hoppe, R. Arch. Mth. Ps. 61 (1877) 438-. cosines of line making given angles with given lines. Tallqvist, H. Helsingf. Acta 17 (1891) 465-. Distance. Cayley, A. B. A. Ep. (1858) (pt. 2) 3. , shortest, of point from straight line or plane, analytic expression. Prouhet, E. N. A. Mth. 3 (1844) 548-. Distances of point attached to fixed centre from fixed plane. Cauchy, A. L. C. E. 16 (1843) 798-. points from straight lines and planes. Schell, W. Grunert Arch. 19 (1852) 79-. . Dostor, G. Arch. Mth. Ps. 57 (1875) 225-. Equation of angle, rectangle, lozenge, etc. Neuberg, J. J. B., & Plateau, J. A. F. (xn) Mathesis 1 (1881) 89. in geometry of straight lines. Muir, T. Mess. Mth. 1 (1872) 150-. of normal. Joachimsthal, F. Crelle J. 56 (1859) 280-. to a plane, Hessian normal form. Lilienthal, R. von. Mth. A. 42 (1893) 497-. Equations represented by 3 systems of circles. Ocagne, M. d\ Z. Mth. Ps. 43 (1898) 269-. systems of three points on curve. Ocagne, M. d'. Acta Mth. 21 (1897) 301-. 427 6430 Methods of Analytical Geometry 6430 Equations, tangential. Laguerre, . Liouv. J. Mth. 17 (1872) 1-. , , new form. Casey, J. Phil. Trans. 167 (1877) 367-. between two variables, general interpretation. Walton, W. Camb. Mth. J. 2 (1841) 103-. , representation. Bjorling, C. F. E. [1874] Stockh. Ak. Hndl. 13 (1876) (No. 4) 40pp. , . Cayley, A. (xii) J. H. Un. Cir. [1] (1882) 210. Equipollences, calculus. Bellavitis, G. A. Sc. Lomb. Ven. 5 (1835) 244-. , (Bellavitis). Hoilel, J. N. A. Mth. 8 (1869) 289-, 337-. Figures, geometrical, algebra. Jarosenko, S. N. Es. S. Nt. Mm. (Mth.) 1 (*1878). Formula for angles and distances, simplifica- tion. Eipert, L. N. A. Mth. 19 (1900) 409-. Formulae, fundamental. Stolz, 0. Mth. A. 43 (1893) 591-. Geometry as a branch of analysis. Heatly, S. G. T. Beng. J. As. S. 12 (1843) 110-. , higher, proofs of fundamental formulae. Cotterill, T. Mess. Mth. 4 (1868) 98-. "Harmonics" of triangle. Hain, E. [1874] Arch. Mth. Ps. 57 (1875) 316-. Hypercycle and theory of polar cycles. Dauthe- ville, . Par. S. Mth.. Bll. 14 (1886) 45-. Imaginary in function theory and in analytic geometry. Schiiler, W. F. D. Nf. B. (*1877) 109-. points. Feldblum, M. Wiad. Mt. 2 (1898) 130- ; Fschr. Mth. (1898) 486. Inclination of lines and planes in space, inter- pretation of formulae. Painvin, L. N. A. Mth. 5 (1866) 337-. plane to horizontal, Lamp's theorem. Trzaska, W. Par. T. Nauk Sc. Pam. 10 (*1878) Art. 9, 3 pp. Inverse figures. Bellavitis, G. A. Sc. Lomb. Ven. 6 (1836) 136- ; Tortolini A. 3 (1860) 60-. "Keys" (Cauchy), geometrical interpretation. Barre de Saint-Venant, . C. E. 36 (1853) 582-. Leibnitz's " Characteristic." Grassmann, H. Leip. Jablon. Preisschr. (No. 1) (1847) 60 pp. Lines, plane, generation. Conti, C. [1828] Padova N. Sag. 2 (1825) 255-. , theory. Milller, G. W. Crelle J. 15 (1836) 229-. Method, general, linking synthetic with analytic geometry. Caspary, F. Bll. Sc. Mth. 13 (1889) 202-. Middle points of transversals. Valat, J. P. F. Bordeaux Ac. Act. 38 (1877) 49-. Multiple centres, theory. Breton, Ph. Liouv. J. Mth. 10 (1845) 430-. Multipliers, indeterminate. Brassinne, E. N. A. Mth. 6 (1847) 381-. Notation, new. O'Brien, M. [1846] Camb. Ph. S. T. 8 (1849) 415-. , Plucker's abridged. Finck,B. N. A. Mth. 3 (1844) 147-. Numerical equations, and analytical geometry in space. Zmurko, L. [1869] Wien D. 30 (1870) (2te Ab.) 217-. Object and image with respect to reflecting circle. Lehmus, . Crelle J. 31 (1846) 85-. Orthogonal axes, systematic treatment. Pelz, C. Wien Ak. Sb. 90 (1885) (Ab. 2) 1060-. coefficients, system, new relations. Hoppe, R. Arch. Mth. Ps. 2 (1885) 413-. Parameter, variable, use in geometry. Bras- sinne, E. N. A. Mth. 6 (1847) 381-. Plane algebra. Macfarlane, A. [1883] Edinb. E. S. P. 12 (1884) 184-. , analytical problem. Dienger, J. Grunert Arch. 11 (1848) 111-. , equation. Schett, W. Schlomilch Z. 1 (1856) 106-. figures, representation and segmentation. Laisant, C. A. Par. S. Mth. Bll. 18 (1890) 123-. Planes, angle between two, in oblique co- ordinates. Junghann, G. Z. Mth. Ps. 12 (1867) 350-. , bisecting, and axes in oblique system. Paterno, F. P. Palermo G. Sc. Nt. 18 (1887) 342-. , position. Littrow, J. J. von. [1813] St Pet. Ac. Sc. Mm. 8 (1817-18) 54-. Points on a circle, 4, relations between distances. Antomari, X. N. A. Mth. 1 (1882) 462-. , , . Cayley, A. Mess. Mth. 17 (1888) 94-. , , , extension of Cayley's theorem. Springer, J. F. J. H. Un. Cir. [9 (1889-90)] 78. with integer coordinates in plane, joins of. Bravais, A. C. E. 5 (1837) 867-. in a plane, property of a system. Lucas, F. Par. S. Mth. Bll. 21 (1893) 109-. ,3, condition for being in straight line. Terquem, O. N. A. Mth. 4 (1845) 462-. ,4 (Euler) . Grunert, J. A . Grunert Arch. 26 (1856) 335-. , , coordinates and angles sub- tended at a 5th point, relation. Grunert, J. A. Grunert Arch. 1 (1841) 89-. , , equation connecting mutual distances. Muir, T. Edinb. Mth. S. P. 3 (1885) 34-. , , problem. Lowry, T. J. Des Moines Anal. 1 (1874) 146-. , , relation between distances. Ciamberlini, C. G. Mt. 31 (1893) 218-. ,5. Cayley, A. Ph. Mg. 29 (1865) 460-. ,6, and their areal product. Eeiss, M. Mth. A. 2 (1870) 397-. , relations between distances. Ovidio, E. d\ G. Mt. 9 (1871) 211-. in space, 3, coordinates, relations (Lagrange). Terquem, 0. N. A. Mth. 1 (1842) 387-. ,5, relation between distances. Salmon , G. QJ. Mth. 3 (1860) 282-. , , . Cayley, A. Mess. Mth. 18 (1889) 100-. , , . Muir, T. Edinb. E. S. P. 16 (1890) 86-. , , . Ciamberlini, C. G. Mt. 34 (1896) 279-. 428 6430 Analytical Geometry Points in space, 5, relation, general. Schu- mann, A. Z. Mth. Ps. 27 (1882) 368-. on a sphere, 5, distances. Antomari, X. N. A. Mth. 1 (1882) 462-. Polar distances of points. Cabreira, A . [1900] Lisb. J. Sc. Mth. 6 (1902) 70-. equations. Midy, . N. A. Mth. 4 (1845) 597-. reciprocation, multiple. Vdlyi, G. Mth. Termt. Ets. 16 (1898) 399-; Mth. Nt. B. Ung. 16 (1899) 50-. surfaces of point and of a plane, theory. Painvin, L. F. [1865] (xn) Lille S. Mm. 2 (1866) 345-. Porisms, algebraical analysis. Davies, T. S. Mathematician 1 (1845) 42-. , analysis for discovering. Babbage, C. [1820] Edinb. K. S. T. 9 (1823) 337-. Problems. Lubbock, J. W. Ph. Mg. 10 (1831) 82-. . Sagajto, A. Par. T. Nauk Sc. Pam. 7 (*1875) Art. _8, 28 pp. rch. 34-. . Delarive, L. Arch. Sc. Ps. Nt. 5 (1881) , solution of two. Tortolini, B. Tortolini A. 1 (1850) 377-. Projection of solid on three coordinate planes. Bottomley, J. Manch. Lt. Ph. S. P. 21 (1882) 188-. Projections on axes of coordinates. Baltzer, E. (ix) Leip. Mth. Ps. B. 25 (1873) 523-. Project! ve geometry and analytical represen- tation. Pasch, M. Mth. A. 30 (1887) 127-. , theorems. Plilcker, J. Crelle J. 10 (1833) 217-,293-; 11 (1834) 26-, 117-, 219-, 356-. Eadicals of form \/.r 2 - y*, Poncelet's geometric approximation. Leaute, H. Par. S. Mth. Bll. 8 (1880) 106-. Eadii vectores, associated, advantages in mathematical physics. Cauchy, A. L. C. E. 38 (1854) 67-. , and geometrical methods. Despeyrous, C. Toul. Mm. Ac. 3 (1871) 217-. Eational polynomial x m + ax m ~ l + ...+l, graphic calculation of value by new coordinate system. Stamm, E. Mil. I. Lomb. Ed. 1 (1864) 203-. Scalar and clinant coordinate geometry. Ellis, A.J. E. S. P. 10 (1859-60) 415-. radical loci, application. Ellis, A. J. E. S. P. 11 (1860-62) 141-, 393-. Signs in analytical geometry. Lucas, E. Mathesis 10 (1890) 5-. Similar figures, analytical treatment. Morley, F. N. Y. Am. Mth. S. Bll. 1 (1895) 235-. Spherical geometry, analytical. Mobius, A. F. Leip. Ab. Jablon. Gs. (1846) 45-. Stewart, Matthew, general theorems (1746). Davies, T. S. Edinb. E. S. T. 15 (1844) 573-. ,, ( ) corrected. Breton [de Champ], P. Liouv. J. Mth. 13 (1848) 281-. , , ( ) deductions. Bonsdorff, E. Helsingf. Ofv. 40 (1898) 36-. Elementary Geometry 6800 Stewart, Matthew, general theorems (1746) demonstrations. Small, E. [1785] Edinb E. S. T. 2 (1790) 112-. . , ( ) - Babbage, C. QJ. Sc. 1 (1816) 6-. Stigmatics. Ellis, A.J. B. A. Ep. 34 (1864) (Sect.) 2 ; E. S. P. 14 (1865) 176- ; 15 (1867) 192. Straight line, formulae. Cayley, A. QJ. Mth. 15 (1878) 169-. and plane, analytical theory. Bret, . Gergonne A. Mth. 5 (1814-15) 329-. , expressions in theory of. Moth, F. X. Baumgartner Z. 4 (1828) 288-; 5 (1829) 17-. lines, imaginary, contained in every equa- tion of first degree. Jimenez Eueda, C. Fschr. Mth. (1893-94) 1104. in space, algebraic representation. Spottisivoode, S. C. E. 76 (1873) 1189-. , equations derived directly from points. Eogner, J. Arch. Mth. Ps. 42 (1864) 95-. Surfaces. Dietrich, M. Arch. Mth. Ps. 44 (1865) 200-. Symbols and undetermined coefficients, use, and new geometric principle. Plilcker, J. Crelle J. 5 (1829) 268-. Symmetrical relations. Boklen, 0. Grunert Arch. 38 (1862) 198-. Synopsis of analytical geometry. Spina, P. Mex. S. "Alzate" Mm. 7 (1893) 57-. Triangle, problem. Grunert, J. A. Grunert Arch. 28 (1857) 344-. in space, area, determinant value. Dostor, G. Arch. Mth. Ps. 58 (1876) 289-. Elementary Geometry. 6800 General. Gruson, J. P. Berl. Ab. (1814-15) (Mth.) 42-. Quetelet, L. A. J. Quetelet Cor. Mth. 1 (1825) 80-. Gudermdnn, C. Crelle J. 6 (1830) 303-. Gruson, J. P. Crelle J. 10 (1833) 275-. Noel, J. N. Liege Mm. S. Sc. 2 (1845-46) 493- ; 10 (1855) 461-. Wolf, E. Grunert Arch. 7 (1846) 440-. Smith, H. J. S. Ashmol. S. T. 2 (1852) 26 pp. Csdnyi, D. [1863] (xn) Mag. Ak. Ets. (Mth. Term.) 5 (1865) 292-. Hoiiel, J. Grunert Arch. 40 (1863) 171-. Analogies of geometry of plane and space. Serret. P. N. A. Mth. 4 (1865) 145-, 193-, 433-. Analogy in geometry. Noel, J. N. Liege Mm. S. Sc. 1 (1843) 1-. Anaritius, commentary of, on Euclid's ele- ments. Mansion, . Brux. S. Sc. A. 24 (1900) (Pt. 1) 47-. Angles, notation. Peirce, J. M. Camb. (M.) Mth. M. 1 (1859) 168-. Angular geometry. Shortrede, E. Beng. J. As. S. 11 (1842) 240-. 429 6800 Elementary Geometry. General 6800 Approximate square roots, methods of Greek geometers for. Buzengeiger, K. H. J. Lindenau Z. 5 (1818) 85-. Area of a surface, definition. Lebesgue, H. G. B. 129 (1899) 870-. Circle, division, apparatus. Anon. Gilbert A. 65 (*1820) 329-. , , (Anon.). Liebherr, J. Gilbert A. 67 (1821) 109-. , , (Liebherr). Eeichenbach, G. E. von. Gilbert A. 68 (1821) 33-. , , (Eeichenbach). Liebherr, J. Gilbert A. 69 (1821) 320-. and sphere. Affolter, F. G. Arch. Mth. Ps. 57 (1875) 1-. , sphere, square, and equilateral triangle. Utting, J. Tilloch Ph. Mg. 59 (1822) 102-. Circles, systems, inscribed in polygons, and spheres in solids. Lotteri, A. Brugnatelli G. 6 (1823) 115-, 361-. Concepts and determinations. Sylvester, J. J. L. Mth. S. P. 16 (1884-85) 201-. Converses of some theorems. Leudesdorf, C. Mess. Mth. 19 (1890) 14-. Demonstrations. Hennessy, J. P. Ph. Mg. 4 (1852) 417-. . Juel, C. S. (xn) Ts. Mth. 2 (1878) Determinants, geometrical application. White, H. S. A. Mth. 1 (1900) 103-. Electrostatic geometrical problem. Malagoli, R. Ev. Sc.-Ind. 29 (1897) 175-, 260-. Equipollences calculus, application to problem. Fran^oise, E. N. A. Mth. 9 (1870) 66-. Euclid and teaching of geometry. Wilson, J. M. G. Mt. 6 (1868) 361-. . Magnus, P. Educ. Times 33 (*1880) 81-. Euclidean teaching, improvement. Walmsley, J. Educ. Times 33 (*1880) 273-. Exercises, geometrical. Riccardi, P. Mod. Ac. Sc. Mm. 16 (1875) 3- ; 17 (1877) 3-. Figures, geometrical, conditions necessary. Catania, S. Ev. Mt. 3 (1893) 101-. , , , number determining. Eraser, A. Y. Edinb. Mth. S. P. 3 (1885) 22-. Forms, geometrical, theory. Battaglini, G. Nap. Ed. 1 (1862) 220- ; G. Mt. 1 (1863) 1-, 41-, 97-, 161-, 227-. Geometrical constructions by compasses only. Delisle, G. N. A. Mth. 19 (1860) 35-. . Colette, J. N. A. Mth. 8 (1889) 512. . Scliatunovsky , S. Fschr. Mth. (1891) 577. . Cesdro, G. Liege S. Sc. Mm. 1 (1899) No. 6, 11 pp. evolution. Degen, C. F. Kiob. Dn. Vd. Selsk. Skr. 2 (1801-2) 217-. Geometrography. Lemoine, E. As. Fr. C. E. (1892) (Pt. 2) 36- ; (1893) (Ft. 2) 113- ; N. A. Mth. 1-3 (1894) 155- ; Edinb. Mth. S. P. 13 (1895) 19-. . Balbin, V. Arg. S. Ci. A. 44 (1897) 110- ; Fschr. Mth. (1897) 439. Geometrography, application to anharmonic ratio. Lemoine, E. As. Fr. C. B. (1894) (Pt. 2) 55-. , comparison of solutions of a problem. Lemoine, E. As. Fr. C. B. (1899) (Pt. 2) 102- ; (1900) (Pt. 2) 74-. of Euclid's problems. Mackay, J. S. Edinb. Mth. S. P. 12 (1894) 2-. and measure of simplicity in construction. Lemoine, E. As. Fr. C. B. (1888) (Pt. 2) 75-; C. E. 107 (1888) 169- ; Mathesis 8 (1888) 217-, 241-. Geometry, applied, propositions. Noel, J. N. Liege Mm. S. Sc. 1 (1843) 208-. with constant radius circle, ancient. Kutta, M. Bb. Mth. (1896) 16. , elementary, as hypothetical deductive sys- tem. Fieri, M. [1899] Tor. Ac. Sc. Mm. 49 (1900) 173-. , , logic of. Adamson, J. (vi Adds.) S. Afr. QJ. 2 (1835) 353-. , , new trend. Ciamberlini, C. Bv. Sc.- Ind. 30 (1898) 99-, 113-, 153-. , modern, elementary exposition. Lentheric, J. [1865-75] Mntp. Mm. Ac. Sect. Sc. 6 (1864-66) (PV.) 35- ; 8 (1872-75) 507-. , note. Cesdro, E. (xn) Mathesis 3 (1883) 73-, 97-, 121-, 144, 176. , notes, miscellaneous. Matzka, W. Grunert Arch. 11 (1848) 432-. , , . Bischqff, J. N. [1862] Crelle J. 61 (1863) 369-. , , . Mackay, J. S. Edinb. Mth. S. P. 3 (1885) 38-. , , . Neuberg, J. As. Fr. C. E. (1894) (Pt. 2) 261-. of the ruler. Brianchon, C. J. Par. EC. Pol. Cor. 2 (1809-13) 383-. . Grunert, J. A. Grunert Arch. 27 (1856) 47-. , metric relations. Mobius, A. F. Crelle J. 4 (1829) 101-. , problems, analytical solution. Gergonne, J. D. Gergonne A. Mth. 7 (1816-17) 325-. Homogeneous body, geometrical concept. Popoff, A. Erman Arch. Bs. 14 (1855) 224-. Imaginary intersections, geometrical repre- sentation. Suhle, . Arch. Mth. Ps. 17 (1900) 244-. points, lines and planes, linear construction for joins and intersections. Grilnwald, J". Z. Mth. Ps. 45 (1900) 10-. Inequalities, geometrical significance. Tsinger, V. Y. [1873] (xn) Bee. Mth. (Moscou) 7 (1874-75) (Pt. 1) 418 [408]- ; (xi) Bll. Sc. Mth. As. 10 (1876) 100-. Infinitesimals. Gergonne, J. D. Gergonne A. Mth. 5 (1814-15) 183-. Lemmas and propositions. Adams, Jas. Thomson A. Ph. 11 (1818) 36-. Line lying outside convex arc is longer than it. Uhde, A. Grunert Arch. 26 (1856) 43-. Locus, class. Euex, P. (xn) Mathesis 1 (1881) 53-. Mansson, Peder, three small treatises attributed to. Enestrom, G. Bb. Mth. (1888) 17-. 430 6800 Elementary Geometry Maxima and minima, geometrical methods. Genese, R. W. Mess. Mth. 5 (1871) 122-. . , problem. Chartres, R. Nt. 37 (1888) 320. Mean proportional. Grunert, J. A. Grunert Arch. 31 (1858) 476-. values, theorems. Crofton, M. W. L.Mth. S. P. 8 (1877) 304-. Mensuration in one, two and three dimensions. Kummell, C. H. A. Mth. 4 (1888) 129-. Organic bodies, representation. Vieth, G. U.A. Gilbert A. 53 (1816) 225-. Pappus on progressions. Mackay, J. S. Edinb. Mth. S. P. 6 (1888) 48-. Polyhedra and polygons, mensuration. Lhui- lier, S. Bb. Un. 18 (1821) 85-. , theorem by Carnot, correction. Griison, J. P. [1826] Berl. Ab. (1831) 97-. Problem. Jiirgensen, C. Mth. Ts. 1 (1859) 81-. Problems. Steichen, . Quetelet Cor. Mth. 6 (1830) 302-. , Euclid's. Breton \de Champ] , P. Liouv. J. Mth. 20 (1855) 209-. , , solution with ruler and circle of fixed radius, by Italian geometers in 16th century. Mackay, J. S. Edinb. Mth. S. P. 5 (1887) 2-. , method, singular, for solution of many. Sixtel, W. Fschr. Mth. (1895) j>61. , solution. Brianchon, C. J J. 10 e cah. (1810) 1-. , by calculation. Heller, H. J. Grunert Arch. 34 (1860) 6-. , ruler and compass. Petersen, J. P. C. Planimetry; Circles 6810 Theorems, method for discovering. Magnus, L. J. Crelle J. 8 (1832) 51-. 6810 Planimetry; straight lines, and circles. Adams's hexagons and circles. Mackay, J. S. Edinb. Mth. S. P. 11 (1893) 104-. ' Angles, conterminal. Bushell, W. D. [1873] Mess. Mth. 3 (1874) 12-. , curvilinear. Miquel, A. Liouv. J. Mth. 9 (1844) 20-; 10 (1845) 347-; 11 (1846) 65. Apollonius, three problems. Chasles, M. Mathematician 3 (1850) 201-. Apparatus for drawing circles and other curves. Cerebotani, L. Em. N. Line. At. 46 (1893) 11-, 86-. Arc of very large radius, construction by two given tangents. Breton, Ph. Les Mondes (1880) 73-. Arcs, wide, parabolic or circular, tracing. Leclert, E. Les Mondes 25 (1871) 18-. Bisection, repeated. Bouniakowsky , V. [1866] St Pet. Ac. Sc. Bll. 11 (1867) 97-. CIECLES. (See also 2880.) (xn) Ts. Mth. 4 (1874) 97-. solvable by ruler and compass, criteria for recognition. Wantzel, L. Liouv. J. Mth. 2 (1837) 366-. , treatment. Mensing, . Grunert Arch. 2 (1842) 341-. , two. Geiser, C. F. [1866] Crelle J. 67 (1867) 78-. Puzzle, geometrical. Darwin, G. H. Mess. Mth. 6 (1877) 87. Kates method, application to geometry. John- son, W. W. Mess. Mth. 4 (1875) 72-. Statical proofs in geometry. Brassinne, E. Toul. Mm. Ac. 2 (1858) 144-. of some theorems. Alison, J. Edinb. Mth. S. P. 4 (1886) 58-. Symbolical calculation with geometrical rela- tions. Eeye, T. Mth. A. 43 (1893) 145-. Symmetrical figures. Mobius, A, F. Leip. B. (1851) 19-. , condition of equality. Weill, G. C. E. 105 (1887) 1237-. , successive. Neuberg, J. As. Fr. C. E. (1890) (Pt. 2) 18-. Symmetry. Fiedler, W. Ziir. Vjschr. 21 (1876) 50-. , its place in elementary geometry. Redard, F. U. Neuch. S. Sc. Bll. 14 (1884) 69-. , plane and in space. Dostor, G. J. N. A. Mth. 6 (1847) 166-. Theorem of Euler in elementary geometry. Fuortes, T. G. Mt. 7 (1869) 378. Crelle, A. L. Crelle J. 14 (1835) 66-. Par. EC. Pol. Mehmke, R. Z. Mth. Ps. 24 (1879) 257-. Bessell, F. Arch. Mth. Ps. 67 (1882) 44-. Tarry, G. As. Fr. C. E. (1891) (Pt. 2) 90-. Approximations, rational. Bankine, W. J. M. Ph. Mg. 29 (1865) 421-. Arc of given length, construction. Eankine, W. J. M. [1864-67] Ph. Mg. 29 (1865) 22- ; 34 (1867) 284-. , rectification. Dostor, G. J. N. A. Mth. 6 (1847) 287-. , resolution. Fuss, N. [1804] St Pet. Ac. Sc. Mm. 2 (1810) 45-. whose tangent is rational. Prouhet, E. Liouv. J. Mth. 1 (1856) 215-. Arcs, connecting two pairs of straight lines. Breton, Ph. Par. A. Pon. Chauss. 20 (1860) 135-. Area of segments. Grunert, J. A. Grunert Arch. 27 (1856) 94-. Chasles's circles. Barisien, E. N. Mathesis 15 (1895) 129-, 158-, 241- ; 16 (1896) 265-. . Droz Farny, A. Mathesis 16 (1896) 193-. , properties. Brocard, H. Mathesis 18 (1898) 61-. Chords through given point. Bilchner, (Dr) . Grunert Arch. 3 (1843) 388-. Circle cutting 2 given circles at given angles. Ponte Horta, F. da. G. Teix. J. Sc. 1 (1878) 180- ; 2 (1880) 7-. . [Schiappa] Monteiro, A. G. Teix. J. Sc. 2 (1880) 54-, 130-, 174- ; 5 (*1883) 143- ; 6 (1885) 103-. 3 circles at equal angles. Mannheim, A. N. A. Mth. 12 (1853) 113-. 431 6810 Apollonian Problem Circle cutting 3 circles isogonally. Wageningen, F. van. N. Arch. Wisk. 2 (*1876) 180-. at given angles. Griffiths, J. L. Mth. S. P. 3 (1869-71) 269- ; 5 (1873-74) 33- ; QJ. Mth. 11 (1871) 366-. derived from seven straight lines. Serret, P. C. B. 119 (1894) 474-, 493- ; 123 (1896) 442-. on diagonal of a rectangle as chord. Locus problem. Siacci,F. G. Arcad. 18 (1860) 166-. as figure of least perimeter for given area. Anon, (vi 525) Gergonne A. Mth. 4 (1813 -14) 338-. . Anon, (vi 545) Gergonne A. Mth. 13 (1822-23) 132-. . Anon, (vi 550) Gergonne A. Mth. 14 (1823-24) 316-. . Anon, (vi 555) Gergonne A. Mth. 15 (1824-25) 257-. , completion of Steiner's proof. Edler, F. Gott. Nr. (1882) 73-. orthotomic to three circles. Whiticorth, W. A. [1864] Mess. Mth. 3 (1866) 26-. through two points cutting another dia- metrally. Clausen, T. Grunert Arch. 15 (1850) 235-. Circles 6810 conjectured Apollonian solution. Flauti, V. Nap. At. Ac. 1 (1819) 71-. construction. Turrell, I. H. Des Moines Anal. 5 (1878) 166-. modern researches. Sqgajto, A. Par. T. Nauk Sc. Pam. 8 (*1876) Art. 6, 7 pp. nature of solutions. Muirhead, R. F. Edinb. Mth. S. P. 14 (1896) 135-. tangencies, problems, solution (new). Fergola, N. [1809] Nap. At. Ac. 1 (1819) 3-. Circle touching straight lines, equations. Cayley, A. N. A. Mth. 18 (1859) 222-. Circles touching a common circle. Allardice, R. E. [1900] Edinb. Mth. S. P. 19 (1901) each other in succession, radii of systems. Seitz, E. B. Des Moines Anal. 1 (1874) 173- ; 2 (1875) 74-. one circle, and orthogonal to another, system. Aubert, P. Edinb. Mth. S. P. 6 (1888) 33-. semicircles, locus of centres. Dostor, G. Arch. Mth. Ps. 66 (1881) 17-. three circles which touch each other. Dostor, G. Arch. Mth. Ps. 51 (1870) 191-. to replace a given curve within a finite . theorem (T 2 = id 1 d 3 ). space, practical tracing. Leaute, H. C. E. Turrell, I. H. Des Moines Anal. 1 (1874) 85 (1877) 1049-. 171-. of similitude of two circles. Nnggerath, given circles, centres and radii, to find. Prange, A. J. A. N. Arch. Wisk. 15 E. J. Grunert Arch. 33 (1859) 329-. Circle touching 3 given circles. (Apollonian problem.) Cauchy, A. L. Par. EC. Pol. Cor. 1 (1804-8) 193-. Gergonne, J. D. [1814-17] Tor. Mm. Ac. 22 (1816) 20*- ; Gergonne A. Mth. 7 (1816-17) 289-. Anon, (vi 547) Gergonne A. Mth. 13 (1822 -23) 193- . Badon Ghijben, J. (vn) Amst. N. Ws. Ntk. Vh. 1 (1844) 164-. Kerz, F. Grunert Arch. 24 (1855) 211- ; 26 (1856) 266-; 28 (1857) 402- ; 35 (1860) 121-. Grunert, J. A. Grunert Arch. 28 (1857) 354-. Kurz, A. Grunert Arch. 37 (1861) 347-. Serret, P. N. A. Mth. 2 (1863) 95-. Barbier, E. N. A. Mth. 4 (1865) 313- ; C. B. 60 (1865)1076-. Stott, . Mth. A. 6 (1873) 613-. Cayley, . QJ. Mth. 25 (1891) 104-. Hioux, V. N. A. Mth. 10 (1891) 399-. Langley, E. M. Mth. Gz. No. 5 (1895) 41. Study, E. Mth. A. 49 (1897) 497-. analytical results. Crelle, A. L. Berl. Ab. (1841) 65- ; Berl. B. (1841) 162-. solution. Strehlke, F. Crelle J. 12 (1834) 359-. . Cayley, A. Ph. Mg. 13 (1857) 507-. . Mertens, F. Z. Mth. Ps. 21 (1876) 443-. Casey's equation and theorem. Cayley, A. QJ. Mth. 8 (1867) 334- ; A. Mt. 1 (1867-68) 132-. (1888) 165- ; Fschr. Mth. (1888) 731. or cutting them at given angle. Fouche, H. N. A. Mth. 11 (1892) 227-, 331-, 404-. , system treated by trilinear coordinates. Caldarera, F. Palermo Cir. Mt. Bd. 4 (1890) 43-. , (Caldarera). Albeggiani, M. L. Palermo Cir. Mt. Bd. 4 (1890) 50-. , 2, relation between. Cayley, A. QJ. Mth. 11 (1871) 82-. , 3, cutting at given angles, system. Cayley, A. Mess. Mth. 17 (1888) 60-. , , , of equations. Cayley, A. Mess. Mth. 17 (1888) 18-. , , mutually tangent, theorems. Muir, T. Edinb. Mth. S. P. 3 (1885) 119-. , , system, deduction from properties. Franklin, F. [1880] (xn) J. H. Un. Cir. [1] (1882) 85. , , with three common tangents, construc- tion. Dewall, (Oberst) von. Arch. Mth. Ps. 41 (1864) 139- ; 42 (1864) 65-, 80-. , 4, in contact, radii. Baur, C. W. Schlo- milch Z. 5 (1860) 365-. , , in piano, and four powers relative to a point in plane, theorem. Lucas, E. N. A. Mth. 15 (1876) 205-. (Lucas). Cassani, P. G. Mt. 14 (1876) 347-. , pairs, through six common points of 3 circles. Griffiths, J. QJ. Mth. 9 (1868) 358- ; 10 (1870) 230-. , 8, touching 3. Casey, J. QJ. Mth. 5 (1862) 318-. 432 6810 Planimetry ; Circles Circumference, descriptive, segmentary, or metrical properties. Monchvt, . C. E. 105 (1887) 602-. Circum-quadrilateral. Steiner, J. Crelle J. 32 (1846) 305-. Coaxal circles. Casey, J. [1860] QJ. Mth. 5 (1862) 43-, 118-. . Schober,K. Mh. Mth. Ps. 5 (1894) 104-. , locus connected with. Holgate, T. F. N. Y. Am. Mth. S. Bll. 4 (1898) 63-; 5 (1899) 135-. , radical axes. Quidde, A. Grunert Arch. 23 (1854) 130-. , . Hopkins, G. H. [1865] Mess. Mth. 3 (1866) 122. , of certain circles in triangle. Tucker, R. [1892] Edinb. Mth. S. P. 11 (1893) 57-. , , circle involved in extended defini- tion of. Mack, L. Arch. Mth. Ps. 62 (1878) 405-. , , line parallel to. Rosters, . Grunert Arch. 19 (1852) 1-. , systems. Lachlan, R. QJ. Mth. 26 (1893) 129-. Concentric circles, two sets of three, having same common difference of radii. Tail, P. G. Edinb. E. S. P. 9 (1878) 533-. , systems, property. Brocard, H. Mathesis 4 (1884) 219-. Condition that three circles should meet in a point. Enneper, A. Z. Mth. Ps. 16 (1871) 257-. Conjugate circles. Mobius, A. F. Leip. B. 10 (1858) 1-. , groups. Cunningliam, (Lt.-Col.) A. B. A. Ep. (1892) 647. Constant ratio circle. Rump, F. H. Grunert Arch. 27 (1856) 30-. Contacts. Paucker, M. G. von. [1827] St Pet. Mm. Sav. Etr. 1 (1831) 503-. and intersections. Steiner, J. Crelle J. 1 (1826) 161-, 252-. . Plilcker, J. Gergonne A. Mth. 18 (1827-28) 29-. of systems of circles. Larmor, A. L. Mth. S. P. 23 (1892) 135-. Contraction caused by arranging circles of different diameters in rows. Kleiber, J. A. Fschr. Ps. (1890) (Ab. 1) 461-. Coplanar circles, properties of triangle formed by. Lachlan, R. QJ. Mth. 21 (1886) 1-. Diameter, to calculate from three contiguous chords. Voll, W. Oken Isis (1826) 670. Division. Gauss, C. F. Par. S. Phlm. Bll. 3 (1802) 102-. . Utenhore, J. M. C. van. Amst. Vh. 4 (1818) 92-. . Valat, . Bordeaux Act. Ac. Sc. (1839) 433-. . Eisenstein, G. Crelle J. 27 (1844) 269-. . Rammelman-Elsevier, W. J. C. (vm) Amst. N. Ws. Ntk. Vh. 1 (1844) 127-. . Kilpper, C. Grunert Arch. 27 (1856) 62-. . Wenckebach, E. Gravenh. I. Ing. Ts. (1870-71) 79-. . Carey, F. S. QJ. Mth. 26 (1893) 322-. 6810 Division into areas of given ratio and same peri- meter. Gergonne, J. D. Gergonne A. Mth. 6 (1815-16) 57-. by compasses. Pfeil, L. von. Arch. Mth. Ps. 41 (1864) 153-. . [Schiappa] Monteiro, A. G. Teix. J. Sc. 4 (1883) 39-. , Gauss's theorems. Jung, G. G. Mt. 6 (1868) 67-. , irreducible equation. Arndt, F. Crelle J. 56 (1859) 178-. , new method. Pellet, A. E. C. E. 93 (1881) 838-. , practical. Housel, . N. A. Mth. 12 (1853) 77-. , . Tempier, . N. A. Mth. 12 (1853) 345-; 13 (1854) 295. , problem. Arndt, F. Grunert Arch. 18 (1852) 461-. , series for. Huguenin, U. Amst. Vh. 5 (1820) 43-. into 5 equal parts. Barbour, L. G. Des Moines Anal. 5 (1878) 180-. 17 parts. A. B. (vi Adds.) Tilloch Ph. Mg. 57 (1821) 172-. , etc. Gruson, J. P. Berl. Ab. (1812-13) (Mth.} 15-. 66537 equal parts. Hermes, J. Gott. Nr. (1894) 170-. Equations. Casey, J. [1878] E. S. P. 27 (1878) 417- ; Ir. Ac. T. 26 (1879) 527-. Faisceaux. Ribaucour, A. C. E. 76 (1873) 830-. Figures composed of circles, examination with a special system of coordinates. Crone, C. (xn) Ts. Mth. 6 (1876) 97-. Grazing of two circles, best value of radii. Fontes, . Toul. Ac. Sc. Mm. 4 (1892) 375-. Groups of circles, certain. Vries, J. de. Amst. Ak. Vs. 6 (1898) 418-. Homocyclic points and circles in collineation, etc. Serret, P. C. E. 123 (1896) 396-. Imaginary circles. Mfibius, A. F. Leip. B. 9 (1857) 38-. Intersections. Gufflet, G. N. A. Mth. 5 (1846) 352-. Inversion, point and tangential. Coelingh, D. N. Arch. Wisk. 16 (1889) 116- ; Fschr. Mth. (1889) 614-. Joachhnsthal's circle. Longchamps, G. de. Mathesis 9 (1889) 153-. Minimum circle enclosing n given points. Chrystal, G. Edinb. Mth. S. P. 3 (1885) 30-. Orthocentroidal circle. Prime, (Mme.) . Mathesis 13 (1893) 33-. Orthogonal systems. Lachlan, R. Mess. Mth. 16 (1887) 98-. trajectories of certain systems. Roberts, R. A. Mess. Mth. 14 (1885) 134-. Plane system, coordinate. Schmite, P. H. Wien Ak. Sb. 94 (1887) (Ab. 2) 786-. , and its representation in space. Thomae, J. Z. Mth. Ps. 29 (1884) 284-. Polar system for circles. Meyer, T. Arch. Mth. Ps. 9 (1890) 18-, vn-. theorems, demonstrations. Zeuthen, H. G. (xn) Ts. Mth. 6 (1882) 75-. VOL. 433 6810 Planimetry. Quadrature of Circle 6810 Polars of points with respect to two circles, theorem. Ploix, E. N. A. Mth. 8 (1849) 152-. . , . Gentil, . N. A. Mth. 8 (1849) 290-. Pothenot's problem. Vieth, G. U. A. Gilbert A. 54 (1816) 311-. . Grunert, J. A. Arch. Mth. Ps. 44 (1865) 184-. . Ldska, V. Prag Sb. (1895) (Mth.-Nt.) No. 17, 5 pp. , algebraic aspect. ScM&milch, 0. Z. Mth. Ps. 9 (1864) 433-. , analytical solution. Grunert, J. A. Grunert Arch. 1 (1841) 446-. , . Fellini, D. Tor. Ac. Sc. At. 32 (1896) 214- or 320-. , theory of errors in. Andrce, . As. Nr. 47 (1858) 193-. Properties. Babbage, C. QJ. Sc. 1 (1816) 6-. . Laguerre, E. N. A. Mth. 2 (1883) 65-. , Grassmann's " Ausdehnungslehre " applied to. Cox, H. QJ. Mth. 25 (1891) 1-. Property. August, E. F. Crelle J. 17 (1837) 387-. . Grunert, J. A. Grunert Arch. 1 (1841) 440- ; 3 (1843) 259-; 25 (1855) 231-. (Grunert). G'opel, A. Grunert Arch. 3 (1843) 403-. . Darboux, G. C. R. 94 (1882) 1108-. , mechanical. Rispal, A. Liouv. J. Mth. 12 (1847) 225-. Ptolemy's and Feuerbach's relations. Krah6, A. Mathesis 18 (1898) 83-. problem, application to astronomy. Hill, G. W. [1900] As. J. 21 (1901) 33-. theorem. Oppermann, L. H. F. (xn) Ts. Mth. 1 (1871) 55-, 124-. , converse. Gerwien, . Crelle J. 11 (1834) 264-. , demonstration. Schnell, (Dr.) . Arch. Mth. Ps. 67 (1882) 225-. , inversion. Forstemann, W. A. Crelle J. 13 (1835) 233-. , . Strehlke, F. Grunert Arch. 2 (1842) 323-. leads to Wallis's and others. Fergola, N. Nap. At. Ac. 1 (1819) 205-. Quadrature of the Circle. Delanges, P'. At. Ac. It. 1 (1810) 157-. Anon, (vi 215) Bb. Un. 3 (1816) 221-. Anon, (vi 1171) Thomson A. Ph. 8 (1816) 13-. Bitnicz, L. Evk. 2 (1832) 151-. Lambert, C. J. N. A. Mth. 12 (1853) 298-. Scheffler, H. Arch. Mth. Ps. 44 (1865) 84-. Pelndf, M. Casopis 6 (*1877) 89-; Fschr. Mth. (*1877) 406-. approximate construction. Boswell, J. W. Nicholson J. 10 (1805) 151- ; 13 (1806) (Boswell). Gough, J. Nicholson J. 11 (1805) 29-. . Willich, C. M. R. S. P. 7 (1854-55) 379. approximate method. Anon, (n 312) Crelle J. 32 (1846) 91-. approximations. Anon, (vi 275) Camb. and Dubl. Mth. J. 7 (1852) App. 1-. . Oiase, P. E. Franklin I. J. 78 (1879) 45, 105 ; 79 (1880) 409- ; 81 (1881) 379. arithmetical. Hill, C. J. D. Stockh. Ak. Hndl. (1854) 405-. elementary. Schlomilch, 0. Grunert Arch. 14 (1850) 101-. Franco of Liege's "De Quadratura Circuli." Winterberg, C. Z. Mth. Ps. 27 (1882) (Sup2>l.) 135-. historical notes, and proof of impossibility. Studnicka, F. J. Casopis 1 (*1872) 35-; Fschr. Mth. (*1872) 254. , . Radio, F. Ziir. Vjschr. 35 (1890) 1-. impossibility. Caluso, (Vabbe) T. V. de. [1801] Mod. Mm. S. It. 9 (1802) 558-. . Rouche, E. N. A. Mth. 2 (1883) 5-. and rectification. Ruffini, P. [1801] Mod. S. It. Mm. 9 (1802) 527-. . Grunert, J.A. Grunert Arch. 6 (1845) 90-. , approximate. Specht, C. G. Crelle J. 3 (1828) 83, 405- ; N. A. Mth. 4 (1845) 495-. , graphic method. Schramm, A . Dingier 226 (1877) 17-. impossible. Snart, J. Tilloch Ph. Mg. 62 (1823) 338-. Viete (Francois) and his work. Ritter, F. As. Fr. C. R. (1879) 143-. Radical and anti-radical circles. Durdn Loriga, J. J. G. Teix. J. Sc. 13 (1897) 33- ; Fschr. Mth. (1896) 398. circles. Durdn Loriga, J. J. G. Teix. J. Sc. 12 (1895) 97-; Mathesis 16 (1896) 105- ; Arch. Mth. Ps. 15 (1897) 117-, 232-. Ratio of Diameter to Circumference. (See also 2920.) Lefaucheux, . [1811] Le Mans S. R. Tr. (1820) 19-. Gergonne, J. D. Gergonne A. Mth. 6 (1815-16) 192- ; 17 (1826-27) 148-. Leger, E. Quetelet Cor. Mth. 9 (1837) 479-. Lehmann, J. W. H. Grunert Arch. 21 (1853) 121-. Didion, [(le gen.) I.]. C. R. 74 (1872) 36-; (xn) Metz Ac. Mm. 53 (1873) 257-. (Didion.) Catalan, E. C. C. R. 74 (1872) 177. Lindemann, C. L. F. C. R. 95 (1882) 72-. Sang, E. Edinb. R. S. P. 15 (1889) 348-. Approximate construction. Grassmann, H. Arch. Mth. Ps. 49 (1869) 1-. Approximation. Herrifield, C. W. Mess. Mth. 3 (1866) 105. Calculation, and circular series. Ma-zzola, G. G. Mt. 2 (1864) 92-, 110-. to 8 decimal places. Drach, S. M. Ph. Mg. 27 (1864) 318-. 200 . Dahse [or Dase], Z. Crelle J. 27 (1844) 198; N. A. Mth. 9 (1850) 12-; 10 (1851) 198. 434 6810 Planimetry ; Circles 6810 Calculation to 200 decimal places (Dase). Schulz van Strasznitzki, L. C. Crelle J. 27 (1844) 198. 250 . Clausen, T. As. Nr. 25 (1847) 207-. 350 . Rutherford, W. N. A. Mth. 14 (1855) 209-. 400 . Eichter, (Prof.) . Grunert Arch. 22 (1854) 473. . Strehlke, F. Grunert Arch. 23 (1854) 475-; 24 (1855) 472. 500 . Eichter, (Prof.) . Grunert Arch. 25 (1855) 471-. 707 . Shanks, W. E. S. P. 21 (1873) 318-. , elementary. Grunert, J. A. Arch. Mth. Ps. 46 (1866) 345-. , , description. Matzka, W. Grunert Arch. 9 (1847) 74-. by logarithms. Chevillier, I. N. A. Mth. 15 (1856) 261-. Determination by equivalent surfaces. Jube, E. N. A. Mth. 7 (1848) 366-. , experimental. Hall, A. [1872] Mess. Mth. 2 (1873) 113-. by isoperimeters. Jube, E. N. A. Mth. 5 (1846) 42-. . Andre, J). N. A. Mth. 13 (1874) 128-. , numerical. Girault, . Eouen Tr. Ac. (1850-51) 63-. by perimeters. Huet, . N. A. Mth. 4 (1845) 156-. . Fouche, M. Par. S. Mth. Bll. 18 (1890) 135-. , Ptolemy's method. Terquem, O. N. A. Mth. 13 (1854) 253-. Dirichlet's formula, proof, etc. Lebesgue, V. A. Liouv. J. Mth. 11 (1846) 76-. Discrepancies in published value of IT. Shanks, W. [1873] E. S. P. 22 (1873-74) 45-. Expression by rapidly convergent series. Lucas, F. C. E. 112 (1891) 1050-. Numerical values of first 12 powers and their reciprocals, etc. Glaisher, J. W. L. L. Mth. S. P. 8 (1876-77) 140-. Powers of w. Broch, 0. J. Sk. Nf . F. 8 (1860) 571-. Eeciprocal of w, calculation to 208 decimal places. Rutherford, W. Phil. Trans. (1841) 281-. Series for w, and general theorem. Gasparis, A.de. Nap. Ed. 7 (1848) 269-. , new. Thomson, (Prof.) James. [1836- 38] E. S. P. 3 (1836) 417- ; Edinb. E. S. T. 14 (1840) 217-. Theorem, of which Wallis's theorem is a case. Grunert, J. A . Grunert Arch. 38 (1862) 466-. Van Ceulen's 35-decimal value and some of his works. Elevens de Haan, D. [1873] (vn) Mess. Mth. 3 (1874) 24-. Wallis's and Brouncker's expressions, ana- lytical investigation. Plana, G. Crelle J. 17 (1837) 1-, 163-, 338-. formula. Collignon, E. Par. EC. Pol. J. 47 cah. (1880) 103-. for 71- , and the area of the curve y = sin k x. Mansion, P. Mathesis 20 (1900) 183-. Wallis's fractions, analysis. Kramp, C. Erfurt N. Acta 1 (1799) 257-. Eectification, approximate. Gergonne, J. D. Gergonne A. Mth. 8 (1817-18) 245-. , new method. Srfttek, J. Casopis 21 (1892) 83-; Fschr. Mth. (1892) 530-. Segment, calculation. Grunert, J. A. Arch. Mth. Ps. 44 (1865) 363-. containing given angle, instrument for de- scribing. Rollet de I'Isle, . Ev. Mar. et Col. 128 (1896) 90-. Segments, Greek approximate formulae (Hero). Tannery, P. Bordeaux S. Sc. Mm. 5 (1883) 347-. Self-polar circle. Grunert, J. A. Arch. Mth. Ps. 41 (1864) 132-. Semicircles, Apollonius's problem. Sangro, G. Nap. At. Ac. 2 (1825) 41-. Shoemaker's knife (a/>/37?Xos) . Mackay, J. S. Edinb. Mth. S. P. 3 (1885) 2-. Stewart's properties, demonstration. Strong, T. [1814] Conn. Mm. Ac. 1 (1816) 393-. Systems of circles. Cayley, A. QJ. Mth. 2 (1858) 83-. . Gehrke, J. N. Ts. Mth. 9 (B) (1898) 11-; Fschr. Mth. (1898) 436. , with application to Dupin's cyclides. Hoist, E. Christiania F. (1885) No. 15, 4 pp.; Fschr. Mth. (1885) 574-. , new method of representation. Fiedler, W. As. Fr. C. E. 10 (1881) 127-. Tangent circles. Guibal, . Nancy Mm. S. Sc. (1843) 67-. Tangents, trapeziums, etc. Hey, R. Phil. Trans. (1814) 348-. Theorem. Miquel, A. Liouv. J. Mth. 3 (1838) 485-. Theorems. Brioschi, F. N. A. Mth. 15 (1856) 462-. . Jefdbek, W. [1880] Arch. Mth. Ps. 66 (1881) 325-. Transformations. Larmor, A. B. A. Ep. (1887) 607. Circular billiards problem. Transon, (Prof.) A. N. A. Mth. 9 (1850) 340-. . Auric, . N. A. Mth. 13 (1894) 215-. Congruent figures, propositions. Petersen, J. N. Ts. Mth. 9 (B) (1898) 1-; Fschr. Mth. (1898) 431. , relative positions. Zeuthen, H. G. Mth. Ts. 5 (1863) 97-. Constructions. Smith, H. J. S. Camb. and Dubl. Mth. J. 7 (1852) 118-. , class. Feldblum, M. Wiad. Mt. 4 (1900) 1-. of 2nd degree, aids to. Adler, A. Wien Ak. Sb. 99 (1891) (Ab. 2a) 846-. founded on the circle. Study, . Bonn Niedr. Gs. Sb. (1897) 80-. , linear. Kirkman, T. P. Manch. Ph. S. Mm. 9 (1851) 279-. , Mascheroni's, theory. Adler, A. Wien Ak. Sb. 99 (1891) (Ab. 2a) 910-. in mensuration. Bangma, 0. S., & Keyser, J. F. Amst. Mengelwerk 2 (1816) 1-. plane and in space with the straight line only. Tilly, J. de. Mathesis 5 (1885) 124-. 435 EE2 6810 Planimetry; Straight Lines and Circles 6810 Curves, pedal and inversion. Fleuri, G. N. S. W. B. S. J. 26 (1892) 60-. on railways, compound. Emch, A. Kan. Un. Q. 5 (1896) 99-. , junction by two arcs of circles. Ocagne, M. d\ N. A. Mth. 17 314-. Diagonals of figures. Dassen, C. C. Arg. S. Ci. A. 42 (1896) 165-, 198-. Discontinuities, on removing certain. Rulf, W. Mh. Mth. Ps. 4 (1893) 173-. Distances, sum of squares of, theorem. Grunert, J. A. Grunert Arch. 20 (1853) 473-. Division of rhomboid. Fuss, N. [1799] St Pet. Ac. Sc. Mm. 3 (1811) 65-. triangles and quadrilaterals, formulae. Le Brun, . C. B. 38 (1854) 752. Euclid, first three books. Wedgwood, H. [1853] Camb. Ph. S. P. 1 (1866) 127-. i, 1 to 32, demonstration. Traill, D. Edinb. Mth. S. P. 2 (1884) 40-. i, 4, Bain's theory. Smith, W. R. [1870] Edinb. B. S. P. 7 (1872) 176-. i, 4, 5, 6, 26. Wolff, F. QJ. Mth. 7 (1866) 76-; 8 (1867) 301-. i, 5. Kendall, E. K. Cn. J. 3 (1858) 318-. i, 32. Fischer, W. Grunert Arch. 28 (1857) 365-. vi, 7. Wolff, J. F. (jun.) QJ. Mth. 11 (1871) 76. vi, 7, simpler demonstration. Walton, W. [1832] QJ. Mth. 9 (1868) 241-. x. Poselger, F. T. Berl. Ab. (1834) 341-. x, axiom, extension (continued subtraction of ^th). Miiller, G. W. Grunert Arch. 1 (1841) 211-. Extreme and mean ratio, expression "to divide a line in", etc. Vincent, A. J. H. N. A. Mth. 3 (1844) 5-. , " ". Terquem, 0. N. A. Mth. 12 (1853) 36-. , "golden section" in square and cube. Hagen, F. W. Dingier 140 (1856) , " "of straight line. Helmes, J. Grunert Arch. 4 (1844) 15-. Fermat's theorem of rectangle on diameter of semi-circle. Blindow, R. Grunert Arch. 31 (1858) 295-; 32 (1859) 124-. , demonstration. Heinen, F. Grunert Arch. 30 (1858) 246-. , generalisation. Krilger, A. Grunert Arch. 31 (1858) 61-. Figure composed of a triangle and parallelo- grams on its sides. Paraira, M. C. N. Arch. Wisk. 9 (*1882) 87-. the squares on its sides, properties. Mackay, J. S. Edinb. Mth S P. 6 (1888) 2-. , geometrical, study. Aubert, P. Edinb. Mth. S. P. 5 (1887) 23-. Figures variable in similar manner. Neuberq, J. L. Mth. S. P. 16 (1884-85) 184-. Graduation, method in which curves are repre- sented by straight lines. Kapteyn, A. P. Cuyper Bv. Un. 40 (1876) 136-. Harmonic centres. Caldarera, F. Palermo Cir. Mt. Bd. 1 (1887) 2-. Homology, centres of. Midy, . N. A. Mth. 3 (1844) 77-. Imaginary solutions in ordinary geometry. Tarry, G. As. Fr. C. B. (1892) (Ft. 2) 132-. Inversion. Liouville, J. N. A. Mth. 13 (1854) 227-. . Mannheim, A. L'l. 28 (1860) 413-. . Grassmann, H. E. Leip. Mth. Ps. B. 29 (1877) 133-. , application to surface envelope of sphere touching three spheres. Mannheim, A. N. A. Mth. 19 (1860) 67-. , circular, and Peaucellier cell. Baker, A. L. [1894] Bochester (N.Y.) Ac. Sc. P. 2 (1895) 256-. , demonstration of theorems by. Casey, J. [1864] QJ. Mth. 7 (1866) 378-. , new method. Coelingh, D. N. A. Mth. 7 (1888) 133-. , propositions obtained by. Bellavitis, G. Tortolini A. 4 (1853) 198-. Isobaric points in space, properties. Ferrari, F. G. Mt. 34 (1896) 73-; 35 (1897) 273-. Lengths, rational relations. Zeuthen, H. G. (xn) Ts. Mth. 6 (1882) 75-. Loci of points, sum of whose distances from given lines or planes is constant. Heis, E. Grunert Arch. 31 (1858) 228-. Locus of points from which two fixed lines subtend equal angles. Stammer, W. Arch. Mth. Ps. 68 (1882) 18-. Lunes, quadrature, fragment of Eudemus. Tannery, P. Bordeaux S. Sc. Mm. 5 (1883) 211-. , , and Hippocrates of Chio. Tannery, P. Bordeaux S. Sc. Mm. 2 (1878) 179-. Mean distance, centres, analytic proofs. Rochat, . Gergonne A. Mth. 3 (1812-13) 286-. , , applications of theory. Gerono, G. C. Gergonne A. Mth. 17 (1826-27) 330-. of field from homestead. Hill, C. J. D. Stockh. Ak. Hndl. (1849) 1-. 2 figures on a plane. Maxwell, J. C. Edinb. B. S. T. 26 (1872) 729-. intersections of variable straight line with three fixed lines, centre. Greiner, M. Arch. Mth. Ps. 69 (1883) 323-. lines from a point. Wetzig, F. Z. Mth. Ps. 11 (1866) 15-. point from system of points. Schlegel, . As. Fr. C. B. (1887) (Pt. 2) 266. line of system of lines in a plane. Ocagne, M. d>. Par. S. Mth. Bll. 12 (1884) 114-. position, centres and lines. Gorton, W. C.L. [1891] A. Mth. 6 (1891-92) 33-. proportionals, two, method of finding. Pendlebury, R. Mess. Mth. 2 (1873) 166-. Medians. Quetelet, E. Brux. Mm. Cour. 4, 25 (1851-53) 32 pp. 436 6810 Planimetry. Polygons Steiner, J. Berl. B. 6810 Minimum distance point. (1837) 144-. . Woldstedt, F. [1858] (vm) Helsingf . Acta 6 (1861) 33-. . Sturm, R. Crelle J. Mth. 97 (1884) 49-. . Ginger, V. Ja. [1891] Kec. Mth. (Moscou) 16 (1893) 317-. . Solovjev, N. M. Mosc. S. Sc. Bll. 93 (No. 1) (1897) 27-; Fschr. Mth. (1897) 441-. , properties. Seydewitz, F. Grunert Arch. 8 (1846) 174-. figures. Ocagne, M. d' . Par. S. Mth. Bll. 12 (1884) 168-. or maximum of sum of positive and negative squares of distances of point from three straight lines in plane. Wetzig, F. Z. Mth. Ps. 12 (1867) 281-. problem. Neovius, E. Gott. Nr. (1887) 407- . sum of distances from three points. Lind- man, C. F. Grunert Arch. 27 (1856) 295-. Mosaics, geometrical, algebraic representation. Mersch, C. F. Lux. I. R. Pb. 19 (1883) 153-. Pappus's theorem. Zahradnik, K. Arch. Mth. Ps. 17 (1900) 79-. Peaucellier cell. Peaucellier, A. N. A. Mth. 12 (1873) 71-. . Lemoine, E. As. Fr. C. B. (1874) 122- ; Par. Ing. Civ. Mm. (1875) 247-. Planimeter, Amsler's. Rateau, A. As. Fr. C. B. (1896) (Pt. 2) 130-. , , "no-rolling" curves. Morley, F. A. Mth. 1 (1900) 21-. , description of new. Fessenden, C. Cn. I. P. 5 (1888) 27-. , polar. Barbarin, P. N. A. Mth. 19 (1880) 212- ; 20 (1881) 266-. , , problem. Echols, W. H. [1889] A. Mth. 5 (1889-90) 9-. for rectangular coordinates. Schmidt, J. Giving. 26 (1880) 437-. Planimeters, report. Henrici, O. B. A. Rp. (1894) 496-. Planimetry, desiderata. Pfeil, L. von. Arch. Mth. Ps. 58 (1876) 369-. , theorems deduced from analogous linear theorems. Mobius, A. F. Leip. B. (1852) 41-. Points, complementary. Vigarie, E. Mathesis 7 (1887) 6-, 57-, 84-, 105-. . , . Ocagne, M. d'. Mathesis 8 (1888) 62-. , two unknown, found from two known. Clausen, T. As. Nr. 18 (1841) 367-. Polygons. (See also 7210.) Cauchy, A. L. [1812] Par. EC. Pol. J. 16" cah. (1813) 87-. Franchini, P. [1819] Lucca At. Ac. 1 (1821) 97-. Cauchy, A. L. C. R. 26 (1848) 489-. Allardice, R. E. Edinb. Mth. S. P. 5 (1887) 78-. Analytical principles. Cuyper,A.C.de. Li6ge Mm. S. Sc. 4 (1848-49) 387-. Analytical theorem. Walton, W. Camb. and Dubl. Mth. J. 5 (1850) 182-. Angles, sum. Liotard, . Gard Mm. Ac. (1847-48) 76-. , . Unferdinger, F. Wien Sb. 57 (1868) (46.2)627-. ' , . Steinhauser, A. Arch. Mth. Ps. 52 (1871) 294-. Areas. Hudson, E. C. Mess. Mth. 24 (1895) 171-. . Schur, F. Dorpat Sb. 10 (1895) 2-. , Mascheroni's theorem. Terquem, 0. N. A. Mth. 7 (1848) 348-. .problem. Ellis, R. L. (vi Adds.) Camb. Mth. J. 2 (1841) 142-. , and sum of angles. Spitzer, S. Haidinger B. 5 (1849) 156-. Circumscriptible polygons. Darboux, G. Bll. Sc. Mth. As. 3 (1879) 64-. , radius and surface. Dostor, G. N. A. Mth. 5 (1866) 73-. Classification. De Morgan, A. QJ. Mth. 2 (1858) 340-. Derived polygons, property. Rubini, R. (vm) A. Mt. 5 (1863) 5-. Determination from angles between sides and diagonals. Hoppe, R. Arch. Mth. Ps. 61 (1877) 439-. Division into equal areas. Oettingen, A. von. Dorpat Sb. 8 (1889) 262-. m-gons. Liouville, J. Liouv. J. Mth. 8 (1843) 391-. , number of ways. Grunert, J. 4. Grunert Arch. 1 (1841) 193-. jj-gons, number of ways. Rou-e, R. C., & Taylor, H. M. L. Mth. S. P. 13 (1881-82) 102-. triangles. Binet, J. P. M. Par. S. Phlm. PV. (1838) 127- ; Liouv. J. Mth. 4 (1839) 79-. . Catalan, E. C. Liouv. J. Mth. 4 (1839) 91-. . Faure, H. N. A. Mth. 17 (1858) 50-. , number of ways. Rodrigues, O. Liouv. J. Mth. 3 (1838) 547-. Equality. Nievengloski, G. H. N. A. Mth. 11 (1852) 462-. . Ponte-Horta, F. da. Lisb. J. Sc. Mth. 1 (1868) 6-. Equiangular and equilateral polygons. Hess, E. Marb. Schr. 10 (Ab. 12, 1874) 611-. . Allardice, R. E. Edinb. Mth. S. P. 5 (1887) 28-. Errors in calculating, estimation. Soulages, E. Arg. S. Ci. A. 36 (1893) 5-; Fschr. Mth. (1893-94) 1842. Even polygons, and odd, property. Allardice, R. E. Edinb. Mth. S. P. 8 (1890) 22-. , properties. Cotterill, T. L. Mth. S. P. 1 (1866) No. 5, 13-. Family formed with given polygon. Ibach, L. F. N. A. Mth. 2 (1883) 226-. Harmonic polygons. Tarry, G., d- Neuberg, J. As. Fr. C. R. (1886) (Pt. 2) 12-. . Casey, J. Mathesis 10 (1890) 96-. ; new method. Simmons, (Rev.) T. C. L. Mth. S. P. 18 (1886-87) 289-. 437 6810 Planimetry. Polygons 6810 Heptagon chord, Diirer's. Drach, S. M. (vn) Ph. Mg. 26 (1863) 408; 27 (1864) 320. , irregular, of Faulhaber. Gilnther, S. Erlang. Ps. Md. S. Sb. 6 (1874) 112-. Hexagons. Dandelin, G. P. Brux. Mm. Ac. Sc. 3 (1826) 1*-. Imaginaries in polygonometry. Spitzer, S. Haidinger B. 4 (1848) 96-. In- and circum-polygon of polygon of same number of sides, construction. Nawrath, . Arch. Mth. Ps. 50 (1869) 1-. polygons of circle. Casinelli, A. (s. L.) Bologna N. Cm. 2 (1836) 241-. . Bouniakmcsky, V. St Pet. Ac. Sc. Mm. 4 (1841) 423-. . Kiirschdk, J. Mth. Termt. fits. 5 (1887) 153- ; Mth. Nt. B. Ung. 5 (1886-87) 77-. polygon of circle. Glaisher, J. W. L. Mess. Mth. 9 (1880) 133. ' with sides passing through fixed points, solution, etc. Mendthal, . Arch. Mth. Ps. 59 (1876) 39-. given points, problem of Pappus. Malfatti, G. F. Verona S. It. Mm. 4 (1788) 201-. polygons of circle. Prouhet, E. N. A. Mth. 9 (1850) 130-. , complete, remarkable straight lines and points. Kantor, S. [1878] Wien Ak. Sb. 78 (1879) (Ab. 2) 193-. , radius and area in terms of sides. Mobius, A. F. Crelle J. 3 (1828) 5-. , remarkable points in. Overeem, M. van (jun.). Amst. Ak. Vh. (Sect. 1) 3 (1896) No. 7, 29 pp.; Fschr. Mth. (1895) 575. polygons. Gardiner, M. [1862] N. S. W. Ph. S. T. 1 (1866) 70-. . Certo, L. G. Mt. 26 (1888) 46-. Inscriptible polygons. Collignon, E, As. Fr. C. B. 9 (1880) 78-. Maximum area with given sides. Umpfenbach, (Dr) . Crelle J. 25 (1843) 184-. . Fasbender, . Crelle J. 26 (1843) 181-. . Molins, H. Toul. Mm. Ac. 2 (1852) 129-. Mensuration. Lhuilier, S. Bb. Un. 25 (1824) 169-. Number of polygons determined by n points taken as vertices. Guibert, A. Liouv. J. Mth. 4 (1839) 392. Pedal polygon of constant area, locus of points for. Combette, E. (xn) Brest S. Ac. Bll. 7 (1871) 1 (bis)-. point. Gardiner, M. [1862] (vn) N. S. W. Ph. S. T. 1 (1866) 63-. polygons, successive. Sctwtten, H. Arch. Mth. Ps. 13 (1895) 65-. Pentagon in and about circle. Grunert, J. A. Grunert Arch. 22 (1854) 357-. , Gauss's theorem, demonstration. Serret, P. N. A. Mth. 7 (1848) 28-. , plane. Clebsch, R.F.A. Mth. A. 4 (1871) 476-. Perimeter of a polygon, double points on. Brunei, G. Bordeaux S. Sc. Mm. 4 (1894) 273-. Perpendiculars on sides, given sum, problem. Mainardi, G. [1849] Mod. S. It. Mm. 25 J1852) 34-. ygon moving similar to itself with 3 sides passing through fixed points. B'oklen, O. Grunert Arch. 36 (1861) 22-. with sides passing through n fixed points, and divided by the points in the same given ratio. Lugli,A. Mil. I. Lomb. Bd. 14 (1881) 564-. Polygons in plane, problem. Moon,R. Camb. and Dubl. Mth. J. 5 (1850) 131-. , (Moon). Thomson, (Sir) W. Camb. and Dubl. Mth. J. 5 (1850) 137-. and quadrilaterals, new propositions. Sparer, B. Arch. Mth. Ps. 7 (1889) 389-. Problem. Berard, . Gergonne A. Mth. 7 (1816-17) 61-. Properties. Laisant, C. A. As. Fr. C. B. (1877) 142-. Regular polygons. Sturm, J. C. F. Gergonne A. Mth. 15 (1824- 25) 250-. Burg, A. Wien Jb. Pol. I. 11 (1827) 99-. Amiot, B. N. A. Mth. 3 (1844) 264-. Anon, (xi 9) A. Cond. Pon. Chauss. 12 (1868) (Pt. 1) 33-. Perron, E. Lux. I. B. Pb. 14 (1874) 36-. Mathet, G. Par. EC. Norm. A. 4 (1875) 303-. Zbikovskij, A. K. Kazan S. Nt. (Pn.-Mth.) P. 2 (1884) 40-. and angles, graphic representation. Schlomilch, O. Giving. 5 (1859) 219-. area. Cabreira, A. [1897] Lisb. J. Sc. Mth. 5 (1898) 7-; Fschr. Mth. (1897) 445. and circle. Lhuilier, S. Bb. Un. 36 (1827) 159-. circumscribed about a similar polygon. Kiist- ner, A. G. [1796] Erfurt N. Acta 1 (1799) 241-. construction. Steczkowski, J. K. Grunert Arch. 24 (1855) 311-. (Staudt-Schroter). A/olter, F. G. Mth. A. 6 (1873) 582-. constructions. Bonsdorff, E. [E. J.~] Helsingf. Acta 10 (1875) 457-. for three. Pascal, E. G. Mt. 25 (1887) 82-. convex and stellate, of same number of sides, property. Muir, T. [1873] (vin) Mess. Mth. 3 (1874) 47-. covering a plane with. Wiiiff-Petersen,A. Ts. Mth. 6 (1889) 182- ; Fschr. Mth. (1888) 524. equation, general. Nieuport, C. F. F. de. [1802] Par. Mm. Sav. Etr. 1 (1806) 659-. equations for sides and diagonals. Buys-Ballot, C. H. D. [1863] Amst. Vs. Ak. 16 (1864) 293-; Arch. Neerl. 1 (1866) 97-; Grunert Arch. 40 (1863) 139-. . Lobatto, R. [1865] Arch. Neerl. 1 (1866) 115- ; Amst. Vs. Ak. 1 (1866) (Ntk.) 33-. 438 6810 Planimetry. Regular Polygons 6810 equations for sides and diagonals. Badon- Ghijben, J. [1865] Amst. Vs. Ak. 1 (1866) (Nt'k.) 293-; Arch. Need. 1 (1866) 131-. , and decomposition of same into equations of lower degrees. Schonborn, W. Arch. Mth. Ps. 46 (1866) 425-. with half the number of sides odd, relations between sides and diameter. Schwarzer, A. Wien Sb. 53 (1866) (Ab. 2) 454-. in- and circum-scribed. Azzarelli, M. Em. N. Line. At. 50 (1897) 69-. inscription. Dickson, L. E. A. Mth. 9 (1894- 95) 73-. isoperimetrical. Ranking, W. J. M. Ph. Mg. 34 (1867) 364-. , and theorems of Chasles. Ter quern, 0. Liouv. J. Mth. 3 (1838) 97-. linkage for describing. Everett, J. D. B. A. Ep. (1894) 559-. mensuration. Hessel, J. F. C. Grunert Arch. 39 (1862) 279-. paving an area with. Levy, L. Par. S. Phlm. Bll. 3 (1891) 46-. perimeter of, equal in area to given polygon. Cigir, G. I. Kazan S. Nt. (Ps.-Mth.) P. 2 (1884) 92-. , (Cigir). Zbikovskij, A. K. Kazan S. Nt. (Ps.-Mth.) P. 4 (1886) 233-. properties. Mensbrugghe, G. van der. Brux. Ac. Bll. 17 (1864) 153-. sides and areas, method of finding. Cabreira, A. Lisb. J. Sc. Mth. 5 (1898) 175-. theorem, general. Wallace, (Prof.) W. Edinb. Ph. J. 11 (1824) 377-. theorems. Fran^ais, J. F. Gergonne A. Mth. 5 (1814-15) 341-. . Reinemund, F. Brux. Ac. Bll. 40 (1875) 801- ; 43(1877)356-. . Meutzner, P. Mth. A. 13 (1878) 566-. . Sparer, B. Z. Mth. Ps. 37 (1892) 25-. 5-gon (Staudt). Barral, H. N. A. Mth. 11 (1852) 388-. .formula. Tortolini,B. A. Mt. 6 (1864) 191-. and ' ' pentalpha ' ' area. Mobius, A. F. As. Nr. 3 (1825) 131-. , theory. Clebsch, R. F. A. Mth. A. 4 (1871) 284-. , 6-gon, and 10-gon in circle, relation between sides. Grunert, J. A. Grunert Arch. 40 (1863) 127-. 6-gon, in- and' circumscribed of circle, etc. Durrande, J. B. Gergonne A. Mth. 14 (1823-24) 29-. 6-gonal facets within given circle. M,, A. J. Mcr. Sc. 6 (1858) 83-. 7-gon. Grunert, J. A. Grunert Arch. 17 (1851) 355-. and circle, fractional values for. Norton, H. Ph. Mg. 27 (1864) 281-. , Eober's construction. Hamilton, (Sir) W. R. [1863] Ph. Mg. 27 (1864) 124-. and 9-gon, approximate construction. Efremoff, D. Fschr. Mth. (1892) 531-. 9-gon. Denys, A. Mathesis 10 (1890) 162-. , inscription in circle. Encontre, D. Mntp. Eec. Bll. 1 (1803) 293-. 15-gon, 30-gon, 60-gon, etc., inscription, and calculation of sides. Dostor, G. Arch. Mth. Ps. 62 (1878) 103-. 20-gon, inscribed, perimeter compared with radius. Hamilton, (Sir) W. R. Ph. Mg. 23 (1862) 267-. 30-gons, inscription of four. Dostor, G. N. A. Mth. 17 (1878) 370-. n- and 2n-gons, relative number, for n even and odd. Dostor, G. [1877] Arch. Mth. Ps. 62 (1878) 148-. Eevolving polygon, curve envelopes of sides, metrical relation. Olsson, 0. Ts. Mth. 4 (1886) 120- ; Fschr. Mth. (1886) 666. Self-inscribed polygons. Brunei, G. Bordeaux S. Sc. PV. (1895-96) 35-. inscription, multiple, polygons with. Brunei, G. Bordeaux S. Sc. PV. (1897-98) 43-. Semi-regular polygons, analytical researches. Pigeon, H. Par. EC. Pol. J. Cah. 41 (1865) 165-. , area. Cabreira, A. [1897] Lisb. J. Sc. Mth. 5 (1898) 43-; Fschr. Mth. (1897) 446. Series. Darboux, G. Bll. Sc. Mth. As. 2 (1878) 298-. . Ocagne, M. d\ Brux. S. Sc. A. 9 (1885) (Pt. 2) 231-. Star polygons. Levy, M. Eouen Tr. Ac. (1824) 40-. . Saigey, J. F. Fe"russac Bll. Sc. Mth. 4 (1825) 201-. . Valat, .. Bordeaux Act. Ac. Sc. (1840) 415-. , area. Hermes, 0. Crelle J. 65 (1866) 174-. , , Jacobi's rule extended to case of multiple points. Hermes, 0. [1865] Crelle J. 65 (1866) 177-. , general theory. Dostor, G. Liouv. J. Mth. 6 (1880) 343-. , geometrical import. Schroder, E. Schlomilch Z. 7 (1862) 55-. , regular, with continuous outline. Azzarelli, M. Em. N. Line. At. 36 (1883) 316-. and rayed polygons. Dostor, G. Arch. Mth. Ps. 59 (1876) 375-. Theorems. Fung, N. [1810] St Pet. Ac. Sc. Mm. 11 (1830) 209-. , and application to arithmetic and trigono- metry. Werner, O. [1857] St Pet. Ac. Sc. Bll. 16 (1858) 1-. Porisms. (See also 4050 and 7210.) Play/air, J. [1792] Edinb. E. S. T. 3 (1794) 154-. Wallace, (Prof.) W. [1796] Edinb. E. S. T. 4 (1798) 107-. Breton [de Champ], P. C. E. 36 (1853) 1008-. Housel, . Liouv. J. Mth. 1 (1856) 193-. (Breton de Champ.) Vincent, A. J. H. Liouv. J. Mth. 4 (1859) 9-. 439 6810 Planimetry. Porisms. Breton [de Chump], P. Liouv. J. Mth. 4 (1859) 153- ; C. B. 51 (1860) 1034-. ( ) Bertrand, J., & Serret, . C. E. 53 (1861) 699-. Circles, in- and circum-scribed, of polygon, relation between distance of centres and radii. Grunert, J. A. Grunert Arch. 32 (1859) 68-. , or circum-scribed, of plane or spherical polygons, equations. Casey, J. As. Fr. C. B. 9 (1880) 230-. , and e-scribed, of triangle, Beltrami's theorem for relation between centres, demon- stration. Schmidt, C. Arch. Mth. Ps. 43 (1865) 238- ; 44 (1865) 120- . , , , circum-circles of. Jeff'ery, H. M. QJ. Mth. 24 (1889) 180-. , porism. Larmor, J. Mess. Mth. 13 (1884) 61-. , porismatic systems. Lachlan, E. Mess. Mth. 16 (1887) 152-. , ring of, touching 2 circles, porism. Taylor, H. M. Mess. Mth. 7 (1878) 148-. , , , . Taylor, W. W. Mess. Mth. 7 (1878) 167-. Euclid's porisms (Chasles), demonstration. Marianini, P. D. Mod. Mm. S. It. 2 (1866) 1-. Fermat's fourth porism. Nagel, C. H. Arch. Mth. Ps. 46 (1866) 11-. Functional symmetry in notation. Graves, J. T. Ph. Mg. 15 (1839) 129-. Polygon porism of two circles. Enneper, A. Schlomilch Z. 7 (1862) 190-. . Weill, M. Liouv. J. Mth. 4 (1878) 265- ; N. A. Mth. 19 (1880) 57-. . Intrigila, C. G. Mt. 21 (1883) 323-. . Dixon, A. C. Mth. Gz. 1 (1900) 121-. - , maxima and minima. Birnbaitm, H. Grunert Arch. 29 (1857) 414-. , relations. DoleZal, E. Arch. Mth. Ps. 15 (1897) 172-. Polygons, irregular symmetrical, both in- and circum-scribed to circle. Fuss, N. [1798] St Pet. Ac. Sc. N. Acta 13 (1802) 166-. , , , Fuss's problem. Mention, J. [1859] St Pet. Ac. Sc. Bll. 1 (1860) 15-, 33-. Problems. Clifford, W. K. [1868] Camb. Ph. S. P. 2 (1876) 120-. Triangles, in-, and circum-scribed, of circles. Gamier, J. G. Quetelet Cor. Mth. 1 (1825) 179-. , , , theorems. Mainardi, G. Brugnatelli G. 9 (1826) 119-, 181-. inscribed in ellipse and circumscribed to concentric circle. Barisien, E. Mathesis 19 (1899) 224-, 247-, 269-; 20 (1900) 84-, 113-, 136-. , two, such that circum-circle of one is in- circle of other, areal relation. Luchterhandt, A. R. Grunert Arch. 9 (1847) 262-. Powers. Arndt, C. F. Grunert Arch. 5 (1844) 113-. of circles and spheres, theorems. Johnson, W. W. B. A. Bp. (1890) 743-. Pythagorean Theorem 6810 Powers of points. Laisant, C. A. As. Fr. C. B. (1875) 139-. Problem. Martini, M. (vm) Bologna Opusc. Sc. N. Col. (1824) 251-. : through given point equidistant from two given lines to draw intercept of given length (Newton). Gergonne, J. D. Gergonne A. Mth. 10 (1819-20) 204-. Font6, . N. A. Mth. 6 (1847) 180-. (Newton). Transon, (Prof.) A. N. A. Mth. 6 (1847) 458-. Problems, solutions of two. Mackay, J. S. Edinb. Mth. S. P. 7 (1889) 42-. on triangle, circle, etc. Paque, A. Liege Mm. S. Sc. 10 (1855) 199-. Pythagorean theorem, MUtter t H. K. F. von. Oken Isis (1826( 763-. Gudermann, C. Crelle J. 42 (1852) 280-. Vincent, A. J. H. N. A. Mth. 11 (1852) 5-. De Morgan, A, QJ. Mth. 1 (1857) 236-. Salmon, G. QJ. Mth. 1 (1857) 237-. Richardson, J. M. Camb. (M.) Mth. M. 2 (1860) 45-. Azzarelli, M. [1873] Bm. N. Line. At. 27 (1874) 66-. Harvey, W. Edinb. Mth. S. P. 4 (1886) 17-. Zahradnik, K. fiasopis 25 (1896) 261- ; Fschr. Mth. (1896) 394. Confirmation. Wittstein, T. Grunert Arch. 11 . (1848) 152-. Demonstration. Buquoy, G. von. Oken Isis 21 (1828) 1-. by dissection. Perigal, H. Mess. Mth. 2 (1873) 103-. Extension to any triangle. Jelinek, V. 6aso- pis 28 (1899) 79-; Fschr. Mth. (1899) 456. Extensions. Anglin, A. H. Edinb. E. S. P. 12 (1884) 703-. Figures for proving. Andre, H. d'. N. A. Mth. 5 (1846) 324-. Generalisation. Umpfenbach, (Dr) . Crelle J. 26 (1843) 92. Puzzles connected with theorem. Brand, E. Bv. Sc. 2 (1894) 274-, 809-. Pythagorean triangles. Grunert, J. A. Grunert Arch. 31 (1858) 472-. . Cantor, M. Schlomilch Z. 4 (1859) 306-. . Lewis, J. Am. Ph. S. P. 9 (1865) 415-. . Whitworth, W. A. Lpool. Lt. Ph. S. P. 29 (1875) 237-. , and application to the division of the circumference. Graeber, . Arch. Mth. Ps. 15 (1897) 337-. , nearly isosceles. Martin, Art. Des Moines Anal. 3 (1876) 47-. , properties. Breton, Ph. Les Mondes 6 (1864) 401-. Eight-angled triangles. Bangma, O. S. Amst. Vh. 4 (1818) 28-. , squares on sides (Hamett). Gergonne, J. D. Gergonne A. Mth. 14 (1823) 334-. 440 6810 Planimetry. Quadrilaterals 6810 Quadrangles, complete. Neuberg, J. Mathesis 11 (1891) 33-, 67-, 81-, 189-. , transformation. Gob, A. As. Fr. C. R. (1894) (Pt. 2) 294-. Quadrilaterals. (See also 7210.) Fraczkieicicz, A. (vi Adds.) Krk. Boczn. Uniwers. 15 (1833) 123-. Fenwick, S. Mathematician 2 (1847) 285-. Baur, C. W. Schlomilch Z. 4 (1859) 236-. Farjon, F. N. A. Mth. 11 (1892) 41-. Area. Dostor, G. J. N. A. Mth. 7 (1848) 69-. with given sides, maximum. Enneper, A. Gott. Nr. (1885) 175-. , minimum. Sturm, R., & Lampe, E. Crelle J. Mth. 96 (1834) 78-. in terms of vertices, determinant expression. Dostor, G. .N. A. Mth. 13 (1874) 559-. Centroid. Commie;-, . N. A. Mth. 2 (1843) 392-. , when external. Matzka, W. Grunert Arch. 18 (1852) 352-. Common or re-entrant quadrilateral, equations of condition. Shortrede, R. Beng. J. As. S. 11 (1842) 28-, 207-. Complete quadrilateral. Schlomilch, O. Leip. B. (1854) 4-. . Mansion, P. [1875] Mess. Mth. 5 (1876) 158-. . Kantor, S. [1877-79] Wien Ak. Sb. 76 (1878) (Ab. 2) 774-; 78 (1879) (Ab. 2) 172- ; 79 (1879) (Ab. 2) 757-. ,6 circles of. Schlomilch, 0. Schlo- milch Z. 2 (1857) 274-. , properties. Ehlert, A. [1882] Arch. Mth. Ps. 69 (1883) 332-. , relating to bisectors. Sancery, L. N. A. Mth. 14 (1875) 145-. and quadrangle, involutory properties, generalisation. Loria, G. Mil. I. Lomb. Ed. 18 (1885) 491-. , theorem. Steiner, J. Gergonne A. Mth. 18 (1827-28) 302-. Cyclic quadrilateral. Greiner, M. Arch. Mth. Ps. 60 (1877) 178-. of two circles, properties. Dostor, G. N. A. Mth. 6 (1867) 57-. (or rather tetrastigm) , properties. Nager, J. Mh. Mth. Ps. 7 (1896) 325-. Deltoid, plane or skew (2 isosceles triangles on common base). Reuschle, C. G. Z. Mth. Ps. 10 (1865) 506-. Doubly-centric quadrilaterals. Schlomilch, 0. X. Z. Mth. Ps. 23 (1878) 193-. . Beyel, C. Z. Mth. Ps. 44 (1899) 237-. Harmonic quadrilateral. Neuberg, J. Mathesis 5 (1885) 202-, 217-, 241-, 265-. In- and circum-quadrilaterals. Schumacher, J. Arch. Mth. Ps. 2 (1885) 383-. . Langr, J. Casopsis 28 (1899) 244- ; Fschr. Mth. (1899) 458. , properties. Durrande, J. B. Ger- gonne A. Mth. 15 (1824-25) 133-. In-quadrilateral with diagonals at right angles. Sancenj, L. N. A. Mth. 10 (1871) 487-. In-quadrilaterals. Grunert, J. A. Grunert Arch. 5 (1844) 428- ; 8 (1846) 335. In-quadrilaterals, Girard's three, having equal sides and inscriptible in same circle. Dostor, G. Arch. Mth. Ps. 66 (1881) 27-. , metric relations. Zimmermann, O. Arch. Mth. Ps. 7 (1889) 64-. Middle points of diagonals in straight line, Gauss's proof. Hall, A. Mess. Mth. 4 (1868) 137. Miquel's theorem, generalisation to n lines. Kantor, S. [1877-79] Wien Ak. Sb. 76 (1878) (Ab. 2) 753-; 78 (1879) (Ab. 2) 789-. Orthogonal quadrilateral, 57 propositions. Beyel, C. Z. Mth. Ps. 34 (1889) 218-, 290-. Parallelograms, most simple, symmetrical about an axis. Tchebitchef, P. L. As. Fr. C. E. (1878) 159-. Properties. Dostor, G. Arch. Mth. Ps. 48 (1868) 245-. Quadrilateral with opposite sides equal. Lu- dicke, M. A. F. Gilbert A. 56 (1817) 198-, 334-, 439- ; 64 (1820) 341-. Quadrilaterals derived from quadrilateral, metrical properties. Bretschneider, C. A. Grunert Arch. 3 (1843) 84-. and triangles, theorems. Vecten, . Ger- gonne A. Mth. 9 (1818-19) 293-. Eational quadrilaterals. Kummer, E. E. Crelle J. 37 (1848) 1-. Sides and diagonals, relation between. Grunert, J. A. Grunert Arch. 38 (1862) 373-. Square with angles situated on four straight lines. Clausen, T. St Pet. Ac. Sc. Bll. 7 (1864) 177-. through four points, construction and generalisation. Crelle, A. L., & Lehmus, . Crelle J. 34 (1847) 280- . Squares connected with two circles, problems, analytical. Mack, L. Arch. Mth. Ps. 64 (1879) 225-. , decomposition into four superposable parts. Brunei, G. Bordeaux S. Sc. PV. (1896-97) 161-. , division into n equal squares, problem. Stmcell, T. P. Des Moines Anal. 2 (If 73-. (1875) , properties. Westcott, 0. S. Am. As. P. (1883) 94-. Trapezium, centroid, construction. Walker, J. J. QJ. Mth. 9 (1868) 338-. , properties. Davies, T. S. Tilloch Ph. Mg. 68 (1826) 116-. Eectilinear figures, cutting into equal portions. Gerwien, . Crelle J. 10 (1833) 228-. , geometry. Neuberg, J. As. Fr. C. E. (1893) (Pt. 2) 26-. , logical order of treatment. End res, . Toul. Mm. Ac. 4 (1860) 339-. , simpler than triangle, and their ap- plication in theory. Matzka, W. Grunert Arch. 8 (1846) 365-. Set square, measurement with false. Grunert, J. A. Grunert Arch. 18 (1852) 477-; 25 (1855) 230-. Similar figures. Grouard, A. (x) Par. S. Phlm. Bll. 2 (1865) 68-, 104- ; 7 (1871) 4-, 51-, 74-, 115- ; 10 (1873) 34-. 441 6810 Planimetry. Straight Lines 6810 Similar figures, associated. Tarry, G. Mathesis 6 (1886) 97-, 148-, 196-. (directly similar), theorems. Schoute, P. H. Delft EC. Pol. A. 6 (1890) 51-. in a plane, system. Laisant, C. A. As. Fr. C. R. (1883) 178-. , theorems. Tarry, . C. R. 112 (1891) 984-. , 3, with extension of Feuerbach's theorem. M'Cay, W. S. [1889] Ir. Ac. T. 29 (1887-92) 303-. , , properties. Tarry, G. (XH) Mathesis 2 (1882) 73-. , , . Neuberg, J. (xn) Mathesis 2 (1882) 76-, 144. , , . Casey, J. Mathesis 7 (1887) 14-, 75. Similitude, general theory. Breton, Ph. [1877] (xn) Isere S. Bll. 8 (1879) 4-, 9, 12-. and inversion. Mackay, J. S. Edinb. Mth. S. P. 6 (1888) 69-. Stewart's theorems, proof of two on circle and polygon. Davies, T. S. Camb. and Dubl. Mth. J. 1 (1846) 229-. STEAIGHT LINES. Pfeil, L. von. Arch. Mth. Ps. 49 (1869) 178-. Hain, E. Arch. Mth. Ps. 64 (1879) 267-. Tarry, G. As. Fr. C. R. (1890) (Ft. 2) 152- ; (1894) (Pt. 2) 184-. Description by link work. Aldis, W. S. N. Z. I. T. 21 (1889) 441-. Drawing a straight line. Kempe, A. B. Nt. 16 (1877) 65-, 86-, 125-, 145-. approximately, by linkwork. Milller, E. D. Nf. Vh. (1897) (Th. 2, Hiilfte 1) 34. Finite lines in plane, number of cases as regards parallelism and situation. Rothe, H. A. Ac. Gas. Leop. N. Acta 12 (1825) 399-. 2 lines making complementary angles with fixed axis, properties. Mensbrugghe, G. van der. Brux. Ac. Bll. 20 (1865) 6CK n lines in plane, metric geometry. Morley, F. N. Y. Am. Mth. S. T. 1 (1900) 97-. , . Loud, F. H. N. Y. Am. Mth. S. T. 1 (1900) 323-. Properties, descriptive, segmentary and metric. Mouchot, A. C. E. 104 (1887) 1053-. Relation of distances from three given points. Ferrers, N. M. QJ. Mth. 2 (1858) 120-. Section of a line. Gergonne, J. D. Gergonne A. Mth/15 (1824-25) 93-. . Wallace, W. Edinb. Mth. S. P. 12 (1894) 76-. . Anderson, R. E. Edinb. Mth. S. P. 15 (1897) 65-. System of concurrent lines, isometrics of a line with reference to. Ocagne, M. d'. Par. S. Mth. Bll. 17 (1889) 171-. straight lines, centre of least squares. Neu- berg, . Brux. S. Sc. A. 23 (1899) (Pt. 1) 27-. , , construction. Rahusen, A. E. N. Arch. Wisk. 3 (1898) 33-. in plane, intercepts. Roberts, S. L. Mth. S. P. 19 (1889) 405-. Systems of straight lines, orientation. Hum- bert, G. Am. J. Mth. 10 (1888) 258-. Theorems. Frankenheim, M. L. Crelle J. 8 (1832) 178-. . Brioschi, F. N. A. Mth. 15 (1856) 462-. Superposition. Kelland, P. [1855-64] Edinb. R. S. T. 21 (1857) 271- ; 23 (1864) 471-. (Kelland). Brodie, R. Edinb. R. S. T. 36 (1892) 307-. by means of dissection. Muirhead, R. F. Edinb. Mth. S. P. 14 (1896) 109-. Theorems, geometrical, derived by inversion, reciprocation, etc. Bellavitis, G. A. Sc. Lomb. Yen. 2 (1832) 250- ; Padova N. Sag. 4 (1838) 243-; Tortolini A. 5 (1854) 31-, 241-, 428-, 473-. Transformation of rectangle into square. Peddie, W. Edinb. Mth. S. P. 4 (1886) 24. by symmetrical straight lines, application to Steiner's problem. Schoute, P. H. Bll. Sc. Mth. As. 7 (1883) (Pt. 1) 314-. Transformations of figures. Gob, A. As. Fr. C. R. (1890) (Pt. 2) 1-. by reciprocal radii, groups. Allardice, R. E. [1887] Edinb. Mth. S. P. 6 (1888) 14-. , (Allardice). Alasia, C. Fschr. Mth. (1900) 530. - vectors. Laguerre, E. N. A. Mth. 1 (1882) 542-. . Ocagne, M. d\ N. A. Mth. 2 (1883) 249-. (Laguerre's transformation). Juel, C. N. Ts. Mth. 3 (B) (1892) 10-; Fschr. Mth. (1892) 578-. , properties of cycles. Laguerre, E. N. A. Mth. 2 (1883) 65-. Translation, method dependent on principle. Maver, D. [1890] Edinb. R. S. P. 17 (1891) 188-. Triangles (including Circles related thereto). (For Conic Sections related to Triangles see 7210.) (Euler.) Terquern, 0. N. A. Mth. 1 (1842) 79-, 196-. Mollmann, B. Grunert Arch. 17 (1851) 373-. Hellwig, C. Grunert Arch. 19 (1852) 14-. Unferdinger, F. Grunert Arch. 27 (1856) 327- ; 29 (1857) 432- ; 33 (1859) 420-. Grunert, J. A. Arch. Mth. Ps. 42 (1864) 229- ; 45 (1866) 429- ; 48 (1868) 465-. Schubert, H. Z. Mth. Ps. 16 (1871) 83-. Hain, E. Arch. Mth. Ps. 60 (1877) 290- ; 61 (1877) 417- ; 62 (1878) 422-. Greiner, M. Arch. Mth. Ps. 61 (1877) 225-. (Euler.) Mackay, J. S. Edinb. Mth. S. P. 4 (1886) 51-. Lemoine, E., <& Neuberg, . Mathesis 6 (1886) 55-, 73-. Lemoine, E. As. Fr. C. R. (1886) (Pt. 2) 83-; (1887) (Pt. 2) 13-; (1888) (Pt. 2) 165- ; (1889) (Pt. 2) 197- ; (1891) (Pt. 2) 130- ; (1892) (Pt. 2) 101- ; (1895) (Pt. 2) 186- ; (1896) (Pt. 2) 58-. Cesdro, E. N. A. Mth. 6 (1887) 215-. 442 Planimetry. Triangles and Connected Circles 6810 Lemoine, E. Edinb. Mth. S. P. 13 (1895) 2-. Perrin, E. As. Fr. C. E. (1897) (Ft. 2) 90-. Candido, G. N. A. Mth. 18 (1899) 31-; 19 (1900) 244-. Analogies in the triangle, rule. Lemoine, E. N. A. Mth. 12 (1893) 20- ; G. Teix. J. Sc. 12 (1895) 105-. , (Lemoine). Michel, C. N. A. Mth. 17 (1898) 24-. Angles, sum. Bertrand, J. C. E. 69 (1869) 1265- ; 70 (1870) 17-. , , demonstration. Petersen, J. P. C. (xn) Ts. Mth. 1 (1883) 3- ; Mth. A. 29 (1887) 238-. , = 180, demonstration. Strong, T. Am. Ac. P. 1 (1848) 129-, 136-. , = , . Hedrick, B. S. Am. As. P. (1874) (Pt. 1) 104-. , = , , Legendre's. Levett, E. [1882] Binn. Ph. S. P. 3 (1883) 226-. , = , without theory of parallels or infinity. Paulet, F. E. S. P. 3 (1836) 404. , = , . Rosa, J. B. da. Coimbra I. 6 (1858) 115-. Annex triangles, etc. Catalan, E. C. Brux. Ac. Bll. 4 (1882) 322-. Antiparallels to sides of triangle. Lemoine, E. Mathesis 4 (1884) 201- ; Par. S. Mth. Bll. 14 (1886) 107- , 163. , use of word. Langley, E. M. Nt. 40 (1889) 460-. Apollonian problem, special case. Hoffmann, K. E. [1880] Arch. Mth. Ps. 66 (1881) 246-. Area. Studnicka, F. Prag Sb. (1872) (pt. 2) , and its dual analogue. Busche, E. Crelle J. Mth. 114 (1895) 1-. , equal division by perpendiculars drawn from a point. Laser, . N. A. Mth. 3 (1884) 332-. of pedal triangle. Clausen, T. Crelle J. 3 (1828) 196-. of triangle formed by joining feet of bisectors of angles. Dostor, G. Arch. Mth. Ps. 64 (1879) 407-. Bisectors of angles. Hain, E. [1875] Arch. Mth. Ps. 58 (1876) 90-. . Mackay, J. S. Edinb. Mth. S. P. 13 (1895) 37-. , construction of triangle from. Berg, F. J. van den. N. Arch. Wisk. 16 (1889) 179- ; 17 (1890) 191- ; Fschr. Mth. 1889) 576; (1890) 590. , . Barbarin, . Par. S. Mth. Bll. 22 (1894) 76-; Mathesis 16 (1896) 143. , . Korselt, A. Z. Mth. Ps. 42 (1897) 304-. , , and bibliographical notes. Baker, M. [1879] Smiths. Misc. Col. 20 (1881) Art. 3, 55-. (Wash. Ph. S. Bll. 3 (1880).) , theorem. Mink, W. Grunert Arch. 15 (1850) 358-. Brocard angle. Fuhrmann, W. Arch. Mth. Ps. 6 (1888) 1-. 6810 Brocard angle and circle. Schoute, P. H. Amst. Ak. Vs. M. 3 (1887) 39-; Fschr. Mth. (1886) 477-. angles, triangles having equal. Neuberg, J. As. Fr. C. E. (1888) (Pt. 2) 135-. and Steiner angles. Avillez, J. F. d\ Lisb. J. Sc. Mth. 5 (1898) 85-. Circles of the Triangle. Kucker, K. Arch. Mth. Ps. 47 (1867) 1-. Duran-Loriga, J. J. As. Fr. C. E. (1897) (Pt. 2) 175- ; (1898) (Pt. 2) 147-. Brocard circle. Brocard, H. As. Fr. C. E. 10 (1881) 138-. and angle. Schoute, P. H. Amst. Ak. Vs. M. 3 (1887) 39-; Fschr. Mth. (1886) 477-. and Brocard line, properties. Lemoine, E. Mathesis 5 (1885) 103-. , generalised, properties. Griffiths, J. L. Mth. S. P. 26 (1895) 173-. and Lemoine point, generalisation of properties. Lemoine, E. N. A. Mth. 4 (1885) 201-. as locus. Langr, J. Casopis 29 (1900) 210- ; Fschr. Mth. (1900) 499. , an( i related systems of conies. Mutter, A. Arch. Mth. Ps. 8 (1890) 337-. , variable seven-points circle analogous to. Griffiths, J. [1893] L. Mth. S. P. 25 (1894) 75. circles of plane triangle generalised, in- version of system. Griffiths, J. L. Mth. S. P. 25 (1894) 376-. and other circles. M'Cay, W. S. [1885] Ir. Ac. T. 28 (*1880-86) 453-. Circle connected with triangle. Cwojdzinski, K. Arch. Mth. Ps. 17 (1900) 238-. Circles connected with triangle formed by cir- cular arcs, properties of some. Lachlan, R. [1890] L. Mth. S. P. 21 (1891) 263-. cutting orthogonally in- and ex-circles of triangle. Hart, H. QJ. Mth. 18 (1882) 363-. touching circles described on sides of tri- angle. Rutherford, W. [1846] Mathema- tician 2 (1847) 69-. sides. Noel, J. N. Quetelet Cor. Mth. 5 (1829) 22-. Circum-circle, power of a point in relation to. Hain, E. [1881] Arch. Mth. Ps. 67 (1882) 106-. , property. Tucker, R. L. Mth. S. P. 22 (1891) 470- ; 25 (1894) 315-. , relation to in- and ex-circles. Chrzasz- czcwski, S. Arch. Mth. Ps. 16 (1898) 119-. , inscribed conic. Laguerre,E. N. A. Mth. 18 (1879) 241-. Groups of three circles related to. Berg, F. J. van den. N. Arch. Wisk. 7 (*1881) 78-. In-circles, equation in trilinears. Russell , W. H. L. [1881-82] E. S. P. 33 (1882) 211- ; 34 (1883) 35-. , etc., theorems. Larenz, N. von. [1878] Arch. Mth. Ps. 63 (1879) 294-; 65 (1880) 212-. , circum-, and ex-circles. Matthes, C. J. Leijd. A. Ac. (1830-31) 66 pp. 443 6810 Planimetry. Malfatti's Problem. Nine-point Circle 6810 In- and ex-circles. Mackay, J. S. Edinb. Mth. S. P. 12 (1894) 86-; 13 (1895) 103-. . Billups, H. B. Mth. Gz. 1 (1900) 177-. (Billups). Mackay, , & Muirhead, . Mth. Gz. 1 (1900) 209. of triangles formed by four straight lines, Steiner's theorem. Mention, J. N. A. Mth. 1 (1862) 16-, 65-. Intersections of circle with triangle, relations. Taylor, H. M. L. Mth. S. P. 15 (*1883-84) 122-. " Intimoscribed " circles. Anderson, E. E. Edinb. Mth. S. P. 10 (1892) 70-. Malfatti's problem. Gergonne, J. D., & Lavernede, . Gergonne A. Mth. 1 (1810-11) 343-. Tedenat, . Gergonne A. Mth. 2 (1811-12) 165. LechmUtz, . Gergonne A. Mth. 10 (1819-20) 289-. Lavernede, T. Gard. Mm. Ac. (1832) 137-. Catalan, E. C. N. A. Mth. 5 (1846) 60-. Ghijben, J. B. Amst. Vs. Ak. 12 (1861) 189-. Talbot, W. H. F. [1865] Edinb. B. S. T. 24 (1867) 127-. Affolter, F. G. Mth. A. 6 (1873) 597-. Simons, P. A. Brux. Ac. Bll. 38 (1874) 88-. Catalan, E. C. Brux. Ac. Bll. 38 (1874) 480-. Pelletreau, . As. Fr. C. R. (1888) (Pt. 2) 99. Neumann, C. Leip. Mth. Ps. B. 41 (1889) 22-. extension. Schellbach, C. H. Crelle J. 45 (1853) 186-. history and complete analytical solution. De- rousseau, J. Liege S. Sc. Mm. 18 (1895) No. 1, 52 pp. and lemma on circles. Adams, C. N. A. Mth. 8 (1849) 62-. solution. Scheffler, H. Grunert Arch. 16 (1851) 424-. . Schellbach, C. H. [1852] N. A. Mth. 12 (1853) 131-. (Schellbach). Cayley, A. QJ. Mth. 1 (1857) 222-. . Lebon, E. Palermo Cir. Mt. Ed. 3 (1889) 120-. solutions, thirteen. Davids, C. Arch. Mth. Ps. 13 (1895)^10-; 14 (1896) 276-. Steiner's construction. Hart, A. S. QJ. Mth. 1 (1857) 219-. , demonstration. Quidde, A. Grunert Arch. 15 (1850) 197-. , . Mendthal, . Arch. Mth. Ps. 55 (1873) 211-. and Steiner's generalisation. Mertens,F. [1875- 76] Wien. Ak. D. 36 (1876) (Ab. 2) 195- ; Z. Mth. Ps. 21 (1876) 297-. . Godt, W. [1877] Crelle J. Mth. 84 (1878) 259-. . Mertens, F. [1893] Krk. Ak. (Mt.- Prz.) Bz. 8 (1895) 67-; Prag Sb. (1894) (Mth.- Nt.) No. 1, 21 pp. Steiner's solution. Schroeter, H. E. Crelle J. Mth. 77 (1874) 230-. Steiner's solution. Petersen, J. Crelle J. Mth. 89 (1880) 127- , demonstration. Zornow, . Crelle J. 10 (1833) 300-. Neuberg's circles. Gob, A. Liege S. Sc. Mm. 16 (1890) No. 3, 14 pp. New circle. Fuhrmann, W. Mathesis 10 (1890) 105-. Nine-point or Feuerbach Circle. Trudi, N. G. Mt. 1 (1863) 29-. Griffiths, J. N. A. Mth. 2 (1863) 339- ; Mess. Mth. 2 (1864) 220- ; QJ. Mth. 8 (1867) 15. Besant, W. H. QJ. Mth. 7 (1866) 302- ; Mess. Mth. 3 (1866) 222-. Taylor, C. Mess. Mth. 4 (1868) 61-. Hain, E. Arch. Mth. Ps. 59 (1876) 323-. Zeuthen, H. G. (xn) Ts. Mth. 1 (1877) 113-. Lemoine, E. N. A. Mth. 5 (1886) 122-. Wilson, (Eev.) J. Edinb. Mth. S. P. 6 (1888) 38. Cayley, A. Mess. Mth. 23 (1894) 23-, 25-. Sparer, B. Wien Ak. Sb. 106 (1897) (Ab. 2a) 739-. analytical treatment. Grunert, J. A. Arch. Mth. Ps. 41 (1864) 121-. circle analogous to, about centroid of perimeter of triangle. Spieker, T. Arch. Mth. Ps. 51 (1870) 10-. Feuerbach's theorem. Lappe, J. [1869] Crelle J. 71 (1870) 387-. . Schroeter, H. E. Mth. A. 7 (1874) 517-. . Slawyk, R. Z. Mth. Ps. 35 (1890) 36-. , demonstration. Leudesdorf, C. Mess. Mth. 13 (1884) 116-. , . Genese, E. W. L. Mth. S. P. 19 (1889) 216- . group of circles related to. Baker, M. Smiths. Misc. Col. 33 (1888) Art. 3, 45-. (Wash. Ph. S. Bll. 8 (1885).) history. Mackay, J. S. Edinb. Mth. S. P. 11 (1893) 19-. properties of triangle, generalisation. Mac- Mahon, P. A. L. Mth. S. P. 14 (1882-83) 129-. Steiner's hypocycloid and the circumscribing hyperbolas of A. Fiedler, W. Ziir. Vjschr. 30 (1885) 390-. and Steiner's quartic. Godt, W. [1895-96] D. Mth. Vr. Jbr. 4 (1897) 161- ; Munch. Ak. Sb. 26 (1897) 119-. Terquem's theorem, extension. Hart, A. S. QJ. Mth. 4 (1861) 260-. theorems. Salmon, G. QJ. Mth. 4 (1861) 152- . . Griffiths, J. QJ. Mth. 6 (1864) 357-; 7 (1866) 46-, 341-. Pedal circle of a point with regard to given triangle. Fries, J. de. [1900] Amst. Ak. Vs. 9 (1901) 249-; Amst. Ak. P. 3 (1901) 323-. 444 6810 Planimetry. Triangles and Connected Circles 6810 Polar circle, and Feuerbach's theorem, note on theorems. Roberts, S. Mess. Mth. 17 (1888) 57-. Relations of circle and triangle. Hain, E. [1876] Arch. Mth. Ps. 60 (1877) 78-. Kings of circles connected with triangle. Taylor, W. W. L. Mth. S. P. 20 (1889) 397-. " Scribed " circles, six, and triangle. Mackay, J. S. [1883] Edinb. Mth. S. P. 1 (1894) 4-, iv. "Sine-triple-angle" circle (Tucker's circle). Tucker, R. Mess. Mth. 16 (1887) 125-. Six circles and triangle. Evans, A. B. Des Moines Anal. 1 (1874) 189-. Geometry of triangle, history. Fuhrmann, . Konigsb. Schr. 40 (1899) [37]-. Harmonic hexagon of a triangle. Casey, J. [1886] Ir. Ac. P. 4 (1884-88) 545-. triangle. Henry, C. Bll. Sc. Mth. As. 5 (1881) 96-. Homologous triangles. Neuberg, J. Mathesis 18 (1898) 155-. In-triangle of circle with sides passing through fixed points, construction, etc. Giordano di Ottaiano, A. Verona Mm. S. It. 4 (1788) given form in given triangle. Allardice, R. E. Edinb. Mth. S. P. 9 (1891) 39-. point circle. Taylor, H. M. Mess. Mth. 11 , inscription. Jenkins, M. Casey, J. QJ. Mth. 4 Prag Sb. (1875) (1882) 177-. , properties. (1861) 245-. Theorems. SallabaSev, I. 66-. Tucker-circles, secondary. Griffiths, J. [1892] L. Mth. S. P. 24 (1893) 121-. or triplicate-ratio circles. Tucker, R. QJ. Mth. 19 (1883) 342-. , group analogous to. Tucker, R. QJ. Mth. 20 (1885) 57-. . and others. Tucker, R. Edinb. Mth. S. P. 12 (1894) 17-. , properties. Tucker, R. L. Mth. S. P. 25 (1894) 389-. , systems analogous to. Third, J.A. Edinb. Mth. S. P. 17 (1899) 70-. Circum-triangle on sides of triangle, construc- tion for vertices. Clausen, T. Crelle J. 4 (1829) 391-. Construction from radii of ex-circles. Griison, J. P. (vi Adds.) Berl. Ab. (1818-19) (Mth.) 37. Correlation. Callegari,P. [1843] Bologna Mm. Ac. 4 (1853) 179-. Covariant geometry. Morley, F. QJ. Mth. 25 (1891) 186-. Dissection of triangles. Muirhead, R. F. Edinb. Mth. S. P. 18 (1900) 5-, 100. Division. Weiss, C. S. Berl. Ab. (1826) 93-. . Dienger, J. Grunert Arch. 17 (1851) 300-. in given ratio. Catalan, E. C. N. A. Mth. 4 (1845) 214-. Equilateral triangles. Hain, E. [1882] Arch. Mth. Ps. 69 (1883) 44-. on sides of triangle, theorem. Tucker, R. Edinb. Mth. S. P. 15 (1897) 98-. Euler's line and circle. Gob, A. Liege S. Sc. Mm. 16 (1890) No. 2, 7 pp. , hyperbola inverse to. Jefdbek, . Mathesis 8 (1888) 81-. , . Neuberg, J. Mathesis 8 (1888) 84-, 115-. , . Fuhrmann, . Mathesis 8 (1888) 115-. Formulae. Catalan, E. Brux. Mm. Cour. 8, 44 (1891) No. 4, 28 pp. Geometry of triangle, bibliography. Vigarie, E. As. Fr. C. E. (1895) (Pt. 2) 50-. , history. Vigarie, E. As. Fr. C. E. (1887) (Pt. 2) 87-; (1889) (Pt. 2) 117-. QJ. Mth. 21 (1886) 84-. , . Allardice, R. E. Edinb. Mth. S. P. 6 (1888) 42-. least perimeter in triangle. Lindeldf, L. L. Helsingf. Ofv. 10 (1868) 31-; 11 (1869) 35-. , maximum equilateral, in given triangle. Malfatti, G. F. [1806] Mod. S. It. Mm. 13 (1807) 247-. of similar triangle, inscription. Hoffmann, H. Grunert Arch. 9 (1847) 280-. In-triangles of circle, special system. Strnad, A. Casopis 24 (1895) 136- ; Fschr. Mth. (1895) 576-. in given triangle, group, etc. Tucker, R. [1892] L. Mth. S. P. 24 (1893) 131-. Inversion of system of n points, application to Ble. Laisant, C. A. As. Fr. C. E. (Pt. 2) 282-. centres. Mackay, J. S. Edinb. Mth. S. P. 15 (1897) 100-. corresponding points. Hain, E. [1876] Arch. Mth. Ps. 60 (1877) 92-. systems in the triangle. Barbarin, P. As. Fr. C. E. (1896) (Pt. 2) 89-. Isogonals of triangle. Mackay, J. S. Edinb. Mth. S. P. 13 (1895) 166-. Isoperimetrical triangles having one side given and satisfying 3 other conditions. Chasles, M. C. E. 84 (1877) 471-, 627-, 1051-. t series satisfying 4 other conditions. Chasles, M. C. E. 84 (1877) 55-. Isosceles triangle, property. Lange, T. Grunert Arch. 13 (1849) 337-; 15 (1850) 221-, 351-. to be proved isosceles when bisectors of base angles are equal. Sylvester, J. J. Ph. Mg. 4 (1852) 366-. triangles. PeliSek, M. Casopis 26 (1897) 181- ; Fschr. Mth. (1897) 442. , problems. Schiappa Monteiro, A. Lisb. J. Sc. Mth. 12 (1888) 57-, 121-. Isoscelian hexagrams. Tucker, R. [1889] L. Mth. S. P. 21 (1891) 4-. Isoscelians. Tucker, R. [1888-91] L. Mth. S. P. 19 (1889) 163- ; 22 (1891) 178-. Isostereans, group. Tucker, R. L. Mth. S. P. 19 (1889) 218-. Least perimeter, triangle of, etc. Gregory, D. F. (vi Adds.) Camb. Mth. J. 1 (1839) 157-. 445 6810 Planimetry. Triangles. Remarkable Points 6810 Lehmus's theorem (angles of triangle equal, when bisectors are equal). Jefdbek, A. Casopis 14 (1885) 20-; Fschr. Mth. (1885) 575. Lines in triangle, class. Franke, . Crelle J. Mth. 99 (1886) 161-. , (Franke). Schroeter, H. Crelle J. Mth. 99 (1886) 233-. Medians, generalisation. Azzarelli, M. Em. N. Line. At. 42 (1889) 59-. Menelaus's theorem, application to Euclid porisms. Serdobinskil, V. E. [1872] (xn) Kec. Mth. (Moscou) 6 (1872-73) (Pt. 2) 3-. Orthocentric triangle. Tucker, E. Edinb. Mth. S. P. 12 (1894) 118-. Orthologic triangles, triangles and conies. Lemoine, E. As. Fr. C. B. (1890) (Pt. 2) 111-. Parallels and anti-parallels of a triangle, some properties. Lemoine, E. Par. S. Mth. Bll. 12 (1884) 72-. from point in plane to sides. Lemoine, E. As. Fr. C. E. (1882) 122-. and triangles, theorems. Denzler, W. Ziir. Mt. 2 (1850-52) 74-. Pedal triangle. Steggall, J. E. A. Edinb. Mth. S. P. 12 (1894) 85. , theorem. Schoute, P. H. Palermo Cir. Mt. Ed. 1 (1887) 380-. Perpendiculars and medians of a triangle, relations between. Pabst, C. Arch. Mth. Ps. 7 (1889) 10-. Pole and polar. Greiner, M. Arch. Mth. Ps. 59 (1876) 351-. Problem. Escher, P. Grunert Arch. 31 (1858) 46-. : In sides AB, AC of triangle to find EF so that BE = EF=FC. Edward, J. Edinb. Mth. S. P. 2 (1884) 5-. = , historical notes. Mackay,J.S. Edinb. Mth. S. P. 8 (1890) 93-. = , solution. Mackay, J. S. Edinb. Mth. S. P. 2 (i.884) 27. : with given vertex to describe a triangle similar to given triangle and having its other angles on two given straight lines. Were- brussow, A. Fschr. Mth. (1897) 441. Problems. Lorenz, N. von. Arch. Mth. Ps. 64 (1879) 253-. . Collignon, E. Edinb. Mth. S. P. 18 (1900) 88. Projections and centra-projections of fixed triangle. Neuberg, J. Brux. Mm. Cour. 8, 44 (1891) No. 2, xvi + 86 pp. Properties. Schulz von Strasnitzki, L. C. Baumgartner Z. 2 (1827) 396-. . Schmidt, J. E. (vm) Amst. N. Ws. Ntk. Vh. 1 (1844) 30-. . Pistoris, de. N. A. Mth. 5 (1846) 451-. . Azzarelli, M. Em. N. Line. At. 44 (1891) 49-. . Molenbroch [? Moleribroek], . N. A. Mth. 11 (1892) 121-, 179-. . Collate, M. L. Liege S. Sc. Mm. 19 (1897) .No. 4, 12 pp. Properties, demonstration. V. [Gregory, I). F.] Camb. Mth. J. 1 (1839) 157-. , extension. Schroter, H. [1867] Crelle J. 68 (1868) 208- . , mean orientation, equisegmentary points. Laisant, C. A. As. Fr. C. E. (1890) (Pt. 2) 23-. , new. Brocard, H. As. Fr. C. E. 12 (1883) 188-. Property. Wallace, (Prof.) W. Camb. Mth. J. 1 (1839) 21-. Proportionals to sides, etc. Albers, J. Arch. Mth. Ps. 68 (1882) 53-. Pseudo-isosceles triangle. Emmerich, A. Ma- thesis 20 (1900) 129-. Quadrilaterals and triangles in connection with progressions. Miksic, M. Casopis 5 (*1876) 134- ; Fschr. Mth. (*1876) 325-. , new property. Brocard, H. [1874] Par. S. Mth. Bll. 3 (1875)^38-. . , series. Collignon, E. As. Fr. C. E. (1891) (Pt. 2) 38-. Quadrisection. Euler, L. [1779] St Pet. Ac. Sc. Mm. 1 (1803-06) 26-, 49-. Eational triangles. Grassmann,H. Arch. Mth. Ps. 49 (1869) 1-. . Simerka, W. [1869] Arch. Mth. Ps. 51 (1870) 196-. . Lehmer, D. N. A. Mth. 1 (1900) 97-. , complete development, with three tables. Rath, H. Arch. Mth. Ps. 56 (1874) 188-. , , (Eath). Curtze, H. Arch. Mth. Ps. 57 (1875) 216-. Eelation between area and sum of angles, sup- posing Euclid's 12th axiom false. Young, G. P. Cn. J. 5 (1860) 341-. Eelations between orthic and median triangles. Pressland, A. J. Edinb. Mth. S. P. 9 (1891) , metrical and of position. Bardelli, G. Q. Mt. 14 (1876) 241-. Eemarkable Points of the Triangle. Lange, J. [1880] Arch. Mth. Ps. 66 (1881) 220- ; 67 (1882) 191-. Lemoine, E. As. Fr. C. E. (1882) 108-. Kapteyn, W. Amst. Ak. Vh. (Sect. 1) 3 (1896) No. 3, 33 pp. ; Fschr. Mth. (1895) 665. Associated points in the plane. Lemoine, E. As. Fr. C. E. (1884) (Pt. 2) 49-. Brocard points. Sweschnikow, P. Fschr. Mth. (1891) 599. and angle. Davis, E. F. Edinb. Mth. S. P. 13 (1895) 28. , construction. Allardice, B. E. Edinb. Mth. S. P. 6 (1888) 88. Centre of equal angular transversals. Brocard, G. Mathesis 16 (1896) 217-. (Brocard). Neuberg, J. Ma- thesis 16 (1896) 221-. Centres of similitude. Arndt, C. F. Grunert Arch. 5 (1844) 113-. of circles connected with triangle. Mackay, J. S. Edinb. Mth. S. P. 2 (1884) 446 6810 Planimetry. Triangles. Remarkable Points 6810 Centres of similitude of triangle of constant Point, the sum of whose distances from vertices form circumscribed to given triangle. Grif- is a minimum. Gergonne, J. D. Gergonne fiths, J. L. Mth. S. P. 24 (1893) 369-. A. Mth. 20 (1829-30) 299-. inscribed in given , . Azzarelli, M. L. Mth. S. 24 triangle. Griffith (1893) 181-. Centroid. Haiti, E. [1875] Arch. Mth. Ps. 58 (1876) 170-. , distances. Dostor, G. N. A. Mth. 2 (1883) Grunert, J. A. and in-circle, distance. Arch. Mth. Ps. 52 (1871) 247-. , intersection of medians and other theorems. Burg, A. Wien Jb. Pol. I. 12 (1828) 4- ; 13 (1828) 223-. , Lemoine, and other points. DurdnLoriga, J. J. G. Teix. J. Sc. 11 (1892) 161- ; Fschr. Mth. (1893-94) 1104. of particular family of triangles, locus. Brocard, H. Mathesis 11 (1891) 153-. , points whose joins are trisected by. Eeuschle, C. G. Z. Mth. Ps. 11 (1866) 475-. Centroids, pedal and symmetrical. Neuberg, J. N. Arch. Wisk. 4 (1899) 192- . "Cosine" orthocentres of a triangle, and a cubic through them. Tucker, R. Mess. Mth. 17 (1888) 97-. Distances of point from vertices. Cayley, A. QJ. Mth. 5 (1862) 381-. . BlazeievsM, E. N. A. Mth. 13 (1894) 28-. points. Thiry, C. Brux. Ac. Bll. 21 (1891) 471-. from each other. Grunert, J. A. Grunert Arch. 36 (1861) 325-. . Lemoine, E. N. A. Mth. 9 (1870) 311-. Eighty-four special points, including Jefabek's. Jefabek, V. Casopis 20 (1891) 141-, 237-; Fschr. Mth. (1891) 603-. Fifth remarkable point. Hochheim,A. Arch. Mth. Ps. 52 (1871) 26-. Four remarkable points (Euler). Grunert, J. A . Grunert Arch. 26 (1856) 343-. , analytically treated. Metzler, C. Arch. Mth. Ps. 47 (1867) 243-. G points of circle with respect to given triangle, finding. Griffiths, J. [1891] L. Mth. S. P. 23 (1892) 96-. Loci. Lindman, C. F. Arch. Mth. Ps. 43 (1865) 350-. . Hochheim, A. Z. Mth. Ps. 15 (1870) 33-. Locus of point whose pedal triangle has given area. Azzarelli, M. Em. N. Line. At. 27 (1874) 333-. points, sum of whose distances from sides is constant (Entf ernungsort) . Grunert, J.A. Grunert Arch. 17 (1851) 361-. , ( ). Emsmann, H. Arch. Mth. Ps. 46 (1866) 121-. Middle points of triangles formed by in- and ex-centres of triangle. Noggerath, E. J. Schlomilch Z. 8 (1863) 394-. Nagel and Gergonne points. Harnischmacher, F. J. Arch. Mth. Ps. 42 (1864) 90-. . Mink, W. Arch. Mth. Ps. 43 (1865) 1-. 447 Em. N. Line. At. 39 (1886) 95-. Eeciprocal trilinear coordinates, properties of points with. Greiner, M. Arch. Mth. Ps. 1 (1884) 130-. Series of points. Poulain, A. Mathesis 10 (1890) 246-. Spieker's point. Hain, E. [1875] Arch. Mth. Ps. 58 (1876) 164-. Steiner's foci of a triangle. Neuberg, J., <& Gob, A. As. Fr. C. E. (1889) (Ft. 2) 179-. point. Neuberg, J. As. Fr. C. E. (1885) (Pt. 2) 89-. Symmedian or Lemoine point. Lemoine, E. N. A. Mth. 12 (1873) 364-; As. Fr. C. E. 2 (1873) 90-; (1874) 1165-. . Picquet, H. As. Fr. C. E. (1874) 1202-. . Hain, E. [1875] Arch. Mth. Ps. 58 (1876) 84-. , early history. Mackay, J. S. Edinb. Mth. S. P. 11 (1893) 92-. point axis of system of triangles. Tucker, R. QJ. Mth. 20 (1885) 167-. Symmetry points. Hoppe, R. Arch. Mth. Ps. 57 (1875) 422-. . Hain, E. Arch. Mth. Ps. 58 (1876) 176-, 385-, 394-; 59 (1876) 415-, 420- ; 60 (1877) 71- ; 64 (1879) 398-. Tarry's point. Neuberg, J. Mathesis 6 (1886) 5-. Three points (Euler). Gentil, . N. A. Mth. 5 (1846) 28-. , certain groups. K$piiiski, S. Prace Mt.-Fiz. 2 (1890) 169-. Two points. Grunert, J. A. Arch. Mth. Ps. 48 (1868) 37-. in plane of a triangle. Lemoine, E. As. Fr. C. E. (1885) (Pt. 2) 23-. Eemarkable triangle. Barisien, (le capit.) E. N. As. Fr. C. E. (1897) (Pt. 2) 107-. Eight angled isosceles triangle, calculation of angles. Voll, W. Oken Isis (1826) 490-. triangles, properties. Dostor, G. Arch. Mth. Ps. 51 (1870) 103-. , theorem. Ramus, C. Crelle J. 20 (1840) 28. , theorems, four. Lilienthal, . Grunert Arch. 21 (1853) 99-. Scalenity. Sharpe, H. J. [1864] Mess. Mth. 3 (1866) 52- ; 4 (1868) 177-. Schroeter, theoremof , andLongchamps's line d. Droz-Farny, A. As. Fr. C. E. (1897) (Pt. 2) 136- . Segmentary properties. Ocagne, M. d'. N. A. Mth. 2 (1883) 497-. Segments of sides by concurrent lines through vertices. Bellavitis, G. A. Sc. Lomb. Yen. 2 (1832) 250-. (Bellavitis). Fusinieri, A. A. Sc. Lomb. Yen. 2 (1832) 254-. , geometry of certain. Ocagne, M. d'. G. Teix. J. Sc. 6 (1885) 125-. 6810 Planimetry. Triangles 6810 Segments of sides by straight line through fixed point. Durege, . Grunert Arch. 30 (1858) 241-. Similar triangles satisfying four conditions, number. Cliasles, M. C. E. 78 (1874) 1599-. , series. Chasles, M. O.K. 78 (1874) 1373- ; 79 (1874) 877-, 1427-. Similitude of triangles belonging to two series. Viaggi, F. G. Mt. 28 (1890) 113-. Simson or Wallace pedal line. Durrande, J. B. Gergonne A. Mth. 7 (1816-17) 253-. . Barbarin, P. (xn) Mathesis 2 (1882) 106-, 122-. . Neuberg, J. J. B. (xn) Mathesis 2 (1882) 108-. . Alison, J. Edinb. Mth. S. P. 3 (1885) 77-. . Strnad, A. Casopis 15 (1886) 114- ; Fschr. Mth. (1886) 471. . Gibson, G. A. Edinb. Mth. S. P. 9 (1891) 50-. . Gnittner, A. Arch. Mth. Ps. 17 (1900) 318-. of cyclic polygon. Quint, N. N. Arch. Wisk. 3 (1898) 153-. , extension of Wallace theorem. Quint, N. N. Arch. Wisk. 3 (1898) 180-. - - , generalisation. Cayley, .4. QJ. Mth. 8 (1867) 264-. , , and application to tetra- hedron. Hunyady, E. von. [1865] Z. Mth. Ps. 11 (1866) 64-. of polygon. S teg gall, . Edinb. Mth. S. P. 14 (1896) 122-. , Steiner's theorem generalised. Beltrami, E. Bologna Ac. Sc. Mm. 7 (1876) 241-. and the Wallace point. Mackay, J. S. Edinb. Mth. S. P. 9 (1891) 83-. lines. Tucker, E. Edinb. Mth. S. P. 14 (1896) 116-. Squares of perpendiculars on sides from a point. Grebe, E. W. Grunert Arch. 9 (1847) 250-. Statics and geometry of a triangle. Saint- Germain, A. de. N. A. Mth. 3 (1884) 37-. Steiner theorem on triangle and ellipse. Droz-Farny, A. Bern Mt. (1900) 135-. theorems, two, and the properties of triangles. Kantor, S. [1877] Wien Ak. Sb. 76 (1878) (Ab. 2) 758-. Steiner 's^axes and Kiepert's hyperbola. Neu- berg, J., d* Gob, A. As. Fr. C. K. (1889) (Pt. 2) 166-. Straight line and triangle, relation. Hain, E. [1875] Arch. Mth. Ps. 59 (1876) 87-. Symmedian. Ocagne, M. d\ N. A. Mth. 2 (1883) 450- ; 3 (1884) 25- ; 4 (1885) 360-. . Azzarelli, M. Em. N. Line. At. 42 (1889) 255-. ' Symmedians. Mackay, J. S. Edinb. Mth. S. P. 14 (1896) 37-. Symmetrical lines in the triangle, and Steiner's curve. Bucking, . Arch. Mth. Ps. 16 (1898) 271-, vii-. properties. Davies, T. S. Ph. Mg. 2 (1827) 26-. Symmetrical properties. Dobson, T. B. A. Ep. (1861) (Pt. 2) 2-. Theorem of Beltrami. Grunert, J. A, Arch. Mth. Ps. 43 (1865) 102-. Mannheim. Catalan, E. Em. N. Line. Mm. 6 (1890) 223-. Theorems on triangles, circles and conies. Lemoine, E. As. Fr. C. E. (1900) (Pt. 2) 79-. Transformation. Lemoine, E. Par. S. Mth. Bll. 19 (1891) 133-. , continuous. Lemoine, E. As. Fr. C. E. (1891) (Pt. 2) 118- ; (1893) (Pt. 2) 132- ; Par. S. Mth. Bll. 19 (1891) 136- ; Mathesis 12 (1892) 58-, 81-, 264. Transformations. Mitirhead, R. F. Edinb. Mth. S. P. 13 (1895) 112-, 129-. of formulae. Lemoine, E. As. Fr. C. E. (1891) (Pt. 2) 118-. Transversal AD through angle A of triangle ABC such that AD*=BD . DC. Jelinek, V. Casopis 21 (1892) 232-; Fschr. Mth. (1892) 526. Transversals, equations relating to three. Kiechl, J. Arch. Mth. Ps. 12 (1894) 411-. , Matthew Stewart's theorem. Bretschneider, C. A. Arch. Mth. Ps. 50 (1869) 11-. , . Mackay, J. S. Edinb. Mth. S. P. 10 (1892) 90-. , properties. Boyman, J. R. Grunert Arch. 13 (1849) 364-, 378-. , reciprocal. Neuberg, . Brux. S. Sc. A. 24 (1900) (Pt. 1) 119-. , remarkable. Gob, A. Mathesis 18 (1898) 129-. , . Zimin, M. Fschr. Mth. (1900) 492. through a remarkable point, characteristic property. Plamenevsky, J. J. Fschr. Mth. (1891) 599. Triangle in which equal transversals cut base angles proportionally. Lange, T. Grunert Arch. 13 (1849) 337-; 15 (1850) 351-. . Grunert, J. A. Grunert Arch. 13 (1849) 341-. . . Mink, W. Grunert Arch. 15 (1850) 358-. . . Baltzer,R. Grunert Arch. 16 (1851) 201-. , lemma. August, E. F. Grunert Arch. 16 (1851) 259-. formed by medians. Grunert, J. A. Arch. Mth. Ps. 46 (1866) 340-. pairs of transversals through each angle of a triangle. Jelinek, V. Casopis 28 (1899) 88-, 145- ; Fschr. Mth. (1899) 466. and point, sum of whose distances from sides is a minimum. Cohen, A. Camb. and Dubl. Mth. J. 8 (1853) 92-. its reciprocal are in perspective, Cayley 's proof. Muir, T. Edinb. E. S. P. 20 (1895) 298-. with side parallel to Euler's line. Avillez, J. F. d' (Vicomte de Reguengo). As. Fr. C. E. (1897) (Pt. 2) 131-. Triangles of arcs of circles. Weyr, E. J. [1873] Bll. Sc. Mth. As. 6 (1874) 89-. 448 6820 Stereometry 6820 Triangles circumscribed in pairs by three circles. Tucker, It. [1892] L. Mth. S. P. 24 (1893) 162- . whose elements can be rationally expressed. Fuss, N. [1808] St Pet. Ac. Sc. Mm. 4 (1813) 240-. formed by common tangents of three circles. Barisien, E. N. Mathesis 16 (1896) 33-, 60-. joining points of contact of inscribed circles in turn, theorem. Lemoine, E. As. Fr. C. E. (1880) 184. having in- and circum-circles common. Boklen, 0. Grunert Arch. 38 (1862) 141-. with rational sides and areas. Breton, Ph. [1867] (xn) Isere S. Bll. 1 (1869) 299-. . Blichfeldt, H. F. A. Mth. 11 (1896-97) 57-. related to a triangle. Azzarelli, M. Km. N. Line. At. 40 (1887) 135-. - whose sides are integers prime to one another, ratio of two angles being an integer. Weill, M. Par. S. Mth. Bll. 10 (1882) 55-. and medians are integers. Grebe, E. W. Grunert Arch. 17 (1851) 463-. are multiples of radius of inscribed circle. Gerono, G. G. N. A. Mth. 17 (1858) 360-. , two, solution of problem of Frenicle's. Pepin, T. Km. N. Line. At. 33 (1880) 284-. Trilinear figures, geometry. Petersen, J. N. Ts. Mth. 9 (B) (1898) 49- ; Fschr. Mth. (1898) 473-. 6820 Stereometry ; straight lines, planes, and spheres ; polyhedra. Bees' cells. Terquem, 0. (vi Adds.) N. A. Mth. 15 (1856) 176-. , construction. Sharpe, S. Ph. Mg. 4 (1828) 19-. , . Powell, B. [1833] Ashmol. S. P. 1 (1844) No. 2, 10-. , . Wright, C. Am. Ac. P. 4 (1857-60) 432- ; Camb. (M.) Mth. M. 2 (1860) 304-. , . Willlich, C. M. Ph. Mg. 18 (1859) 427- ; C. R. 51 (1860) 633-. , . Hultman, F. W. Ts. Mt. Fys. 1 (1868) 197-. , . Glaisher, J. W. L. Ph. Mg. 46 (1873) 103-. , . Jonquiere, A. Bern Mt. (1884) (Heft 1, Ab.) 71-. , . Hennessy, H. R. S. P. 39 (1886) 253- ; 41 (1887) 442- ; 42 (1887) 176-. , . Goodicin, (Et. Rev.) H. Nt. 43 (1891) 295. , . Busmann, . [1895] Westf. Vr. Jbr. (1895-96) 194-. Cells, forms. Hennurn, J. 0. Arch. Mth. Ntvd. 9 (1884) 301-. Centre of gravity in stereometry. T.andre, C. L. Grunert Arch. 39 (1862) 361-. Cone, frustum. Grunert, J. A. Grunert Arch. 22 (1854) 343-. Cone, right. Bigourdan, E. N. A. Mth. 8 (1849) 9-. , right circular. Woepcke, F. Liouv. J. Mth. 6 (1861) 231-. , ("Woepcke's theorems). Steen, A. Mth. Ts. 4 (1862) 81-. , . Tat/lor, C. Mess. Mth. 4 (1875) 145- ; 5 (1876) 189-. , , angle-property. Taylor, C. [1871] Mess. Mth. 1 (1872) 67-. , and frustum with spherical base. Milller, J. H. T. Grunert Arch. 16 (1851) 462-. , scalene. Taylor, C. [1879] Mess. Mth. 9 (1880) 33-. Constructions in plane and in space with straight lineonly. Tilly, J.de. Mathesis 5 (1885)124-. Crystallographic derivations in the 1st system, table. Cesdro, E. Bll. Sc.Mth. 12 (1888) 270-. Division of frustum of cone or pyramid. Fli'tgl, J. Grunert Arch. 12 (1849) 423-. space into infinitesimal cubes. Tait, . [1892] Edinb. R. S. P. 19 (1893) 193-. surfaces and bodies, theorems. Olivier, T. Quetelet Cor. Mth. 5 (1829) 324-, 386-. Duality, principle. Gergonne, J. D. Ger- gonne A. Mth. 19 (1828-29) 114-. Euclid xi, 4. Grunert, J. A. Grunert Arch. 10 (1847) 260- ; 26 (1856) 106-. xi, 28. Hoffmann, J. J. I. Grunert Arch. 10 (1847) 77-. Figures, derivation by Bellavitis's method. Padula, F. Tortolini A. 5 (1854) 286-. , plane or spherical, of equal perimeter or surface. Thibault, . N. A. Mth. 2 (1843) 480-. Finite decomposition of two equivalent figures into wiperposable elements. Brunei, G. Bordeaux S. Sc. PV. (1896-97) 18-. Gauss's pentagramma mirificum. Cayley, A. Ph. Mg. 42 (1871) 311-. . Ahlborn, H. Hamb. Mth. Gs. Mt. 2 (1890) (Festschr., Tl. 2) 69-. . Schlesinger, L. Mth. Termt. Ets. 17 (1899) 526- ; Mth. Nt. B. Ung. 17 (1901) 20-. point in system of n planes. Neuberg, . Brux. S. Sc. A. 17 (1893) (Pt. 1) 16-., Geometrography in space. Lemoine, E. As. Fr. C. R, (1900) (Pt. 2) 60-; C. R. 131 (1900) 937-. Geometry of plane and space, analogies. Serret, P. Liouv. J. Mth. 7 (1862) 377-. , solid. Hachette, J. N. P. Par. S. Phlm. Bll. (1816) 201-. , . Quetelet, L. A. J. [1826] Brux. Ac. Sc. Mm. 4 (1827) 51-. , . Barsotti, G. [1829] Lucca At. Ac. 11 (1842) 113-. , , combination with plane geometry in teaching. Candido, G. Ens. Mth. 1 (1899) 204-. _, _ ( problems. Prediger, C. Halle Z. Nw. 12 (1858) 24-. Hexagonal pile of balls. Bach, . N. A. Mth. 4 (1845) 196-. Infinitesimal mensuration of circle, pyramid, etc. Noel, J. N. [1865] Liege Mm. S. Sc. 1 (1866) 51-. VOL. I. 449 6820 Stereometry. Mensuration 6820 Locus of intersection of 2 lines through given points, equally inclined to plane. King, L. [1881] Arch. Mth. Ps. 67 (1882) 330-. Macles (twin crystals). Cesdro, G. Brux. Mm. Cour. 4, 53 (1893-94) No. 5, 47 pp. Mensuration of solids. Delanges, P. [1804] Mod. S. It. Mm. 12 (1805) 1-. Mensuration of Volumes. Eossi-Amatis, J. [1805] Turin Mm. Ac. (1809-10) (pte. 2) 98-. Bordoni, A. Mod. Mm. S. It. 19 (1821) 527-. Ligoivski, W. Grunert Arch. 26 (1856) 204- ; 32 (1859) 241- ; 36 (1861) 181-. Steen, A. Mth. Ts. 4 (1862) 1-. Soufflet, . Les Mondes 17 (1868) 110-. Baillarge', C. Queb. T. 9 (1872) 73-. Circle and inscribed square, ratio, applications to measurement of timber. Taylor, C. Madras J. 6 (1837) 301-. Cone and cylinder, right circular. Frank, A. von. Arch. Mth. Ps. 59 (1876) 76-. , right, frustum, formula. Almeida, C. A. M. de. Lisb. J. Sc. Mth. 5 (1876) 208-. , segment. Grunert, J. A. Grunert Arch. 15 (1850) 356-. Conicoid of revolution. August, E. F. Crelle J. 45 (1853) 239-. Crystals, tesseral, cubic content of certain. Fttrst, J. Casopis 19 (1890) 20-; Fschr. Mth. (1890] 599-. Earth and wall work, solid contents. Puller, . Hann. Archt.-Vr. Z. 39 (1893) 549-. Earthwork and cuttings. Wojciechowski, L. Par. T. Nauk Sc. Pam. 5 (M874) Art. 3, 48 pp. ; 9 (*1877) Art. 5, 15 pp. in street sections, geometry of. Baggi, V. Tor. Ac. Sc. At. 33 (1897) 471- or 701-. Formula. Lombard, C. N. A. Mth. 16 (1857) 131-. Gauging of casks. Eytelwein, J. A. (vi Adds.) Bert. Ab. (1803) 93-. . Beeck-Calkoen, J. F. van. Eot. N. Vh. 5 (1810) 185-. . Lubbock, J. W. Ph. Mg. 4 (1834) 326-. . Stampfer, S. Wien SB. (1849) 227-. . Grunert, J. A. Grunert Arch. 20 (1853) '301- ; 23 (1854) 207-. and buoys, formula. Schroder, C. A. der Hydrog. 23 (1895) 459-. , doubly elliptic. Bruschetti, G. Brugnatelli G. 5 (1822) 165-. , or elliptic. Cossali, P. [1814] Mod. Mm. S. It. 17 (1815) 237-. , formulae. Mansion, P. Mathesis 12 (1892) 14-. , problem by Kepler, general solution. Monteiro da Eocha, J. Lisb. Ac. Sc. Mm. 1 (1797) 1-. , Prussian method. Stampfer, S. Wien SB. (1849) 291-. , Young's method. Wiseman, W. Tilloch Ph. Mg. 63 (1824) 415-. Guldin's theorem, and truncated prism, demon- stration. Lentheric, . Mntp. Ac. Sc. Mm. 2 (1851-54) 109-. Obelisks. Grunert, J. A. Grunert Arch. 9 (1847) 82-. Polygons, areas, and polyhedra, volumes. Bellavitis, G. A. Sc. Lomb. Yen. 4 (1834) 256-. , , , . Staudt, G. K. C. von. Crelle J. 24 (1842) 252-. Polyhedra. Lhuilier, S. Bb. Un. 37 (1828) 256-. . Mdbius, A. F. Leip. B. 17 (1865) 31-. bounded by triangles, volumes expressed by coordinates of vertices. Freuchen, P. (xn) Ts. Mth. 6 (1870) 189-. , regular. Schultze, F. Crelle J. 28 (1844) 108-. , . Dellrnann, F. Schlomilch Z. 8 (1863) 460-. , , and Poinsot's, and their volumes by determinants. Lowe, 0. Arch. Mth. Ps. 57 (1875) 392-. , symmetrical, equality of volumes. Ampere, A. M. Par. EC. Pol. Cor. 1 (1804-08) 184-. Polyhedron with trapezoidal faces. Koppe, C. N. A. Mth. 8 (1849) 108-. , volume. Newson, H. B. A. Mth. 1 (1900) 108-. Prismatoid. Grebe, E. W. Grunert Arch. 39 (1862) 93-. Prismoidal formula. Wright, C. Camb. (M.) Mth. M. 1 (1859) 21-, 53-. . Davis, J. W. V. Nost. Eng. Mg. 20 (1879) 410-. . Knibbs, G. H. N. S. W. E. S. J. 33 (1899) 129-. , limits. Hyde, E. W. Des Moines Anal. 3 (1876) 113-. formulae. Taylor, T. U. Texas Ac. Sc. T. 1 (No. 5) (1897) 33-. , two-term. Halsted, G. B. Texas Ac. Sc. T. 1 (No. 5) (1897) 19-. volumes. Grunert, J. A. Grunert Arch. 26 (1856) 341- ; 30 (1858) 118-, 453-. Prismoids. Kinkelin, H. Grunert Arch. 39 (1862) 181-. . Sinram, T. Arch. Mth. Ps. 63 (1879) 440-. Prisms. Matzka, W. Grunert Arch. 6 (1845) 113-. , truncated. Faa de Bruno, F. Tortolini A. 4 (1853) 71-. Problem. Grunert, J. A. Arch. Mth. Ps. 43 (1865) 108-. Pyramid and sphere. Havllcek, T. V. Casopis 18 (1889) 76- ; Fschr. Mth. (1889) 577. , triangular. Hessel, J. F. C. Grunert Arch. 14 (1850) 162-. , . Grunert, J. A. Grunert Arch. 18 (1852) 239-. , . Heinemann, P. G. K. Grunert Arch. 23 (1854) 361-. , . Faa de Bruno, F. Tortolini A. 8 (1857) 77-. , truncated. Anon, (vi 565) Gergonne A. Mth. 20 (1829-30) 288-. 450 6820 Mensuration Pyramid, truncated. Dietzel, F. Grunert Arch. 13 (1849) 447-. , , n sides, volume = h(B + b+ \/Bb). Flesch, J. Grunert Arch. 3 (1842) 444-. , volume, theorem by Pappus extended to. Paraira, M. C. N. Arch. Wisk. 9 (*1882) 96-. , , . Boguslavskij, A. I, Mosc. S. Sc. Bll. 73 (No. 1) (1891) 27-; Fschr. Mth. (1891) 614. Simpson's rule. Lodge, A. Mth. Gz. No. 6 (1895) 58-. rules, and Woolley's rule, value in ship- building calculations. Pur kiss, H. J. [1865] QJ. Mth. 7 (1866) 235-. Solids with two parallel faces, theorems. Steiner, J. Crelle J. 23 (1842) 275-. , volume, in terms of transverse sections. Knibbs, G. H. N. S. W. B. S. J. 34 (1900) 36-. Sphere. Wittstein, T. Grunert Arch. 39 (1862) . 'Saltel, L. [1880] Bordeaux S. Sc. Mm. 4 (1882) 375-. . Postnikov, M. V. Kazan S. Nt. (Ps.-Mth.) P. 5 (1887) 124-. , portion bounded by planes not passing through centre, volume and area. Somov, I. I. (xn) Eec. Mth. (Moscou) 3 (1868 (Pt. 1) 79-. , portions. Woodhouse, R. Phil. Trans. (1801) 153-. and portions of sphere, volume and area. Gergonne, J. D. Gergonne A. Mth. 13 (1822-23) 343-. Spherical segment. Bary, . Gergonne A. Mth. 21 (1830-31) 326-. . Guimaraes, R. Lisb. J. Sc. Mth. 5 (1898) 172- ; Fschr. Mth. (1898) 444. between two arbitrary planes, volume and area. Czuber, E. Crelle J. Mth. 105 (1889) 180. Tetrahedra. Gergonne, J. D. Gergonne A. Mth. 19 (1828-29) 151-. . Lhuilier, S. Bb. Un. 45 (1830) 208-. . Hunyady, E. von. Z. Mth. Ps. 11 (1866) 163-. . Dostor, G. N. A. Mth. 6 (1867) 410-. . Gunther,S. Arch. Mth. Ps. 56 (1874) 17-. . Faure, H. N. A. Mth. 20 (1881) 338-. , class. Hill, M. J. M. [1895] L. Mth. S. P. 27 (1896) 39-. , polar reciprocal, volumes, relation. Luch- terhandt, A. R. Grunert Arch. 10 (1847) 198-. , relation between volume, area of faces and area of projection on a plane, theorem by Eoberts, generalisation. Lebesgue, V. A. N. A. Mth. 20 (1861) 63-. , volumes, from equations of bounding elements. Studnitka, F. Prag Sb. (1872) (pt. 2) 45-. , and normal, relation. Sylvester, J. J. N. A. Mth. 13 (1854) 203-. , product of 16 algebraic values of surface. Sylvester, J. J. Camb. and Dubl. Mth. J. 8 (1853) 171-. Stereometry 6820 Tetrahedra, volumes, theorem. Grunert, J. A. Arch. Mth. Ps. 45 (1866) 66-. . , , Staudt's. Gentil, . N. A Mth 19 (1860) 218-. , , and theorem by Steiner. Virieu, J. de. N. A. Mth. 5 (1866) 316-. , , . Sartiaux, A. N. A. Mth 5 (1866) 317-. i triangles, areas, in terms of sides, Joachimsthal's formulae. Droz, A. N. A Mth. 20 (1881) 411-. > i , , theorems. Gretschel, H. Arch. Mth. Ps. 45 (1866) 194-. Tetrahedron, etc., theorems. Staudt, G. K. C. von. Crelle J. 57 (1860) 88-. Theorems, elementary. Koppe, C. N. A. Mth. 6 (1847) 480-. , , proof. Hessel, J. F. C. Grunert Arch. 7 (1846) 284-. Tortoise, approximate valuation of volume and surface. Maurel, , & Rey-Pailhade, . Toul. S. H. Nt. Bll. 33 (1900) 79-, 88. Transformation, fundamental, in stereometry. Bordoni, A. (vi Adds.) Opusc. Mt. Fis. 1 (1832) 14-. Trapezoidal volumes. Giard, A. N. A. Mth. 6 (1867) 408-. Varignon's theorem, extension to volumes of three cones. Hocevar, F. Wien Az. 17 (1880) 196-. Minimum distance problem. Talbot, W. H. F. Gergonne A. Mth. 13 (1822) 329-. Obelisks, formation, etc. Grunert, J. A. Grunert Arch. 9 (1847) 87-. , , (Grunert). Brix, A. F. W. Grunert Arch. 11 (1848) 339-. , Koppe's. Bretschneider, C. A. Grunert Arch. 36 (1861) 18-. , and prismatoid. Schlomilch, 0. Z. Mth. Ps. 13 (1868) 160-. , reality, problem. Schellen, . Grunert Arch. 11 (1848) 341-. , , . Matzka, W. Grunert Arch. 11 (1848) 377-. Perpendicular to given plane through two given straight lines. Grunert, J. A. Grunert Arch. 37 (1861) 445-. Planes, seven, problem. Poudra, . N. A. Mth. 15 (1856) 161- . Point from which 3 other given points sub- tend given angles, determination. Bessel, F. W. Zach M. Cor. 27 (1813) 222-; As. Nr. 3 (1825) 193-, 221-. , 2 straight lines and plane, problem. Flauti, V. Nap. At. Ac. 3 (1832) 35-. Points in space, regular assemblages, properties. Bravais, A. C. E. 28 (1848) 601- ; Par. S. Phlm. PV. (1848) 69-; Par. EC. Pol. J. 33" cah. (1850) 1-. Polar trihedral angles and polar triangles. Harst, A. D. van der. N. Arch. Wisk. 17 (1890) 188- ; Fschr. Mth. (1890) 600. Polygons, skew and plane. Sturm, J. C. F. Gergonne A. Mth. 15 (1824-25) 309-. , plane and spherical, theorems. Bernardi, G. G. Mt. 29 (1891) 63-, 173-, 378. 451 FF2 6820 Stereometry. Polyhedra POLYHEDEA. (See also 8075.) Cauchy, A. L. Par. EC. Pol. Cor. 2 (1809-13) 253- ; Par. EC. Pol. J. 16 cah. (1813) 68-, 87-. Anon, (vi 531) Gergonne A. Mth. 9 (1818-19) 321-. Poinsot, L. C. E. 46 (1858) 65-. (Descartes.) Prouket, E. C. E. 50 (1860) 779-. Catalan, E. C. [1862] Par. EC. Pol. J. Cah. 41 (1865) 1-. Kirkman, T. P. Phil. Trans. (1862) 121- ; E. S. P. 12 (1862-63) 341-. Jordan, C. C. E. 60 (1865) 400- ; 61 (1865) 205- ; 62 (1866) 1339- ; Crelle J. 66 (1866) 22-; 68 (1868) 297-. Becker, J. C. Z. Mth. Ps. 14 (1869) 65-, 337-; 18 (1873) 328-. Landre, C. L. N. Arch. Wisk. 4 (*1878) 194-. Lippich, F. [1881] Wien Ak. Sb. 84 (1882) (Ab. 2) 20-. Bourlet, C. N. A. Mth. 8 (1889) 366-. Brunei, . Bordeaux S. Sc. Mm. 3 (1893) xxxvii-. Bricard, R. N. A. Mth. 15 (1896) 331-. Hermes, 0. Crelle J. Mth. 120 (1899) 27-, 305- ; 122 (1900) 124-. Angles, new theorems. Brianchon, C. J. Par. EC. Pol. J. 25 e cah. (1837) 317-. Archimedean polyhedra of higher kind, four. Hess, E. Marb. Schr. 11 (Ab. 4) (1878) 12pp. Articulated octahedron. Bricard, R. Liouv. J. Mth. 3 (1897) 113-. (Bricard). Mannheim, A. Liouv. J. Mth. 3 (1897) 149- . Auto-polar polyhedra. Kirkman, T. P. Phil. Trans. (1857) 183-. , solution and constructions up to p = 10. Kirkman, T. P. Lpool. Lt. Ph. S. P. 33 (1879) 133-. Axial properties. Cayley, A. [1863] (VH) QJ. Mth. 7 (1866) 304-. Axis of ternary symmetry. Viola, C. [1900] Pisa S. Tosc. At. (PV.) 12 (1899-1901) 75-, 138-. Central polyhedron. Wolf, E. Bern Mt. (1847) 93-. Classification. Breton, Ph. C. E. 51 (1860) 722-. ' . Fasci,A. Fr. Cg. Sc. 33 (1866) (pt. 2) 87-. . Eberhard,V. D. Mth. Vr. Jbr. 1 (1892) 50-. Congruent polyhedra. Dehn, M. Gott. Nr. (1900) 345-. Constants, number, and Euler's theorem. Schubert, H. C. H. Arch. Mth. Ps. 63 (1879) 97-. Construction. Kirkman, T. P. Lpool. Lt. Ph. S. P. 32 (1878) 217-. of special polyhedra. Brunei, . Bordeaux S. Sc. Mm. 5 (1895) xliv-. Convex polyhedra. Minkoivski, H. Gott. Nr. (1897) 198-. , angular relations. Lalanne, L. C. E. 74 (1872) 602-. 6820 Crystallography, and analytical theory of poly- hedra. Baptista, I. E. Lisb. A. 2 (1858) 257-. Deformable polyhedra. Pizzetti, P. Em. E. Ac. Line. Ed. 7 (1898) (Sem. 2) 19-. Diagonals, number. Binder, H. Grunert Arch. 8 (1846) 221-. , . Prouhet, E. N. A. Mth. 2 (1863) 77-. , in polygons and polyhedra. Vauthier, . Toul. Mm. Ac. 2 (1830) 95-. Divisions, centroids, etc. Padula, F. Nap. At. I. Inc. 1 (1864) 37-. Eberhard's hexagonoids. Schoenflies, A. Gott. Nr. (1894) 316-. Equality of similar polyhedra. Cauchy, A. L. Par. S. Phlm. Bll. 3 (1812) 66-. , etc., theorems. Thibault, . N. A. Mth. 2 (1843) 163-. Equiangular polyhedra from crystallographic standpoint. Krejci, J. Prag Sb. (1885) (Mth.-Nt.) 120- ; Fschr. Ps. (1885) (Ab. 1) 178. Euler's polyhedra. Bertini, E. [1869] Pisa A. Scuola Norm. 1 (1871) 89-. . Feil, M. Wien Ak. Sb. 93 (1886) (Ab. 2) 869-. relation. De Morgan, A. (\iAdds.) Ph. Mg. 12 (1838) 323-. theorem. Gergonne, J. D. Gergonne A. Mth. 19 (1828-29) 333-. . Baltzer, R. Berl. Mb. (1861) 1043-. . Schdffer, H. Z. Mth. Ps. 9 (1864) 365-. . Bugaev, N. V. (xn) Eec. Mth. (Moscou) 2 (1867) (Pt. 2) 87-. . Crone, C. Ts. Mth. 3 (1885) 44-; Fschr. Mth. (1885) 571-. . Jonquieres, de. C. E. 110 (1890) 110-, 169-. . Weill, E. N. A. Mth. 17 (1898) 120-. , apparent limitations. Matthiessen, L. Schlomilch Z. 8 (1863) 449-. , corollary. Hessel, J. F. C. Crelle J. 8 (1832) 13-. , demonstration. Rothe, H. A. Kastner Arch. Ntl. 4 (1825) 301-. , . Steiner, J. Crelle J. 1 (1826) 364-. , . Pollak, (Dr) . [1849] Haidinger B. 6 (1850) 93-. , . Thiel, J. M. N. Arch. Wisk. 19 (1892) 98-. , and exceptions. Lhuilier, S. St Pet. Ac. Sc. Mm. 3 (1811) 271-. , exceptions. Schulz von Strasznitzki, C. Crelle J. 14 (1835) 83-. , generalisation. Listing, J. B. Gott. Ab. 10 (1861-62) (Mth.) 97-; D. Nf. Tbl. (*1878) 77, 224-. , . Hoppe, R. D. Nf. Tbl. (*1878) 39 ; Arch. Mth. Ps. 63 (1879) 100-. , . Perrin, R. C. E. 110 (1890) 273-. , . Poincare, H. C. E. 117 (1893) 144-. Existence of certain polyhedra. Cesdro, E. N. A. Mth. 2 (1883) 46-. 452 6820 Stereometry. Polyhedra 6820 General laws. Gergonne, J. D. Gergonne A. Mth. 15 (1824-25) 157-. Harmonic polygons and polyhedra. Tarry, G., & Neuberg, J. As. Fr. C. E. (1886) (Pt. 2) 12-. Hexahedron inscribed in sphere. Merrifield, C. W. Ph. Mg. 22 (1861) 382-. In- and circum-scribed polyhedra. Forsyth, A. E. [1882] L. Mth. S. P. 14 (1883) 35-. Inscriptible polygons and polyhedra, properties. Brassinne, E. Toul. Mm. Ac. 3 (1847) 209-. Isosceles polyhedra. Badoureau, A. C. E. 87 (1878) 823- ; Par. EC. Pol. J. call. 49 (1881) 47-. Maxima, theorem. Berner, T. [1865] Z. Mth. Ps. 11 (1866) 81-. Maximum and minimum, cubes and regular tetrahedra as. Sturm, E. Crelle J. Mth. 97 (1884) 1-. polyhedra. Lindelof, L. Helsingf. Acta 24 (1899) No. 8, 47 pp. ' Mensuration. Collalto, A. Padova Mm. Ac. (1809) 247-. . Franc s ais, J. F. Gergonne A. Mth. 3 (1812-13) 189-. . Lhuilier, S. Bb. Un. 37 (1828) 249-. . Koppe, C. [1837] Crelle J. 18 (1838) 275-. . Becker, J. K. Grunert Arch. 38 (1862) 345-; 40 (1863) 12-. , theorem, elementary proof. Martini, M. Palomba Eac. 1 (1845) 84-. , theorems. Lhuilier, S. [1799] Par. Mm. Sav. Etr. 1 (1806) 264-. Mobius's theory. Eeinhardt, C. Leip. Mth. Ps. B. 37 (1885) 106-. Parallelepipedal system, conditions of perpen- dicularity. Smith, H. J. S. [1876] L. Mth. S. P. 8 (1876-77) 83-. Parallelepipeds, diagonals. Smith, Arch. Camb. Mth. J. 1 (1839) 94-. , equality, demonstration. Matzka, W. Grunert Arch. 4 (1844) 363-. and parallelograms. Gergonne, J. D. Gergonne A. Mth. 9 (1818-19) 51-. , partition into tetrahedra. Brown, C. Edinb. E, S. T. 37 (1895) 711-. , rectangular, theorem. Mann, F. Grunert Arch. 34 (1860) 116-. , skew. Littrow, J. J. von. Lindenau Z. 3 (1817) 42-. Pentahedron. Le Paige, C. Brux. Ac. Bll. 13 (1887) 488-. of given volume and minimum surface. Sucksdor/, C. G. Liouv. J. Mth. 2 (1857) 91-. Polar properties. Olivier, T. Quetelet Cor. Mth. 3 (1827) 187-. Polyacrons, A-faced. Cayley, A. Manch. S. Mm. 1 (1862) 248-. Polygons and polyhedra. Poinsot, L. [1809] Par. Mm. Sav. Etr. 2 (1811) 552-. . Timmermans, J. A. Gergonne A. Mth. 18 (1827-28) 217-. . Cauehy, A. L. C. E. 26 (1848) Polygons and polyhedra. Mainardi, G. Em. At. N. Line. 26 (1873) 80-. , polyhedra and polygrams. Hertzer, (Dr) . Z. Mth. Ps. 11 (1866) 244-. and polyhedra, theorems. Bellavitis, G. A. Sc. Lomb. Yen. 2 (1832) 186-, 241-. , . Sylvester, J. J. Ph. Mg. 4 (1852) 335-. Polyhedra with both angles and faces equal. Hess, E. Marb. Schr. 11 (AT). 1) (1876) 95pp. congruent with their reflexions, but not symmetrical. Juel, C. N. Ts. Mth. 6 (B) (1895) 12-. with minimum surface for given volume. Kotter, E. Crelle J. Mth. 110 (1892) 198-. superposable on their images. Cesdro, G. [1896-97] Brux. Ac. Mm. 53 (1895-98) (No. B) 40pp., (No. 5) 15pp. , tetrahedron, and circum-sphere. Brassinne, E. Toul. Mm. Ac. 3 (1847) 209-. with trihedral summits and having neither triangle nor quadrilateral. Kirkman, T. P. [1882] Lpool. Lt. Ph. S. P. 37 (1883) , n sides and (n-l)-gonal base. Kirkman, T. P. [1855] Phil. Trans. (1856) 399-. Prismoids and trapezoids. August, E. F. Crelle J. 60 (1862) 377-. Prisms, cones, and rhombs, acrometric relations. Eiedl von Leuenstern, J. [1850] Haidinger Ab. 4 (1851) (Ab. 2) 49-. Problem, general solution. Kirkman, T. P. [1858] Manch. Ph. S. Mm. 15 (1860) 92-. Properties. Gough, J. Nicholson J. 25 (1810) 321-. Property. Minich, S. B. A. Sc. Lomb. Yen. 2^(1832) 15-. , general, demonstration. Andreev, K. A. (xn) Eec. Mth. (Moscou) 6 (1872-73) (Pt. 1) 457-. Pyramid of Cheops and geometrical structures. Zhbikovskil, A. K. [1882] (xn) Kazan S. Nt. (Ps.-Mth.) P. 1 (1883) [No. 5] 15-. , Great. Piani, D. Bologna Mm. Ac. Sc. 10 (1859) 391-. , , angle of. Willich, C. M. Les Mondes 10 (1866) 479. , pentahedral, on trapezium as base. Ter- quem, 0. N. A. Mth. 1 (1842) 396-. Pyramids, analysis. Svanberg, A. F. Stockh. Ak. Hndl. (1829) 55-. , centroid, and other theorems. Ferroni, P. [1812] Mod. S. It. Mm. 16 (1813) 347-. or cones, solid angle at vertex. Kramp, C. Gergonne A. Mth. 4 (1813-14) 187-. , corners, theorem. Miiller, J. H. F. Grunert Arch. 2 (1842) 113-. with equal bases and heights. Hessel, J. F. C. Arch. Mth. Ps. 47 (1867) 433-. and frustum. Tortolini, B. (vm) A. Mt. 4 (1861) 175. polygons, analysis. Svanberg, A. F. Quetelet Cor. Mth. 9 (1837) 72-. , problem : to find 3 equal pyramids together equal to portion of earth. Moolenaar, A. Leijd. A. Ac. (1817-18) 9 pp. 453 6820 Stereometry. Regular Polyhedra 6820 Pyramids, sum of solid angles. Leuenstern, J. R. von. [1849] Haidinger Ab. 3 (1850) (Ab. 2) 87-. , theorem. Catalan, E. C. N. A. Mth. 6 (1847) 353-. Regular Polyhedra. Legendre, A.M., & Mains, . Par. S. Phlm. Bll. 2 (1811) 325-. Lhuilier, S. Gergonne A. Mth. 3 (1812-13) 233-. Gamier, J. G. Quetelet Cor. Mth. 1 (1825) 39-, 64-. Gilbert, D. Ph. Mg. 3 (1828) 161-. Fischer, . Grunert Arch. 11 (1848) 159-. Bertrand, J. C. E. 46 (1858) 79-. Steichen, . Brux. Mm. Cour. 8, 9 (1859) 32pp. Bjorling, E. G. Ts. Mt. Fys. 3 (1870) 145-. Cayley, A. QJ. Mth. 15 (1878) 127-. Azzarelli, M. Bm. N. Line. Mm. 4 (1888) 123-. Steggall, . Edinb. Mth. S. P. 7 (1889) 66. Fiedler, W. Ziir. Vjschr. 35 (1890) 343-. Angles between faces, and volumes. Verdam, G. J. Groningen A. Ac. (1822-23) 79 pp. Conjugate polyhedra. Dostor, G. [1877] Arch. Mth. Ps. 62 (1878) 285-. Construction on one face. Krinicyn, P. K. Kazan S. Ps.-Mth. Bll. 1 (1891) (Prot.) 68. of the 5 regular solids. Sohncke, L. A. Arch. Mth. Ps. 47 (1867) 39-. Convex polyhedra, new properties. Dostor, G. Arch. Mth. Ps. 59 (1876) 50-. Group theory. Schoute, P. H. N. Arch. Wisk. 20 (1893) 100-. In- and circum-spheres, radii. Sommer, B. Grunert Arch. 32 (1859) 289-. Maxima. Babinet, J. C. B. 61 (1865) 827-, 973. Mensuration. Zbikovskij, A. K. Kazan S. Nt. (Ps.-Mth.) P. 4 (1886) 80-. Properties. Cauchy,A.L. C. E. 26 (1848) 517-. analogous to Stewart's of regular polygons. Jeffery, H. M. [1882] E. S. P. 34 (1883) 105-. Property. Todhunter, I. Mess. Mth. 3 (1866) 142-. Eegular polyhedra obtained by folding paper. Wiener, . Z. Nw. 61 (1888) 631-. Eelations between angles. Drach, S. M. Mess. Mth. 3 (1874) 6. Space configurations, derivable. Fries, J. de. Wien Ak. Sb. 100 (1891) (Ab. la) 822-. Sphere, property of, and regular polyhedra. Lhuilier, S. Bb. Un. 36 (1827) 260-. Ten polyhedra, table of principal elements. Barbier, E. C. E. 101 (1885) 562-. Theorems. Mollame, V. G. Mt. 9 (1871) 64-. . Akerlund,J. R. N. Ts. Mth. 9 (A) (1898) 1-. 6-hedra, cube, division into 4 equal parts. Willich, C. M. C. E. 56 (1863) 100-. , , partitions, etc. Willich, C. M. Ph. Mg. 33 (1867) 27-. , Eonayne's cubes (one passing through the other). Hennessy, H. B. A. Ep. (1894) 8-hedra, Levy's theorem. Davies, T. S. Tilloch Ph. Mg. 67 (1826) 52-. 9-hedra, 9-acral, enumeration and construction. Kirkman, T. P. Lpool. Lt. Ph. S. P. 32 (1878) 177-. 12-hedra, description. Barbier, E. C. E. 95 (1882) 560-. , janal. Kirkman, T. P. Lpool. Lt. Ph. S. P. 29 (1875) 251-. 14-hedra, janal, 14-acral. Kirkman, T. P. Lpool. Lt. Ph. S. P. 30 (1876) 271-. 20-hedra, convex, sixteen nets of planes of. He'nard, E. C. E. 101 (1885) 232-. , , . Barbier, E. C. E. 101 (1885) 304-, 540. and star 12-hedra, construction and represen- tation. Schonemann, P. [1872] Z. Mth. Ps. 18 (1873) 387-. 12-hedra. Wicke, C. Grunert Arch. 25 (1855) 131-. , relations in projections. Schubert, J. Z. Mth. Ps. 19 (1874) 460-. , theorems. Akerlund, J. R. N. Ts. Mth. 8 (A) (1897) 17-. Eegular and semi-regular polyhedra. Babinet, J. C. E. 26 (1848) 530-. Eepresentation. Kirkman, T. P. [1855] Phil. Trans. (1856) 413-. and enumeration. Kirkman, T. P. [1853] Manch. Ph. S. Mm. 12 (1855) 47-. Ehomboidal bodies, especially rhomboidal tri- acontahedra. Rothe, H. A. Kastner Arch. Ntl. 4 (1825) 129-, 257-. Self -con jugate polygons and polyhedra. John- son, A. R. QJ. Mth. 22 (1887) 158-. Semi-regular polyhedra of Archimedes. Valat, . Bordeaux Mm. S. Sc. 5 (1867) 319-. , etc., series complementary to. Valat, . Bordeaux Act. Ac. Sc. 28 (1866) 385-. , convex. Terrier, L. [1875] Neuch. S. Sc. Bll. 10 (1876) 201-. , resume of theory. Catalan, E. C. As. Fr. C. E. 1 (1872) 147-. , some. Hayicard, R. B. [1897] Mth. Gz. 1 (1900) 73-. with 6 squares and 8 triangles as faces or 12 pentagons and 20 triangles. Wezel, L. Louvain A. Ac. 6 (1826) 60 pp. Seven-point polyhedron, theorem. Anon. (vi 310) Crelle J. 3 (1828) 199-. Similitude of solid figures. Dcllac, H. Mars. Fac. Sc. A. 9 (1899) Fasc. 1, 108pp. Star polygons and polyhedra. Poinsot, L. N. A. Mth. 8 (1849) 68-, 132-. (Poinsot). Dienger, J. Grunert Arch. 13 (1849) 434-. polyhedra. Cauchy, A. L. N. A. Mth. 8 (1849) 304-. (Poinsot). Cayley, A. Ph. Mg. 17 (1859) 123-, 209-. . Valat, . As. Fr. C. E. 1 (1872) 155. , regular. Wiener, C. Z. Mth. Ps. 12 (1867) 174-. , , general properties. Dostor, G. Liouv. J. Mth. 5 (1879) 209-. , , three spheres of. Dostor, G. [1877] Arch. Mth. Ps. 62 (1878) 78-. 454 6820 Stereometry. Tetrahedra 6820 Surface area. Grunert, J, A. Arch. Mth. Ps. Maximum tetrahedron with faces of given area. 53 (1871) 482-. Painvin, L. N. A. Mth. 1 (1862) 267-, Surfaces, fundamental theorem, new proof, 353-, 414-. etc. Becker, J.C. Z. Mth. Ps. 19 (1874) 459-. . BorcMrdt, C. W. Symmetrical polyhedra. Bravais, A. Liouv. Berl. Ab. (1865) (Nth.) 1-. J. Mth. 14 (1849) 137-, 141-. . Le Besgue, V. A. C. , division. Miibius, A. F. (vi Adds.) Crelle J. 4 (1829) 296-. Symmetry of non-Eulerian polyhedra. Jordan, C. Crelle J. 66 (1866) 86-. , inverse. Jordan, C. Crelle J. 68 (1868) 350-. in polyhedra. Cesdro,G. Brux. Mm. Cour. 4, 53 (1893-94) No. 4, 34 pp. Tetrahedra. L., J. (vi Adds.) Gergonne A. Mth. 1 (1810- 11) 353-. (Joachimsthal.) Liersemann, H. Grunert Arch. 32 (1859) 107-. Junghan, G. Grunert Arch. 34 (1860) 369-. Torry, A. F. Mess. Mth. 1 (1862) 156-. Cayley, A. Mess. Mth. 3 (1866) 8-. Transon, A. N. A. Mth. 12 (1873) 519-. Pellet, A. E. As. Fr. C. B. (1881) 170-. Thieme, H. Z. Mth. Ps. 27 (1882) 56-. Intrigila, C. Nap. Ed. 22 (1883) 69-. Neuberg, J. Brux. Mm. Cour. 8, 37 (1886) No. 1, 72pp. Strnad, A. Casopis 29 (1900) 146- ; Fschr. Mth. (1900) 506. Angle, 6th, in terms of other five. Clausen, T. Crelle J. 8 (1832) 138-. Application of symmetric functions. Reiss, M. Quetelet Cor. Mth. 10 (1838) 229-. Cube in tetrahedron and square in triangle. Ferriot, . Gergonne A. Mth. 2 (1811-12) 180-. Desmic systems of three tetrahedra. Steplianos, C. Bll. Sc. Mth. As. 3 (1879) (Pt. 1) 424-. tetrahedra. Caspary,F. Mth. A. 29 (1887) 581-. , construction of three. Schroeter,' H. Crelle J. Mth. 109 (1892) 341-. Dihedral angles. Clausen, T. N. A. Mth. 5 (1846) 374-. Distance of summits from plane, theorem. Faure, H. N. A. Mth. 20 (1861) 222-. Doubly homologous tetrahedra. Gallucci, G. [1898] G. Mt. 37 (1899) 1-. Edges, opposite, inclination in terms of edges. Ferrers, N. J/. QJ. Mth. 3 (1860) 145-. Equilateral tetrahedra. Hoppe, E. Arch. Mth. Ps. 12 (1894) 327-. and orthocentric tetrahedra. Hoppe, R. Arch. Mth. Ps. 16 (1898) 257-, 333-. Expression by coordinates of vertices. Baltzer, R. Leip. B. 22 (1870) 97-. Geometric principle of resultants, application. Chelini, D. Bologna Ac. Sc. Mm. 7 (1867) 79-. Irregular tetrahedron, problems. Kuijper[s], T. Leijd. A. Ac. (1835-36) 32pp. Locus problem. Maur, A. Grunert Arch. 19 (1852) 121-. R. 66 (1868) 248-. . Mertens, F. Crelle J. Mth. 83 (1877) 180-. Minimum tetrahedra and sections of trihedra. Bermann, . Arch. Mth. Ps. 6 (1888) 76-, vn. theorem. Steiner, J. N. A. Mth. 12 (1853) 47-. Mobius's tetrahedra. Neuberg, J. Liege S. Sc. Mm. 11 (1885) No. 7, 14pp. Nine point circle, analogues in space of three dimensions. Roberts, S. L. Mth. S. P. 19 (1889) 152- . Orthocentric tetrahedra. Longchamps, G. de. Mathesis 10 (1890) 49-, 77-. with rational edges. Hoppe, R. Arch. Mth. Ps. 9 (1890) 434-. Pairs of tetrahedra. Muth, P. Z. Mth. Ps. 37 (1892) 117-. Parallelogram, property, analogues in space. Walker, J. J. [1866] Mess. Mth. 4 (1868) 144-. Perpendicular planes bisecting edges, theorem. Flemming, C. G. Grunert Arch. 10 (1847) 326-. Perpendiculars on faces from any point, relation between. Gilbert, P. Brux. Ac. Bll. 28 (1869) 535-. Points. Hermes, O. [1865] Crelle J. 65 (1866) 293-. of symmetry. Hain, E. [1876] Arch. Mth. Ps. 60 (1877) 304-. Polar tetrahedra, properties. Poudra, N. G. N. A. Mth. 4 (1865) 496-. Positions, special, of two tetrahedra. Schur, F. Mth. A. 19 (1882) 429-. Problems. Baltzer, R. Grunert Arch. 16 (1851) 125-. . Azzarelli, M. Em. N. Line. At. 29 (1876) 126-. Properties. Lankeren-Matthes, D. ran. Leijd. A. Ac. (1835-36) 33 pp. -. Ferrers, N. M. [1861] QJ. Mth. 5 (1862) 167-. . Gellenthin, H. Arch. Mth. Ps. 3 (1886) 52-. , generalisation of certain. Brill, J. Camb. Ph. S. P. 9 (1898) 98-. Property. Dostor, G. Arch. Mth. Ps. 56 (1874) 245-. Pyramid, triangular. Monge, G. Par. EC. Pol. Cor. 1 (1804-08) 440- ; 2 (1809-13) 1-, 263-. , . Hachette, J. N. P. Par. EC. Pol. Cor. 2 (1809-13) 332-. . , . Schulz von Strasznitzki, L. C. Baumgartner Z. 2 (1827) 530-. , . Bretschneider, C. A. Grunert Arch. 1 (1841) 1-. , . Clausen, T. As. Nr. 19 (1842) 389-. , . Grunert, J. A. Grunert Arch. 21 (1853) 352-; 23 (1854) 284-. , , formula. Hoppe, R. Grunert Arch. 3 (1843) 213-. 455 6820 Stereometry. Tetrahedra Pyramid, triangular, formulae. Grunert, J. A. Grunert Arch. 36 (1861) 356-. , , and inscribed spheres. Unferdinger, F. Grunert Arch. 28 (1857) 97-. , , problem. Hochette, J. N. P. Quetelet Cor. Mth. 2 (1826) 211-. , , problems. Flauti, V. Nap. At. Ac. 1 (1819) 51-. , , solution. Hachette, J. N. P. Par. EC. Pol. Cor. 1 (1804-08) 41-, 273-. Quadrilaterals and tetrahedra, theorems, demonstration. Scheerer, T. Crelle J. 6 (1830) 98-. Eational tetrahedra. Schwering, K. Crelle J. Mth. 115 (1895) 301-. Eeciprocal triangles and tetrahedra. Ferrers, N. M. QJ. Mth. 1 (1857) 191-. Solid angles and tetrahedra. Hellwig, C. [1875] Arch. Mth. Ps. 58 (1876) 180-. , with application of determinants. Dostor, G. Arch. Mth. Ps. 57 (1875) 113-. Species. Hill, M. J. M. B. A. Up. (1895) 619-. Spheres, circum-scribed. Brassinne, E. Toul. Mm. Ac. 3 (1847) 209-. , , radius in terms of edges. Dostor, G. N. A. Mth. 13 (1874) 523-. , e-scribed, radii. Unferdinger, F. Arch. Mth. Ps. 50 (1869) 110-. , in- and circum-scribed, radii. Dostor, G. N. A. Mth. 12 (1873) 367-, 370-. , e-scribed, radii, Format's theorem. Stammer, W. Arch. Mth. Ps. 50 (1869) 111-. , inscription. Hermary, . Par. S. Mth. Bll. 7 (1879) 138-. , tetrahedra always inscriptible in, theorem. Baudrimont, A. Bordeaux Mm. S. Sc. 3 (cah. 2) (1865) 445-. Surfaces, tetrahedral, two classes of surfaces analogous to. Ra/y, L. Par. S. Mth. Bll. 24 (1896) 2-. Tetrahedra bounded by similar triangles. Schoute, P. H. Amst. Ak. Vs. M. 6 (1889) 462-; Fschr. Mth. (1889) 578-. with 6 edges touching sphere. JungJian, G. Grunert Arch. 40 (1863) 447-. . Dostor, G. N. A. Mth. 13 (1874) 563-. in- and circum-scribed to each other. Mobius, A. F. Crelle J. 3 (1828) 273-. . Bauer, G. Munch. Ak. Sb. 27 (1898) 359-. with opposite edges equal in pairs. Lemoine, E. As. Fr. C. E. (1875) 173-. , and solution of problem. Lemoine, E. N. A. Mth. 19 (1880) 133-. perpendicular, theorem. Wolsten- holme, J. QJ. Mth. 3 (1860) 89. at right angles. Lewis, T. C. Mess. Mth. 10 (1881) 36-. . Temperley, E. [1881] Mess. Mtb. 11 (1882) 114-. related to 4 spheres meeting in point. Roberts, S. L. Mth. S. P. 12 (1880-81) 117-. of same base and altitude, equivalence. Gergonne, J. D. Gergonne A. Mth. 12 (1821-22) 362-. Tetrahedra of same base and altitude, equiva- lence. Querret, . Gergonne A. Mth. 12 (1821-22) 364-. with sid es some or all of which are equal, etc. August, F. Arch. Mth. Ps. 17 (1900) 65-. Tetrahedron of 4 congruent triangles. Schmidt, A. Z. Mth. Ps. 29 (1884) 321-. with 4 faces passing through 4 given lines, and 3 vertices on 3 other lines, theorem by Chasles, correction. Lange, E. Z. Mth. Ps. 26 (1881) 98-. , --- , (Lange)'. Schroeter, H. E. Z. Mth. Ps. 26 (1881) 270-. Theorem analogous to Pythagoras's. (Euc. i, 47.) Enestrom, G. Bb. Mth. (1898) 113-. Theorems. Clarke, A. E. Mathematician 3 (1850) 182-. . Heis, E. Grunert Arch. 31 (1858) 41-. . Staudt, G. K. C. von. N. A. Mth. 18 (1859) 441. . Marmier, G. N. A. Mth. 5 (1866) 268-. . Schroder, . D. Nf. Tbl. (*1879) 177. , two. Miiller, J. H. T. Grunert Arch. 9 (1847) 319-. Transformation, continuous, application. Le- moine, E. As. Fr. C. E, (1893) (Ft. 2) 146-. Transversals in plane triangle or tetrahedron. Seipp, H. Arch. Mth. Ps. 9 (1890) 375-. Triangles and tetrahedra, analogies. Ferriot, . Gergonne A. Mth. 2 (1811-12) 133- ; Fr. Cg. Sc. 24 (1857) 474-. --- , ; 9-point circle, and 12-point sphere. Prouhet, E. N. A. Mth. 2 (1863) 132-. -- , theorems. Fregier, . Gergonne A. Mth. 7 (1816-17) 167-. Twelve-point sphere and quadrics. Ripert, L. Mathesis 18 (1898) 218-. Volume and angles in terms of edges. Unfer- dinger, F. Arch. Mth. Ps. 51 (1870) 353-. Theorem analogous to Cotterill's on polygons. Clifford, W. K. [1872] L. Mth. S. P. 4 (1871-73) 178-. of Euclid generalised. Fedoroff, E. Bll. Sc. Mth. 18 (1894) 59-. Theorems. Meyer, A. Brux. Ac. Bll. 15 (1848) 261-. of Descartes, etc. Minich, S. R. Ven. At. (1859-60) 939-. Uniaxal polyhedra of minimum surface with constant volume. Hoppe, R. Arch. Mth. Ps. 13 (1895) 69-. Polymerisation of carbon. Brunei, G. Bor- deaux S. Sc. PV. (1894-95) 8-. Propositions in piano, and analogues in three dimensions. Hearn, G. W. Mathematician 2 (1847) 117-. Quadrilaterals, analogues in space to two properties of. Weddle, T. Camb. and Dubl. Mth. J. 5 (1850) 238-. and pentahedra, correspondence. Hermes, O. [1857] Crelle J. 56 (1859) 247-. , plane and spherical, inscribed in circle. Gueneau d'Aumont, . Gergonne A. Mth. 12 (1821-22) 269-. 456 6820 Stereometry. Spheres 6820 Quadrilaterals, plane and spherical, inscribed in circle. Baur, C. W. Schlomilch Z. 6 (1861) 221-. , , theorems, two. Strehlke, F. Grunert Arch. 2 (1842) 323-. , skew. Studerende. (Pseud.) Mth. Ts. 1 (1859) 85-. , , section and form. Beyel, C. Z. Mth. Ps. 32 (1887) 301-. , , trigonometric properties. Wasteels, J. Mathesis 18 (1898) 85-. and tetragons, complete, analogous systems in space. Weddle, T. Camb. and Dubl. Mth. J. 7 (1852) 251-. , theorem extended to space. Hermes, 0. Crelle J. 56 (1859) 204-. Eelations in space, graphic representation. Eitter, C. Berl. Ab. (1828) (Ph.) 213-. Eesultant angle of two angles. Fontene, G. N. A. Mth. 18 (1899) 407-. Eotation surface (urn), diameter of cross- section. Besch, C. Konigsb. Schr. 14 (1873) 66-. 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Approximations, accurate determination of when sin or cos c/> nearly 1. Grunert, J. A. Grunert Arch. 1 (1841) 73-. , arcs in terms of sine or cosine. Yanishevskil, E. P. (xii) Kazan Un. Mm. (1855) (Bk. 4) 119-. , . Lampe, E. Mathesis 17 (1897) 129-, 153-, 183-. Approximations, circular sector, construction of triangle of nearly equal area. Collignon , E. N. A. Mth. 13 (1874) 389-. , Euler's formula. Glaisher, J. W. L. Mess. Mth. 7 (1878) 190-. , , demonstration. Realis, S. N. A. Mth. 9 (1870) 12-. , hyperbolic formulae, algebraic signs. Macloskie, G. A. Mth. 8 (1893-94) 53-. , inaccuracy of formulae in certain cases, correction. Nell, A. M. Arch. Mth. Ps. 49 (1869) 104-. , Poncelet's formulae, elementary demonstra- tion. Fort, 0. Schlomilch Z. 2 (1857) 412-. , sin = = tan 0, error in assuming, when small. Matzka, W. Grunert Arch. 13 (1849) 138-. , sin > - ^ 3 , demonstration. Raicson, R. Mess. Mth. 3 (1866) 101-. , sin0<0- , geometrical demonstration. Jo/roy, J. N. A. Mth. 14 (1875) 171-. Arc of ecliptic and equator, problem of Begio- montanus. Delambre, J. B. J. Con. des Temps (1818) 284-. and sine, difference. Deladereere, A. N. A. Mth. 2 (1843) 494-. , (Pappus). Terquem, O. N. A. Mth. 3 (1844) 49-. , , limit of error. Lionnet, E. N. A. Mth. 2 (1843) 216-. Arc = tan = sin for small angles. Wolfers, J. P. Grunert Arch. 30 (1858) 359-. Arcs of first quadrant, lines, calculation. Gelin, E. Mathesis 8 (1888) (Suppl. 3) 1-. , multiplication, application of formula. Goursat, E. N. A. Mth. 14 (1895) 245-. , small, computation. Wurm,J.F. Lindenau Z. 5 (1818) 188-. trisecting circle, property of tangents. Maskelyne, N. Phil. Trans. (1808) 122-. Area and sum of angles for triangle and quadri- lateral, geometrical theorems. Veltmann, W. Arch. Mth. Ps. 17 (1900) 442-. Cayley's theorem. Airy, G. B. Ph. Mg. 17 (1859) 176-. Chords in circle, theorem. Le Cointe, L. A. N. A. Mth. 1 (1842) 508-. .values (Ptolemy). Arnott,G.A.W. Tilloch Ph. Mg. 52 (1818) 454-. Circle and trigonometry, general geometry. Tarry, G. As. Fr. C. E. (1891) (Pt. 2) 90-. Circles in and about quadrilateral, radii. Milinowski, . Z. Mth. Ps. 23 (1878) 139-. , tracing on uneven ground. Hinard, . A. Cond. Pon. Chauss. 14 (1870) 117-. Circular functions of 30 and 50. Anglin, A. H. Edinb. B. S. P. 21 (1897) 453-. Circumference, approximate construction. Lemoine, E. Par. S. Mth. Bll. 23 (1895) 242-. Clouds, heights, determination. Besson, L. Bv. Sc. 48 (1891) 58-. Colour analogy, trigonometrical. PlaSil, J. Casopis 5 (*1876) 32-; Fschr. Mth. (*1876) 333. 461 6830 Trigonometry. Angle-division. Trisection 6830 Compass, variation, graphical method of deter- mination. SalicJwn, (le It.) H. Bv. Mar. et Col. 124 (1895) 341-. Cotes's property. Brinkley, J. [1797] Ir. Ac. T. 7 (1800) 151-. theorem deduced from formula in cosines of multiple arcs. Fergola, N, Nap. At. Ac. 1 (1819) 249-. and De Moivre's. Arndt, F. Grunert Arch. 11 (1848) 181-. , theorems related to. Gregory, D. F. (vi Adds.) Camb. Mth. J. 3 (1843) 144-. Decimal division of angles and time. Yvon- Villarceau, A. C. B. 70 (1870) 1233- ; 71 (1870) 362-. . Mendizdbal Tamborrel, J. de. Mex. S. "Alzate" Mm. 10 (1896) 285-. . circles, etc., and remarks on tables. Sang, E. Edinb. B. S. P. 12 (1884) 533-. quadrant. Abbadie, A. d'. C. B. 71 (1870) 335-. right angle. L., 0. Cztg. Opt. 21 (1900) 54-. Degree, Fernel's measure. De Morgan, A. (vi Adds.) Ph. Mg. 19 (1841) 445-; 20 (1842) 116-, 408- ; 21 (1842) 22-. Distance of moving ship, determination. Wollin, L. Stockh. Ak. Hndl. 24 (1803) 231-. , . Bergsten, N. J. Stockh. Ak. Hndl. (1833) 193-. 2 places whose situations relative to a third are known. Littrow, J. J. von. Crelle J. 1 (1826) 232-. tower from observer at 3 points. Bretschneider, C. A. Grunert Arch. 2 (1842) 431-. Division of Angles and Circular Arcs. Nieuport, C. F. F. de. [1802] Par.. Mm. Sav. Etr. 1 (1806) 659-. Gough, J. Nicholson J. 12 (1805) 225-. Cerquero, J. S. Cadiz Period. M. Ci. 1 (1848) 76-. Tate, T. Ph. Mg. 19 (1860) 261-. Wasserschleben, (Maj.) von. Arch. Mth. Ps. 56 (1874) 335-. Radicke, A. [1878] Arch. Mth. Ps. 63 (1879) 328-. Horst, E. Z. Mth. Ps. 24 (1879) 407-. Nicholson, J. W. Des Moines Anal. 10 (1883) Murphy, I. J. Belfast NH. S. Ep. & P. (1892-93) 47-. Matter, F. Dresden Isis Sb. (1899) 24-. Approximate division. Hill, C. J. D. Skandia 3 (1834) 268-. , and hints for reducing error. Kummell, C. H. Des Moines Anal. 5 (1878) 172-. by rule and compass. Pellet, A. C. B. 105 (1887) 1119- ; Par. S. Mth. Bll. 16 (1888) 113-. Bisection. Sirjaev, V. A. Kazan S. Nt. (Ps.- Mth.) P. 5 (1887) 1-. Division of angles, and inscriptibility of poly- gons. Speelman, E. Brux. Ac. Bll. 25 (1868) 665-. Division of angles, and multiplication of cube. Soymie, E. Les Mondes 22 (1870) 348-. given angle. Rauch, F. Kosmos (Lw.) 20 (1895) 77-. Equations. Biehler, C. N. A. Mth. 8 (1889) 552-. Formulae. Ramus, C. Kiob. Ov. (1856) 239-. Multisection. Mariantoni, , & Palatini, . N. A. Mth. 18 (1899) 126-. by cycloid. Wilkie, D. E. Viet. T. B. S. 5 (1860) 217-. epicycloid. Leeuwen, J. H. van. N. Arch. Wisk. 7 (*1881) 213. Practical methods. Michelotti, I. [1797] Turin Mm. Ac. 6 (1792-1800) 105-. . Glotin, . Bordeaux Mm. S. Sc. Ps. 2 (1861-63) 252-. Eight angle, division, mechanical. Herschel, A. S. QJ. Mth. 4 (1861) 315-. Trisection. Winthrop, J. Bost. Mm. Am. Ac. 2 (1793) Mansfield, J. Conn. Mm. Ac. 1 (1810) 119-. Newton, P. Tilloch Ph. Mg. 62 (1823) 10-. Fusinieri, A. Bb. It. 40 (1825) 285-. Querret, . Gergonne A. Mth. 16 (1825-26) 108-. Gamier, J. G. Quetfilet Cor. Mth. 2 (1826) 271-. Hill, C. J. D. Crelle J. 2 (1827) 304- ; 7 (1831) 44-. Masters, W. Beng. As. S. J. 1 (1832) 501-. Barre, . Lille Mm. S. (1833) 44-. Morrieson, N. Beng. J. As. S. 2 (1833) Hudson, C. I. Ev. 2 (1838) 399. Le Monnier, J. F. B. Cherb. Mm. S. Ac. (1838) 320-. Otto, . Grunert Arch. 4 (1844) 223-. Fusinieri, A. [1845] Mod. S. It. Mm. 23 (1846) 294-. Ferrari, S. II Cim. 6 (1855) 536-. Poudra, . N. A. Mth. 15 (1856) 381-. Tietz, J. Grunert Arch. 30 (1858) 114-. Walter, F. Grunert Arch. 34 (1860) 295-. Kuhlmey, . Grunert Arch. 36 (1861) 123-. Richter, . Z. Bauw. 12 (1862) 284-. (Bichter.) Franz, . Z. Bauw. 12 (1862) 287-. O'Donoghue, J. Franklin I. J. 45 (1863) 382-. Veriot, E. C. E. 66 (1868) 619-, 730-. D., G. Ts. Mt. Fys. 4 (1871) 199-. Genese, R. W. Mess. Mth. 1 (1872) 103, 181. Hillhouse, W. [U.S.A.'] Des Moines Anal. 3 (1876) 151-. Leeuwen, J. H. van. N. Arch. Wisk. 2 (*1876) 177-. Garbieri, G. G. Mt. 15 (1877) 111-. Rusch, M. Arch. Mth. Ps. 68 (1882) 444-. Sparer, B. Arch. Mth. Ps. 69 (1883) 224. Moser, C. Bern Mt. (1884) (Heft 3, Ab.) 462 6830 Trigonometry. Angle- trisection. Formulae; As. Fr. C. E. (1887) (Pt. 2) 6830 Collignon, E. 141-. Cadiz, M. Chili S. Sc. Act. 3 (1893) Ixxxviii. Candy, A. L. [1893] Kan. Un. Q. 2 (1894) 35-. Obrecht, . Chili S. Sc. Act. 3 (1893) xcvi-, cxviii. Crawford, L. Edinb. Mth. S. P. 16 (1898) 42-. Mardones, F. Santiago de Chile Un. A. 101 Durell, ' C. V., & Beard, W. F. Mth. Gz. 1 (1900) 232. apparatus. Schweder, G. (jun.). Eiga Cor.- Bl. 37 (1894) 46-. application of property of equilateral hyper- bola. SAretuen, S. J. (xn) Ts. Mth. 4 (1868) 1-. approximate. Bjorling, C. F. E-. Arch. Mth. Ps. 15 (1897) 223-. , by compasses and rule. Lampe, E. Crelle J. Mth. 100 (1887) 364. Baratta's suggestion. Schiaparelli, G. V. Mil. I. Lomb. Ed. 2 (1869) 1083-. classification of solutions. Baker, M. Smiths. Misc. Col. 33 (1888) Art. 4, 96-. (Wash. Ph. S. Bll. 10 (1888).) by compasses. Lucas, E. C. E. 81 (1875) 368-. compasses for, theorem leading to. Toscani, C. Tortolini A. 3 (1852) 276-. and conchoids. Sidler, G. Bern Mt. (1873) (Ab.) 31-. cube, multiplication, etc., solutions. Werneburg, J. F. C. Oken Isis (1819) 1519-. by curve of 3rd degree. Boyman, J. E. Grunert Arch. 15 (1850) 205-. 4th degree. Bichot, . Fr. Cg. Sc. (1835) 87-. curves for. Allen, W. Silliman J. 4 (1822) 343-. by fixed conies. Panzerbieter, W. Arch. Mth. Ps. 11 (1892) 408-. , arbitrary. Glaser, S. Arch. Mth. Ps. 12 (1894) 367-. hyperbola. Panzerbieter, W. Arch. Mth. Ps. 10 (1891) 333-. geometrical principles connected with. Walker, John. Thomson A. Ph. 7 (1824) 356-. graphic. Sarrus, F. Gergonne A. Mth. 10 (1819-20) 189-. . Anon, (vi 231) Bb. Un. 27 (1824) 162-. . Lhuilier, S. Bb. Un. 28 (1825) 3-. by hyperbola and circle. Teixeira, A. J. Coimbra I. 6 (1858) 121-, 133, 134-, 177-, 191-. , equilateral. Neuberg, J. J. B. (xn) Mathesis 3 (1883) 130-. and inscribed quadrilateral. Broager, P. D. (xn) Ts. Mth. 4 (1868) 113-. mechanical. Laurent, P. Nancy Mm. S. Sc. (1833-34) 58. methods. Hudson, C. I. Ev. 2 (1838) 332-, 396-. and multisection. Roche, . Quetelet Cor. Mth. 6 (1830) 375-. new curve for. Hillhouse, W. [U.S.A.] [1878] Des Moines Anal. 9 (1882) 181-. optical solution. Hill, C. J. D. Grunert Arch. 1 (1841) 215-. radiatrix for. M., N. (vi Adds.) Thomson A. Ph. 6 (1815) 154-. rectilinear geometric method. Lyesn...skii, I. (xn) Mosc. Un. Mm. 9 (1835) 40-. without trigonometry. Dippe, M. C. Grunert Arch. 7 (1846) 108-. n- and (re + l)-division of angle and circle. Grinten, A. van der. Arch. Mth. Ps. 70 (1884) 393-. Double-altitude problem, solution. Borgen, C. A. der Hydrog. 28 (1900) 84-. Elimination of sines and cosines. M'Laren, (Lord). Edinb. E. S. P. 20 (1895) 145-. Eliminations in trigonometry. Meyer, F. D. Mth. Vr. Jbr. 4 (1897) 92-. Equation giving tan , when tan a is known. Biehler i C. N. A. Mth. 6 (1887) 5-. t&nx-x = a, solution. Gough, J. Nichol- son J. 21 (1808) 1-. 2 cosr0=0, solution. Gregory, D. F. (vi Adds.) Camb. Mth. J. 1 (1839) 44-. tan x m tan a. Girault, C. Caen Ac. Mm. (1856) 22-. cos a 2 sin 4 - 2 sin a cos a sin tf> cos + sin a 2 cos < 4 = 0. Grunert, J. A. Arch. Mth. Ps. 45 (1866) 417-. TO sin 4 z = sin (z-q), table for numerical solution. Gasparis, A. de. Nap. Ed. 21 (1882) 94-. 2c arc tan x = k ^ , complex integers applied to solve in rational numbers x, c, k. Stfrmer, C. Arch. Mth. Ntvd. 19 (1897) No. 3, 96 pp. Equations, trigonometrical, roots determined by ruler and compasses. Florow, P. Fschr. Mth. (1897) 455. _ f _, solution by series. Chauvenet, W. Gould As. J. 3 (1854) 161. , , transformation, cases. Hop/, L. Nr. 57 (1862) 137-. Finite variations calculus, application. Lorgna A. M. Verona S. It. Mm. 7 (1794) 346-. Formula of Plane Trigonometry. Zahradnik, K. Arch. Mth. Ps. 6 (1888) 415-. Cosine of angle between lines. Grunert, J. A. Grunert Arch. 8 (1846) 194-. Derivation from coordinate geometry. Eiese, (Prof.) von. Grunert Arch. 30 (1858) 143-. of formulae. Matzka, W. Grunert Arch. 13 (1849) 73-. As. 463 6830 Formula of Plane Trigonometry Extension to system of 3 non-intersecting straight lines. Grunert, J. A. Grunert Arch. 35 (1860) 1-. Formulae deduced from definite integrals. Girault, C. Caen Ac. Mm. (1856) 19-. regular polygons in-, or circum-scribed to circle. Kroutil, J. Casopis 19 (1890) 70-; Fschr. Mth. (1891) 265. a=bcosC + ccosB. Egidi, G. Em. N. Line. At. 43 (1890) 147-. Fundamental formulae (Legendre). Osorio, R. G. Coimbra I. 3 (1855) 234-. Half angles. Redl, F. Ens. Mth. 2 (1900) 201-. Hero's polygon formulae, are they trigono- metrical? Schmidt, W. Bb. Mth. 1 (1900) 319-. Multiple angles from equation of circle. Rudberg, F. Stockh. Ak. Hndl. (1823) 229-. Ozanam's formula for approximate solution of triangle. Brocard, H. Mathesis 9 (1889) 161-. . . Mansion, P. Mathesis 9 (1889) 162-, 181-, 265-. , ascribed to W. Snell. Le Paige, . Mathesis 10 (1890) 34-. Quadrilaterals, area, given sum of 2 opposite angles. Anglin, A. H. H. QJ. Mth. 19 (1883) 138-. , , Lefevre's formula. Verhulst, P. F. Quetelet Cor. Mth. 6 (1830) 120-. Series derived from sin 2 x + cos 2 a; = l. Lemon- nier, H. N. A. Mth. 4 (1845) 606-. 6830 Wallis's formula, and Stirling's, proved by hyperbola. Mansion, . Brux. S. Sc. A. 23 (1899) (Pt. 1) 1-. (2 cos x) m , formulae. Anon. (vr558) Gergonne A. Mth. 16 (1825-26) 254-. 1 - cos n(f>, factorisation, and series for . jr etc. Huguenin, U. (vn) Amst. N. Ws. Ntk. Vh. 1 (1844) 1-. Cos n x in sines or cosines of multiples of x. Dirksen, E. H. Berl. B. (1845) 261-. Cos- in terms of cos a, note on equation. A. N. A. Mth. 4 (1845) 52-, Guilmin, 122- . for tangent, cotangent, etc. Herschel, (Sir) J. F. W. Nicholson J. 31 (r ' ' x n - 2 cos na + , factorisation. Airy, (Sir) G. B. [1868] Camb. Ph. S. T. 11 (pt. 2) (1869) 426-. x n + - 2 cos n a , factorisation. Adams, (Prof.) J. C. [1868] Camb. Ph. S. T. 11 (pt. 2) (1869) 444-. 360 Cos , when rational. Hessel,J.F.C. Arch. Mth. Ps. 48 (1868) 81-. Cos n in powers of cos . Heine, E. Mth. A. 2 (1870) 187-. Cos ma, sin ma as functions of sin a or cos a, demonstration. Desboves, A. N. A. Mth. 14 (1875) 385-. x n - 2 cos n6 4 , factorisation. Ferrers, N. M. - cos - g cos 20 + etc. = log ( Blissard, J. (1812) 133. etc. Mess. Mth. 5 (1876) 6. Cos 2 a + cos 2 ft + cos 2 -y + 2 cos a cos cos y - 1 = 0. Zahradnlk, K. Arch. Mth. Ps. 62 (1878) Mess. Mth. 1 (1862) 124-. 330 ~' . Substitution of 1-* for A, etc. Dostor, G. = &>BC + ccoBB, substitution of (J+-) for 2cosJ5, etc. Cayley, A. (xn) Un. Cir. [1] (1882) 241. N. A. Mth. 19 (1880) 362-. Sums and products. Wiegers, C. Grunert Arch. 33 (1859) 338-. System connecting sines and cosines of A, B, C and their differences. Glaisher, J. W. L. [1880] Mess. Mth. 10 (1881) 73-. Systems, complete, in plane trigonometry. Meyer, W. F. D. Mth. Vr. Jbr. 5 (1901) (Heft 1) 61-. Triangle, altitude. Dufour, C. [1877] Laus. S. Vd. Bll. 15 (1878) (PV.) 49-. , area. Wallace, (Prof.) W. Camb. Mth. J. 2 (1841) 35-. , . Baker, M. [1885] A. Mth. 1 (1884-85) 134- ; 2 (1885-86) 11-. , , geometric derivation of formulae. Hultman, F. W. Ts. Mt. Fys. 2 (1869) , formulae, examination question. Gerono, G. C. N. A. Mth. 16 (1857) 76-. Triangles, right angled, sides. Slop de Caden- berg, G. Mod. S. It. Mm. 13 (1807) 285-. , , . AzzarelU, M. Em. At. N. Line. 26 (1873) 43-. Wallis's formula, deduction. Herschel, (Sir) J. F. W. Nicholson J. 32 (1812) 13. J. H. Cos , quadratic radical expression for. Steg- gall, . Edinb. Mth. S. P. 7 (1889) 4-. Cos n in terms of cosines of multiples of 6. Candida, G. Par. S. Phlm. Bll. 1 (1899) 5-. Sin nA and other formulae. Cagnoli, A . Verona Mm. S. It. 7 (1794) 1-. Sin n 0, cos n and sinn0, cosnO, extension to hyperbolic areas. Brinkley, J. [1797] Ir. Ac. T. 7 (1800) 27-. Sin (A B) and cos (A B) deduced from Ptolemy's theorem. Mollweide, C. Zach M. Cor. 26 (1812) 601-. , method of obtaining formulas. Strong, T. Silliman J. 1 (1818) 424-. . Thibault, . N. A. Mth. 2 (1843) 309-. , construction. Midy, . N. A. Mth. 3 (1844) 374-. . Arndl, F. Grunert Arch. 6 (1845) 95-. . Bianchi, G. Palomba Eac. 1 (1845) 325-. 464 6830 Plane Trigonometry. Formulas 6830 and cos (A B), demonstration. Rispal, A. N. A. Mth. 5 (1846) 84-. deduced from triangle. Grunert, J. A. Grunert Arch. 21 (1853) 237-. , generality of, demonstration. Spitz, C. Grunert Arch. 32 (1859) 293-. , demonstration. Schreder, E. Grunert Arch. 36 (1861) 447-. . Barbe, P. A. Cond. Pon. Chauss. 9 (1865) 182-. deduced from Ptolemy's theorem. Tucker, R. [1865] Mess. Mth. 3 (1866) 123. . Tucker, R. [1865] Mess. Mth. 4 (1868) 31-. . Lyetnikov, A. F. (xn) Bee. Mth. (Moscou) 2 (1867) (Pt. 2) 96-. . Lemonnier, H. N. A. Mth. 10 (1871) 26-. . Gravelaar, A. W. N. Arch. Wisk. 5 (*1879) 187-. . Zejliger, D. N. Kazan S. Ps.-Mth. Bll. 2 (1893) (Prot.) 23. , and other formulae, geometrical demonstration. Butters, J. W. Edinb. Mth. S. P. 15 (1897) 86-. - deduced from triangle. Bochow, . Arch. Mth. Ps. 17 (1900) 97-. Sin x = I*-*, n tfb r j"l^ . D U Bourguet, . o TO Gergonne A. Mth. 3 (1812-13) 19-. Sin"*0 and cos m in series of sines and cosines of multiples of 6. Pagani, G. M. Ferussac Bll. Sc. Mth. 5 (1826) 4-. Sin m x and cos m x for all values of m. Gamier, J. G. Brux. Ac. Bll. 6 (1839) (pte. 2) 375-, 488-. Sinm0 and cosmB in terms of sin0 and cos#. Moon, R. Camb. Mth. J. 4 (1845) 250- . Sinx and cos a;, imaginary expressions for. Y. Mathematician 1 (1845) 232-. Sin (xy) or COS(XT/), simple demonstration. Astrand, J. J. Grunert Arch. 18 (1852) 479-. Sin and cos . Sturm, J. B. Grunert Arch. 24 (1855) 112-. Sin"0 and cos n 0, expansion in sines and cosines of multiples of 0. Wolfers, J. P. Grunert Arch. 24 (1855) 303-. Sin - , - , - , etc. , and cos , etc. Gerono, G. C. N. A. Mth. 17 (1858) 46-. Sin(0 + J + a + ... + n ) and cos (9 + ... + 6J, expansion, and related theorem of numbers. Unferdinger, F. Grunert Arch. 34 (1860) 72-. Sin fj.9 and cos /j.6, expansion. Gylden,H. As. Nr. 69 (1867) 193 -. Sin (na + z), and cos (na + z), sin a, and cos n a, expansion. Andre, D. N. A.Mth.lO(1871)359-. Sin (A +B + ...) or cos (A + B + ...). Legge, A. di. G. Mt. 9 (1871) 377-. Sin oo and cos OD . Walton, W. QJ. Mth. 11 (1871) 326-. Sin- 1 and cos" 1 (x + iy), values. Schlomilch, 0. Z. Mth. Ps. 17 (1872) 245-. Sin no. and cos wo, expansion in powers of 2 cos a and 2 sin a. Le Besgue, V. A. N. A. Mth. 12 (1873) 425-. Sin ma and cos?a in terms of sin a or cos a. Mourgue, . N. A. Mth. 12 (1873) 408-. , calculation. Ocagne, M. d\ N. Arch. Wisk. 17 (1890) 229-. lsin(2|3 + l) u. Cayley, A. Mess. Mth. 5 (1876) 7-. sin0 sin 20 sin 30 2 ~~p" 2 ' 3 ' g enera l ls ation. Stormer, C. Acta Mth. 19 (1895) 341-. Tan (A 1 + A 2 + etc.) in terms of tan^ 1; tan A 2 , etc., demonstration. Stainville, de. Par. EC. Pol. Cor. 2 (1809-13) 432-. Tan z, continued fraction for. Schlomilch, O. Z. Mth. Ps. 16 (1871) 259-. Tan nx, continued fraction for. Glaisher, J. W. L. [1873-74] (vn) Mess. Mth. 3 (1874) 137; (x) 4 (1875) 65-. Tan ( So, ), tan n0, and other formulae. An- glin, A . H. [1884] Ir. Ac. P. 4 (1884-88) 259-. Sarc tana r . Anglin, A. H. Arch. Mth. Ps. 2 1 (1885) 407-. = ncot. ( J, sign. Stern, M. A. Liouv. J. Mth. 5 (1840) 216-. b + c , B+C I , B-C :- = tan jr / tan - , geometric demon- stration. Vignal, . N. A. Mth. 3 (1844) 456-. Foundations of trigonometry. Brisse, C. N. A. Mth. 16 (1877) 49-. Functions, developments and properties. Gamier, J. G. Quetelet Cor. Mth. 11 (1839) 283-. Goniometry. Lindman, C. F. Arch. Mth. Ps. 53 (1871) 112-. , problem. Cotterill, T. [1865] QJ. Mth. 7 (1866) 259-. , (Cotterill). Cayley, A. QJ. Mth. 15 (1878) 196-. , theorems. Schlomilch, O. Grunert Arch. 8 (1846) 453-; 9 (1847) 1-. Graphic tables, use in solution of problems in cosmography and spherical trigonometry. Collignon, E. N. A. Mth. 18 (1879) 179- ; 1 (1882) 490-. History. Suter, H. Bb. Mth. (1893) 1-. Identities, algebraic, in plane trigonometry. Meyer, W. F. Fschr. Mth. (1898) 441. , trigonometrical. Fouret, G. N. A. Mth. 2 (1883) 262-. Identity. Hermite, C. N. A. Mth. 4 (1885) 57-. , case. Glaisher, J. W. L. As. Fr. C. B. 9 (1880) 222-. involving products of four sines. Glaisher, J. W. L. B. A. Ep. (1880) 484-. , arc tan - = S arc tan where A j a Glaisher, J. W. L. [1877-78] As. Fr. C. B. 6 (1877) 211- ; QJ. Mth. 15 (1878) 151- ; 16 (1879) 329-. 465 6830 Trigonometry Plane Triangles 6830 T , , 1 1 , TT , Non-Euclidean trigonometry, plane, of 3-di- Hentity, m arc tang- + n arc tang - = k j , solu- mensional homogeneous space. Rethy, M. tion in integers. Stormer, C. Par. S. Mth. Bll. 27 (1899) 160-. , cos 36 - cos 72 = , proof. Volpicelli, P. Em. At. 16 (1862-63) 85. , if S cos x r = 0, 2 cos (x r + a) = 0, then shall 2cos(av + 0) = 0. Laisant, C. A. Par. S. Mth. Bll. 15 (1887) 198-. Inaccessible points, distance between 2. Boy- man, J. R. Grunert Arch. 18 (1852) 452-. , . Ladame, H. (jun.) Neuch. Bll. 7 (pt. 2) (1866) 403-. ., 3. Lemoine, E. As. Fr. C. E. (1884) (Pt. 2) 47-. straight line, problem. Gelin, E. Mathesis 9 (1889) 93-. Inequalities, trigonometrical. Femvick, E. Mth. Gz. 1 (1900) 107-. Inequality, case. Genese, R. TF. As. Fr. C. E. (1894) (Pt. 2) 134-. Isoperimetric figures, simple demonstrations. Steiner, J. Berl. Ab. (1836) 117-. Landen's substitution, geometrical representa- tion. Schellbach, C. H. Crelle J. Mth. 91 (1881) 347-. Logarithmic sines and tangents of small arcs, finding. Burgess, J. Edinb. E. S. P. 22 (1900) 265-. Logarithms, graphic notation. Haro, . As. Fr. C. E. (1887) (Pt. 2) 128-. Maximum and minimum, problems. Burg, A. Wien Jb. Pol. I. 8 (1826) 213-. product of tangents of n angles whose sum is given. (a + 6<90, etc.) Gerono, G. C. N. A. Mth. 17 (1858) 360-. de Moivre's theorem. Barfuss, F. W. Grunert Arch. 7 (1846) 3-. , analogue in plane point system. Hyde, E. W. Science 4 (1896) 344; A. Mth. 11 (1896-97) 129-. , demonstration, elementary. Davidov, A. Y. (xn) Eec. Mth. (Moscou) 2 (1867) (Pt. 2) 20-' , without imaginaries. Gopel, A. Grunert Arch. 6 (1845) 102-. (xn) Mag. Tud. Ak. Etk. (Mth.) 4 (1877) (No. 7) 25 pp. Parallelograms formed from any three elements. Cebysev, P. [1879] St Pet. Ac. Sc. Mm. (Rs.) 36 (*1880) (Suppl.) No. 3, 40pp. symmetric about axis. Cebysev, P. [1878] St Pet. Ac. Sc. Mm. (Rs.) 34 (* 1879) (Suppl.) No. 3, 16 pp. Partial fraction and product, developments for trigonometric functions. Schroter, H. Z. Mth. Ps. 13 (1868) 254-. Plane Triangles. Ambiguous case. Grunert, J. A. Grunert Arch. 2 (1842) 333-. Angles, construction for difference of two. Walker, J. J. QJ. Mth. 8 (1867) 323-. , property. Arndt, F. Grunert Arch. 6 (1845) 333-; 8 (1846) 111-. , relations between trigonometrical ratios. Dostor, G. Arch. Mth. Ps. 65 (1880) 188-. in terms of sides. Borenius, H. G. [1839] Helsingf. Acta 1 (1840) 159-. , Newton's Arithmetica Univer- salis, prob. xi., solution and generalisation. Leblais, A. Presse Sc. 2 (1863) 687-. Area. Tweedie, C. [1900] Edinb. Mth. S. P. 19 (1901) 2-. , determinant value. Dostor, G. Arch. Mth. Ps. 57 (1875) 204-. of indeterminate triangle. Gyory, S. [1860] (xn) Mag. Ak. Ets. (Mth. Term.) 2 (1861- 62) 59-. in terms of bisectors. Sacchi, G. N. A. Mth. 1 (1862) 332-. sides. Gerono, G. C. Gergonne A. Mth. 15 (1824-25) 305-. , Hero's theorem. Hultsch, F. Z. Mth. Ps. 9 (1864) 225-. , . Panek, A. Casopis 20 (1891) 310- ; Fschr. Mth. (1891) 607. , proof of the " Three Brothers." Kinkelin, H. Grunert Arch. 39 ' 187-. , that e xi = cos x + i sin x. Bauer, K.L. . < Ownert J. A . Arch. Mth. Ps. 46 (1866) 355-. , symbolic use. Dupuis, N. F. [1891-95] Cn. E: S. P. & T. 9 (1892) (Sect. 3) 43-; 2 (1896) (Sect. 3) 167-. Navigation problems, graphical solutions. Anon. Nt. 47 (1892-93) 547-. Negative quantities, use. Waille, J. [1876] (xn) Doubs S. Mm. 1 (1877) 33-. Networks, trigonometrical, of low order. Zrzavy, F. Prag Sb. (1872) 107-. Nomenclature. Bjorling, E. G. Stockh. Ak. Hndl. (1846) 275-, 299-. Non-Euclidean trigonometry. Story, W. E. Am. J. Mth. 4 (1881) 332-; 5 (1882) 180-. , fundamental equations derived by ele- mentary method. Rethy, M. Arch. Mth. Ps. 58 (1876) 416-. Grunert Arch. 40 (1863) 134-. . , synthetic demonstration. Degen, C. F. [1821] Kiob. Dn. Vd. Selsk. Afh. 1 (1824) 129-. Hypotenuse, approximation to. Buquoy, G. von. Oken Isis (1826) 570-, 662-. Mensuration, notes on. Durdn Loriga, J. J. G. Teix. J. Sc. 12 (1895) 45-; Fschr. Mth. (1893-94) 928. Pothenot's problem. Clausen, T. As. Nr. 20 (1843) 69-. , Lehmann's theorem. Hartner, . Wien SB. (1849) (Ab. 2) 216-. Problems, important, solution. Korselt, A. Arch. Mth. Ps. 17 (1900) 275-. Eational right-angled triangles, incommen- surability of angles. Berdelle, C. As. Fr. C. E. (1890) (Pt. 2) 186-. 466 6830 Plane Triangles Sides and angles, elimination problem. Diek- mtnni, J. Arch. Mth. Ps. 63 (1879) 267-. in terms of altitudes. Huet, . N. A. Mth. 5 (1846) 225-. bisectors of angles. Anon, (xi 54) N. A. Mth. 18 (1879) 311-. . Gerono, C. C. N. A. Mth. 18 (1879) 313-. Similar triangles, solution of difficulty in theory. Suremain-de-Missery, A. Gergonne A. Mth. 1 (1810-11) 129-. Solution. Franchini, P. [1831] Lucca At. Ac. 8 (1835) 1-. . Vercaet, P. J. Casopis 5 (*1876) 57-; Fschr. Mth. (1876) 327. , 3rd case. Seidelin, C. J. L. (xn) Ts. Mth. 5 (1869) 8-. , given a, b, C ; c found more directly. Wallace, (Prof.) W. [1823] Edinb. E. S. T. 10 (1826) 168- . , , , , find c. Schnlten, N. G.af (fil.). [1838-43] Helsingf. Acta 1 (1840) 151- ; Grunert Arch. 3 (1843) 1-. , , , , . Czuber (or Cuber), E. [1877] Arch. Mth. Ps. 62 (1878) 222-. , 2 sides and contained angle. Girault, , Ferussac Bll. Sc. Mth. 15 (1831) 159-. without logarithmic tables. Olbers, W. Zach M. Cor. 16 (1807) 539-. Tangents of angles, property. Garrard, W. Phil. Trans. (1808) 120-. Theorems. Hain, E. Arch. Mth. Ps. 55 (1873) 331-. Triangle with sides in arithmetical progression. Lilrickf, A. Casopis 27 (1898) 141-, 220- ; Fschr. Mth. (1898) 442. Vertex transversals. Helwig, P. I. N. Arch. Wisk. 17 (1890) 217- ; 19 (1892) 216- ; Fschr. Mth. (1890) 581 ; (1892) 527-. Plane trigonometry, basis. Unferdinger, F. Grunert Arch. 33 (1859) 429-. , and spherical, analogies. Ferroni, P. Mod. S. It. Mm. 12 (1805) 106-. Points, determination from measured directions. Zrzav$, F. Prag Sb. (1882) 12-. , 4 in plane, relation. Luchterhandt, A. R. Grunert Arch. 2 (1842) 62-. Porismatic equations, systems. Wolstenholme, J. L. Mth. S. P. 4 (1871-73) 312-. Porisms, trigonometrical. Bromwich, T. J. Pa. Mth. Gz. 1 (1900) 331-. Powers of angular functions. Crelle, A. L. Crelle J. 7 (1831) 253-, 314-. Problem, geometrical, solution. Aubert, P. Edinb. Mth. S. P. 5 (1887) 23-. , trigonometrical. Gardie, V. A. Cond. Pon. Chauss. 20 (1876) (!'' Pt.) 52-, 113. Problems , solution by continued approximations. Andrew*, H. (jun.). V. Nost. Eng. Mg. 31 (1884)54-. Prosthapheric method instead of logarithms. Braunmiihl, A. von. Bb. Mth. (1896) 105-. Ptolemy's problem. Hill, G. W. [1900] As. J. 21 (1901) 33-. Trigonometry 6830 Ptolemy's theorem and formula for cos (A - B). Zanten, L. van. N. Arch. Wisk. 11 (1884) 188-. plane trigonometry. Quidde, A. Grunert Arch. 23 (1854) 238-. Quadrilateral with sines of angles in given ratio. Euler, L. [1780] St Pet. Ac. Sc. Mm. 5 (1815) 73-. Quadrilaterals, cyclic, areas, simple investiga- tion. Terquem, O. N. A. Mth. 3 (1844) 219-. , trigonometry. Bretschneider, C. A. Gru- nert Arch. 2 (1842) 225-. , . Catalan, E. C. Liege S. Sc. Mm. 15 (1888) No. 1, 167-. Eadius of vision, calculation. Ramsauer, F. D. Alpvr. Z. 29 (1898) 81-. Eational trigonometry. Caluso, (I'abbe) T. V. de. Turin Mm. Ac. (1809-10) 179-. Eatios, generalised forms. Hatliaway, A. S. [1880] (xii) J. H. Un. Cir. [1] (1882) 48. Eelations between sides of triangle whose angles satisfy a linear relation. Bang, A. S. Ts. Mth. 2 (1884) 53-. , trigonometrical, connected with cyclotomy. Sparer, B. Arch. Mth. Ps. 16 (1898) 68-. Ship's position, application of altitude lines. Delafon, (le It.) R. Ev. Mar. et Col. 120 (1894) 509-. , formulae. Ocagne, M. d\ Ev. Mar. et Col. 100 (1889) 305-. , movements of current or another ship, calculation. Ja/re, P. Ev. Mar. et Col. 136 (1898) 461-. Signs of trigonometrical lines (ratios). Homer, W. G. Ph. Mg. 6 (1835) 86-. Sine and cosine of infinity. Earwhaw, S. [1844] Camb. Ph. S. T. 8 (1849) 255-. of half arc. Aynard, . N. A. Mth. 5 (1846) 399-. Sines, calculation. Bach, . N. A. Mth. 12 (1853) 108- . and cosines of angle and half angles. Bouvier, L. C. Gergonne A. Mth. 15 (1824- 25) 56-. , etc., calculation. Bakewell, T. If. Franklin I. J. 7 (1844) 221-. of irrational angles. Gylden, H. [1867] Helsingf. Acta 9 (1871) 73-. multiple and sub-multiple angles. Pezzi, F. [1803] Mod. S. It. Mm. 11 (1804) 10-. , products, expansion. Verdon, R. [1877] Mess. Mth. 7 (1878) 122-. or cosines, products of four, theorems. Glaisher, J. W. L. Mess. Mth. 10 (1881) 26-. and cosines, tables, calculation. Grunert, J. A. Grunert Arch. 6 (1845) 205-. in terms of arc, new approximations. Buquoy, G. von. Oken Isis 21 (1828) 1088. for every 3 degrees, table. Lambert, C. J. N. A. Mth. 12 (1853) 284-. of first quadrant from degree to degree, pro- duct. Laisant, C. A. As. Fr. C. E. (1875) 159-. , fundamental theorem. Encontre, D. Mntp. Eec. Bll. 2 (1805) 219-. 467 GG 2 6830 Sines of higher orders. Yvon- Villar.ceau, A.J.F. C. E. 86 (1878) 1160-, 1216-, 1287- ; 91 (1880) 195-. . Farkas, J. C. B. 91 (1880) 209-, 278-, 544-. . West, E. C. E. 92 (1881) 1279-. , higher, of same order, algebraic relations. Rouyaux, J. A. C. E. 92 (1881) 1276-. , natural, limit of error. Finch, B. N. A. Mth. 2 (1843) 329-. , , tables, construction and degree of accuracy. Vincent, A. J. H. N. A. Mth. 1 (1842) 272-. , etc. , relations. Brehmer, . Grunert Arch. 4 (1844) 236-. , tangents, and secants, rational expressions for. Gould, D. Silliman J. 22 (1832) 392-. Solid angles. Schulten, N. G. af (fil.). [1839] Helsingf. Acta 1 (1840) 101-. . Stammer, W. Grunert Arch. 27 (1856) 123-. , four, formed by 4 straight lines from a point, relation between sines. Dostor, P. Arch. Mth. Ps. 58 (1876) 1-. , measurement. Riedl von Leuenstern, J. [1847] Haidinger Ab. 2 (1848) (Ab. 2) 1-. , new solution by projection. Klekler, K. [1876] Arch. Mth. Ps. 61 (1877) 337-. of pyramid, sums. Riedl von Leuenstern, J. Haidinger Ab. 3 (1850) (Ab. 2) 87-. , theorems, two. Sturm, J. B. Grunert Arch. 24 (1855) 112-. , trihedral. Barbel, . N. A. Mth. 6 (1847) 104-. , , and lines related to them. Grunert, J. A. Arch. Mth. Ps. 42 (1864) 377-. , , sines, geometric significance. Timer, A. Z. Mth. Ps. 28 (1883) 249-. , , , properties. Dostor, G. Arch. Mth. Ps. 56 (1874) 247-. SPHEEICAL TEIGONOMETEY. Puissant, L. Par. fie. Pol. Cor. 3 (1814-16) 60-. Rube, K. (vi Adds.) Krk. Eoczn. Uniwers. 5 (1820) 290-. Sorlin, A. N. J. Gergonne A. Mth. 15 (1824- 25) 273-. Barsotti, G. [1829] Lucca At. Ac. 11 (1842) 113-. Ribeiro de Sousa Pinto, R. Coimbra I. 3 (1855) 130-, 185-. Unferdinger, F. Grunert Arch. 29 (1857) 479-; 33 (1859) 14-; Arch. Mth. Ps. 42 (1864) 453-. Meissel, E. Mth. A. 15 (1879) 380 ; 16 (1880) 529-; As. Nr. 95 (1879) 69-; 96 (1880) Allardice, R. E. Edinb. Mth. S. P. 2 (1884) 53-. Study, E. D. Mth. Vr. Jbr. 2 (1893) 72-. Azimuth, calculation. Le Blanc, C. M. Ev. Mar. et Col. 118 (1893) 326-. Concurrent quadrants, theorem. Lasala, A. Fschr. Mth. (1892) 551. Equations. Heppel, G. Mth. Gz. No. 3 (1894) Spherical Trigonometry. Formulas Formulae of Spherical Trigonometry. Approximation formulae. KlUgel, G. S. Bode As. Jb. (1803) 215-. Cosine, etc. Matzka, W. Grunert Arch. 13 (1849) 88-. , . Franke, T. Grunert Arch. 17 (1851) 309-. formula, incorrect demonstration in Verdam's spherical trigonometry. Berg, F. J. van den. N. Arch. Wisk. 10 (1884) 193-. Deduction of chief formulas. Cordeiro-Feyo, J. Lisb. Mm. Ac. Sc. 10 (1830) (pte. 2) 208-. Delambre's or Gauss's analogies. Feldt, L. Crelle J. 7 (1831) 68-. . Arndt, F. Grunert Arch. 13 (1849) 159-. . Oiartres, R. Nt. 40 (1889) 644. , mnemonic rule. Dostor, G. J. Grunert Arch. 30 (1858) 467- ; N. A. Mth. 5 (1866) 417-. , , note. Sylvester, J. J. Ph. Mg. 32 (1866) 436-. , and Napier's. Crelle, A. L. Crelle J. 12 (1834) 348-. , . Crelle, A. L. N. A. Mth. 7 (1848) 232-. , . Grunert, J. A. Grunert Arch. 17 (1851) 259-. , . Crofton, M. W. [1869] L. Mth. S. P. 3 (1869-71) 13-. , . Potter, A. Ens. Mth. 2 (1900) 303-. , , demonstration by construc- tion. Essen, E. Grunert Arch. 27 (1856) 38-. , other formulae. Gergonne, J. D. Gergonne A. Mth. 3 (1812-13) 348-. , . Werner, 0. Grunert Arch. 24 (1855) 55-. , . Essen, E. Grunert Arch. 27 (1856) 158-. Determination of signs. Stolz, 0. [1870] Z. Mth. Ps. 16 (1871) 168-. Formulae derived from plane figure. Unfer- dinger, F. Grunert Arch. 26 (1856) 436-; 27 (1856) 300-. - involving | - 45, 45 - $ (S - A), etc. Unferdinger, F. Wien Sb. 58 (1868) (Ab. 2) 30-. , two. Arrest, H. d\ As. Nr. 35 (1853) 281-. Fundamental formula and proof of polar triangle theorem. Schreder, E. Grunert Arch. 37 (1861) 438-. formulae, easy demonstration. Grunert, J. A. Grunert Arch. 16 (1851) 194-. . a-b . A-B sin sin Gauss'sformulffi -,etc. Ser- vois, J. F. 84-. Gergonne A. Mth. 2 (1811-12) 468 6830 Formulae General demonstration. Harst, A, D. van der. N. Arch. Wisk. 17 (1890) 176- ; Fschr. Mth. (1890) 600. Generalisation. Chauvenet, IF. Gould As. J. 3 (1854) 50-, 137. Interpretation of double sign in certain formulae. Spherical Trigonometry 6830 Fundamental principle. Vincent, A. J. H. N. A. Mth. 1 (1842) 33-. . Crova, (Prof.) . (vi Adds.) Perpi- gnan Bll. S. Ag. Pyr. Orient. 7 (1848) 255. . Foucaut, f C. N. A. Mth. 8 (1849) 58. . Patry, E. N. A. Mth. 17 (1858) 208-. Thomson, (Prof.) James. Ph. Mg. 10 (1837) Grea t c irc Delation betw^n spkerical 18-. Walton, W. QJ. Lagrange's formulae, deduction from theorem in determinants. Studnicka, F. J. Prag Sb. (1875) 1-. Napier's analogies. Gerono, G. C. N. A. Mth. 2 (1843) 222-. . Grunert, J. A. Grunert Arch. 3 (1843) 104-. . Cortazar, J. N. A. Mth. 6 (1847) 218-. . Werner, O. Grunert Arch. 24 (1855)95-. Ptolemy and formulae of spherical trigono- metry. Schubert, F. T. [1796] St Pet. Ac. Sc. N. Acta 12 (1801) 165-. Significance in case of complex argument. Schilling, F. Gott. Nr. (1891) 188-. Spherical excess. Grunert, J. A. Grunert Arch. 16 (1851) 483-. , formulae, derivation. Werner, O. [1857] St Pet. Ac. Sc. Bll. 16 (1858) 11-. , Lhuilier's formula. Gent, . Grunert \-~-j-v ~^J_ j ac j e Arch. 20 (1853) 358-. 16 (1898) 40-. j Prouhet, E. N. A. Mth. 15 .' Lovett, E. O. N. Y. Am. Mth. (1856) 91-. g_ EH 4 (I 8 g 8 \ 552-. . Ligowski, W. [1875] Arch. , graphic proof . Herschel, A. S. Mth. Ps. 58 (1876) 96-. Nt . 5 ( 1 87 2)24, 141-. , .demonstration. Lobatto, R. (vm) , partly anticipated. De Morgan, Arch. Mth. Ps. 39 (1862) 240. j. Ph. Mg. 22 (1843) 350-. of quadrilateral in small circle, and radius Orthogonal substitutions, elliptic functions, etc. tances from three points. Mth. 4 (1861) 328-. Hexagon, rectangular skew, trigonometry. Burnside, W. Mess. Mth. 28 (1899) 92-. Identity, case. Cayley, A. Mess. Mth. 1 (1872) 145. Mechanical solution of problems. Rambaut, A. A. Dubl. S. Sc. P. 5 (1886-87) 642-. Method, new. Pessuti, G. Mod. S. It. Mm. 15 (1811) 197-. Mnemonics. Chasles, M., & Forestier, . N. A. Mth. 12 (1853) 312-. Napier's rules of circular parts. Bowditch, N. Am. Ac. Mm. 3 (1809) 33-. . Hemert, W. J. J. van, & Schroder, J. C. Amst. Mengelwerk 2 (1816) 105-. . Ivory, J. Tilloch Ph. Mg. 58 (1821) 255-. Edinb. Mth. S. P. of small circle circumscribing quadrilateral. Hart, H. Mess. Mth. 4 (1875) 181-. polygon, area. Quetelet, L. A. J. [1820-32] Brux. Ac. Sc. Mm. 2 (1822) 105- ; Quetelet Cor. Mth. 7 (1832) 278-. triangles. Bretschneider, C. A. Crelle J. 13 (1835) 85-, 145-. . Dienger, J. Grunert Arch. 7 (1846) 225-. , cotangent formulae. Unferdinger, F. Arch. Mth. Ps. 50 (1869) 107-. Supplements of angles, introduction. Grebe, E. W. Grunert Arch. 39 (1862) 226-. ABC sin A + sin B + sin C = 4 cos generalisation. (1852) 34-. A A "2 ' Arrest, H. d'. Leip. B. sin , cos , etc., geometric demonstration. Barbier, E. N. A. Mth. 4 (1865) 421-. sin a cos B = cos b sin c - sin b cos c cos A. Hertz, K. [1869] (xn) Eec. Mth. (Moscou) 4 (1869-70) (Pt. 2) 94-. tan - - tan - (cos A + i sin A) 1 + tan - tan - (cos A + i sin A) Cayley, A. L. Mth. S. P. 11 (1879-80) 48-. Study, E. Leip. Mth. Ps. Ab. 20 (1893) 85-. Polar triangles, and polar trihedral angles. Harst, A. D. van der. N. Arch. Wisk. 17 (1890) 188- ; Fschr. Mth. (1890) 600. in spherical trigonometry, history of. Braunmuhl, A. von. Bb. Mth. (1898) 65-. Projection, stereographic, application to spheri- cal trigonometry. Grunert, J. A. Grunert Arch. 39 (1862) 318-. Quadratic binary forms, relation of system of three, to spherical trigonometry. Stephanos, C. Par. S. Mth. Bll. 10 (1882) 134-. Eational normal configurations M\ of S 6 . Young, G. C. Tor. Ac. Sc. At. 34 (1898) 429- or 587-. Eesultants, concept. Meyer, F. Crelle J. Mth. 115 (1895) 209-. Simplification of spherical trigonometry. Keogh, C., <& Lebesgue, V. A. N. A. Mth. 12 (1853) 304-. Spherical excess. Grunert, J. A. Grunert Arch. 38 (1862) 220-. . Bacaloglo, E. Grunert Arch. 39 (1862) 237-. , construction. Terquem, 0. N. A. Mth. 5 (1846) 17-. , Gauss's theorem. Clausen, T. As. Nr. 20 (1843) 33-. oblong, area. Gudermann, C. Crelle J. 42 (1852) 280-. polygonometry. Raabe, J. L. Crelle J. 2 (1827) 9-. 469 6830 Spherical Trigonometry. Triangles 6830 Spherical polygons, area. Barbet, . N. A. Mth. 9 (1850) 183-. , regular. Sornin, J. Toul. Mm. Ac. 6 (1856) 76-. , . Faure, H. N. A. Mth. 19 (1860) 421-. , , relations between sides and diameter. Schwarzer, A. Wien Sb. 52 (1866) (Ab. 2) 363-. , star, area. Dostor, G. Arch. Mth. Ps. 63 (1879) 433-. quadrangle, problem. Palmer, J. C. Mth. Gz. 1 (1900) 231-. quadrilaterals. Sohncke, L. A. Grunert Arch. 4 (1844) 447-. , area. Konig, J. F. Grunert Arch. 34 (1860) 12-, 355-. , . Strehlke, F. Grunert Arch. 35 (1860) 104-, 447-. , birectangular. Kramp, C. Gergonne A. Mth. 1 (1810-11) 161-. , inscribed. Collete, J. N. A. Mth. 8 (1849) 435-. , , with sides given. Kambly, . Grunert Arch. 40 (1863) 440-. in small circle. Anon, (vi 317) Dijon Se. Ac. (1821) 106-. , theorem: if two diagonals are quadrants, so is third. Colson, C. G. Mess. Mth. 5 (1876) 161-. Spherical Triangles. Lagrange, J. L. (Comte) de. Par. EC. Pol. J. 6 e cafe. (1799) 270-. Schmeisser, . Crelle J. 10 (1833) 129-. Maas, A. J. Brux. Ac. Bll. 6 (1839) (pte. 2) 271-. Angelitti, F. Nap. Ac. Pont. At. 25 (1895) No. 9, 24 pp. Ambiguous cases. Lentheric, . N. A. Mth. 2 (1843) 32-. . Carbonnelle, J. Brux. Ac. Bll. 19 (1852) (pte. 3) 71-. . Jenkins, M. Mess. Mth. 2 (1873) 150-. Area. Gerling, C. L. Zach M. Cor. 27 (1813) 297-. . Tedenat, . Gergonne A. Mth. 6 (1815- 16) 46-. . Buzengeiger, K. H. J. Lindenau Z. 6 (1818) 3H-. . Lebesgue, V. A. N. A. Mth. 16 (1857) 319-. , and spherical ellipse. Fournier-Vanmon , . N. A. Mth. 7 (1848) 14-, 51-. , theorem, notes. Juel, C. N. Ts. Mth. 10 (A) (1899) 13-. of two spherical triangles, equality. Grunert, J. A. Grunert Arch. 32 (1859) 118-. Areas of plane and spherical triangles. Slwrt- rede, B. Beng. J. As. S. 11 (1842) 776-. . Tillot [Tillol], J. N. A. Mth. 9 (1850) 406-. -: . Lebesgue, V. A. N. A. Mth. 15 (1856) 352. , two theorems. Terquem, 0. N. A. Mth. 9 (1850) 278. "Associated" triangles, properties. Femcick, S. Mathematician 1 (1845) 18-. Cases. Crossicell, W. Bost. Mm. Am. Ac. 3 (1809) 38-. Centre of mean distances. Lhuilier, S. Ger- gonne A. Mth. 2 (1811-12) 72-. Centroid. Besge, V. A. Liouv. J. Mth. 7 (1842) 516. (Besge and Giulio) . Grunert, J. A . Grunert Arch. 4 (1844) 75-. . Matzka, W. Grunert Arch. 4 (1844) 359-. . Hansen, P. A. Leip. B. 22 (1870) 71-. Chordal triangle and spherical triangle, Grunert's theorem, demonstration. Unfer- dinger, F. Grunert Arch. 33 (1859) 89-.' , relations. Simonelli, R. Tortolini A. 5 (1854) 186-. Circles, circum-,in-ande-scribed. Colson, C. G. Edinb. E. S. P. 8 (1875) 589-. , and in-scribed. Clausen, T. Crelle J. 6 (1830) 84-, 90. , , radii. Unferdinger, F. Grunert Arch. 29 (1857) 238. , in- and e-scribed, circle touching. Salmon, G. QJ. Mth. 6 (1863) 67-. Coexistence of three spherical triangles, with two sides equal in pairs. Berg, F. J. van den. N. Arch. Wisk. 14 (1888) 78- ; Fschr. Mth. (1887) 565. Congruence theorems for fully denned spherical triangles. Peter sen, J. N. Ts. Mth. 11 (B) (1900) 77- ; Fschr. Mth. (1900) 504. Differentials of sides and angles, etc. Wolfers, J. P. Grunert Arch. 10 (1847) 431-. Gauss's pentagramma mirificum, extension to any spherical triangle. Dziobek, . Arch. Mth. Ps. 16 (1898) 320-. Graphical solution. Littrow, K. L. von. Wien Sb. 42 (1860) 203-. . Penrose, F. C. As. S. M. Not. 37 (1877) 403-. . Janse, L. N. Arch. Wisk. 11 (1884) 1- ; 12 (1886) 113-. by projection, history. Haller, S. Bb. Mth. (1899) 71-. Large geodetic triangles, resolution. Saija, G. Catania Ac. Gioen. At. 12 (1899) Mem. 3, 58 pp. Legendre's theorem. Gauss, C. F. Crelle J. 22 (1841) 96. (Gauss). Grunert, J. A. Grunert Arch. 1 (1841) 436-. . Winckler, A. Liouv. J. Mth. 16 (1851) 375- ; Crelle J. 44 (1852) 273-. . Catalan, E. C. N. A. Mth. 11 (1852) 406-. . Caldarera, F. Palermo G. Sc. Nt. 1 (1866) 126-. , demonstration. Ivory, J. Tilloch Ph. Mg. 58 (1821) 270-. , . Grunert, J. A. Grunert Arch. 23 (1854) 111-. , . Tissot, A. N. A. Mth. 1 (1862) 5-. , and Serret's. Bouche, E. N. A. Mth. 15 (1856) 354-. , (Bouche). Serret, P. N. A. Mth. 16 (1857) 53-. 470 6830 Spherical Trigonometry. Triangles 6830 Locus of vertex of spherical triangle of given base and area (Lexell's theorem). Barbier, E. Les Mondes 4 (1864) 42-. Malfatti's problem for spherical triangle. Mertens, F. [1872] Crelle J. 76 (1873) 92-. Nine-points circle of spherical triangle. Cayley, . QJ. Mth. 27 (1895) 35-. Perimeter, inner side of. Mobiits, A. F. Leip. B. 12 (1860) 51-. Perpendiculars. Grunert, J. A. Grunert Arch. 22 (1854) 478-. Plane construction. Birkbeck, W. L. Thomson A. Ph. 1 (1821) 259-. and spherical triangles. Danione,D. Brug- natelli G. 5 (1822) 1-. . , analogies. Domenico, D. Brug- natelli G. 4 (1821) 259-. . , , and analytical treatment. Tralles, J. G. Berl. Ab. (1816-17) 65-. , comparison. Grunert, J. A. Grunert Arch. 1 (1841) 110-. , , degree of approximation of results on taking one for the other. Valente do Couto, M. Lisb. Mm. Ac. Sc. 3 (pte. 2) (1814) 119-. , problem, solution. Steiner, J. Crelle J. 28 (1844) 375-. Problems. Meissel, E. Arch. Mth. Ps. 65 (1880) 429-. Rational spherical triangles. Bessell,F. Arch. Mth. Ps. 65 (1880) 363-. Eelations. Moth, F. X. As. Nr. 6 (1828) 217-. between two triangles, plane or spherical. Bretschneider, C. A. Grunert Arch. 2 (1842) 132-. Right angled spherical triangles. Adams, Jas. Thomson A. Ph. 3 (1822) 422-. . Grunert, J. A. GrunertArch.il (1848) 225-. , problems solved by spherical 5-gon. Prestel, M. A. F. GrunertArch.il (1848) 56-. , properties. Dostor, G. Arch. Mth. Ps. 51 (1870) 109-. triangles, spherical and plane, analogies. Lhuilier, S. Gergonne A. Mth. 1 (1810-11) 197-. , , . Buzengeiger, K. H. J. Lindenau Z. 3 (1817) 198- . Small-circle spherical triangle. Hudson, E. C. QJ. Mth. 27 (1895) 378-. spherical triangle compared with plane. Buzengeiger, K. H. J. Lindenau Z. 6 (1818) 264-. . Grunert, J. A. As. Nr. 52 (1860) 49-. triangles. Burg, A. Wien Jb. Pol. I. 7 (1825) 86-. . Grunert, J. A. Grunert Arch. 9 (1847) 8-. . Caldarera, F. Palermo At. 2 (1853) 14 pp. Solution. Unferdinger, F. Arch. Mth. Ps. 41 (1864) 142-. . Eagona, D. Mod. Ac. Sc. Mm. 5 (1887) 53-. Solution of all triangles, spherical and plane, by one formula. Steffenelli, A. von. Munch. D. (1813) 343-. , case (a, b, C). Mollweide, C. Lindenau Z. 1 (1816) 459-. , ( , , ). Sang, E. Edinb. N. Ph. J. 14 (1833) 311-. .constructive. Steinheil, C. A. von. Munch. Sb. (1869) (II) 369-. , given altitudes. Unferdinger, F. Wien Sb. 51 (1865) (Ab. 2) 97-. , . Hunyady, E. von. Z. Mth. Ps. 12 (1867) 91-. , perpendiculars. Skrivan, G. Grunert Arch. 28 (1857) 471-. , side, opposite angle, and one other element. Matzka, W. Grunert Arch. 11 (1848) 300-. , sides. Thomson, (Prof.) James. Edinb. J. Sc. 4 (1831) 124-. , , in plane or spherical triangles. Bretschneider, C. A. Arch. Mth. Ps. 52 (1871) 371-. , (a, b, A) or (A, B, a). Lhuilier, S. Gergonne A. Mth. 2 (1811-12) 257-. by plane triangles on gnomonic projection. Glotin, P. Bordeaux S. Sc. Mm. 3 (1880) 395-. , special cases. Saporetti, A. Bologna Ac. Sc. Mm. 7 (1897) 213-. , and stereographic projection. Martinetti, V. Mil. I. Lomb. Rd. 24 (1891) 830-, 976-. Spherical triangle, one side small, solution. Puissant, L. Par. EC. Pol. Cor. 2 (1809-13) 236-. , ABC, angles between OA and chord BC. Grunert, J. A. Grunert Arch. 26 (1856) 113-. with C = A + B. Unferdinger, F. Arch. Mth. Ps. 53 (1871) 344-. Substitution of chords for sides. Nagel, A. Schlomilch Z. 1 (1856) 257-. Surface. So-rnin, J. Toul. Mm. Ac. 5 (1855) 351-. Transversals bisecting sides. Schawen, P. von. Z. Mth. Ps. 27 (1882) 126-. joining mid-points of sides, and applications. Unferdinger, F. Wien Sb. 52 (1866) (Ab. 2) 323-. Unlimited spherical triangles. Chaurenet, W. Gould As. J. 1 (1851) 39-, 41-. Substitution groups, Napier's rules in particular. Pund, 0. [1897] Hamb. Mth. Gs. Mt. 3 (1900) 290-. Transformation of elliptic integrals, application to spherical trigonometry. Russell, R. QJ. Mth. 20 (1885) 378-. Transversals in spherical triangles and quadri- laterals. Rogner, J. Arch. Mth. Ps. 45 (1866) 318-. Spheroidal triangle, small, reduction of angles to angles of corresponding plane or spherical triangle. Hansen, P. A. Leip. B. 21 (1869) 138- ; Leip. Ab. Mth. Ps. 9 (1871) 289-. 471 6830 Trigonometry Spheroidal trigonometry. Oriani, B. Zach M. Cor. 10 (1804) 244-; 11 (1805) 551- ; Bo- logna Mm. I. It. 1 (1806) 119- ; 2 (1808) 1-; (pte. 2) (1810) 1- ; Mil. Mm. I. Lomb. Yen. 4 (1832) 325- ; Mil. Ephem. As. (1827) 3- ; (1828) 3-; (1829) 3-. . Cayley, A. [1876] As. S. M. Not. 37 (1877) 92. , formula. Bonsdorff,A. [1885] StPet. Ac. Sc. Bll. 30 (1886) 205-. Stadia measurements, theory. Winslow, A. V. Nost. Eng. Mg. 30 (1884) 313-. Straight lines, and plane trigonometry. Schroder, J. C. Mengelwerk Wisk. Vh. 2 (1816) 119-. Sum 2 sin (p + xq) etc. Fuss, N. [1796] St Pet! Ac. Sc. N. Acta 12 (1801) 125-. . Reusch, E. Z. Mth. Ps. 11 (1866) 536-. Sums of like powers of sines and cosines of divisions of circumference. Lentheric, . Gergonne A. Mth. 16 (1825-26) 39-. Surds, trigonometrical quadratic, values. Gray, P. Mess. Mth. 6 (1877) 105-. Survey, trigonometrical, of Europe. Eeitz, F. H. Hamb. Mth. Gs. Mt. 1 (1889) 39-. Surveying, approximately concurrent lines, to find intersection. Sanchez, P. C. Mex. S. "Alzate" Mm. 13 (1900) 43-. , errors. Perez , E. Mex. S. ' ' Alzate ' ' Mm. 8 (1894) 135-. , , probable. Gauss, C. F. As. Nr. 1 (1823) 81-. , importance of verticality in instruments. Sanchez, P. C. Mex. S. " Alzate" Mm. 12 (1898) 467-. , plane, problem. Hoppe, E. Arch. Mth. Ps. 17 (1900) 269-. , Pothenot's problem and modern geometry. Binder, W. Wien Ak. Sb. 83 (1881) (Ab. 2) 659-. , problem of 3 points in. Campbell, John. Madras J. 4 (1836) 321-. , simplification of formula in. Driencourt, . A. Hydrog. 20 (1899) 68-. , trigono telemeter, use. Antequeda, C. Arg. S. Ci. A. 45 (1898) 390- ; 46 (1898) 5-. Tables. Lindman, C. F. Stockh. Ofv. 11 (1854) 183- ; Grunert Arch. 25 (1855) 284-. for calculating radius of an arc, length, angle at centre, etc., when chord and height of the arc are given. Sigaud, . (vm) A. Cond. Pon. Chauss. 1 (1857) 132-. Tachymetry (tacheometry or rapid surveying). Teischinger, E. Hann. Archt.-Vr. Z. 31 (1885) 215-. ( ) , methods. Jordan, W. Hann . Archt.-Vr. Z. 30 (1884) 453-. ( ), theory, and logarithmic tachy- meter. Fischer, G. Cztg. Opt. 6 (1885) 169-, 181-, 200-. Tangent (45 w>). Parkhurst, H. M. Gould As. J. 6 (1861) 103. Tangents and anti-tangents, calculation. Weddle, T. Mathematician 2 (1847) 135-. Descriptive Geometry 6840 Teaching of trigonometry. Hoilel, J. G. Mt. 13 (1875) 72-. . Aragon, A. Mex. S. " Alzate " Mm. 5 (1891) 69-. . Fehr, H. Ens. Mth. 1 (1899) 45-. , various points. Pfeil,L.von. Arch. Mth. Ps. 58 (1876) 319-. Theorem : sin a sin b sin c sin d + cos a cos b cos c cos d = sin a' sin b' sin c' sin d' + cos a' cos b' cos c' cos d'. Glaisher, J. W. L. Camb. Ph. S. P. 3 (1880) 319-, 383-. Transformation, trigonometrical. Cayley, A. Camb. and Dubl. Mth. J. 9 (1854) 61-. , , used by Hansen in theory of perturba- tions. Baillaud, B. Bll. Sc. Mth. As. 2 (1878) 292-. Trapezium, calculation from sides. Bret- schneider, C. A. Arch. Mth. Ps. 52 (1871) 24-. Trigonometry in absolute geometry. Simon, M. Crelle J. Mth. 109 (1892) 187-. Viete, Francois, trigonometry of. Ritter, F. As. Fr. C. E. (1892) (Pt. 2) 208-. 6840 Descriptive geometry; perspective. Analytical treatment of descriptive geometry and perspective. Doppler, C. Bohm Gs. Ab. 1 (1837-40) 78 pp. , four lectures. Hesse, L. O. Z. Mth. Ps. 11 (1866) 369-. Anharmonic or Cross-Ratio. Davies, T. S. Mathematician 1 (1845) 169-, 243-; 2 (1847) 91-, 199-, 310- ; 3 (1850) Suppl. 35-. De Morgan, A. Ph. Mg. 35 (1849) 165-. Terquem, O. N. A. Mth. 12 (1853) 272-, 358-. Clifford, W. K. [1866] L. Mth. S. P. 2 (1869) 3-. Busche, E. Mth. A. 41 (1893) 591-. Autonne, L. N. A. Mth. 18 (1899) 341-. Analogues in solid geometry. Heis, E. Grunert Arch. 31 (1858) 37-. Anharmonic configurations, interchangeable- ness of elements. Hofmann, F. [1882] Arch. Mth. Ps. 69 (1883) 214-. ratio equation. Cayley, A. Mess. Mth. 17 (1888) 95-. of four points in plane. Lucas, F. C. E. 77 (1873) 1463- . , three being fixed, variations in. Laisant, C. A. Par. S. Mth. Bll. 17 (1889) 169-. and invariant theory of complex quantities. Schick, J. [1900] Munch. Ak. Sb. 30 (1901) 249-. Pappus's problem. Fiedler, W. Schlomilch Z. 5 (1860) 377-. sextic. Walker, J. J. QJ. Mth. 10 (1870) 53-. (Walker). Cayley, A. QJ. Mth. 10 (1870) 56-. 472 6840 Anharmonic Ratio Anharmonic ratio sextic of pair of conies. Walker, J. J. QJ. Mth. 10 (1870) 162-. and sine generalisations. Folie, F., & Le Paige, C. Brux. Ac. Bll. 48 (1879) 41-. ratios of m points in plane. Mobius, A. F. Leip. B. (1853) 14-. Doubly anharmonic relation. Battaglini, G. G. Mt. 1 (1863) 321-. Equi-anharmonic range, construction. Schroeter, H. E. Mth. A. 10 (1876) 420-. ranges, upon 2 straight lines, theorems. SpottisiKoode, W. Camb. and Dubl. Mth. J. 8 (1853) 103-. relations. Battaglini, G. Nap. Ed. 2 (1863) 88-. Geometrography, application of. Lemoine, E. As. Fr. C. E. (1894) (Pt. 2) 55-. Notation. Johnson, W. W. Des Moines Anal. 9 (1882) 185- ; 10 (1883) 76-. Pencil of 5 straight lines in space, anharmonic ratio. Merrifield, C. W. Mess. Mth. 5 (1876) 94-. Pencils of straight lines, anharmonic properties. Terquem, 0. N. A. Mth. 12 (1853) 90-. Problem. Baltrusch, S. E. Grunert Arch. 16 (1851) 245-. . Ponte Horta, F. da. [1881] Lisb. J. Sc. Mth. 8 (1882) 137-. Projective ranges. Jacobi, [A. non] C. G. J. Crelle J. 31 (1846) 40-, 93-. , construction. Sacheri, G. [1873] (xi) Tor. Ac. Sc. At. 9 (1873-74) 76-. , theorem. Pasch, M. Crelle J. Mth. 91 (1881) 349- ; Z. Mth. Ps. 27 (1882) 124-. Eange of points in a straight line. Eetali, V. [1882] Bologna Ed. (1883) 38-. Eelation of 1st degree. Battaglini, G. Nap. Ed. 2 (1863) 122-. von Staudt's casts (Wiirfe), theory. Sturm, E. Mth. A. 9 (1876) 333-. (),. Schroder, E. Mth. A. 10 (1876) 289-. ( ), , and its extension. Lilroth, J. Gott. Nr. (1873) 767-. (-),, . Kohn, G. [1894] Mth. A. 46 (1895) 285-. ( ), , , and imaginaries in geometry. Lilroth, J. [1874-76] Mth. A. 8 (1875) 145- ; 11 (1877) 84-. Theorem. Liouville, J. Liouv. J. Mth. 17 (1852) 391-. of higher geometry. Tsinger, V. Y. [1869] (xn) Eec. Mth. (Moscou) 4 (1869-70) (Pt. 2) Lie. Nazimov, P. Kazan S. Ps.-Mth. Bll. 7 (1898) 92-. Theorems, extension. Heis,E. Grunert Arch. 31 (1858) 39-. Axonometry. Schlomilch, 0. Giving. 2 (1856) 196- ; 5 (1859) Weislach, J. Giving. 2 (1856) 236- ; 3 (1857) 61-, 102-. Mann, F. Schlomilch Z. 4 (1859) 284-. Schlomilch, 0. Schlomilch Z. 4 (1859) 361-. Axonometry 6840 Vecchi, S. II Polit. 18 (1870) 857- ; 19 (1871) 466-, 541-, 592-, 717- ; 20 (1872) 98-, 216-. Eeis, J. Forster Al. Bauztg. 37 (1872) 69-. Hauck, G. Z. Mth. Ps. 21 (1876) 81-. Seidelin, C. N. Ts. Mth. 1 (B) (1890) 49-. Sobotka, J. Prag Sb. (1900) (Mth.-Nt.) No. 33, 20pp. Angles, locus of planes on which projection is constant. Delarive, L. [1882] Gen. S. Ps. Mm. 28 (1883-84) No. 2, 76 pp. Axes of affinity, application to graphic deter- mination of plane. Fiedler, W. Schlomilch Z. 6 (1861) 76-. projections of circle, direct determination. Morstadt, E. Wien Ak. Sb. 56 (1867) (Ab. 2) 92-. Axis of homology of triangle and its Brocard's triangle. Tarry, G. Mathesis 15 (1895) 5-. Change of coordinate planes. N., F. G. Mt. 12 (1874) 154-. Clinogonal representation of surfaces of revolu- tion. Pelz, C. Prag Sb. (1895) (Mth.-Nt.) No. 7, 15 pp. Clinographic parallel perspective, graphic solu- tion. Koutny, E. Giving. 18 (1872) 265-. Drawing and axonometry, geometrical princi- ples. Sella, O. Arch. Mth. Ps. 43 (1865) 245-. Oblique free projection. Peschka, G. A. von. Wien Ak. Sb. 75 (1877) (Ab. 2) 917-. and orthogonal parallel projection. Neelsen, C. Giving. 14 (1868) 461-. projection, identity of constructions. Staudigl, E. Wien Sb. 64 (1871) (Ab. 2) 490-. parallel projection. Burmester,L. Z. Mth. Ps. 16 (1871) 449-. projections. Deschwanden, J. W. von. Ziir. Vjschr. 6 (1861) 254-; 7 (1862) 159-. Orthogonal axonometric foreshortened circle. Tesaf , J. Wien Ak. Sb. 81 (1880) (A b. 2) 453-. axonometry. Streissler, J. Arch. Mth. Ps. 65 (1880) 208-. , scientific treatment. Pelz, C. Wien Ak. Sb. 81 (1880) (Ab. 2) 300- ; 83 (1881) (Ab.2) 375- ; Prag Sb. (1885) (Mth.-Nt.) 648-. projection, axonometry as. Weiler, A. Z. Mth. Ps. 33 (1888) 257-. on two planes not at right angles as founda- tion for perspective. Skuhersky, E. [1858] Bohm. Gs. Ab. 10 (1859) 51-. projections. Maszkowski, K. Par. T. Nauk Sc. Pam. 5 (*1874) Art. 1, 38pp. in metric-projective geometry. Ovidio, E. d\ Tor. Ac. Sc. At. 11 (1875-76) 830-. Pohlke's theorem. Schwarz, H. A. [1863] (vm) Crelle J. 63 (1864) 309-. . Deschwanden, J. W. von. Ziir. Vjschr. 10 (1865) 384-. . Eeye, T. Ziir. Vjschr. 11 (1866) 350-. . Pelz, C. [1877] Wien Ak. Sb. 76 (1878) (Ab. 2) 123-. . Peschka, G. A. von. [1878] Wien Ak. Sb. 78 (1879) (Ab. 2) 1043-. . Schur, F. Mth. A. 25 (1885) 596-. . Mandl, J. Wien Ak. Sb. 94 (1887) (Ab. 2) 60-. 473 6840 Axonometry Pohlke's theorem. Kilpper, C. Mth. A. 33 (1889) 474-. . Ruth, F. Wien Ak. Sb. 100 (1891) (Ab. 2a) 1088-. . Schur, F. Crelle J. Mth. 117 (1897) 24-. Problem, fundamental. Beck, A. Crelle J. Mth. 106 (1890) 121-. Problems, graphic solution. Deschtvanden, J. W. von. Ziir. Vjschr. 9 (1864) 223-. Projection or shadow of a circle, solution of 2 problems. Motta Pegado, L. P. da. Lisb. J. Sc. Mth. 6 (1878) 217-. Bilinear forms, three pairs of principal tri- angles connected with. Igel, B. Mh. Mth. Ps. 5 (1894) 289-. Centric figures, graphical properties. Reyes y Prosper, V. Mth. A. 32 (1888) 157-. Complex elements in geometry, analysis. Klein , C. F. Gott. Nr. (1872) 373-. Correspondence, elementary presentation of, etc. Essen, E. Grunert Arch. 29 (1857) 77-. Drawing, compulsory laws of error. Haddon, A. L. Nt. 48 (1893) 402-. , geometrical. Paulus, C. Grunert Arch. 23 (1854) 364-. from nature, instrument for. B., T. C. Nicholson J. 10 (1805) 111-. , programme for systematic teaching of. Gobin, A. Lyon S. Ag. A. 5 (1882) 131-. , solid and perspective, teaching. La Gour- nerie, de. A. Cons. Arts et M6t. 10 (*1873) 260-. Figures, mutual dependence. Battaglini, G. Nap. Ac. Sc. Mm. 2 (1855-57) 175-. , rectilinear. Olivier, L. Crelle J. 1 (1826) 227-. Forms, geometrical, of second species. Batta- glini, G. G. Mt. 3 (1865) 298- ; 4 (1866) 96-, Harmonic axes and centres. Cassani, P. (ix) Yen. Aten. At. 3 (1866) 147- ; (vn) 5 (1868) 207-. of system of straight lines and planes. Cassani, P. G. Mt. 4 (1866) 128-. configuration 24 4 . Victor, A. [1884] Frei- burg B. 8 (1882-85) 206-. division, generalisation. Laisant, C. A. As. Fr. C. R. (1878) 135-. mean-centres (Poncelet). Cauchy, A. L. Gergonne A. Mth. 16 (1825-26) 349-. pencils. Chevillard, A. J. N. A. Mth. 1 (1842) 312-. -. Euler, J. Fschr. Mth. (1891) 584. plane of point with reference to tetrahedron. Hain, E. [1876] Arch. Mth. Ps. 60 (1877) 302. pole of points in a line. King, L. Mh. Mth. Ps. 8 (1897) 361-. range, to find pair of points forming, with 2 given pairs. Grelle, F. Z. Mth. Ps. 13 (1868) 148-. ranges. Dostor, G. J. N. A. Mth. 9 (1850) 118- ; 10(1851)73-. Descriptive Geometry 6840 Harmonic relation of two lines or points. Cayley, A. Ph. Mg. 6 (1853) 105-. section, illustrations. Browning, H. H. Edinb. Mth. S. P. 2 (1884) 6-. on straight line. Bohannan, R. D. A. Mth. 1 (1884-85) 121-. 6-gon and 8-gon. Heger, R. Z. Mth. Ps. 18 (1873) 307-. Homographic division, imaginary. Battaglini, G. G. Mt. 2 (1864) 142-. of 2 straight lines (de la Gournerie). [Schiappa] Monteiro, A. G. Teix. J. Sc. 1 (1878) 117-. . Bohanntin, R. D. A. Mth. I (1884-85) 109-. figures and projective systems of planes. Reye, T. Crelle J. Mth. 104 (1889) 211- ; 106 (1890) 30-, 315- ; 107 (1891) 162- ; 108 (1891) 89-. in space, properties. Jonquieres, E, de. N. A. Mth. 17 (1858) 51-. , self -corresponding points. Deirulf, E. N. A. Mth. 17 (1878) 265-. , volumes of 2. Amigues, E. N. A. Mth. 12 (1873) 374-. Homographies and rotations, correspondence. Cayley, A. Mth. A. 15 (1879) 238-. Homography. Poudra, . N. A. Mth. 19 (1860) 108-. , double elements of homographies and their relation to curves of 3rd order and class. Weyr, E. Prag Sb. (1869) 3-. , in projective systems. Broden, T. Ts. Mth. 6 (1889) 6-; Fschr. Mth. (1888) 585. in geometry. Chasles,M. Brux. Mm. Cour. II (1837) 575-. and homology of figures in 3 dimensions. Dewulf, E. Bll. Sc. Mth. As. 1 (1877) 137-. , Newtonian transformation applied to. Lentheric, . (vi Adds.) Mntp. Ac. Sc. Mm. 5 (1861-63) 205-. and perspective. Housel, . N. A. Mth. 8 (1869) 492-. Homologous figures, theory. Seiliger, D. Kazan Un. Mm. (1894) (Pt. 4) 127-. tetrahedra. Hermes, O. [1857] Crelle J. 56 (1859) 218-. triangles. Muir, T. [1872] Mess. Mth. 2 (1873) 55-, 99-. , similar. Jefabek, V. Mathesis 16 (1896) 81-. in s.tudy of pentahedron. Gallucci, G. Nap. Rd. 38 (1899) 131-. , triply homologous. Aubel, H. van. Mathesis 17 (1897) 53-. , , related to conies. Ripert, L. As. Fr. C. R. (1900) (Pt. 2) 112- ; Mathesis 20 (1900) 226-. , , . Third, J. A. Mathesis 20 (1900) 153-. Homology. Cassani, P. G. Mt. 3 (1865) 150-. , centres, faisceaux, and envelopes of. Martin, H. Edinb. R. S. T. 24 (1867) 591-. , principles, application to perspective. Regis, D. G. Mt. 8 (1870) 222-. 474 6840 Involution Homology, principles, application to perspec- tive. Zona, T. Palermo G. Sc. Nt. 15 (1882) 152-. , trihedral sheafs. Eocha Peixoto, A. F. da. G. Teix. J. Sc. 5 (*1883) 17-. Imaginaries, geometry of. Tarry, G. As. Fr. C. E. (1887) (Pt. 2) 168- ; (1888) (Pt. 2) 31-. , . Retail, V. Bologna Ac. Sc. Mm. 9 (1888) 259-. in project! ve geometry. Schur, F. Mth. A. 39 (1891) 113-. , introduction of. Bonola, R. G. Mt. 38 (1900) 105-. Imaginary elements, pairs of, in synthetic pro- jective geometry. Segre, C. Tor. Ac. Sc. Mm. 38 (1888) 3-. projection. Servais, C, Brux. Mm. Cour. 8, 52 (1894-95) No. 3, 51 pp. of conic sections. Retail, V. Bologna Ac. Sc. Mm. 7 (1886) 601-. Infinitely distant elements in geometry. Weyr, E. Casopis 1 (*1872) 161- ; 2 (*1873) 105 ; Fschr. Mth. (*1872) 328-; (*1873) 382. Involution. Terquem, O. N. A. Mth. 12 (1853) 24-. Hesse, L. 0. [1863] (vn) Crelle J. 63 (1864) 179-. Haiti, E. Arch. Mth. Ps. 63 (1879) 407-. Korteweg, D. J. N. Arch. Wisk. 18 (1891) 131-. Configurations, and application to conies. Seydewitz, F. Grunert Arch. 4 (1844) 246- ; 5 (1844) 225-, 331-. , uniform, elements of rank. Paulus, C. Grunert Arch. 21 (1853) 175-. Elementary property of involutions. Pomey, J. B. N. A. Mth. 4 (1885) 489-. Hesse's theorem and adjoined involutions. Roger, R. [1899] Hamb. Mth. Gs. Mt. 3 (1900) 390-. Hexagon, involutions connected with. Boger, R. [1899] Hamb. Mth. Gs. Mt. 3 (1900) 387-. Homography and involution. Fiedler, W. Schlomilch Z. 6 (1861) 1-. Involution common to group of 5 lines and system of 9 planes. Serret, P. C. R. 113 (1891) 326-, 347-. of different orders in systems of 2nd species. Battaglini, G. (ix) Nap. Ac. At. 2 (1865) A*b. 19, 15 pp. Plane involution. Poudra, N. G. N. A. Mth. 3 (1864) 498-. relatively to sphere. Poudra, N. G. N. A. Mth. 4 (1865) 493-. Points. Mobius, A. F. Leip. B. 7 (1855) 33-. in plane. Mobius, A. F. Leip. B. (1853) 176-. , 6, systems lying in three ways. Maschke, H. A. Mth. 10 (1895-96) 22-. , 8, in plane, involutions formed by. Le Pont, H. G. Teix. J. Sc. 7 (1886) 91-. Poles of pencil in involution. Gribodo t G. [1880] Tor. Ac. Sc. At. 15 (1879) 521-. Protective involutions. Paolis, R. de. Bm. R. Ac. Line. Rd. 2 (1886) (Sem. 2) 335-. Perspective 6840 Projective involutions, and projective, pyra- midal, triangular and other relations. Ter- quem, O. N. A. Mth. 6 (1847) 68-. Ranges. Sudden, E. Mth. Gz. 1 (1900) 86. Straight line, involution on, theorem. Jon- quieres, E de. N. A. Mth. 19 (1860) 316-. Magnitude, theorems independent of. Seyde- witz, F. Grunert Arch. 10 (1847) 59-. Panoramas. Petit, G. Rv. Sc. 43 (1889) 558-. Pencils, intersections of four line and two line. Cayley, A. Ph. Mg. 29 (1865) 501-. of lines, theorems. Spottiswoode, W. Camb. and Dubl. Mth. J. 6 (1851) 160-. , projective, determination of skew relation of two in plane. Consentius, R. 0. Z. Mth. Ps. 25 (1880) 122-. and ranges. Ramisch, A. Arch. Mth. Ps. 69 (1883) 54-. , projective. Gtipel, A. Grunert Arch. 3 (1843) 93-. PERSPECTIVE. Tacani, F. Bb. It. 40 (1825) 417-. Lermier, J. Bordeaux Ac. Sc. Se. Pbl. (1829) 40-. Fagnoli, G. Bologna N. Cm. 9 (1849) 553-. Grunert, J. A. Grunert Arch. 32 (1859) 361-. Larmor, A. QJ. Mth. 21 (1886) 339-. PeliSek, M. Prag Sb. (1886) (Mth.-Nt.) 290-. Rees, J. van. Mbl. Nt. (1888) 73-. (Newton.) Graham, R. H. Nt. 41 (1890) 439-. PeliSek, M. Prag Sb. (1890) 1 (Mth.-Nt.) 175-. Affinity of two spaces. Beck, A. Z. Mth. Ps. 44 (1899) 85-. Analytical perspective. Mossbrugger,L. Gru- nert Arch. 11 (1848) 113-. . Kahrer, G. Wien Jbr. Ober-Realsch. Inn. Stadt 2 (1860) 84-. Anamorphosis and distortion. Hachette, J. N. P. Par, EC. Pol. Cor. 1 (1804-08) 361-. new constructions for reflexion images forplane and curved surfaces. Niemtschik, R. Wien Sb. 53 (1866) (Ab. 2) 577-. Architectural plans, effects of perspective with reference to. Menzel,C.A. Dingier 58 (1835) 89-. Barycentric perspective, application to trans- formation of structures. Rankine, W. J. M. (vin) Ph. Mg. 26 (1863) 387-. Bounding plane in linear perspective. Anton, H. Wien Sb. 54 (1866) (Ab. 2) 230-. Circle, perspective of. Lubbock, J. W. Ph. Mg. 5 (1829) 420-. _, . Chapuis, . J. Ge"n. Civ. 8 (1830) 131-. , . Sobotka, J. Mh. Mth. Ps. 10 (1899) 289-. , , and axes of quadrics. Sobotka, J. Wien Ak. Sb. 109 (1900) (Ab. 2a) 583-. , , theorems. Hey, R. Phil. Trans. (1814) 348-. 475 6840 Perspective Conic, projection of, is a conic, demonstration. Lean, . Par. S. Phlm. Bll. 1 (1899) 87-. Construction, new. Leib, L. Arch. Mth. Ps. 11 (1892) 1-, vn. Coordinate planes. De Morgan, A. Camb. Mth. J. 2 (1841) 92-. Curved surfaces. Hachette, J. N. P. Par. S. Phlm. Bll. (1823) 147- ; Crelle J. 1 (1826) 371-. Distance relations of ground-lines, preservation. Chevillard, A. J. N. A. Mth. 13 (1854) 91-. Distortion. Anger, G. T. Danzig N. Schr. 4 (1851) (Heft 4) 20 pp. in perspective representation of plane figures. Scheffers, G. Leip. Mth. Ps. B. 44 (1892) 162-. Double perspective. Brasseur, J. B. Liege Mm. S. Sc. 3 (1873) 731-. Eye, quality of, in relation to perspective. Hiern, W. P. (m) Mess. Mth. 2 (1864) 30-. History of theory. Viehweger, . Giving. 35 (1889) 165-. Interference curves depending on perspective. Mallock, A. [1899] Nt. 61 (1899-1900) 29-. Isometric perspective. Parish, W. [1820] Camb. Ph. S. T. 1 (1822) 1-. Linear perspective. Gergonne, J. D. Ger- gonne A. Mth. 13 (1822-23) 181-. . Marstaller, E. Hann. Z. Archt. Vr. 14 (1868) 395-. , elements. Mazzetti, G. M. G. Arcad. 46 (1830) 272- ; 47 (1830) 32-, 300- ; 48 (1830) 125- ; 50(1831) 1-. by method of vanishing points. Hachette, J. N. P. Par. EC. Pol. Cor. 1 (1804-08) 313-. , principles (Taylor). Uglieni, M. G. Mt. 3 (1865) 338-. , . Hauck, G. Z. Mth. Ps. 26 (1881) 273- ; 27 (1882) 236-. , and application. Gamier, J. G. Quetelet Cor. Mth. 2 (1826) 3-. Metric relations. Schlomilch, O. Grunert Arch. 7 (1846) 274-. Monodimetric and anisometric methods. Weis- bach, J. Pol. Mt. 1 (1844) 125-. Orthographic perspective. Skuhersky, R. Wien SB. (1850) (Ab. 2) 326-. Paintings, theory of perspective in. Danders, . Amst. Ak. Vs. M. 1 (1885) 318-, 383-. Perspective drawing, history. RouchS, E. A. Cons. Arts et Met. 3 (1891) 357-. , methods. Williams-Ellis, J. C. [1872] Camb. Ph. S. P. 2 (1876) 276-. ellipse of sphere. Gergonne, J. D. Gard Not. Tr. Ac. (1807) 192-. projections. Hain, E. Arch. Mth. Ps. 70 (1884) 281-. . Thornton, W. M. A. Mth. 1 (1884-85) 12-. . Echols, W. H. A. Mth. 4 (1888) 93- ; 5 (1889-90) 15-. representations. King, L, (xn) Mag. Tud. Ak. Etk. (Mth.) 9 (No. 11) (1882) 10 pp. . PeliSek, M. Prag Sb. (1886) (Mth.- Nt.) 360-. . Prochdzka, B. Casopis 26 (1897) 45-; Fschr. Mth. (1897) 462. Photographic perspective. Mallock. A. Nt. 43 (1891) 517-. Descriptive Geometry 6840 Plane point-systems. Voss, (Dr) A. Z. Mth. Ps. 17 (1872) 375-. Proportional perspective. Stix, A. Forster Al. Bauztg. 35 (1870) 31-. Belief-perspective (Poudra). Chasles, M. C. E. 37 (1853) 880-. . Anger, C. T. As. Nr. 38 (1854) 289-. . Burghardt, . Grunert Arch. 36 (1861) 437-. . Pelitek, M. Prag Sb. (1886) (Mth.-Nt.) 434- ; Casopis 18 (1889) 78-, 107. , especially of sphere. Morstadt, R. Z. Mth. Ps. 12 (1867) 326-. Rows of trees, perspective of. Gergonne, J. D. Gard Not. Tr. Ac. (1807) 168-. Similarly divided lines. Skuhersky, R. Wien SB. (1851) (Ab. 2) 471-. Stereoscopic drawing apparatus. Jensen, J. [1870] Arch. f. Ap. 4 (1871) 233-. perspective. Stevens, W. Le C. Am. As. P. (1894) 109. Survey maps, perspective drawing in. Laus- sedat, (le col.) A. A. Cons. Arts et Met. 2 (1890) 291- ; 3 (1891) 313- ; 5 (1893) 283-; 6 (1894) 1-, 81-. Triangles in perspective. Rosanes, J. Mth. A. 2 (1870) 549-. . Schroter, H. Mth. A. 2 (1870) 553-. . Hain, E. [1880] Arch. Mth. Ps. 66 (1881) 282-. , Desargues's theorem. Hunyady, E. von. [1879] Crelle J. Mth. 89 (1880) 79-. , proved by descriptive geometry. Sucharda, A. Casopis 29 (1900) 42-; Fschr. Mth. (1900) 517. , , space analogies. Vdlyi, G. Mth. Termt. Ets. 11 (1893) 30-; Mth. Nt. B. Ung. 13 (1897) 166-. , multiple, groups. Vdlyi, J. Mli. Mth. Ps. 9 (1898) 169-. , plane, relations of position. Wede- kind, L. Mth. A. 16 (1880) 209-. , triply. Third, J. A. [1900] Edinb. Mth. S. P. 19 (1901) 10-. and tetrahedra in perspective. Hess, E. Mth. A. 28 (1887) 167-. with their reciprocals. Brioschi, F. QJ. Mth. 1 (1857) 368-. Vanishing points. Benteli,A. Bern Mt. (1875) 229-. Photogrammetric constructions. Prochdzka, B. fiasopis 27 (1898) 1-, 101- ; Fschr. Mth. (1898) 449. Photogrammetry. Finsterwalder, S. D. Nf. Vh. (1897) (Th. 2, Hdlfte 1) 32. , problem in photographic geometry. Pro- chdzka, B. 6asopis 25 (1896) 341- ; Fschr. Mth. (1896) 416. Polar quadrilateral. Brill, A. Mth. A. 20 (1882) 531-. systems, theorem of Cremona's. Jung, G. G. Mt. 14 (1876) 139-. triangles, of a circle, two are in perspective. Affolter, F. G. [1872] G. Mt. 11 (1873) 110-. 476 6840 Practical Solid Geometry 6840 Poristic tetrahedra. Bobillier, . Gergonne A. Mth. 18 (1827-28) 320-. (Bobillier). Gergonne, J. D. Gergonne A. Mth. 19 (1828-29) 32-, 220-. Practical Solid Geometry. Mange, G. Par. Se. EC. Norm. 1 (1800) 49-, 280-, 413- ; (pte. 2) 28-, 63-, 144- ; 2 (1800) 149-, 338-; 3 (1800) 61-, 332-; 4 (1800) 87-, 291-. Hachette, J. X. P. Quetelet Cor. Mth. 7 (1832) 360-. Dames, T. S. Camb. and Dubl. Mth. J. 1 (1846) 227-; 2 (1847) 252-. Fiedler, W. Ziir. Vjschr. 25 (1880) 217-. Mtiller, E. Konigsb. Schr. 40 (1899) [23]-. Apparatus for illustrating descriptive geometry. Hoza, F. Arch. Mth. Ps. 53 (1871) 188-. Applications. Brasseur, J. B. [1853] Brux. Ac. Sc. Mm. 29 (1855) 148 pp. Cone, foci of section obtained by tangential spheres. Dandelin, G. P. Par. S. Phlm. N. Bll. (1826) 3. Constructions. Burali-Forti, C. Rv. Mt. 2 (1892) 96-. , execution. Bardin, . N. A. Mth. 10 (1851) 32-; 11 (1852) 268-. , problem. Breton, Phil. Mathesis 9 (1889) 73-. Cylinder, right, cut by another at right angles, surface area intercepted. Lommel,E. Grunert Arch. 34 (1860) 286-. Domes. Hausding,A. Forster Al. Bauztg. 33- 34 (1868-69) 305-. Families of curves, plane and "relief" repre- sentation. Graberg, F. Ziir. Vjschr. 34 (1889) 209-. Focal planes of plane curve with one or more axes of symmetry. Schoute, P. H. C. R. 125 (1897) 931- . Forms, geometrical, representation. Nicodemi, JR. Nap. Ac. Pont. At. 25 (1895) No. 3, 24 pp. ; 28 (1898) No. 3, 14 pp. Geometrography, application. Lemoine, E. As. Fr. C. R. (1894) (Pt. 2) 82-. History. Bache, L. Mth. Ts. 2 (1860) 88-, 129-. Intersection of planes with cones and cylinders, etc., etc., general method of representation. Niemtschik, R. Wien Sb. 63 (1871) (Ab. 2) 571. Maximum brilliancy, points of. Hachette, J. N. P., d- Mange, . Par. EC. Pol. Cor. 1 (1804-08) 295-. , . Schoute, P. H. Wien Ak. Sb. 90 (1885) (Ab. 2) 983-. Methods. Fiedler, W. Schlomilch Z. 8 (1863) 444-. New projection method. Fiedler, W. Ziir. Vjschr. 24 (1879) 205-. Practical solid geometry and projective geo- metry. Fiedler, W. Wien Sb. 55 (1867) (Ab. 2) 659-. Problem, note. Buffini, F. P. (xi) Mod. Ac. Sc. Mm. 7 (1866) 17-. Projection, estimation of errors. Burg, A. Wien Jb. Pol. I. 7 (1825) 99-; 8 (1826) 213-. Projection, horizontal, etc., of system of sur- faces, representation of points. Caron, J. N. A. Mth. 2 (1883) 161-. Rectangle on sphere. Jochmann, E. Z. Mth. Ps. 14 (1869) 532-. Revolution quadrics, projection of intersection curve. Gros, . N. A. Mth. 19 (1860) 29-. surface on plane, best projection. Markov, A. A. St Pet. Ac. Sc. Bll. 2 (1895) 177-. Shadow of fillet on base of column. Belli, G. Brugnatelli G. 10 (1817) 287-. hemispherical vault. Bordoni, A. Brug- natelli G. 8 (1815) 60-. theory, apparent contours, and separation of light and shadow. Hachette, J. N. P. Par. S. Phlm. Bll. (1833) 2-. , application to machine-drawing. Hachette, J. N. P. Par. EC. Pol. Cor. 2 (1809-13) 13-. , constructive. Dupuis, . Par. EC. Pol. J. (1794-95) 167-. , , formulae. Fischer, F. W. N. Arch. Wisk. 10 (1884) 172-. , , fundamental theorem. Streissler, J. Arch. Mth. Ps. 69 (1883) 144-. , , (Streissler). Pelz, C. Arch. Mth. Ps. 69 (1883) 437-. , penumbra, contour. Bordoni, A. Brug- natelli G. 5 (1822) 255-, 341-. , photometric bases in brushwork. Chretien, L. V. A. Cond. Pon. Chauss. 17 (1873) 29-, 41-. , sphere, shading in brushwork. Breton, Phil. [1872] (xn) Isere S. Bll. 4 (1875) 259-, 321-. Shadows, application of theorem. Bruno, G. (VH) A. Mt. 5 (1863) 18-. , theory of envelopes. Meyer, C. T. Grunert Arch. 9 (1847) 45-. , architectural ornaments. Bordoni, A. Brugnatelli G. 8 (1815) 284-. , and cast shadows. Bonnet, O. N. A. Mth. 5 (1866) 71-. , loci of centres of boundary curves. Moss- brugger, L. Grunert Arch. 6 (1845) 7-. of quadrics, construction, for central illumi- nation. Pelz, C. Prag Sb. (1879) 514-. rotation surfaces. La Gournerie, J. de. Par. EC. Pol. J. 35" cah. (1853) 29-. in parallel light. Koutny, E. Wien Sb. 55 (1867) (Ab. 2) 215-. screw surfaces. La Gournerie, J. de. Par. EC. Pol. J. 34 cah. (1851) 1-. , screw of Saint-Gilles. Vialla, B. Par. EC. Pol. J. cah. 37 (1858) 191-. , triangular thread screw, contour. Frane/iis, J. F. Par. EC. Pol. Cor. 2 (1809- 13) 69-. _ , , in parallel light. Hachette, J. N. P. Par. EC. Pol. Cor. 2 (1809-13) 447-. Solution of problems by synthetic geometry. Cardinaal, J. Delft EC. Pol. A. 3 (1887) 151-. Straight line meeting four lines. Chevillard, A. J. N. A. Mth. 15 (1856) 306-. 477 6840 Practical Solid and Projective Geometry- 6840 Straight lines, plan and elevation of which coin- cide. Fries, H. de. N. Arch. Wisk. 1 (1895) 107- ; Fschr. Mth. (1893-94) 950-. , . Waelsch, E. Mh. Mth. Ps. 3 (1892) 92-. , . Fiedler, W. Mh. Mth. Ps. 3 (1892) 193-. Teaching of descriptive geometry. Verstraeten, T. (xn) Mathesis 3 (1883) 5-. , first lesson. Budelot, G. Ens. Mth. 1 (1899) 286-. , lacuna in. Tilly, (le maj.) de. Brux. S. Sc. A. 9 (1885) (Pt. 2) 95-. Tetrahedron, construction from data. Gantier, . Toul. Mm. Ac. 3 (1834) 111-. Transformations, in geometry of representation (practical solid). Fiedler, W. Z. Mth. Ps. 9 (1864) 331-. , theory of, history. Torelli, G. G. Mt. 13 (1875) 352-. Trihedral angle, section of given species. Bruno, F. Nap. At. Ac. 2 (1825) 29-. , (Bruno). Hachette, J. N. P. Par. S. Phlm. N. Bll. (1826) 81-. Vaulted roofs for constructions with trapezoidal ground plan. Staudigl, R. Z. Mth. Ps. 14 (1869) 97-. and lunes, construction. Riccati, G. Verona S. It. Mm. 5 (1790) 48-. Vaults, figure and construction. Bossut, (I'abbe) C. (xn) Lille S. Mm. 6 (1860) 217-. Projective Geometry. Framais, J. F. Gergonne A. Mth. 4 (1813-14) 61-. Gueneau d'Aumont, . Dijon Se. Ac. (1816) Valloi, J. N. Bordeaux Act. AC. Sc. (1842) 1-. Beazeley, A. Allelod. S. T. (1848) 15-. Brioschi, F. Tortolini A. 6 (1855) 209-. Mainardi, G. Mil. G. I. Lomb. 9 (1856) 89-. Sattelberger, . Schlomilch Z. 6 (1861) 81-. Battaglini, G. [1873-75] Nap. Ac. At. 6 (1875) #0. 6, 10 pp., No. 12, 21 pp.; 7 (1878) No. 5, 21 pp. Ovidio, E. d\ (x) A. Mt. 6 (1873-75) 72-; 7 (1875-76) 25-. Lucas, E. Ev. Sc. 17 (1879) 154- ; 18 (1880) 948-; 19 (1880) 36-, 375-; 1 (1881) 408-, 783-; 2 (1882) 365-. Veronese, -G. G. Mt. 17 (1879) 172-. Andreev, K. A. (xn) Kharkov Mth. S. Com. (1880) 139-. Goedseels, . Brux. S. Sc. A. 17 (Pt. 1) 50-. Fieri, M. [1895-97] Tor. Ac. Sc. At. 30 (1895) 607- ; 31 (1895) 267- or 381-, 315- or 457-; Tor. Ac. Sc. Mm. 48 (1899) 1-. Pasch, M. Mth. A. 48 (1897) 111-. (Von Staudt.) Scott, C. A. Mth. Gz. 1 (1900) 307-, 323-, 363-. Algebraic theory and application to rotation of rigid body. Pagani, G. M. Brux. Ac. Sc. Mm. 7 (1832) 20 pp. Analytical representation and protective geo- metry. Pasch, M. Mth. A. 30 (1887) 127-. Analytical solution of conies, problem. Moss- brugger, L. Grunert Arch. 16 (1851) 138-. theory. Giorgini, G. [1819] Lucca At. Ac. 1 (1821) 29-. Angle on horizontal plane, projection. Elides, A. N. A. Mth 4 (1845) 360-. Applications. Noel, J. N. Quetelet Cor. Mth. 8 (1834) 238-. Central projection derived from orthogonal projection. 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(1897) 470. theorems. Paolis, R. de. Em. E. Ac. Line. Mm. 9 (1881) 489-. . Le Paige, C., (& Deruyts, E. Brux. Ac. Bll. 15 (1888) 335-. . Fano, G. Palermo Cir. Mt. Ed. 9 (1895) 79-, 84-. . Enriques, F. Palermo Cir. Mt. Ed. 9 (1895) 82-. , two. Montesano, D. G. Mt. 23 (1885) 285-. Gnomonic projection. Zona, T. Palermo G. Sc. Nt. 15 (1882) 79-. Grassmann's method in projective geometry. Burali-Forti, C. Palermo Cir. Mt. Ed. 10 (1896) 177-, 254; 11 (1897) 54-. Harmonic and general projection, new deriva- tion of first law. Wiener, C. Z. Mth. Ps. 11 (1866) 1-. Higher geometry, introduction. Nazimov,P.S. Kazan Un. Mm. (1891) (Pt. 3) App., (Pt. 5) App., (Pt. Q)App.\ (1892) (Pt. 6) App.; (1893) (Pt. 2) App. ; Fschr. Mth. (1892) 566. Imaginary points, geometrical definition. Stephanos, C. Bll. Sc. Mth. As. 7 (1883) (Pt. 1) 204- . Invariants of a quadrangle by projective trans- formation. Lovett, E. 0. A. Mth. 12 (1898-99) 79-. Isotomic projection of quadrilateral, centres. Eueda, C. J. Fschr. Mth. (1897) 445. Maclaurin's " Geometrica Organica." Jon- quieres, E. de. Liouv. J. Mth. 2 (1857) 153-. Map projection, graphic processes for. Drien- cmirt, . A. Hydrog. 20 (1899) 78-. Maximum projection of system of plane figures in space, and of spherical triangle. Lhuilier, S, Gergonne A. Mth. 2 (1811-12) 49-. Mechanics, relation of projective geometry to. StepJianos, C. B. A. Ep. (1900) 644. ' Metric properties of figures, projection. Faure, H. A. (xn) Isere S. Bll. 1 (1869) 243-; 2 (1870) 381- ; 4 (1875) 129-. relations, projections, conic and orthogonal. Poncelet, P. V. [1831] Quetelet Cor. Mth. 7 (1832) 118-, 141-. Mine, inclination of passages. Lefroy, (Ing ) J. Mines 9 (1798-99) 339-. Modern geometry and projections. Schlomilch, 0. Grunert Arch. 1 (1841) 248-. methods. Brent, (Dr) . N. Z I T 3 (1870) (P.) 59-. n dimension geometry, new theorem. Meyer, . D. Nf. Tbl. (1884) 141. Notation of geometric position. Henry, L. d'. (xn) Lille S. Mm. 10 (1882) 5-. Orthogonal projections, certain. Magistrini, D. [1846] Bologna N. Cm. 10 (1849) 195-. Pappus's theorem on triangle, sides of which pass through fixed collinear points, two vertices move on fixed lines. Zahradnik, K. Arch. Mth. Ps. 56 (1874) 11-. Planes and lines. Koppe, C. Crelle J. 14 (1835) 70-. Polygons with pairs of axial and parallel sides. Matzka, W. Prag Sb. (1864) (Pt. 2) Polyhedra, projections. Chevillard, A. J. N. A. Mth. 15 (1856) 197-. Problem : to place two projective straight lines so that they shall generate a circle (Steiner). Kramer, A. Crelle J. 18 (1838) 185-. Projections of special dimensions, mathematical basis. Weisz, J. A. (xn) Mag. Ak. Ets. (Mth. Term.) 1 (1860) 67-. Projective geometry in plane. Grassmann, H. [1851] Crelle J. 42 (1852) 193-, 204- . Projectivity in space, problem. Sturm, R. Mth. A. 6 (1873) 513-. , , simplification. Sturm, R. Mth. A. 15 (1879) 407-. Quadrilateral, complete, mid-points of 3 diago- nals are collinear. Collins, M. Arch. Mth. Ps. 52 (1871) 243-. , theorem and extension. Sachse, A, Z. Mth. Ps. 27 (1882) 380-. Skew projection. Transon, A. N. A. Mth. 4 (1865) 385- ; 5 (1866) 63-. , case. Fonteng, G. N. A. Mth. 15 (1896) 369-. vault, section. Mannheim, A. N. A. Mth. 16 (1897) 404-. Solutions by projections. Coste, L. M. P. Gergonne A. Mth. 7 (1816-17) 304-. Synthetic geometry. Esson, W. L. Mth. S. P. 28 (1897) 491-. , development since Monge. Kotter, E. [1898-1901] D. Mth. Vr. Jbr. 5 (1901) (Heft 2) xxvni + 486 pp. , theorem in, and some corollaries. Del Re, A. Palermo Cir. Mt. Ed. 2 (1888) 128-. and projective geometry, need of, in analytical geometry. Davidescu, D. A. Bucarest S. Sc. Bll. 9 (1900) 19-. 479 6840 Projective Geometry Systems of trilaterals and quadrilaterals. Folie, F. Liege S. Sc. Mm. 7 (1878) No. 3, 109.pp. Teaching, introduction to projective geometry. Amodeo, F. G. Mt. 34 (1896) 22-. Theorem. Amodeo, F. [1879] G. Mt. 18 (1880) 15-. Theorems. Bauer, G. Crelle J. 19 (1839) 205-. of Cremona's, proofs of two. Dewulf, E. G. Mt. 13 (1875) 168-. , introductory. Asted, A. N. Ts. Mth. 6 (A) (1895) 1-. , various. Cayley, A. Camb. Mth. J. 2 (1841) 267-; Crelle J. 31 (1846) 213- ; 34 (1847) 270- ; 38 (1849) 97-; 41 (1851) 66-, 84. , . Jacobi, C. G. J. Crelle J. 31 (1846) 40-, 93-, 359-. Topographical problems, solution by "one plane descriptive geometry." Richardson, J. M. Franklin I. J. 37 (1859) 148-. Triangle traced by plane of projection on the coordinate planes, sides. Pantanelli, D. G. Mt. 8 (1870) 161. Trilinear relationship, new constructions in perspective, etc. Hauck, G. Crelle J. Mth. 95 (1883) 1-. Vectors. Kotellnikov, A. Kazan S. Ps.-Mth. Bll. 8 (1899) (App.)\ 9 (1900) (App.) 317 + 40 pp. ; Fschr. Mth. (1899) 501-. Proportional axes. Tawaststjerna, C. G. Hel- singf. Acta 1 (1840) 489-. planes. Tawaststjerna, C. G. [1844] Hel- singf. Acta 2 (1847) 777-. Relations, geometrical, generalisation of certain. TStdssy, B. von. Mth. Termt. Ets. 18 (1900) 233-; Mth. Nt. B. Ung. 18 (1903) 427. between segments got by dividing a line in 3, 4, 5 ... points. Teixeira, A. J. Fschr. Mth. (1892) 671. of 2nd degree (between planes and pencils). Eeye, T. [1865] Z. Mth. Ps. 11 (1866) 280-. Representation, elementary methods, history and theory. Fiedler, W. Ziir. Vjschr. 27 (1882) 125-. of planes on planes. Miihll, K. von der. Crelle J. 69 (1868) 264-. Stenographic Projections, and other Projections of the Sphere. (See also 8840 and Geography 83.) Orthogonal projection in spherical geometry. Lodge, A. Mth. Gz. 1 (1900) 97-. Sphere, perspective. Gergonne, J. D. Ger- gonne A. Mth. 7 (1816-17) 311-. , , conical. Ocagne, M. d\ N. A. Mth. 17 (1898) 44-. , projection on plane. Landr6, C. L. N. Projections of Sphere 6840 Sphere, projective geometry. Allardice, JR. E. Edinb. Mth. S. P. 4 (1886) 56-. Stereographic projection. Heawood, P. J. Mth. Gz. 1 (1900) 102-. of circle, proof of Francceur's problem. Brocard, H. Fschr. Mth. (1892) 551. , converse. Jeffery, H. M. L. Mth. S. P. 17 (1887) 379-. , equatorial, of sphere, application of Peaucellier cell. Phillips, A. W. B. A. Rp. (1884) 649. , history. Gilnther, . D. Nf. Tbl. (1887) 227-. and theorems of Quetelet and Dandelin. Pelz, C. Prag Sb. (1898) (Mth.-Nt.) No. 31, 4pp. , use in geometry. Dandelin, G. P. Gergonne A. Mth. 16 (1825-26) 322- ; Brux. Mm. Ac. Sc. 4 (1827) 13-. projections of point and its inverse, relation between. Tu-eedie, C. Edinb. Mth. S. P. 10 (1892) 46-. Arch. Wisk. 6 (*1880) 203-. ,_ of 2/3. James, (Sir) H. (vm) R. E. Pp.7 (1858) 134. Straight lines joining vertices of triangles or of skew quadrilaterals, theorems. Steiner, J. Crelle J. 1 (1826) 38-. Transformation of figures. Anger, C. T. Grunert Arch. 4 (1844) 281-. . Lentheric, . (vi Adds.) Mntp. Ac. Sc. Mm. 5 (1861-63) 440-, 446-, 449. metric properties of figures. Fanre, H. N. A. Mth. 17 (1858) 276-, 381- ; 18 (1859) 181-, 381- ; 19 (1860) 189-. plane figures, Newtonian. Lentheric, . Mntp. Ac. Sc. Mm. 4 (1858-60) 145-, 351-. Transversals. Ferriot, . Gergonne A. Mth. 17 (1826-27) 141-. Gamier, J. G. Quetelet Cor. Mth. 4 (1828) 143-. Cousinery, . N. A. Mth. 2 (1843) 66-. Killp, L. Arch. Mth. Ps. 56 (1874) 437-. Cesdro, E. G. Mt. 22 (1884) 240-. Candy, A. L. A. Mth. 10 (1895-96) 175- ; 11 (1896-97) 10-. Analytical problems. Bergsten, N. J. Stockh. Ac. Hndl. (1820) 1-. Application of geometrical theorem. Bump, F. H. Grunert Arch. 27 (1856) 332-. Carnot's theorem, application to polygons. Lerch, M. Prag Sb. (1882) 132-. , tangents. Terquem, 0. N. A. Mth. 18 (1859) 347-. , demonstration. Walton, W. Camb. and Dubl. Mth. J. 5 (1850) 92-. Circle, transversals, theorems. Lefaivre, P. N. A. Mth. 2 (1843) 362-. Complete polygons and polyhedra, tranversals. Vanecek, M. N. Prag Sb. (1881) 274-. Metrical relations. PeliSek, M. Casopis 27 (1898) 26-, 81-, 165- ; Fschr. Mth. (1898) 438. Parallel transversals, equal. Hain,E. [1874-77] Arch. Mth. Ps. 57 (1875) 441; 61 (1877) 177-. 480 7200 Geometry of Conies and Quadrics. General 7200 Parallel transversals of triangle. Hain, E. [1874] Arch. Mth. Ps. 57 (1875) 438-. Polygons, transversals. Becker, J. K. Grunert Arch. 38 (1862) 342-. Problems. F. Camb. Mth. J. 1 (1839) 87-. Straight lines, 4, in space, determination of transversals. Schlomilch, 0. Leip. B. 7 (1855) 39-. System of 2 points. Hain, E. Arch. Mth. Ps. 66 (1881) 280-. Theorem. Delacour, V. N. A. Mth. 2 (1843) 444-. . Terquem, O. N. A. Mth. 7 (1848) 459-. . Hyde, E. W. Des Moines Anal. 5 (1878) 113-. . Lamia, A. Fschr. Mth. (1891) 584. . Krahe, A. Fschr. Mth. (1900) 499. Theorems. Ferrari, F. N. A. Mth. 14 (1895) 41-. , two, analytical proof. Sturm, J. C. F. Gergonne A. Mth. 14 (1823-24) 225-. Transversals, two, cut proportionally. Chevil- lard, A.J. N. A. Mth. 1 (1842) 449-. Triangles, plane, transversals. Skrivan, G. Wien Jbr. Ober-Kealsch. Inn. Stadt 4 (1862) and pyramid, transversals. Most, R. Arch. Mth. 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Classification of conies and quadrics. Pouasart, A. Ens. Mth. 1 (1899) 279-. . Bromwich, T. J. PA. Camb. Ph. S. P. 10 (1900) 358-. Concomitants of systems of conies and quadrics, etc. Johnson, A. R. [1890] L. Mth. S. P. 21 (1891) 432-. Conditions for determining conic or quadric surface, number of. Anon, (vi 563) Ger- gonne A. Mth. 20 (1829-30) 185-. Cone and sections, general properties. Ciiasles, M. Liouv. J. Mth. 3 (1838) 102-. Confocal conies or quadrics, properties. Koenigs, G. N. A. Mth. 19 (1880) 74-. Conies, description. Pelosi, A. Tortolini A. 3 (1852) 252-. , . Kilpper, C. Grunert Arch. 28 (1857) 100-. , by homography. Cassani, P. Ven. Aten. At. 12 (1875) 119-. , tangents. Gergonne, J. D. Ger- gonne A. Mth. 5 (1814-15) 49-. , general method of solving questions on. Chaste, M. C. B. 59 (1864) 7-, 209-. , geometrical, elementary sketch. Zeuthen, H. G. (xn) Ts. Mth. 2 (1878) 33-, 65-, 109-, 132- ; 6 (1882) 13-. , , principles. Taylor, C. Mess. Mth. 5 (1871) 141-, 214-. with infinite branches. Page, C. E. N. A. Mth. 1 (1842) 21-, 61-. , new treatment. Homer, J. K. Zach Cor. 4 (1820) 440-. and perspective in the teaching of analytical geometry. Chelini, D. Bologna Rd. (1864) 152-. quadrics dividing given segments har- monically. Fuortes, T. G. Mt. 10 (1872) 98-. which reciprocate two given conies or quadrics. Ovidio, E. d\ G. Mt. 10 (1872) 313. revolution quadrics. Tucci, F. P. [1812] Nap. At. S. Pont. 3 (1819) 131-. in solid geometry. Quetelet, L.A.J. [1820] Brux. Ac. Sc. Mm. 2 (1822) 123-. . Bobillier, . Quetelet Cor. Mth. 3 (1827) 270-. . Geer, P. van. N. Arch. Wisk. 13 (1887) 58- ; Fschr. Mth. (1886) 754. , synthetic geometry. Ponte Horta, F. da. Lisb. Ac. Sc. Mm. 3 (Pt. 2) (*1865) No. 1, 59 + 8 pp. , theorems derived from general equation. Giorgini, G. [1823] Lucca At. Ac. 3 (1827) 63-. , use in geometrical problems. Penny, W. G. Atlantis 2 (1859) 435-. Conjugate conies and quadrics. Cremona, L. Tortolini A. 3 (1860) 257-. diameters. Camus, . N. A. Mth. 1 (1842) 300-. _ _. Fenwick, S. Mathematician 1 (1845) 38-. . Brassinne, E. N. A. Mth. 6 (1847) 38 1-. Conjugation-point and its application to graphic constructions. Loud, F. H. A. Mth. 7 (1892-93) 77-. Conoid circumscribed to conic, sections of. MottaPegado,L.P.da. [1875] Lisb. J. Sc. Mth. 5 (1876) 65-. Construction on cone for finding parameter of given section. Berkhout, J. J. T. van. (vi Adds.) Arch Wisk. Gn. 2 (1863) 299-. VOL. I. 481 HH Geometry of Conies and Quadrics. General 7200 Construction of conies and quadrics. Dupin , C. Par. EC. Pol. J. cah. 14 (1808) 45-. walls of parabolic shape. Conard, J. Brux. A. Tr. Pbl. 50 (1893) 359-. Constructions by means of cone and cylinder. Staudigl, R. Wien Sb. 58 (1868) (Ab. 2) 960-. Dandelin's theorem, extension, etc. Sauze, . N. A. Mth. 17 (1858) 33-. , sphere inscribed to cone touches plane of section at focus, etc. Transon,A. N. A. Mth. 12 (1873) 21-. Determination of conies and quadrics. Hun- yady, J. (xii) Mag. Tud. Ak. Etk. (Mth.) 1 (1881) (No. 18) 39 pp. Discontinuity of constructions caused by imaginary elements. Wiener, C. Z. Mth. Ps. 12 (1867) 375-. in locus question connected with ellipse. Goodwin, H. Camb. Mth. J. 4 (1845) 190-. Discriminants, application to conies and quad- rics. Dostor, G. Arch. Mth. Ps. 58 (1876) 5-. Ellipse and ellipsoid, inscribed. Girault, C. Caen Ac. Mm. (1884) 3-. , maximum inscribed triangle and tetrahedron. Unferdinger, F. Arch. Mth. Ps. 51 (1870) 127-. , and minimum polygons and poly- hedra. Mainardi, G. Tortolini A. 1 (1850) 348-. 7200 Coordinates, tetrahedral or triangular. Clwlini, D. Bologna Mm. Ac. Sc. 3 (1863) 3-. Determination of conies and quadrics. Halphen , G. H. (x) Par. S. Mth. Bll. 1 (1873) 130-, 226- ; 2 (1874) 11-. Geometrical construction. Berard, . Ger- gonne A. Mth. 6 (1815-16) 157-. interpretation of coefficients. Faure, H. N. A. Mth. 5 (1866) 1-. significance. Schulten, N. G. of (fil.). [1832] St Pet. Mm. Sav. Etr. 2 (1835) 313-. Eeduction. Boole, G. Camb. Mth. J. 2 (1841) 64-. Eepresentation. Ctiambers, C. [1877] Manch. Lt. Ph. S. P. 17 (1878) 58-. Tangential and quadriplanar coordinates, appli- cation of Salmon's notation. Versluys, J. Arch. Mth. Ps. 52 (1871) 377-. Tangents to conies or quadrics, equation. Verdam, G. J. Grunert Arch. 2 (1842) 188-. Transformation. Lemonnier, H. N. A. Mth. 14 (1875) 396-. of coordinates, factor. Greiner, M. Arch. Mth. Ps. 57 (1875) 337-. Focal properties of conies and quadrics. Booth, J. Ph. Mg. 27 (1845) 538-. , minimum, round triangle and tetrahe- . Walker, John. Camb. and dron. Liourille, J. Liouv. J. Mth. 7 (1842) 190-. , , . Borsch, A. Z. Mth. Ps. 25 (1880) 59-. , normals. JoacMmsthai, F. Crelle J. 26 (1843) 172-. , properties. Rocliat, . Gergonne A. Mth. 3 (1812-13) 25-. , . Durrande, J. B. Gergonne A. Mth. 12 (1821-22) 223-. .tracing. Maurin,H. (vm) A. Cond. Pon. Chauss. 4 (1860) 106-. Equations relating to Conies and Quadrics. Gergonne, J. D. Gergonne A. Mth. 5 (1814 -15) 61-. Raabe, J. L. Crelle J. 2 (1827) 182-. Lentheric, . Mntp. Ac. Sc. Mm. 2 (1851-54) 455- ; 3 (1855-57) 449-. Grunert, J. A. Arch. Mth. Ps. 51 (1870) 326-. Versluys, J. Arch. Mth. Ps. 52 (1871) 278-. Axes, equations, rapid notation. Dostor, G. J. Grunert Arch. 30 (1858) 202-. - in oblique coordinates. Bobillier, . Quetelet Cor. Mth. 4 (1828) 216-. Centres, equations for, in absolute geometry. Frank, A. von. Steierm. Mt. (1874) 47-. Characteristics of conies given by equation. Ehlert, A. Arch. Mth. Ps. 1 (1884) 51-. Conditions for circle or sphere. Ferrers, N. M. QJ. Mth. 2 (1858) 267-. Conies, plane and spherical, in Boothian tan- gential coordinates. Jeffery, H. M. QJ. Mth. 9 (1868) 189-. , , 3-point tangential coordinates. Jeffery, H. M. [1867] QJ. Mth. 9 (1868) Dubl. Mth. J. 7 (1852) 16-. . Ravier, . N. A. Mth. 9 (1890) 233-. Foci and poles of conies and quadrics. Cauchy, A. L. C. E. 16 (1843) 809-. Generation of conies and quadrics. Breton [de Champ] , P. N. A. Mth. 3 (1844) 289-. (Jacobi). Waille, J. [1874] (xn) Doubs S. Mm. 9 (1875) 123-. Homographic ray systems, theorem of Chasles. Bohannan, R. D. A. Mth. 1 (1884-85) 139-. Hyperbola and hyperboloid, rectangular. KUp- per, C. Schlomilch Z. 3 (1858) 119-. Imaginary projection of conies, quadrics and twisted quartic curves. Wiener, . D. Nf. Tbl. (1885) 354-. Indices (powers) of points with regard to conies and quadrics. Faure, H. N. A. Mth. 11 (1872) 261-, 385-. Intersections of lines and surfaces. Lame, G. Gergonne A. Mth. 7 (1816-17) 229-. Invariants of conies and quadrics given by motion and similitude groups. Lovett,E. 0. A. Mth. 11 (1896-97) 33-. Inverse of conies and quadrics. Van&tek, J. S. Prag Sb. (1883) (Mth. Vortr.) 28-. from centre. Rice, M. E. Kan. Ac. Sc. T. 14 (1896) 14-. Involutions arising from conies and quadrics. Bischoff, J. N. [1874] A. Mt. 6 (1875) 232. . Cardinaal, J. Arnst. Ak. Vs. 8 (1900) 271- ; Amst. Ak. P. 2 (1900) 234-. Loci, some, derived from conies and quadrics, cases. Azzarelli, M. Em. N. Line. At. 27 (1874) 216-. , quadratic, absolute classification. Story, W. C. Am. J. Mth. 6 (1884) 222-. 482 7200 Conies and Quadrics Oblique circular cones, plane sections. Pistoye, de. (xi) Par. S. Mth. fill. 1 (1873) 117-. Parameters, variation. Rutherford, W. Mathe- matician 1 (1845) 116-. Plane sections of cone of revolution, theorem. Rocha Peixoto, A. F. G. Teix. J. Sc. 3 (1881) 46-. , method. Dostor, G. J. Grunert Arch. 30 (1858) 185-. Polar triangle and tetrahedron for conies and quadrics. Fiedler, W. Schlomilch Z. 6 (1861) 140-. . Humbert, G. N. A. Mth. 2 (1883) 167-. . Heister, K. Z. Mth. Ps. 31 (1886) 321-. Polars, theory for conies and quadrics. Len- theric, . Mntp. Ac. Sc. Mm. 1 (1847-50) 279-, 427-. Poles of conies and quadrics, theory. Ger- gonne, J. D. Gergonne A. Mth. 3 (1812-13) 293-. Polygons and polyhedra, maximum and mini- mum, in quadric curves and surfaces. Lindelof, L. L. Helsingf. Ofv. 11 Porisms, etc. Brougham and Vaux, (Lord). Phil. Trans. (1798) 378-. Projective geometry of conies and quadrics. Del Re, A. Bv. Mt. 2 (1892) 138-. Properties of conies and quadrics. Picquet, H. (xi) Par. S. Phlm. Bll. 2 (1865) 196-. , new. Booth, J. Ph. Mg. 27 (1845) 538-. curves and surfaces of 1st and 2nd degree. Cauchy, A. L. C. E. 16 (1843) 805-. reciprocal polar of conic and of quadric. Ter- quem, O. N. A. Mth. 16 (1857) 264-. propositions. Serret, P. Liouv. J. Mth. 6 (1861) 9-. Representation in space of the ellipses drawn by an ellipsograph. Blake, E. M. A. Mth. 11 (1896-97) 195-. Segments of ellipse, ellipsoid, and hyperboloid. Unferdinger, F. Grunert Arch. 28 (1857) 52-. parabola and elliptic paraboloid. Unfer- dinger, F. Grunert Arch. 29 (1857) 209-. Simultaneous transformation of two quadratic functions. Campbell, J. E. Mess. Mth. 21 (1892) 78-. System of related conies and quadrics. Heiler- mann, . Schlomilch Z. 6 (1861) 353-. Teaching conies. Grunert, J. A. Grunert Arch. 17 (1851) 54-. , comments on syllabus. Taylor, C. Mth. Gz. No. 5 (1895) 37-. Theorems on conies and quadrics. Steiner, J. Crelle J. 1 (1826) 38-. . Faure, H. N. A. Mth. 5 (1866) 299-. . Catalan, E. Brux. Ac. Bll. 23 (1892) 500-. . Barbarin, . Mathesis 13 (1893) 60-. Triangle, trihedral angle and tetrahedron. Chailes, M. Gergonne A. Mth. 19 (1828-29) 65-. Properties of Conies 7210 7210 Metrical and Projective Properties of Conies. Sturm, J. C. F. Gergonne A. Mth. 16 (1825 -26) 265- ; 17 (1826) 173-. Franchini, P. [1831] Lucca At. Ac. 8 (1835) Terquem, 0. Liouv. J. Mth. 2 (1837) 36. Page, C. E. N. A. Mth. 1 (1842) 223-. Obici, P. Pisa A. Un. Tosc. Sc. Cosm. 1 (1846) 361-. Le Poivre, . Hain. Mm. S. 1 (1853) 146-. Grunert, J. A. Grunert Arch. 24 (1855) 370-. Maurin, H. (vm) A. Cond. Pon. Chauss. 6 (1862) 29-, 53-. Taylor, C. Mess. Mth. 1 (1862) 250- ; 2 (1864) 64, 65. Cremona, L. G. Mt. 1 (1863) 225-; 2 (1864) Michaelis, J. P. Lux. S. Sc. Nt. 6 (1863) 65-. Burmide, W. S. [1865] QJ. Mth. 7 (1866) Chasles, M. Em. At. N. Line. 18 (1865) 55-. Bauer, G. [1867] Crelle J. 69 (1868) 293-. Ovidio, E. d\ G. Mt. 6 (1868) 46-, 190-, 259- : 7 (1869) 1-. Bdcklund, A. V. Stockh. Ofv. 28 (1871) 323-. Morel, A. N. A. Mth. 10 (1871) 269-, 305-, 537 . Mittelacher, C. Z. Mth. Ps. 18 (1873) 1-. Taylor, C. Mess. Mth. 2 (1873) 97-. Lucas, E. N. A. Mth. 14 (1875) 265-, 458-. Gundelfaiger, S. N. A. Mth. 3 (1884) 7-. Maleyx, L. N. A. Mth. 9 (1890) 240-, 318-, 424-, 481-, 596- ; 10 (1891) 37-, 91-, 125-, 163-. Lerch, M. Casopis 23 (1894) 151-, 223-, 288-; Fschr. Mth. (1893-94) 1107-. Algebraic identity with 2 geometrical applica- tions. Bromwich, T. J. PA. Mess. Mth. 29 (1900) 184-. Analytical methods, and solution of some problems. Poncelet, J. V. Gergonne A. Mth. 8 (1817-18) 141-. theory. Grunert, J. A. Grunert Arch. 31 (1858) 67-. . Schaar, M. Brux. Ac. Bll. 6 (1859) 42-. (1872) . Cayley,A. Phil. Trans. (1862) . Genese, R. W. Mess. Mth. 1 104. . Hesse, L. 0. Z. Mth. Ps. 19 (1874) 1-; 21 (1876) 1-. , . Jarolimek, C. fiasopis 19 (1890) 14-, 76-; Fschr. Mth. (1890) 723. Anharmonic property. Graves, C. B. A. Bp. (1838) (pt. 2) 1-. ratio, sectional and tangential. Battaglini, G. Em. At. E. Ac. 26 (1873) 565-. of 4 straight lines through a point and touching two conies. Weyr,E. [1871] Crelle J. 75 (1873) 67-. Arches, parabolic and elliptic. Haase, H. Forster Al. Bauztg. 48 (1883) 75-, 89-. Asymptotes, circular, as coordinate axes. Lerch, M. Prag Sb. (1884) (Mth.-Nt.) 90-. 483 HH 2 7210 Properties of Conies 7210 Axes of conies. Lolli,C. G. Mt. 28 (1890) 193-. . Jefdbek, V. Casopis 28 (1899) 351- ; Fschr. Mth. (1899) 481-. , determination. Pelz, C. Wien Ak. Sb. 73 (1876) (Ab. 2) 379-. , formulae. Rochat, . Gergonne A. Mth. 2 (1811-12) 331-. , (Eochat, addition to). Gergonne, J. D. Gergonne A. Mth. 2 (1811-12) 335-. , (Kochat and Dubourquet). Bret, . Gergonne A. Mth. 3 (1812-13) 31-. , minor axis, determination from major axis and perimeter. Clement, L. A. Cond. Pon. Chauss. 8 (1864) 305-. Central conies, trigonometrical formulae. Egidi, G. Em. N. Line. At. 37 (1884) 295-. Centres of similitude and radical axes. Collins, M. Camb. (M.) Mth. M. 1 (1859) 268-. , theorems. Terquem, O. N. A. Mth. 4 (1845) 260-, 304- , 322-, 371-. Chasles, application of a theorem of. Balitrand, . Mathesis 14 (1894) 62-, 81-. Chords, common, of conic and circle of curva- ture. Grunert, J. A. Arch. Mth. Ps. 50 (1869) 69-. , , zero radius. Ocagne, M. d\ Edinb. Mth. S. P. 5 (1887) 84-. of conies and of parabola. A modeo, F. A. Mth. 4 (1888) 92. at right angles, note. Steiner, J. N. A. Mth. 9 (1850) 407-. , supplemental. Ponte-Horta, F. da. Lisb. J. Sc. Mth. 2 (1870) 169-. Conies determined by given Points or Tangents, and Methods of Determination.. Conies enveloped by rays joining homologous points, discrimination. Chrystal, G. Edinb. Mth. S. P. 2 (1884) 47-. , given centre and in-, circum-, or polar triangle, properties. Schroeter, H. Z. Mth. Ps. 29 (1884) 160-. by a diameter and conjugate chord, relation of straight lines to. Barchanek, C. Wien Ak. Sb. 79 (1879) (Ab. 2) 712-. imaginary points and tangents, con- struction. Staudigl, R. Wien Sb. 61 (1870) (Ab. 2) 607-. points or tangents. Jonquieres, E. de. N. A. Mth. 18 (1859) 215-. through certain points, theorem. Valeri, D. Mod. Ac. Sc. Mm. 7 (1890) 181-. touching 4 lines. Arndt, F. Grunert Arch. 8 (1846) 342-. Criteria for species of conic. Hart, H. Mess. Mth. 10 (1881) 90-. determined from points or tangents. Hunyady, J. (xn) Mag. Tud. Ak. Etk. (Mth.) 7 (1881) (No. 25) 17 pp. Determination of conies. Dillner, G. Ups. Arsk. (1863) (Mth.) 45 pp. 9 line conic. Ladd, (Miss) C. Des Moines Anal. 7 (1880) 147-. and 9 point conic. Cassani, P. G. Mt. 7(1869)369-; 8(1870)374-. Number determined by points, tangents and normals. Wiman, A. Z. Mth. Ps. 40 (1895) 296-. 1 point and 3 tangents given, conies with. Cayley, A. QJ. Mth. 8 (1867) 220- . 2 points and 2 tangents given, conies with. Cayley, A. QJ. Mth. 8 (1867) 211-. 3 points and centre given, Steiner's criterion. Hunyady, J. [E. von]. [1880] (xn) Mag. Tud. Ak. Etk. (Mth.) 1 (1881) (No. 24) 13 pp. (x) Crelle J. Mth. 91 (1881) 248-. , . Halphen, G. H. [1881] B. A. Ep. (1881) 532- ; Par. S. Phlm. Bll. 6 (1882) 19-. focus given, construction (Halley's problem). Housel, . N. A. Mth. 14 (1855) 425-. , . Grunert, J. A. Arch. Mth. Ps. 54 (1872) 99-. , theorems, proof. Jonquieres, E. de. N. A. Mth. 14 (1855) 440-. 1 tangent given, conies with. Cayley, A. (i) QJ. Mth. 6 (1864) 24-. 2 tangents given, conies with. Bruno, G. Tor. Ac. Sc. At. 17 (1881) 29-. 4 points and 1 tangent given, Newton's method. Grunert, J. A. Arch. Mth. Ps. 65 (1880) 1-. 2 tangents given, conies with. Cayley, A. [1863] (vn) QJ. Mth. 8 (1867) 162-. 5 point conic. Grunert, J. A. Grunert Arch. 9 (1847) 293-; 24 (1855) 330- ; 27 (1856) 178-. . Schlimilch, 0. Leip. B. 7 (1855) 1-. , Newton's 1st method. Grunert, J. A. Arch. Mth. Ps. 64 (1879) 337-. or 5 line conic, Mobius's criteria. Hunyady, J. [E. von]. [1879] (xn) Mag. Tud. Ak. Etk. (Mth.) 7 (1881) (No. 6) 15pp. ; (x) Crelle J. 89 (1880) 70-. , for species. Durege, H. [1880] Wien Ak. Sb. 82 (1881) (Ab. 2) 123-. points, conic through any, theorem. Budden, E. Mth. Gz. 1 (1900) 145-. 9 point conic. Beltrami, E. Bologna Mm. Ao. 2 (1862) 361- ; G. Mt. 1 (1863) 109-, 208-, 354-. . Grunert, J. A. Arch. Mth. Ps. 43 (1865) 54-. . Bocher, M. A. Mth. 6 (1891-92) 132, 178. . Boger, R. [1898] Hamb. Mth. Gs. Mt. 3 (1900) 346-. , generalisation of Feuerbach and Steiner's theorems. Beltrami, E. Bologna Ac. Sc. Mm. 5 (1874) 543-. and its stereometric sequel. Gretschel, H. Arch. Mth. Ps. 43 (1865) 293-. Conies, given, to obtain from given cone. Ponte Horta, F. da. Lisb. Ac. Sc. Mm. 3 (Pt. 2) (*1865) No. 2, 14 pp. , and quadratic and cubic binary forms. Pittarelli, G. G. Mt. 21 (1883) 19-. right cone, contributions to theory of. Ruth, F. Arch. Mth. Ps. 8 (1890) 1-. satisfying a certain condition. Bauer, G. [1867] Crelle J. 68 (1868) 293-. 484 7210 Conies. Contacts. Curvature 7210 Conies satisfying 5 conditions, construction. Lentheric, . (vi Adds.) Mntp. Ac. Sc. Mm. 5 (1861-63) 438. ---- , . Chasles, M. C. K. 58 (1864) 297-, 1167-. , ; and applications of higher geometry. Jonquieres, E. de. N. A. Mth. 16 (1857) 116-. Conjoint lines. Chasles, M. (1838) 385-. Conjugate diameters, determination. Vauthier, . Toul. Mm. Ac. 3 (1834) 117-. -- 5 properties. Dostor, G. N. JL Mth. 8 -- , theorems in perspective, and focal pro- perties of oblique cone. Chasles, M. Liouv. J. Mth. 2 (1837) 388-. pairs of points. Eosanes, J. Z. Mth. Ps. 17 (1872) 174-. Constant angle, staight lines cutting conies at. Schoute, P. H. Mathesis 7 (1887) 38-. Construction by circles of curvature. Meyer, M. H. Grunert Arch. 24 (1855) 3-. and general equation of conies. Schmitt, C. Arch. Mth. Ps. 43 (1865) 365-. from imaginary elements. Circles, 2, and ellipse, tri-tangential system. Avillez, J. F. d>. Lisb. J. Sc. Mth. 4 (1897) 166- ; Fschr. Mth. (1896) 434. Conies having double contact. Steiner, J. Crelle J. 45 (1853) 212-. ---- . Cayley, A. QJ. Mth. 3 (1860) 246-. ---- with another conic. Lefebure de Fourcy, R. N. A. Mth. 13 (1874) 513-. Liouv. J. Mth. 3 ----- two conies. Amodeo, F. G. Mt. 24 (1886) 346-. ------ fixed conies, locus of their foci. Roberts, R. A. N. Y. Am. Mth. S. Bll. 2 (1896) 98-. -------- , systems. Bonnesen, T. N. Ts. Mth. 9 (B) (1898) 25-. and pencil of lines, contacts. Jonquieres, E. de. Liouv. J. Mth. 4 (1859) 49-. through 3 points and having double contact with given circle. Laguerre, E. N. A. Mth. 19 (1880) 347-. touching given conies. Lachlan, R. Mess. Mth. 16 (1887) 140-. -- 2, 3, or 4 conies, construction. Niemt- schik, R. Wien Ak. Sb. 69 (1874) (Ab. 2) Mh. Mh. Mth. Euth, F. Mth. Ps. 3 (1892) 81-. --- , with ruler. Spath, F. Ps. 1 (1890) 237-. ---- , -- . Machovec, F. Mh. Mth. Ps. 4 (1893) 95-. , mechanical. Guest, J. J. Cn. E. S. P. & T. 2 (1896) (Sect. 3) 25-. , methods. Jiirges, W. Z. Mth. Ps. 38 (1893) 350-. , new method. Transon, A (1873) 5-. N. A. Mth. 12 Contact of 2nd and 3rd orders. Retali, V. Mathesis 12 (1892) 178-, 219-. -- 3rd order, porismatic property of conies having. Sylvester, J. J. (vi Adds.) Ph. Mg. 37 (1850) 438-. property of eleven-point conic. Allardice, R. E. Edinb. Mth. S. P. 10 (1892) 37-. relations of systems of circles and conies. Orr, W. McF. [1891] Camb. Ph. S. T. 16 (1898) 95-. Double and four-point contact of two conies. Retali, V. Liege S. Sc. Mm. 18 (1895) A T o. 8, from points, by means of planaltimetry. 35 pp. Rossi Re, V. de. Km. N. Line. At. 29 (1876) Generalised problem. Jeffery, H. M. QJ. Mth. 23 (1889) 358-. Koutny, E. Wien Curvature of Conies. (See also 8430.) 240-. and tangents. Sb. 57 (1868) (Ab. 2) 469-. , problem. Jonquieres, E. de. N. A. Mth. 18 (1859) 404-. Constructions. Fiedler, W. Ziir. Vjschr. 35 (1890) 322-. , and solutions of graphic problems. Genty, . N. A. Mth. 19 (1880) 216-. Contacts. Pliicker, J. Gergonne A. Mth. 17 (1826-27) 37-. (Chasles.) Cayley, A. B. A. Kp. 34 (1864) (Sect.) 1. Wasserschleben, (Maj.) von. Arch. Mth. Ps. 60 (1877) 410-. Apollonian problem, extension, and conies. Fiedler. W. Ziir. Vjschr. 24 (1879) 190- ; 25 (1880) 217-. Circles bitangent to conic. Heilermann, . Crelle J. 56 (1859) 365-. . Le Besgue, V. A. N. A. Mth. 4 (1865) 161-. . Sparer, B. Z. Mth. Ps. 41 (1896) 210-. conies, theorem, etc. Steiner, J. Crelle J. 45 (1853) 189-. Servais, C. Brux. Ac. Bll. 19 (1890) 231-, 529-; 23 (1892) 243-. Centres of curvature. Boschi, P. Bologna Ac. Sc. Mm. 3 (1881) 579-. . Machovec, F. Prag Sb. (1884) (Mth.- Nt.) 345- ; Fschr. Mth. (1885) 625-. , construction. Schirek, C. Arch. Mth. Ps. 3 (1886) 318-. , _. Janisch, E. Mh. Mth. Ps. 10 (1899) 182-. , parallels to tangent through, theorem. Siacci, F. G. Arcad. 18 (1860) 166-. Circle of curvature of ellipse, theorems. Janni, V. (vra) Tortolini A. 4 (1861) 199-. through 3 points of ellipse and circle of curvature. Grunert, J. A. Grunert Arch. 37 (1861) 255-. Circles of curvature. Weiler, A. Z. Mth. Ps. 34 (1889) 1-, 177-, 282-. , Lamarle's construction. Grunert, J. A. Grunert Arch. 30 (1858) 296-. Concurrent chords of curvature. Servais, C. Mathesis 17 (1897) 222-. 485 7210 Properties of Conies. Ellipses 7210 Conic through 5 points, radius of curvature, construction. Rohn, K. Leip. Mth. Ps. B. 52 (1900) 17-. Conies inscribed in triangle of reference, curva- ture. Longchamps, G. de. As. Fr. C. E. (1891) (Ft. 2) 11-. Curvature properties of conies. Sobotka, J. Prag Sb. (1894) (Mth.-Nt.) No. 42, 9 pp. treated dynamically. Collignon, E. As. Fr. C. E. (1898) (Pt. 2) 34-. Formula (Neuberg). Gob, A. As. Fr. C. E. (1893) (Pt. 2) 254-. Eadii of curvature. Strehlke, F. Crelle J. 2 (1827) 380-. . Transon, (Prof.) A. Liouv. J. Mth. 1 (1836) 191-. . Gregory, D. F. (vi Adds.) Camb. Mth. J. 1 (1839) 61-. . Schlamilch, O. Schlomilch Z. 2 (1857) 187-. . Ribaucaur, A. N. A. Mth. 7 (1868) 171-. , case. Mannheim, A. Par. S. Mth. Bll. 18 (1890) 155-. , . Gob, A. Mathesis 14 (1894) 133-. , construction, geometrical. Azzarelli, M. Em. N. Line. Mm. 11 (1895) 87-. , , simple. Brix, A. F. W. Grunert Arch. 9 (1847) 316-. , constructions as corollaries of a theorem of Steiner's. Pelz, C. Prag Sb. (1879) 205- ; (1882) 3-. , evaluation. Taylor, C. Mess. Mth. 3 (1866) 97- ; 2 (1873) 142-. , property. Steiner, J. Crelle J. 30 (1846) 271-. Curve resulting from binocular union of straight line with circular arc, or of 2 circular arcs, form. Rogers, W. B. Edinb. N. Ph. J. 3 (1856) 210-. lindric Cylindrical projection of conies. Glaser, S. Arch. Mth. Ps. 13 (1895) 113-. Degenerate conjugate conies. Retail, V. Bologna Ed. (1886-87) 88-. forms. Hirst, T. A. L. Mth. S. P. 2 (1869) 166-. Demonstration, new, of properties. Bobillier, . Gergonne A. Mth. 18 (1827-28) 359-. , , -- . Olivier, T. Par. S. Phlm. PV. (1847) 36. Derivation from any one conic. Azzarelli, M. Em. N. Line. At. 43 (1890) 103-. Diameter, constructions for. Taylor, C. Mess. Mth. 3 (1866) 5-. Diameters. Painvin,L. N. A. Mth. 4(1865)498-. , elementary geometrical proofs. Zeuthen, H. G. (XH) Ts. Mth. 4 (1868) 145-. Director (or orthoptic) circle. Lemoine, E. N. A. Mth. 17 (1858) 240-. ( -- ) . Laguerre, E. N. A. Mth. 18 (1879) 204-. ( -- ) of inscribed conic. Rouse, E. P. Mth. Gz. No. 2 (1894) 12-. ( -- ) , locus of vertices of right angles circumscribed to ellipse. Forcade-Prunet, G. N. A. Mth. 13 (1854) 192-. Director (or orthoptic) circles of linear system of conies. Vries, J. de. Amst. Ak. Vs. 7 (1899) 371- ; Amst. Ak. P. 1 (1899) 305-. Eccentric circle of Boscovich. Langley, E. M. Mth. Gz. No. 1 (1894) 1-; No. 3 (1894) 17-. Ellipses. Noel, J. N. Liege Mm. S. Sc. 2 (1845-46) Weddle, T. Mathematician 2 (1847) 179-. (Sinram.) Jefdbek, W. [1879] Arch. Mth. Ps. 65 (1880) 215-. Axes, construction. Tucker, R. [1865] Mess. Mth. 3 (1866) 151-. , from 2 conjugate diameters. Broch, O. J. Crelle J. 40 (1850) 233-. , . Seidelin, C. J. L. (xn) Ts. Mth. 6 (1870) 20-, 22- ; 6 (1876) 63-. , . Millar, J. B. Manch. Lt. Ph. S. P. 14 (1875) 89-, 112-. , (Millar's method). Rawson, R. Manch. Lt. Ph. S. P. 14 (1875) 98-. , . Mannheim, A. N. A. Mth. 17 (1878) 529-; Mess. Mth. 11 (1882) 175-. , when considered as a Lissajou's curve. Janmchke, H. Arch. Mth. Ps. 60 (1877) 222-. , determination. Fialkoicski, N. Wien SB. 20 (1856) 167-. and radii-vectores. Crocchi, L. G. Mt. 12 (1874) 375-. Brocard's ellipse. Catalan, E. [1890] Brux. Ac. Mm. 49 (1890-93) (No. 3) 20 pp. Circle and ellipse. Kilbinger, G. Ens. Mth. 1 (1899) 452-. through feet of 3 normals to ellipse. Weill, M. N. A. Mth. 19 (1880) 60-. 3 points of ellipse, theorem. Joachims- thai, F. Crelle J. 39 (1850) 138- . , theorem of Euler, generalisation and ex- tension to ellipse. Dostor, G. Arch. Mth. Ps. 51 (1870) 106-. Circles, system of 286, connected with ellipse. Boklen, 0. Z. Mth. Ps. 28 (1883) 300-. Conjugate diameters, problem. Berg, F. J. van den. Amst. Arch. Wisk. Gn. 2 (1860- 63) 129-, 152-, 189-. , solution of problem (claim of priority for Le Poivre). Le Paige, C. Main. S. Mm. 10 (1888) 587-. Construction. Faconnet, . (vn) A. Cond. Pon. Chauss. 3 (1859) 381- ; 5 (1861) 187-. of ellipse and of circle. Fialkowski, N. Wien SB. 16 (1855) 9-. Constructions. Statidigl, R. Wien Sb. 59 (1869) (Ab. 2) 189-. Cotes's theorem extended to ellipse. Grunert, J. A. Grunert Arch. 30 (1858) 104-. "Deviation" angle. Ocagne, M. d\ N. A. Mth. 7 (1888) 268-. in the ellipse. Ocagne, M. d\ N. A. Mth. 5 (1886) 370-. . Mannheim, A. N. A. Mth. 16 (1897) 249-. Ellipses of equal probability. Brtgtr, P. Ev. Artl. 10 (1877) 82-. 486 7210 Ellipses Properties of Conies 7210 Ellipses, 2, or ellipse and parabola, with common focus. Bratt, (Prof.) . Stockh. Ac. Hndl. 24 (1803) 108-. Envelope of perpendiculars to radii. Ayttard, . N, A. Mth. 5 (1846) 540-, 582-. Fermat, theorem of, extended to ellipse. Schlomilch, O. X. Z. Mth. Ps. 19 (1874) 462. In- or circum-triangles and quadrilaterals, area. Grunert, J. A. Grunert Arch. 30 (1858) 11-. In-ellipse of circle with centre and tangent given, construction. Niemtschik, R. Wien Sb. 68 (1873) (Ab. 2) 377-. Leonardo da Vinci's construction. Rulf, W. Mh. Mth. Ps. 9 (1898) 30-. Maximum ellipse touching 4 lines. Seydeicitz, F. Grunert Arch. 12 (1849) 44-. or minimum ellipse through 2 points and touching 2 lines. Seydeicitz, F. Grunert Arch. 14 (1850) 364-. Mean radius vector. Dubois, E. A. Gen. Civ. 2 (1863) 193- ; Les Mondes 3 (1863) 330-. Minimum ellipse through 3 given points. Dantscher, V. von. Wien Ak. Sb. 99 (1891) (Ab. 2a) 10- ; 104 (1895) (Ab. 2a) 301-. 4 given points. Hearn, G. W. Mathe- matician 2 (1847) 315-. . Anger, C. T. As. Nr. 26 (1848) 35-. New property. Auria, L. d\ Franklin I. J. 123 (1887) 27-. , and its application to the "oval chuck". Leavitt, F. M. Franklin I. J. 81 (1881) 114-. Parallel and different ellipses impossible, proof. Hyde, E. W. Des Moines Anal. 2 (1875) 144. Parallels. Games Teixeira, F. Brux. Mm. Cour. 8, 58 (1898-99) No. 5, 39 pp. Point-circle and ellipse. Davis, R. F. Edinb. Mth. S. P. 13 (1895) 105-. Projection, application. Ferriot, . Gergonne A. Mth. 2 (1811-12) 240-. of circle, ellipse considered as, etc. Jullien, A. N. A. Mth. 14 (1875) 324-, 359-. , , . Kantor, S. Z. Mth. Ps. 23 (1878) 414- ; 24 (1879) 54-; 25 (1880) Properties. Bratseur, L. N. A. Mth. 3 (1864) 111-. of ellipse, and trisection of an angle. Boklen, O. Grunert Arch. 30 (1858) 434-. Ptolemy's theorem extended to ellipse. Grunert, J. A. Grunert Arch. 30 (1858) 109- . Radial. Tucker, R. QJ. Mth. 18 (1882) 311-. Radius of curvature, construction, Tran&on, (Prof.) A. Par. S. Phlm. PV. (1844) 85-. Rectilinear figure, in- or circum-scribed. Lind- nuin, C. F. Ts. Mt. Fys. 2 (1869) 119-. Relation between points of intersection of tan- gents and of normals at ends of a chord. Bruno, G. Tor. At. Ac. Sc. 8 (1872-73) 90-. Remarkable points, four. Osengu, G. Palom- ba Rac. 3 (1847) 353-. Sectors and segments with reference to con- jugate diameters. Zehme, W. Schlomilch Z. 3 (1858) 311-. Singular case. Fuss, N. [1798] St Pet. Ac. Sc. N. Acta 15 (1806) 71-. Theorems. Steiner, J. Crelle J. 32 (1846) 300-. (Steiner). Clausen, T. Grunert Arch. 9 (1847) 259-. Theory deduced from a transformation of circle. Lentheric, . (vi Adds.) Mntp. Ac. Sc. Mm. 5 (1861-63) 25-. Trigonometry of ellipse. Malagoli, R., d~ Mannei, E. G. Mt. 27 (1889) 60-. Envelope, case. Wolstenholme, J. [1883] L.Mth. S. P. 15 (*1883-84) 48-. of chord of circle, conies as. Unferdinger, F. Grunert Arch. 34 (1860) 406-." subtending constant angle. Breton [de Champ], P. N. A. Mth. 4 (1845) 300-. chords of constant length. Trudi, N. Nap. Rd. 1 (1862) 7-. . Weyr,E. [1871] Z. Mth. Ps. 17 (1872) 164-. circle whose diameter is focal radius of conic. Peter sen, J. P. C. (xn) Ts. Mth. 3 (1867) 24- ; 5 (1869) 140- ; 2 (1878) 178-. line cut by two circles harmonically. Cazamian, A. N. A. Mth. 13 (1894) 218-. cutting off intercepts whose sum is constant on two lines at right angles. Wol- fers, J. P. Grunert Arch. 25 (1855) 109-. of given length inscribed in angle. JoachimstiMl, F. N. A. Mth. 6 (1847) 260- . Envelopes of lines making constant angle with tangents to conic. Del Re, A. G. Mt. 22 (1884) 75-. variable lines. Ocagne, M. + M*). Baker, H. J. Mth. A. 42 (1893) 601-. (F = KG + M*). Scott, (Miss) C. A. Mth. A. 52 (1899) 593-. notion of plane curve of given degree. Cayley, A. Mess. Mth. 20 (1891) 148-. parameter, apparent difficulty in theory. Cayley, A. Mess. Mth. 14 (1885) 12-. plane, anonymous treatise. Loria, G. Bb. Mth. (1899) 10-. , general questions. Jonquieres, E. de. Crelle J. 59 (1861) 313-. , theorems on simplest forms. Kneser, A. Mth. A. 41 (1893) 349-. , history of investigation. Loria, G. Wiad. Mt. 2 (1898) 203-. , imaginary, representation. Saussure, R. de. J. H. Un. Cir. [15 (1895-96)] 39-. , intersection-figures. Study, E. Mth. A. 36 (1890) 216-. or skew, construction of tangents at multiple point. Olivier, T. Par. EC. Pol. J. 2l'cah. (1832) 302-. , theory, general. Mahler, E. [1880] Arch. Mth. Ps. 66 (1881) 365-. , , geometric. Kotter, E. Berl. Ak. Ab. (1887) (Anh., Mth.) 303 pp. , , projective. Cremona, L. Bologna Mm. Ac. 12 (1861) 305-. , 3rd degree, in space, characteristics. Schubert, H. C. H. Gott. Nr. (1874) 267-. relation SP"=0 for curve defined by N-l conditions. Serret, P. C. E. 86 (1878) 39-. similitude. Pirondini, G. A. Mt. 15 (1887- 88) 53-. study of curve near one of its points. Cosserat, E. Toul. Fac. Sc. A. 4 (1890) 0, 16 synthetic geometry, Kotter's. Fraser, N. Mth. S. P. 7 (1889) 46-. theory. Jacobi, C. G. J. Crelle J. 14 (1835) 56-. . Miquel, A. Liouv. J. Mth. 3 (1838) 202-. . Cayley, A. Camb. Mth. J. 4 (1845) 102-. . Bugaev, N. V. [1889] Eec. Mth. (Moscou) 15 (1891) 58-. , analytical, principles. Hoppe, B. Arch. Mth. Ps. 56 (1874) 41-. , and elimination. Cayley, A. [1863-64] (vn) Crelle J. 63 (1864) 34-; 64 (1865) 167-. , general. Mainardi, G. Em. At. N. Line. 26 (1873) 86-. , . Puzyna, J. Krk. Ak. (Mt.-Prz.) Ez. 2 (1892) 1-; Crc. Ac. Sc. Bll. (1891) 62-. , geometric. Grassmann, H. Crelle J. 31 (1846) 111-. Diameters. Le Cointe, J. L. A. C. E. 60 (1865) 1083-. Equations, algebraic, geometric relations be- tween two systems of points defined by. Mangeot, S. C. E. 129 (1899) 464-. , curves with trinomial, and surfaces with quadrinomial. Euzet, . N. A. Mth. 13 (1854) 193-. , differential, 1st order and degree, algebraic integration. Poincare, H. Palermo Cir. Mt. Ed. 11 (1897) 193-. 515 KK2 7600 Algebraic Curves and Surfaces. General 7600 Equations, homogeneous and central, solutions. M'Laren, (Lord). [1888] Edinb. R. S. T. 35 (1890) 1043-. in n variables, geometric interpretation. Peclet, E. Gergonne A. Mth. 14 (1823- 24) 65-. " Equipollence " method, application to find- ing certain loci. Minich, S. E. A. Sc. Lomb. Yen. 7 (1837) 57-, 171-. , (Minich). Bellavitis, G. A. Sc. Lomb. Yen. 7 (1837) 165-. Focal curves and surfaces. Schoute, P. H. Amst. Ak. Vs. 6 (1898] 404-. Generation of forms of 3rd degree by projective forms of 1st and 2nd degree. Certo, L. G. Mt. 26 (1888) 41-. , mechanical, curves and surfaces capable of. Puchta, A. [1883] Wien Ak. Sb. 88 (1884) (Ab. 2) 571-. of polar elements for curves and surfaces by projective method. Geisenheimer, L. Z. Mth. Ps. 31 (1886) 193- ; 32 (1887) 127-. by projective forms. Joerres, . Crelle J. 72 (1870) 327-. of surfaces and skew curves. Vanecek, M. N. C. E. 94 (1882) 210-. , method analogous to MacLaurin's. Vanecek, M. N. Liouv. J. Mth. 9 (1883) 269-. Intersection of curves and surfaces, centres of mean distances of points. Roberts, S. QJ. Mth. 9 (1868) 63-. or surfaces, relations between points. Jacobi, C. G. J. Crelle J. 15 (1836) 285-. and surfaces by straight lines. Sharp, W. J. C. Mess. Mth. 9 (1880) 49-. Loci, classification. Clifford, W. K. Phil. Trans. 169 (1878) 663-. Maximum and minimum distance of points on 2 curves or surfaces. Odin, A. A. Laus. S. Vd. Bll. 23 (1888) 108-. distances between points, curves, or surfaces, taken in pairs. Tucci, F. P. (xi) Nap. Ac. At. 2 (1865) No. 14, 13 pp. Method, new, application. Stubbs, J. W. Ph. Mg. 23 (1843) 338-. Normals. Stuyvaert, . Mathesis 18 (1898) 105-. , common, distance between lines and sur- faces with. Bordoni, A. Brugnatelli G. 6 (1823) 1-. and normal planes, construction. Schone- mann, P. (xi) Berl. Ak. Mb. (1855) 255-. Plane enveloping sphere, property, and exten- sion to any surface. Ocagne,M.d\ L. Mth. S. P. 18 (1886-87) 361-. Polar groups of systems of points. Stuyvaert , . Mathesis 18 (1898) 20-. reciprocal figures, transformation. Vanelek, J. S. Liege S. Sc. Mm. 11 (1885) No. 2, 21 pp. theory of curves and surfaces, 3rd degree. Milinowski, . Crelle J. Mth. 89 (1880) 136-. Polars of a straight line relative to curves and surfaces. Laguerre, E. Par. S. Mth. Bll. 3 (1875) 174-. Polynomials, geometric application of a theorem. Laisant, C. A. As. Fr. C. R. (1898) (Ft. 2) 73-. Priority claim for various theorems. Mainardi, G. Rm. At. N. Line. 26 (1873) 88-. Product of secants through 2 points. Jube, E. N. A. Mth. 5 (1846) 340-. Properties, central. Breton \de Champ}, P. Liouv. J. Mth. 11 (1846) 153- ; N. A. Mth. 7 (1848) 187-. , general. Terquem, 0. N. A. Mth. 16 (1857) 266-. Quadric and cubic, polar n-hedra. Johnson, A. R. Mess. Mth. 16 (1887) 63-. Segmentary theorems. Terquem, O. N. A. Mth. 18 (1859) 111-. Similar curves and surfaces. Crelle, A. L. Berl. Ab. (1828) 21-. Similarity of curves and solids. Brenner, . Grunert Arch. 13 (1849) 182-. Similarly variable figures, and loci derived from them. Reveille, J. N. A. Mth. 12 (1893) 297-. Similitude and symmetry of 2 curves or sur- faces. Mangeot, S. Par. EC. Norm. A. 15 (1898) 385-. Singularities, decomposition. Korteweg, D. J. Mth. A. 41 (1893) 286-. Surface generated by conic doubly secant to given conic. Demartres, G. C. R. 106 (1888) 340-. Surfaces and curves of form Sa r z r "=0. Pellet, A. C. R. 115 (1892) 498-. given by equation S = 0, where u r denotes line or plane. Hunyady, E. von. C. R. 64 (1867) 218- ; 65 (1867) 497-. with only one side, and singular points of plane curves. Delaunay, N. Par. S. Mth. Bll. 26 (1898) 43-. Symmetric products in relation to curves and surfaces. Johnson, A. R. QJ. Mth. 22 (1887) 325-. Symmetry of 2 algebraic figures with refer- ence to a point. Mangeot, S. N. A. Mth. 19 (1900) 451-. , tetrahedral, curves and surfaces with. Jamet, V. Par. EC. Norm. A. 4 (1887) (Svppl.) 78 pp. Tangents of equidistant curves and surfaces, construction. Schirek, C. Z. Mth. Ps. 28 (1883) 183-. Theorems, various. Frtgier, . Gergonne A. Mth. 9 (1818-19) 241-. , . Serret, P. (xi) Par. S. Phlm. Bll. 3 (1866) 98-. 516 7610 Algebraic Plane Curves of Higher Degree 7610 Metrical and projective properties of algebraic plane curves of degree higher than the second. (See also 8030, 8075, 8430.) Bergsten, N. J. Stockh. Ac. Hndl. 34 (1813) 22-. Le Frangois, (Prof.) . Quetelet Cor. Mth. 6 (1830) 315-. Eeiss, M. Quetelet Cor. Mth. 9 (1837) 249-. Steiner,J. [1848] Berl. B. (1848) 310- ; Crelle J. 47 (1854) 1-. Terquem, 0. N. A. Mth. 9 (1850) 283-. Schwarz, F. S. H. [1856] Halle Ab. Nt. Vr. 1 (1860) 149-. Woepcke, F. Crelle J. 53 (1857) 260-. Steen, A. Mth. Ts. 1 (1859) 65-. Beltrami, E. G. Mt. 4 (1866) 76-. Olivier, A. [1868] Crelle J. 70 (1869) 156-. Laguerre, E. (x) Par. S. Phlm. Bll. 5 (1868) 112- ; 7 (1871) 241- ; C. E. 80 (1875) 1218-. Azzarelli, M. Bm. N. Line. Mm. 8 (1892) 7-. Michell, J. H. Aust. As. Bp. (1892) 257. Algebraic curve cannot have 'point d'arret.' Serret, P. N. A. Mth. 4 (1865) 311-. Anallagmatic curves and surfaces, application of general properties of curves. Laguerre, E. (x) Par. S. Phlm. Bll. 5 (1868) 112-. Analytical geometry, new method. Story, W. E. Am. J. Mth. 9 (1887) 38-. Anharmonic curves. Bioche, C. Par. S. Phlm. Bll. 5 (1893) (C. R.) (No. 10) 2. Asymptote is always tangent to 2 parts of the curve, proof. Terquem, 0. N. A. Mth. 8 (1849) 288-. Asymptotes. Lentheric, . Gergonne A. Mth. 17 (1826-27) 79-. . Fournier-Vannson, . N. A. Mth. 2 (1843) 398-. . Gregory, D. F. Camb. Mth. J. 4 (1845) 42-. . Serret, P. N. A. Mth. 13 (1854) 144- ; 1 (1862) 23-. . Stolz, 0. Mth. A. 11 (1877) 41-. . Casorati, F. Palermo Cir. Mt. Bd. 3 (1889) 49-. , curvilinear. Haskell, M. W. [1897] Calif. Ac. P. (Mth. Ps.) 1 (1903) 25-. , equations. Casorati, F. Mil. I. Lomb. Bd. 12 (1879) 117-. , Newton's theorem. Lebesgue, V. A. N. A. Mth. 7 (1848) 385-. , , proof, and theorem on multiple points. Terquem, 0. N. A. Mth. 7 (1848) 422-. , property, and points at infinity. Transon, A. N. A. Mth. 12 (1873) 289-. , real, to imaginary branches of curves. Walton, W. Camb. Mth. J. 2 (1841) 236-. , rectilinear. Smith, E. B. A. Mth. 1 (1884-85) 11-, 72. 7610 iptotes, rectilinear. Himstedt, A. Arch. 1th. Ps. 12 (1894) 357-. Asymptotic curves. Serret, J. A. N. A. Mth. 6 (1847) 217-. points, theory. Schi/ner, F. Arch. Mth. Axes, conditions that curve may have given number. Mangeot, S. Par. S. Mth. Bll. 25 (1897) 54-, 268. , harmonic. Chasles, M. C. B. 73 (1871) 229-, 1241-, 1289-, 1405- ; 74 (1872) 21-. of symmetry, curves with several. St. Germain, A. de. N. A. Mth. 19 (1880) 350-. Bicursal curves. Cayley, A. L. Mth. S. P. 4 (1871-73) 347-. Bifocal curves. Onnen, H. N. Arch. Wisk. 17 (1890) 77-. Bitangential curve. Heal, W. E. Des Moines Anal. 8 (1881) 171-. Bitangents. Cayley, A. Phil. Trans. (1859) 193-. . Holditch,H. QJ. Mth. 3 (1860)289-; 4 (1861) 28-. . Eetali, V. [1890] Bologna Bd. (1890- 91) 35-. . Dowling, L. W. [1897] N. Y. Am. Mth. S. Bll. 4 (1898) 2, 4. with coordinates rational functions of a parameter, number. Schwering, K. Z. Mth. Ps. 21 (1876) 130- . , determination of points of contact. Salmon, G. QJ. Mth. 3 (1860) 317-. Boundary curves, envelope of xn + ym = mn (m? + P = s*>) . Meyer, C. W. Arch. Mth. Ps. 16 (1898) 150-. Branches of curves, number. Harnack, A. Mth. A. 10 (1876) 189-. , imaginary, associated with possible magni- tudes. Walton, W. [1864] QJ. Mth. 7 (1866) 232-. , infinite. Haillecourt, A. N. A. Mth. 6 (1847) 372-. , . Bellavitis, G. [1879] Em. E. Ac. Line. Mm. 5 (1880) 43-. ,, forms. Frost, P. QJ.Mth. 3 (1860) 164-. and points, imaginary. Loud, F. H. A. Mth. 8 (1893-94) 29-. , real. Hilbert, D. Mth. A. 38 (1891) 115-. Carnot's theorem generalised. Gob, A. As. Fr. C. B. (1893) (Pt. 2) 258-. Centre of curve. Taylor, H. M. QJ. Mth. 24 (1890) 55-. . Baker, H. F. QJ. Mth. 24 (1890) 338-. mean distances. Liouville, J. Liouv. J. Mth. 9 (1844) 337-. of points of rational curve. Weill, M. Par. S. Mth. Bll. 10 (1882) 137-. , theorem of Newton, generalisation. Delens, P. N. A. Mth. 12 (1893) 407-. Centres. Eoguet,C. N. A. Mth. 2 (1843) 210-. of harmonic means. Poncelet, J. V. Crelle J. 3 (1828) 213-. , 2 theorems. Schoute, P. H. Par. S. Mth. Bll. 10 (1882) 219-. 517 7610 Algebraic Plane Curves of Higher Degree 7610 Centroid of certain points of curves. Ocagne, M. d\ C. E. 99 (1884) 779-. intersections of curves with circle of fixed centre. Ocagne, M. d\ N. A. Mth. 5 (1886) 295-. straight line with curve, Steiner's theorems. Sparer, B. Z. Mth. Ps. 37 (1892) 65-, 340-. points common to 2 curves. Sporer, B. Arch. Mth. Ps. 3 (1886) 84-. Class of curve, lowering. Painvin, L. N. A. Mth. 6 (1867) 113-. , owing to presence of cusp. Painvin, L. Bll. Sc. Mth. As. 4 (1873) 131-. Concentric parallel curves and complementary inverse curves. Chamousset, (I'abbe) F. [1868] (xn) Sav. Ac. Mm. 11 (1869) LXXXVI. Conditions that 4 curves have 2 common points, order. Roberts, S. QJ. Mth. 9 (1868) 176-. Construction of curves. Bellavitis, G. [1878] Em. S. It. Mm. 3 (1879) No. 4, 43 pp. Contact of curves. Spottiswoode, W. [1861-66] Phil. Trans. (1862) 41- ; QJ. Mth. 7 (1866) , double, curves having. Martin, J. W. Cn. J. 5 (1860) 331-. of higher order. Laurent, H. N. A. Mth. 16 (1877) 26-. poir point, lines making constant angle with tangent at. Chasles.M. C.E.74(1872) 1146-. Coordinates, bi-polar. Fries, J. de. Amst. Ak. Vs. 4 (1896) 219- ; Fschr. Mth. (1896) 491. of plane curve in space. Tanner, H. W. L. L. Mth. S. P. 13 (1881-82) 125-. , polar, symmetry in. Lefevre, J. N. A. Mth. 11 (1892) 302-, 353-. , tangential. Amstein, H. Laus. S. Vd. Bll. 15 (1878) 393-. , trilinear, of circular points at infinity. Mansion, P. [1875] Mess. Mth. 5 (1876) 158-. , , and higher curves. Foglini, G. Em. N. Line. Mm. 1 (1887) 173-. Cotangent theorems. Eckardt, F. E, Z. Mth. Ps. 10 (1865) 503- ; 12 (1867) 352-. Cotes's theorem. Walker, J. J. L. Mth. S. P. 10 (1878-79) 180-. Co variant of 9th order of ternary cubic form, geometric significance. Gerbaldi, F. Tor. Ac. Sc. At. 15 (1879) 707-. Covariants, 2, applied to construction of certain curves. Mangeot, S. N. A. Mth. 16 (1897) 76-. of ternary forms. Gundelfinger, S. A. Mt. 5 (1871-73) 223-. Cubic and biquadratic problems, solution by geometry. Smith, H. J. S. A. Mt. 3 (1869- 70) 112-, 218-. Curvature conies of plane curves. Berg, F. J. van den. Amst. Ak. Vs. M. 9 (1892) 85- ; Fschr. Mth. (1892) 654-. of curves. Servais, C. Brux. Ac. Bll. 21 (1891) 587-; 22 (1891) 512-. Curve generated by 2 projective pencils of curves, degree. Olivier, A. [1869] CrelleJ. 71 (1870) 195-. Curve, polar of which with respect to given circle is congruent to curve itself. Genocchi, A. N. A. Mth. 14 (1855) 248. with tangents meeting at constant angle, problem. Ivory, J. Thomson A. Ph. 8 (1816) 272-. Curves adjoint to algebraic curve. Biermann, 0. Mh. Mth. Ps. 10 (1899) 373-. with branches in several planes, existence. (Geometrical interpretation of \/^l and fractional powers.) Gregory, D. F. Camb. Mth. J. 1 (1839) 259-. coordinates rational functions of a parameter. Clebsch, E. F. A. [1864] Crelle J. 64 (1865) 43-. generated by projective relations on curves. Schroter, H. E. Crelle J. 54 (1857) 31-. section of cone by plane. Gergonne, J. D. Gergonne A. Mth. 12 (1821-22) 113-. in polar coordinates. Gourieff, S. [1797] St Pet. Ac. Sc. N. Acta 12 (1801) 176-. referred to centre as origin. Billelaut, . N. A. Mth. 5 (1846) 228-. subtending right angle from points on circle. Levdnen, S. Helsingf. Ofv. 34 (1892) 22-. , 3rd class. Laguerre, E. Liouv. J. Mth. 4 (1878) 213-. , , classification. Bellavitis, G. [1851] Mod. Mm. S. It. 25 (1855) 1-. , , with given foci. Kapteyn, W. Amst. Ak. Vs. 5 (1897) 146- ; Fschr. Mth. (1896) 487. , , polar conies, theorems. Smith, T.StL. QJ. Mth. 2 (1858) 327-. __, , rational. H., G. D. Ts. Mth. 6 (1889) 193- ; Fschr. Mth. (1888) 715. , , types. Breyer, H. G. Amst. Ak. Vs. [2] (1894) 60-, 81. , 4th class, with 3 bitangents. Le Paige, C. Prag Sb. (1884) (Mth.-Nt.) 45-. , , quadruple foci. Jeffery, H. M. B. A. Ep. (1880) 478- ; QJ. Mth. 18 (1882) , , , loops. Jeffery, H. M. [1881] QJ. Mth. 18 (1882) 158-. , , singularities. Laguerre, E. Liouv. J. Mth. 1 (1875) 265-. , , with a triple and a single focus. Jeffery, H. M. B. A. Ep. (1883) 412- ; QJ. Mth. 20 (1885) 273-. , 6th class, symmetric. Crone, C. Acta Mth. 2 (1883) 81-. , 9th class, adjoint to general curve of 4th degree. Kohn, G. Wien Ak. Sb. 95 (1887) (Ab. 2) 338-. , nth class, foci. Serret, P. C. E. 86 (1878) 385-. , , orthoptic curve, theorem of Steiner. Dewulf, . N. A. Mth. 18 (1859) 174-. , , , . Salmon, G. N. A. Mth. 18 (1859) 314-. , even degree, theory. Mahler, E. [1882] Arch. Mth. Ps. 69 (1883) 108-. 518 7610 Higher Plane Curves. Cubics 7610 Curves, 2nd and 3rd degre Clausen, T. Grunert Arch. 15 (1850) 345-. canonical forms. Roberts, R. A . [1889] L. Mth. S. P. 21 (1891) 62-. circum-polygons of conic, and in-polygons of cubic. Roberts, R. A. L. Mth. S. P. 17 (1887) 158-. parametric determination of points. Juel, C. Mth. A. 47 (1896) 72-. tactinvariant of a conic and a cubic. Linde- mann, F. B. A. Ep. (1884) 647-. in trilinear coordinates. Jettmar, H. von. Arch. Mth. Ps. 10 (1891) 13-. Curves, 3rd degree. Clausen, T. Crelle J. 11 (1834) 402- ; As. Nr. 21 (1844) 209-. Cayley, A. Liouv. J. Mth. 9 (1844) 285-; 10 (1845) 102-. Chasles, M. N. A. Mth. 9 (1850) 382-. Cotterill, T. Camb. and Dubl. Mth. J. 7 (1852) 14-. Bjerknes, C. A. Crelle J. 55 (1858) 310- ; N. Mg. Ntvd. 10 (1859) 191-. Transon, (Prof.) A. N. A. Mth. 18 (1859) 266-. [Salvatore-]Dino, N. G. Mt. 1 (1863) 187-, 334-. Hunyady, J. (xn) Mag. Ak. Ets. (Mth. Term.) 4 (1863) 19- ; 5 (1865) 66-. Sartiaux, A. N. A. Mth. 6 (1867) 68-. Eckardt, F. E. Z. Mth. Ps. 13 (1868) 263-. Bdcklund, A. V. Stockh. Ofv. 28 (1871) 715- ; 29 (No. 2) (1872) 109-. Koehler, C. N. A. Mth. 11 (1872) 21-, 66-, 122-. Schroter, H. [1872] Mth. A. 6 (1873) 85-. Milinowski, . Crelle J. Mth. 78 (1874) 177-. Sharp, W. J. C. QJ. Mth. 16 (1879) 186-, 298-. Folie, F.,(&Le Paige, C. [1880-81] Brux. Ac. Mm. 43 (1882) No. 7, 43 pp. ; Brux. Ac. Bll. 1 (1881) 610- . Arcais, F. d\ Yen. I. At. 8 (1882) 673-. Boije of Gennas, C. 0. Des Moines Anal. 9 (1882) 108-. Franklin, F. Am. J. Mth. 5 (1882) 212-. Le Paige, C. M. M. H. H. [1882] Brux. Ac. Bll. 4 (1882) 334- ; Prag Sb. (1882) 200- ; Liege S. Sc. Mm. 10 (1883) (No. 2) 103 pp. Folie, F., & Le Paige, C. Brux. Ac. Mm. 45 (1884) (No. 1) 45 pp. Springer, J. F. [1889] J. H. Un. Cir. [9 (1889-90)] 31. Loud, F. H. [1891] A. Mth. 6 (1891-92) 5-. Sauve, A. Em. N. Line. At. 53 (1900) 129-. through 7 angular points of complete quadri- lateral. Machovec, F. Casopis 16 (1887) 113- ; Fschr. Mth. (1887) 629. anharmonic property. Salmon, G. Crelle J. 42 (1852) 274-. ratio of a curve. Jamet, . Par. S. Mth. Bll. 15 (1887) 35-. axes, determination. Mangeot, S. Par. EC. Norm. A. 10 (1893) 43-. branches, principal and secondary. Grass- mann, H. Gott. Nr. (1872) 505-. circumscribed to complete quadrilateral . Ovidio , E. d\ G. Mt. 10 (1872) 16-. classification. Plilcker, J. (vm) Crelle J. 10 (1833) 98-. . Cayley, A. [1864] Camb. Ph. S. T. 11 (pt. 1) (1866) 81-. . Cremona, L. G. Mt. 2 (1864) 78-. . Jeffery, H. M. [1878] QJ. Mth. 16 (1879) by nature of tangents from any point of curve. Durege, H. [1871] Crelle J. 75 (1873) 153- ; 76 (1873) 59-. , Newton's. Ball, W. W. R. Bb. Mth. (1891) 35-. through 3 collinear points, and touching pairs of lines. Roberts, R. A. Mess. Mth. 15 (1886) 152-. and cones, 3rd degree. Cayley, A. [1864] Camb. Ph. S. T. 11 (pt. 1) (1866) 129-. 27 conies of closest contact. Sylvester, J. J. Ph. Mg. 11 (1856) 463-. conies touching in 3 distinct points. Hesse, L. 0. Crelle J. 36 (1848) 143-. conjugate points, triangles, etc. Caporali, E. Em. E. Ac. Line. T. 1 (1877) 232-. connected with isogonal triangles. Barbarin, P. As. Fr. C. E. (1896) (Pt. 2) 89-. construction. Grave, P. Kazan Un. Mm. (1898) (Pt. 3) 121-, (Pt. 4) 63-, (Pts. 5 & 6) 1-, (Pts. 7 & 8) 63-, (Pt. 9) 51-, (Pt. 10) 1-, (Pt. 11) 1- ; (1899) (Pt. 2) 1- ; Fschr. Mth. (1898) 508-. from conjugate points. Heger, R. Z. Mth. Ps. 25 (1880) 100-. contact, 6-point, with conic, 27 points of. Steiner, J. Berl. B. (1845) 386-. , 9-point. Hart, A. S. Ir. Ac. T. 25 (1875) 559-. contacts, conies in 5, cubics in 8 consecutive points. Salmon, G. Phil. Trans. (1858) 535-. coordinates. Hermite, C. [1876] Crelle J. Mth. 82 (1877) 343-. eovariant conies and cubics of plane cubic. White, H. S. N. Y. Am. Mth. S. T. 1 (1900) 1-. locus of vertex of pencil of tangents to cubic in involution. Walker, J. J. [1881] L. Mth. S. P. 13 (1881-82) 66-. covariants. Gordan, P. N. Y. Am. Mth. S. T. 1 (1900) 9-. curve as sum of 4 cubes. Walker, J. J. L. Mth. S. P. 10 (1878-79) 180-. of 3rd class touching in 9 points. Halphen, G. H. [1880] Par. S. Mth. Bll. 9 (1881) 96-. as curves of involution. Weyr, E. J. Prag Sb. (1877) 131-. and curves of 3rd class. Hesse, L. 0. Crelle J. 38 (1849) 241-. cyclic polygons on, application of elliptic func- tions. Fries, J. de. Amst. Ak. Vs. M. 7 (1890) 430- ; Arch. Neerl. 25 (1892) 1-. decomposition of curve into conic and straight line. Brioschi, Fr. Em. E. Ac. Line. At. 3 (1876) (Pte. 2) 89-. 3 straight lines. Thaer, A. [1878] Mth. A. 14 (1879) 545-. 519 7610 Higher Plane Curves. Cubics 7610 derivative points, algebraic theory. Sylvester, J. J. Ph. Mg. 16 (1858) 116-. diameters. Walker, J. J. [1887] Phil. Trans. (A) 179 (1889) 151-. with double point. Weyr, Em. [1870] Prag Sb. (1870) 43- ; Mth. A. 3 (1871) 235-. . Igel, B. Mth. A. 6 (1873) 633-. . Anelli, P. G. Mt. 16 (1878) 364-. . Pittarelli, G. Nap. Ed. 24 (1885) 111-. . Dingeldey, F. Mth. A. 27 (1886) 272-. . Torelli, G. G. Mt. 26 (1888) 172-. . Berzolari, L. Mil. I. Lomb. Bd. 25 (1892) 1025-. , and curves of 4th degree with 3 double points. Juel, G. S. (xn) Ts. Mth. 1 (1877) 17-. , equation. Walker, J. J. QJ. Mth. 14 (1877) 242-. , metrical properties. Stiner, G. Mh. Mth. Ps. 4 (1893) 99-. , series of tangents. Durege, H. Mth. A. 1 (1869) 509-. , Steiner's polygons in. Loria, G. Prag Sb. (1896) (Mth.-Nt.) No. 36, 4 pp. , synthetic theory. Drasch, H. Wien Ak. Sb. 85 (1882) (Ab. 2) 534-. , tangents drawn from any point to. Roberts, R. A. L. Mth. S. P. 12 (1880-81) 99-. , theorems. Weyr, Em. Wien Sb. 61 (1870) (Ab. 2) 731-, 819-. , . Zimmermann, H. E. M. 0. Z. Mth. Ps. 28 (1883) 56-. 2 double points. Gordan, P. Mth. A. 3 (1871) 631-. and ellipse, theorems. Steiner, J. Crelle J. 32 (1846) 300-. , (Steiner). Clausen, T. Grunert Arch. 9 (1847) 259-. enumeration. Commines De Marsilly, (le gen.) A. de. As. Fr. C. E. (1886) (Ft. 2) 110-. envelope derived from curve. Schoute, P. H. N. Arch. Wisk. 3 (1898) 30-. as envelopes of conies. Kttpper, G. Prag Sb. (1871) (pt. 2) 63-. family, critical centres for. Borisov, E. V. [1894] Eec. Mth. (Moscou) 17 (1895) 461- ; Fschr. Mth. (1893-94) 1135. figure formed by 16 cotangential chords of a curve. Roberts, S. QJ. Mth. 9 (1868) 232-. figures. Disteli, M. Z. Mth. Ps. 36 (1891) 138-. fundamental forms. Mobius, A. F. Leip. B. 2 (1848) 12-; Leip. Ab. Mth. Ps. 1 (1852) 1-. general properties, contravariant (Cayleyan), etc. Cayley, A. [1856] Phil. Trans. (1857) 415-. generation. Weyr, Em. Wien Sb. 58 (Ab. 2) 633-. generation by conic pencil and ray pencil, syn- thetic proof. Milinowski, . Z. Mth. Ps. 23 (1878) 327-. , . Schur,F. Z.Mth. Ps. 24 (1879) 119-. , linear. Grassmann, H. Crelle J. 36 (1848) 177- ; 52 (1856) 254-. , (Grassmann). Schroeter, H. Crelle J. Mth. 104 (1889) 62-. by symmetric elemental systems of 2nd degree. Weyr, E. J. Wien Ak. Sb. 69 (1874) (Ab. 2) 784-. Hessian and Cayleyan, construction. Scott, (Miss) C. A. [1893] Phil. Trans. (A) 185 (1895) 247-. Hessians and Cayleyans. Gordan, P. N. Y. Am. Mth. S. T. 1 (1900) 402-. with infinitely distant stationary tangent, and their occurrence in statics. Solin, J. M. [1877] Prag Ab. 9 (1878) (Mth.) (No. 2) 32 pp. inflexional lines, triplets and triangles. White, H. S. N. Y. Am. Mth. S. Bll. 4 (1898) 258-. tangents. Clebsch, A. [1860] Crelle J. 58 (1861) 229-. . Wirtinger, W. Mh. Mth. Ps. 4 (1893) 395-. (Wirtinger). Kohn, G. Mh. Mth. Ps. 4 (1893) 398. of a curve, and conies touched by them. Ferrers, N. M. Mess. Mth. 24 (1895) 77-. intersections. Chasles, M. C. E. 41 (1855) 1190-. . Hart, A. S. [1878] Ir. Ac. T. 26 (1879) 449-. of 2. Taylor, H. M. L. Mth. S. P. 29 (1898) 265-. with straight lines. Laguerre, E. [1875] Par. S. Mth. Bll. 4 (1876) 110-. 3 straight lines ; if 6 lie on conic, other 3 are on straight line. Poncelet, J. V, Quetelet Cor. Mth. 7 (1832) 79-. of tangents through 2 points on a curve. Roberts, S. QJ. Mth. 3 (1860) 118-. involution. Cayley, A. [1864] Camb. Ph. S. T. 11 (pt. 1) (1866)39-. irrational covariant cubics of plane cubics. White, H. S. N. Y. Am. Mth. S. T. 1 (1900) 170-, 508. linear constructions. Harnack, A. Z. Mth. Ps. 22 (1877) 38-. orthogonal invariants. Thomae,J. Leip. Mth. Ps. B. 51 (1899) (Mth.) 317-. in pencil of 6 lines. Cayley, A. QJ. Mth. 9 (1868) 210-. plane and skew. Mangeot, S. Par. S. Mth. Bll. 21 (1893) 44-. 9th point determined by 8. Weddle, T. Camb. and Dubl. Mth. J. 6 (1851) 83-. . Hart, A. S. Camb. and Dubl. . Schroter, H. [1871] Mth. A. 5 (1872) 50-. Mth. J. 6 (1851) 181-. . Clebsch, R. F. A. Mth. A. 5 (1872) 422-. . Q Quetelet, E. Brux. Ac. Bll. 5 Cayley, A. QJ. Mth. 5 (1862) 5 (1872) . Tognoli, 0. G. Mt. 11 (1873) 378-. (1858) 15-. (Schroter). Hurwitz, A. Crelle J. Mth. 107 (1891) 141-. 222-. . loachimescu, A. G. [Bucarest S. Sc. Bl. . London, F. Mth. A. 36 (1890) 6 (1897)] 152-. 585-. 520 7610 9th point determined by 8. Kierboe, T. N. Ts. Mth. 7 (B) (1896) 53-; Fschr. Mth. (1896) 437-. depending on system of conies. Smith, H. J. S. A. Mt. 2 (1868-69) 318-. , , Salmon's theorem. Goursat, E. N. A. Mth. 15 (1896) 20-. point, polar conic of which is a circle. Stuy- vaert, . N. A. Mth. 18 (1899) 275-. through 9 points, condition for double point. Thomae, J. Leip. Mth. Ps. B. 47 (1895) 515-. , construction. Chasles, M. C. E. 36 (1853) 943-. , . Tslnger, V. Y. [1867] (xn) Eec. Mth. (Moscou) 3 (1868) (Pt. 1) 290-. , . Beyel, G. Z. Mth. Ps. 40 (1895) 99-. points of inflexion. Hesse, L. 0. Crelle J. 28 (1844) 97- ; 38 (1849) 257-. . Salmon, G. Crelle J. 39 (1850) 365-. . Sartiaux, A. [1867] (xn) Lille S. Mm. 5 (1868) 429-. . Clebsch, R.F. A. [1869] Mth. A. 2 (1870) 382-. . St Germain, A. de. N. A. Mth. 12 (1873) 356-. . Servais, C. Brux. Ac. Bll. 20 (1890) 453-. . Wiman,A. N. Ts. Mth. 5 (B) (1894) , and cusps of curves of 3rd class. Painvin, L. F. [1862] (xn) Lille S. Mm. 10 (1864) 103-. , generalisation of theorem. Legoux, A. N. A. Mth. 2 (1883) 77-. , position. Disteli, M. Ziir. Vjschr. 35 (1890) 145-. , and stationary tangents of curves of 3rd class. Hesse, L. O. Crelle J. 38 (1849) 257-. , theorem. Bussell, W. H. L. [1882] E. S. P. 34 (1883) 35-. , (Eussell). Spottiswoode, W. [1882] E. S. P. 34 (1883) 37-. 9 points of intersection of 2 curves. Schottky, F. Crelle J. Mth. 119 (1898) 72-. points, ruler construction for, given point P on curve and points of contact of 4 tangents from P. Thomson, F. D. [1868] Mess. Mth. 5 (1871) 27-. polar conies, theorems. Smith, T. St L. QJ. Mth. 2 (1858) 208- . figures. London, F. Mth. A. 36 (1890) 535-. . Scorza, G. Mth. A. 51 (1899) 154-. reciprocal of curve with respect to cubic. Russell, W. H. L. E. S. P. 44 (1888) 388-. of syzygy /-/)A-0. Gerbaldi,F. Palermo Cir. Mt. Ed. 7 (1893) 19-. system of a curve. Bonsdorff, E. \E. J.} [1877] St Pet. Ac. Sc. Bll. 24 (1878) 409-. polygon of 2n sides inscribed to. Torelli, G. G. Mt. 26 (1888) 327-. polygons in- and circum-scribed to same curve. Weyr, E. Mh. Mth. Ps. 4 (1893) 120, 154. , Steiner's, on. Czuber, E. Crelle J. Mth. 114 (1895) 312-. and poristic polygons. Steiner, J. Berl. B. (1845) 386-. Higher Plane Curves. Cubics 7610 problem on two. (Construction of conic through the 5 unknown points, 4 being given.) Poudra, . N. A. Mth. 15 (1856) 24-. projective generation, rational curves, etc. Kotter, E. Mth. A. 38 (1891) 287-. rational. Weyr, Em. G. Mt. 9 (1871) 145-. . Zahradnik, K. Arch. Mth. Ps. 56 (1874) 131- ; 58 (1876) 23-; 61 (1877) 1-. . Astor, A. N. A. Mth. 11 (1892) 276-; 13 (1894) 184-. . Cazamian, A. N. A. Mth. 14 (1895) 297-. , application of polar reciprocals. Cazamian, A. N. A.Mth. 13 (1894) 300-. , construction. Schoute, P. H. As. Fr. C. E. (1885) (Pt. 2) 169-. , . Willig, H. Arch. Mth. Ps. 10 (1891) , of Hessian curve. Dingeldey, F. Mth. A. 28 (1887) 81-. , projectivities and involutions on. Weyr, E. J. Wien Ak. Sb. 81 (1880) (Ab. 2) 169-. referred to tetrad of corresponding points. Jeffery, H. M. QJ. Mth. 15 (1878) 198-. relations on general curve. Weyr, E. J. Wien Ak. Sb. 87 (1883) (Ab. 2) 837-. representation on circle. Durege, H. Z. Mth. Ps. 17 (1872) 433-. by elliptic functions. Picquet, . Par. EC. Pol. J. 54 (1884) 31-. ruler constructions in connexion with. Russell, R. [1893] Ir. Ac. T. 30 (1892-96) 295. satellite of line. Roberts, W. R. W. [1880] Ir. Ac. P. 3 (1883) 293-. . Walker, J. J. [1890] L. Mth. S. P. 21 (1891) 247-. series of pentagons. Serret, P. C. E. 115 (1892) 406-, 436-. (Serret). Rouquet, V. Toul. Ac. Sc. Mm. 5 (1893) 113-. singularities. Gundelfinger, S. Mth. A. 4 (1871) 561-. and surfaces of 2nd degree, geometrical con- structions for. Engel, J. H. Ziir. Vjschr. 34 (1889) 299-. synthetic treatment. Milinowski, . Z. Mth. Ps. 21 (1876) 427-; 23 (1878) 343-. system of linear equations in connexion with a curve. Rosanes, J. Mth. A. 36 (1890) 316-. tangential of point on. Cayley, A. Phil. Trans. (1858) 461-. 6 tangents, conic through points of contact. Pliicker, J. Crelle J. 34 (1847) 329-. tangents, forming pencil in involution. Roberts, R. A. [1881] L. Mth. S. P. 13 (1881-82) 25-. 6 tangents from point, problem. Russell, W. H. L. [1881] E. S. P. 33 (1882) 211-. tangents, projective proof of anharmonic pro- perty. Dixon, A. C. Mess. Mth. 26 (1897) 53-. , real and imaginary. Sturm, R. Crelle J. Mth. 90 (1881) 85-. 'tertians,' diametral or centric. Newman, F. W. B. A. Ep. 39 (1869) (Sect.) 10-. 521 7610 Higher Plane Curves. 3rd and 4th Degrees 7610 and theorem of Steiner on polygons. Clebsch, B. F. A. [1863] (vn) Crelle J. 63 (1864) 94-. transformation of equation to canonical form. Dingeldey, F. Mth. A. 31 (1888) 177-. triameters. Mainardi, G. A. Sc. Lomb. Ven. 7 (1837) 118-. triangle of points of inflexion. Gundelfinger, S. [1871] Mth. A. 5 (1872) 442-. triangles in- and ex-scribable to general curve. Sylvester, J. J. [1880] (xn) J. H. Un. Cir. [1] (1882) 49. of tangents. Beiss, M. Quetelet Cor. Mth. 9 (1837) 249-. in trilinear coordinates. Whitworth, W. A. (yi) Mess. Mth. 2 (1864) 116-, 129-. and 3rd class. Zahradnik, K. Prag Sb. (1873) 298-. --- , a geometrical relationship. Zahradnik, K. Wien Ak. Sb. 75 (1877) (Ab. 2) 437-. , with a single and a double focus. Jeffery, H. M. QJ. Mth. 14 (1877) 359-. --- , 3 single foci. Jeffery, H. M. B. A. Ep. (1877) (Sect.) 26-; QJ. Mth. 16 (1879) 65-, 348-. --- , triple foci. Jeffery, H. M. \& Cayley, A.] [1876] QJ. Mth. 14 (1877) Curves, 4th degree. Zahradnik, K. Prag Sb. (1873) 4th class. 310-. . Pittarelli, G. Nap. Ed. 24 (1885) --- , conies of triple contact with. Weyr, E. J. [1879] Wien Ak. Sb. 80 (1880) (Ab. 2) 1040-. 6th class. Vdlyi, G. Mth. Termt. Ets. 8 (1890)23-; 9(1891)18-; 10(1892)2-; Mth. Nt. B. Ung. 8 (1891) 69-; 9 (1892) 143- ; 10 (1893) 168-. Curves, 3rd and 4th degrees. Chasles, M. Quetelet Cor. Mth. 5 (1829) 231- ; C. E. 37 (1853) 272-. Kiipper, C. Prag Ab. 5 (1872) 19 pp. Zeuthen, H. G. (xn) Ts. Mth. 3 (1873) 97-; 4 (1874) 14-. Amanzio, D. G. Mt. 14 (1876) 48-. construction. Valentiner, H. N. Ts. Mth. 3 (B) (1892) 33-; Fschr. Mth. (1892) 599-. constructions, Chasles's. Brioschi, F. Tor- tolini A. 6 (1855) 380-. intersections of conies with. Bohn, K. Dresden Isis Sb. (1898) 24. rational. Ameseder, A. [1879] Wien Ak. Sb. 80 (Ab. 2) (1880) 487-. , absolute systems of elements on. Binder, W. Z. Mth. Ps. 36 (1891) 78-. , generation. Bienayme, A. N. A. Mth. 13 (1894) 144-. reduction of equations to line coordinates. Hesse, L. 0. Crelle J. 40 (1850) 316-. systems of tangents to. Walker, J. J. L. Mth. S. P. 3 (1869-71) 202-. Curves, 3rd and 5th degrees. interpolation by. Dupin, C. C. E. 25 (1847) 769. Brioschi, Fr. A. Mt. 7 (1876) 202-. Postnikov, M. V. Kazan S. Nt. (Ps.-Mth.) P. 4 (1886) 238 ; 7 (1889 118-) ; N. A. Mth. 13 (1894) 348-. bitangents. Hesse, L. 0. Crelle J. 49 (1855) 279-; 55 (1858) 83-. . Cayley, A. Phil. Trans. (1861) 357-. . Aronhold, . Berl. Mb. (1864) 499-. . Boch, G. [1864] Crelle J. 66 (1866) 97-. . Cayley, A. [1867] Crelle J. 68(1868)176-. . Geiser, C. F. Mth. A. 1 (1869) 129- ; Crelle J. 72 (1870) 370-. . Crone, C. (xn) Ts. Mth. 5 (1875) 161- ; 1 (1877) 97-, 151- ; (rx) Mth. A. 12 (1877) 561-. . Cayley, A. Crelle J. Mth. 94 (1883) 93-. . Frobenius, G. Crelle J. Mth. 99 (1886) 275-. . Hossfeld, L. Z. Mth. Ps. 31 (1886) 1-. . Ciani,E. Mil. I. Lomb. Ed. 31 (1898)310-. , algebraic solution. Aeschlimann, (Dr) U. Ziir. Vjschr. 25 (1880) 365-. , classification by. Zeuthen, H. G. C. E. 77 (1873) 270- ; Mth. A. 7 (1874) 410-. and contact-conies. Kohn, G. Wien Ak. Sb. 97 (1889) (Ab. 2a) 325-, 1381- ; Crelle J. Mth. 107 (1891) 1-. , double transformation of 3rd degree, first genus. Paolis,B.de. Em. E. Ac. Line. Mm. 2 (1878) 851-. , equation for point of contact. Freyberg, J. Mth. A. 17 (1880) 329-. , Galois group. Weber, H. Mth. A. 23 (1884) 489-. , grouping. Timerding, H. E. Crelle J. Mth. 122 (1900) 209-. , Hesse's method. Bosati, C. G. Mt. 38 (1900) 165-. , mutual relations. Steiner, J. C. E. 37 (1853) 121-. , properties relative to. Steiner, J. [1852] Crelle J. 49 (1855) 265-. studied through Kummer's configuration. Ciani, E. Mil. I. Lomb. Ed. 31 (1898) 312- ; A. Mt. 2 (1899) 53-. , and surface of 4th degree. Humbert, G. Liouv. J. Mth. 6 (1890) 423-. , systems of conies related to. Noether, M. Mth. A. 46 (1895) 545-. classified according to nature of infinite branches. Plilcker, J. Liouv. J. Mth. 1 (1836) 229-. shape. Cayley, A. [1864] Ph. Mg. 29 (1865) 105-. conditions for curve being repeated conic. Brioschi, Fr. Em. E. Ac. Line. At. 3 (1876) (Pte. 2) 91-. confocal, geometric generation. Berner, T. Z. Mth. Ps. 9 (1864) 369-. conic of 4-point contact. Zeuthen, H. G. (xn) Ts. Mth. 5 (1875) 190-. conies inscribed in. Humbert, G. Toul. Fac. Sc. A. 4 (1890) L, 8 pp. , intersections of which with quartic can be constructed by rule and compass. Crone, C. N. Ts. Mth. 7 (B) (1896) 81- ; Fschr. Mth. (1896) 441. 522 7610 Higher Plane Curves. Quart ics 7610 contact conies etc. Ameseder, A. [1882] Wien Ak. Sb. 86 (1883) (Ab. 2) 396-; 87 (1883) (Ab, 2) 15-. , Jacobian covariants of. Frobenius, G. Crelle J. Mth. 103 (1888) 139-. , relations between systems. Kohn, G. Mb.. Mth. Ps. 1 (1890) 71-, 129-. curves. Noether, M. Munch. Ak. Ab. 17 (1892) 103-. covariant S, theorem. Scorza, G. Mth. A. 52 (1899) 457-. , . Ciani, E. Palermo Cir. Mt. Ed. 14 (1900) 16-. determined by 14 points, description. Jon- quieres, E. de. Liouv. J. Mth. 1 (1856) 411- ; 2 (1857) 267-. with double point. Brill, A. Mth. A. 17 (1880) 103-, 517-. , bitangents. Brioschi, F. Mth. A. 4 (1871) 95-. , (Brioschi). Cremona, L. Mth. A. 4 (1871) 99-. , . Brill, A. [1872] Mth. A. 6 (1873) 66-. double points coinciding with those of Co- variant curves. Je/ery, H. M. [1889] QJ. Mth. 24 (1890) 250-. with 2 double points. Zeuthen, H. G. Kjob. Ov. (1879) 89-; (Resume, 15-). . Genese, R. W. B. A. Bp. (1881) 538. . Davis, E. W. (xn) J. H. Un. Cir. [2] (1883) 144. . Humbert, G. C. E. 97 (1883) 1287-. . Fries, J. de. N. Arch. Wisk. 14 (1888) 193- ; Fschr. Mth. (1887) 631. . Hjelmman, A.L. Helsingf . 6fv. 39 (1897) 222-. , analytical proof of properties. Story, W. E. [1881] (xn) J. H. Un. Cir. [1] (1882) 178. , classification. Jeffery, H. M. [1891] L. Mth. S. P. 23 (1892) 18-. , generation by conies. Jeffery, H. M. [1890] L. Mth. S. P. 21 (1891) 287-. , and graphical representation of elliptic functions. Cayley, A. Camb. Ph. S. T. 14 (1889) 484-. 3 double points. Cayley, A. Camb. and Dubl. Mth. J. 5 (1850) 148-. . Ameseder, A. Wien Ak. Sb. 79 (1879) (Ab. 2) 241-. . Ferrers, N. M. QJ. Mth. 18 (1882) 73-. . Schoute, P. H. Amst. Ak. Vs. M. 19 (1884) 420- ; Bll. Sc. Mth. As. 8 (1884) 278-. . Vanecek, J. S., & Vanecek, M. N. Prag Sb. (1885) (Mth.-Nt.) 96-, 223-. . Vidal, . [1885] I. Egypt. Bll. 6 (1886) 133-. . Motet, J. C. Mathesis 9 (1889) 89-. . Vries, J.de. [1899-1900] Amst. Ak. Vs. 7 (1899 340- ; Amst. Ak. P. 1 (1899) 263- ; Haarl. Ms. Teyl. Arch. 7 (1902) 1-. with 3 double points, bitangents. Durege, H. [1875] Wien Ak. Sb. 72 (1876) (Ab. 2) 495. , (Durege). Kantor, S. [1882] Wien Ak. Sb. 86 (1883) (Ab. 2) 1051-. , construction. Ameseder, A. Prag Sb. (1880) 3-. double tangential node and double point. Beyel, C. Ziir. Vjschr. 31 (1886) 178-. equation of 24th degree giving inflexions of general curve. Gerbaldi, F. Palermo Cir. Mt. Ed. 7 (1893) 178-. equations of form Za^f,-^, where ^x^. Schmidt, E. (xn) Ts. Mth. 5 (1881) 145-. , when one part is a double line. Zeuthtn, H. G. C. E. 75 (1872) 950-. generation, linear. Grassmann, H. [1851] Crelle J. 44 (1852) 1-. by 2 pencils of conies. Chasles, M. C. E. 37 (1853) 372-, 437-. ray pencil and ray system of 2nd class. Ameseder, A. Arch. Mth. Ps. 64 (1879) 109-. genus 1. Liebmann, H. Z. Mth. Ps. 41 (1896) 85-. , geometry. Fiedler, W. Ziir. Vjschr. 29 1884) 332-. 2. Bobek, K. Wien Ak. D. 53 (1887) (Ab. 2) 119-. , construction by ruler and compass. Bjarling, C. F. E. Stockh. Ofv. (1887) 19-. geometric theory. Cardinaal, J. Crelle J. Mth. 102 (1888) 160-. group of coordinated conies. Pascal, G. Em. E. Ac. Line. Ed. 1 (1892) (Sem. 2) 385-, 417- ; 2 (1893) (Sem. 1) 8-. Hessian with double point. Geiser, C. F. A. Mt. 9 (1878-79) 35-. intersections of curve with conic. Jung, G. Mil. I. Lomb. Ed. 8 (1875) 698-. 2 invariant curves. Ciani, E. A. Mt. 20 (1892-93) 257-. multiple tangents of the Cayleyan. Bertini, E. Tor. Ac. Sc. At. 32 (1896) 32-. pentagon not always inscribable. Liiroth, J. Mth. A. 13 (1878) 548-. penultimate. Cayley, A. Mess. Mth. 1 (1872) 178-. polar figures. Scorza, G. A. Mt. 2 (1899) 155-. polars. Clebsch, A. [1860] Crelle J. 59 (1861) 125-. problem. Jonquieres, E. de. N. A. Mth. 15 (1856) 370-. as projections of intersection of 2 conical surfaces. Hachette, J. N. P. Par. EC. Pol. Cor. 1 (1804-08) 368-. quadratic series of conies enveloping. Ciani, E. Mil. I. Lomb. Ed. 28 (1895) 659-. quadritangential conies. Ameseder, A. Wien Ak. Sb. 80 (1880) (Ab. 2) 187- ; 83 (1881) (Ab. 2) 829-. . Andoyer, . Toul. Fac. Sc. A. 3 (1889) D, 6 pp. , degeneration of the 63 systems. Fontene", G. Par. S. Mth. Bll. 27 (1899) 229-. 'quartan,' with 1 diameter. Newman, F. W. B. A. Ep. 42 (1872) (Sect.) 23-. 523 7610 Higher Plane Curves, Degrees 4-n 7610 ' quartan,' with 2 diameters. Newman, F. W. B. A. Bp. 41 (1871) (Sect.) 20-. , 3 diameters. Newman, F. W. B. A. Ep. 42 (1872) (Sect.) 22-. , or 4 diameters. Newman, F. W. B. A. Ep. 42 (1872) (Sect.) 23. rational. Weyr, Em. Wien Sb. 63 (1871) (Ab. 2) 493-. . Brill, A. Mth. A. 12 (1877) 90-. . Bobek,K. [1879] Wien Ak. Sb. 80 (1880) (Ab. 2) 361-. . Hossfeld, C. [1882] Z. Mth. Ps. 28 (1883) 296-. . Astor, A. N. A. Mth. 3 (1884) 181-. . Roberts, R. A. L. Mth. S. P. 16 (1884- 85) 44-. . Kohn, G. Wien Ak. Sb. 95 (1887) (Ab. 2) 318-. . Stahl, W. Crelle J. Mth. 101 (1887) 300- ; 104 (1889) 302-. , algebraic generation. Meyer, F. Mth. A. 31 (1888) 96-. , and associated conies. Roberts, R. A. L. Mth. S. P. 16 (1884-85) 9-. , bitangents. Binder, W. Z. Mth. Ps. 35 (1890) 25-. , double points. Nagel, K. [1881] Mth. A. 19 (1882) 433-. , with 2 double points and through 7 points. Dewulf, E., & Schoute, P. H. As. Fr. C. E. 8 (1879) 249-; Bll. Sc. Mth. As. 3 (1879) 383-. , . Dewulf, E. [1880] N. A. Mth. 20 (1881) 401-. , and new method of transformation. Leinekugel, G. N. A. Mth. 14 (1895) 391-. , system of tangential points. Binder, W. Z. Mth. Ps. 34 (1889) 272-. , with tangents at double point becoming inflexional tangents. Weyr, Em. Wien Sb. 67 (1873) (Ab. 2) 286-. , . Ameseder, A. Wien Ak. Sb. 79 (1879) (Ab. 2) 472-. reciprocal of cubic whose S invariant = 0. Mathews, G. B. L. Mth. S. P. 22 (1891) 173-. representation by 5 fourth powers. Lilroth, J. [1868] Mth. A. 1 (1869) 37-. , Clebsch's theorem, extension. Sylvester, J. J. C. E. 102 (1886) 1532-. singularities. Beer, A. Grunert Arch. 14 (1850) 318-. synthetic treatment. Milinowski, . Z. Mth. Ps. 23 (1878) 85-, 211-. with tacnode at infinity at which line at in- finity is multiple tangent. Jeffery, H. M. L. Mth. S. P. 13 (1881-82) 86-. triangle in- and circum- scribed to curve. Cayley, A. Ph. Mg. 30 (1865) 340-. with triple point. Vidal, . [1884] I. Egypt. Bll. 5 (1885) 102-. , construction and properties, fcazarski, M. Krk. Ak. (Mt.-Prz.) Ez. 15 (1887) 224-; Fschr. Mth. (1887) 631-. and 3rd class. Battaglini, G. Nap. Ed. 4 (1865) 399-. , generation. Frahn [Frahm], W. Z. Mth. Ps. 18 (1873) 363-. and 3rd class, generation by motion of a point. Siebeck, H. Crelle J. 66 (1866) 344-. , Steiner's curve, and Feuerbach's circle. Godt, W. [1896] Munch. Ak. Sb. 26 (1897) 119-. 4th class, and porism of in- and circum- polygon. Roberts, R. A. L. Mth. S. P. 23 (1892) 202-. Curves, 5th degree. bitangential of. Heal, W. E. [1889-91] A. Mth. 5 (1889-90) 33-; 6 (1891-92) 64-. bitangents. Maisano, G. Mth. A. 29 (1887) 431-. . Cayley, A. A. Mth. 5 (1889-90) 109-. and inflexional tangents. Maisano, G. Palermo Cir. Mt. Ed. 1 (1887) 86-. with 4 double points. Fries, J. de. [1894] Wien Ak. Sb. 104 (1895) (Ab. 2a) 46-. , involutions. Fries, J. de. [1894] Amst. Ak. Vs. 3 (1895) 115-. equation. Frattini, G. G. Mt. 16 (1878) 377. points of inflexion. Basset, A. B. B. A. Ep. (1900) 647-. rational, simple linear construction. Rohn, K. Mth. A. 25 (1885) 598-. Curves, 5th and 6th degrees. with multiple points, construction. Koehler, C. [1872] (x) Par. S. Mth. Bll. 1 (1873) 27-. Curves, 6th degree. in astronomy. Jonquieres, E. de. Tortolini A. 1 (1858) 112-. with 9 double points. Halphen, G. H. Par. S. Mth. Bll. 10 (1882) 162-. 2 triple points. Hjelmman, A. L. Helsingf. Ofv. 41 (1900) 26-. and 4th class, family. Longchamps, G. G. de. N. A. Mth. 5 (1866) 118-. Curves, 14t/i degree. cutting curve of 4th degree, transformation. Hesse, L. 0. Crelle J. 52 (1856) 97-. Curves, mth degree. C m , protective generation. Kilpper, C. Prag Sb. (1896) (Mth.-Nt.) No. 1, 16pp.; Mth. A. 48 (1897) 401-. Curves, nth degree. bitangents. Dersch, (Dr.) 0. Mth. A. 7 (1874) 497-. , number. Jacobi, C. G. J. Crelle J. 40 (1850) 237-. , (Jacobi). Rubini, R. Tortolini A. 2 (1851) 463-. _, _ (_). Clebsch, R. F. A. [1863] (vn) Crelle J. Mth. 63 (1864) 186-. with I (n - 3) double points. Clebsch, R. F. A . C. E. 60 (1865) 68-. Mth. A. 2 (1870) 515-. 524 7610 Higher Plane Curves. Degree n 7610 ... n-l . n-2 with 2 double points, expression of coordinates by parameter. Schivarz, C. H. A. Liouv. J. Mth. 6 (1880) 111-. and index N. Jonquieres, E. de. Liouv. J. Mth. 6 (1861) 113- ; 8 (1863) 71-. with multiple point of order n-l. Niewen- glowski, B. C. E. 80 (1875) 1067- ; N. A. Mth. 15 (1876) 126-. . Fouret,G. C. E. 80 (1875) 1158-. , construction with ruler, and reciprocal problem. Weyr, Em. Prag Sb. (1873) 198-. ri 2, new property. Bertini, E. Rm. E. Ac. Line. T. 1 (1877) 92-. number of points of maximum and minimum curvature in. Brunei, G. Bordeaux S. Sc. Mm. 3 (1893) 167-. needed to determine a curve. Machovec, F. 6asopis 16 (1887) 225- ; Fschr. Mth. (1887) 685. Curves, nth and pth degrees. intersections, theorem. Schoute, P. H. Amst. Ak. Vs. [1] (1893) 53-, 62-. Cusps. Schubert, F. T. [1818] St Pet. Ac. Sc. Mm. 8 (1822) 176-. Degeneration of curves into straight lines. Gordan, P. Mth. A. 45 (1894) 410-. Dependence, mutual, of curves, and general properties. Steiner, J. Liouv. J. Mth. 18 (1853) 309-. Desargues's theorem, generalisation. Saltel, L. Brux. Ac. Bll. 41 (1876) 594. Diameters. Wantzel, L. Par. S. Phlm. PV. (1847) 98-; Liouv. J. Mth. 14 (1849) 111-. . Fenwick, S. Mathematician 3 (1850) 156-. , bisecting. Breton [de Champ], P. N. A. Mth. 14 (1855) 7-. , properties. Chasles, M. C. E. 72 (1871) 794-. , theory. Lucas, F. Liouv. J. Mth. 8 (1863) 145-. Diametrical curves. Walton, W. Camb. Mth. J. 2 (1841) 76-. Directrices. Raabe, J. L. Crelle J. 2 (1827) 330-. Discontinuities in curves. Vogel, P. Z. Mth. Ps. 26 (1881) 391-. Discontinuous curves, geometry. Boidi, G. Bologna Opusc. Sc. 3 (1819) 276-. Double point of plane curve, resolution in 3- dimensional space, and related mechanical problem. Schlegel, S. F.V. Z. Mth. Ps. 28 (1883) 105-. points. Oppenheimer, H. Z. Mth. Ps. 41 (1896) 305-. , curve with 6. Godefroy, A. N. N. Arch. Wisk. 18 (1891) 151-. , group of curves with m 2 . Fries, J. de. Amst. Ak. Vs. 3 (1895) 139-. Elements at infinity in geometric figures, deter- mination. Weyr, E. J. [1873] (xi) Bll. Sc. Mth. As. 6 (1874) 93-. Elimination theory, with application to curves. Saltel, L. N. A. Mth. 20 (1881) 546-; 2 (1883) 554-. Elliptic curves, equation giving points of in- flexion. Brioschi, F. Mil. I. Lomb. Ed. 2 (1869) 559-. Envelope of diameters. Walker, J. J. L. Mth. S. P. 10 (1878-79) 180-. -- family of curves. Landre, C. L. N. Arch. Wisk. 5 (*1879) 205-. Envelopes. Weill, . N. A. Mth. 3 (1884) 376-. Equation, algebraic, curves whose intersections give imaginary roots of. Sprague, T. B. [1882] Edinb. E. S. T. 30 (1883) 467-. in point coordinates obtained from tangential equation. Turquan, . N. A. Mth. 4 (1845) 410-. of tangent. Purkiss, H. J. Mess. Mth. 3 (1866) 19-. --- , magical. Hiern, W. P. OJ. Mth. 6 (1863) 31-. , tangential polar, of a curve. Ferrers, N. M. QJ. Mth. 1 (1857) 210-. 2 fSi = (X) . application. Beltrami, E. BlognaAc. Sc. Mm. 10 (1879) 233-. Equations of curve in polar coordinates, number for same axis. Forestier, C. Toul. Ac. Sc. Mm. 1 (1879) 250-. , intrinsic, of curves. Onnen, H. N. Arch. Wisk. 1 (*1875) 1-; 5 (*1879) 1-; (x) Arch. Neerl. 14 (1879) 1-. , , -- , especially cycloids. Onnen, H. N. Arch. Wisk. 4 (*1878) 30-. .higher, construction. Hill,C.J.D. Stockh. Ofv. 19 (1862) 307-. , , geometric representation of numerical system. Spitz er, S. Wien SB. 6 (1851) 152-; 7 (1851) 455-. Equidistant curves. Frolov, (le gen.) M. As. Fr. C. E. (1894) (Pt. 2) 235-. Focal and normal properties. Laguerre, (le cap.) . C. E. 60 (1865) 70-. Foci. Plucker, J. [1832] Crelle J. 10 (1833) 84-. . Roguet, C. N. A. Mth. 1 (1842) 131-. . Vachette, A. N. A. Mth. 1 (1842) 417-. . Laguerre-Verly, E., & Sacchi, J. N. A. Mth. 12 (1853) 225-. . Quetelet, E. Brux. Mm. Cour. 4, 26 (1854-55) 44 pp. . Roberts, S. QJ. Mth. 2 (1858) 195-. . Terquem, O. N. A. Mth. 18 (1859) 399-. . Gibert, E., & Niewenglowski, B. C. E. 82 (1876) 913-. . Amigues, E. N. A. Mth. 11 (1892) 163-. of curves and focals of cones. Laguerre, E. N. A. Mth. 18 (1879) 57-. , new analytic projective treatment. Siebeck, H. Crelle J. 64 (1865) 175-. Generation of curves. Terquem, 0. N. A. Mth. 12 (1853) 220-. --- . Jonquieres, E. de. [1857] Par. Mm Sav. Etr. 16 (1862) 159-. 525 7610 Higher Algebraic Plane Curves Chasles, 7610 Generation of curves (de Jonquieres) . M. C. B. 45 (1857) 318-. . H&rtenberger,G. Crelle J. 58 (1860) 54-. . Olivier, A. [1869] Crelle J. 71 (1870) 1-. . Vanecek, J. S., & Vanetek, M. N. Prag Sb. (1885) (Mth.-Nt.) 202-. the curves determined by unknown points of intersection of given curves. Olivier, A. Z. Mth. Ps. 14 (1869) 209-. curves, linear. Grassmann, H. Crelle J. 42 (1852) 187-. by many valued elements. Weyr, Em. [1870] Prag Ab. 4 (1871) 32 pp. _, mechanical. Schlegel, V. [1872] Mth. A. 6 (1873) 321-. , projective. Bobek, K. Mth. A. 25 (1885) 448-. by projective involutions. Weyr, Em. [1870] Mth. A. 3 (1871) 34-. __ projective curved configurations. Mili- nowski, . Z. Mth. Ps. 18 (1873) 288-. straight lines by curves. Steen, A. Mth. Ts. 4 (1862) 123-. Geometrical transformation (pseudo-Newton- ian). Brocard,H. Liege S. Sc. Mm. 1 (1899) No. 3, 24 pp. (_). Retail, V. Liege S. Sc. Mm. 2 (1900) No. 4, 24 pp. Hessian. Chrystal, . [1885] Edinb. E. S. T. 32 (1887) 645-. and Cayleyan of curve, common tangents. Maisano, G. [1885] Palermo Cir. Mt. Bd. I (1887) 46-. curve. Brill, A. [1877] Mth. A. 13 (1878) 175-. . Pezzo, P. del. Nap. Ed. 22 (1883) 203-. , geometric treatment. Hotter, E. Mth. A. 34 (1889) 123-. and Steinerian. Steiner, J. [1848] Crelle J. 47 (1854) 1-. (Steiner). Clebsch, E. F. A. [1864] Crelle J. 64 (1865) 288-. Hessians, Steinerians, and Cayleyans, curves analogous to. Maisano, G. Palermo Cir. Mt. Ed. 1 (1887) 66-. and Steinerians of higher orders in 1 -dimen- sional geometry. Newson, H. B. A. Mth. II (1896-97) 121-. Intersection of curve with straight line. Routh, E. J. QJ. Mth. 24 (1890) 257-. curves. Burrow, R. As. Eesearches 1 (1788) 330-. . Cayley, A. Camb. Mth. J. 3 (1843) 211- ; Mth. A. 30 (1887) 85-. 2 curves. Painvin, L. Bll. Sc. Mth. As. 5 (1873) 138-. in a singular point. Halphen,G.H. Par. Sc. Mth. Bll. 3 (1875) 76-. , curvilinear, theory of loci. Walton, W. Camb. Mth. J. 2 (1841) 85-. of lines with curves of odd degree. Dawson, H. G. Mess. Mth. 19 (1890) 108-. n tangents where radius vector cuts curve C^, problem of Steiner. Schoute, P. H. Bll. Sc. Mth. 10 (1886) 242-. 526 Intersection of n tangents where radius vector cuts curve C^, problem of Steiner. Zeuthen, H. G. Bll.lte. Mth. 11 (1887) 82-. Intersections of curves, theorems. Valentiner, H. (xn) Ts. Mth. 5 (1881) 1-. , of Humbert and Fouret. Borel, E. N. A. Mth. 9 (1890) 123-. 2 curves. Franklin, F. Am. J. Mth. 3 (1880) 192. , theorem of Jacobi, proof. Peter sen, J. P. C. (xn) Ts. Mth. 2 (1878) 14-. Inversion, classification of curves with reference to. Johnson, W. W. Des Moines Anal. 4 (1877) 42-. Kinematic determination of curves. Ocagne, M. d'. N. A. Mth. 1 (1882) 40-; 2 (1883) 189-. geometry, applications. Mannheim, A. Prag Sb. (1888) (Mth.-Nt.) 87-. Least chord through given point. Emsmann, H. Grunert Arch. 23 (1854) 460-. Loci, applications of law of decomposition. Saltel, L. Brux. Ac. Bll. 42 (1876) 586-. , degree and class. Halphen, G. H. C. E. 83 (1876) 705-. and envelopes. Chasles, M. C. E. 82 (1876) 1463- ; 85 (1877) 362-. , theorem of Chasles, applications. Serret, P. C. E. 86 (1878) 116-. , general law. Saltel, L. Brux. Ac. Bll. 41 (1876) 595-. , node- and cusp-, enveloped by tangents at the cusps. Hill, M. J. M. L. Mth. S. P. 22 (1891) 216-. , orthoptic, degree. Taylor, C. Mess. Mth. 16 (1887) 1-. , and isoptic, degree. Taylor, C. E. S. P. 37 (1884) 138-. , various. Brasseur, J. B. Brux. Ac. Bll. 13 (1846) 311- ; 18 (1851) 372-; Brux. Mm. Cour. 4, 21 (1846) 26 pp. Locus of contact points of net of curves touching fixed curve. Faure, H. N. A. Mth. 18 (1859) 237-. , degree k, of curves of pencil with those of o> * linear system. Franchis, M. de. Palermo Cir. Mt. Ed. 10 (1896) 118- ; 11 (1897) 12-. defined by algebraic conditions, method of determining degree. Saltel, L. Brux. Ac. Bll. 42 (1876) 300-. , number of double points. Saltel, L. C. E. 88 (1879) 329-. , determination of degree without calculation. Saltel, L. [1874] Brux. Mm. Cour. 8, 24 (1875) No. 5, 33 pp. of harmonic centres on radii vectores. Palatini, F. G. Mt. 33 (1895) 210-. intersection of tangents at ends of chord of constant length in a conic. Schlomilch, 0. Z. Mth. Ps. 14 (1869) 158-. point in chord of constant length in ellipse. Enneper,A. Schlomilch Z. 7 (1862) 200-. points whose distances from curves fulfil given relation, symbolic formula for degree. Fouret, G. G. E. 83 (1876) 605-. 7610 Higher Algebraic Plane Curves 7610 Locus of points, tangents from which to 2 curves are equal. Laguerre, E. [1876] Par. S. Mth. Bll. 5 (1877) 25-. of 3 -point contact of curve pencil with curve net. Bagnera, G. Palermo Cir. Mt. Ed. 10 (1896) 81-. problem. Gergonne, J. D. Gergonne A. Mth. 8 (1817-18) 312-. . Schmitz, P. J. Louvain A. Ac. 7 (1825) 44 pp. of projections of point on tangents to curve. Eichard, . N. A. Mth. 2 (1843) 436-. vertex of constant angle circumscribed to curve of nih class. Jonquieres, E. de. N. A. Mth. 20 (1861) 206-. yx 2 +bx + c=0. Bergis, . N. A. Mth. 17 (1858) 118-. Maclaurin's " Descriptio Linearum Curvarum." Vanecek, J. S., & Vanecek, M. N. Prag Sb. (1884) (Mth.-Nt.) 374-. method for conies, curves obtained by exten- sion of . Roberts, S. L. Mth. S. P. 14 (1882- 83) 110-. Mechanical and geometrical relations of curves. Bardelli, G. Mil. I. Lomb. Ed. 13 (1880) 56-. Mechanics, principles applied to properties of curves. Fabri, R. Em. At. 6 (1852-53) 73-. Metrical geometry, general theorems. Michel, C. N. A. Mth. 19 (1900) 169-. properties. Humbert, G. N. A. Mth. 6 (1887) 526-. , synthetic methods for study of. Hoist, E. Christiania F. (1882) No. 11, 113pp.; (1883) No. 13, 6 pp. theorem in conies, and its generalisation. Schwarz, A. Mh. Mth. Ps. 9 (1898) 346-. theorems. Hoist, E. [1876] Mth. A. 11 (1877) 341-, 575. Motion on curves with separate branches, relation to motion on one branched curve. Staude, . Dorpat Sb. 8 (1889) 336-. Multiple points. Walton, W. Camb. Mth. J. 2 (1841) 155- . . Padula, F. Tortolini A. 3 (1852) 211-. , curves with 3. Jonquieres, E. de. Tortolini A. 1 (1858) 110-. , determination of curves with. Chasles, M. C. E. 62 (1866) 1354-. and equal roots, theorem. Sylvester, J. J. Ph. Mg. 3 (1852) 375-. at infinity. Painvin, L. N. A. Mth. 3 (1864) 145-, 193-, 241-. , curves with 2. Michel, F. As. Fr. C. E. (1900) (Ft. 2) 250-. , number of, and rational curves. Pellet, A.E. N. A. Mth. 20 (1881) 444-. Multiplicity, new theory. Sylvester, J. J. Ph. Mg. 3 (1852) 460-. New properties. Mannheim, A. Par. S. Mth. Bll. 4 (1876) 158-. Newton's organic description of curves. Taylor, G. Camb. Ph. S. P. 3 (1880) 381-. theorem , extension and applications . Gob , A . Brux. Ac. Bll. 28 (1894) 57-. , various consequences. Demoulin, A. Brux. Mm. Cour. 8, 45 (1891) No. 2, 18 Node-cusp. Cayley,A. (i) QJ. Mth. 6 "' 74-. Normal, construction, in certain mode of genera- tion of curves. Ocagne, M. d\ N. A. Mth. 20 (1881) 197-. Normals. Backlund, A. V. Lund Acta Un. 6 (1869) (Mth.) No. 2, 38pp. , curve through feet of, theorem of Steiner. Dewulf, . N. A. Mth. 18 (1859) 174-. to curves, and evolutes. Steiner, J. Crelle J. 49 (1855) 333-. , generalisation of theory by substituting for each normal a pencil of straight lines. C. E. 80 (1875) 505-. Collignan, E. N. A. Mth. 19 Dewulf, . N. A. Mth. 16 Chasles, M. , problems. (1900) 433-. , theorem. (1857) 464. , . Birkeland, K. Mh. Mth. Ps. 1 (1890) Number of points common to 2 curves. Saltel, L. Brux. Ac. Bll. 43 (1877) 266-. determining curves. Plilcker, J. Gergonne A. Mth. 19 (1828-29) 97-. Obliques of curve, theorems. Chasles, M. C. E. 74 (1872) 1277-. Parallel of curve, degree and class. Leaute, H. [1875] Toul. Ac. Sc. Mm. 8 (1876) 59-. , singularities. Roberts, S. L. Mth. S. P. 3 (1869-71) 209-. Penultimate forms of curves. Cayley, A. C. E. 74 (1872) 708-. Perspectives, skew, of curves. Halphen, G. H. C. E. 80 (1875) 638-. Plane or skew curves whose points can be determined individually. Chasles, M. C. E. 62 (1866) 579-. Point cluster of 7ith degree with base on curve of mth degree. Chasles, M. C. E. 45 (1857) 393-. Points, collinear. Bellavitis, G. Yen. Mm. I. 8 (1859) 161-. , conjugate, theory. Young, J. R. Ph. Mg. 27 (1845) 91-. of inflexion. Eliot, Z. Bll. Sc. Mth. As. 2 (1878) 216-. , curves with. Bammert, (Dr) . Z. Mth. Ps. 10 (1865) 165-. of curves in polar coordinates. Isely, J. P. Neuch. S. Sc. Bll. 18 (1890) 56-. , Hesse's equation. Lemonnier, H. N. A. Mth. 9 (1870) 531-. , . Cretin, . N. A. Mth. 20 (1881) 131-. and other points. Cayley, A. Crelle J. 34 (1847) 30-. intersection of 2 curves, multiplicity. Stolz, O. Innsb. Nt. Md. B. 8 (1879) (Heft 3) vii-; Mth. A. 15 (1879) 122-. and tangents, construction. Zeuthen, H. G. Mth. Ts. 3 (1861) 12-. Polar curves, general theory. Paolis, R. de. Em. E. Ac. Line. Mm. 3 (1886) 265-. , theorem, proof. Jonquieres, E. de. N. A. Mth. 16 (1857) 347-. properties, geometric derivation. Schur, F. Z. Mth. Ps. 22 (1877) 220-. reciprocal of curve, degree. Gergonne, J. D. Gergonne A. Mth. 19 (1828-29) 218-. 527 7610 Higher Algebraic Plane Curves 7610 Polar reciprocals, theory. Lentheric, . N. A. Mth. 8 (1849) 252-. theory. Schwarz, A. Wien Ak. Sb. 95 (1887) (Ab. 2) 42-. Polars. Gaertner, E. Arch. Mth. Ps. 7 (1889) 180-. , 1st, condition for 2 curves of same degree having same. Bertini, E. Tor. Ac. Sc. At. 33 (1897) 23-. .generalised. Clifford, W. K. [1868] L. Mth. S. P. 2 (1869) 116-. , inclined, theory. Dewulf,E. Bll. Sc.Mth. As. 2 (1878) 41-, 372-. with respect to plane curves, construction. Andreev, K. A. (xn) Kharkov Mth. S. Com. (1879) 51-. , successive, theory. Le Paige, C. M. M. H. H, Brux. Ac. Bll. 1 (1881) 134-. Poles of inversion, principal, curve having a geometric locus of. Fouret, G. N. A. Mth. 2 (1883) 259-. and polars. Jonquieres, E. de. Liouv. J. Mth. 2 (1857) 249-. . Fiedler, W. Schlomilch Z. 4 (1859) 91-. . Polar reciprocals. Hunyady, J. (xn) Mag. Tud. Ak. Etk. (Mth.) [1] (No. 2) (1867) 30 pp. Polygons in- or circum-scribed to curves. Chasles, M. C. B. 78 (1874) 922-. and circum-scribed to same curve. Weill, M. Par. S. Mth. Bll. 10 (1882) 127-. , inscribed, 2 conjugate, generalisation of Pappus's theorem. Puzyna, J. Krk. Ak. (Mt.-Prz.) Pam. 17 (1890) 24- ; Ore. Ac. Sc. Bll. (1889) No. 4, xxxvii-. , , of maximum area. Hathaway, A. S. (xn) J. H. Un. Cir. [2] (1883) 48. , porismatic. Brioschi, F. Tortolini A. 8 (1857) 119-. , Steiner's, generalisation. August, F. [1875] Arch. Mth. Ps. 59 (1876) 1-. Porismatic problems, especially polygons in curves. Hurwitz, A. [1878] Mth. A. 15 (1879) 8-. "Power" of point with respect to curve. Petersen, J. P. C. (xn) Ts. Mth. 5 (1869) 106-. Projective metrics. Ovidio, E. d'. Tor. Ac. Sc. At. 28 (1893) 566-. properties. Juel, C. [1899] Kj0b. Dn. Vd. Selsk. Skr. 10 (1899-1902) 1-. of curves and surfaces. Jouquieres, E. de. Crelle J. 66 (1866) 289-. , application of transversals. Poncelet, J. V. Crelle J. 8 (1832) 21-, 117-, 213-, 370-. elements of curves. Scholz,P. Z.Mth. Ps. 13 (1868) 267-, 355-. Quadrilaterals with their 6 vertices on curve of 3rd class. Weill, . N. A. Mth. 3 (1884) 401-. Radii vectores, generalisation of theorems ex- pressed in. Chasles, M. C. B. 82 (1876) 1463-. Bational curves. Weyr, Em. Z. Mth. Ps. 16 (1871) 80- ; Prag Sb. (1872) (pt. 2) 9-. . Painvin, L. C. B. 78 (1874) 1194-. Bational curves. Pasch, M. Mth. A. 18 (1881) 91-. . Humbert, G. Par. S. Mth. Bll. 13 (1885) 49-, 89-. . Marin, E. Arg. S. Ci. A. 32 (1891) 15-. . Laisant, C. A. As. Fr. C. B. (1892) (Pt. 2) 25-. . Houston, W. A. Mess. Mth. 28 (1899) 187-. , class. Mansion, P. (xn) Ts. Mth. 6 (1876) 64-. , . Darboux, G. C. B. 94 (1882) 930-. , construction by points and tangents. Schoute, P. H. N. Arch. Wisk. 12 (1886) 1-; Arch. Neerl. 20 (1886) 49-. , degeneration. Bissing, G. [1883] (xn) J. H. Un. Cir. [3] (1884) 58. of even degree, generation by means of projective series of circles and spheres. Tognoli, 0. G. Mt. 12 (1874) 161-. , double points and tangents. Weltzien, C. Mth. A. 26 (1886) 516-. 1 generation. Stahl, W. Mth. A. 38 (1891) 561-. , , new mode. Fouret, G. C. B. 101 (1885) 1241-. , , theorems, two. Tognoli, O. G. Mt. 12 (1874) 136-. , geometric constructions for, especially quintic with 5 given double points. Jon- quieres, E. de. Palermo Cir. Mt. Ed. 2 (1888) 118-. with imaginary double points, special class. Fuchs, L. Berl. Ak. Sb. (1900) 74-. , linear constructions. Vanecek, M. N. Prag Sb. (1882) 75-. for which multiple points can be assigned arbitrarily. Bertini, E. G.Mt.l5(1877)329-. , and new algebraic theorem. Garbieri, G. G. Mt. 16 (1878) 1-, 108-. , plane and skew. Eoberts, E.A. L.Mth. S. P. 17 (1887) 25-. , , in- and circum-polygons. Zecca, G. G. Mt. 25 (1887) 333-. , systems. Story, W. E. J. H. Un. Cir. [6] (1886-87) 77. , theorem, proof. Lttroth, J. [1875] Mth. A. 9 (1876) 163-. , theory, application. Hermite,C. L.Mth. S. P. 4 (1871-73) 343-. treated by inversion. Newson, H. B. [1892] Kan. Un. Q. 1 (1893) 47-. , 3rd class. Laguerre, E. [1877] Par. S. Mth. Bll. 6 (1878) 54-. Beciprocal of curve, equation. Salmon, G. Crelle J. 42 (1852) 277-. curves of foci. Bjorling, C. F. E. Stockh. Ofv. 33 (1876) No. 1, 17-. Eeciprocity in plane, and application to curves. Beyel, C. Ziir. Vjschr. 31 161-. Eeducible curves. Noether, M. Erlang. Ps. Md. S. Sb. 17 (1885) 13-; Acta Mth. 8 (1886) 161-. Eeflexions of points on circles, or inversion of normal problem. Eollner, F. [1880] Wien Ak. Sb. 82 (1881) (Ab. 2) 1129-. 528 7610 Higher Algebraic Plane Curves 7610 Secants of curves, sum of squares. Minding, E.F. A. Crelle J. 11 (1834) 20-. , Maclaurin's theorem. Malet, J. C. Ir. Ac. T. 28 (*1880-86) 759-. "Sectorial" curves. Jezek, O. Prag Sb. (1882) 117- ; (1883) (Mth. Vortr.) 116-. Segments of lines by curves. Ladd, (Miss) C. Am. J. Mth. 4 (1881) 272. on tangents and normals, loci derived from, and enumerative theorems. Chasles, M. C. B. 81 (1875) 253-, 355-, 643-, 993-, 1221- ; 82 (1876) 1399- ; 83 (1876) 97-, 467-, 495-, 519-, 589-, 641-, 757-, 867-, 1123-, 1195- . Series of points given by recurring series, and connected curves. Segar, H. W. Mess. Mth. 22 (1893) 161-. Singular points. Poisson, S. D. Par. EC. Pol. J. 14 e call. (1808) 130-. . Briot, C. Liouv. J. Mth. 10 (1845) 368-. . Choquet, C. N. A. Mth. 9 (1850) 260-. . De Morgan, A. [1855] (ix) Camb. Ph. S. P. 1 (1866) 155-. . Boussinesq, J. Les Mondes 45 (1878) 575-. . Halphen, G. H. Par. Mm. Sav. Etr. 26 (1879) No. 2, 112 pp. . Biehler, C. N. A. Mth. 19 (1880) 492-; 20 (1881) 97-, 489-, 537-. of algebraic functions and curves. Stolz, 0. Mth. A. 8 (1875) 415-. curve, and singular systems of values of algebraic function. Noether, M. Mth. A. 9 (1876) 166-. , new, discovered by Plateau. Mansion, P. (xn) Mathesis 3 (1883) 193-. Singularities of curves. Clebsch, R.F.A. [1864] Crelle J. 64 (1865) 98-. . Kostlin,W. Z. Mth. Ps. 41 (1896)1-. , analytical determination. Brioschi, Fr. Bm. B. Ac. Line. At. 2 (1875) xliv-. , discriminants and resultants of equa- tions for. Meyer, F. Mth. A. 38 (1891) 369-. , Pliicker's first equation connecting. Knott, C. G. Edinb. Mth. S. P. 1 (1894) 130-. , higher. Cayley, A. Crelle J. 64 (1865) 369-; QJ. Mth. 7 (1866) 212-. , . La Gournerie, J. de. Liouv. J. Mth. 14 (1869) 425- ; 15 (1870) 1-. , . Scott, (Miss) C. A. Am. J. Mth. 14 (1892) 301-. , , equivalents. Bjorling, C. F. E. Stockh. Ofv. 35 (1878) No. 7, 33- ; Ups. S. Sc. N. Acta 10 (1879) (No. 17) 25 pp. , , (Bjorling). Zeuthen, H. G. (xn) Ts. Mth. 3 (1879) 74-. Steiner's, Cayley's and other covariant curves, behaviour at singular points of original curve. Wdlffing, E. Z. Mth. Ps. 40 (1895) 31-. Straight lines cutting plane curve at same angle. Fouret, G. Palermo Cir. Mt. Bd. 5 (1891) 75-. Supplementary curves, obtained by "ideal chords." Finlay, R. [1853] Manch. Ph. S. Mm. 11 (1854) 169-. System of 2 curves, properties. Demoulin, A. Brux. Ac. Bll. 23 (1892) 527-. Tangent to certain curves, construction. Lisle- ferine, H. de. (xn) Mathesis 3 (1883) 206-. constructions. Hurwitz, A. Mth. A. 22 (1883) 230-. to curve, segments intercepted on line by it and a parallel. Cremona, L. Tortolini A. 3 (1860) 169-. property, curve determined from. Unfer- dinger, F. Arch. Mth. Ps. 52 (1871) 361-. , . Brocard, H. [1875] Par. S. Mth. Bll. 4 (1876) 42-. Tangents. Bret, . Gergonne A. Mth. 8 (1817-18) 285-. . Reid, J. Mathematician 2 (1847) 83. . Bischo/, J. N. Crelle J. 56 (1859) 166-. . Gundelfinger, S. [1870] Crelle J. 73 (1871) 171-. . Woodward, C. M. Am. As. P. (1886) 72-. and centres of curvature of curves, construc- tion. Machovec, F. Prag Sb. (1888) (Mth.- Nt.) 169-. , . Sauve, A. Bm. N. Line. Mm. 14 (1898) 57-. to certain curves, construction. Wiegers, C. Grunert Arch. 33 (1859) 166-. , conjugate. Hoza, F. Arch. Mth. Ps. 61 (1877) 218-. to curve from multiple point. Caporali, E. Nap. Bd. 20 (1881) 143-. 1 point, application of equation of squares of differences of roots. Joachimsthal, F. N. A. Mth. 9 (1850) 98-. , determination. Colard, H. N. A. Mth. 1 (1842) 268-. , formula. Joachimsthal, F. N. A. Mth. 12 (1853) 323-. , general and analytical method of drawing. Gruson, J. P. Berl. Ab. (1820-21) 39-. to polar curves. Rispal, A. - N. A. Mth. 2 (1843) 511-. projections of curves, construction. Binet, J. P. M. Par. EC. Pol. Cor. 3 (1814- 16) 199-. , . Chasles, M. Liouv. J. Mth. 2 (1837) 293-. and radii of curvature, new graphical con- struction. Chasles, M. [1833] Quetelet Cor. Mth. 8 (1834) 58-. , Boberval's method for. Geer, P. van. N. Arch. Wisk. 11 (1884) 28-; Fschr. Ps. (1884) (Ab. 1) 217-. and tangent planes, parallel, theorem of Chasles ; point of mean distance. Terquem, O. N. A. Mth. 4 (1845) 153-, 178-. , theory. Doedes, L. Utr. A. Ac. (1829-30) 149 pp. , n (n - 1), from a point to curve of nth degree. Zeuthen, H. G. (xn) Ts. Mth. 2 (1872) 65-. Theorems of Maclaurin and Jacobi. Terquem, 0. N. A. Mth. 9 (1850) 440-. , various. Ascoli, G. G. Mt. 5 (1867) 358-, 365-. Tracing of curve from its equation, asymptotes, etc. Biehler, C. N. A. Mth. 2 (1883) 354-, 397-. VOL. I. 529 LL 7630 Special Plane Algebraic Curves. Cartesians 7630 Transversal, cotangent, theorem. Hunyody, E. von. [1865] Z. Mth. Ps. 11 (1866) 77-. directions corresponding to axes of conies. Steplianos, G. Par. S. Mth. Bll. 9 (1881) 49-. theory for plane curves. Gundelfinger, S. Z. Mth. Ps. 19 (1874) 68-. Transversals. Steiner, J. Berl. B. (1851) 302- ; Crelle J. 47 (1854) 7-. Triangle, generalised form of analytical. Story, W. E. [1879] (xn) J. H. Un. Cir. [1] (1882) 34. Triangles, 2, circumscribing conic, theorem ex- tended to j^gons. Dixon, A. G. QJ. Mth. 26 (1893) 212-. Undulations of curves C^. Binder, W. Wien Ak. Sb. 106 (1897) (Ab. 2a) 295-; 107 (1898) (Ab. 2a) 23-. (in, n) homographies, and double elements. Weyr, Em. Prag Sb. (1869) (pt. 2) 33-. 7630 Special plane algebraic curves. (See also 8030.) Wesely, J. Arch. Mth. Ps. 9 (1890) 420-. Aberrancy curve of plane cubic. Mukho- padhyay, A. Beng. As. S. J. 59 (Pt. 2) (1891) 61-. Anallagmatic curves. Darboux, G. Par. EC. Norm. A. 4 (1867) 81-. . Picquet, H. As. Fr. C. E. (1878) 95- ; C. R. 87 (1878) 460-. . Laquiere, E. M. As. Fr. C. R. (1880) 121-. Aplanatic lines. Chasles, M. Quetelet Cor. Mth. 5 (1829) 116-, 188-. , etc. Quetelet, L. A. J. Quetelet Cor. Mth. 5 (1829) 190-. Araneids, division of angles by. Heymann, W. Z. Mth. Ps. 44 (1899) 263-. Astroids derived from conies. Ameseder, A. Arch. Mth. Ps. 64 (1879) 177-. " Atriphthaloid," r 2 (r - h) + fc 2 see* u = 0. Townsend, (Rev.) R. [1882] Ir. Ac. P. 4 (*1884-88) 62-. Ballistic quatrefoil, (x* + y 2 ) 3 =c 2 x' 2 y*. Horta, F. Lisb. A. 1 (1857) 525-. Barycentrides, locus of centroid of variable arc of curve. Borgnet, . Rouen Ac. Tr. (1839) 113-. Cardioid. .Purkiss, H. J. (v) Mess. Mth. 2 (1864) 241-. . Laguerre, E. N. A. Mth. 17 (1878) 55-. . Weill, M. N. A. Mth. 20 (1881) 160-. , connection between pole and centroid of triangle of contact. Zahradnik, K. Prag Sb. (1877) 188-. , generation by 2 unequal circles. Brandli, H. Bern Mt. (1858) 68-. , involutory angular relations. Weyr, Em. Prag Sb. (1871) (pt. 2) 69-. , locus of intersection of perpendicular tan- gents. Wolstenholme, J. L. Mth. S. P. 4 (1871-73) 327-. and Maclaurin's trisectrix, relation between. Longchamps, G. de. Prag Sb. (1887) (Mth.- Nt.) 601-. Cardioid, and its mechanical construction. Pleyl, J. Arch. Mth. Ps. 68 (1882) 166-. , metrical property. Weyr, Em. Mil. I. Lomb. Rd. 5 (1872) 204-. , pole of constant triangle of contact. Zahradnik, K. Prag Sb. (1877) 184-. , relation to ellipse. Jopling, J. R. S. P. 6 (1852) 267-. , theory. Zahradnik, K. Prag Sb. (1874) 180- ; Arch. Mth. Ps. 59 (1876) 337-. Cartesian ovals. Genocchi, A. N. A. Mth. 14 (1855) 202-, 260-. Cayley, A. QJ. Mth. 1 (1857) 320-. Strebor [Roberts, W.]. N. A. Mth. 17 (1858) 234-. Roberts, S. [1870] L. Mth. S. P. 3 (1869-71) 106-. Tait, P. G. [1871] Edinb. R. S. P. 7 (1872) 434-. Cayley, A. QJ. Mth. 12 (1873) 16-. Haentzschel, E. Arch. Mth. Ps. 69 (1883) 395-. Dixon, A. C. QJ. Mth. 28 (1896) 375-. bibliography. Liguine, V. Bll. Sc. Mth. As. 6 (1882) (Pt. 1) 40-. bipolar equations. Johnson, W. W. Des Moines Anal. 2 (1875) 106-. confocal, correspondence with right lines of hyperboloid. Cayley, A. Mess. Mth. 18 (1889) 128-. construction. Cayley, A. QJ. Mth. 15 (1878) 34. generation, Chasles's method. Ocagne, M. d'. C. R. 97 (1883) 1424-. and hyperboloid of one sheet. Fries, J. de. Amst. Ak. Vs. 4 (1896) 252-; Fschr. Mth. (1896) 491. with 2 imaginary axial foci. Cayley, A. [1870] L.Mth. S. P. 3 (1869-71) 181-. mechanical description. Cayley, A. L. Mth. S. P. 6 (1875) 81- ; QJ. Mth. 13 (1875) 328-. . Hammond, J. Am. J. Mth. 1 (1878) 283. by string. Zeuthen,H. G. (xn) Ts.Mth. 6 (1882) 145-. and other focal curves, mechanical description. Maxwell, J. C. [1846] Edinb. R. S. P. 2 (1851) 89-. quartics, conjugate functions. Greenhill, A. G. [1881] Camb. Ph. S. P. 4 (1883) 77-. properties. Crofton, M. W. L. Mth. S. P. 1 (1866) M. 6, 5-. regarded as conic intersections. Merrifield, C. W. Mess. Mth. 4 (1875) 148. projected intersections of quadric surfaces. Merrifield, C. W. Mess. Mth. 3 (1874) 141-. and related curves as sections of the anchor- ring. Brown, W. V. A. Mth. 6 (1891-92) 161-. system through 4 points on circle. Roberts, R. A. L. Mth. S. P. 12 (1880-81) 200-. triple focus. Engberg, C. C. [1898] A. Mth. 12 (1898-99) 54-. Cassini's oval. Zeuthen, H. G. Mth. Ts. 5 (1863) 157-. . Cayley, A. Mess. Mth. 4 (1875) 187-. 530 7630 Special Plane Algebraic Curves. Cublcs 7630 Cassini's oval. Hart, H. Mess. Mth. 6 (1877) 169-. . MacMahon, P. A. [1882] Mess. Mth. 12 (1883) 118-. derived from section of rotation surface. Seidl, A. Wien SB. 18 (1855) 311-. ovals. Huber, G. Bern Mt. (1888) 93-. . Bigler, U. Arch. Mth. Ps. 7 (1889) 311-. , centre of curvature. Friedrich, J. Casopis 27 (1898) 96-; Fschr. Mth. (1898) 511. and lemniscates, generation. Garlin, . N. A. Mth. 14 (1855) 305-. , plane and spherical. Laguerre, E. (x) Par. S. Phlm. Bll. 5 (1868) 40-. Catenaries through 2 fixed points. Hancock, H. A. Mth. 10 (1895-96) 159-. Cissoid. Schulz von Strasznitzki,L. C. Baum- gartner Z. 8 (1830) 179-. . Greiner, M. [1881] Arch. Mth. Ps. 69 (1883) 313-. , . Pdnek.A. Casopis 16 (1887) 33- ; Fschr. Mth. (1887) 734. , locus of points having a constant triangle of contact. Zahradnlk, K. Prag Sb. (1877) 125-. , property. Petersen, J. Mth. Ts. 4 (1862) 40. and strophoid. Servais, C. Mathesis 10 (1890) 9-. , tangents. Walker, J. J. L. Mth. S. P. 2 (1869) 161-. , theorem. Marcmi, J. N. A. Mth. 2 (1843) 486- ; 3 (1844) 472-. , theory. Zahradnlk, K. Arch. Mth. Ps. 59 (1876) 335-. , based on a rational parameter. Zahradnlk, K. Prag Sb. (1873) 221-. Cissoidal curves. Zahradnik, K. Casopis 2 (*1873) 183- ; Arch. Mth. Ps. 56 (1874) 8-. Cissoidometry. Werneburg, J. F. C. Oken Isis (1819) 209-. Conchalic curves, trajectories, and cissoids of 4th degree. Huber, G. Mh. Mth. Ps. 6 (1895) 157-. Conchoid. Midy, . N. A. Mth. 2 (1843) 281-. with circular base applied to solution of Delian problem. Clausen, T. GrunertArch. 2 (1842) 196-. , generalisation. Machovec, F. Casopis 14 (1885) 15-; Fschr. Mth. (1885) 714-. of E. de Sluse. Loria, G. Mathesis 17 (1897) 5-. , theory. A/olter, F. G. Arch. Mth. Ps. 55 (1873) 175-. traced by parabola. Andre, H. d'. N. A. Mth. 5 (1846) 122-, 208. Conchoids, class. Kulp, L. Arch. Mth. Ps. 48 (1868) 97-. and conchoidal surfaces. Minich, S. E. Padova Kv. Period. 10 (1862) 135-. , generation. Schoentjes, H. Mathesis 4 (1884) 145-. Conies, curves depending on. Oddi, G. Bologna Opusc. Sc. 2 (1818) 293-. Conjugate curves. Schlesinger, O. Mth. A. 30 (1887)453-; 33(1889)315-. Constructions in connection with special curves. Stuyuaert, . Mathesis 19 (1899) 33. Convolutes, unipolar. London, (Rev.) H. Manch. Lt. Ph. S. P. 24 (1885) 38-. Critic centres. Morley, F. Am. J. Mth. 10 (1888) 141-. Cubic centres of a line with respect to 3 lines and a line. Cayley, A. Ph. Mg. 20 (1860) 418- ; 22 (1861) 433-. Curve with 3 foci. Hachette, J. N. P. [1831] Quetelet Cor. Mth. 7 (1832) 32-. . Eeiss, M. Quetelet Cor. Mth. 7 (1832) 123-. , 4th class, metric theorems. Loud, F. H. Am. As. P. (1900) 43. Curves with centres on family of cubics. Schoute, P. H. N. Arch. Wisk. 13 (1887) 1-; Fschr. Mth. (1886) 692-. , and properties of curves generally. Steiner, J. Berl. B. (1851) 302- ; Crelle J. 47 (1854) 7-. , Steiner's. Bobek, K. Wien Ak. Sb. 98 (1890) (Ab. 2a) 5-, 394-, 526-. of constant "power" with respect to points in their plane. Euffini, F. P. Bologna Ac. Sc. Mm. 10 (1889) 337-. containing foci and poles of surface of 2nd degree. Cauchy, A. L. C. E. 16 (1843) 885. with harmonic pole and straight line, and deductions. Gussfeldt, P. [1868] Mth. A. 2 (1870) 65-. infinite branches. Bergsten, N. J. Stockh. Ac. Hndl. 31 (1810) 17-. pedals having "power" at every point of plane. Euffini, F. P. Bologna Ac. Sc. Mm. 3 (1892) 277-; 4 (1894) 235-. traced in plane of conic by foci and poles. Cauchy, A. L. C. E. 16 (1843) 818-. with trinomial equations. Gilbert, P. N. A. Mth. 9 (1870) 370-. Curves, 3rd degree. circular. Eckardt, F. E. Z. Mth. Ps. 10 (1865) 321-. . Durege, H. Z. Mth. Ps. 14 (1869) 368-. . Consensus, E. 0. Z. Mth. Ps. 25 (1880) 119-. . Jefdbek, V. Mathesis 18 (1898) 224-. . Eetali, V. Mathesis 19 (1899) 87-. , with double point. Morley, F. Am. J. Mth. 11 (1889) 307-. , . Servais, C. N. A. Mth. 8 (1889) 197-. _, . Beyel, C. Z. Mth. Ps. 42 (1897) 281-. _, , Quetelet's. Weyr, Em. G. Mt. 9 (1871) 259-. _; , . Pelz, C. [1880] Wien Ak. Sb. 82 (1881) (Ab. 2) 1207-. , focal curve, construction from 6 points. Milller, E. Z. Mth. Ps. 40 (1895) 337-. 531 LL 2 7630 Special Plane Curves. Degrees 3, 4 7630 Sporer, B. Z. Mth. circular, of genus 1, foci. Schoute, . Amst. Ak. Vs. 5 (1897) 261-. , inverses of lemniscate with respect to its vertices. Wolstenholme, (Rev.) J. L. Mth. S. P. 7 (1875-76) 91-. , involution system, and description by points when double focus is on curve. Cotterill, T. L. Mth, S. P. 1 (1866) No. 8, 14-. , metric properties, theorems of Jakob Steiner. Disteli, M. Ziir. Vischr. 36 (1891) 255-. , rational, construction. Longchamps, G. de. As. Fr. C. E. (1885) (Pt. 2) 131-. , , points of inflexion on. Longchamps, G. de. As. Fr. C. E. (1886) (Pt. 2) 5-. connected with reflexion of bright point in a sphere. Nicodemi, R. Nap. Ac. Pont. At. 17 (1887) 109-. triangle. Steggall, . Edinb. Mth. S. P. 7 (1889) 66-. triangles in perspective. Roberts, S. L. Mth. S. P. 28 (1897) 448-. and curves of 3rd class. Spc Ps. 40 (1895) 159-. with cusp. Greiner, M. Arch. Mth. Ps. 68 (1882) 1-. . Bioche, C. N. A. Mth. 12 (1893) 294-. . Zahradnik, C. N. A. Mth. 18 (1899) 389-. , circum-polygons. Roberts, R. A. L. Mth. S. P. 13 (1881-82) 148-. , construction by points and tangents. Ocagne, M. d'. N. A. Mth. 11 (1892) 386-. , 13 degenerate forms and fundamental numbers of. Schubert, H. C. H. Gott. Nr. (1875) 359-. , properties. Drasch, H. Mh. Mth. Ps. 4 (1893) 81-. cyclic. Andreasi, A. G. Mt. 30 (1892) 241- . degeneration. Taylor, H. M. L. Mth. S. P. 28 (1897) 545-. equianharmonic. Scott, (Miss) C. A. Mess. Mth. 25 (1896) 180- . and hyperboloid, various theorems. Oeking- haus, E. Arch. Mth. Ps. 70 (1884) 370-. inflecting on crossing their asymptotes. Morley, F. Mess. Mth. 17 (1888) 51-. locus of foci of tangential pencil of conies. Schroter, H. [1871] Mth. A. 5 (1872) 50-. . Durege, H. [1871] Mth. A. 5 (1872) 83-. whence 2 lines subtend equal angles. Hermes, O. Crelle J. Mth. 97 (1884) 177-. . Schoute, P. H. Crelle J. Mth. 99 (1886) 98-. 16-point, of triangle. Thomson, F. D. [1864] Mess. Mth. 3 (1866) 15-. related to 2 circles. Mention, J. [1859] St Pet. Ac. Sc. Bll. 1 (1860) 233-. through vertices of complete quadrilateral. Roberts, S. [1861] QJ. Mth. 5 (1862) 54-. . Roberts, M. QJ. Mth. 5 (1862) 365-. and 4th class, "evolution" in. Weyr, E. J. Brux. Ac. Bll. 49 (1880) 7-. 532 Curves, 3rd and 4th degrees. application of conic section coordinates. Oppenheimer, H. Arch. Mth. Ps. 13 (1895) 84-. circular. Czuber, E. Z. Mth. Ps. 32 (1887) 257-. , mechanical description. Roberts, S. L. Mth. S. P. 2 (1869) 125-. Curves, 4th degree. apolar to conies. Grassi, A. G. Mt. 38 (1900) 244-. arising from quadratic transformation of conic. Leuch, A. Bern Mt. (1889) 1-. Caporali's quartic. Ciani, E. Nap. Ed. 35 (1896) 126-. case. Jefdbek, V. Mathesis 19 (1899) 15-. central, axes of symmetry. Schiff, V. 0. Kharkov Mth. S. Com. 3 (1893) 163- ; Fschr. Mth. (1892) 600. circular. Casey, J. [1867] Ir. Ac. T. 24 (1871) 457-. . Cayley, A. Phil. Trans. 167 (1877) 441-. . Loria, G. QJ. Mth. 22 (1887) 44-. , with collinear foci. Jeffery, H. M. [1880- 83] L. Mth. S.P. 12 (1880-81) 17-; 14(1882- 83) 239-. , confocal. Franklin, F. Am. J. Mth. 12 (1890) 323-. , , and "cyclides", analogy to Ivory's theorem. Dixon, A. L. L. Mth. S. P. 24 (1893) 306-. , construction. Cayley, A. L. Mth. S. P. 5 (1874) 29-. , "cyclics" and "cyclides", generation. Saltel, L. Par. S. Mth. Bll. 3 (1875) 95-, 99-. , , foci. Saltel, L. Par. S. Mth. Bll. 3 (1875) 100-. , and "cyclides", curves connected with double normals of. Smyly, J. G. [1899] Ir. Ac. P. 5 (1898-1900) 370-. , double foci of, and nodal focal curves of a cyclide. Lachlan, R. L. Mth. S. P. 27 (1896) 71-. , with double point, description. Cayley, A. [1870] L. Mth. S. P. 3 (1869-71) 100-. , 3 double points and surface of elasticity. Saltel, L. Brux. Ac. Bll. 35 (1873) 46-. , focal conies. Hart, H. L. Mth. S. P. 11 (1879-80) 143-. , geometrical treatment. Purser, F. B. A. Ep. (1878) 465. , mechanical description. Hart, H. L. Mth. S. P. 14 (1882-83) 199-. , proofs of theorems by elementary geometry. Crone, C. (xn) Ts. Mth. 3 (1879) 81-. , properties. Crofton, M. W. [1867] L. Mth. S. P. 2 (1869) 33-. , property. Jessop, C. M. QJ. Mth. 23 (1889) 371-. , quadritangential conies. Richter, O. Z. Mth. Ps. 35 (1890) (Suppl.) Ill pp. and skew, theorem. Lachlan, R. [1890] L. Mth. S. P. 21 (1891) 274-. , special involution of 4th order applied to. La Gournerie, J. de. C. E. 66 (1868) 832-. 7630 Special Plane Curves. Degrees 4-6. Lemniscates 7630 circular, theorems. Lachlan, R. Camb. Ph. S. P. 7 (1892) 87-. with 3 common double points, series. Taylor, H. M. L. Mth. S. P. 28 (1897) 316-. cusp. Richmond, H. W. QJ. Mth. 26 (1893) 5-. at infinity at which line at infinity is a tangent. Jeffery, H. M. [1882] L. Mth. S. P. 14 (1882-83) 71-. 3 cusps, circum-polygons. Roberts, R. A. [1882] L. Mth. S. P. 14 (1882-83) 56-. of 1st species. Padula, F. Nap. Mm. Ac. Sc. 1 (1852-54) 31-. double points of inflexion. Beyel, C. Z. Mth. Ps. 30 (1885) 1-, 65-. . Schoute, P. H. G. Teix. J. Sc. 13 (1897) 10-. , particularly the "Kreuz- curve". Balitrand,. Mathesis 11 (1891) 241- ; 13 (1893)5-. , lemniscate. Laguerre, E. N. A. Mth. 17 (1878) 337-. double tangential node and double point. Beyel, C. Ziir. Vjschr. 31 (1886) 178-. envelope derived from conic. Hall, A. [1871] Smiths. Misc. Coll. 20 (1881) Art. 1, 30-. (Wash. Ph. S. Bll. 1 (1874).) Klein's quartic is its own (uj) being functions of trigonometrical ratios of w. Fouret, G. N. A. Mth. 19 (1880) 28-. y m + < m _ 2 (x , y) = , properties . Weill , M. N . A. Mth. 20 (1881) 498-. Karlinski, F. (xn) Krk. Ak. (Mt.-Prz.) Ez. & Sp. 10 (1883) xi-. 535 7640 Higher Algebraic Surfaces. Contacts 7640 # 2 3 x s + xJx-. + , 3 a; 2 , covariant theory. Brioschi, F. Mil. I. Lomb. Ed. 17 (1884) 401- ; Km. E. Ac. Line. T. 8 (1884) 164-. (2y - Sb) (x 2 + i/ 2 ) + b 3 = 0. Almeida Lima, Joao d>. G. Teix. J. Sc. 6 (1885) 13-; Fschr. Mth. (1884) 644-. ax 3 + 3by 2 z + 3cz*x = Q, concomitants. Walker, J. J. L. Mth. S. P. 20 (1889) 382-, Contacts, and Tangent Lines and Planes. (See also 8450.) Contact of conies with surfaces. Spottiswoode, W. Phil. Trans. 160 (1870) 289-. straight lines with surfaces. Salmon, G. QJ. Mth. 1 (1857) 329-. r"=a n sin nO. Iversen, I. M. N. Ts. Mth. 4 ,4. and 5-point. Gordan, P (B) (1893) 59-; Fschr. Mth. (1893-94) 1154. />=2acosn0. Pirondini, G. Mathesis 14 (1894) 14-. p=R sinjtw, algebraic if /x is integral, other- wise transcendental. Loria, G. Fschr. Mth. (1899) 515. 7640 Algebraic surfaces of degree higher than the second. (See also 8040.) (Miiller.) Petzval, J. Wien SB. 41 (1860) 735-. Cremona, L. Bologna Ac. Sc. Mm. 6 (1866) 91- ; 7 (1867) 29-. Backlund, A. V. Stockh. Ak. Hndl. 9 (1870) No. 9, 65 pp. Mainardi, G. Em. At. N. Line. 26 (1873) 384-. Beltrami, E. Bologna Ac. Sc. Mm. 10 (1879) 233-. Ocagne, M. d'. C. E. 99 (1884) 744-. Asymptotes, curvature and singular points in surfaces of 2nd and 3rd degree. Beer, A. Grunert Arch. 17 (1851) 329-. Asymptotic plane and cylinders of surface. Biehler, C. N. A. Mth. 8 (1889) 536-. planes and surfaces. Walton, W. Camb. and Dubl. Mth. J. 3 (1848) 28-. straight lines, planes, cones, and cylinders. Weddle, T. Ph. Mg. 31 (1847) 425-. Axes, 3 rectangular, cutting surface, theorems. Haillecourt, A. N. A. Mth. 12 (1853) 398-. Cones, algebraic, properties. Laguerre, E. [1870] (x) Par. S. Phlm. Bll. 7 (1871) 139-. , 3rd degree. Cayley, A. Ph. Mg. 18 (1859) 439-. , with degree, having vertices in straight line, and 2 plane intersection curves, theorem. Serret, P." N. A. Mth. 1 (1862) 21-. Conical points on surfaces. Korteweg, . N. Arch. Wisk. 18 (1891) 153-. Conjugate lines analogous to conjugate points. Walton, W. Camb. and Dubl. Mth. J. 9 (1854) 238-. Construction of surfaces from given points. Escherich, G. von. Wien Ak. Sb. 90 (1885) (Ab. 2) 1036-. by means of reciprocal pencils of surfaces. Escherich, G. von. Wien Ak. Sb. 85 (1882) (Ab. 2) 526-. plane sections. Cardinaal, J. Amst. Ak. Vs. M. 6 (1889) 198-; Fschr. Mth. (1889) 665-. Z. Mth. Ps. 12 (1867) 495-. , various kinds. Clebsch, A. Crelle J. 58 (1861) 93-. surfaces. Spottiswoode, W. Phil. Trans. 162 (1872) 259-. (Spottiswoode 's problems). Clifford, W. K. E. S. P. 21 (1873) 425. . HalpJim, G. H. [1874] Par. S. Mth. Bll. 3 (1875) 28-. ,2, having common generator, funda- mental principle. Mansion, P. Brux. Ac. Bll. 3 (1882) 753-. , , order of condition for. Roberts, S. QJ. Mth. 12 (1873) 229-. , , , criteria. Plucker, J. Crelle J. 4 (1829) 349-. Normals and tangents to algebraic surfaces and lines. Bardelli, G. Mil. I. Lomb. Ed. 5 (1872) 167-. Points where straight line has 5-point contact. Clebsch, E.F.A. [1863] (vn) Crelle J. 63 (1864) 14-. Problem: if + -\ = 1 is a tangent plane, x l y l 2j and makes f(x, y, z) a maximum or minimum, then /(a;, y, z) is constant, e. [Ellis, R. L.] (vi Adds.) Camb. Mth. J. 4 (1845) 47-. Eotation surfaces, tangent planes, construction. Matzek, F. Wien Sb. 58 (1868) (Ab. 2) Euled surfaces, 2, touching along generator, condition. Ch., J. F. N. A. Mth. 2 (1863 262-. Skew surfaces, method of drawing tangents to. Marchand, J. Par. S. Mth. Bll. 13 (1885) Surfaces, 3rd degree, contact of 4th order with surface. Moutard, T. (x) Par. S. Phlm. Bll. 2 (1865) 23-. Tangent planes. Reid, J. Mathematician 2 (1847) 187-. to algebraic surface through multiple line of surface, number. Fouret,G. Palermo Cir. Mt. Ed. 8 (1894) 202-. certain surfaces, arrangement. Weyr, E. [1878] Bordeaux S. Sc. Mm. 3 (1880) 191-. , higher, of algebraic surfaces. Totossy, B. Mth. Termt. Ets. 16 (1898) 127-. and normals to skew surfaces. Chasles, M. Liouv. J. Mth. 2 (1837) 413-. at point on surfaces. Laisant, A. N. A. Mth. 7 (1868) 116-. to skew surfaces. Monin, T. Prag Sb. (1888) (Mth.-Nt.) 210-. , theorems of Hachette and Chasles. Goedseels, E. (xii) Mathesis 3 (1883) 49-. 536 7640 Tangent planes to skew surfaces, theorems of Hachette and Chasles. Mansion, P. (xn) Mathesis 3 (1883) 52-. touching surface along curve, condition for. Greatheed, S. S. (vi Adds.) Camb. Mth. J. 2 (1841) 22-. , . Salmon, G. Camb. and Dubl. Mth. J. 3 (1848) 44-. Tangential hyperboloid of certain ruled quartic surface. Janisch, E. Mh. Mth. Ps. 10 (1899) 223-. Tangents at multiple point of curve of inter- section. Olivier, T. Par. S. Phlm. N. Bll. (1832) 189-. to surface or single system of surfaces, some numerical determinations for. Bonsdorff, E. [E. J.] [1881] Helsingf. Acta 12 (1883) 95-. touching surface in two points. Boyd, J, H. A. Mth. 5 (1889-90) 109. (Boyd). Loud, F. H. A. Mth. 5 (1889-90) 229. Higher Algebraic Surfaces 7640 Cylinders, oblique. Euler, L. [1778] St Pet. Ac. Sc. N. Acta 12 (1801) 91-. Decomposition due to presence of multiple lines, and paradox. Saltel, L. Brux. Ac. Bll. 47 (1879) 184-. Determination of degree. Cayley, A. [1882] Mess. Mth. 12 (1883) 29-. Developable surfaces, double lines of. Kiinzler, G. Arch. Sc. Ps. Nt. 6 (1898) 370-. Diametral planes of algebraic surfaces. Breton [de Champ], P. C. E. 34 (1852) 170-. surfaces. Binet, J. P. M. Par. EC. Pol. Cor. 2 (1809-13) 74-. Equation, homogeneous, of plane section of geometrical surface. Walker, J. J. L. Mth. S. P. 15 (*1883-84) 224-. , reduced, special forms. Dumont, F. Par. S. Mth. Bll. 26 (1898) 124-. of surface PS - QE = 0, number of constants. Cayley, A. (xn) Ts. Mth. 4 (1880) 145-. Equations for lines on developable surfaces. Enneper, A. Gott. Nr. (1867) 73-. , plane coordinates. Pliicker, J. Crelle J. 9 (1832) 124-. , stereometric, of 3rd degree, and resulting surfaces. Grassmann, H. [1852] Crelle J. 49 (1855) 47-. as sum of homogeneous functions. Car- michael, E. Camb. and Dubl. Mth. J. 9 Generation, projective, of general surface of 3rd, 4th, and any degree, by systems of surfaces of lower degree. Eeye, T. Mth. A. 1 (1869) 455-. of surfaces. Blavette, . Eure Eec. S. Ag. 5 (1834) 71-. . Vantcek, J. S., & Vanecek, M. N. C. B. 97 (1883) 1473-, 1548-. . Jonquieres, E. de. C. B. 105 (1887) 1203-. . Sayre, H. A. [1898] A. Mth. 12 (1898-99) 118-. by quadrilinear sheaves of planes. Le Paige, C. Brux. Ac. Bll. 8 (1884) 238-. Generators, singular, in algebraic ruled surfaces. Bjorling, G. F. E. Stockh. Ofv. (1888) 587-. , , bi-nonnal and principal normal sur- faces. Bjorling, C. F. E. Stockh. Ak. Hndl. Bh. 15 (Afd. 1) (1890) No. 8, 18 pp. Harmonic pole and plane. Lebesgue, V. A. N. A. Mth. 8 (1849) 22-. . Tognoli,0. G. Mt. 8 (1870) 166-. Hessians of cubic and quartic, certain sin- gularities. Heal,W.E. A. Mth. 4 (1888) 37-. , nodes. Clebsch, A. Crelle J. 59 (1861) 193-. in relation to multiple points. Eohn, K. Mth. A. 23 (1884) 82-. of surfaces. Voss, (Dr) A. Gott. Nr. (1873) 418-. , degree, geometrical determination. Schubert, H. Z. Mth. Ps. 15 (1870) 126-. Intersection curve of two surfaces, degree. Terquem, O. N. A. Mth. 9 (1850) 342-. of surface by plane, equation. Housel, . N. A. Mth. 7 (1868) 277-. 2 surfaces, construction by means of spheres and rotation surfaces. Niemtschik, E. Wien Sb. 64 (1871) (Ab. 2) 117-. 3 surfaces, points, and curve of inter- section of 2 surfaces. Tognoli, 0. G. Mt. 9 (1871) 68-. Loci arising from three figures, project! vely related. Del Re, A. Palermo Cir. Mt. Ed. 1 (1887) 272-, 398. Locus of common apex of pyramids, surfaces as. Dufilhol, . Eouen Ac. Tr. (1814) 44-. points, such that lines, joining each to 3 points, cut given plane in triangle of constant area. Sturm, J. C. F. Gergonne A. Mth. 14 (1823-24) 302-. , sum of whose distances from 2 lines is constant. Picart, A. Par. EC. Norm. A. (1854) 55-. of surfaces, 2nd and 3rd degrees, geometric - (1864} 285- meaning of coefficients. Mossbrugger, L. and class of Grunert Arch. 1(1841) 337-. surfaces. Mannheim, A. Bll. Sc. Mth. As. , tangential. Laguerre, E. (x) Par. S. 3 (1872) 119- Phlm. Bll. 8 (1872) 64-. Maximum and 'minimum distances of point from surface, etc. Mayer, A. C. G. Leip. Mth. Ps. B. 33 (1881) 28-. Metrical properties. Craig, T. Am. J. Mth. 4 (1881) 297-. Multiple points of surface. Umpfenbach, (Dr) . Crelle J. 28 (1844) 380. , specialising. Eohn, K. Leip. Mth. Ps. B. 36 (1884) 1-. General properties. Lardner, D. [1824] Ir. Ac. T. 14 (1825) 69-. . Dietrich, M. Schlomilch Z. 7 (1862) 398-. survey of progress. Bertrand, J. J. Sav. (1867) 644-. Generation, linear, of all algebraic surfaces. Grassmann, H. [1852-55] Crelle J. 48 (1854) 417- ; 49 (1855) 1-, 21-. 537 7640 Higher Algebraic Surfaces 7640 Multiple points, surface with 2, theorem. Saltel, L. Brux. Ac. Bll. 34 (1872) 51-. Newton's segment theorem. Terquem, O. N. A. Mth. 9 (1850) 342-. Normals. Sturm, E. Mth. A. 7 (1874) 567-. . Schoute,P.H. N.Arch.Wisk.6(*1880)19-. . Eeina, V. Nap. Kd. 29 (1890) 75-. common to two surfaces. Rindi, S. Palermo Cir. Mt. Rd. 5 (1891) 106-. (Rindi). Fieri, . Palermo Cir. Mt. Rd. 5 (1891) 323. , double, to algebraic surfaces. Fieri, M. Rm. R. Ac. Line. Rd. 2 (1886) (Sem. 2) 40-. , , . Rindi, S. As. Fr. C. R. (1887) (Ft. 2) 216-. , locus, when sum of squares is constant. Laisant, G. A. [1889] Par. S. Mth. Bll. 18 (1890) 141-. to surface from given point. August, F. [1867J Crelle J. 68 (1868) 242-. , number. Terquem, O. Liouv. J. Mth. 4 (1839) 175-. , . Salmon, G. Camb. and Dubl. Mth. J. 3 (1848) 46-. , theorems. Fouret, G. As. Fr. C. R. 6 (1877) 205-. Paradox arising in construction of surface through given points, solution. Schur, F. Leip. Mth. Ps. B. 35 (1883) 59-. Pedal and director surfaces, surfaces in relation to. Raabe, J. L. [1854] Crelle J. 50 (1855) 194-. surfaces. Mossbrugger, L. Grunert Arch. 22 (1854) 139-. Rotation surfaces, note. Steen, A. (xn) Ts. Mth. 1 (1871) 50-. , oblique. Walton, W. QJ. Mth. 6 (1863) 134-. , plane sections. Sobotka, J. Wien Ak. Sb. 105 (1896) (Ab. 2a) 371-. , properties. Beltrami, E. A. Mt. 6 (1864) 271-. , singularities. Workman, W. P. QJ. Mth. 25 (1891) 89-. Ruled surfaces. Cliasles, M. Quetelet Cor. Mth. 11 (1839) 49-. . Pirondini, G. G. Teix. J. Sc. 12 (1895) 19-. Singular lines. Halphen, H. G. [1877] A. Mt. 9 (1878-79) 68-. points. Saltel, L. [1875] Brux. Mm. Cour. 8, 27 (1877) No. 5, 66 pp. , class. Zeuthen, H. G. Mth. A. 9 (1876) 321-. , Newton's parallelogram generalised. Wolffing, E. Z. Mth, Ps. 42 (1897) 14-. Singularities. Cayley, A. Camb. and Dubl. Mth. J. 7 (1852) 166-. . Jonquieres, E. de. Liouv. J. Mth. 7 (1862) 409- ; N. A. Mth. 3 (1864) 5-. . Painvin, L. Bll. Sc. Mth. As. 4 (1873) 91-. , flecnodal and spinodal lines on surfaces, intersections. Korteiveg, . Amst. Ak. Vs. M. 5 (1889) 402- . , fleflecnodal planes. Cayley, A. QJ. Mth. 15 (1878) 49-. of Jacobian of 4 surfaces. Levi, A. Tor. Ac. Sc. At. 31 (1895) 358- or 502-. Plane sections, method of discussing. Dixon, . Gerbaldi, F. Palermo Cir. A. C. Camb. Ph. S. P. 9 (1898) 198-. Points at infinity. Painvin, L. G. R. 59 (1864) 666-; Crelle J. 65 (1866) 112- , 198-. Polar reciprocal of surface with multiple points, determination of double curve. Saltel, L. Brux. Ac. Bll. 40 (1875) 27-. surfaces, revision and extension of numerical formulas in theory . Zeuthen ,H.G. Mth. A. 10 (1876) 446-. , theorems. Hamilton, (Sir) W. R. Ir. Ac. P. 4 (1850) 192-. surfaces of plane, theory. Painvin, L. F. [1865] (xn) Lille S. Mm. 2 (1866) 345-. theory, extension. Reye, T. Crelle J. Mth. 78 (1874) 97-. Projective generation, points of intersection, and sections, of surfaces. Reye, T. [1869] Mth. A. 2 (1870) 475-. surface of centres. Voss, A. Munch. Ak. Ab. 16 (1888) 243-. Relations and affinities of surfaces. Peterson, K. M. [1865] (xn) Rec. Mth. (Moscou) [1] (1866) 391-. Rotation surfaces. Pirondini, G. N. A. Mth. 7 (1888) 486-; A. Mt. 18 (1890) 165-. , conditions for surfaces of any degree. Mangeot, S. N. A. Mth. 16 (1897) 408-. , discrimination, and determination of axis. Mangeot, S. Par. EC. Norm. A. 14 (1897) 191-. , intersection, descriptive geometry. [Schiappa] Monteiro, A. G. Teix. J. Sc. 1 Mt. Rd. 10 (1896) 158-. , pinch points. Cayley, A. QJ. Mth. 9 (1868) 332-. , spinodal lines at conical point on surface. Korteweg, . Amst. Ak. Vs. M. 8 (1891) 385-. , surfaces with any, property. Guccia, G. T>. Rm. R. Ac. Line. Rd. 5 (1889) (Sem. 1) 349-, 490-. Skew surfaces. Olivier, T. Par. S. Phlm. N. Bll. (1832) 185- ; Par. S. Phlm. PV. (1838) 84-. . Catalan, E. C. Par. S. Phlm. PV. (1848) 67-. . Cayley, A. Camb. and Dubl. Mth. J. 7 (1852) 171- ; Phil. Trans. 153 (1863) 453-; 154 (1864) 559-; 159 (1869) 111-. (Cayley, 1864). Cremona, L. Mil. I. Lomb. Rd. 1 (1868) 109-. . Vanecek, J. S., & Vanecek, M. N. Prag Sb. (1883) (Mth. Vortr.) 449-. . Machovec, F. Casopis 13 (1884) 32- ; Fschr. Mth. (1884) 603. . Zejliger, D. Kazan S. Ps.-Mth. Bll. 7 (1898) 122- ; Fschr. Mth. (1897) 553. , classification. Fats, A. Bologna Rd. (1883) 170-. , construction of paper models. Schone- mann, P. Z. Mth. Ps. 28 (1883) 243-. , hyperbolic involutions of points on generators. Zajqczkoivski, W. Krk. Ak. (Mt.-Prz.) Rz. 12 (1896) 279-; Crc. Ac. Sc. Bll. (1896) 111-. 538 7640 Higher Algebraic Surfaces. Configurations 7640 Skew surfaces, hyperboloid generated by tangent lines. Bruno, G. Tor. At. Ac. Sc. 7 (1871-72) 235-. , (Bruno's theorem). Dewulf, E. Bll. Sc. Mth. As. 7 (1874) 142-. and orthogonal cone. Beyel, C. Z. Mth. Ps. 32 (1887) 321-. , osculating hyberboloid to. Sobotka, J. Prag Sb. (1893) (Mth.-Nt.) No. 14, 16pp. Surface generated by solution of isotropic solid. Bottomley, J. Manch. Lt. Ph. S. Mm. & P. 2 (1889) 154-, vin. with only one straight line, equation. Breton [de Champ], P. N. A. Mth. 8 (1849) 61-, 130-. Surfaces apolar to one another. Eeye, T. Crelle J. Mth. 79 (1875) 159-. of even degree, theory. Mahler, E. Arch. Mth. Ps. 70 (1884) 313-. passing through certain number of points. Plilcker, J. Gergonne A. Mth. 19 (1828-29) 129-. intersection of several surfaces. Delassus, E. Bll. Sc. Mth. 21 (1897) 59-. Surfaces, 2nd and 3rd degrees. Geiser, C. F. Crelle J. 69 (1868) 197-. Surfaces, 2nd, 3rd, and 4th degrees. Vanecek, J. 8. [1882] St Pet. Ac. Sc. Bll. 28 (1883) 287-. Surfaces, 3rd degree. Sartiaux, A. [1867] (xn) Lille S. Mm. 5 (1868) 441-. Gayley,A. [1868] Phil. Trans. 159 (1869) 231-. Cremona, L. Crelle J. 68 (1868) 1-. Klein, F. Mth. A. 6 (1873) 551- ; Erlang. Sb. Ps. Md. S. 5 (1873) 95-. Zeuthen, H. G. (xn) Ts. Mth. 4 (1874) 53-. Sturm, R. Crelle J. Mth. 88 (1880) 213-. Caporali, E. Nap. Ed. 20 (1881) 122-. Paolis, E. de. Em. E. Ac. Line. Mm. 10 (1881) 123-. Le Paige, C. M. M. H. H. Acta Math. 3 (1883) 181-, 381 ; C. E. 98 (1884) 971-. Vanecek, M. N. Liege S. Sc. Mm. 11 (1885) No. 3, 5 pp. Ascione, E. Nap. Ed. 32 (1893) 39-. Chasles's theorems for plane cubics, analogues. Dumont, F. Par. S. Mth. Bll. 25 (1897) 235-. Classification. Rodenberg, C. Mth. A. 14 (1879) 46-. . Bobek, K. Wien Ak. Sb. 96 (1888) (Ab. 2) 355-. with reference to singular points and reality of lines. Schlafli, L. Phil. Trans. (1863) 193-. Configurations. (See also 8075.) Brioschi, F. Tortolini A. 6 (1855) 374-. Double-sixers. Cayley, A. QJ. Mth. 10 (1870) 58-; Camb. Ph. S. T. 12 (1879) 366-. Hexad of lines on surface, case. Kohn, G. Mh. Mth. Ps. 2 (1891) 293-. Lines which can be placed on surface of 3rd class or 3rd degree. G., E. N. A. Mth. 13 (1894) 138-. Pascal's and Brianchon's theorems, extension to cubic surfaces. Folie, F. Liege Mm. S. Sc. 3 (1873) 663-. theorem, extension to cubic surfaces. Petot, A. C. E. 102 (1886) 737-. Pentad of co-tangential points, surface referred to. Jeffery, H. M. B. A. Ep. (1878) 491-. Pentahedra. Gordan,P. Mth. A. 5 (1872) 341-. , complete, inscribed in surface. Zeuthen, H. G. Acta Mth. 5 (1884) 203-. Polar hexahedra. Cremona, L. D. Nf. B. (*1877) 93-; Mth. A. 13 (1878) 301-. 27 real straight lines, forms of surfaces contain- ing. Blythe, W. H. Camb. Ph. S. P. 9 (1898) 6-. straight lines. Steiner, J. Berl. Mb. (1856) 50-. . Jonquieres, E. de. N. A. Mth. 18 (1859) 129-. .- Sylvester, J. J. C. E. 52 (1861) 977-. . Schroter, H. E. Crelle J. 62 (1863) 265-. . Cremona, L. Mil. I. Lomb. Ed. 3 (1870) 209-. . Affolter, F. G. Arch. Mth. Ps. 56 (1874) 113-. . Sturm, R. Mth. A. 23 (1884) 289-. , delineation. Caron, J. Par. S. Mth. Bll. 8 (1880) 73-. , determination and classification of surfaces with respect to. Schlfifii, L. QJ. Mth. 2 (1858) 55-, 110-. , equation. Jordan, C. Liouv. J. Mth. 14 (1869) 147- ; C. E. 68 (1869) 865-. , determining, resolution. Klein, F. Liouv. J. Mth. 4 (1888) 169-. , groups of substitutions connected with. Pascal, E. A. Mt. 20 (1892-93) 269- ; 21 (1893) 85-. , new combination. Jordan, C. C. E. 70 (1870) 326-. , and parabolic curve. Bauer, G. Munch. Ak. Sb. 13 (1883) 320-. , representation on plane. Schoute, P. H. Amst. Ak. Vs. [1] (1893) 143-. and 45 triple tangent planes. Bertini, E. A. Mt. 12 (1883-84) 301-. and 36 double-sixers, construction of models showing lines. Frost, P. QJ. Mth. 18 (1882) 89-. Triple tangent planes. Cayley, A. Camb. and Dubl. Mth. J. 4 (1849) 118-. . Salmon, G. Camb. and Dubl. Mth. J. 4 (1849) 252-. . Spottisicoode, W. C. E. 77 (1873) 1181-. , Cayley's theorem, proof. Kohn, G. Mh. Mth. Ps. 2 (1891) 343-. , polyhedral configurations. Bertini, E. Mil. I. Lomb. Ed. 17 (1884) 478-, 712-. , property. Brioschi, F. Em. E. Ac. Line. At. 3 (1876) (Pte. 2) 257- Construction, mechanical. Cayley, A. [1872] L. Mth. S. P. 4 (1871-73) 175-. 539 7640 Higher Algebraic Surfaces 7640 Construction from 19 points. Pannelli, M. A. Mt. 22 (1894) 237-. by projective rays. Jonquieres, E. de. C. B. 106 (1888) 526-, 907-, 1038 ; 107 (1888) 209-. , synthetic. Thieme, H. Mth. A. 28 (1887) 133-. Constructions, linear. Schroeter, H. Crelle J. Mth. 96 (1884) 282-. Cubic surface of Grassmann, construction, and reciprocal of 3rd degree. Haft, F. Arg. S. Ci. A. 29 (1890) 157-. Cubics, geometry of surfaces derived from. Russell, E. [1889] Ir. Ac. P. 5 (1898-1900) 462-. Equation, general, special form, and diagram of 27 straight lines. Taylor, H. M. [1893] Phil. Trans. (A) 185 (1895) 37-. , pentahedral. Beltrami, E. Mil. I. Lomb. Ed. 12 (1879) 24-. of ruled surfaces. Godefroy, A. N. N. Arch. Wisk. 1 (1895) 137- ; Fschr. Mth. (1893-94) 1259-. Generation. Picquet, H. Par. S. Mth. fill. 4 (1876) 128-. . Deruyts, F. Liege S. Sc. Mm. 14 (1888) No. 4, 12 pp. , homographic. Le Paige, C. Acta Mth. 5 (1884) 195-. , (Le Paige). Schur, F. Leip. Mth. Ps. B. 36 (1884) 128-. , method. Deruyts, F. Brux. Ac. Bll. 22 (1891) 35-. , new mode. Kohn, G. Wien Ak. Sb. 99 (1891) (Ab. 2a) 683-. Hessian of Hessian. Bauer, G. Munch. Ak. Ab. 14 (1883) (Ab. 3) 77-. Hessians, cubic surfaces which form parts of Hessian of another cubic surface. Ciani, E. Mil. I. Lomb. Bd. 27 (1894) 222-. , determination of surfaces by. Dumont, F. N. A. Mth. 15 (1896) 312-. of surfaces. Le Paige, C. Wien Ak. Sb. 91 (1885) (Ab. 2) 981-. . Ciani, E. Mil. I. Lomb. Bd. 26 (1893) 498-, 523-, 557-. , property. Eckardt, F. E. Z. Mth. Ps. 19 (1874) 259-. Invariants. Bobek, K. Wien Ak. Sb. 103 (1894) (Ab. 2a) 887-; Mh. Mth. Ps. 8 (1897) 145-. Models, construction. Blythe, W. H. QJ. Mth. 29 (1898) 206-. by Bodenburg. Korteweg, D. J. N. Arch. Wisk. 20 (1893) 63- ; Fschr. Mth. (1893-94) 83. Polar system, construction. Thieme, H. Mth. A. 20 (1882) 144-. Portion not cut by any real plane. Schlafti, L. A. Mt. 5 (1871-73) 289-; 7 (1875-76) 193-. Properties of position. Zeuthen, H. G. Mth. A. 8 (1875) 1-. Quadrilinear form, and cubic surfaces. Le Paige, C. Brux. Ac. Bll. 8 (1884) 555-. Bepresentation on plane. Clebsch, E. F. A. Mth. A. 5 (1872) 419-. Sections, parabolic and equilateral. Picquet, H. Par. S. Mth. Bll. 4 (1876) 153-. Surface arising from pencil of quadrics. Cassani, P. G. Mt. 22 (1884) 51-. Surfaces, 3rd and 4th degrees. Congruences connected with. Schur, F. Crelle J. Mth. 95 (1883) 207-. Surfaces, 4th degree. Cayley, A. L. Mth. S. P. 3 (1869-71) 19-, 198-, 234-. Durrande, H. N. A. Mth. 9 (1870) 410-, 440-. Case. Hunyady, J. (xn) Mag. Tud. Ak. Etk. (Mth.) 8 (1881) (No. 12) 20 pp. Developable, reciprocation. Cayley, A. [1865] QJ. Mth. 7 (1866) 87-. Hessian. Cayley, A. QJ. Mth. 15 (1878) 141-. Buled surfaces. Eouyer, L. Toul. Fac. Sc. A. 2 (1900) 163-. , different kinds. Eohn, K. Mth. A. 28 (1887) 284-. with double curve of 3rd degree, repre- sentation on plane. Clebsch, E. F. A. Mth. A. 2 (1870) 445-. Section of surface by bitangent plane. Juhel- Eenoy, . N. A. Mth. 7 (1888) 282-. Surface and circumscribed sextic cones, theorem. Cayley, A. L. Mth. S. P. 3 (1869-71) 220-. derived from surface of nth degree. Painvin, L. N. A. Mth. 10 (1871) 481- ; 11 (1872) 376-. Surfaces with 8 to 16 nodes. Eohn, K. Leip. Mth. Ps. B. 36 (1884) 52-. Surfaces, 4th and 5th degrees. Becent researches. Cayley, A. [1870] L. Mth. S. P. 3 (1869-71) 186-. Surfaces, 5th degree. Developables, demonstration of theorems. Ovidio, E. d'. G. Mt. 3 (1865) 107-, 184-, 214-. Feet of all axes of quadric through arbitrary point lie on quintic surface. Kloboucek, J. Casopis 26 (1897) 156- ; Fschr. Mth. (1897) 490. Surfaces, (m + n)th degree. Degeneration when every plane section de- generates. Cayley, A. Ph. Mg. 25 (1863) 61-. Surfaces, nth degree. Cone circumscribing surface. Salmon, G. Camb. and Dubl. Mth. J. 4 (1849) 188-. Construction. Schur, F. Mth. A. 23 (1884) 437-. Singularities of general surface. Sturm, E. Crelle J. 72 (1870) 350-. Tangent singularities. Schubert, H. C. H. Mth. A. 11 (1877) 347-. Tangents, 5-pointic, and related problems, solution. Schubert, H. C. H. Gott. Nr. (1876) 89-. Umbilici, number. Voss, A. E. Mth. A. 9 (1876) 241-. 540 7650 Special Algebraic Surfaces 7650 7650 Special algebraic surfaces. Anallagmatic surfaces. (See also 8020.) Circular sections. Laguerre, E. (x) Par. S. Phlm. Bll. 5 (1868) 48-. Cyclides. Godart, A. N. A. Mth. 5 (1866) 218-. . Maxwell, J. C. QJ. Mth. 9 (1868) 111-. . Darboux, G. C. E. 68 (1869) 1311- ; Bordeaux Mm. S. Sc. 8 (1870) 291- ; 9 (1873) 1-. . Lemonnier, H. N. A. Mth. 9 (1870) 514- . . Laguerre, E. (x) Par. S. Phlm. Bll. 7 (1871) 209-. . Darboux, G. Par. EC. Norm. A. 1 (1872) 273-. . Cayley, A. QJ. Mth. 12 (1873) 148-. . Elliot, Z. Bll. Sc. Mth. As. 3 (1879) 238-. . Humbert, G. Par. EC. Pol. J. 55 (1885) 127-. _. Neuberg, J. As. Fr. C. E. (1885) (Pt. 2) 79-. . Johnson, A. R. Mess. Mth. 16 (1887) 33-. . Bortniker, (Mile.) . C. E. 106 (1888) 824-. , focal quadrics. Hart, H. Mess. Mth. 14 (1885) 1-. , generation. Saltel, L. Par. S. Mth. Bll. 3 (1875) 95-, 99-. , plane sections. Laqutere, E. M. As. Fr. C. E. (1880) 121-. and sphero-quartics. Casey, J. Phil. Trans. 161 (1871) 585-. upin's cycli Dupin's cyclides. Enneper, A. Schlomilch Z. 7 (1862) 198-. . Neuberg, J. Liege S. Sc. Mm. 10 (1883) (No. 9) 10 pp. _ _. Cosserat, E. Toul. Fac. Sc. A. 6 (1892) F, 7 pp. . Pellet, A. As. Fr. C. E. (1899) (Pt. 1) 138. and Dupuis's theorem. Marchand, E. N. A. Mth. 9 (1890) 98-, 182-. , nodes in. Snyder, V. [1896] A. Mth. 11 (1896-97) 137-. , principal poles of inversion. Fouret, G. N. A. Mth. 7 (1888) 113-. - , systems of circles applied to. Hoist, E. Christiania F. (1885) No. 15, 4 pp. ; Fschr. Mth. (1885) 574-. , and theory of curvature, elementary account. HolzmUller, G. Z. Mth. Ps. 44 (1899) 194-. " Girocy elides " of 4th degree. Jamet, V. N. A. Mth. 20 (1881) 344-, 385-, 434-. Properties. Laguerre, E. (x) Par. S. Phlm. Bll. 5 (1868) 17-. Sphero-cyclides. Je/ery, H. M. [1884] L. Mth. S. P. 16 (1884-85) 109-. Surfaces with infinite number of inversions. Hadamard, J. Bll. Sc. Mth. 12 (1888) 118-. , 4th degree. Moutard, T. N. A. Mth. 3 (1864) 536-. Annular surfaces, case. Eossi, V. A. (xn) Nap. Ac. Pont. At. 5 (1853) 53-. with directrix cone. Rossi, V. A. [1849] (xn) Nap. Ac. Pont. At. 6 (1854) 53-. enveloping ruled surfaces, etc. Rossi, V. A. Nap. Ms. 4 (1844) 19-. , 1st and 2nd classes. Rossi, V. A. Nap. Ms. 5 (1845) 56-, 322-; 6 (1845) 248-; 8 (1846) 174. Apsidal surfaces. Cayley, A. QJ. Mth. 16 (1879) 109 -. . Mannheim, A. C. E. 122 (1896) 1396-. Asymptotic lines, surfaces with skew cubics for. Blutel, . C. E. 117 (1893) 722. , . Bioche, C. C. E. 125 (1897) 15-; Par. S. Mth. Bll. 26 (1898) 217-. , , 3rd degree. Bioche, C. Par. S. Mth. Bll. 27 (1899) 96-. Cardioido-hyperboloid surface. Vanecek, J. S. Prag Ceske Ak. Fr. Jos. Ez. (Tfida 2) 4 (1895) Art. 30, 5 pp. Cartesian ovals, analogues in space. Sylvester, J. J. Ph. Mg. 31 (1866) 287-, 380-. Cassini's ovals, surface analogues. Roberts, W. (vm) A. Mt. 4 (1861) 133-. Centro-surface of ellipsoid. Cayley, A. [1870] Camb. Ph. S. T. 12 (1879) 319-. . stahl, W. Crelle J. Mth. 101 (1887) 73-. , Eudio's inverse. Cayley, . QJ. Mth. 22 (1889) 156-. Clebsch's diagonal surface. Ciani,E. Em. E. Ac. Line. Ed. 7 (1891) (Sem. 1) 227-. . Bioche, C. N. A. Mth. 17 (1898) 111-. Cono-cunei. Pabst, C. Arch. Mth. Ps. 2 (1885) 281-, 337-. Cono-cuneus, Wallis's, plane sections. Rossi, V. A. [1835] II Progresso 17 (1837) 74-. Conoidal screw propellers, system of variants. Moshammer, K. Wien Sb. 51 (1865) (Ab. 2) 49-. surface with conic as curvilinear directrix. Bruno, G. [1866] Tor. Mm. Ac. Sc. 24 (1868) 327- two straight lines as directrices. Bruno, G. [1866] Tor. Mm. Ac. Sc. 24 (1868) 317-. surfaces, section by plane. Fodor, L. Orv.- Termt. Ets. (Termt. Szak) (1886) 113-. Conoids circumscribing plane curve. Schiappa Monteiro, A. Lisb. J. Sc. Mth. 12 (1888) 19-, , and especially Plucker's. Mannheim, A. C. E. 106 (1888) 820-, 1038. , plane sections. Poselger, F. T. Berl. Ab. (1825) 97-. , Plucker's. Picquet, H. Par. S. Mth. Bll. 14 (1886) 68-. , , new use. Mannheim, A. C. E. 119 (1894) 394-. , , transformation. Michel, F. As. Fr. C. E. (1893) (Pt. 2) 184-. , , . Mannheim, (le col.) A. As. Fr. C. E. (1894) (Pt. 2) 207. , shadow cast on themselves. Cahen, G. P. Par. S. Mth. Bll. 4 (1876) 88-. 541 7650 Special Algebraic Surfaces. Kummer's Surface 7650 Conoids, 3rd and 4th degrees, certain. Seipka, E. Mh. Mth. Ps. 9 (1898) 1-. Cylindrical surfaces, application of determin- ants. Dostor, G. Arch. Mth. Ps. 58 (1876) 293-. Cylindroid. Lewis, T. C. Mess. Mth. 9 (1880) 1-. , properties. Goebel, J. B. Z. Mth. Ps. 25 (1880) 281-. , . Ball, E. S. [1885] Ir. Ac. P. 4 (1884- 88) 518-. , property, characteristic. Appell, P. Par. S. Mth. Bll. 28 (1900) 261-. , Wallis's. Flauti, V. [1828] Nap. At. Ac. 4 (1839) 1-. , , problem, solution. Sangro, G. Nap. At. Ac. 1 (1819) 83-. Desmian surface of 4th degree. Humbert, G. Liouv. J. Mth. 7 (1891) 353-. Developables arising from 2 quadrics. Cayley, A. Camb. and Dubl. Mth. J. 5 (1850) 46-. circumscribing 2 quadrics. Chasles, M. C. E. 54 (1862) 715-. . Painvin, L. C. K. 67 (1868) 816- ; (xn) Lille S. Mm. 9 (1871) 97-; 11 (1873) 81-; 13(1874)329-. . Cdyley,A. Mess. Mth. 1 (1872) 111-. quadrics and rotation surfaces, construc- tion. Sobotka, J. Prag Sb. (1893) (Mth.- Nt.) No. 2, 16 pp. depending on elliptic functions. Cayley, A. QJ. Mth. 14 (1877) 235-. derived from quartic, nodal curve. Cayley, A. Ph. Mg. 27 (1864) 437-. quintic curve. Cayley, A. Camb. and Dubl. Mth. J. 5 (1850) 152-. , Prohessians, etc., and note- on sextic. Cayley, A. (i) QJ. Mth. 6 (1864) 108- ; (vn) 7 (1866) 105- . , singularities. Cayley, A. Liouv. J. Mth. 10 (1845) 245- ; Camb. and Dubl. Mth. J. 5 (1850) 18-; QJ. Mth. 7 (1866) 110-. , 4th class, quadrics inscribed to. Cremona, L. Tortolini A. 2 (1859) 65-. , degree and 3rd class, conies inscribed to. Cremona, L. Tortolini A. 2 (1859) 201-. , 5th degree. Cremona, L. C. B. 54 (1862) 604-. , . [Salvatore-] Dino, N. G. Mt. 3 (1865) 100-, 133-. , 6th degree, derived from quartic. Cayley, A. QJ. Mth. 9 (1868) 129- ; A. Mt. 2 (1868- 69) 99-, 219.-. , , , transformation. Cayley, A. QJ. Mth. 9 (1868) 373-. , , with edge quadri-quadric curve. Cayley, A. QJ. Mth. 14 (1877) 229-. , , edge of regression. Sucharda, A. Prag Sb. (1891) (Mth.-Nt.) 369-. , , with excubo-quartic curve. Cayley, A. [1869] (rx) Camb. Ph. S. T. 11 (1871) 507-. Ellipse revolving about straight line in its plane, problem. Boklen, 0. Grunert Arch. 36 (1861) 22-. Ellipsoid, negative pedal surface. Tortolini, B. Palomba Bac. 2 (1846) 140-. Ellipsoid, negative pedal surface. Cayley, A. B. S. P. 9 (1857-59) 171-. , . Craig, T. Am. J. Mth. 4 (1881) 358- ; 5 (1882) 76-. Envelope of quadric revolving about straight line. La Gournerie, J. de. Liouv. J. Mth. 10 (1865) 33-. sphere with centre on conic. Cayley, A. [1864] Ph. Mg. 29 (1865) 19-. Equatorial surfaces, Pliicker's models. Cayley, A. L. Mth. S. P. 3 (1869-71) 281-. Flat ("aplatie") quartic. Cayley, A. C. B. 74 (1872) 1393-. Glaisher's surface, construction of normal. Mannheim, A. QJ. Mth. 17 (1881) 147-. Irrational quartic surfaces of genus zero. Franchis, M. de. Palermo Cir. Mt. Bd. 14 (1900) 33-. Isogonal surfaces. Heffter, L. CrelleJ. Mth. 115 (1895) 1-. Kummer's surface. (See also 8060 and 8080.) Kummer, E. E. Berl. Mb. (1864) 246-, 495-. Cayley, A. Crelle J. 73 (1871) 292-; 84 (1878) 238-; 94 (1883) 270-. Eeye, T. Crelle J. Mth. 86 (1879) 84-. Brioschi, F. C. B. 92 (1881) 944-. Humbert, G. Liouv. J. Mth. 10 (1894) 473-. Associated line complexes, etc. Klein, F. Gott. Nr. (1871) 44-. Birational transformations. Hutchinson, J. I. [1900] N. Y. Am. Mth. S. Bll. 7 (1901) Configurations, in- and circum-scribed. Klein, F. Mth. A. 27 (1886) 106-. Contact sextics. Pascal, E. A. Mt. 19 (1891- 92) 159-. Curves of principal tangents. Klein, F., <& Lie, S. Berl. Mb. (1870) 891-. . Segre, C. Crelle J. Mth. 98 (1885) 301-. Elliptic surfaces. Humbert, G. Am. J. Mth. 16 (1894) 221-. Equation, new form. Cayley, A. Camb. Ph. S. P. 4 (1883) 321. , rational. Pascal, E. A. Mt. 18 (1890) 227-. , 16th degree, connected with Kummer's surface. Jordan, C. Crelle J. 70 (1869) 182-. Forms, various. Rohn, K. Mth. A. 18 (1881) 99-. Generation. Humbert, G. Palermo Cir. Mt. Bd. 11 (1897) 1-. Models. Kummer, E. E. Berl. Mb. (1866) 216-. Particular case. Segre, C. Leip. Mth. Ps. B 36 (1884) 132-. cases. Eohn, K. Leip. Mth. Ps. B. 36 (1884) 10-. Project! ve generation of plane quartic curve with node, relation to surface. Wirtinger, W. D. Mth. Vr. Jbr. 4 (1897) 97-. Properties, some. Paolis, de. Bm. B. Ac. Line. Ed. 6 (1890) (Seat. 2) 3-. 542 7650 Singular points and planes. Caporali, E. Km. R. Ac. Line. Mm. 2 (1878) 791-. Skew hexagon and Rummer's surface. Geiser, C. F. Ziir. Vjschr. 41 (1896) (Festschr., Th. 2) 24-. 16 straight lines of surface connected with Rummer's lines. Sucharda, A. Casopis 15 (1886) 149- ; Fschr. Mth. (1886) 769-. Surface with 9 portions, note. Klein, C. F. Mth. A. 18 (1881) 160. S of 6th degree connected with Rummer's surface. Humbert, G. C. E. 120 (1895) 365-, 425-. , relation to Rummer's surface and curves of 4th class. Humbert, G. C. E. 120 (1895) 863-. Systems through lemniscate. Schjerning, W. Arch. Mth. Ps. 7 (1889) 113-. Tetrahedroid, equation. Hofmann, F. Crelle J. Mth. 98 (1885) 264. , note. Hutchinson, J. I. N. Y. Am. Mth. S. Bll. 4 (1898) 327-. as particular case. Cay ley, A. Crelle J. 65 (1866) 285-; 87 (1879) 161-. Theta functions with 2 variables. Cayley, A. Crelle J. Mth. 83 (1877) 210-, 220-. . Weber, H. Crelle J. Mth. 84 (1878) 332-. . Darboux, G. C. E. 92 (1881) 685-, 1493-. , Gopel's biquadratic relation between 4, representation of surface by. BorcMrdt, C. W. Crelle J. Mth. 83 (1877) 234-. Locus of centres of quadrics through 7 points of surface of 4th degree. Taylor, H. M. Mess. Mth. 13 (1884) 145-. (Taylor). Frost, P. Mess. Mth. 14 (1885) 97-. point, product of whose distances from 3 coplanar lines is constant. Czuber, E. Mh. Mth. Ps. 3 (1892) 217-. Nodal points, biplanar and uniplanar, theory. Rohn, K. Mth. A. 22 (1883) 124-. Normal developable of quadric cone. Jaroli- mek, C. Casopis 8 (*1879) 247-; Fschr. Mth. (*1879) 574-. surfaces, certain. Sucharda, A. Prag Sb. (1889) 2 (Mth.-Nt.) 385-. One-sided surface ("marrowbone"). Schoute, P. H. Edinb. E. S. P. 19 (1893) 208-. surfaces, class. Feldblmn, M. Wiad. Mt. 1 (1897) 101- ; Fschr. Mth. (1897) 579. Parallel projection, and surfaces. Matzka, W. [1874] Prag Ab. 7 (1875) (Mth.) 70 pp. surface of ellipsoid. Cayley, A. Tortolini A. 3 (1860) 345-. . Fiedler, W. Grunert Arch. 39 (1862) 19-. . Craig, T. Crelle J. Mth. 93 (1882) 251-. surfaces. Hoppe, R. Arch. Mth. Ps. 1 (1884) 280-. of conicoids and conies. Roberts, S. [1872] L. Mth. S. P. 4 (1871-73) 57-. Special Algebraic Surfaces 7650 Parallel surfaces, cylindrical. Dienger, J. Grunert Arch. 10 (1847) 54-. , and lines on surface. Bordoni, A. Mil. G. I. Lomb. 5 (1853) 70-. Polar surfaces of certain ring-surfaces. Su- charda, A. Casopis 13 (1884) 1-, 161- ; Fschr. Mth. (1884) 708-. Quadricuspidal surfaces. La Gournerie, J. de. Liouv. J. Mth. 15 (1870) 264-. Rational surfaces, case. Brambilla, A. Pa- lermo Cir. Mt. Ed. 2 (1888) 176-. connected with theory of complex vari- able. Pincherle, S. Mil. I. Lomb. Ed. 24 (1891) 354-. , ruled, 4th degree, general properties. Ameseder, A. [1879] Arch. Mth. Ps. 65 (1880) 239-. , skew. Genty, . Par. S. Mth. Bll. 19 (1891) 70-. , 4th degree. Noether, M. Mth. A. 33 (1889) 546-. , 9th degree, double curve of special case. Brambilla, A. G. Mt. 32 (1894) 133-. Eeciprocal to surface of centres of central quadric. Roberts, S. QJ. Mth. 13 (1875) 188-. surfaces of same degree as their polars, and with same singularities. Rummer, E, E. Berl. Ak. Mb. (1878) 25-. Eeciprocity in central surfaces of 4th degree, case. Sucharda, A. Wien Ak. Sb. 101 (1892) (Ab. 2a) 585-. Eotation surface with algebraic geodesies. Tannery, J. Par. S. Phlm. Bll. 4 (1892) 85-. ' serving as type of pseudo-spherical sur- faces. Beltrami, E. G. Mt. 10 (1872) 147-. surfaces. Mann, F. Grunert Arch. 29 (1857) 446-. generated by conies. La Gournerie, J. de. Par. EC. Pol. J. 40 e cah. (1863) 1- ; Liouv. J. Mth. 8 (1863) 52-. , plane sections. Benedict, . Silliman J. 31 (1837) 258-. with generator known graphically, ele- ments. Andre, L. Ev. Artl. 5 (1875) 546-. lemniscate for meridian. Levi, D. G. Mt. 21 (1883) 379-. Ruled surfaces, general theorem. Darboux, G. Bll. Sc. Mth. As. 2 (1871) 301- . generated by straight line resting on 3 directrix curves. Salmon, G. Camb. and Dubl. Mth. J. 8 (1853) 45-. , genus p = 0. Clebsch, R. F. A. [1871] Mth. A. 5 (1872) 1-. , new (La Gournerie). Chasles, M. C. R. 63 (1866) 254-. , 3rd degree. Bioche, C. Par. S. Phlm. Bll. 6 (1894) (C. R., No. 13) 1-. , , case. Moshammer, K. WienAk. Sb. 73 (1876) (Ab. 2) 143-. , , model. La Gournerie, J. de. Par. A. Cons. 6 (1866) 205-. , and 4th degrees, obtained from circle. Godefroy, A. N. N. Arch. Wisk. 20 (1893) 108-. 543 7650 Special Algebraic Surfaces. Steiner's Surface 7650 Kuled surfaces, 4th degree, case. Totossy, B. [1882] (xn) Mth. Term. Ets. 1 (1883) 12-; Mth. Nt. B. Ung. 1 (1882-83) 25-. , , generated by 2 projective sheaves of planes of 2nd degree. Holgate, T. F. Am. J. Mth. 15 (1893) 344-; 22 (1900) 27-. , , with generators forming groups of "quadruples." Ameseder, A. Wien Ak. Sb. 81 (1880) (Ab. 2) 615-. , , nodal conic. Ameseder, A. Wien Ak. Sb. 81 (1880) (Ab. 2) 271-. , , 2 nodal lines. Ameseder, A. Arch. Mth. Ps. 65 (1880) 73-. , 6th degree, classification. Bergstedt, J. Fschr. Mth. (1886) 775. , 8th degree, with generators all touching 4 quadric cones, case. La Gournerie, J. A. R. M. de. (x) Par. S. Phlm. Bll. 2 (1865) 124-. , , 4 nodal conies, case. La Gournerie, J. de. C. B. 60 (1865) 1178-. , , 5 nodal quartics, case. La Gournerie, J. de. C. B. 61 (1865) 116-. Self -reciprocal surface. Cayley, A. Berl. Ak. Mb. (1878) 309-. , generation. Cayley, A. [1868] L. Mth. S. P. 2 (1869) 61-. Skew surfaces with director plane. Catalan, E. C. Par. EC. Pol. J. 29" cah. (1843) 121-. , level lines of which are conchoid. Catalan, E. C. Liege S. Sc. Mm. 15 (1888) No. 1, 17-. with 2 rectilinear directrices and director conic. Drasch, H. Wien Ak. Sb. 101 (1892) (Ab. 2a) 171-. , 3rd degree. Cremona, L. Mil. At. I. Lomb. 2 (1860) 291- ; Crelle J. 60 (1862) 313-. , . Cayley, A. Ph. Mg. 24 (1862) 514-. , (Cayley). Steen, A. Mth. Ts. 5 (1863) 86-; Ph. Mg. 25 (1863) 546-. , . Nicodemi, E. G. Mt. 21 (1883) 270-. , , analytical and projective study. Pittarelli, G. G. Mt. 32 (1894) 141-. , , delineation. Cayley, A. Ph. Mg. 25 (1863) 528-. , , first kind, geometry. Ferry, F. C. Arch. Mth. Ntvd. 21 (1899) No. 3, 60 pp. , , polar surfaces. Hochheim, A. Z. Mth. Ps. 23 (1878) 308-, 345-; 24 (1879) 18-. , 4th degree. Cremona, L. Bologna Ac. Sc. Mm. 8 (1868) 235-. , . Cardinaal, J. Amst. Ak. Vs. M. 5 (1889) 447-; Fschr. Mth. (1888) 648. , , case. Nicodemi, R. (xn) Nap. Ac. Pont. At. 15 (1883) (Pt. 2) 179-. , , generated by two plane move- ments. Blake, E. M. N. Y. Am. Mth. S. T. 1 (1900) 421-. , , with 3 nodal lines. Segen, D. Crelle J. Mth. 112 (1893) 39-. ,z = Ay*lx2. Hochheim, A. Arch. Mth. Ps. 53 (1871) 350-. , z = My*x. Hochheim, A. Arch. Mth. Ps. 55 (1873) 38-. Steiner's surface. Kummer, E. E. Berl. Mb. (1863) 324-. Weierstrass, C. Berl. Mb. (1863) 337-. Schroter, H. E. Berl. Mb. (1863) 520-. Cayley, A. [1864] Crelle J. 64 (1865) 172-. Moutard, T. (x) Par. S. Phlm. Bll. 2 (1865) 66-. Clebsch, R. F. A. [1866] Crelle J. 67 (1867) Sturm, R. [1869] Mth. A. 3 (1871) 76-. Eckardt, F. E. [1871] Mth. A. 5 (1872) 30-. Laguerre, E. (x) Par. S. Phlm. Bll. 8 (1872) Cayley , A. [1873] (DC) L. Mth. S. P. 5 (1874) 14-. Amigues, E. C. B. 86 (1878) 38-. Voss, A. Mth. A. 27 (1886) 357-. Kttpper, K. Prag Ab. 2 (1888) No. 10, llpp. Vahlen, K. T. Acta Mth. 19 (1895) 199-. Schoute, P. H. Amst. Ak. Vs. 4 (1896) 224-; Fschr. Mth. (1896) 531. Brambilla, A. Nap. Ed. 37 (1898) 19-. Lacour, E. N. A. Mth. 17 (1898) 437-. Montesano, D. Nap. Ed. 38 (1899) 88-. Asymptotic lines. Darboux, G. (ix) Par. S. Phlm. Bll. 10 (1873) 37. Coordinates of point of, in terms of two parameters. Lacour, E. N. A. Mth. 17 (1898) 499-. Curves on. Schoute, . Amst. Ak. Vs. 4 (1896) 272-; Fschr. Mth. (1896) 531. Locus of poles of fixed plane with reference to conies on, theorem. Lie, M. S. Arch. Mth. Ntvd. 3 (1878) 84-. , . Koenigs.G. [1887] Par. S. Mth. Bll. 16 (1888) 15-. Nodal cones of quadrinodal cubics, and zomal conies of tetrazomal quartics. Toivnsend, R. [1869] QJ. Mth. 10 (1870) 264-. Plane sections, and three intersecting conies. Koenigs, G. Bll. Sc. Mth. 12 (1888) 28-. Property, extension. Brambilla, A. Nap. Ed. 37 (1898) 300-. Surface, 3rd degree, reciprocal of Steiner's surface. Laguerre, . N. A. Mth. 11 (1872) 319-, 337-, 418- ; 12 (1873) 55-. Surfaces of Veronese and Steiner. Rosati, C. Tor. Ac. Sc. At. 35 (1900) 12-. Theorem of Eckardt, extension. Brambilla, A. Nap. Ed. 38 (1899) 144-. Surface of equal slope circumscribing conic. Cremona, L. N. A. Mth. 4 (1865) 271-. generated by apparent directions of star owing to aberration. Wittstein, A. Arch. Mth. Ps. 69 (1883) 195-. circular arc through ellipse and two straight lines. Camuri, A. (ix) Mod. Ac. Sc. Mm. 10 (1869) 3-. Surfaces of constant power with respect to points in space. Rufflni, F. P. Bologna Ac. Sc. Mm. 1 (1890) 235-. containing families of conies. Weyr, E. Mh. Mth. Ps. 2 (1891) 351-. 544 7650 Special Algebraic Surfaces. Degrees 3, 4 7650 Surfaces containing infinite number of conies, tangent planes of. Machovec, F. Prag Ceske Ak. Fr. Jos. Ez. (TfidaZ) 2 (1893) Art. 10, 1-. with continuous transitive group of trans- formations. Fano, G. Palermo Cir. Mt. Ed. 10 (1896) 1-. derived from given surface by composition of two projections, equations and properties. Bottomley, J. (ix) Manch. Lt. Ph. S. Mm. 8 (1884) 218-. quadrics. Andries, C. Brux. A. Un. (1845) 247-. , case. Malet, J. C. [1878] Ir. Ac. T. 26 (1879) 456-. doubly generated by circles, and surfaces doubly enveloped by quadrics. Koenigs, G. C. E. 109 (1889) 364-. with finite number of straight lines, es- pecially of 4th and 5th degrees. Sturm, E. Mth. A. 4 (1871) 249-. hyperelliptic canonical curves. Enriques, F. Em. E. Ac. Line. Ed. 5 (1896) (Sam. 1) 191-. inscribed in scalene cone, special cases. Waf elaer, P. F. Louvain A. Ac. 5 (1823) 45 pp. whose plane sections are all unicursal. Picard, E. Par. S. Phlm. Bll. 2 (1878) 127- ; Crelle J. Mth. 100 (1887) 71-. (Picard). Guccia, G. B. Palermo Cir. Mt. Ed. 1 (1887) 165-. sections are curves of genus three. Castelnuovo, G. Tor. Ac. Sc. At. 25 (1890) 695-. perpendicular to one axis are similar ellipses. Niemtschik, R. Wien Sb. 57 (1868) (Ab. 2) 246-. , unicursal, transformation. Cayley, A. [1870] Mth. A. 3 (1871) 469-. , 6th class, two special. Kantor, S. Wien Ak. Sb. 79 (1879) (Ab. 2) 768-. Surfaces, 3rrf degree. with axis of ternary symmetry, and surfaces with indicatrices of 3rd degree. Dumont, . Par. S. Mth. Bll. 28 (1900) 117-. case considered by Bertrand. Anon, (vi 885) N. A. Mth. 13 (1854) 413-. with conical point, equations. Richmond, H. W. QJ. Mth. 23 (1889) 170-. equation, canonical form for. Dumont, . N. A. Mth. 17 (1898) 503-. Hessian of which breaks up. Ciani, E. Em. E. Ac. Line. Ed. 6 (1890) (Son. 2) 55-. with infinite number of umbilici. Saint- Germain, A. de. C. E. 101 (1885) 1246-. nodal points. Kohn, G. Wien Ak. Sb. 96 (1888) (Ab. 2) 1298-. 4 nodal points. Eckardt, F. E. [1871] Mth. A. 5 (1872) 30-. . Machovec, F. Casopis 16 (1887) 82-; Fschr. Mth. (1887) 645-. , case. Thieme, H. Z. Mth. Ps. 40 (1895) 362-. self-polar with reference to quadric. Dumont, F. Par. S. Mth. Bll. 25 (1897) 74-. 3 straight lines on which meet in point. Eckardt, F. E. [1875] Mth. A. 10 (1876) 227-. Surfaces, 3rd and 4th degrees. with nodal conic. KUpper, . Z. Mth. Ps. 34 (1889) 129-. related to tetrahedron. Jettmar, H. von. Arch. Mth. Ps. 10 (1891) 398-. Surfaces, 4th degree. case. Vanecek, J. S. Prag Sb. (1883) (Mth. Vortr.) 231-. class generated by collineations. Schur, F. Mth. A. 20 (1882) 254-. connected with 2 tetrahedra. Bauer, G. Munch. Ak. Sb. 18 (1889) 337-. with cuspidal conies. [Totossy, B.] Bela, T. [1881] Mth. A. 19 (1882) 291-. . Segre, C. Mth. A. 24 (1884) 313-. giving value of Jx* + y* approximately, case. Poncelet, J. V. N. A. Mth. 7 (1848) 39-. with integral of total differential of 1st kind. Berry, A. C. E. 129 (1899) 449-. 2 intersecting double pairs, construction. Korndorfer, G. Mth. A. 3 (1871) 496-. nodal conic. Clebsch, R. F. A. Berl. Mb. (1868) 284-; Crelle J. 69 (1868) 142-. . Geiser,C.F. Crelle J. 70 (1869) 249-. . Cremona, L. Mil. I. Lomb. Ed. 4 (1871) 140-, 159-. . Zeuthen, H. G. [1879] A. Mt. 14 (*1886-87) 31-. . Berzolari, L. A. Mt. 13 (1885) 81-. . Bobek, K. Wien Ak. Sb. 90 (1885) (Ab. 2) 923-, 1168-. . Del Re, A. Em. E. Ac. Line. Ed. 2 (1893) (Sem. 1) 211-. consisting of two indefinitely near straight lines, construction. Korndorfer, G. Mth. A. 4 (1871) 117-. , construction from quadric systems. Cardinaal, J. Amst. Ak. Vs. M. 8 (1891) 88-; Fschr. Mth. (1891) 847-. and nodes, representation on plane. Korndorfer, G. [1868] Mth. A. 1 (1869) 592-; 2 (1870) 41-. , properties. Juel, C. S. (xn) Ts. Mth. 4 (1880) 81-, 113-. , special case. Rudio, F. Crelle J. Mth. 104 (1889) 85-. , and 16 straight lines. KUpper, K. Prag Ab. 2 (1888) No. 3, 36 pp. , , construction. Del Re, A. Mod. S. Nt. At. 12 (1893) 234-. or cuspidal conic. Weiler, A. Z. Mth. Ps. 30 (1885) 170-. line consisting of circle at infinity. Loria, G. Tor. Ac. Sc. Mm. 36 (1885) 199-. lines, generation from quadric surfaces. Cardinaal, J. Amst. Ak. Vh. (Sect. 1) 1 (1893) No. 6, 63 pp.; Fschr. Mth. (1892) 765-. 2 nodal lines. Fontene, G. N. A. Mth. 19 (1900) 400-. node. Vries, J. de. N. Arch. Wisk. 3 (1898) 158- ; Fschr. Mth. (1897) 532. nodes. Rohn, K. Leip. Jablon. Preisschr. 26 (1886) 58 pp. , construction. Jonquteres, de. C. E. 107 (1888) 430-. VOL. I. 545 7650 Higher Algebraic Surfaces Tore 7650 with 4 nodes, case. Cayley, A. B. A. Rp. (1886) 540-. 12 nodes. Cayley, A. QJ. Mth. 14 (1877) 103-. 14, 15, and 16 nodes. Jessop, C, M. QJ. Mth. 31 (1900) 354-. obtained by projection, case. Pannetti, M. G. Mt. 29 (1891) 133-. passing through 8 points, generation by homo- graphic ruled quadrics. Poudra, . N. A. Mth. 17 (1858) 158-. with properties similar to quadrics. Zmurko, W. Krk. Ak. (Mt.-Prz.) Pam. 14 (1888) 208-; Fschr. Mth. (1888) 827. special case. Sucharda, A. Casopis 24 (1895) 286-; Fschr. Mth. (1895) 627. surface cut along 2 conies by each of its tangent planes. Cremona, L. Crelle J. 63 (1864) 315-. and 2 quadrics, giving points whence 2 conies project into circles. Schiappa Mon- teiro, A. Lisb. J. Sc. Mth. 5 (1898) 221-. with triple point. Eohn, K. Mth. A. 24 (1884) 55-. Surfaces, 5th degree. with nodal cubic and triple point. Del Be, A . Em. E. Ac. Line. Ed. 1 (1892) (San. 2) 170-, 203-, 343-, 378-, [464]. 5 triple points. Del Re, A. Em. E. Ac. Line. Ed. 2 (1893) (Sem. 2) 99-, 138- ; 3 (1894) (Sem. 2) 11-. quartic. Murer, V. Yen. I. At. (1886- 87) 1223-. quintic. Caporali, E. [1875] A. Mt. 7 (1876) 149-. . Del Re, A. Em. E. Ac. Line. Ed. 6 (1890) (Sem. 2) 221-. , extension of configuration of 10 straight lines. Pascal, E. Em. E. Ac. Line. Ed. 2 (1893) (Sem. 1) 65-. , new construction. Del Re, A. Nap. Ed. 25 (1886) 272-. , projective generation. Del Re, A. Tor. Ac. Sc. At. 28 (1893) 420-. simple and nodal lines and one triple line. Del Re, A. Em. E. Ac. Line. Ed. 7 (1891) (Sem. 2) 11-, 111-. 2 or 3 triple points. Del Re, A. Mod. Ac. Sc. Mm. 9 (1893) 331-. ' Surfaces, 6th degree. with 9 nodal lines, case. Del Pezzo, P. Nap. Ed. 36 (1897) 196-. 6 straight lines. Bordiga, G. C. E. 102 (1886) 743-. triple cubic curve. Weyr, E. J. Wien Ak. Sb. 85 (1882) (Ab. 2) 513-. triple tangents of certain. Fieri, M. Tor. Ac. So. At. 24 (1889) 514-. Surfaces, 1th degree. three. Hill, J. E. [1896-97] N. Y. Am. Mth. S. Bll. 3 (1897) 2, 5-; Am. J. Mth. 19 (1897) Surfaces, 8th degree. case. Cayley, A. Mth. A. 4 (1871) 558-. . Battaglini, G. Em. E. Ac. Line. At. 2 (1875) 244-. cases. Cayley, A. QJ. Mth. 14 (1877) 249-. family. Brambilla, A. G. Mt. 35 (1897) 1-. remarkable case. Picquet, H. [1875] Par. S. Mth. Bll. 4 (1876) 45-. Surfaces, nth degree. passing through given cubic curve. Cayley, A. Mess. Mth. 23 (1894) 79-. with (n 1) -pie point, homaloidal system formed by. Paolis,R.de. G. Mt. 13 (1875) 226-, 282-. (n - 2) -fold straight line, generation. Murer, V. Palermo Cir. Mt. Ed. 2 (1888) 107-. l. stem, A. Mth. Ts. 2 (1860) a" b n 101-. Surfaces, (n + 2)th degree. with double curve of degree 1 + n (n - l)/2. Lo Piano, D. Nap. Ed. 39 (1900) 130-. Symmetrical surfaces. Goursat, E. Par. EC. Norm. A. 4 (1887) 159-, 241-, 317-. . Lecornu, L . Acta Mth . 10 (1887) 201-. . Ciani, E. Em. E. Ac. Line. Ed. 6 (1890) (Sem. 1) 399-. . Mangeot, S. C. E. 112 (1891) 1497- ; Par. S. Mth. Bll. 20 (1892) 84-. Dumont,. N. A. Mth. 16 (1897) 463-. , 3rd degree, connected with quadric. Mangeot, S. N. A. Mth. 10 (1891) 235-. Systems of conies and quadrics, locus of centres, etc. Geiser,C.F. Zur.Vjschr. 10 (1865)219-. Tangents common to two confocal surfaces. Chasles, M. Liouv. J. Mth. 11 (1846) 105-. Tetrahedral surfaces, umbilici. Lebon, E. Palermo Cir. Mt. Ed. 4 (1890) 115-. symmetrical surfaces. Hunyady, E. von. Z. Mth. Ps. 11 (1866) 356-. . La Gournerie, J. de. C. E. 62 (1866) 78-. (La Gournerie). Cayley, A. [1867] Camb. Ph. S. T. 11 (pt. 2) (1869) 277-. (Cayley). Sucharda, A. Wien Ak. Sb. 99 (1891) (Ab. 2a) 549-. , 4th degree, with 48 straight lines, case. Schur,F. Liege S. Sc. Mm. 11 (1885) No. 12, 7 pp. Tore or anchor-ring. Cayley, A. QJ. Mth. 13 (1875) 127-. Vries, J. de. N. Arch. Wisk. 3 (1898) 72-. and bitangent sphere. Monniot, . N. A. Mth. 13 (1874) 383-. curve of intersection with sphere, and new property. Fuortes.T. G. Mt. 12 (1874) 146-. cylinder circumscribing rotation tore. Dunesme, M. C. E. 45 (1857) 527-. intersection with quadric. Mangeot, S. N. A. Mth. 11 (1892) 519-. inversion. Taylor, H. M. L. Mth. S. P. 5 (1873-74) 105-. 546 7650 Higher Algebraic Surfaces. Wave Surface 7650 problem : to draw from given point lines doubly tangent to tore. Janin, , < Mannheim, A. (x) Par. S. Phlm. Bll. 2 (1865) 54-. properties. Woodward, C. M. Am. As. P. (1886) 71-. section by plane through centre and tangent at opposite sides. Hyde, E. W, Des Moines Anal. 3 (1876) 78-. tangent plane through centre. Anon. (vi 896) N. A. Mth. 18 (1859) 258-. sections. Garlin, . N. A. Mth. 13 (1854) 415-; 18 (1859) 319-. . Cornu, A. N. A. Mth. 20 (1861) 101-. . Darboux, G. N. A. Mth. 3 (1864) 156-. , bicircular, theorem. Yvon-Villarceau, A. J. F. N. A. Mth. 7 (1848) 345-. , , Yvon-Villarceau's theorem. Allocati,A. G. Mt. 1 (1863) 284-. , , . Barbier, E. Les Mondes 3 (1863) 675-, 703-. , , . Vianna, P. A. G. Teix. J. Sc. 1 (1878) 84-. by bitangent plane. Godart, (Prof.) A. N. A. Mth. 3 (1864) 350-. . Dyri(m,L. N. A. Mth. 4 (1865) 348. . Gerono, G. C. N. A. Mth. 4 (1865) 349-. . Eckardt, F. E. Z. Mth. Ps. 12 (1867) 183. , and by sphere. Nombel, J. N. A. Mth. 4 (1865) 346-. , circular. Godart, A. N. A. Mth. 4 (1865) 159-. , . ricaire,E. Par. S. Mth. Bll. 19 (1891) 46-. , , of tore, etc. Breton [de Champ], P. N. A. Mth. 15 (1856) 40-. , plane. Fuortes, T. G. Mt. 10 (1872) 97. , , constructions for tangent and radius of curvature. Laquiere, E. M. N. A. Mth. 1 (1882) 561-. shadow cast on itself. Cahen, C. P. Par. S. Mth. Bll. 4 (1876) 87-. Wave surface. Cauchy, A. L. C. E. 13 (1841) 184-. Cayley, A. Liouv. J. Mth. 11 (1846) 291-. Prescott, J. E. QJ. Mth. 2 (1858) 1-. Cayley, A. QJ. Mth. 3 (1860) 16-, 142-. Durrande, H. N. A. Mth. 20 (1861) 456-; 2 (1863) 193-, 252-. Mathieu, E. Liouv. J. Mth. 11 (1866) 298-. Turnbull, W. P. Mess. Mth. 3 (1866) 153-, 205-. Catalan, E. C. [1868] Brux. Ac. Sc. Mm. 38 (1871) 64 pp. Mannheim, A. As. Fr. C. E. (1874) 1168- ; (1875) 167- ; (1877) 167- ; (1878) 63-; C. E. 88 (1879) 1248- ; E. S. P. 32 (1881) 447-. (Mannheim's researches.) Niven, C. QJ. Mth. 15 (1878) 242-. Cayley, A. A. Mt. 20 (1892-93) 1-. Lacour, . N. A. Mth. 17 (1898) 266-. Boklen, 0. Z. Mth. Ps. 44 (1899) 289-. More, L. T. N. Y. Am. Mth. S. Bll. 6 (1900) 3, 12-. asymptotic lines. Darboux, G. C. E. 97 (1883) 1039-. Cartesian equation. Tait, P. G. QJ. Mth. 3 (1860) 269-. Catalan's memoir on, and a geometric trans- formation, report. Gilbert, P. Brux. Ac. Bll. 27 (1869) 129-. corresponding surface of elasticity, etc. Lacour, E. N. A. Mth. 19 (1900) 362-. definition, new. Darboux, G. C. E. 92 (1881) 446-. equation in elliptic coordinates. Cayley, A. Mess. Mth. 8 (1879) 190-. , forms. Roberts, S. QJ. Mth. 17 (1881) 319-. equi-radial curve. Walton, W. QJ. Mth. 6 (1863) 144-. wave-cone. Walton, W. QJ. Mth. 6 (1863) 78-. family of surfaces including. Walton, W. Camb. and Dubl. Mth. J. 7 (1852) 105-. Fresnel's wave surface and related surfaces. Booth, W. [1896] Dubl. S. Sc. T. 6 (1898) 205-. generation by intersection of 2 rotation sur- faces. Walton, W. [1860] QJ. Mth. 4 (1861) 310-. , new, and various constructions. Mannheim, A. C. E. 90 (1880) 1333-. Hamilton's singular points. Booth, W. [1896] Dubl. S. Sc. P. 8 (1893-98) 381-. modelling, practical method. Hicks, W. M. Mess. Mth. 5 (1876) 183-. normals. Mannheim, A. As. Fr. C. E. (1877) 175-. , curvatures, and lines of curvature. Mann- heim, A. C. E. 64 (1867) 170-, 268-. optical properties. Mannheim, A. As. Fr. C. E. (1875) 231- ; C. E. 82 (1876) 368-; 122 (1896) 708-. properties. Niven, C. QJ. Mth. 15 (1878)257-. property. Townsend, It. [1872] Mess. Mth. 2 (1873) 28-. , generalisation. Anon, (xi 55) N. A. Mth. 1 (1882) 29-. , geometrical. Walton, W. QJ. Mth. 4 (1861) 151-. representation by elliptic functions. Weber, H. Ziir. Vjschr. 41 (1896) (Festschr., Th. 2) 82-. as surface limit. Mannheim, A. C. E. 90 (1880) 971-. tangent planes and radii, relation to ellipsoid of construction. Walton, W. QJ. Mth. 5 (1862) 285-. , singular, and sections by planes parallel to them. Mannheim, A. As. Fr. C. E. (1877) 125-. theorems. Niven, C. QJ. Mth. 9 (1868) 22-. _, 2. Mannheim, A. C. E. 78 (1874) 839-. triple system of confocal quartics comprising. Legoux, A. N. A. Mth. 4 (1885) 393-. umbilici, geometric determination. Mannheim, A. C. E. 88 (1879) 902-. Weddle's quartic surface. Cayley, A. C. E. 52 (1861) 1216-. . Poudra, . N. A. Mth. 2 (1863) 307-. . Hierholzer,C. Mth. A. 4 (1871) 172-. 547 MM 2 7660 Skew Algebraic Curves. Cubics 7660 Weddle's quartic surface. Laguerre, E. (x) ee's quartic surface. Lague Par. S. Mth. Bll. 1 (1873) 71-. --- . Hunyady, E. von. [1881] Crelle J. Mth. 92 (1882) 304-. --- . Hutchinson, J. I. A. Mth. 11 (1896-97) 158-. --- , analogue. Seydewitz, F. Grunert Arch. 10 (1847) 202-. --- , generalisation. Cayley, A. [1872] L. Mth. S. P. 4 (1871-73) 11-. zy 2 + Bxy + Cz 2 + Dy + Ex + F= 0. Terquem, 0. N. A. Mth. 5 (1846) 370-. x n + y n + z n =l. Burhenne, H. Grunert Arch. 21 (1853) 35-. ^ + ? + - = l. Boklen, 0. Grunert Arch. a o c 35 (1860) 93-. X^ + /w 2 + vr 3 = 0. Cayley, A. Ph. Mg. 21 (1861) 491-. Ax 3 + By* + QCRST=Q. Cayley, A. Ph. Mg. 27 (1864) 493-. (*) (U, V, JF) 2 =0. Cayley, A. QJ. Mth. 10 (1870) 24- ; 11 (1871) 15-. 2 ='/'X. etc. Kummer, E. E. Berl. Mb. (1872) 474-. V = 0. Cayley, A. QJ. Mth. 14 (1877) 46-. xyz = l 3 , property. Floquet, G. C. B. 105 (1887) 854-. 2xyz -ax 2 - ay* - az* + a 3 = 0. Allardice, E. E. Edinb. Mth. S. P. 10 (1892) 59-. 7660 Skew algebraic curves. (See also 7240, 7660, 8030, 8050.) Bischoff, J. N. Crelle J. 58 (1861) 179- ; 59 (1861) 394-. Halphen, G. C. B. 70 (1870) 380- ; 76 (1873) 558. Eckardt, F. E. [1871] Mth. A. 5 (1872) 30-. Weyr, E. [1873] (xi) Prag Ab. 6 (1874) (Mth.) 27pp. Schoute, P. H. Amst. Ak. Vs. M. 14 (1879) 251-. Sturm, R. B. A. Bp. (1881) 440-. Noether, M. Berl. Ak. Ab. (1882) 120 pp. Valentiner, H. Acta Mth. 2 (1883) 136-. Rados, G. Mth. Termt. Ets. 8 (1890) 46-; Mth. Nt. B. Ung. 8 (1891) 90-. Anisimov, V. A. Bee. Mth. (Moscou) 17 (1895) 447-. Chords of curve through given point. Fries, H. de. N. Arch. Wisk. 1 (1895) 127- ; Fschr. Mth. (1893-94) 1232. Classification. Salmon, G. Camb. and Dubl. Mth. J. 5 (1850) 23-. . Halphen, G. H. Par. EC. Pol. J. Cah. 52 (1882) 1-. by monoid surfaces. Cayley, A. C. B. 54 (1862) 55-, 396-, 672-; 58 (1864) 994-. Closed curves, Gaussian integral giving number of knots of 2. Thomae, C. J. Freiburg B. 7 (1880) 18-. , parallel. Hoppe, . D. Nf. Tbl. (*1880) 174-. Closed curves, representation on sphere. Flye Sainte-Marie, C. (rx) Par. S. Mth. Bll. 1 (1873) 82-. Common bisecants. Weyr, E. J. Wien Ak. Sb. 85 (1882) (Ab. 2) 840-. Construction, linear. Seydewitz, F. Grunert Arch. 10 (1847) 203-. Curves and developables. Cayley, A. Liouv. J. Mth. 10 (1845) 245-; Camb. and Dubl. Mth. J. 5 (1850) 18-; QJ. Mth. 7 (1866) 110- ; 11 (1871) 294-. , polar planes. Bdcklund, A. V. Lund Acta Un. 8 (1871) (Mth.) No. 4, 50 pp. on quadric whose osculating planes touch a quadric. Stahl,W. Crelle J. Mth. 99 (1886) 154-. Curves, 3rd degree. Cremona, L. Tortolini A. 1 (1858) 164-, 278-; 2 (1859) 19-; N. A. Mth. 19 (I860) 356-; 1 (1862) 287-, 366-, 436- ; Crelle J. 60 (1862) 188- ; Bologna Mm. Ac. Sc. 3 (1863) 385-; G. Mt. 1 (1863) 278-. Drach, C. A. van. Z. Mth. Ps. 12 (1867) (Suppl.) 73-. Beltrami, E. Mil. I. Lomb. Ed. 1 (1868) 130-, 407-. Schroder, . D. Nf. Tbl. (*1879) 176. Catania, S. Catania Ac. Gioen. At. 19 (1886) 85-. Cantone, A. Palermo Cir. Mt. Bd. 1 (1887) 72-. Cardinaal, J. Crelle J. Mth. 101 (1887) 142-. Caspary, F. Bll. Sc. Mth. 11 (1887) 222-. Balitrand, . N. A. Mth. 8 (1889) 520-. Dixon, A. C. QJ. Mth. 23 (1889) 343-; 24 (1890) 30-. Bioche, C. Edinb. Mth. S. P. 13 (1895) 146-. Stuyvaert, . Brux. Ac. Bll. (1900) 820-. analogue of helicoidal motion. Appell, P. C. B. 82 (1876) 70-; Par. EC. Norm. A. 5 (1876) 245-. application of binary cubic and quadratic forms. Pittarelli, G. G. Mt. 17 (1879) 260-. forms. Ovidio, E. d\ [1879] Tor. Ac. Sc. Mm. 32 (1880) 1-. imaginaries in geometry. Servais, C. Brux. Mm. Cour. 8, 49 (1896) No. 3, 64 pp. quaternions. Hamilton, (Sir) W. R. [1863] Ir. Ac. P. 8 (1864) 331-. resultant of 2 equations. Waelsch, E. Mh. Mth. Ps. 2 (1891) 421-. automorphic for octahedral and ikosahedral groups, two. Kohn, G. Wien Ak. Sb. 108 (1899) (Ab. 2a) 58-. binary cubic and quartic forms on. L. Palermo Cir. Mt. Bd. forms on. Sturm, R. Crelle (1879) 116-. of order 5 on. Waelsch, E. Wien Ak. Sb. 100 (1891) (Ab. 2a) 803-. case. Hurwitz,A. Mth. A. 30 (1887) 291-. centre. Geisenheimer, L. Z. Mth. Ps. 27 (1882) 321-. cones through a given curve. Cayley, A. Ph. Mg. 12 (1856) 20-. congruences associated with. Kriiger, H. Z. Mth. Ps. 40 (1895) 193-. 548 7660 Skew Algebraic Curves. Degrees 3, 4 7660 48 coordinates. Spottiswoode, W. Phil. Trans. 172 (1882) 375-. correspondence with conic. Weyr, Em. Mil. I. Lomb. Ed. 4 (1871) 636-. and developables, class. Lelieuvre, . C. B. 117 (1893) 537-, 616-. elimination problem. Pittarelli, G. G. Mt. 17 (1879) 244-. equiangular curve. Krilger, H. Z. Mth. Ps. 38 (1893) 344-. focal curve of skew parabola. Wirtinger, W. Wien Ak. Sb. 94 (1887) (Ab. 2) 302-. system. Servais, C. Brux. Mm. Cour. 8, 52 (1894-95) No. 2, 11 pp. generation, elements and characteristics. Sturm, B. Crelle J. Mth. 79 (1875) 99-; 80 (1875) 128-, 334. through 5 given points. Koenigs, G. N. A. Mth. 2 (1883) 301-. 6 given points, graphic construction. Poudra, N. G. N. A. Mth. 5 (1866) 313-. historical notes. Reye, T. Hamb. Mth. Gs. Mt. 2 (1890) (Festschr., Th. 2) 43-. hyperboloidal projection. Cremona, L. (vn) A. Mt. 5 (1863) 227-. inscribed tetrahedra. Gallucci, G. Nap. Bd. 37 (1898) 207-. involution on a curve, and resulting complex. Wetter, A. Z. Mth. Ps. 24 (1879) 159-. metrical properties. Valeri, D. Mod. Ac. Sc. Mm. 8 (1892) 385-. . Heinrichs, E. Z. Mth. Ps. 39 (1894) 213-, 273-. . Sturm, R. Z. Mth. Ps. 40 (1895) 1-. . Mehmke, R. Z. Mth. Ps. 40 (1895) 211-. number of secants common to 2. Sturm, R. A. Mt. 3 (1869-70) 28-. osculating plane. Tannery, J. Bll. Sc. Mth. As. 11 (1876) 183-. at infinity, metrical properties. Loria, G. Nap. Bd. 24 (1885) 299-. perspective relations to plane curves. Weyr, Em. Prag Sb. (1869) 22-. plane and skew. Mangeot, S. Par. S. Mth. Bll. 21 (1893) 44-. 4, 5 and 6 point configurations. Guichard, C. Par. EC. Norm. A. 3 (1886) 259-. with 2 points, tangents and osculating planes common, pencil. Stuyvaert, . N. A. Mth. 19 (1900) 548-. polar reciprocal of cubic curve. Zeeman, P. (Gz.) Fschr. Mth. (1900) 626. projections into conies. Bioche, C. N. A. Mth. 17 (1898) 541-. family of conies. Fabry, C. N. A. Mth. 8 (1889) 56-. properties. Chasles, M. Liouv. J. Mth. 2 (1857) 397-. . Lee, T. B. L. Mth. S. P. 24 (1893) 352-. . Cazamian, A. N. A. Mth. 14 (1895) 108-. on quadric surfaces. Milller, Hub. Mth. A. 1 (1869) 407-. . Cayley, A. Mess. Mth. 14 (1885) 129-. surfaces and curves circumscribing polyhedra formed by osculating planes. Cremona, L. Mil. I. Lomb. Bd. 12 (1879) 347-. 4 tangents. Voss, A. E. Mth. A. 13 (1878) 168-. theorem. Hurwitz, A. Mth. A. 20 (1882) 135-. . Cantone, A. G. Mt. 25 (1887) 42-, 182. of Chasles. Cayley, A. Liouv. J. Mth. 10 (1845) 383-. . Stuyvaert, . Mathesis 20 (1900) 181. theorems derived from involution. Cantone, A. Nap. Ed. 25 (1886) 181-. touching tangent planes of skew cubic in 4, 5 or 6 points. Kohn, G. Wien Ak. Sb. 105 (1896) (Ab. 2a) 1035- ; Mth. A. 52 (1899) 293-. and 3rd class. Schroter, H. E. Crelle J. 56 (1859) 27-. (Schroter). Joachimsthal, F. Crelle J. 56 (1859) 44-. . Cremona, L. Crelle J. 58 (1861) 138-. Curves, 3rd and 4/t degrees. description. Chasles, M. C. B. 53 (1861) 767-. polarity in. Stuyvaert, . Brux. Ac. Bll. (1900) 87-. theorems, new. Folie, F. Brux. Ac. Bll. 38 (1874) 65-, 465-. Curves, th degree. elliptic. Chasles, M. C. E. 54 (1862) 317-, 418-. . Laguerre, E. (x) Par. S. Phlm. Bll. 5 (1868) 65-. . Schroeter, H. E. Crelle J. Mth. 93 (1882) 132-. . Spottiswoode, W. [1882] L. Mth. S. P. 14 (1882-83) 15-. . Milinowski, . Crelle J. Mth. 97 (1884) 277-. . Gegenbauer, L. Wien Ak. Sb. 93 (1886) (Ab. 2) 790-. , application of elliptic functions. Mathews, G. B. L. Mth. S. P. 19 (1889) 507-. f . Loria, G. Bm. B. Ac. Line. Ed. 6 (1890) (Sem. 2) 179-. _, . Vdlyi, G. Mth. Termt. Ets. 10 (1892) 244-; Mth. Nt. B. Ung. 10 (1893) 181-. _, . Kluyver, J. C. Am. J. Mth. 19 (1897) 319-. , Chasles' s generation method. Laguerre, . Liouv. J. Mth. 15 (1870) 193-. , configurations on. Ameseder, A. Wien Ak. Sb. 87 (1883) (Ab. 2) 1179-. , coordinates of tangent line. Roberts, W. , with cusp. Weyr (Ab. 2) 400-, B. W. L. Mth. S. P. 12 (1880-81) 143-. . Weyr, E. J. Wien Ak. Sb. 71 (1875) (Ab. 2) 400-. , double point. Weyr, E. J. Wien Ak. Sb. 75 (1877) (Ab. 2) 168-. , , principal polygons. Brambilla, A. Nap. Ac. At. 9 (1899) No. 10, 33 pp. , double points in projection. Zeuthen, H. G. Ts. Mth. 2 (1884) 60-; Fschr. Mth. (1884) 520. 549 7660 Skew Algebraic Curves. Degrees 5 and higher 7660 elliptic, inflexion tangents. Kluyver, J. C. Amst. Ak. Vs. M. 8 (1891) 346-; Fschr. Mth. (1891) 849-. , intersection of sphere with quadrie. Laguerre, E. (x) Par. S. Phlm. Bll. 4 (1867) 51-. , and intersections with quadrie surface. Eeye,T. [1868] A. Mt. 2 (1868-69) 129-.222-. , plane bicircular quartics, theorem. Lachlan, R. L. Mth. S. P. 21 (1891) 274-. , properties. Duporcq, E. N. A. Mth. 15 (1896) 266-. , and 2 quadratic forms. Westphal, G. Mth. A. 13 (1878) 1-. , rational. Eberhard, V. Z. Mth. Ps. 32 (1887) 65-, 129-. , secant systems, representation by elliptic functions. Harnack, A. Mth. A. 12 (1877) 47-; 13 (1878)49-. , 16 stationary points. Lange, E. Z. Mth. Ps. 28 (1883) 1-, 65-. , tetrahedra in multiple perspective on. Vdlyi, G. Mth. Termt. Ets. 11 (1893) 322-; Mth. Nt. B. Ung. 13 (1897) 183-. models. Rohn, G. D. Mth. Vr. Jbr. 1 (1892) 43-. through 8 points, intersections with plane through 2. Serret, P. C. B. 82 (1876) 322-, 370-. and quadrie surfaces, imaginary projection. Retali, V, [1885] Bologna Bd. (1885-86) 21-. rational. Cremona, L. (vn) A. Mt. 4 (1861) 71-. . Bertini, E. Mil. I. Lomb. Bd. 5 (1872) 622-. Armenante, A. [1873] G. Mt. 11 (1873) 221- ; 12 (1874)250-. . Weyr, E. J. Prag Sb. (1874) 164-. . Appell, P. C. B. 83 (1876) 1209- ; Arch. Mth. Ps. 62 (1878) 175-. . Adler, A. [1882] Wien Ak. Sb. 86 (1883) (Ab. 2) 919-, 1201-, 1212-. . Roberts, R. A. L. Mth. S. P. 14 (1882- 83) 22-, 308-. . Brambilla, A. Ven. I. At. (1884-85) 1471-. . Wirtinger, W. [1885] Wien Ak. Sb. 93 (1886) (Ab. 2) 28-. . Study, E. Leip. Mth. Ps. B. 38 (1886) 3-. . Meyer, F. Mth. A. 29 (1887) 447-. . Berzolari, L. Mil. I. Lomb. Bd. 23 (1890) 96-. . Forsyth, A. R. QJ. Mth. 27 (1895) 247-. . Griinwald, J. Wien Ak. Sb. 108 (1899) (Ab. 2a) 1009-. , case. Cremona, L. Mil. I. Lomb. Bd. 1 (1868) 199-. , cases. Brambilla, A. Nap. Bd. 24 (1885) 279-. , combinants of binary forms associated with. Berzolari, L. A. Mt. 20 (1892-93) 101-. , with double point. Brambilla, A. Mil. I. Lomb. Bd. 17 (1884) 857-. , hyperboloids attached to. Berzolari, L. Mil. I. Lomb. Bd. 25 (1892) 950-. , lines of striction of hyperboloids as. Migotti, A. [1879] Wien Ak. Sb. 80 (1880) (Ab. 2) 1023-. rational, number through 8 points. Cayley, A. As. Fr. C. B. 9 (1880) 135-. , point systems on. Weyr, E. J. Wien Ak. Sb. 75 (1877) (Ab. 2) 458-. , stationary planes coinciding in pairs. Weyr, Em. Mil. I. Lomb. Bd. 4 (1871) 144-. Curves, 5th and higher degrees. 5th degree, genus 1. Fries, J. de. Amst. Ak. Vs. 8 (1900) 451- ; Amst. Ak. P. 2 (1900) 374-. , rational. Berzolari, L. Bm. B. Ac. Line. Mm. 7 (1891) 305-. 6th degree, classification. Weyr, Ed. C. B. 76 (1873) 424-, 475-, 555-. , genus 1. Petot, A. C. B. 102 (1886) 805-. , 4, triple tangent planes. Pascal, E. Bm. B. Ac. Line. Ed. 2 (1893) (Sem. 1) 204-, 239-. , rational, quadrisecants. Deruyts, F. Brux. Ac. Bll. 35 (1898) 421-. 7th degree. Weyr, E. Wien Ak. Sb. 69 (1874) (Ab. 2) 399-. , case. Schoute, P. H. N. Arch. Wisk. 10 (1884) 54-. nth degree, with (n - 2) -fold secants. Bobek, K. Wien Ak. Sb. 95 (1887) (Ab. 2) 349-. pih and qth degrees, intersections. Picquet, H. C. B, 77 (1873) 474-. Cyclics. Darboux, G. C. B. 68 (1869) 1311- ; Bordeaux Mm. S. Sc. 8 (1870) 291- ; 9 (1873) 1-. Degree, theorems. Saltel, L. Brux. Ac. Bll. 43 (1877) 29-. Differential covariants. Gwyther, R. F. B. S. P. 56 (1894) 272-. Double normals, number. Fieri, M. Bm. B. Ac. Line. Bd. 2 (1886) (Sem. 1) 327-. points, apparent. Brunn, H. [1893] D. Mth. Vr. Jbr. 3 (1894) 84-. Generation by variable figures. Caspary, F. Crelle J. Mth. 100 (1887) 405-. (Caspary). Carvallo, . Par. S. Mth. Bll. 15 (1887) 158-. Halphen's characteristic n. Cayley, A. Crelle J. Mth. Ill (1893) 347-. Hyperboloid of one sheet, curves on. Chasles, M. C. B. 53 (1861) 985-. Inflexional tangent planes. Clebsch, R. F. A. [1862] (vii) Crelle J. 63 (1864) 1-. Inscribed polygons. Deruyts, F. Brux. Ac. Bll. 36 (1898) 553-. Intersection of 2 cones, construction of tan- gents at double points. Juel, C. N. Ts. Mth. 8 (B) (1897) 28-; Fschr. Mth. (1897) 490. projects into Cartesian oval. Trenquelleon, C. de. N. A. Mth. 3 (1864) 539-. Involutions on skew curves. Deruyts, F. Brux. Ac. Bll. 36 (1898) 194- . Isographic figures, and generation of skew curves. Jonquieres, E. de. G. Mt. 23 (1885) 48-. 550 7660 Skew Algebraic Curves Arch. Mth. Ps. Knotted curve. Hoppe, R. 64 (1879) 224. , Hoppe's. Durege, H. [1880] Wien Ak. Sb. 82 (1881) (Ab. 2) 135-. curves. Brill, A. Mth. A. 18 (1881) 95-. Line geometry of skew curves. Voss, A. E. Mth. A. 13 (1878) 232-. Multiple secants of curve in 3- and 4-dimen- sional space. Berzolari, L. Palermo Cir. Mt. Ed. 9 (1895) 186-. Number of constants involved. Rohn, K. [1896] D. Mth. Vr. Jbr. 5 (1901) (Heft 1) points determining curves as intersection of S m , S n . Terquem, O. (vi Adds.) N. A. Mth. 15 (1856) 357-. Numbers analogous to Pliicker's for skew curves. Kluyver, J. C. Amst. Ak. Vs. M. 7 (1890) 121- ; Fschr. Mth. (1890) 683-. Osculating plane of intersection of 2 surfaces, reduction of degree by 2. Hesse, L. 0. Crelle J. 40 (1850) 316-. planes. Hesse, L. 0. Crelle J. 41 (1851) 272-. Points on 2 curves, problem. Grunert, J. A. Wien SB. 17 (1855) 3-. Projections. Wiener, . D. Nf. Tbl. (*1879) Properties. Steiner, J. Berl. B. (1839) 76-. . Halphen, G. H. Par. S. Mth. Bll. 2 (1874) 69-. Eational curves. Korndorfer, G. [1870] Mth. A. 3 (1871) 415-. . Weyr, Em. G. Mt. 9 (1871) 217- ; Z. Mth. Ps. 16 (1871) 354-. . Tognoli, 0. G. Mt. 12 (1874) 220-. . Genty, . Par. S. Mth. Bll. 9 (1881) 115. . Weyr, E. J. Mil. I. Lomb. Ed. 15 (1882) 250- ; Prag Sb. (1882) 158-. . Schoute, P. H. Amst. Ak. Vs. 8 (1900) 548-; Amst. Ak. P. 2 (1900) 421-. and apolarity. Meyer, F. W. Mth. A. 21 (1883) 125-. , generation. Stahl, W. Mth. A. 40 (1892) 1-. , multiple secants. Deruyts, F. Brux. Ac. Bll. 35 (1898) 287-. , plane and skew. Roberts, R. A. L. Mth. S. P. 17 (1887) 25-. , , in- and circum-polygons. Zecca, G. G. Mt. 25 (1887) 333-. and ruled surfaces. Brill, A. Munch. Ak. Sb. 15 (1886) 276-. , theorem. Appell, P. Arch. Mth. Ps. 60 (1877) 125-. Eectangular trihedra, whose edges are normals to quadric, locus of vertices. Painvin, L. N. A. Mth. 10 (1871) 337-. Eepresentation by cones. Cayley, A. QJ. Mth. 3 (1860) 225-; 5 (1862) 81-. of curves, and a formula of Halphen. Autonne, L. C. E. 119 (1894) 845-. , number of conditions determining them. Autonne, L. Lyon Un. A. [20] 7660 Loria, G. Par. S. (1896) 37 pp. Euled surfaces of 3rd and 4th degrees, curves on. Chasles, M. G. E. 53 (1861) 884-. Self-conjugate curves. Mth. Bll. 23 (1895) 4-. Singular points. Spottiswoode, W. Crelle J. 42 (1852) 372-. . Halphen, G. H. As. Fr. C. B. 6 (1877) 132-. , classification. Chome, F. Mathesis 18 (1898) 177-. , theory. Meder, A. Crelle J. Mth. 121 (1900) 230-. , types. Meder, A. Crelle J. Mth. 116 (1896) 50-, 247-. Singularities. Halphen, G. H. [1877] Par. S. Mth. Bll. 6 (1878) 10-. . Bjarling, C. F. E. Arch. Mth. PB. 8 (1890) 83-; 10(1891) vn. . Meyer, F. Gott. Nr. (1891) 88- ; D. Mth. Vr. Jbr. 2 (1893) 62-. , discriminants of equations for. Meyer, F. Mh. Mth. Ps. 4 (1893) 229-, 331-. , reduction. Pannelli, M. Mil. I. Lomb. Ed. 26 (1893) 216-. Spherical curve of 4th degree. Gudermann, C. [1845] Crelle J. 43 (1852) 93-. curves. Davies, T. S. [1832] Edinb. B. S. T. 12 (1834) 259-, 379-. . Mobius, A. F. N. A. Mth. 12 (1853) 238-. with no singular points, form. Mobius, A. F. Leip. B. 2 (1848) 179-. , 3rd class, with double foci and double cyclic arcs. Jeffery, H. M. B. A. Bp. (1876) (Sect.) 19-; QJ. Mth. 15 (1878) 131-. , , with 3 single foci. Jeffery, H. M. [1880] QJ. Mth. 17 (1881) 104-. , 4th class, with quadruple foci. Jeffery, H. M. [1881] QJ. Mth. 18 (1882) 270-. , 3rd degree, with triple foci. Jeffery, H. M. B. A. Ep. (1875) (Sect.) 16-. , 4th degree. Casey, J. Phil. Trans. 161 (1871) 585-. , , foci. Jeffery, H. M. QJ. Mth. 21 (1886) 130-. , , with quadruple cyclic arc, and triple focus. Jeffery, H. M. L. Mth. S. P. 12 (1880-81) 168-. cyclics (intersections of quadric and sphere). Saltel, L. Par. S. Mth. Bll. 3 (1875) 95-, 99-. lemniscate arcs. Ctiasles, M. C. E. 21 (1845) 199-. (Chasles). Roberts, W. Liouv. J. Mth. 10 (1845) 451-. parabola. Arrest, H. d'. Leip. B. (1853) 58-. and plane parabola, analogies. Mehmke, R. [1876] Arch. Mth. Ps. 60 (1877) 215-. polygons and their polars, theorems. Lowenstern, S. Crelle J. 13 (1835) 79-, 159-. Spherics, analytical. Gudermann, C. Crelle J. 6 (1830) 244-; 13 (1835) 262-. , elementary. Gudermann, C. Crelle J. 8 (1832) 363-. Sphero-conic is locus of points equidistant from 2 small circles. Gudermann, C. Crelle J. 8 (1832) 160-. Sphero-conics. Cayley, A. QJ. Mth. 3 (1860) 53. 551 8000 Transformations Sphero-conics. Cremona, L. N. A. Mth. 19 (1860) 269-. . Heilermann, . Schlomilch Z. 6 (1861) 153-. . Sykes, G. S. [1877] Am. Ac. P. 13 (1878) 375-. . Morley, F. Mth. Gz. 1 (1900) 249-. , application of elliptic functions. Enneper, A. Z. Mth. Ps. 22 (1877) 244-. , homogeneous spherical coordinates. Cassani, P. G. Mt. 6 (1868) 81-. , confocal. Chasles, M. Liouv. J. Mth. 5 (1860) 425-. , . Fournier-Vannson, . N. A. Mth. 19 (1860) 197-. , . Heilermann, . Schlomilch Z. 6 (1861) 326-. and developable formed by tangents. Huber, G. Z. Mth. Ps. 45 (1900) 86-. as envelopes of systems of circles. Czuber, E. Arch. Mth. Ps. 7 (1889) 143-. , focal circles. Graves, (Rt. Rev.) C. [1889] Ir. Ac. T. 29 (1887-92) 285-. in- and circum-scribed to quadrangle. Jeffery, H. M. QJ. Mth. 13 (1875) 350-. , mechanical description. Hart, H. L. Mth. S. P. 6 (1874-75) 136-. , properties. CJiasles, M. Brux. Ac. Sc. Mm. 6 (1830) 46 pp. Sphero- and plane conies, properties. Mehmke, R. Z. Mth. Ps. 23 (1878) 255-. Surface of triple secants ; number of quadruple secants. Picquet, H. (xi) Par. S. Mth. Bll. 1 (1873) 260-. ; (Picquet). Guccia, G. B. Palermo Cir. Mt. Rd. 1 (1887) 26-. Surfaces passing through skew curves, "ex- cess." Kohn, G. [1880] Wien Ak. Sb. 82 (1881) (Ab. 2) 755-. Tangent, construction. Anon, (vi 533) Ger- gonne A. Mth. 10 (1819-20) 89-. Tangents and osculating planes of curves. Czuber, E. Mh. Mth. Ps. 3 (1892) 337-. to projections of curves. Hachette, J. N. P. Par. EC. Pol. Cor. 3 (1814-16) 197-. Transformations and General Methods for Algebraic Con- figurations. (References to papers dealing with enu- merative geometry may be found in sections 8000 to 8100.) 8000 General. Centrals, theory. Grassmann, H. Crelle J. 24 (1842) 262-, 372-; 25 (1843) 57-. Configurations, definition by construction of their polar systems. Thieme, H. Z. Mth. Ps. 24 (1879) 221-, 276-. Continuous correspondences between the points of r groups. Paolis, R. de. A. Mt. 18 (1890) 93-. Collineation ; duality 8010 Envelopes of Sm^ (cue,- + py f + yz t - p) 2 = 0. Ruffini, F. P. Bologna Ac. Sc. Mm. 4 (1882) 123-. Equations of geometry. Jordan, C. C. R. 68 (1869) 656-. Higher geometry. Dillner, G. Ts. Mt. Fys. 1 (1868) 27-, 124-, 169-, 209-, 270- ; 2 (1869) 82-, 137- ; 3 (1870) 30-, 111-, 176- ; Ups. N. Acta S. Sc. 8 (1873) No. 9, 136 pp. Homogeneous project! ve coordinates in plane and space. Aschieri, F. Mil. I. Lomb. Rd. 25 (1892) 381-. Imaginary points; antiprojective correspond- ence. Segre, C. Tor. Ac. Sc. At. 25 (1890) 276-, 430-, 592- ; 26 (1891) 35-. Locus theorems in which generating point divides variable linear segment in constant ratio, generalisation. Roberts, S. Am. J. Mth. 3 (1880) 336-. Point system, harmonic mean point. Kohn, G. [1883] Wien Ak. Sb. 88 (1884) (Ab. 2) 424-. systems, equations representable by 3. Ocagne, M. d' . Par. S. Phlm. Bll. 5 (1893) (C. R., No. 17) 2-; C. R. 123 (1896) 988-, 1098, 1254-. Projective relations of geometric systems. Franke, J. N. Par. T. Nauk Sc. Pam. 1 (*1871) 101-. Singular points on algebraic plane curves. Tichomandrickij, M. A. Kharkov Mth. S. Com. 2 (1891) 114- ; Fschr. Mth. (1891) 757. Transformation in geometry and in mathe- matical physics. Haton de la Goupilliere, J. N. C. R. 58 (1864) 1001- ; Par. EC. Pol. J. 25 (1867) 153-. principle. Hesse, L. 0. [1865] Crelle J. 66 (1866) 15-. . Cayley, A. [1866] Crelle J. 67 (1867) 95-. sequence. Muirhead, R. F. Edinb. Mth. S. P. 16 (1898) 70-. Transformations, circular. Emch, A. [1898] A. Mth. 12 (1898-99) 141-. of geometric propositions, examples. Zen- then, J. C. G. N. A. Mth. 3 (1864) 297-. , theory. Servais, C. Mathesis 8 (1888) 105-. Trigraphy, applications. Russell, J. W. L. Mth. S. P. 26 (1895) 446-. 8010 Collineation; duality. Affinity of geometric figures, general kind. Mobius, A. F. Crelle J. 12 (1834) 109-. , special case, applications. Korteweg, D. J. Z. Mth. Ps. 21 (1876) 28-. . tyP 6 - Timerding, H. E. Arch. Mth. Ps. 17 (1900) 60-. Anharmonic ratio, analogues. Brill, J. QJ. Mth. 29 (1898) 286-. , conical, of 10 points. Mozat, . C. R. 118 (1894) 790-. 552 8010 Collineation ; duality 8010 Automorphic curves and surfaces, family. Klein, F., & Lie, S. C. E. 70 (1870) 1222-, 1275-. infinite transformations of a surface. En- riques, F. Ven. I. At. (1892-93) 1590- ; (1893-94) 638-. transformations of space, groups. Schon- flles, A. Mth. A. 34 (1889) 172-. Birational transformations in geometry of posi- tion. Fiedler, W. Ziir. Vjschr. 21 (1876) 369-. Collinear figures. Frischauf, J. Steierm. Mt. 2 (Heft 1) (1869) 85-. forms, united elements. Nicoli, F. Mod. Ac. Sc. Mm. 7 (1890) 253-. fundamental configurations. Reye, T. [1870] Crelle J. 74 (1872) 1-. , curves and surfaces generated by. Schur, F. Mth. A. 18 (1881) 1-. forms of 2nd and 3rd species having united elements. Boschi, P. Bologna Ac. Sc. Mm. 2 (1880) 507-. planes, metrical relations between. Ise, E. Nap. Ac. Pont. At. 24 (1894) No. 3, 15 pp. ray-systems, 2, generation of curves of 4th and 3rd degrees by. Heger, R. [1873] Z. Mth. Ps. 19 (1874) 170-. and reciprocal forms in ordinary geometry. Aschieri,F. [1879] Bologna Ac. Sc. Mm. 1 (1880) 145-. spaces, cyclic-projective point-quadruples in two. Schroeter, H. E. Mth. A. 20 (1882) 231-. , quintuple position. Ameseder, A. Wien Ak. Sb. 98 (1890) (Ab. 2a) 588-. , series of points, planes, etc., in. Sturm, R. Mth. A. 28 (1887) 261-. systems. Genty, M. Par. S. Mth. Bll. 21 (1893) 131-. transformation of foci of conic. King, L. Arch. Mth. Ps. 6 (1888) 88-. Collineation. Sturm, R. Mth. A. 10 (1876) and affinity, identity of projection with. Spitzer, S. Grunert Arch. 9 (1847) 345-. , central, nth order, and plane Collineation of nth class. Beyel, C. Ziir. Vjschr. 31 (1886) 1-. , , , by similar points of circles. Beyel, C. Ziir. Vjschr. 26 (1881) 297-. Collineation, new theory. Mehmke,R. Fschr. Mth. (1888) 861-. between non-ruled quadric and system of spheres, locus of centre. Jefdbek, W. Arch. Mth. Ps. 65 (1880) 161-. , paradox in theory. Hofmann, F. Z. Mth. Ps. 28 (1883) 318-. of space, decomposition into 3 skew re- flections. Wilson, E, B. N. Y. Am. Mth. S. T. 1 (1900) 193-. in space, like and unlike. Hauck, G. Z. Mth. Ps. 24 (1879) 381-. , perspective and projective, axono- metric theory. Hauck, G. Z. Mth. Ps. 21 (1876) 402-. , spherical. Laroche, F. N. A. Mth. 8 (1849) 415. of two 1-dimensional geometric forms. Neumayr, E. Innsb. Nt. Md. B. 9 (1879) 144-. Collineations, automorphic, cubic surfaces with. Bobek, K. Mh. Mth. Ps. 10 (1899) 122-, 307-. , class of quartic surfaces generated by. Schur, F. Mth. A. 20 (1882) 254-. , construction. Newson, H. B. Kan. Un. Q. 9 (1900) 65-. , cyclic, in plane and space. Schroter, . D. Nf. Tbl. (*1874) 121. , involutoric, in plane and space. Emch, A. Kan. Un. Q. 4 (1896) 205-. keeping 5 points fixed, invariant scrolls in. Snyder, V. Am. J. Mth. 22 (1900) 253-. , Klein's group of (n + 1) ! n-ary. Moore, E. H. Am. J. Mth. 22 (1900) 336-. , most general. Kantor, S. Wien Ak. D. 46 (1883) (Ab. 2) 83-. , perspective, projective groups. Emch, A. Kan. Un. Q. 5 (1896) 1-. , plane. White, H. S. [1897] N. Y. Am. Mth. S. Bll. 4 (1898) 17-. , , covariants. Muth, P. Mth. A. 40 (1892) 89-. , , finite groups with irreducible quartic invariant. Ciani, E. Mil. I. Lomb. Ed. 33 (1900) 1170-. , , geometric interpretation of certain in- variant in. Kraus,J. Mth. A. 29 (1887) 234-. , , simple group of 360. Wiman, A. Mth. A. 47 (1896) 531-. , construction of conies from points and , , . Gerbaldi, F. Mth. A. 50 tangents by. Pelz, C. Prag Sb. (1875) 117-. and duality. Sturm, R. Mth. A. 22 (1883) 569-; 28 (1887)268-. . Pasch, M. Mth. A. 23 (1884) 419-. , curvature and torsion in. Servais, C. Brux. Mm. Cour. 8, 58 (1898-99) No. 2, 48 pp. of fundamental configurations of 2nd rank. Dantscher von Kollesberg, V. Innsb. Nt. Md. B. 18 (1889) 86-. 3rd rank, special. Totdssy, B. [1883] (xn) Mth. Term. Ets. 2 (1884) 50-; Mth. Nt. B. Ung. 2 (1883-84) 237-. groups, discontinuity in certain. Fricke, R. Mth. A. 47 (1896) 557-. as mode of motion. Bocher, M. N. Y. Mth. S. Bll. 1 (1892) 225-. (1898) 473-; Palermo Cir. Mt. Ed. 12 (1898) 23-; 13(1899) 161- ; 14 (1900) 66-. and reciprocities, geometric theory. Amese- der, A. Mh. Mth. Ps. 1 (1890) 371-. , invariants. Muth, P. Mth. A. 33 (1889) 493-. , related to quadrics or skew cubic curves. Sturm, R. Mth. A. 26 (1886) 465-. , space, with automorphic quadric. Wood, (Miss) R. G. [1900] N. Y. Am. Mth. S. Bll. 7 (1901) 202, 207. , , invariable forms in. Mertens, F. Mh. Mth. Ps. 1 (1890) 13-. Conic, transformation of circle into. Barani- ecki, M. A. Krk. Ak. (Mt.-Prz.) Ez. 13 (1886) 172-. 553 8010 Collineation ; duality 8010 Conies as collinearly related to circles. Schlo- milch, 0. Schlomilch Z. 1 (1856) 1-. Correlation of 2 planes. Hirst, T. A. L. Mth. S. P. 5 (1873-74) 40-; 8 (1876-77) 262-; Em. E. Ac. Line. T. 1 (1877) 86-. , simplest, between 2 regions of space. Eichelot, F. J. [1868] Crelle J. 70 (1869) 137-. of 2 spaces. Hirst, T. A. [1874] L. Mth. S. P. 6 (1874-75) 7-. Correlations, finite groups. Kantor, S. Crelle J. Mth. 116 (1896) 171-. , metrical properties. Segre, C. G. Mt. 25 (1887) 20-. , polar, for which twisted cubic is self-con- jugate. Montesano, D. Em. E. Ac. Line. Mm. 3 (1886) 105-. Correlative pencils. Sturm, JR. L. Mth. S. P. 7 (1875-76) 175- ; Mth. A. 12 (1877) 254-. systems in plane. Le Paige, C. Brux. Ac. Bll. 12 (1886) 422-. Correspondence of spaces, special. Biltzberger, F. Z. Mth. Ps. 38 (1893) 1-. between straight lines and points. Poulain, A. Mathesis 11 (1891) 88-. Doubly harmonic linear elements, determina- tion. Raffy,L. Liouv. J. Mth. 10 (1894) 331-. Duality. Poncelet, J. V. Gergonne A. Mth. 18 (1827-28) 125-. . Booth, J. Lpool. Lt. Ph. S. P. 1 (pt. 4) (1844-45) 117-. . Gergonne, . Mntp. Ac. Sc. Mm. 1 (1847-50) 61-. . Zeuthen, H. G. (xn) Ts. Mth. 1 (1871) 1-, 129-, 161- ; 2 (1872) 161-. . Study, E. Leip. Mth. Ps. B. 48 (1896) 199-. . Brill, J. Mess. Mth. 26 (1897) 134-. and collineation in space. Alle, M. Wien Ak. Sb. 85 (1882) (Ab. 2) 1021-. of figures in space. Mobius, A. F. Crelle J. 10 (1833) 317-. , history. Milller, J. H. T. Grunert Arch. 34 (1860) 1-. and homography. Chasles, M. Brux. Mm. Cour. 11 (1837) 575-. applied to metric properties of space. Ripert, L. N. A. Mth. 18 (1899) 306-. in plane geometry. Ripert, L. N. A. Mth. 17 (1898) 446-. , a principle of, in geometry of space. Rethy, M. , Gott. Nr. (1873) 6-. of quadrilateral, spherical and plane. Beez, . Schlomilch Z. 1 (1862) 129-. Equipollent figures. Tarry, G. N. A. Mth. 11 (1892) 251-. Homogeneous protective coordinates for ele- ments of ordinary space. Aschieri, F. Mil. I. Lomb. Ed. 18 (1885) 569-. Homographic or correlative systems, relations between infinitesimal elements. Demmdin, A. C. E. 126 (1898) 390-. correspondence between elements of 2 linear spaces. Deruyts, F. Brux. Mm. Cour. 4, 52 (1890-93) .Vo. 7, 40 pp. figures, focal properties. Smith, H. J. S. L. Mth. S. P. 2 (1869) 196-. Homographic figures, focal properties. Reye, T. Mth. A. 46 (1895) 423-. , theorems. Petersen, L. (xn) Ts. Mth. 5 (1875) 21-. pencils HZ*, singularities of cubic surface generated by. Versluys, W. A. Liege S. Sc. Mm. 20 (1898) No. 6, 61 pp. relation of 2 figures. Swellengrebel, J. G.H. Utr. Aant. Prv. Gn. (1853-54) 82-. transformation. Painvin, L. N. A. Mth. 9 (1870) 97-. of any linear space, general. Aschieri, F. Mil. I. Lomb. Ed. 18 (1885) 989-. metric properties of plane figures. Brocard, G. N. A. Mth. 15 (1896) 426-; 16 (1897) 293-. , particular case. Antomari, X. N. A. Mth. 17 (1898) 489-. of plane figure, continuous. Mollame, V. G. Mt. 9 (1871) 269-. transformations, plane. Lucas, F. Liouv. J. Mth. 6 (1861) 137-. , special kind. Bortniker, (Mile.) L. C. E. 104 (1887) 771-. , . Darboux, G. C. E. 104 (1887) 773-. Homographies, binary, and their harmonic groups. Segre, C. Crelle J. Mth. 100 (1887) 317-. , , products. Aschieri, F. Mil. I. Lomb. Ed. 24 (1891) 964-. , , systems of, and geometric representa- tion of imaginaries. Amodeo, F, G. Mt. 26 (1888) 363-. and correlations, with special reference to 3-dimensional space. Segre, C. Tor. Ac. Sc. Mm. 37 (1886) 395-. of any linear space, construction. Bertini, E. Mil. I. Lomb. Ed. 20 (1887) 650-. , resolution into homologies. Bertini, E. Tor. Ac. Sc. At. 22 (1886-87) 865-. , ternary, united real elements. Amodeo, F. G. Mt. 27 (1889) 40-. Homography. Battaglini, G. Giamb. Vico 1 (1857) 121- ; 2 (1857) 272-. , general theory. Predella, P. Tor. Ac. Sc. At. 27 (1892) 270-. , problem. Weyr, E. Prag Ceske Ak. Fr. Jos. Ez. (Tflda 2) 8 (1899) Art. 24, 8 pp.; Fschr. Mth. (1899) 478. of 2 solid bodies. Sylvester, J. J. C. E. 101 (1885) 35-, 139-. 3rd order. Le Paige, C. M. M. H. H. Em. N. Line. At. 36 (1883) 22; Brux. Ac. Bll. 5 (1883) 85-. higher order. Appell, P. [1879] Par. S. Phlm. Bll. 4 (1880) 18-. Homology and polar reciprocals, curves and surfaces corresponding by. Fouret, G. Par. S. Mth. Bll. 14 (1886) 18-. represented by space of conies in a plane. Certo, L. G. Mt. 20 (1882) 321-. Involution, collinear, of pairs of points in plane and space. Mobius, A. F. Leip. B. 8 (1856) 143-. , generalisation of concept. Schroeter, . D. Nf. Tbl. (*1868) 94. 554 8010 Involution and homography, extension of theories. Le Paige, G. M. M. H. H. Brux. Ac. Bll. 44 (1877) 546-. of kth rank and (2fc + l)th degree, neutral group. Le Paige, C. M. M. H. H. [1882] Wien Ak. Sb. 86 (1883) (Ab. 2) 104-. Involutions, harmonic, on plane and sphere. Vries, J. de. Haarl. Ms. Teyl. Arch. 4 (1896) 119-. , higher order. Mobius, A. F. Leip. B. 7 (1855) 123-. Involutory automorphic substitutions of ra- tional function, parametric representation. Hofmann, F. Arch. Mth. Ps. 8 (1890) 225-. configurations, protective. Stahl, W. Crelle J. Mth. 107 (1891) 179-. correspondence, system of conies in. Retail, V. Bologna Ac. Sc. Mm. 5 (1883) 373-. space transformations determined by line complex of Hirst. Pieri, M. Mil. I. Lomb. Bd. 25 (1892) 1037-. transformation of straight line. Emch, A. [1895] Kan. Un. Q. 4 (1896) 111-. transformations, 2, with applications. Stiner, G. Ziir. Vjschr. 40 (1895) 317-. Linear group of transformation in plane, fun- damental property. Emch, A. [1895] A. Mth. 10 (1895-96) 3-. substitutions, new theory. Laurent, H. Liouv. J. Mth. 4 (1898) 75-. transformation of geometric figures. Er- makov, V.P. Bee. Mth. (Moscou) 13 (1886) 749-. transformations. Kantor, S. [1880] Wien Ak. Sb. 82 (1881) (Ab. 2) 34-. between 2 quadrics. Taber, H. L. Mth. S. P. 24 (1893) 290-. , successive. Kantor, S. [1880] Wien Ak. Sb. 82 (1881) (Ab. 2) 39-. Lines and polar surfaces, theory. Lazzeri, G. Mil. I. Lomb. Bd. 24 (1891) 1021-. Loci arising from 3 figures, protectively related. Del Re, A. Palermo Cir. Mt. Ed. 1 (1887) 272-, 398. Metrical properties of involutions and con- ditions for double elements. Boguslavskij, A. I. Mosc. S. Sc. Bll. 65 (No. 2) (1890) 52-; Fschr. Mth. (1892) 572. Null system, special case. Sucharda, A. Casopis29 (1900) 89-; Fschr. Mth. (1900) 528. Plane curves, polar transformation. Ocagne, M. d\ G. Teix. J. Sc. 6 (1885) 3-. transformable into one another by group of transformations. Klein, F., < Lie, S. Mth. A. 4 (1871) 50-. transformations, conditions of rationality of double planes. Castelnuovo, G., <& Enriques, F. Palermo Cir. Mt. Bd. 14 (1900) 290-. Point plane systems, general. Voss, A. Mth. A. 23 (1884) 45-. , rational algebraic. Voss, A. Mth. A. 23 (1884) 359-. Polar plane systems; and ternary quadratic correspondences. Del Re, A. G. Mt. 28 (1890) 257-. reciprocal figures, contact of inverse figures with. Vanecek , J. S., & Vanecek, M. N. [1883] St Pet. Ac. Sc. Bll. 29 (*1884) 281-. Collineation ; duality 8010 Polar reciprocal figures, transformation. Vanecek, J. S. Liege S. Sc. Mm. 11 (1885) No. 2, 21 pp. of surface with multiple points, determi- nation of double curve. Saltel, L. Brux. Ac. Bll. 40 (1875) 27-. surfaces, theorems. Hamilton, (Sir) W. R. Ir. Ac. P. 4 (1850) 192-. , theory, revision and extension of numerical formulae. Zeuthen, H. G. Mth. A. 10 (1876) 446-. reciprocals, transformation of barycentric properties by. Ocagne, M. d'. Mathesis 5 (1885) 170-. surfaces of plane, theory. Painvin, L. F. [1865] (xii) Lille S. Mm. 2 (1866) 345-. systems. Romero, A. del. (xn) Barcel. Ac. Mm. 1 (1878) 133-. . Jung, G. Mil. I. Lomb. Bd. 12 (1879) 169-, 218-, 535-. (points, lines, planes), construction. Schroder, . D. Nf. Tbl. (*1879) 176-. in space, polar pentagons and hexagons of. Reye, T. Crelle J. Mth. 77 (1874) 269-. theory of algebraic surfaces, extension. Reye, T. Crelle J. Mth. 78 (1874) 97-. Poles and polars. Fenwick, S. Mathematician 1 (1845) 125-, 235-, 281-. Projective connection of systems of curves. Fiedler, W. Ziir. Vjschr. 24 (1879) 180-. construction of rational curves. Schoute, P. H. N. Arch. Wisk. 12 (1886) 1-; Arch. Neerl. 20 (1886) 49-. correspondence, plane, double points in. Broden, T. Arch. Mth. Ps. 9 (1890) 225-. correspondences, degenerate. Del Prete, G. Mil. I. Lomb. Bd. 30 (1897) 400-, 464-. in fundamental geometric forms of 1st kind. Paolis, R. de. Tor. Ac. Sc. Mm. 42 (1892) 495-. between 2 planes and between 2 spaces. Loria, G. G. Mt. 22 (1884) 1-, 117. fundamental forms, surfaces arising from. Cardinaal, J. N. Arch. Wisk. 18 (1891) 123-. geometry. Killing, W. Mth. A. 43 (1893) 569-. metrics. Ovidio, E. d'. Tor. Ac. Sc. At. 28 (1893) 566-. properties of curves and surfaces. Jon- quieres, E. de. Crelle J. 66 (1866) 289-. , application of transversals. Poncelet, J. V. Crelle J. 8 (1832) 21-, 117-, 213-, 370-. reflection, and space collineation. Bocher, M. Mth. A. 43 (1893) 598-. relations given by 4 straight lines in space. Krieg von Hochfelden, F. (Frhr.) Wien Ak. Sb. 97 (1889) (Ab. 2a) 806-. relationships and theory of higher curves. Seydewitz, F. Grunert Arch. 7 (1846) 113-; 8 (1846) 1-. substitutions leaving a circle unchanged. Schlesinger, L. Orv.-Termt. Ets. (Termt. Szak) (1898) 14-, (Rv.) 15- ; Crelle J. Mth. 121 (1900) 168-. transformation. Genese, R. W. B. A. Bp. (1886) 538-. 555 8010 Collineation ; duality Protective transformation. Lovett, E. 0. A. Mth. 10 (1895-96) 5-. transformations, normal forms. Newson, H. B. Kan. Un. Q. 7 (1898) 125-. in plane and space. Newson, H. B. Kan. Un. Q. 6 (1896) 63-. of plane, 5 types. Newson, H. B. Kan. Un. Q. 8 (1899) 43-. , properties. Delaunay, N. B. Bee. Mth. (Moscou) 19 (1897) 387-; Fschr. Mth. (1896) 476. Quadrilateral and associated quadrangle, five- plane and associated pentagon. Kohn, G. Wien Ak. Sb. 93 (1886) (Ab. 2) 314-. , quadrangle, and projective correspondence in plane. Pasch, M . Mth. A. 26 (1886) 211-. ^Reciprocal and allied relations. Sturm, JR. Mth. A. 19 (1882) 461-. (Sturm). Kantor, S. Mth. A. 20 (1882) 297-. coincident configurations. Schroeter, H. E. Crelle J. Mth. 77 (1874) 105-. of curve, equation. Salmon, G. Crelle J. 42 (1852) 277-. curves, property. Pasch, M. [1871] Crelle J. 74 (1872) 92-. and surfaces. Bacaloglo, E. Grunert Arch. 36 (1861) 1-. derivability of 2 functions by polars. Perna, A. G. Mt. 34 (1896) 110-. plane systems. Sucharda, A. fiasopis 29 (1900) 1-; Fschr. Mth. (1900) 529. relations, 2, representation of curves of 3rd degree and class by. Beyel, G. Z. Mth. Ps. 38 (1893) 65-. relationship. Rosanes, J. Crelle J. Mth. 90 (1881) 303-. and dependent point systems. Rosanes, J. Crelle J. Mth. 95 (1883) 247-. systems, elements doubly conjugate to one another. Keller, J. Ziir.Vjschr. 25 (1880)1-. , plane. Krok, J. M. Lund Acta Un. 9 (1872) (Mth.) No. 5, 28pp. transformations and theorems of Ceva and Menelaos. Emch, A. N. Y. Am. Mth. S. Bll. 5 (1899) 143-. Beciprocity, geometrical. Cayley, A. Camb. and Dubl. Mth. J. 3 (1848) 173-. , plane, and application to theory of curves. Beyel, C. Z. Mth. Ps. 31 (1886) 147-. Belation of spaces by projective plane pencils. Krieg, F. (Frhr.) von. Z. Mth. Ps. 29 (1884) (Suppl.) 38-. Belationship, geometric, of systems in space. Silldorf, (Dr) . Z. Mth. Ps. 18 (1873) 523-. of plane figures, new. Mobius, A . F. Leip. B. (1853) 14-. Belationships in linear geometry. Hunyady, J. (xn) Mag. Tud. Ak. Ets. 16 (No. 2) (1882) 39-. Buled surfaces and envelopes of straight lines joining corresponding points of 2 plane curves. Chizzoni, F. Em. B. Ac. Line. Mm. 3 (1879) 69-. Similar systems, and a cubic surface. Mos- tiammer, K. [1876] Wien Ak. Sb. 74 (1877) (Ab. 2) 131-. Similarly variable figures, and loci derived there- from. Reveille, J. N. A. Mth. 12 (1893) 297-. Other Transformations 8020 Similitude of 2 similar curves, centre. Monge, G. Par. EC. Pol. Cor. 3 (1814-16) 4-. Similitudes in real plane, group. Del Re, A. Bv. Mt. 2 (1892) 99-. Skew curves on quadric, whose osculating planes touch a quadric. Stahl, W. Crelle J. Mth. 99 (1886) 154-. Space transformation depending on quadric. Schoute, P. H. Palermo Cir. Mt. Ed. 1 (1887) 47, 379-. . Capelli,A. Palermo Cir. Mt. Ed. 1 (1887) 48-, 53-. Surface and polar reciprocal in given homo- graphy, condition. Del Re, A. Palermo Cir. Mt. Ed. 2 (1888) 37-. Symmetry of 2 algebraic figures with reference to a point. Mangeot, S. N. A. Mth. 19 (1900) 451-. System of 6 positions of plane figure. Stephanos, C. As. Fr. C. B. (1883) 175-. Tangential correspondence, decomposition. Fontene, G. Par. S. Mth. Bll. 25 (1897) 247-. transformation in plane. Naetsch, E. Dresden Isis Sb. (1896) 15-. Trilinear relation between 3 fundamental con- figurations of 1st rank. Schubert, H. C. H. Mth. A. 17 (1880) 457-. . London, F. Mth. A. 44 (1894) 37.5-. of plane systems. Hauck, G. Crelle J. Mth. 97 (1884) 261- ; 98 (1885) 304; 108 (1891) 25-; 111 (1893) 207-. between 3 variables, reduced forms. Fontene, G. N. A. Mth. 19 (1900) 494-. Trilinearly related fields in space. Schmid, T. Mh. Mth. Ps. 6 (1895) 99-; 7 (1896) 180-. (1, 1) point transformation of plane. Czuber, E. Mh. Mth. Ps. 5 (1894) 267-. 8020 Other algebraic transforma- tions. (See also 6810, 7650.) Adjoint curves. Scott, (Miss) C. A. QJ. Mth. 28 (1896) 377-. Algebraic curves, transformation. Bertini, E. Ev. Mt. 1 (1891) 22-. ,_. Simart, . C. E. 116 (1893) 1047-. , . Goursat, E. Am. J. Mth. 16 (1894) 291-. , , and theorems of Steiner. Sparer, B. Z. Mth. Ps. 36 (1891) 339-. Anallagmatic curves and surfaces. Picquet, H. As. Fr. C. E. (1878) 95-. , application of general properties of curves. Laguerre, E. (x) Par. S. Phlm. Bll. 5 (1868) 112-. , new species. Picquet, H. C. B. 87 (1878) 460-. Arguesian curves and surfaces, properties de- rived from. Crocchi,L. G.Mt.l2(1874)378-. transformation. Saltel, L. [1870] Brux. Mm. Cour. 8, 23 (1873) (No. 4) 112 pp. , application to curves and surfaces. Saltel, L. [1870] Brux. Mm. Cour. 8, 22 (1872) (No. 5) 53 pp. , extension to skew curves. Saltel, L. Brux. Ac. Bll. 46 (1878) 90-. 556 8020 Other Algebraic Transformations 8020 Arguesian transformations, SaltePs. Mansion, P. Brux. Ac. Bll. 36 (1873) 625-. Arguesians, theorem. Saltel, L. Brux. Ac. Bll. 43 (1877) 265-. Bifocal transformations. Burmester, L. Mth. A. 16 (1880) 89-. Birational correspondence between 2 spaces, transformation by. Levi, B. Tor. Ac. Sc. At. 35 (1900) 20-. plane transformation, multiple transforma- tions associated with every. Pannelli, M. G. Mt. 29 (1891) 154-. , transformation of genus p and degree n+1 associated with every. Jung, (?. Mil. I. Lomb. Ed. 19 (1886) 812-. transformations. Guccia, G. B. C. E. 101 (1885) 808-, 980. quadratic transformation. Ocagne, M. d'. N. A. Mth. 12 (1893) 337-. , geometric interpretation. Servais, C, Mathesis 7 (1887) 129-, 187-. , reciprocal. Ocagne, M. d'. Liege S. Sc. Mm. 16 (1890) No. 7, 10 pp. , special cases. Third, J. A. Edinb. Mth. S. P. 18 (1900) 11-. transformations. Mansion, P. Mathesis 7 (1887) 110-. reciprocities in plane. Lazzeri, G. Em. E. Ac. Line. Ed. 2 (1886) (Sem. 2) 61-. , null. Del Re, A. Nap. Ed. 28 (1889) 101-. space. Lazzeri, G. Em. E. Ac. Line. Ed. 2 (1886) (-Sem. 2) 73-. transformation, 2 multiple transformations associated with every. Jung, G. Em. E. Ac. Line. Ed. 2 (1886) (Sem. 2) 339-. of 6th degree in 3-dimensional space. Cremona, L. L. Mth. S. P. 15 (1883-84) 242-. transformations of 3 geometric forms of 2nd kind. Jung, G. Mil. I. Lomb. Ed. 19 (1886) 161-. with integral coefficients. Kantor, S. C. B. 126 (1898) 946-. , surfaces generated by. Jung, G. Em. E. Ac. Line. Ed. 2 (1886) (Sem. 1) 85-. of nth order. Jonquieres, de. C. E. 101 (1885) 720-, 848. Central transformation, case. Longchamps, G. de. Mathesis 15 (1895) 186-. ,_. Meurice,L. Mathesis 15 (1895) 191-. transformations of plane curves. Ocagne, M. d'. Mathesis 4 (1884) 73-, 97-. Circles touching 4 isotropic planes, and surfaces with double circular generation. Cosserat, E. C. E. 130 (1900) 385-. Conic, space transformations depending on. Fieri, M. Palermo Cir. Mt. Ed. 7 (1893) 296-. Conical transformations of space. Del Pezzo, P. Nap. Ed. 35 (1896) 288-. Conies, transformation theory. Verbessem, . Mathesis 14 (1894) 184-. Correspondence of 2 plane figures, quadratic and rational. Duporcq,E. C. E. 126 (1898) 1405-. points collinear with fixed origin. Cotterill, T. L. Mth. S. P. 6 (1875) 200. Correspondences, invariant formations of. Peano, G. G. Mt. 20 (1882) 79-. Cremona cubic groups. Autonne, L. C. E. 101 (1885) 53-; 105 (1887) 267-; Liouv. J. Mth. 2 (1886) 49-. group with any number of variables, com- plete groups contained in. Del Re, A. Em. E. Ac. Line. Ed. 6 (1890) (-Sem. 2) 271-. quadratic groups. Autonne, L. C. E. 98 (1884) 565-; 103 (1886) 1176- ; 104 (1887) 767-, 1422- ; 105 (1887) 929-; 106 (1888) 262-; Liouv. J. Mth. 1 (1885) 431-; 4 (1888) 177-, 407-. transformation. Dickson, L. E. [1898] Calif. Ac. P. (Mth. Ps.) 1 (1903) 13-. defined by conic. Dickson, L. E. Palermo Cir. Mt. Ed. 9 (1895) 256-. transformations, cross-ratio group of 120. Slaught, H. E. Science 8 (1898) 121- ; Am/ J. Mth. 22 (1900) 343-. transformation, involutions contained by. Doehlemann, K. Z. Mth. Ps. 36 (1891) 356-. between 2 planes. Roberts, S. [1872] L. Mth. S. P. 4 (1871-73) 121-. , solution of equations of condition. Ruffini, F. P. Bologna Ac. Sc. Mm. 8 (1877) 457-. , . Palatini, F. Yen. I. At. (1896-97) 1555-. transformations. Clebsch, R. F. A. Mth. A. 4 (1871) 490-. . Guccia, G. B. C. E. 101 (1885) 866-. . Sturm, R. Mth. A. 26 (1886) 304-. . Guccia, G. B. Palermo Cir. Mt. Ed. 1 (1887) 17-, 20-, 24-, 50-, 56-, 119-, 398. admitting fixed curve. Castelnuovo, G. Em. E. Ac. Line. Ed. 1 (1892) (Sem. 1) , continuous groups. Enriques, F. Em. E. Ac. Line. Ed. 2 (1893) (Sem. 1) 468-. , . Noether, M. D. Nf. Vh. (1896) (Th. 2, Hdlfte 1) 11-. , . Fano, G. Mb. Mth. Ps. 9 (1898) 17-. , , and certain projective transfor- mation groups. Fano, G. Palermo Cir. Mt. Ed. 10 (1896) 16-. , indeterminate analysis relating to. Jonquieres, (I'amiral) E. de. G. E. 101 (1885) 857-, 1082. Cubic through 8 points, geometrical transfor- mation by. Deruyts, F. Liege S. Sc. Mm. 14 (1888) No. 5, 14 pp. transformation and line complex arising therefrom. Machovec, F. Prag Sb. (1888) (Mth.-Nt.) 104-. in space. Schoute, P. H. N. Arch. Wisk. 4 (1899) 90-. transformations. Casey, J. Ir. Ac. Cunningham Mm. 1 (*1880) 140 pp. Cyclic groups, number in a Cremona trans- formation. Kantor, S. [1880] A. Mt. 10 (1880-82) 71-. , quadratic plane transformation. Kantor, S. [1880] A. Mt. 10 (1880-82) 64-. , space transformation. Kantor, S. C. E. 90 (1880) 1156-. Double cubic transformation of 1st genus, and quartic curves. Paolis, R. de. Em. E. Ac. Line. Mm. 2 (1878) 851-. 557 8020 Double quadratic plane transformation, and application to non-Euclidean geometry. Paolis, E. de. [1877] Em. E. Ac. Line. Mm. 2 (1878) 31-. space transformation, and application to non-Euclidean geometry. Aschieri, F. Mil. I. Lomb. Bd. 14 (1881) 673- ; 15 (1882) 66-, 147-, 247-. transformations of figures. Aoust, L. C. B. 56 (1863) 906-. Enneads of lines and points. Cayley, A. B. A. Ep. 40 (1870) (Sect.) 10. Figures generated by 2 fundamental forms of 2nd kind with (1. v) correspondence of nth degree. Visalli, P. Em. E. Ac. Line. Ed. 3 (1887) (Sem. 1) 124-. , transformation, application to construction of quadric through nine points. Darboux, G. Par. EC. Norm. A. 6 (1869) 61-. Generalised Jonquieres groups. Fano, G. Tor. Ac. Sc. Mm. 48 (1899) 221-. Generation of curves and surfaces by corre- sponding elements. Weyr, Em. [1871] Prag Ab. 5 (1872) 54 pp. Geometric transformation with application in special case to kinematics. Dewulf, E. Par. EC. Norm. A. 3 (1886) 405-. Groups of transformation in 3-dimensional space, class. Blichfeldt, H. E. Am. J. Mth. 22 (1900) 113-. Higher singularities of plane curves. Scott, (Miss) C. A. Am. J. Mth. 14 (1892) 301-. Homaloidal transformation of quadrics. Michel, P. N. A. Mth. 12 (1893) 192-. Homographies, higher. Saltel, L. [1875] Brux. Mm. Cour. 8, 27 (1877) No. 4, 32 pp. Hyperbolic and other transformations. Schia- parelli, G. V. [1861] Tor. Ac. Mm. 21 (1864) 227-. Inverse curves. Verdam, G. J. Amst. N. Vh. 12 (1846) 67-. and surfaces, mutual relations. Booth, J. B. A. Bp. (1858) (pt. 2) 3. figures. Tortolini, B. Tortolini A. 2 (1859) 189-. Inversion. Hart, A. S. Camb. and Dubl. Mth. J. 8 (1853) 47-. . Boklen, 0. Grunert Arch. 32 (1859) 83-. . Moutard, T. N. A. Mth. 3 (1864) 306-. . Habich, E. N. A. Mth. 5 (1866) 399-. . Newenglowski, (Prof.) . Par. EC. Norm. A. 2 (1873) 133-. . Bianchi, L. [1878] G. Mt. 17 (1879) 40-. . Vanecek, J. S. As. Fr. C. E. (1883) 127-. . Pirondini, G. G. Mt. 27 (1889) 168-. . Laisant, . Mathesis 10 (1890) 224-. . Giacomini, A. G, Mt. 35 (1897) 125-. of curves. Ferrers, N. M. OJ. Mth. 1 (1857) 32-. , dual, of Cartesian and Boothian coordinates. Jeffery, H. M. [1880] QJ. Mth. 17 (1881) 311-. , general. Vanecek, J. S. [1882] C. B. 94 (1882) 1042- ; Munch. Ak. Sb. 12 (1882) 463-; Prag Sb. (1882) 203- ; E. S. P. 34 (1883) 29-. , . Vanecek, J. S., & Vanecek, M. N. Prag Sb. (1883) (Mth. Vortr.) 345-; Fschr. Mth. (1884) 603-. Other Algebraic Transformations 8020 Inversion, general. Gaertner, E. Arch. Mth. Ps. 10 (1891) 337-. , method. Taylor, H. M. Mess. Mth. 3 (1866) 228-. , quadratic, of plane curves. Hirst, T. A. E. S. P. 14 (1865) 91-. . Series of quadrangles. Gob, A. As. Fr. C. B. (1891) (Pt. 2) 232-. Involution in plane. Castelnuovo, G. Yen. I. At. (1885-86) 1559-. Involutory birational transformation. Schoute, P. H. N. Arch. Wisk. 9 (*1882) 117- ; Bll. Sc. Mth. As. 6 (1882) (Pt. 1) 152-, 174-. multiple transformations of 2 spaces. Pannelli, M. Nap. Ed. 26 (1887) 153-. quadratic relation, type. Kilbinger, . Z. Mth. Ps. 33 (1888) 14-. transformations. Schoute, P. H. As. Fr. C. B. (1884) (Pt. 2) 42-. transformations of space. Martinetti, V. Mil. I. Lomb. Bd. 18 (1885) 132-. . Montesano, D. Mil. I. Lomb. Bd. 21 (1888) 688-; Em. E. Ac. Line. Ed. 5 (1889) (Sem. 2) 123-. Isogonal transformation, Eoberts's. Ocagne, M. d'. G. Teix. J. Sc. 9 (1889) 109-. Linear quasi-inversion, applications to oscu- lating curves. Servais, C. Mathesis 8 (1888) 28-. Linkwork and point transformations, simpler cases. Burgatti, P. Palermo Cir. Mt. Bd. 14 (1900) 192-. Metamorphic methods of transformation, and corresponding points. Terquem, 0. N. A. Mth. 5 (1846) 497-. Method of transformation in geometry. Zeuthen, H. G. Mth. Ts. 5 (1863) 129- . Gohierre de Longchamps, . Par. EC. Norm. A. 3 (1866) 321-. . Allardice,R.E. Edinb. Mth. S. P. 6 (1888) 14-. , miscellaneous applications. Collignon, E. As. Fr. C. E. (1894) (Pt. 2) 14-. Multiform relation of 2 plane figures. Wiener, C. [1869] Mth. A. 3 (1871) 11-. transformation of plane. Bjorling, C. F. E. Stockh. Ak. Hndl. Bh. 13 (Afd. 1) (1888) No. 8, 19 pp. Multiple correspondence between two 3-dimen- sional spaces. Tognoli, 0. [1871] G. Mt. 10 (1872) 1-, 141-. plane transformations. Jung, G. Em. E. Ac. Line. Ed. 2 (1886) (Sem. 2) 302-. . Borghese, F. G. Mt. 32 (1894) 58-. of minimum order. Jung, G. Mil. I. Lomb. Ed. 20 (1887) 370-. Non-isogonal plane transformations. Laisant, C. A. Par. S. Mth. Bll. 15 (1887) 103-. Orthogonal transformation, geometrical mean- ing of a form. Hill, M. J. M. L. Mth. S. P. 26 (1895) 339-. Orthotangential transformation in plane and space. Longchamps, G. de. Prag Sb. (1888) (Mth.-Nt.) 241-. Periodic cubic transformations. Kantor, S. Am. J. Mth. 19 (1897) 1-. transformations. Kantor, S. C. B. 100 (1885) 95-. 558 8020 Periodic uniform transformations, method of treatment. Kantor, S. G. E. 100 (1885) 42-. Plane curves, transformation. Cremona, L. B. A. Ep. 34 (1864) (Sect.) 3-. , . Cayley, A. [1865] L. Mth. S. P. 1 (1866) No. 3, 11 pp. , . Scott, (Miss) C. A. As. Fr. C. E. (1897) (Pt. 2) 50-; QJ. Mth. 29 (1898) 329-. figures, transformation, and mode of generat- ing skew curves. Jonquieres, E. de. N. A. Mth. 3 (1864) 97-. .transformations. Cremona, L. [1864-73] Bologna Ac. Sc. Mm. 5 (1865) 3- ; Bll. Sc. Mth. As. 5 (1873) 206-. Point transformation of algebraic curves. Jamet, V. N. A. Mth. 19 (1900) 506-. transformations, general properties. Brill, J. QJ. Mth. 27 (1895) 356-. in plane. Brill, J. Mess. Mth. 24 (1895) 113-. . Lovett, E. 0. [1897] N. Y. Am. Mth. S. Bll. 4 (1898) 97-. of space leaving volumes unaltered. Carda, K. Mh. Mth. Ps. 8 (1897) 170-. Polar reciprocal of curve with respect to cubic. Russell, W. H. L. E. S. P. 44 (1888) 3_88-. Polynomial F (x, y) for which x, y, \JF are rational functions of 2 parameters. Bottari, A. A. Mt. 2 (1899) 277-. Porismatic (2, 2) relation, algebraic theory. Pasch, M., & Eosanes, J. [1868] Crelle J. 70 (1869) 169-. Projection, double, by means of quadric. Fame, H. A. (xn) Isere S. Bll. 2 (1870) 260-. , stereoscopic. Cassani, P. Ven. I. At. (1884-85) 1835-. , . Emilio, R. d'. Ven. Aten. 2 (1886) 71-; 2 (1887) 75-; (1890) 607-. Quadratic plane transformations. Retail, V. Bologna Ac. Sc. Mm. 10 (1889) 653-. , construction by tri-homologous tri- angles. Ferrari, F. Par. S. Phlm. Bll. 1 (1899) 93-. , degenerate forms of 1st rank. Schubert, H. C. H. Hamb. Mth. Gs. Mt. 1 (1889) 31-. , successive. Kantor, S. [1880] Wien Ak. Sb. 82 (1881) (Ab. 2) 237-. transformation. Hirst, T. A. N. A. Mth. 5 (1866) 213-. . Igel, B. Z. Mth. Ps. 17 (1872) 516-. . Paraira, M. C. N. Arch. Wisk. 2 (*1876) 62-. . Hirst, T. A. QJ. Mth. 17 (1881) 301-. . Rychlicki, S. Par. T. Nauk Sc. Pam. 12 (*1882) Art. 2, 19 pp. . Petersen, J. Ts. Mth. 6 (1889) 1-. . Neuberg, J. Liege S. Sc. Mm. 16 (1890) No. 8, 12 pp. of conic, quartic curves arising from. Leuch, A. Bern Mt. (1889) 1-. between 2 planes, cases. Cayley, A. (xn) J. H. Un. Cir. [1] (1882) 178-. of planes of space into system of quadrics. Timerding, H. E. A. Mt. 1 (1898) 95-. transformations. Amigues, E. N. A. Mth. 16 (1877) 422-, 451-, 496-, 529-. Other Algebraic Transformations 8020 Quadratic transformations by means of pencils in involution. Neuberg, J. Mathesis 8 (1888) ranges, etc., in involution. Eipert, L. Mathesis 19 (1899) 185-, 217-. and rational surfaces. Eeye, T. Mth. A. 48 (1897) 113-. , reduction of point-singularities by. Levi, B. A. Mt. 26 (1897) 219-. , transformations in n-dimensional space compounded of. Kantor, S. Am. J. Mth. 18 (1896) 219-. Quartic transformation in 3-dimensional space. Cremona, L. [1884] Ir. Ac. T. 28 (1880-86 279-. Eational curves in 2 dimensions. Autonne, L. C. E. 121 (1895) 673-. 3 dimensions. Autonne, . C. E. 121 (1895) 881-, 1129-. and involutory space transformations. Montesano, D. G. Mt. 31 (1893) 36-; Mil. I. Lomb. Ed. 30 (1897) 563-. quadratic space transformation whose in- verse is of 4th degree. Cremona, L. Bologna Ac. Sc. Mm. 1 (1871) 365-. transformation. Hofmann, F. Z. Mth. Ps. 31 (1886) 283-. transformations. Haskell, M. W. [1898] Calif. Ac. P. (Mth. Ps.) 1 (1903) 1-. space transformations. Cremona, L. Mil. I. Lomb. Ed. 4 (1871) 269-, 315- ; A. Mt. 5 (1871-73) 131-. , classification. Loria, G. Mil. I. Lomb. Ed. 23 (1890) 824-. transformation between 2 planes. Cayley, A. L. Mth. S. P. 3 (1869-71) 196-. spaces. Cayley, A. [1869] L. Mth. S. P. 3 (1869-71) 127-. transformations in 3-dimensional space, class. Aschieri, F. [1879] Mil. I. Lomb. Mm. 14 (1881) 251-. , theorem, and a transformation of Hirst. Saltel, L. Brux. Ac. Bll. 34 (1872) 51-. Bay systems and focal surfaces of 2 related surfaces. Voss, A. Mth. A. 30 (1887) 227-. Eeciprocal Cremonian systems of degree m, surfaces generated by 2. Jung, G. Em. E. Ac. Line. Ed. 1 (1885) 762-, 773-, 810-, 884. directions, transformation in space by. Laguerre, E. C. E. 92 (1881) 71-. , (Laguerre). Smith, P. F. A. Mth. 1 (1900) 153-. radii vectores, interpretation of correspon- dences by. Amaldi, I. Nap. Ed. 30 (1891) 238-. ' Eelationship, geometric. Swellengrebel, J. G. H. [1850] Crelle J. 43 (1852) 245-. .quadratic. Geiser,C.F. Bern Mt. (1865) 97-. , . Weyr,Ed. Z. Mth. Ps. 14 (1869) 445-. , . Cassani, P. [1897] Yen. I. At. (1897- 98) 30-. , , reciprocal. Milinowski, . Crelle J. Mth. 79 (1875) 140-. of 5th degree. Milller, Hub. Mth. A. 2 (1870) 281-. Eepresentable surfaces : algebraic surfaces uni- formly representable on plane. Nother, M. Gott. Nr. (1870) 1-. 559 8020 Other Transformations Eepresentable surfaces, equation. Brill, A. [1871] Mth. A. 5 (1872) 401-. Eepresentation, plane, of algebraic surfaces with cuspidal curves. Cremona,, L. Bologna Ac. Sc. Mm. 2 (1872) 117-. , , multiple curve. Tognoli, O. G. Mt. 14 (1876) 263-, 378-. , , reciprocal of Steiner's surface. Laguerre, E. [1872] (x) Par. S. Mth. Bll. 1 (1873) 21-. , , ruled surfaces of 4th degree, with double curve of 3rd degree. Clebsch, R.F. A. Mth. A. 2 (1870) 445-. , , surface of 3rd degree. Clebsch, E. F. A. Mth. A. 5 (1872) 419-. , , , as affected by singulari- ties. Diekmann, J. Mth. A. 4 (1871) 443-. of surface of 4th degree. Frahm, W. [1873] (ix) Mth. A. 7 (1874) 512-. Scalene transformation of plane curve. Cayley, A. QJ. Mth. 13 (1875) 321-. Simply rational transformations of surfaces. Painleve, P. C. B. 110 (1890) 184-. , and class of differential equations. Painleve, P. C.B. 110 (1890) 226-. Solid figures, transformation, 2 methods. Amigues, E. N. A. Mth. 18 (1879) 548-; 19 (1880) 433-, 481-. Straight lines and circles through one point, systems. Fuortes, T. G. Mt. 16 (1878) 91-. , transformations. Lovett, E. 0. C. B. 129 (1899) 20-, 144-, 358. Substitution, surf aces obtained from plane curves by. Eckardt, F. E. (vn) Mth. A. 7 (1874) 590-. Substitutions, rational, permitting rational in- version. Eosanes, J. [1870] Crelle J. 73 (1871) 97-. , regular non-linear. Autonne, . C. B. 122 (1896) 1043-. Surfaces, transformations by planes through normals. Enneper, A. Gott. Nr. (1877) 369- ; (1881) 305- ; Mth. A. 21 (1883) 267-. Symmetrical (n, n) point transformations and Poncelet tetrastigms. Schober, K. Mh. Mth. Ps. 9 (1898) 89-. systems of elements. Weyr, E. J. [1874] Bordeaux S. Sc. Mm. 10 (1875) 329-. System of quadrics with common generator and related projective system. Cardinaal, J. Crelle J. Mth. Ill (1893) 31-. Tangential transformation. Genese, E. W. B. A. Ep., (1891) 571-. Transformations which have no fundamental curves of 1st kind. Kantor, S. Acta Mth. 21 (1897) 1-. Transmutations of curves. Cayley, A. Liouv. J. Mth. 14 (1849) 40- ; 15 (1850) 351-. Uniform automorphic transformations, alge- braic configurations with. Hurwitz,A. Mth. A. 32 (1888) 290- ; 41 (1893) 403-. cubic space transformation, simple. Dohle- mann, K. Munch. Ak. Sb. 24 (1895) 41-. transformations, construction. Bertini, E. Mil. I. Lomb. Ed. 15 (1882) 154-. plane and space transformations. Bianchi, L. G. Mt. 16 (1878) 263-. Point-groups 8030 Uniform plane transformations. Nother, M. Mth. A. 5 (1872) 635-. quadratic plane transformations. Timerding, H. E. Mth. A. 53 (1900) 193-. space transformations, and application to representation of algebraic surfaces. Nother, M. [1870] Mth. A. 3 (1871) 547-. transformations, 2. Schoute, P. H. As. Fr. C. B. (1883) 131-. , , geometric representation. Le Paige, C. M. M. H. H. Brux. Ac. Bll. 3 (1882) 760- ; 4 (1882) 415-. (1, 2) plane transformations. Noether, M. Erlang. Ps. Md. S. Sb. 10 (1878) 81-. ( , ) relation between elements of funda- mental configuration of 1st rank. Schubert, H. C. H. Crelle J. Mth. 88 (1880) 311-. ( , ) fundamental configurations of 2nd rank. Schwarz, A. Wien Ak. Sb. 94 (1878) (Ab. 2) 310-. ( , ) space transformations. Paolis, E. de. Em. E. Ac. Line. Mm. 1 (1885) 576-. ( , ) . Wimmer, B. Z. Mth. Ps. 36 (1891) 214-. (1, 3) representation of cubic surface. Kantor, S. [1881-83] Crelle J. Mth. 95 (1883) 147-. (1, 4) correspondence. Cotterill, T. [1868] L. Mth. S. P. 2 (1869) 119-. ( , ) relationship between 2 planes. Alekseev, V. G. [1888] Bee. Mth. (Moscou) 14 (1890) 223- ; Fschr. Mth. (1891) 647-. (2, 2) relationship. Heger, E. Z. Mth. Ps. 17 (1872) 71-. ( , ) , extension of Poncelet' s polygons. Fontene, G. N. A. Mth. 16 (1897) 437-. ( , ) transformation. Visalli, P. Palermo Cir. Mt. Ed. 3 (1889) 165-. ( , ) transformations obtained by 2 quadric transformations. Burali-Forti, C. Palermo Cir. Mt. Ed. 5 (1891) 91-. (3, 3) transformation. Ascione, E. G. Mt. 31 (1893) 55-. (m, n) relation on curve, Stephanos 's theorem generalised. Kohn, G. Wien Ak. Sb. 90 (1885) (Ab. 2) 226-. 8030 Groups of points on an alge- braic curve; genus of curves; principle of correspondence. (See also 7610, 7630, 7660.) Algebraic functions. Brill, A., & Nother, M. [1873] Gott. Nr. (1873) 116- ; Mth. A. 7 (1874) 269-. . Noether, M. Mth. A. 15 (1879) 507-. (Noether). Weiss, W. Mh. Mth. Ps. 7 (1896) 321-. , development of theory. Brill, A., < Noether, M. [1893] D. Mth. Vr. Jbr. 3 (1894) 107-. and singularities of curves. Noether, M. Mth. A. 9 (1876) 166-. , extension of theorem of Noether's. Del Pezzo, P. Palermo Cir. Mt. Ed. 2 (1888) 139-. 560 8030 Groups of Points. Correspondence Conies through several points of given curves. Berzolari, L. Mil. I. Lomb. Ed. 33 (1900) 664-, 809-. Constructions of 3rd and 4th degrees, by straight line and cubic curve. London, F. Z. Mth. Ps. 41 (1896) 129-. Correspondence, Cremonian, in plane and space. Aschieri, F. Mil. I. Lomb. Bd. 14 (1881) 21-. , quadratic, between elements of 2 line spaces. Aschieri, F. Mil. T. Lomb. Bd. 14 (1881) 219-. of 2 curves (Steiner's theorems, Crelle J. 1846). Padova, E. G. Mt. 5 (1867) 240-. on curves. Kiipper, C. Prag Sb. (1892) (Mth.-Nt.) 257-. , formula. Brill, A. Mth. A. 7 (1874) 607-. , multiform, application to curves of 3rd degree. Fries, J. de. N. Arch. Wisk. 12 (1886) 82-; Fschr. Mth. (1885) 629. of 2 points on curve. Cayley, A. L. Mth. S. P. 1 (1866) No. 7, 7 pp. between points of 2 plane curves, special. Pirondini, G. G. Mt. 38 (1900) 92-. of points in relation to 2 tetrahedra. Cayley, A. L. Mth. S. P. 4 (1873) 396-. , quadratic, between 2 simply infinite systems. Weyr, Em. A. Mt. 4 (1870-71) 272-. between straight line and curve, theorem of Liiroth and Gordan. Netto, E. Mth. A. 46 (1895) 310-. of straight lines ; correspondence of points. Aschieri, F. G. Mt. 13 (1875) 328-. systems of points on curve. Brill, A. Gott. Nr. (1871) 507- ; Mth. A. 6 (1873) 33-. (1, 1) between focal cubics, metrical pro- perties. Bricard, R. Par. S. Mth. Bll. 28 (1900) 39-. ( , ), involutional. Genese, R. W. B. A. Bp. (1881) 539. (1, 2), algebraic and geometric study. Pittarelli, G. Bm. B. Ac. Line. Mm. 3 (1886) 375-. ( , - ) on cubics with double point. Pittarelli, G. Bm. B. Ac. Line. Mm. 3 (1886) 401-. (I. 4) between points on curves of surfaces of 3rd degree. Waelsch, E. [1894] D. Mth. Vr. Jbr. 4 (1897) 113-. (2, 2). Cayley, A. QJ. Mth. 11 (1871) 83-; 12 (1873) 197-. ( , ), and curves of genus 1. Weyr, JB. J. Wien Ak. Sb. 87 (1883) (Ab. 2) 592-. . ( , ) , form / (x 2 i/ 2 ) and its invariants and covariants. Capelli, A. G. Mt. 17 (1879) 69-. (m, n). Retail, V. Mil. I. Lomb. Bd. 32 (1899) 1051-. Correspondences, algebraic. Brill, A. Mth. A. 31 (1888) 374-; 36 (1890) 321-. , , and the generalised principle. Hurwitz, A. Leip. Mth. Ps. B. 38 (1886) 10-; Mth. A. 28 (1887) 561-. , , between r points of a linear space. Berzolari, L. Bm. E. Ac. Line. Ed. 4 (1895) (Sem. 2) 148-. 8030 Correspondences on elliptic curves. Broden, T. Stockh. Ofv. (1893) 45-, 213-. . Kantor, S. Tor. Ac. Sc. At. 29 (1894) 9-. , uniform. Segre, C. Tor. Ac. Sc. At. 24 (1889) 734-. between points of 2 curves. Halphen, G. H. [1876] Par. S. Mth. Bll. 5 (1877) 7-. , symmetrical, of odd degree. Kohn, G. Wien Ak. Sb. 106 (1897) (Ab. 2a) 488-. , uniform, between groups of p points on curve of genus p. Castelnuovo, G. Mil. I. Lomb. Ed. 25 (1892) 1189-. , , hyperelliptic curves having. Kantor, S. Palermo Cir. Mt. Ed. 9 (1895) 65-. , , singular, of harmonic elliptic curves. Amodeo, F. A. Mt. 19 (1891-92) 145-. (1, 2) and (1, 3). Piazza, S. Tor. Ac. Sc. At. 17 (1881) 431-. (Pi P) on curves of genus p with general moduli. Scorza, G. Tor. Ac. Sc. At. 35 (1900) 285- or 443-. Corresponding points on 2 curves, theorems. Zeuthen, H. G. [1870] Mth. A. 3 (1871) 150-, 323-. , Zeuthen's theorem. Weiss, W. Mth. A. 29 (1887) 382-. , , deduction from. Brodtn, T. Stockh. Ofv. (1893) 345-. Cremona transformations with fixed curves. Doehlemann, K. Mth. A. 39 (1891) 567-. Curve, genus 0, binary forms represented on. Maisano, G. Palermo Cir. Mt. Ed. 1 (1887) 14-. , 1, cyclic relations of points on. Gegen- bauer, L. Mh. Mth. Ps. 4 (1893) 330. , 3, and uniform functions of 2 variables. Picard, E. C. E. 93 (1881) 835-. , p, 2q points on, problem. Ptaszycki, J. C. E. 130 (1900) 105-. Curves, any genus, linear systems. Jung, G. [1887] A. Mt. 15 (1887-88) 277-. , genus 0, algebraic forms in theory. Igel, B. Wien Ak. Sb. 89 (1884) (Ab. 2) 218-. _, , families. Lerch, M. Mh. Mth. Ps. 2 (1891) 465-. , 1. Humbert, G. C. E. 97 (1883) 989-, 1042-, 1136-. , . Schlesinger, 0. C. B. 107 (1888)224-. , , involutions on. Weyr, E. Wien Ak. Sb. 101 (1892) (Ab. 2a) 1506-. , , , and Steiner's polygons. Weyr, E. Wien Ak. Sb. 101 (1892) (Ab. 2a) 1457-, , , parametric representation. Duarte Leite, . G. Teix. J. Sc. 9 (1889) 3-; Fschr. Mth. (1888) 716. , , symbolical calculus on. Czuber, E. [1894] D. Mth. Vr. Jbr. 4 (1897) 100-. , , . Weyr, E. Wien Ak. Sb. 193 (1894) (Ab. 2a) 365-. , , systems. Martinetti, V. Mil. I. Lomb. Bd. 20 (1887) 264-, [854]. , >1, hyperelliptic. K-ilpper, K. Prag Sb. (1896) (Mth.-Nt.) No. 43, 11 pp. t 2, proof of theorem. Esclaibes, (le rev. pere) d\ (xn) Brux. S. Sc. A. 5 (1881) (Pt. 2) 34-. VOL. I. 561 8030 Correspondence Curves, genus ^>, normal and subnormal, and bi- a-ational and uniform transformation. Arme- nante, A., & Jung, G. G. Mt. 7 (1869) 235-. , genera 4, 5, 6, allowing automorphic trans- formation. Wiman, A. [1895] Stockh. Ak. Hndl. Bh. 21 (Afd. 1) (1896) No. 3, 41 pp. of maximum genus, plane and skew. Kupper, K. Prag Sb. (1888) (Mth.-Nt.) 265-. , plane, decomposition of singularities, and genus. Bertini, E. Mil. I. Lomb. Bd. 21 (1888) 326-, 413-. , , general theorem. Guccia, G. B. C. B. 107 (1888) 903-, 1030. , 2, related point to point projectively, are of same genus p, proof. Bertini, E. G. Mt. 7 (1869) 105-. , Biemann's surfaces in relation to. Del Pezzo, P. Palermo Cir. Mt. Ed. 6 (1892) 115-. and surfaces, points in theory. Humbert, G. Liouv. J. Mth. 10 (1894) 169-. , (Humbert). Painleve, P. Liouv. J. Mth. 10 (1894) 203-. , 3rd degree, curves of mth degree through 3m -1 points. Halphen, G. H. Mth. A. 15 (1879) 359-. , , rational derivation on. Story, W. E. Am. J. Mth. 3 (1880) 356-. , , recurring series of pentagons in. Serret, P. C. B. 115 (1892) 406-, 436-. , 4th degree, genus 3, parametric expression of coordinates. Cayley, A. L. Mth. S. P. 6 (1875) 81-. , 7ith degree. Kiipper, K. Prag Ab. 3 (1890) No. 7, 25 pp. , , maximum basis B for /t-fold variety. Kiipper, C. Mh. Mth. Ps. 6 (1895) 5-. "Curves," points on. Segre, C. A. Mt. 22 (1894) 41-. Cyclic curves, curvature, and projective geo- metry. Schwarz, A. Mh. Mth. Ps. 10 (1899) 250- ; 11 (1900) 71-. Cyclic projective groups of points. Liiroth, J. Mth. A. 13 (1878) 305-. systems. Bonsdorff, E. [E. J.~\ [1879] Helsingf. Acta 11 (1880) 327-. Double planes which can be represented on single plane. Noether, M. Mth. A. 33 (1889) 525-. Elliptic curyes, geometry on. Castelnuovo, G. Tor. Ac. Sc. At. 24 (1889) 4-. Genus and adjoint curves of algebraic curve, method of determining. Tikhomandritzky, . Bll. Sc. Mth. 17 (1893) 51- ; Par. EC. Norm. A. 10 (1893) 151-. , conservation in uniform transformations. Halphen, G. H. [1875] Par. S. Mth. Bll. 4 (1876) 29-. , (1, 1) transformation of plane curve. Schubert, H. C. H. Mth. A. 16 (1880) 180-. of curves. Baltzer, R. (rx) Leip. Mth. Ps. B. 25 (1873) 523-. . Halphen, G. H. As. Fr. C. B. 4 (1875) 237-. Groups of Points 8030 Genus of curves corresponding to points in plane involutions of any class. Martinetti, V. Palermo Cir. Mt. Bd. 4 (1890) 30-, 126-. , etc., of curves, Pliicker's symbols for. Zahradnik, K. Casopis 2 (1873) 172-, 266-; Fschr. Mth. (*1873) 414. of plane triangular curves. Jamet, V. Par. S. Mth. Bll. 16 (1888) 132-. set of curves. Zeuthen, H. G. Palermo Cir. Mt. Bd. 3 (1889) 171-. skew curves. Salvatore-Dino, N. Nap. Ed. 18 (1879) 133-. Groups of points. Paolis, R. de. Em. E. Ac. Line. Mm. 1 (1877) 511-. Castelnuovo, G. Tor. Ac. Sc. At. 24 346-. Bertini, E. Tor. Ac. Sc. At. 26 (1891) 118-. Valentiner, H. Sk. Nf. F. (1892) 347-; Fschr. Mth. (1892) 660. Kiipper, C. Mh. Mth. Ps. 6 (1895) 127-. Macaulay, F. S. L. Mth. S. P. 26 (1895) 495-. Hardcastle, (Miss) F. B. A. Ep. (1900) 121-. (de Paolis, 1877.) Scott, (Miss) C. A. [1900] N. Y. Am. Mth. S. Bll. 7 (1901) 24-. with bibliography. Hardcastle, (Miss) F. N. Y. Am. Mth. S. Bll. 4 (1898) 390-. Cay ley-Brill theorem, new proof. Sporer, B. Z. Mth. Ps. 39 (1894) 228-. on circle. Kantor, S. [1878] Mth. A. 14 (1879) 323-. curves of 3rd degree. Weyr, Em. Z. Mth. Ps. 15 (1870) 344-. , 8th point not determined by 7. Cayley, A. Mess. Mth. 21 (1892) 132-. , relation between 10 points. Weddle, T. Camb. and Dubl. Mth. J. 5 (1850) 236-. , . Reiss, M. [1867] Mth. A. 2 (1870) 413-. mth degree, genus p, cut by oo 2 lines. Kiipper, K. Prag Sb. (1887) (Mth.-Nt.) 477-, 609-. determination. Chasles, M. C. E. 73 (1871) 927-, 970-. harmonic, on rational curves of 3rd and 4th degrees. Zahradnik, K. Prag Sb. (1874) 12-. involution triplets. Weyr, E. J. [1883] Wien Ak. Sb. 88 (1884) (Ab. 2) 436-. involutory. Vries, J. de. Amst. Ak. Vs. M. 6 (1889) 92- ; Arch. N6erl. 23 (1889) 355-. series, irrational. Amodeo, F. A. Mt. 20 (1892-93) 227-. of 1st order, anharmonic ratio of 4 pairs of points. Klein, B. Z. Mth. Ps. 28 (1883) 252-. irrational series. Castelnuovo, G. Em. E. Ac. Line. Ed. 7 (1891) (Sem. 2) 294-. linear. Rosanes, J. Crelle J. Mth. 88 (1880) 241-. series. Bertini, E. A. Mt. 22 (1894) 1-. , multiples. Castelnuovo, G. Palermo Cir. Mt. Ed. 7 (1893) 89-. 562 8030 Correspondence. Groups of Points Involutions 8030 minimum abnormal, on curve C" 1 . Kiipper, C. Prag Sb. (1892) (Mth.-Nt.) 403-. primitive and imprimitive, on curve C p n . Kiipper, K. Prag Sb. (1897) (Mth.-Nt.) No. 31, 14 pp. protective, on same straight line. Sidler, G. Ziir. Vjschr. 9 (1864) 217-. on rational curves. Weyr, Em. Prag Sb. (1873) 70-. and reduction of algebraic integral. Cayley, A. (xn) J. H. Un. Cir. [1] (1882) 209. remarkable. Kraus, L. [1879] Mth. A. 16 (1880) 245-. on skew curve intersected by surfaces, com- parison with those of residual curves. Rohn, K. Leip. Mth. Ps. B. 49 (1897) 627-. special systems, on particular type of base- curve. Hardcastle, (Miss) F. [1897] L. Mth. S. P. 29 (1898) 132-. on straight line, and generation of plane curves. Schumacher, E. Crelle J. Mth. 110 (1892) 230- ; 111 (1893) 254-. 7ith order, with base on curve of mth degree. Chasles, M. C. B. 45 (1857) 393-. G., 1 , on sextic curve with 5 double points. Cayley, A. [1874] Mth. A. 8 (1875) 359-. g < 2 ' on curves C p n . Kiipper, C. Prag Sb. (1892) (Mth.-Nt.) 264-. (m, n) homographies, and double elements. Weyr, Em. Prag Sb. (1869) (pt. 2) 33-. Homography of 2nd kind. Castelnuovo, G. Ven. I. At. (1886-87) 1041-. on skew cubic. Deruyts, F. Brux. Ac. Bll. 17 (1889) 312-. . Aschieri, F. Mil. I. Lomb. Ed. 23 (1890) 312-. Intersection of 2 cones, and problem of Steiner's on plane curves. Beck, A. Ziir. Vjschr. 38 (1893) 199-, 266-. curve with non-adjoint curves, systems of points. Noether, M. Mth. A. 15 (1879) 507-. 2 curves at a singular point. Guccia, G. B. C. E. 107 (1888) 656-, 766. curves, systems of points. Bacharach, J. [1879] Erlang. Ps. Md. S. Sb. 12 (1880) 12-. Intersections of curves. Krey, H. Z. Mth. Ps. 22 (1877) 396-. of 3rd and 4th degrees with conies. Rohn, K. Dresden Isis Sb. (1898) 24. plane curves. Scott, (Miss) C. A. N. Y. Am. Mth. S. Bll. 4 (1898) 260-. . Macaulay, F. S. N. Y. Am. Mth. S. Bll. 4 (1898) 540-. , multiplicities in. Segre, C. G. Mt. 36 (1898) 1-. Involutions. Hearn, G. W. Mathematician 2 (1847) 63-. Terquem, 0. N. A. Mth. 12 (1853) 24-. any type. Genty, M. Par. S. Mth. Bll. 20 (1892) 106-. biquadratic. Le Paige, C. C. E. 98 (1884) 285-. . Fairon, J. Brux. Ac. Bll. (1900) 950-. 563 biquadratic, degenerate forms, and 7 systems of conies having 4-point contact with curve of 4th degree. Weyr, E. J. Wien Ak. Sb. 83 (1881) (Ab. 2) 807-. , 1st rank. Weyr, E. J. Wien Ak. Sb. 83 (1881) (Ab. 2) 300-. , 2nd rank, and their typical curves. Weyr, E. J. Wien Ak. Sb. 81 (1880) (Ab. 2) 1007-. , 3rd rank, and application to curves of 4th degree. Le Paige, C. M. M. H. H. Prag Sb. (1881) 61-. conjugate. Le Paige, C. M. M. H. H. [1881] Wien Ak. Sb. 84 (1882) (Ab. 2) 233-; 85 (1882) (Ab. 2) 844-. cubic. Le Paige, G. M. M. H. H. Wien Ak. Sb. 81 (1880) (Ab. 2) 845- ; 83 (1881) (Ab. 2) 351- ; Liege S. Sc. Mm. 11 (1885) No. 1, 19pp. . Berzolari, L. Nap. Ed. 30 (1891) 35-, 71-. . Vries, J. de. Amst. Ak. Vs. 3 (1895) . Klug, L. Orv.-Termt. Ets. (Termt. Szak) (1898) 95-, (Rv.) 42-. and biquadratic. Milinowski, . Z. Mth. Ps. 19 (1874) 205-. , for conic. Kantor, S. Prag Sb. (1878) 312-. , conjugate. Servais, C. Brux. Ac. Bll. 20 (1890) 272-. , fundamental problem. Weyr, Em. Prag Sb. (1872) 28-. and higher. Vries, J. de. Mh. Mth. Ps. 5 (1894) 1-. , involving anharmonic properties. Weyr., Em. Mil. I. Lomb. Ed. 4 (1871) 206-. , in plane of complex variable. Vries, J. de. Palermo Cir. Mt. Ed. 5 (1891) 289-. , relation between singular elements. Le Paige, C. M. M. H. H. Wien Ak. Sb. 81 (1880) (Ab. 2) 159-. , (Le Paige). Weyr, E. J. Wien Ak. Sb. 81 (1880) (Ab: 2) 162-. , , projective. Weyr, E. J. Wien Ak. Sb. 73 (1876) (Ab. 2) 654-. , on skew curves of 3rd degree. Vries, J. de. N. Arch. Wisk. 4 (1899) 101-. , theory. Weyr, E. J. [1874] Prag Ab. 7 (1875) (Mth.) 54 pp. on curves of 3rd degree. Weyr, E. Casopis 9 (*1880) 145- ; Fschr. Mth. (*1880) 484. . Kiipper, G. Prag Ab. 12 (1885) (Mth.) No. 1, 18 pp. . Martinetti, V. G. Mt. 23 (1885) 37-. 4th degree. Le Paige, C. Brux. S. Sc. A. 8 (1884) (Pt. 2) 87-. fundamental, on rational curves. Stahl, W. Crelle J. Mth. 104 (1889) 38-. generalisation of theory, geometrical applica- tions. Jonquieres, E. de. Tortolini A. 2 (1859) 86-. higher. Weyr, Em. Crelle J. 72 (1870) 285-; Prag Sb. (1870) 14-; Wien Sb. 61 (1870) (Ab. 2) 600- ; Z. Mth. Ps. 16 (1871) 353-. . August, F. Arch. Mth. Ps. 55 (1873) 337-. . Eggers, H. Arch. Mth. Ps. 55 (1873) 341-. NN 2 8030 Involutions higher. Le Paige, C. M. M. H. H. [1877] (xn) Brux. S. Sc. A. 2 (1878) (Ft. 1) 54- ; (Pt. 2) 25-. , groups of points common to 2. Le Paige, C. Brux. Ac. Bll. 11 (1886) 121-. , and invariants of binary forms. Le Paige, C. M. M. H. H. Brux. Ac. Bll. 44 (1877) 231-. , multiple points. Le Paige, C. M. M. H. H. Brux. Ac. Bll. 46 (1878) 247-. , represented on common basis. Weyr, E. J. Liege S. Sc. Mm. 10 (1883) (A T o. 3) 8 pp. , on skew curves of genus 1. Weyr, E. Wien Ak. Sb. 100 (1891) (Ab. 2a) 589-. and homographies, higher. Le Paige, C. G. Teix. J. Sc. 5 (*1883) 27-, 77-. multiple. Vdlyi, G. Mth. Termt. Ets. 12 (1894) 394- ; 13 (1895) 408- ; 14 (1896) 381- ; Mth. Nt. B. Ung. 13 (1897) 244-, 259-, 343-. multiply infinite, belonging to curve, linearity. Castelnuovo, G. Tor. Ac. Sc. At. 28 (1893) 727- with ordinary singularities. Guccia, G. B. Palermo Cir. Mt. Ed. 7 (1893) 49- ; 8 (1894) 227-. plane. Bertini, E. [1882] Mil. I. Lomb. Ed. 16 (1883) 89-, 190-. , new theorem. Berzolari, L. Palermo Cir. Mt. Ed. 3 (1889) 145-. quadruple, on curves of 4th degree. Fries, J. de. Amst. Ak. Vs. M. 4 (1888) 307- ; Arch. Neerl. 23 (1889) 93-. rational, number on curve of given genus. Castelnuovo, G. Em. E. Ac. Line. Ed. 5 (1889) (Sem. 2) 130-. on skew curves of 4th degree. Le Paige, C. C. E. 98 (1884) 353-. surfaces. Weyr, E. J. Brux. Ac. Bll. 3 (1882) 472-. theory. Cayley, A. [1864] Camb. Ph. S. T. 11 (1871) 21-. . Tognali, 0. G. Mt. 20 (1882) 270-. . Berzolari, L. Nap. Ed. 30 (1891) 35-. various orders, in systems of 1st species. Battaglini, G. Nap. Ac. At. 1 (1863) No. 12, 14pp. , 2nd species. Battaglini, G. Nap. Ed. 4 (1865) 226-. 3rd and 4th classes. Martinetti, V. A. Mt. 12 (*1883-84) 73-. nth degree. Weyr, Em. G. Mt. 10 (1872) 165-. and fcth rank. Weyr, E. J. Wien Ak. Sb. 79 (1879) (Ab. 2) 680-. , number of groups in. Lerch, M. Prag Sb. (1885) (Mth.-Nt.) 597-. 2nd rank. Weyr, E. J. Wien Ak. Sb. 83 (1881) (Ab. 2) 349-. Involutory pairs in multiform configurations, number. Weyr, Em. [1871] Crelle J. 74 (1872) 189-. plane transformations. Martinetti, V. A. Mt. 13 (1885) 53-. , double. Bertini, E. Mil. I. Lomb. Ed. 22 (1889) 771-. Principle of Correspondence 8030 Involutory plane transformations, uniform. Bertini, E. [1876] A. Mt. 8 (1877) 11-, 146, , . Caporali, E. Nap. Ed. 18 (1879) 212-. , . Bertini, E. Mil. I. Lomb. Ed. 13 (1880) 443-. , . Bobek, K. Wien Ak. Sb. 91 (1885) (Ab. 2) 476-. , , application to determine involu- tions of 5th class. Berzolari, L. A. Mt. 16 (1888-89) 191-, 335. relation arising from 2 curves. Fouret, G. C. E. 96 (1883) 1213-. Moduli of class of algebraic equations. Brill, A. Mth. A. 1 (1869) 401- ; 2 (1870) 471-. equations or curves of given genus, number. Casorati, F., & Cremona, L. Mil. I. Lomb. Ed. 2 (1869) 620-. Normal curves for p = 5, 6, 7. Noether, M. Mth. A. 26 (1886) 143-. space and maximum genus for curves of with degree. Amodeo, F. Nap. Ed. 37 (1898) 433. trigonal curves, plane. Amodeo, F. C. E. 130 (1900) 1744- ; 131 (1900) 400. Null system, significance for cubic involutions. Weyr, E. J. [1881] Wien Ak. Sb. 84 (1882) (Ab. 2) 1264-. , 2i] " d degree. Ameseder, A. Wien Ak. Sb. 83 (1881) (Ab. 2) 385-. , , Ameseder's, applications. Oppen- heimer, H. Arch. Mth. Ps. 13 (1895) 268-. , , general. Ameseder, A. Crelle J. Mth. 97 (1884) 62-. Polar curves. PlUcker, J. Crelle J. 12 (1834) 105-. groups. Weyr, E. J. Wien Ak. Sb. 81 (1880) (Ab. 2) 841-. , geometric representation of theory. Waelsch, E. [1883] Wien Ak. Sb. 88 (1884) (Ab. 2) 418-. Polygonal curves. Amodeo, F. A. Mt. 21 (1893) 221- ; 24 (1896) 1- ; Nap. Ac. At. 9 (1899) No. 4, 21 pp.; Nap. Ed. 39 (1900) , nth degree, genus p. Kupper, C. Prag Sb. (1895) (Mth.-Nt.) No. 25, 16 pp.; (1896) (Mth.-Nt.) No. 23, 9 pp. , , (21)- Kilpper, C. Mh. Mth. Ps. 8 (1897) 57-. and skew curves, relations between. Kilpper, C. Prag Sb. (1896) (Mth.-Nt.) No. 4, 11 pp. PEINCIPLE OF COEEESPONDENCE. Chasles, M. C. E. 62 (1866) 579-. Cayley, A. [1867] Phil. Trans. 158 (1868) 145-. Chasles, M. C. E. 72 (1871) 577-. Zeuthen, H. G. Bll. Sc. Mth. As. 5 (1873) 186-. Cayley, A. QJ. Mth. 15 (1878) 32-. Burkhardt, H. C. E. 126 (1898) 1854-. application. Vahlen, K. T. Crelle J. Mth. 118 (1897) 251-. 564 8030 Principle of Correspondence 8030 application to analytical solutions. Saltel, L. C. E. 82 (1876) 324-. to finding class of evolute and caustic of curve. Chasles, M. C. E. 72 (1871) 394-. degree of envelope of curve or surface. Saltel, L. C. E. 83 (1876) 608-. locus defined by algebraic con- ditions. Saltel, L. C. E. 82 (1876) 63- ; 83 (1876) 529-. number of points of intersection of 2 curves. Chasles, M. C. E. 75 (1872) 736- ; 76 (1873) 126-. 3 surfaces, or of skew curve with a surface. Fouret, G. (rx) Par. S. Mth. Bll. 1 (1873) 122-, 258-. plane in which 3 curves of 3 given systems touch. Fouret, G. Par. S. Mth. Bll. 5 (1877) 130-. solutions of n algebraic equations in n unknowns. Fouret, G. C. E. 78 (1874) 183-. , new developments due to. Saltel, L. Brux. Ac. Bll. 45 (1878) 102-. to proof of Bezout's theorem. Saltel, L. C. E. 81 (1875) 884-. segments. Chasles, M. C. E. 83 (1876) 467-, 495-, 519-. on tangents. Chasles, M. C. E. 81 (1875) 253-. Cayley and Brill's. Zeuthen, H. G. Mth. A. 40 (1892) 99-. character. Chasles, M. C. E. 78 (1874) 577-. Chasles's, analytical extension. Saltel, L. C. E. 80 (1875) 1064-. complementary theorem applied to determina- tion, without calculation, of order of multi- plicity of point on given locus. Saltel, L. C. E. 81 (1875) 1047-. definition, and applications. Saltel, L. [1879] Bordeaux S. Sc. Mm. 4 (1882) 1-. degree of multiplicity of solutions, and appli- cation to Pliicker's relations. Zeuthen, H. G. N. A. Mth. 6 (1867) 200-. generalisation, applied to elimination. Saltel, L. N. A. Mth. 12 (1873) 565-. generalised. Bobek, K. Wien Ak. Sb. 93 (1886) (Ab. 2) 899-. for groups of n points and n rays. Schubert, H. C. H. Mth. A. 12 (1877) 180-. and law of decomposition applied to skew curves, in particular to locus of centres of osculating spheres of given skew curve. Saltel, L. Brux. Ac. Bll. 43 (1877) 266-. of plane and space. Zeuthen, H. G. C. E. 78 (1874) 1553-. and systems of curves. Segre, C. Bb. Mth. (1892) 33-. theory of characteristics. Legoux,A. Toul. Ac. Sc. Mm. 8 (1886) 208-. of 2 variables. Chasles, M. C. E. 41 (1855) 1097-. Protective generation of groups of curves C*+". Kilpper, K. Prag Sb. (1897) (Mth.- Nt.) No. 5, 15 pp. Projective properties of curves. Juel, C. [1899] Kjtfb. Dn. Vd. Selsk. Skr. 10 (1899-1902) 1-. [With French resuml.] Eational curve, plane, 3rd degree, representa- tion on conic. Weyr, E. J. Wien Ak. Sb. 79 (1879) (Ab. 2) 429-. , skew, 4th degree. Rohn, K. Leip. Mth. Ps. B. 42 (1890) 208- ; 43 (1891) 1-. , , , automorphic transforma- tion. Brambilla, A. Mil. I. Lomb. Ed. 20 (1887) 780-. , , , harmonic centre of "quad- ruple" given by. Weyr, E. J. Wien Ak. Sb. 81 (1880) (Ab. 2) 1218-. , , , representation on conic. Weyr, E. J. [1875-76] Wien Ak. Sb. 72 (1876) (Ab. 2) 686-; 73 (1876) (Ab. 2) 203-. curves. Davis, E. W. J. H. Un. Cir. [3] (1884) 123. with double point or cusp, Abel's theorem. Johansson, N. A. Fschr. Mth. (1885) 405. in 7i-flat space. Bissing, G. J. H. Un. Cir. [3] (1884) 123. , generation. Jonquieres, E. de. C. E. 105 (1887) 1148-. , plane. Weyr, E. Casopis 8 (*1879) 193- ; Fschr. Mth. (*1879) 488. , , representations on one another. Weyr, E. J. Wien Az. 15 (1878) 228-. , , degrees n and n-3, intersections. Fabry, E. Par. EC. Norm. A. 13 (1896) 107-. , skew, singularities of 2nd order. Weyr, Em. A. Mt. 4 (1870-71) 328-. Eeducible curves. Noether,M. Erlang. Ps. Md. S. Sb. 17 (1885) 13-; Acta Mth. 8 (1886) 161-. Eesiduation in regard to curve of 3rd degree. Cayley, A. [1873] (vn) Mess. Mth. 3 (1874) 62-. theorem (intersections of plane curves at multiple points). Macaulay, F. S. L. Mth. S. P. 31 (1900) 381-. , Noether's theorem, and the Eiemann- Eoch theorem. Macaulay, F. S. [1899] L. Mth. S. P. 31 (1900) 15-. Eesidues in canonical series of adjoint curves of degree m-3-a. Amodeo, F. Em. E. Ac. Line. Ed. 2 (1893) (Sem. 1) 528-. relative to asymptotes. Quadrature of curves dependent on genus. Marie, M. C. E. 76 (1873) 943-. Singular points. PlUcker, J. [1836] Liouv. J. Mth. 2 (1837) 11-. and tangents, influence on degree and class of plane curves, tazarski, M. Krk. Ak. (Mt.-Prz.) Ez. 15 (1887) 278-; Fschr. Mth. (1887) 607. Singularities of plane curves, higher. Smith, H. J. S. L. Mth. S. P. 6 (1874-75) 153-. , and new species of curve. Brill, A. [1879] Mth. A. 16 (1880) 348-. Skew curve, 4th degree, with double point, representation on conic. Weyr, E. J. [1878] Wien Ak. Sb. 78 (1879) (Ab. 2) 891-. , , property of 9 points, etc. Petot, A. C. E. 98 (1884) 1245-. , 5th degree, genus 1. Montesano, D. Nap. Ed. 27 (1888) 181-. 565 8030 Groups of Points and Curves on Surfaces. Genus 8040 Skew curve, 6th degree, genus 1, arising from 8040 Groups of CUTV6S and points 45* (i8"4) r 54l a ! ions - London ' F ' Mth ' A ' on an algebraic surface ; genus curves. Kilpper, K. Prag Ab. 4 (1892) O f surfaces. (See also 7640.) No. 5, 16 pp., No. 7, 15 pp. v , genus 0. Tognoli, 0. G. Mt. 11 (1873) 180-. , . Saltel, L. C. R. 80 (1875) 1324-. , , double points. Brill, A. Mth. A. 3 (1871) 456-. of given degree, maximum genus. Bobek, K. Wien Ak. Sb. 93 (1886) (Ab. 2) 13-. , 5th degree, genus 1. Weyr, E. Wien Ak. Sb. 90 (1885) (Ab. 2) 206- ; 92 (1886) (Ab. 2) 498-; 97 (1889) (Ab. 2a) 592-. , 6th degree, genus 1. Weyr, E. Wien Ak. Sb. 99 (1891) (Ab. 2a) 932- ; 100 (1891) (Ab. 2a) 457-. Steiner's polygons on curve of 3rd degree. Kilpper, C. [1873-84] Prag Ab. 6 (1874) 39 pp. ; Mth. A. 24 (1884) 1-. . Martinetti, V. Palermo Cir. Mt. Ed. 5 (1891) 109-. porismatic polygons determined by (2, 2) relationship. Weyr, E. [18691 Crelle J. 71 (1870) 18-; 73(1871) 87-. System of 3 doubly quadratic relations between 3 variables. Fontene, G., & Bricard, R. N. A. Mth. 18 (1899) 437-. Transformation of curve to one without mul- tiple points. Fieri, M. Rv. Mt. 4 (1894) 40-. . Levi, B. Rm. R. Ac. Line. Rd. 7 (1898) (Sem. 1) 111-. systems in plane, uniform periodic. Kantor, S. Acta Mth. 19 (1895) 115-. method, reduction of singularities. Vessiot, E. Par. S. Mth. Bll. 22 (1894) 208-. of plane curves, uniform. Voss, (Dr) A. Gott. Nr. (1873) 414-. Transformations of curves, birational. Poincare, H. C. R. 117 (1893) 18-. in plane, birational, finite groups. Wiman, A. Mth. A. 48 (1897) 195-. , uniform automorphic, curves admitting. Noether, M. Mth. A. 20 (1882) 59-; 21 (1883) 138-. , periodic, foundations for theory. Kantor, S. Nap. Ac. At. 1 (1888) No. 1, 264 pp. ; 4 (1891) No. 2, 74 pp. Trigonal curves. Cavazzoni, L. Mil. I. Lomb. Rd. 32 (1899) 777-. Triplets of points on cissoid. Zahradnik, K. Casopis 12 (*1883) 105- ; Arch. Mth. Ps. 6 (1888) 392-. Castelnuovo, G. Mil. I. Lomb. Rd. 24 (1891) 127-, 307-. Enriques, F. [1893-96] Tor. Ac. Sc. Mm. 44 (1894) 171- ; Rm. S. It. Mm. 10 (1896) 1-. Castelnuovo, G., & Enriques, F. Mth. A. 48 (1897) 241-. Asymptotic curves on skew surfaces. Jung, V. Casopis 20 (1891) 294-; Fschr.Mth. (1891)824. Class of surfaces. Humbert, G. Palermo Cir. Mt. Rd. 3 (1889) 277-; 4 (1890) 54-, 71, 72. (Humbert). Castelnuovo, G. Palermo Cir. Mt. Rd. 4 (1890) 69-, 71-, 195-. (Castelnuovo). Guccia, . Palermo Cir. Mt. Rd. 4 (1890) 71. . Noether, M. Palermo Cir. Mt. Rd. 4 (1890) 299-. Collineation groups on Kummer's surfaces. Kantor, S. Am. J. Mth. 19 (1897) 86-. Continuous series of curves on regular surface. Enriques, F. Palermo Cir. Mt. Rd. 13 (1899) 95-. Correspondence, point to point, class of surfaces representable on plane by. Brambilla, A. Mil. I. Lomb. Rd. 21 (1888) 334-, 511-. between points of plane and those of elliptic paraboloid. Ciamberlini,C. G. Mt. 32 (1894) 69-. , projective, between forms of 2 dimensions. Paolis, E. de. Rm. R. Ac. Line. Rd. 3 (1894) (Sem. 2) 225-. , uniform, of algebraic configurations. Noether, M. Mth. A. 2 (1870) 293-; 8 (1875) 495-. , , surfaces, 2 criteria. Nother, M. Gott. Nr. (1873) 248-. , , , and differential equations, 2nd order, with fixed critical points. Painleve, P. C. R. 117 (1893) 611-. Correspondences, uniform, curves and surfaces admitting. Kantor, S. C. R. 100 (1885) 343-. (1, 4) between points on curves of surfaces of 3rd degree. Waelsch, G. [1894] D. Mth. Vr. Jbr. 4 (1897) 113-. Curve of 4-point contact on surface. Voss, 'A. E. Mth. A. 9 (1876) 483-. Curves on cones, genus. Sturm, R. Mth. A. 19 (1882) 487-. surfaces. Brunei, . Bordeaux S. Sc. Mm. 3 (1893) iv. and surfaces, points in theory. Humbert, G. Liouv. J. Mth. 10 (1894) 169-. Uniform functions connected by an algebraic ^ ' ' _ (Humbert).' Painleve, P. relation, property. Picard, E. Bll. Sc. Mth. As. 4 (1880) (Pt. 1) 416-. Weierstrass's points on algebraic curve. Haure, M. Par. EC. Norm. A. 13 (1896) 115-. . Segre, C. Rm. R. Ac. Line. Rd. 8 (1899) (Sem. 2) 89-. Liouv. J. Mth. 10 (1894) 203-. Developables, curves on. Kunzler, G. Sch. Nf. Gs. Vh. (1898) 47. Double planes of genus 1. Enriques, F. Rm. S. It. Mm. 10 (1896) 201-. Envelope with n parameters connected by n - 2 relations, degree. Saltel, L. C. R. 83 (1876) 894- ; Brux. Ac. Bll. 43 (1877) 24-, 31-. Genus of certain surfaces. Cayley, A. Mth. A. 3 (1871) 526-. 566 8040 Groups of Curves and Points on Surfaces. Genus 8040 Groups of points in involution marked on surface. Le Paige, C. M. M. H. H. C. B. 99 (1884) 537-. plane, and effect in determining curves. Macaulay, F. S. L. Mth. S. P. 29 (1898) 673-. . on surface, impossibility of some series. Picard, E. C. E. 126 (1898) 1383-. straight lines on surfaces of higher degree. A/olter, F. G. Mth. A. 27 (1886) 277-; 29 (1887) 1-. Homaloidal surfaces containing groups of conies. Montesano, D. Mil. I. Lomb. Ed. 24 (1891) 889-. , 4th and 5th degrees, with no multiple lines. Montesano, D. Nap. Ed. 39 (1900) 158-. Hyperelliptic curves, surfaces containing net of. Castelnuovo, G. Em. E. Ac. Line. Ed. 3 (1894) (Sem. 1) 473-. surfaces, curves on. Humbert, G. Liouv. J. Mth. 9 (1893) 29-, 361-. Intersection of 3 surfaces in a singular point. Guccia, G. B. Em. E. Ac. Line. Ed. 5 (1889) (Sem. 1) 456-. Intersections of 3 surfaces. Berzolari, L. A. Mt. 24 (1896) 165-. Invariant number in theory of surfaces. Picard, E. C. E. 116 (1893) 285-. numbers in theory of surfaces. Picard, E. C. E. 119 (1894) 1169- ; 122 (1896) 101-. Involution, generalised. Cayley, A. Camb. and Dubl. Mth. J. 2 (1847) 52-. of higher dimensions. Vanecek, J. S., & Vanecek, M. N. C. E. 99 (1884) 742-, 856-, 909-. and rational homography, theory. Deruyts, F. Liege S. Sc. Mm. 17 (1892) No. 3, 2UH pp. Involutions, plane. Chizzoni, F. Em. E. Ac. Line. Mm. 19 (1884) 301-. , , rationality. Castelnuovo, G. Em. E. Ac. Line. Ed. 2 (1893) (Sem. 2) 205- ; Mth. A. 44 (1894) 125-. Involutory transformations, groups. Monte- sano, D. Yen. I. At. (1887-88) 1425-. Linear genus of surface and classification to which it leads. Castelnuovo, G. Em. E. Ac. Line. Ed. 6 (1897) (Sem. 1) 372-, 406-. systems of curves on surface. Castelnuovo, G. Em. S. It. Mm. 10 (1896) 82-; A. Mt. 25 (1897) 235-. lines on surface. Picard, E. Palermo Cir. Mt. Ed. 13 (1899) 344-. Moduli of class of surfaces, number. Noether, M. Berl. Ak. Sb. (1888) 123-. Multiple curves and developables of certain surface. Lazzeri, G. Yen. I. At. (1887-88) 171-. of surfaces. Ndther, M. [1871] A. Mt. 5 (1871-73) 163-. Eational curves, surfaces with systems. Nother, M. [1870] Mth. A. 3 (1871) 161-. Eeciprocal of surface, degree. Salmon, G. [1847-55] Camb. and Dubl. Mth. J. 2 (1847) 65-; Ir. Ac. T. 23 (1859) 461-. surfaces. Cayley, A. Ir. Ac. P. 7 (1858) 20-; Phil. Trans. 159 (1869) 201- ; 162 (1872) 83-. Eeciprocal surfaces. Zeuthen, H. G. Mth. A. 4 (1871) 633-. Eiemann-Eoch theorem extended to surfaces. Noether, M. C. E. 103 (1886) 734-. Euled surfaces, double curve on. Wiman, A. Acta Mth. 19 (1895) 63-. of any genus. Segre, C. Tor. Ac. Sc. At. 22 (1886-87) 362-. with rational double curves. Weyr, E. J. [1881] Wien Ak. Sb. 84 (1882) (Ab. 2) 691-. , 4th degree, curves on. Williams, F. B. [1900] Am. Ac. P. 36 (1901) 17-. , 5th degree. Schwarz, H. A. [1866] Crelle J. 67 (1867) 23-. Singular lines of surfaces, generic points. Levi, B. A. Mt. 2 (1899) 127-. points. Guccia, G. B. C. E. 105 (1887) 741-. . Del Pezzo, P. Palermo Cir. Mt. Ed. 6 (1892) 139-. , decomposition. Segre, C. Tor. Ac. Sc. At. 32 (1896) 521- or 781- ; 33 (1897) 19-; A. Mt. 25 (1897) 1-. , (Segre). Del Pezzo, P. Nap. Ac. Pont. At. 27 (1897) No. 4. 13 pp., No. 10, 7 pp. , resolution. Levi, B. Tor. Ac. Sc. At. 33 (1897) 56- or 66-. Singularities, reduction by birational trans- formations of space. Pannelli, M. A. Mt. 25 (1897) 67-. Skew curve, 4th degree, automorphic trans- formation, and correlation with developable of osculating planes. Del Re, A. Tor. Ac. Sc. At. 22 (1886-87) 901-. , , correlations changing into de- velopable of bitangent planes. Del Re, A. Nap. Ed. 26 (1887) 167-. , , developable quartics and sextics occurring in automorphic transformation of. Del Re, A. Nap. Ed. 27 (1888) 37-. curves on hyperboloid of one sheet. Cremona, L. C. E. 52 (1861) 1319-. . Chasles, M. C. E. 53 (1861) 1077-, 1203-. surface of 3rd degree. Clebsch, R. F. A. C. E. 62 (1866) 1114-. _ surfaces. Jonquieres, E. de. [1862] (vm) A. Mt. 5 (1863) 24-. of 3rd and 4th degrees. Rohn, K. Leip. Mth. Ps. B. 46 (1894) 84-; 49 (1897) 631-. surfaces of given degree, maximum genus. Bobek, K. Wien Ak. Sb. 96 (1888) (Ab. 2) 1024-. , 2nd degree, homographic systems of points on. Towmend, R. [1867] QJ. Mth. 9 (1868) 249-, 296-. Surface, 3rd degree, representation on plane. Dunumt, F. N. A. Mth. 15 (1896) 318-. , 6th degree, genus - 1. Noether, M. Mth. A. 21 (1883) 399-. , 8th degree, genus 0. Systems of straight lines passing through given straight line. Aschieri, F. G. Mt. 12 (1874) 129-. > (n-4)th degree, passing through double curve of surface, constancy of number of coefficients. Clebsch, R. F. A. C. E. 67 (1868) 1238-. 567 8040 Groups of Points Surface z 2 =f(xy), double planes with pW = l. Enriques, F. Bm. B. Ac. Line. Bd. 7 (1898) (Sem. 1) 234-, 253-. Surfaces admitting doubly infinite system of plane reducible sections. Castelnuovo, G. Em. E. Ac. Line. Bd. 3 (1894) (Sem. 1) 22-. and algebraic groups, theory. Picard, E. C. E. 120 (1895) 658-. containing curves of certain genus. Castel- nuovo, G., <& Enriques, F. C. B. 131 (1900) 739-. with double circular generation, determina- tion. Cosserat, E. C. B. 130 (1900) 311-. on which families of conies lie. Wetter, A. Zur. Vjschr. 29 (1884) 223-. , genus 0. Tognoli, 0. G. Mt. 11 (1873) 180-. , . Castelnuovo, G. Em. S. It. Mm. 10 (1896) 103-. , , special configurations on. Brunei, G. [1899] Bordeaux S. Sc. PV. (1899- 1900) 2-. , p = 0, representation on plane. Arme- nante, A. [1869] A. Mt. 4 (1870-71) 50-. , linear genus pW = 2. Enriques, F. Em. B. Ac. Line. Ed. 6 (1897) (Sem. 1) 139-. , pl. Franchis, M. de. Palermo Cir. Mt. Ed. 13 (1899) 130-. Linearity of systems belonging to algebraic surface. Enriques, F. Em. E. Ac. Line. Ed. 2 (1893) (Sem. 2) 3-. Loci, geometric, for systems. Ekama, H. Arch. Mth. Ps. 12 (1894) 23-. Locus of contact-points, order k, of curve of pencil, with those of cc k linear system. Franchis, M. de. Palermo Cir. Mt. Ed. 10 (1896) 118- ; 11 (1897) 12-. points of 3-point contact of curve pencil with curve net. Bagnera, G. Palermo Cir. Mt. Ed. 10 (1896) 81-. "Moires" (systems of curves formed by inter- sections of 2 systems of lines on surfaces). Anon, (vi 562) Gergonne A. Mth. 19 (1828- 29) 371-. Net of curves, contact of curve with. Spottis- woode, W. C. E. 83 (1876) 627-. of 4th degree. Sardi, C. G. Mt. 6 (1868) 217-. ?ith degree, involutory correspon- dences in plane determined by. Steinmetz, C. P. Am. J. Mth. 14 (1892) 39-. Nets of curves cutting given curve at given angle. Fouret, G. C. E. 83 (1876) 633-. of 3rd degree. Reye, T. Z. Mth. Ps. 13 (1868) 521-. 4th degree. Kttpper, C. Prag Ab. 1 (1886) No. 7, 25 pp. ; 3 (1890) No. 5, 11 pp. generated by homographic relations. Boguslavskil, A. I. [1881] (xn) Eec. Mth. (Moscou) 11 (1883-84) 313-. , geometry. Moore, E. H. (jun.) Am. J. Mth. 10 (1888) 243-. of plane curves. Castelnuovo, G. Tor. Ac. Sc. Mm. 42 (1892) 3-. . Scheffers, G. Leip. Mth. Ps. B. 50 (1898) (Mth.) 261-. , genus 2, redundant. Franchis, M. de. Palermo Cir. Mt. Ed. 13 (1899) 200-. , singular. Laguerre, E. [1873] (x) Par. S. Mth. Bll. 6 (1878) 129-. , principle of theory. Folie, F. Brux. Ac. Bll. 46 (1878) 193-. Noether's fundamental theorem (F=K + M\f/). Baker, H. J. Mth. A. 42 (1893) 601-. ( = ). Scott, (Miss) C. A. Mth. A. 52 (1899) 593-. Pencil of curves of 3rd degree, relation deter- mined by. Alekseev, V. G. [1891] Eec. Mth. (Moscou) 16 (1893) 256-. , differential equation. Casorati, F. Mil. I. Lomb. Ed. 10 (1877) 422-. , locus of points of inflexion. Bobek, K. Casopis 11 (*1882) 283-; Fschr. Mth. (*1882) 580. of plane curves, tangential, locus of foci. Humbert, G. C. E. 105 (1887) 54-. rational curves, surfaces containing. Enriques, F. Em. E. Ac. Line. Ed. 7 (1898) (Sem. 2) 281-, 344-; Mth. A. 52 (1899) 449- Pencils. Woepcke, F. Liouv. J. Mth. 4 (1859) 329-. of curves determining involution on any straight line. Scheffers, G. Leip. Mth. Ps. B. 44 (1892) 269-. , mutually tangential. Bischoff, J. N. Crelle J. 64 (1865) 185-. , projective. Gruss, G. Prag Sb. (1879) 287-. , , degree of curve generated by 2. Olivier, A. [1869] Crelle J. 71 (1870) 195-. , simple or double contact in. Berzo- lari, L. Tor. Ac. Sc. At. 31 (1895) 332- or 476-. of 3rd degree. Caporali, E. Em. E. Ac. Line. T. 1 (1877) 236. 584 8090 Systems of Curves Pencils of curves of 3rd and 4th degrees. Kan- tor, S. Wien Ak. Sb. 79 (1879) (Ab. 2) 787-. nth degree. Weyr, Em. Wien Sb. 61 (1870) (Ab. 2) 82-. , intersections. Dewulf, E. N. A. Mth. 11 (1872) 297-. and nets of plane curves, singularities. Ruffini, F. P. Bologna Ac. Sc. Mm. 1 (1880) 367. Pentagon, complete, and connected systems of curves. Kantor, S. [1878] Wien Ak. Sb. 78 (1879) (Ab. 2) 165-, 797-. Plane curves. Beltrami, E. (vn) A. Mt. 4 (1861) 102-. . Tognoli, 0. G. Mt. 13 (1875) 359-. . Krey, H. Acta Mth. 7 (1885) 49-. , isometrics of straight line with reference to systems. Ocagne, M. d'. Par. S. Mth. Bll. 13 (1885) 71-. , properties of systems, and characteristics of those of 4th degree. Zeuthen, H. G. [1873] (xi) Kjob. Dn. Vd. Selsk. Skr. 10 (1875) 285-. (With French resume.) , singularities. Franchis, H. de. Palermo Cir. Mt. Bd. 11 (1897) 104-. , 3rd degree. Schoute, P. H. C. B. 101 (1885) 736-, 805-. Polar curves, definition. Guccia,G.B. Palermo Cir. Mt. Bd. 7 (1893) 263-. , . Kohn, G. Palermo Cir. Mt. Bd. 7 (1893) 307-. , first. Koehler, C. (x) Par. S. Mth. Bll. 1 (1873) 124-. systems, real, having double contact with given real polar systems. Del Re, A. Nap. Bd. 27 (1888) 424-. Property of systems. Weill, . Par. S. Mth. Bll. 16 (1888) 155-. Bational curves, surfaces having non-linear system. Kantor, S. C. B. 131 (1900) 791-. Bepresentation of systems in plane. Vanecek, J. S., & Vanecek, M. N. Prag Sb. (1884) (Mth.-Nt.) 354-; Fschr. Mth. (1885) 602-. Skew curves. Jonquieres, E. de. G. Mt. 4 (1866) 210-. , system dual or reciprocal to itself. Fiedler, W. Ziir. Vjschr. 20 (1875) 173-. , 3rd degree, and developables of 3rd class. Gerbaldi, F. Tor. Ac. Sc. Mm. 32 (1880) 309-. , 4th degree, on cubic surface. Brambilla, A. Nap. Bd. 35 (1896) 171- ; 36 (1897) 203-. Stationary curves. Zeuthen, H. G. (xn) Ts. Mth. 6 (1882) 5-. Surfaces containing systems of curves. Hum- bert, G. C. B. 117 (1893) 361-. Tangents and normals to systems. Schlomilch, O. X. Z. Mth. Ps. 23 (1878) 337-. Triangles, parabolas, etc., systems generated by similar point-series. Stegemann, W. Arch. Mth. Ps. 10 (1891) 225-. SYSTEMS OF SUBFACES. Algebraic or transcendental surfaces, systems defined by 2 characteristics. Fouret, G. C. B. 79 (1874) 467-. Systems of Surfaces 8090 Algebraic or transcendental surfaces, systems, notion deduced from that of implexes. Fouret, G. C. B. 80 (1875) 167-. Contact problem in theory of linear systems, solution. Bdcklund, A. V. (xn) Ts. Mth. 2 (1878) 97-. of systems of surfaces, transformations. Fouret, G. C. B. 85 (1877) 1224-. Families of surfaces. Merrifield, C. W. Mess. Mth. 1 (1872) 81-. Family comprising new family of cyclides. Chizzoni, F. Bm. B. Ac. Line. Bd. 2 (1886) (Sem. l\ 476-. of developable surfaces passing through given skew curve. Levy, L. C. B. 97 (1883) 986-. homaloidal surfaces. Montesano, D. Palermo Cir. Mt. Bd. 2 (1888) 131-. Generation of surfaces by projective nets. Padova, E. G. Mt. 9 (1871) 148-. and skew curves by systems of surfaces. VanZcek, J. S., <& Vanecek, M. N. Bm. B. Ac. Line. Bd. 1 (1885) 130- ; A. Mt. 14 (1886-87) 73-. Implex and system of surfaces. Fouret, G. C. B. 80 (1875) 805-. Jacobian of 4 surfaces, singularities. Levi, A. G. Mt. 34 (1896) 215-. Linear system of surfaces of nth degree, in- volution defined by. Steinmetz, C. Z. Mth. Ps. 35 (1890) 219-, 272-, 354-. systems. Enriques, F. Bm. B. Ac. Line. Bd. 2 (1893) (Sem. 2) 281- ; 3 (1894) (Sem. 1) 481-, 536- ; Mth. A. 46 (1895) 179-. of cones. Fieri, M. Bv. Mt. 3 (1893) 44-. monoids. Fieri, M. G. Mt. 31 (1893) 151-. surfaces with certain singularities. Guccia, G. B. Palermo Cir. Mt. Bd. 1 (1887) 338-. Locus of singular points and lines in connection with ultimate intersections of system of sur- faces. Hill, M. J. M. [1891] Phil. Trans. (A) 183 (1893) 141-. Method of deducing surface from plane figure, example. Cremona,, L. Edinb. B. S. T. 32 (1887) 411-. Net, conjugate, of surf aces derived from quadrics. Mangeot, S. C. B. 125 (1897) 1083-. of surfaces, locus of parabolic points. Lo Monaco- Aprile, L. Palermo Cir. Mt. Bd. 12 (1898) 141-. of 2nd degree. Toeplitz, E. [1876] Mth. A. 11 (1877) 434-. Normal variety formed by simple rational series of planes. Segre, C. Tor. Ac. Sc. At. 21 (1885) 95-. Number of surfaces of net, having triple con- tact with intersection curve of 2 surfaces. Tognoli, 0. G. Mt. 9 (1871) 188-. Pencil of surfaces, double points. Guccia, G. B. C. E. 120 (1895) 896-. Pencils of surfaces, contact. Vanecek, J. S., & Vanecek, M. N. A. Mt. 15 (1887-88) 73-. Problems relating to 2 systems. Combescure, E. [1872] A. Mt. 5 (1871-73) 236-. Properties. Jonquiereg, E. de. C. B. 58 (1864) 567-; 61 (1865) 440-. 585 8100 Algebraic Configurations in Hyperspace Properties of surfaces and systems of surfaces. Rindi, S. G. Mt. 24 (1886) 94-. Rational systems. Murer, V. G. Mt. 24 (1886) 106-. ' Reciprocal linear systems. Escherich, G. von, Wien Ak. Sb. 75 (1877) (Ab. 2) 523-. relationship of .F 2 -systems and 4> 2 -systems (Gewebe), and quadratic F 2 -systems of 8th rank. Beye, T. Crelle J. Mth. 82 (1877) 173-. Rotation surfaces, 2nd degree, with common curve of intersection. Gretschel, H. Arch. Mth. Ps. 44 (1865) 124-. Series of surfaces (Gebiische). Murer, V. G. Mt. 26 (1888) 178-. (pencils and nets). Murer, V. G. Mt. 25 (1887) 363-. Surfaces with common curve. End, W. Mth. A. 35 (1890) 82-. through corners of given solid. Palatini, F. Yen. I. At. (1898) 187-. enveloped by quadric cones, where contact is along a circle. Blutel, E. C. R. 103 (1886) 687-. represen table point for point on plane. Visalli, P. Rm. R. Ac. Line. Rd. 2 (1886) (San. 2) 80-, 84-. several times generated by conies, deter- mination of all. Koenigs, G. Par. EC. Norm. A. 5 (1888) 177-. of various degrees of freedom. Reye, T. Crelle J. Mth. 82 (1877) 1-. Systems depending on 1 parameter. Legoux, A. Toul. Ac. Sc. Mm. 9 (1887) 325-. Translation-surfaces of 4th degree, singularities of class. Sucharda, A. Wien Ak. Sb. 97 (1889) (Ab. 2a) 1083-. Triple system. Legoux, A. Toul. Ac. Sc. Mm. 2 (1890) 440-. x a y ft z yu s + kw e =0. Legmix, A. N. A. Mth. 2 (1883) 233-. 8100 Algebraic configurations in hyperspace. Cassani, P. Yen. I. At. (1885-86) 227-. Kilhne, H. Arch. Mth. Ps. 11 (1892) 353-. Del Pezzo, P. G. Mt. 31 (1893) 1-. Abelian integrals and curves in hyperspace. White, H. S. Ac. Nt. C. N. Acta 57 (1892) Algebraic theorem proved by geometry of n dimensions. Veronese, G. [1883] Ven. I. At. (* 1883-84) 137-. Analytical geometry. Aschieri, F. Mod. Ac. Sc. Mm. 11 (1895) 301-. Binary forms and correlations in hyperspace. Waelsch, E. Mh. Mth. Ps. 6 (1895) 261-, 375-. Canonical curves of linear space and their quartic covariants. Scorza, G. Tor. Ac. Sc. At. 35 (1900) 517- or 765-. " Characteristic " problem for linear space. Schubert, H. Hamb. Mth. Gs. Mt. 1 (1889) 134-. Characteristics of linear spaces. Schubert, H. Mth. A. 38 (1891) 598-. 8100 "Chords" of algebraic "curves." Levi, B. [1898] Tor. Ac. Sc. Mm. 48 (1899) 83-. Clifford-Klein forms of space. Killing, W. Mth. A. 39 (1891) 257-. Complexes and congruences. Bordiga, G. A. Ven. I. At. (1885-86) 163-. ,line. Kantor,S. Crelle J. Mth. 118(1897)74-. , R t -, in R r , n moments. Kantor, S. Munch. Ak. Sb. 26 (1897) 531-. Congruences, orthogonal, in any variety. Ricci, G. Rm. R. Ac. Line. Mm. 2 (1895) 276-. "Conies" and "quadrics." Schubert, H. Hamb. Mth. Gs. Mt. 1 (1889) 290-. Coordinates, homogeneous, interpretation. Kohn, G. Wien Ak. Sb. 104 (1895) (Ab. 2a) 1167-. Correlation, involutory polar, in any linear space. Deruyts, F. Liege S. Sc. Mm. 17 (1892) No. 1, 16 pp. Correspondence, algebraic, between 2 ruled spaces. Fieri, M. Tor. Ac. Sc. At. 25 (1890) 365-. between 2 hyperspaces. Giudice, F. G. Mt. 29 (1891) 163-. in linear space, principle. Fieri, M. Rm. R. Ac. Line. Rd. 3 (1887) (Sem. 1) 196-. Correspondences, class. Del Re, A. G. Mt. 26 (1888) 348-. , linear reciprocal, in linear space of any species. Aschieri, F. Mil. I. Lomb. Rd. 19 (1886) 167-. Cross-ratio of certain Cremona transformations. Moore, E. H. [1896] Am. J. Mth. 22 (1900) 279-. curves of nth degree. Schoute, P. H. [1900] Amst. Ak. Vs. 9 (1901) 268-; Amst. Ak. P. 3 (1901) 255-. Curves. Jordan, C. C. R. 79 (1874) 795-. of given degree and maximum genus in any space. Fano, G. [1893] Tor. Ac. Sc. Mm. 44 (1894) 335-. in linear space. Piccioli, E. G. Mt. 36 (1898) 271-. and ruled surfaces. Segre, C. Mth. A. 30 (1887) 203- ; 34 (1889) 1-. "Curves," points on. Segre, C. A. Mt. 22 (1894) 41-. Cyclical surfaces of 4th degree. Snyder, V. N. Y. Am. Mth. S. Bll. 6 (1900) 194-. Cylinder in homogeneous spaces. Quensen, C. Arch. Mth. Ps. 3 (1886) 45-. Developable spaces, the most general. Puchta, A. Wien Ak. Sb. 101 (1892) (Ab. 2a) 355-. Displacement, general. Schoute, P. H. Amst. Ak. Vs. M. 9 (1892) 416- ; Delft EC. Pol. A. 7 (1892) 139-. Enumeration for "conies." Schubert, H. Hamb. Mth. Gs. Mt. 2 (1890) (Festschr., Tl. 2) 172-. quadrics. Schubert, H. Mth. A. 45 (1894) 153-. Enumerative geometry, generalisations. Schu- bert, H. Mth. A. 26 (1886) 26-. Figures with + 2 inversions. Schoute, P. H. Haarl. Ms. Teyl. Arch. 5 (1898) 159-, 241- ; 6 (1900) 163-. Fresnel surface. Vahlen, . N. A. Mth. 14 (1895) 344-. 586 8100 Algebraic Configurations in Hyperspace 8100 Fresnel surface. Schoute, P. H. N. Arch. Wisk. 3 (1898) 239-; Fschr. Mth. (1897) 587-; Edinb. Mth. S. P. 16 (1898) 51-. Homographic pencils, properties in linear spaces. Enriques, F. Rm. R. Ac. Line. Bd. 6 (1890) (Sem. 2) 63-. Homographies. Predella, P. A. Mt. 17 (1889- 90) 113-. , binary and ternary. Aschieri, F. Mil. I. Lomb. Ed. 24 (1891) 278-. , cyclic. Enriques, F. G. Mt. 30 (1892) 311-. , harmonic, in linear spaces. Enriques, F. G. Mt. 30 (1892) 319-. and interchangeable relations. Del Prete, G. G. Mt. 37 (1899) 107- ; 38 (1900) 40-. , involutory, in linear space. Bertini, E. Mil. I. Lomb. Ed. 19 (1886) 176-. in linear space, theory and classification. Segre, C. Em. E. Ac. Line. Mm. 19 (1884) 127-. Homography. Bordiga, G. Yen. I. At. (1896- 97) 1091-. , fundamental spaces. Segre, C. Em. E. Ac. Line. Ed. 2 (1886) (Sem. 1) 325-. + 2 hyperspheres, circumscribing prismatoids formed by n + 2 hyperplanes, meet in point. Kuhne, H. Crelle J. Mth. 119 (1898) 186-. Hypersurfaces of S r with same 1st polars, systems. Bertini, E. Em. E. Ac. Line. Rd. 7 (1898) (Sem. 2) 217-, 275-, 357. Indeterminate equations and n-dimensional geometry. Brunei, G. Bordeaux S. Sc. Mm. 2 (1886) 129-. Intersection of 2 varieties, order. Fieri, M. G. Mt. 26 (1888) 251-. in a simply infinite variety of space. Levi, B. Tor. Ac. Sc. At. 34(1898) 581-or745-. Involutes of helices, generalisation. Piccioli, H. N. A. Mth. 18 (1899) 132-. Involution. Castelnuovo, G. Ven. I. At. (1885- 86) 1167-. in connection with algebraic forms. Deruyts, F. Brux. Ac. BH. 14 (1887) 650-. , neutral elements. Deruyts, F. Brux. Ac. Bll. 36 (1898) 187-. , , and their application to skew curves. Deruyts, F. Brux. Ac. Bll. 35 (1898) 885-. on rational base, neutral groups with multiple elements in. Severi, F. Em. R. Ac. Line. Rd. 9 (1900) (-Sem. 1) 379-. Involutions, groups of elements common to 2. Deruyts, F. Brux. Ac. Bll. 31 (1896) 664-. , points common to 2. Le Paige, C. Brux. Ac. Bll. 11 (1886) 121-. , neutral elements. Le Paige, C. Brux. Ac. Bll. 14 (1887) 211-. , common to 2, groups. Deruyts, F. Brux. Ac. Bll. 26 (1893) 232-. , any number, groups. Deruyts, F. Brux. Ac. Bll. 27 (1894) 495-. , unicursal, neutral groups of elements. Deruyts, F. Brux. Ac. Bll. 35 (1898) 196-. , , representation. Deruyts, F. Brux. Ac. Bll. 14 (1887) 322-. Jacobian locus in hyper-geometry. Schoute, P. H. L. Mth. S. P. 30 (1899) 322-. Rummer's surface, analogue tor p = 3. Wirt- inger, W. Gott. Nr. (1889) 474-. Line geometry. Schubert, H. Hamb. Mth. Gs. Mt. 3 (1900) 86-. Linear spaces, geometry. Bertini, E. Mil. I. Lomb. Rd. 19 (1886) 855-. of 4th species, spaces composed of. Aschieri, F. Mil. I. Lomb. Rd. 19 (1886) 614-, 698-. systems. Bertini, E. [1880] Mil. I. Lomb. Rd. 15 (1882) 24-. , order of variety arising from several. Fieri, M. Palermo Cir. Mt. Rd. 11 (1897) 58. of varieties in relation to birational trans- formations. Palatini, F. G. Mt. 38 (1900) 315-. varieties, complexes, congruences. Zindler, K. Wien Ak. Sb. 101 (1892) (Ab. 2a) 215-. , synthetic construction. Zindler, K. Crelle J. Mth. Ill (1893) 303-. Lines, surfaces, and higher configurations in multiply extended Gaussian and Riemannian spaces. Schering, E. Gott. Nr. (1873) 13-. Multiple tangents. Schubert, H. Mth. A. 26 (1886) 52-. Normal curves of genus p of various spaces. Segre, C. Mil. I. Lomb. Ed. 21 (1888) 523-. linear spaces, property common to. Deruyts, F. Brux. Ac. Bll. 17 (1889) 545-. lines of spaces of odd number of dimensions, general theorem. Cassani, P. Em. R. Ac. Line. Rd. 2 (1886) (Sem. 1) 482-. Pencil of quadrics, linear spaces of maximum dimension associated with. Eosati, C. Mil. I. Lomb. Rd. 32 (1899) 1267-. of space S n , quartic base, representation in S n _ 2 . Rosati, C. A. Mt. 1 (1898) 25-. Pencils of quadrics. Bertini, E. Rm. R. Ac. Line. Rd. 2 (1886) (Sem. 2) 208-. Point aggregates in continuous space. Cantor, G. Acta Mth. 2 (1883) 409- ; 7 (1885) 105-. groups of linear space. Chizzoni, F. Nap. Ac. At. 5 (1893) No. 1, 28 pp. , regular, and the appropriate linear sub- stitutions. Biermann, O. Wien Ak. Sb. 95 (1887) (Ab. 2) 523-. pairs, oo n algebraic series. Pieri, M. Palermo Cir. Mt. Rd. 5 (1891) 252-. 1 oo (*+i) (-*), coincidences in r-dimensional space. Severi, F. Rm. R. Ac. Line. Rd. 9 (1900) (Sem. 2) 321-. Polar properties of algebraic varieties, geometric method. Rodenberg, C. Mth. A. 26 (1886) 557-. triangles, 2, lie in perspective, extension of theorem. Schlafli, L. [1865] Crelle J. 65 (1866) 189-. Polarity. Bordiga, G. A. Ven. I. At. (1884- 85) 2097-. Polars, theorem. Brambilla, A. Tor. Ac. Sc. At. 22 (1886-87) 787-. Poncelet-Zeuthen polygons inscribed in p-ary quadric. Voss, A. Mth. A. 26 (1886) 231-. Projection of 4-fold figures upon a three-flat. Hall, T. P. Am. J. Mth. 15 (1893) 179-. Projections of surface and variety of n dimen- sions. Del Pezzo, P. Nap. Rd. 25 (1886) 205-. 587 Algebraic Configurations in Hyperspace 8100 Projective geometry, fundamental postulates. Amodeo, F. Tor. Ac. Sc. At. 26 (1891) 741-. in linear space, fundamental postulates. Fano, G. G. Mt. 30 (1892) 106-. groups, classification. Marotte, F. C. E. 129 (1899) 580-. Quadric in hyperspace, representation of ordi- nary space on plane by. Aschieri, F. Mil. I. Lomb. Ed. 18 (1885) 494-. Quadrics, confocal. Meyer, W. F. Konigsb. Schr. 41 (1900) [24]-. , differential equations. Berzolari, L. Em. E. Ac. Line. Ed. 5 (1896) (Sem. 1) 247-. of even number of dimensions. Bertini, E. Tor. Ac. Sc. At. 30 (1895) 350-. , generation by reciprocation. Da Porto, A. G. Mt. 37 (1899) 124-. in linear space. Segre, C. [1883] Tor. Ac. Sc. Mm. 36 (*1885) 3-. , foci. Concina, U. G. Mt. 38 (1900) 129-. reciprocally polar to themselves. Del Pezzo, P. Nap. Ed. 24 (1885) 186-. Eational curves. Davis, E. W. J. H. Un. Cir. [3] (1884) 123. in n-flat space. Bissing, G. J. H. Un. Cir. [3] (1884) 123. : curves of degree (n + 1). Biiring, G. J. H. Un. Cir. [3] (1884) 77. linear space. Brambilla, A. Mil. I. Lomb. Ed. 19 (1886) 326-. . Berzolari, L. A. Mt. 21 (1893) 1-. normal curves. Loria, G. G. Mt. 26 (1888) 334-. surfaces and varieties, class. Brambilla, A. Nap. Ac. At. 9 (1899) No. 14, 28 pp. Eegular figures. Stringham, W. I. Am. J. Mth. 3 (1880) 1-. . Biermann, O. Wien Ak. Sb. 90 (1885) (Ab. 2) 144-. , convex, analytical determination. Puchta, A. Wien Ak. Sb. 90 (1885) (Ab. 2) 168-. , and semi-regular. Gosset, T. Mess. Mth. 29 (1900) 43-. Euled space, geometry. Aschieri, F. Mil. I. Lomb. Mm. 15 (1885) 75-, 263-. surfaces. Bordiga, G. Yen. I. At. (1885- 86) 1085-. , elliptic. Segre, C. Tor. Ac. Sc. At. 21 (1885) 868-. Simplicissima. Sharp, W. J. C. L. Mth. S. P. 18 (1886-87) 325-; 19 (1889) 423-; 21 (1891) 316-. Singular points of algebraic curves, obtained by projection from hyperspace. Del Pezzo, P. Nap. Ed. 32 (1893) 15-, 45-. Singularities of curves of multiple curvature. Fine, H. B. Am. J. Mth. 9 (1887) 180-. discriminant surface. Hilbert, D. Mth. A. 30 (1887) 437-. Spaces cutting several lines, number. Castel- nuovo, G. Em. E. Ac. Line. Ed. 5 (1889) (Sem. 2) 71-. touching surface or variety. Del Pezzo, P. Nap. Ed. 25 (1886) 176-. 8100 Straight line, geometry, deduced from that of quadrics in n-dimensional space. Aschieri, F. Mil. I. Lomb. Ed. 17 (1884) 756-. lines, systems. Segre, C. Palermo Cir. Mt. Ed. 2 (1888) 148-. Surfaces, analysis situs, and integrals of total differentials. Picard, E. C. E. 124 (1897) 532-. of 2 dimensions, new groups. Bordiga, G. C. E. 102 (1886) 1442-. , general theory, extension to varieties. Mahler, . D. Nf. Tbl. (*1882) 147-. , normal equation. Pellet, A. C. E. 128 (1899) 233-. , plane section unicursal. Moore, E. H. (jun.) Am. J. Mth. 10 (1888) 17-. , theorems, extension. Hoppe, E. Arch. Mth. Ps. 3 (1886) 277-. and varieties. Del Pezzo, P. Nap. Ed. 26 (1887) 40-. , property. Segre, C. Tor. Ac. Sc. At. 31 (1895) 341- or 485-. , sections of which are normal curves, genus p. Del Pezzo, P. A. Mt. 15 (1887- 88) 115-. of zero curvature in elliptic geometry. Bianchi, L. A. Mt. 24 (1896) 93-. spaces of constant curvature. Bianchi, L. Tor. Ac. Sc. At. 30 (1895) 743-. , nth degree. Del Pezzo, P. Palermo Cir. Mt. Ed. 1 (1887) 241-. Theorem, 3-dimensional, generalisations. Schubert, H. Hamb. Mth. Gs. Mt. 1 (1889) 87. Theorems of Clifford and Cayley, extensions. Moore, E. H. [1885] Conn. Ac. T. 7 (1885- 88) 9-. Transformations, birational, between 2 spaces. Carrone, C. Catania Ac. Gioen. At. 11 (1898) Mem. 8, 62 pp. , involutory linear, complete groups of 3. Del Re, A. Em. E. Ac.. Line. Ed. 6 (1890) (Sem. 1) 57-. , quadratic, characteristics. Kantor, S. Mil. I. Lomb. Ed. 27 (1894) 477-. t _, periodic. Kantor, S. Mil. I. Lomb. Ed. 27 (1894) 712-, 749-; 28 (1895) 249-, 298-. , successive linear. Kantor, S. Par. S. Mth. Bll. 8 (1880) 208-. of surfaces. Newcomb, S. Am. J. Mth. 1 (1878) 1-. , uniform automorphic, of elliptic curves. Segre, C. Mth. A. 27 (1886) 296-. Varieties composed of simply infinite series of spaces. Segre, C. Em. E. Ac. Line. Ed. 3 (1887) (-Sem. 2) 149-. with continuous non-integrable group of projective automorphic transformations. Fano, G. Tor. Ac. Sc. Mm. 46 (1896) 187-. , cubic, Hessian vanishing identically. Perazzo, U. G. Mt. 38 (1900) 337-. of different dimensions, representation on each other. Lilroth, J. Erlang. Ps. Md. S. Sb. 31 (1900) 87-. , unicursal. Autonne, L. Par. S. Mth. Bll. 27 (1899) 263-. 588 8100 Hyperspace Variety, quadric, successive projection on itself. DelRe,A. Km. E.Ac. Line. Ed. 5 (1896) (Sem. 2)365-. ^-dimensional space. collinearities and correlations, ordinary and special. Visalli, P. Mil. I. Lomb. Ed. 29 (1896) 351-, 439-, 521-, 559-. complex of 1st order. Ascione, E. Em. E. Ac. Line. Ed. 6 (1897) (Sem. 1) 162-. complexes. Bordiga, G. Yen. I. At. (1887- 88) 919-. . Palatini, F. Yen. I. At. (1899-1900) (Pt. 2) 861-. condition that 5 straight lines should lie on quadric. Bichmond, H. W. Camb. Ph. S. P. 10 (1900) 210-. configuration of lines. Bordiga, G. Ven. I. At. (1896-97) 420-. congruence, 3rd order and 6th class. Castel- nuovo, G. Ven. I. At. (1886-87) 1249- ; (1887-88) 525-. , 4th order and 2nd class. Bordiga, G. Ven. I. At. (1893-94) 1605-. Cremonian transformation. Del Pezzo, P. Nap. Ed. 35 (1896) 336-; 36 (1897) 33-. cubic varieties. Segre, C. Tor. Ac. Sc. Mm. 39 (1889) 3-. . Enriques, F. G. Mt. 31 (1893) 31-. variety with 10 double points. Segre, C. Tor. Ac. Sc. At. 22 (1886-87) 791-. elliptic curve of 5th degree. Meyer, F. Mth. A. 26 (1886) 154-. figure formed from 6 points. Bichmond, H. W. Mth. A. 53 (1900) 161-. focal properties of quadrics. Sommer, J. Mth. A. 53 (1900) 113-. geometric interpretation of linear equation. Nicoli, F. Mod. Ac. Sc. Mm. 7 (1890) 205- ; 8 (1892) 489-. homaloidal systems of quadrics. Del Pezzo, P. Nap. Ed. 34 (1895) 133-. hyperquadrics. Schoute, P. H. [1900] Amst. Ak. Vh. (Sect. 1) 7 (1901) No. 4, 66 pp. ; Haarl. Ms. Teyl. Arch. 7 (1902) 117-. line geometry. Castelnuovo, G. Ven. I. At. (1890-91) 855-. linear, birational transformation of ordinary ruled space in. Pannelli, M. Em. E. Ac. Line. Ed. 6 (1890) (Sem. 1) 479-. transformations. Hathaway, A. S. [1898] N. Y. Am. Mth. S. Bll. 5 (1899) 88, 93-. normal curve. Aschieri, F. Em. E. Ac. Line. Mm. 4 (1887) 172-. of 4th degree. Bordiga, G. A. Ven. I. At. (1885-86) 503-. plane, lines of triple curvature. Pirondini, G. G. Mt. 28 (1890) 219-. point pairs. Frege, . Jena. Sb. (*1883) 98-. points of, and straight lines of ordinary space, one-one correspondence between. Chizzoni, F. Catania Ac. Gioen. At. 20 (1888) 273-. primitive groups of transformations. Page, J. M. Am. J. Mth. 10 (1888) 293-. regular convex figures. Puchta, A. Wien Ak. Sb. 89 (1884) (Ab. 2) 806-. 2 representations of ruled space on linear form. Loria, G. G. Mt. 27 (1889) 224-. Infinitesimal Geometry 8400 ruled surface of 4th degree. Caldarera, G. Acireale Ac. At. 7 (1896) (Sc.) 141-. surface of 6th degree, with 10 straight lines, and its projections in ordinary space. Bordiga, G. Em. E. Ac. Line. Mm. 4 (1887) 182-. surfaces whose triple secants form complexes of 1st order. Ascione, E. Em. E. Ac. Line. Ed. 6 (1897) (Sem. 1) 240-. syzygies connecting quadratic relations of line coordinates. Young, W. H. Tor. Ac. Sc^ At. 34 (1898) 438- or 596-. tetratops ( = 4-dimensional angle bounded by four spaces) and polytops. Hoppe, B. Arch. Mth. Ps. 67 (1882) 412-. use in study of surfaces including several series of conies. Cosserat,E. C. E. 124 (1897) 1004-. varieties with continuous integrable group of projective automorphic transformations. Fano, G. Ven. I. At. (1895-96) 1069-. variety of 27th order. Brambilla, A. Nap. Ac. At. 9 (1899) No. 2, 37 pp. 5-dimensional space, homaloidal surface of 2 dimensions and 4th degree. Veronese, G. Em. E. Ac. Line. Mm. 19 (1884) 344-. , surface of 7th degree. Bordiga, G. Ven. I. At. (1886-87) 1397-. , systems of straight lines considered as surfaces. Fano, G. A. Mt. 21 (1893) 141-. 6-dimensional space, surfaces of 5th and 6th degrees. Bordiga, G. Ven. I. At. (1885-86) 1461-. n - 1 dimensional space, axes of quadric surface. Story, W. E. J. H. Un. Cir. [3] (1884) 122. , involutions of degree n and species 1. Morale, M. Palermo Cir. Mt. Ed. 13 (1899) 274-. , quadrics having n-fold common polars. Burali-Forti, C. Palermo Cir. Mt. Ed. 4 (1890) 118-. , rational curves of nth degree. Loria, G. Palermo Cir. Mt. Ed. 2 (1888) 201-. , uniform correspondences in elliptic curves of nth degree. Amodeo, F. A. Mt. 19 (1891-92) 1-. n + 1 dimensional space, lines in ruled surfaces of nth degree. Palatini, F. G. Mt. 36 (1898) 368-. , projective invariants of system of m + 1 points. Lovett, E. O. Bll. Sc. Mth. 23 (1899) 10-. , strip elements and collections of elements. Kowalewski, G. Leip. Mth. Ps. B. 52 (1900) 91-. , surfaces of nth degree. Del Pezzo, P. Nap. Ed. 24 (1885) 212-. Infinitesimal geometry ; applica- tions of differential and integral calculus to geometry. 8400 General. Area of surface, definition. Peano, G. Em. B. Ac. Line. Ed. 6 (1890) (Sem. 1) 54-. Consecutive surfaces. N., M. N. Camb. Mth. J. 1 (1839) 187-. 589 8400 Infinitesimal Geometry 8410 Construction of tangents to plane curves. Gode- froy, B. C. B. 102 (1886) 604-. Contact of lines and surfaces. Chasles, M. Liouv. J. Mth. 2 (1837) 299-. lines of surfaces with circumscribed cones. Valles, F. Gergonne A. Mth. 16 (1825-26) 315-. Correspondence of tangent planes in trans- Method of tangents, Eoberval's. Duhamel, J. M. C. Par. Mm. Sav. Etr. 5 (1838) 257-. Multiple integrals in tangential and in ruled space, and principle of duality. Cartan, E. Par. S. Mth. Bll. 24 (1896) 140-. Orthogonal symmetry with reference to any cylinder. Pirondini, G. N. A. Mth. 19 (1900) 107-. formation of surfaces by symmetry curve. line Pirondini G G Mt Mangeot, S. C. B. 114 (1892) 1463-. 35 (1897) 181- ; 37 (1899) 212-. ' Curvature propositions. Seiliger, D. N. Fschr. surface of revolution. Pirondini, Mth. (1892) 582. Curved surfaces, general properties. Lilienthal, von. Crelle J. Mth. 98 (1885) 131-. , theory. Lipschitz, B. Acta Mth. 10 (1887) 131-. , , general. Bibaucour, A. Liouv. J. Mth. 7 (1891) 5-, 219-. Curves, properties demonstrated by theory of limits. Azzarelli, M. Em. N. Line. At. 42 (1889) 160-. and surfaces, theory. Catalan, E. C. [1874] Brux. Mm. Cour. 8, 24 (1875) (No. 2) 48 pp. Cylinders, cones, and developable surfaces. Tanner, H. W. L. Mess. Mth. 6 (1877) 145-. Density of chords of surfaces and plane curves. Hoppe, B. Arch. Mth. Ps. 7 (1889) 165-. Differential relations in geometry, and synectic integrals of differential equations. Cauchy, A. L. C. E. 40 (1855) 445-. Essential equations of plane curves, especially cycloids. Onnen, H. N. Arch. Wisk. 4 (1878) 30-. Formation of surfaces by straight line. Hube, K. (vi Adds.) Krk. Eoczn. Uniwers. 11 (1826) 23-. Geometrical increments, with applications to geodesies, orthogonal surfaces etc. Lafon, A . As. Fr. C. E. (1876) 104-. significance of =0. Tychsen, C. (xn) Ts. Mth. 1 (1865) 127-. Infinitesimal geometry. Mannheim, A. Torto- lini A. 2 (1859) 208-. , problem. Gilbert, L. P. (xn) Mathesis 1 (1881) 97-. , theorem. Koenigs, G. C. E. 116 (1893) Infinitesimals, theorem, in conies. M'Vicker, C. E. Mth. Gz. 1 (1900) 209-. Integral calculus and geometry, reciprocal relations. Meyer, F. Konigsb. Schr. 39 (1898) [44]-. Line and surface integrals, generalisation of Stokes's theorem. Gilbert, . Brux. S. Sc. A. 16 (1892) (Pt. 1) 2-. Maximum and minimum, fallacy hi certain proofs. Alison, J. Edinb. Mth. S. P. 15 (1897) 83-. Mean direction of curved line in plane. Lind- hagen, A. N. Ts. Mth. 1 (B) (1890) 80-; Fschr. Mth. (1890) 701-- Method of tangents, inverse, problems. Eider, L. [1781] St Pet. Ac. Sc. Mm. 10 (1826) 16. G. A. Mt. 22 (1894) 213- ; 25 (1897) 61-. Osculating circles and surfaces to conies and quadrics. Ducoroy, G. N. A. Mth. 19 (1860) 118-. Spheres touching two surfaces. Pellet, A. As. Fr. C. E. (1887) (Pt. 2) 116-. Sum of distances of point from given points. Schlegel, V. N. Y. Am. Mth. S. Bll. 1 (1895) 33-. Surfaces referred to their lines of zero length. Cosserat, E. C. E. 125 (1897) 159-. Transformation of coordinates from rectangular to polar in differential expressions. Smith, Arch. Camb. Mth. J. 1 (1839) 121-. 8410 Principles of infinitesimal geometry. Genty, . N. A. Mth. 2 (1883) 237-. Cesdro, E. Mathesis 7 (1887) 25-. Analytic geometry. Bordoni, A. Mil. G. I. Lomb. 6 (1854) 287-, 420-. Areas described in motion of plane figure. Koenigs, G. G. E. 118 (1894) 965-. Circles or spheres related to envelope. Serret, P. C. E. 117 (1893) 400-, 435-, 480-. Curvature, problem. Seiliger, D. Kazan S. Ps. -Mth. Bll. 10 (1901) 262-. Curves, ternary operator in analysis. Walker, J. J. [1878-87] L. Mth. S. P. 9 (1877-78) 226- ; 16 (1884-85) 215- ; 19 (1889) 483-. Differential geometry, use of rectangular tri- hedron. Schoenfties, A. Gott. Nr. (1898) Envelopes. Colard, H. N. A. Mth. 1 (1842) Geometrical theorems connected with Poisson's equation =*(; -2 2 ) * Maclay, J. Mth. S. Bll. 7 (1901) [1900] N. Y. Am. 202, 208-. Infinitesimal deformations, use of V-l and alternate numbers i, j, k in. Perna, A. Palermo Cir. Mt. Ed. 12 (1898) 322-. elements, projective relation between. Oliver, J. E. [1884] A. Mth. 1 (1884-85) 40-. properties of curves, auxiliary curves. Ocagne, M. d\ Am. J. Mth. 11 (1889) 55- ; 15 (1893) 389-. transformations, geometry. Zorawski, K. Krk. 'Ak. (Mt.-Prz.) Ez. 14 (1899) 218- ; 17 (1900) 154- ; Crc. Ac. Sc. Bll. (1897) 365- ; (1899) 81-. Intrinsic geometry, questions. Cesdro, E. Mathesis 20 (1900) 5-, 37-, 57-. 590 8420 Kinematic Geometry Isogonal and isologic transformation of plane figures. Tranum, A. N. A. Mth. 8 (1869) 222-. Orthogonal trajectories on plane and sphere. Roberts, M. Liouv. J. Mth. 10 (1845) 251-. Osculation function, Zmurko's. Mertens, F. (xn) Krk. Ak. (Mt.-Prz.) Pam. 2 (1876) 124-. Question of limit in theory of surfaces. Lecornu, L. Bll. Sc. Mth. 16 (1892) 307-. Real infinitesimals and infinite segments. Veronese, G, Mth. A. 47 (1896) 423-. Surface elements of higher order in space of 3 dimensions. Weber, E. von. Mth. A. 44 (1894) 458-. dx* + dy* + dz* = \ (da 2 + dp? + dy*) . Liouville, J. Liouv. J. Mth. 15 (1850) 103. 8420 Kinematic geometry. (See also Mechanics 0420.) Mannheim, A. Camb. Ph. S. P. 6 (1889) 32-. Speckel, C. N. A. Mth. 11 (1892) 268-. Nobile, V. N. A. Mth. 18 (1899) 218-. Angle sliding along line, problem. Lamarle, E. Brux. Ac. Bll. 9 (1860) 127-, 361-. touching two curves, locus of vertex. En- neper, A. Z. Mth. Ps. 16 (1871) 342-. Astroids, construction for tangents. Sucharda, A. Arch. Mth. Ps. 66 (1881) 321-. , . Stammer, W. Arch. Mth. Ps. 67 (1882) 222-. , . Stall, (Dr) . Arch. Mth. Ps. 67 (1882) 447-. , . Ameseder, A. Casopis 14 (1885) 46 ; Fschr. Mth. (1885) 713. , . Sucharda, A. Casopis 14 (1885) 138- ; Fschr. Mth. (1885) 714. , curves parallel to. Loria, G. Mathesis 20 (1900) 244-. , . Neuberg, J. Mathesis 20 (1900) 247-. 3-bar curves, triple generation. Clifford, W. K. L. Mth. S. P. 9 (1877-78) 26-. linkage, curves described by. Jensen, P. V. (xn) Ts. Mth. 6 (1882) 154-. motion in plane. Roberts, S. [1875] L. Mth. S. P. 7 (1875-76) 14-. 7-bar pantograph of Delaunay. Somov, P. 0. [1895] Vars. S. Nt. Tr. (1895-96) (C.R., Ps. C.) No. 2, 5 pp. Cartesian oval treated kinematically. Cavalli, E. Tor. Ac. Sc. At. 20 (1885) 1143-. Centre of curvature of Cassini's oval, expo- nential curve, catenary, tractrix, logarithmic and hyperbolic spirals, kinematic construc- tion. Prochdzka, B. Prag Cesk6 Ak. Fr. Jos. Bz. (Tfida 2) 4 (1895) Art. 8, 8 pp. ; Fschr. Mth. (1896) 794. curves produced by motion of in- variable figure in a plane. Seidelin, C.J.L. (xn) Ts. Mth. 6 (1876) 57-. Descartes's oval, kinematic con- struction for. ProcMzka, B. Casopis 24 (1895) 230- ; Fschr. Mth. (1895) 621. Curve of Pursuit 8420 Centres of curvature of curve rolling on straight line. Mannheim, A. Liouv. J. Mth. 4 (1859) 93-. curves described by points in dis- placed plane. Bobek, K. Prag Sb. (1880) 56-. and tangents of conic sections, kine- matic construction for. Prochdzka, B. Prag Cesk6 Ak. Fr. Jos. Ez. (Tfida 2) 3 (1894) Art. 19, 5 pp. ; Fschr. Mth. (1893-94) 955. of trajectories of points in moving figure. Marcolongo, R. Palermo Cir. Mt. Bd. 7 (1893) 29-. Circle as roulette, curve represented by elliptic functions. Lieblein,J. [1869] Prag Ab. 3 (1870) 14 pp. Compound and reverse curves. Thornton, W. M. A. Mth. 3 (1887) 39-. Cone rolling on plane, curve traced by plane section. Schwering, K. Z. Mth. Ps. 25 (1880) 25-. Coordinates, "ideal" (for moving systems). Hansen, P. A. Leip. B. (1851) 39-. Curvature of paths of points of rigid system moving about fixed point. Minich, S. R. Yen. At. (1859-60) 183-. plane curves treated kinematically. Neuberg, J. Brux. Ac. Bll. 25 (1893) 374-, 848. surface generated by point of solid moving under conditions. Reveille, J. C. B. 105 (1887) 159-. surfaces. Suslow, G. Fschr. Mth. (1890) 868-. Curve of flight. Lehmus, . Crelle J. 1 (1826) 61-. , locus of intersection of 2 lines rotating about 2 fixed points. Le Francois, (Prof.) . Quetelet Cor. Mth. 5 (1829) 379-. Curve of pursuit. Querret, . Gergonne A. Mth. 13 (1822-23) 389-. Burg, A. Wien Jb. Pol. I. 4 (1823) 508-. Gosselin, (le col.) . Metz Mm. Ac. 40 (1858- 59) 459-. Tychsen, C. Mth. Ts. 3 (1861) 33-. Ingalls, J. M. Des Moines Anal. 7 (1880) 89-, 117-. Cesdro, E. N. A. Mth. 2 (1883) 85-. application. Dienger, J. Grunert Arch. 15 (1850) 335-. centre of curvature. Ocagne, M. d\ Par. S. Mth. Bll. 11 (1883) 134- . equation, ratio of velocities being constant. Anon, (vi 882) N. A. Mth. 8 (1849) 91-. generalisation. Barbour, L. G. Des Moines Anal. 6 (1879) 108-. length. Ficklin, J. Am. As. P. 20 (1871) 63-. solution. Saint-Laurent, T. de. Gergonne A. Mth. 13 (1822-23) 145-. Sturm, J. C. F., & Saint-Laurent, T. de. Gergonne A. Mth. 13 (1822-23) 289-. 591 8420 Kinematic Geometry. Cycloids. Displacement 8420 Curve rolling on straight line. Resal, H. A. Liouv. J. Mth. 6 (1880) 115-. traced by focus of ellipse rolling on straight line, property. Spitzer, S. Arch. Mth. Ps. 48 (1868) 235-. Curves circumscribing rotating polygons with reference to drilled holes. Mattock, A. E. S. P. 35 (1883) 319-. generated by motion of " reptation." Prouhet, E. N. A. Mth. 13 (1854) 274-, 335-. traced by end of moving bar, paradox. Terquem, O. N. A. Mth. 12 (1853) 107-. _ O val chuck. Wilkens, . Kiob. 6v. (1843) 61-. Cyclic curves. Boklen, 0. Grunert Arch. 37 (1861) 119-. , special kind. Durege, H. Ziir. Vjschr. 8 (1863) 127- ; Z. Mth. Ps. 9 (1864) 209-. Cycloidal curves. Fabri, R. Em. At. 10 (1856-57) 225-, 387-; 11 (1857-58) 399-. . Stuuture, R. de. Am. J. Mth. 17 (1895) 269-. , algebraic. Hoist, E. [1881] Arch. Mth. Ntvd. 6 (1882) 125-. , common tangents. Morley, F. N. Y. Am. Mth. S. Bll. 2 (1896) 111-. , plane and spherical. Ekama,H. Arch. Mth. Ps. 7 (1889) 207-. , spherical. Ekama, H. N. Arch. Wisk. 16 (1889) 32- ; Fschr. Mth. (1889) 805-. and other curves, method of construction. Hele-Shaw, H. S. B. A. Ep. (1898) 792-. on sphere. Fuss, N. [1817] St Pet. Ac. Sc. Mm. 8 (1822) 161-. system having base-line and one cusp common, orthogonal trajectories. Baur, G. W. [1871] Z. Mth. Ps. 17 (1872) 424-. as tautochrones. Mechanical discussion by ancient geometrical method. Lehmann, J. W. H. Crelle J. 6 (1830) 49-. theory. Schjellerup, H. C. F. C. Mth. Ts. 1 (1859) 73-. , geometrical. Du Bourguet, . Gergonne A. Mth. 6 (1815-16) 29-. Deformation, infinitesimal, of a medium. Beltrami, E. C. E. 108 (1889) 502-. Deformations due to elongation of rectangular parallelepiped in one direction only. Tresca, H. E. C. E. 97 (1883) 928-. Determination of contour of skew helicoidal surface. Tesaf, J. [1882] Wien Ak. Sb. 86 (1883) (Ab. 2) 377-. Displacement. continuous, of solid. Mannheim, A. C. E. 63 (1866) 1386-. of curve touching fixed curve. Combescure, E. Crelle J. 63 (1864) 332-. double cone. Mannheim, A. C. E. Ill (1890) 634-, 817-. figure of invariable form. Mannheim, A. C. E. 66 (1868) 591- ; Par. Mm. Sav. Etr. 20 (1872) 1-. trochoidal curves, adjustable. Morley, F. . Darboux, G. C. E. 92 (1881) Am. J. Mth. 16 (1894) 188-. Cycloids. Vincent, A. J. H. Lille Mm. S. (1841) 5-; Eouen Tr. Ac. (1853-54) 129-. Gibbes, L. JR. Camb. (M.) Mth. M. 1 (1859) Weisz, J. A. (xn) Mag. Ak. Ets. (1859) (Suppl., Mth. Term.) 81-. Holtz, . [1883] N.-Vorp. Mt. 15 (*1884) xm-. construction. Kracht, H. Dingier 289 (1893) 108-. elementary demonstrations. Schulz von Strasznitzki, L. C. Grunert Arch. 13 (1849) 272-. treatment. HolzmUller, F. G. Z. Mth. Ps. 21 (1876) 128-. and epicycloids and hypocycloids. Cardinal, . Dingier 35 (1830) 329-. , rectification and quadrature. Lentheric, . Mntp. Ac. Sc. Mm. 2 (1851-54) 1-. equilibrium curve in mining. Sanchez, P. C. Mex. S. "Alzate" Mm. 9 (1895) 107-. evolutes. Lang, R. Grunert Arch. 30 (1858) 319-. lie solution of problems by. Guimaraes, , G. Teix. J. Sc. 6 (1885) 85-; Fschr. Mth. (1884) 492. known in 16th century? Gilnther, S. Bb. Mth. (1887) 8-. polar circle pairs. Victor, A. Z. Mth. Ps. 25 (1880) 263-. 118-. in its plane. Dewulf, E. C. E. 92 (1881) 1091-. , planes of which pass through fixed points. Mannheim, A. C. E. 112 (1891) 283-. figures. Brisse, C. Liouv. J. Mth. 15 (1870) 281- . finite, conjugate axes in, generalisation of theorem of Chasles. Ligin, V. N. (xn) Eec. Mth. (Moscou) 6 (1872-73) (Pt. 1) 193-. , of figure of invariable form. Brisse, C. Liouv. J. Mth. 19 (1874) 221- ; 1 (1875) 141-. , in space. Chasles, M. C. E. 51 (1860) 855-, 905- ; 52 (1861) 77-, 189-, 487-. , rigid system, reciprocal figures obtained by, and reciprocal plane diagrams deduced from them. Padelletti, D. Em. E. Ac. Line. T. 3 (1879) 120-. helicoidal, of quadric touching fixed plane. Gontarev, D. A. Vars. S. Nt. Tr. (1895-96) (C. R., Ps. C.) No. 5, 6 pp. infinitely small, of algebraic surface. Mann- heim, A. C. E. 70 (1870) 1025-. , under 4 conditions. Mannheim, A. C. E. 73 (1871) 1096-. , of dihedron of invariable size. Mann- heim, A. C. E. 84 (1877) 1373-. , , proof of a theorem, and applica- tion. Mannheim, A. [1877] Par. S. Mth. Bll. 6 (1878) 5-. 592 8420 Displacement Kinematic Geometry Epicycloids 8420 of a jointed hyperboloid. Bricard, R. Par. S. Mth. Bll. 25 (1897) 98-. most general, of solid in space, treatment. Crofton, M. W. L. Mth. S. P. 5 (1874) 25-. , straight line, points of which describe spherical trajectories. Duporcq, E. C. E. 125 (1897) 762-. of plane figure, curvature of curves described during. Servais, C. Brux. Ac. Bll. 19 (1890) 241-. in its plane. Chasles, M. C. E. 80 (1875) 346-. , theorem. Leaute, H. Par. S. Mth. Bll. 6 (1878) 170-. ,2 points of which slide on any 2 curves. Chasles, M. C. E. 82 (1876) 431-. , all points of which describe spherical lines. Bricard, E. C. E. 125 (1897) 1024-. remarkable case. Bricard, R. C. E. 123 (1896) 939-. of rigid system in elliptic space, linear trans- formation corresponding to. Ball, R. S. [1885] Ir. Ac. P. 4 (1884-88) 532-. systems. Kiipper, C. Schlomilch Z. 6 (1861) 12-. solid of invariable form. Halphen, G. H. [1873] (x) Par. S. Mth. Bll. 2 (1874) 56-. , new method of normals to curves or surfaces described during. Mannheim, A. (x) Par. S. Phlm. Bll. 3 (1866) 79-. , singular cases. Halphen, G. H. [1879] Par. S. Mth. Bll. 8 (1880) 18-. system in space, and resulting variation of coordinates. Rodrigues, 0. Liouv. J. Mth. 5 (1840) 380-. straight line such that 3 of its points remain on faces of rectangular trihedron. Mann- heim, A. Bll. Sc. Mth. 9 (1885) 137-. system of points, differential parameters con- nected with. Durrande, H. C. E. 78 (1874) 1036-. theory. Halphen, G. H. N. A. Mth. 1 (1882) 296-. Double cone rolling on 2 converging lines. Azzarelli, M. Tortolini A. 7 (1856) 317-. Ellipse sliding against 2 straight lines, locus problem. Ferrier, L. N. A. Mth. 3 (1844) 352-, 438. Ellipse-glissette, eliminant of equations. M'Laren, (Lord). Edinb. E. S. P. 19 (1893) 89-. , elimination problem. Muir, T. Edinb. E. S. P. 19 (1893) 25-. , . Nanson, E. J. Edinb. E. S. P. 22 (1900) 158-. , , symmetric solution. McLaren, (Lord). Edinb. E. S. P. 22 (1900) 379-. Ellipses described by points in moving line. Breton [de Champ}, P. N. A. Mth. 5 (1846) 591-. as glissettes, and circular surface of 8th order. Vanecek, J. S., <& Vanecek, M. N. Par. S. Mth. Bll. 11 (1883) 76-. Envelope of cone rolling on 2 developables. Blutel, . C. E. 108 (1889) 496-; Par. EC. Norm. A. 7 (1890) 155-. Envelope of sphere rolling on another sphere. SveSnikov, P. Kazan S. Ps.-Mth. Bll. 5 (1896) 141-. Epicycloids. Johnson, E. F. Silliman J. 21 (1832) 280-. Anon, (vi 879) N. A. Mth. 4 (1845) 83-. Drach, S. M. Ph. Mg. 35 (1849) 487-. Dieu, T. N. A. Mth. 19 (1860) 125-. Fouret,G. (ix) Par. S. Phlm. Bll. 5 (1868) 80-. Collignon, E. As. Fr. C. E. 6 (1877) 92-. Svesnikov, P. I. Kazan S. Nt. (Ps.-Mth.) P. 8 (1890) 374-. Morley, F. Am. J. Mth. 13 (1891) 179-. and binomial theorem. Drach, S. M. Ph. Mg. 34 (1849) 444-, 520-. caustics by reflexion in circle etc. Quetelet, L.A.J. Quetelet Cor. Mth. 5 (1829) 190-. centre of curvature. Mannheim, A. N. A. Mth. 18 (1859) 371-. contact of loops. Sang, E. [1865] Edinb. E. S. T. 24 (1867) 121-. , solution of Perigal's problem. Sang, E. [1865] Edinb. E. S. P. 5 (1866) 338-. double generation. Fouret, (le It.) . N. A. Mth. 8 (1869) 162-. general theory. Raabe, J. L. Crelle J. 1 (1826) 289-. and hypocycloids. Eckardt, F. E. Z. Mth. Ps. 15 (1870) 129- ; 18 (1873) 319-. . Wolstenholme, J. L. Mth. S. P. 4 (1871-73) 321-. , curves derived from. Barisien, E. N. G. Teix. J. Sc. 14 (1900) 121-. and derived curves. Kiepert, L. Z. Mth. Ps. 17 (1872) 129-. , polar reciprocals. Jefdbek, V. Mathesis 19 (1899) 105-. , quadrature. Baur, C. W. Schlomilch Z. 4 (1859) 311-. , tangential property. Jeffery, H. M. [1882] E. S. P. 34 (1883) 105-. intersecting at constant angles, locus of vertices of tangents to. Loucheur, . N. A. Mth. 11 (1892) 374-. multisection of angle by. Leeuwen, J. H. van. N. Arch. Wisk. 7 (*1881) 213. plane and spherical, centres of curvature. Olivier, T. Par. EC. Pol. J. cah. 23 (1834) 85-. in space. Vanecek, J. S. [1880] Wien Ak. Sb. 83 (1881) (Ab. 2) 69-. spherical. Hachette, J. N. P. Par. EC. Pol. Cor. 2 (1809-13) 22-. . Ruiz de Cardenas, A. G. Mt. 12 (1874) 313-. , rectifiable. Jeffery, H. M. B. A. Ep. (1882) 453-. , and their tangents. Hachette, J. N. P. Par. EC. Pol. Cor. 2 (1809-13) 87-. tangent problems. Binder, W. Wien Ak. Sb. 107 (1898) (Ab. 2a) 362-. Epi-ellipside (analogue to epicycloid), equation. Biermann, O. Mh. Mth. Ps. 8 U897) 87-. Epi- and hypo-trochoids and allied curves, Pliickerian characteristics. Roberts, S. L. Mth. S. P. 4 (1871-73) 353-. VOL. I. 593 PP 8420 Kinematic Geometry Hypocycloids 8420 Epitrochoid. Cayley,A. Mess. Mth. 20 (1891) 150-. Epitrochoidal surfaces, and generalisation of their properties. Svesnikov, P. Kazan S. Ps.-Mth. Bll. 1 (1891) 166-, 178- ; Fschr. Mth. (1891) 861. Equations of configurations in n-dimensional space. Hatzidakis, N. J. C. B. 130 (1900) 557-. curves, determination by kinematics. Verdam, G. J. Grunert Arch. 11 (1848) 13-. Fan, construction and properties of new sort. Sylvester, J. J. L. Mth. S. P. 6 (1874-75) 196-. Formation of figures in affinity by similarly variable systems. Geisenheimer, L. Z. Mth. Ps. 24 (1879) 345-. Generation of surfaces and volumes. Barbier, E. N. A. Mth. 19 (1860) 463-. Geometrical problem solved by mechanics. Breton, P. Les Mondes 24 (1871) 339-. and statical theorems on pyramid, etc. Grunert, J. A. Grunert Arch. 9 (1847) 353-. transformation with application in special case to kinematics. Dewulf, E. Par. EC. Norm. A. 3 (1886) 405-. Glissette, centre of curvature. Mannheim, A. Par. EC. Pol. J. 37" cah. (1858) 179-. , curvature. Chasles, M. Liouv. J. Mth. 10 (1845) 204-. derived from circle and straight line. Dewall, (Oberst) von. Arch. Mth. Ps. 42 (1864) 65-. , rod sliding against 2 fixed lines. Unfer- dinger.F. [1861] Wien SB. 45 (Ab. 2) (1862) 251-. of sides of square with two angular points describing straight lines. Pomey, J. B. N. A. Mth. 5 (1886) 520-. 2-term oval + ~ = 1. M'Laren, a n b n (Lord). Edinb. B. S. P. 18 (1892) 83-. Glissettes of ellipse and hyperbola. Tait, . Edinb. B. S. P. 17 (1891) 2-. , formula. Besant, W. H. [1870] QJ. Mth. 11 (1871) 38-. Groups of movements. Jordan, C. C. B. 65 (1867) 229- ; A. Mt. 2 (1868-69) 167-, 322-. , continuous, of any configuration in 3 dimensions. Eicci, G. C. B. 127 (1898) 344-, 360r, 390. Helicoidal velocities of solid, connections of which are defined by inequalities. Somov, P. O. Vars. S. Nt. Tr. (1895-96) (C. E., Ps. C.) No. 8, 5 pp. Herpolhode in case of quadric surface. Sparre, (le comte) de. Brux. S. Sc. A. 9 (1885) (Pt. 2) 249-. of Poinsot, and an apparatus of Darboux and Koenigs. Gilbert, P. Brux. S. Sc. A. 14 (1890) (Pt. 2) 25-. Hinged duplicator. Somov, P. O. Vars. S. Nt. Tr. (1900) (C. E., Ps. C.) No. 2, 1-. Hyperboloid of jointed rods, and de Sparre's theorem. Mannheim, A. C. B. 102 (1886) 501-. " Hypercycles " (geometry of direction). La- guerre, E. C. B. 94 (1882) 778-, 832-, 933-, 1033-, 1160-. Hypocycloids. Eoot, 0. Camb. (M.) Mth. M. 1 (1859) 133-. with 3 cusps. Steiner, J. Berl. Mb. (1856) 1-. . Cremona, L. [1864] Crelle J. 64 (1865) 101-. . Clebsch, E.F.A. [1864] Crelle J. 64 (1865) 124-. . Painvin, L. [1865] N. A. Mth. 9 (1870) 202-, 256-. . Milinowski, . Z. Mth. Ps. 19 (1874) 115-. . Cahen, C. P. N. A. Mth. 14 (1875) 21-. . Kantor, S. Bll. Sc. Mth. As. 3 (1879) (Pt. 1) 136-. . Laguerre, E. Par. S. Mth. Bll. 7 (1879) 108-. . MacMaJwn, P. A. [1882] Mess. Mth. 12 (1883) 151-. . Intrigila, C. G. Mt. 23 (1885) 263. . Serret, P. C. B. 125 (1897) 404-, 423-, 445-, 459-. . Ruffi.nl, F. P. [1900] Bologna Bd. 5 (1901) 13-. , connected with family of parabolas. Eupp, O. Mh. Mth. Ps. 4 (1893) 135-. , construction. Stiner, G. Mh. Mth. Ps. 6 (1895) 372-. , envelope of pedal line of triangle, Ferrers' theorem. Serret, P. N. A. Mth. 9 (1870) 73-. , Simson's line. Brocard, H. [1876] Par. S. Mth. Bll. 5 (1877) 18-. , . Badoureau, A. N. A. Mth. 18 (1879) 33-. , , proof of Steiner's theorem. Brocard, H. (ix) Par. S. Mth. Bll. 1 (1873) 224-. , Feuerbach's circle, and the circum- scribing hyperbolas of A. Fiedler, W. Ziir. Vjschr. 30 (1885) 390-. , Steiner's problem, extension. Mac Mahon, P. A. [1882] Mess. Mth. 12 (1883) 138-. , tangential geometry. Kantor, S. [1878] Wien Ak. Sb. 78 (1879) (Ab. 2) 204-. 4 cusps. Breton [de Champ], P. N. A. Mth. 4 (1845) 135-. . Wolfers, J. P. Grunert Arch. 25 (1855) 109-. . M., J. Mathesis 14 (1894) 129-. , evolute. Graves, E. H. A. Mth. 4 (1888) 36. , and other curves. Bbklen, 0. Grunert Arch. 37 (1861) 105-. Infinitesimal angle between adjacent normals and application to rolling of surfaces. Eesal, H. C. B. 114 (1892) 381-, 700. Instantaneous centre and its geometrical appli- cations. Habich, E. Les Mondes 18 (1868) 709-, 750- ; 19 (1869) 29-, 121-. 594 8420 Kinematic Geometry Motion 8420 Joints, simple. Chebuishev, P. L. [1879] (xn) Eec. Mth. (Moscou) 9 (1878-81) (Pt. 1) 340-. Kinematic determination of curves. Ocagne, M. d>. N. A. Mth. 1 (1882) 40-; 2 (1883) 189-. geometry, applications. Mannheim, A. Prag Sb. (1888) (Mth.-Nt.) 87-. of space, fundamental theorem in. Geiser, C. F. [1880] Crelle J. Mth. 90 (1881) 39-. and metrical properties of conjugate quadri- laterals. Stephanos, C. C. E. 95 (1882) 677-. Kinematics applied to y modern geometry. Prochdzka, B. Prag Cesk6 Ak. Fr. Jos. Ez. (Tfida 2) 5 (1896) Art. 40, 19 pp.; Fschr. Mth. (1896) 415-. of figures on surfaces. Levy, M. C. E. 86 (1878) 812-. jointed frameworks. Somov, P. 0. Vars. S. Nt. Tr. (1900) (C. R., Ps. C.) No. 5, 46 pp. plane. Cayley, A. QJ. Mth. 16 (1879) 1-. , reciprocal figures and quadratic transforma- tion in. Cavalli, E. Nap. Ac. At. 9 (1899) No. 12, 28 pp. of variable configurations with relations of similarity or affinity. Schumann, A. Z. Mth. Ps. 26 (1881) 157-. or rigid systems, cases. Burmester, L. Z. Mth. Ps. 23 (1878) 108-. Limacon, mechanical construction and parallel motion deduced therefrom. Hart, H. Mess. Mth. 5 (1876) 35-. Lines, 2, revolving in same plane about 2 fixed points, curves of intersection. Mensbrugghe, G. van der. [1863] (vm) Brux. Mm. Cour. 8, 16 (1864) 22 pp. , , with uniform angular velocity about point, appearances. Plateau, J. A. F. Que- telet Cor. Mth. 4 (1828) 393-. Linkage, every algebraic condition imposed on motion of body is realisable by. Koenigs, G. C. E. 120 (1895) 981-. C. E. 120 (1895) 861-. Linkwork, expression of curves generated by, as irreducible determinant. Sylvester, J. J. L. Mth. S. P. 6 (1874-75) 196-. Lissajous's curves. Braun, W. Mth. A. 8 (1875) 565-. . Himstedt, . Arch. Mth. Ps. 70 (1884) 337-. . Ekama, H. Arch. Mth. Ps. 6 (1888) 39-. Mechanical construction of curves. Painvin. L. N. A. Mth. 15 (1856) 139-. . Russell, W. H. L. [1869] E. S. P. 18 (1870) 72-. , 4th degree. Cayley, A. [1872] L. Mth. S. P. 4 (1871-73) 186-. sphero-conics. Hart, H. L. Mth. S. P. 6 (1874-75) 136-. Method of tangents. Johnson, W. W. A. Mth. 1 (1884-85) 131-. Motion. apparent, of heavy particle at earth's surface. Gilbert, L. P. [1877] (xn) Brux. S. Sc. A. 2 (1878) (Pt. 1) 61-, (Pt. 2) 49-. of body remaining homothetic to itself during displacement and deformation. Fouret, G. C. E. 88 (1879) 227-. conchoidal. Habich,E. Les Mondes 11 (1866) 307-. under 5 conditions, case. Rouquet, V. Toul. Fac. Sc. A. 10 (1896) F, 23 pp. , curves and surfaces described in. Schoenflies, . Bll. Sc. Mth. 12 (1888) 18-. of connecting rod, curves generated by, rule for substitution in given arc. Barbier, E. C. E. 96 (1883) 1648. continuous .material system, differential invariants. Zorawski, K. Krk. Ak. (Mt.- Prz.) Ez. 18 (1901) 353-; Crc. Ac. Sc. Bll. (1900) 367-. 2 convicts chained together, problem. Col- lignon, E. As. Fr. C. E. (1893) (Pt. 2) 47-. curve on another equal curve. Minozzi, A. G. Mt. 14 (1876) 190-. developable cylinders. Bottomley,J. Manch. Lt. Ph. S. P. 20 (1881) 128-. elliptic, theorem of Mobius. Mansion, . [1900] Brux. S. Sc. A. 25 (1901) (Pt. 1) 71-. , of unresisted projectiles, constructions con- nected with. Tait, P. G. Edinb. E. S. P. 5 (1866) 565-. of figure in its plane. Ribaucour, A. (xi) Par. S. Phlm. Bll. 6 (1869) 12-. figures in plane and space. Jordan, C. (x) Par. S. Mth. Bll. 1 (1873) 144-. remaining similar to themselves. For- menti, C. G. Mt. 17 (1879) 232-. during displacement in space, case. Grouard, A. (x) Par. S. Phlm. Bll. 10 (1873) 47-. general, of solid. Seiliger, D. N. Kazan S. Ps.-Mth. Bll. 7 (1898) 131-. of heavy spherical segment on inclined plane. Didion, . C. E. 77 (1873) 167-. helicoidal, of solid. Appell, P. Par. EC. Norm. hyperbolic. Hall, A. [1867] Mess. Mth. 4 (1868) 106-. of invariable figure, areas, arcs, etc. traced, properties. Darboux, G. Bll. Sc. Mth. As. 2 (1878) 333-. in naval manoeuvres. Tournier, (le It.) E. Ev. Mar. et Col. 125 (1895) 430- ; 130 (1896) 209-. . Reveille, J. Ev. Mar. et Col. 134 (1897) 465-. . Tournier, (le It.) E. Ev. Mar. et Col. 135 (1897) 51-. . May, (Capt.) H. J. Ev. Mar. et Col. 136 (1898) 523-; 137 (1898) 77-. . Reveille, J. Ev. Mar. et Col. 139 (1898) 186-. non-Euclidean geometry. Newson, H. B. N. Y. Am. Mth. S. Bll. 4 (1898) 2, 5-. 1l861) 37 pp.' of plane under certain conditions. Roberts, S. L. Mth. S. P. 3 (1869-71) 286-; 7 (1875-76) 216-. plane, under conditions, ruled surfaces gene- rated by. Blake, E. M. [1898] Am. J. Mth. 21 (1899) 257-. 595 pp2 8420 Motion Kinematic Geometry 8420 plane, curvature. Bresse, J. A. C. Par. EC. Pol. J. 35 e cah. (1853) 89-. of plane figure, formulae. Haton de la Gou- pilliere, J. N. Par. EC. Pol. J. Cah. 45 (1878) 105-. in its plane. Habich, E. Les Mondes 14 (1867) 71-, 402- ; 15 (1867)24-; 16 (1868) 540-, 600-, 677-. . Pellet, A. C. B. 120 (1895) 1204-. remaining similar to itself. Laisant, C. A. As. Fr. C. R. (1882) 106-. . Nicoli, F. Mod. Ac. Sc. Mm. 1 (1883) 59-, 171-. , trajectories connected with. Pro- chdzka, B. Casopis 25 (1896) 81-, 161- ; Fschr. Mth. (1896) 577. plane, of 1 surface over another, problem. Eecen, A. N. Arch. Wisk. 15 (1888) 67-. of plane variable systems with relations of affinity or collinearity. Burmester, L. Z. Mth. Ps. 19 (1874) 465-. similarity. Burmester, L. Z. Mth. Ps. 19 (1874) 154-. point, diagrams, general method of obtain- ing. Padelletti, D. Mil. I. Lomb. Rd. 10 (1877) 134-. on ellipsoid. Schellbach, E. H. Crelle J. 54 (1857) 380-. given surface. Ruffini, F. P. Bologna Ac. Sc. Mm. 5 (1883) 211-. , properties. Liguine, V. N. A. Mth. 1 on surface. Haag, . Par. S. Mth. Bll. 5 (1877) 166-. , application of Hamilton's method. Padova, E. G. Mt. 7 (1870) 90-. relative, expansion of coordinates of point in, and curvature of orthogonal lines. Bali- trand, . Par. S. Mth. Bll. 23 (1895) 26-. , normals and centres of curvature found in case of. Gilbert, L. P. [1878] (xn) Brux. S. Sc. A. 3 (1879) (Pt. 1) 55-; (Pt. 2) 81-. , of ship and wind. Collignon, E. As. Fr. C. R. (1885) (Pt. 2) 6-. , surfaces of revolution touching one another. Zimmermann, H. Z. Mth. Ps. 19 (1874) 242-. to any systems. [Barre de] Saint- Venant, . N. A. Mth. 8 (1849) 326-. and reversions. Study, E. Mth. A. 39 (1891) of rigid body, geometrical investigation. Neu- mann, C. Mth. A. 1 (1869) 195-. or project! vely variable systems, curves and surfaces described in. Rodenberg, C. Gott. Nr. (1888) 176-. rolling body. Beltrami, E. G. Mt. 10 (1872) 103-. ship towards given point in given direction. Corrard, A. Rv. Mar. et Col. 136 (1898) 433-. in uniform current. Collignon, E. As. Fr. C. R. (1887) (Pt. 2) 42-. similarly varying systems. Geisenheimer, L. Z. Mth. Ps. 24 (1879) 129-. of skew quadrilateral, 2 angles of which de- scribe circles. PeliSek, M. Prag Cesk6 Ak. Fr. Jos. Rz. (Tfida 2) 8 (1899) Art. 39, 13 pp.; Fschr. Mth. (1899) 492. solid remaining similar to itself, 2 cases. Nicoli, F. Mod. Ac. Sc. Mm. 1 (1883) 249-. straight line. Halphen, G. H. (x) Par. S. Mth. Bll. 1 (1873) 114-. in plane. Habich, E. C. R. 89 (1879) 405-. surface about fixed point. Floquet, G. C. R. 105 (1887) 746-. always touching fixed surface. Gautero, G. G. Mt. 20 (1882) 168-. system of points collinearly variable. Somov, P. O. Vars. S. Nt. Tr. (1890-91) (C. R., Ps. C.) No. 8, 1-. tangent rolling on curve, properties. Cha- mousset, (I'abbe) F. [1868] (xn) Sav. Ac. Mm. 11 (1869) cvi-. triangle in plane, theorem. Serret, P. N. A. Mth. 12 (1853) 68-. Moving curve, curves described by point of. Collins, E. [1813] St Pet. Ac. Sc. Mm. 6 (1813-14) 133-. curves, locus of intersections of 2. Nico- laides, N. N. A. Mth. 3 (1864) 39-. line, orthogonal trajectories. Pirondini, G. N. A.Mth. 9 (1890) 297-. pencil of planes, properties. Mannheim, A. As. Fr. C. R. (1874) 1176-. plane, orthogonal trajectories. Serret, J. A. C. R. 41 (1855) 1253-. sphere, orthogonal trajectories. Serret, J. A. C. R. 42 (1856) 105-. straight line, curves described by. La Gournerie, J. de. Liouv. J. Mth. 14 (1849) 417-. , generation of curves by. Le Francois, (Prof.) . Quetelet Cor. Mth. 4 (1828) 296-. Orthogonal web of jointed rods. Sylvester, J. J. L. Mth. S. P. 6 (1874-75) 197. Plane kinematic geometry, applications. Ocagne, M. d'. N. A. Mth. 19 (1880) 264-, 289-. trigonometry, theorem suggested by Gauss's curvatura Integra. Thomson, (Sir) W. Ph. Mg. 32 (1891) 471-. Polar reciprocal curves, and hodograph. Hir- man, S. N. Vars. S. Nt. Tr. (1891-92) (C. R., Ps. C.) No. 7, 4-. Problem. Geiser, C. F. Sch. Nf. Gs. Vh. 59 (1876) 236-. . Prochdzka, B. Casopis 24 (1895) 291- ; Fschr. Mth. (1895) 621. , application of non-Euclidean geometry to. Ball, R. S. B. A. Rp. (1881) 535-. on geometry of circle. Sobotka, J. Mh. Mth. Ps. 7 (1896) 347-. Projective properties of surfaces generated by curves on moving plane. Schlesinger, J. Wien Sb. 58 (1868) (Ab. 2) 435-. variable plane systems. Burmester, L. Mth. A. 14 (1879) 472-. Pseudo-trochoids. Wolffing, E. Z. Mth. Ps. 44 (1899) 139-. 596 8420 Kinematic Geometry Roulettes 8420 Quadrangle, differential relation between sides. Cayley, A. Mess. Mth. 6 (1877) 99-. "Kadial" curves (derived from curve and its evolute). Tucker, R. [1865] L. Mth. S. P. 1 (1866) No. 5, 7 pp. Eadius of curvature of curve described by point of moving figure. Saint Loup, L. N. A. Mth. 12 (1873) 113-. Eeal surfaces denned by J0 , 9 = 0. Mmt- a j u au-S cheuil, de. Par. S. Mth. Bll. 27 (1899) 114-. Rolling cone, envelope of connected cone. Hennessy, H. Ir. Ac. P. 7 (1861) 361-. motion of spheres, solution of problem. Siverly, W. Des Moines Anal. 2 (1875) 195-. , . Adcock, R. J. Des Moines Anal. 3 (1876) 61-, 129-. , . Hill,G.W. Des Moines Anal. 3 (1876) 92-. of plane curves. Lundberg, E. Ups. Arsk. (1874) (Mth.) (No. 3) 31 pp. solid bounded by cylinder on horizontal plane. Hoppe, R. Arch. Mth. Ps. 66 (1881) 213-. tangent surface on horizontal plane. Hoppe, R. Arch. Mth. Ps. 66 (1881) 373-. surfaces. Resal, H. C. E. 100 (1885) 260-. on one another. Hoppe, R. Arch. Mth. Ps. 60 (1877) 159-. . Cosserat, E. C. E. 121 (1895) 935-. Eotation axes, loci in theory. Zancevskij, I. N. Es. S. Nt. Mm. (Mth.) 14 (1892) i-, 1-. , geometrical theory. Donkin, W. F. Ph. Mg. 1 (1851) 187-. , infinitely small, proof of theorems by. Mannheim, A. C. E. 78 (1874) 633-. , representation without projecting on plane perpendicular to axis. Fialkou-ski, N. Wien SB. 21 (1856) 181-. Eotations and homographies, correspondence. Cayley, A. Mth. A. 15 (1879) 238-. , spherical. Stephanos, C. Mth. A. 22 (1883) 299-. Roulettes. Holditch, H. [1838] Camb. Ph. S. T. 7 (1842) 61-. Lentheric, . Mntp. Ac. PV. (1850-51) 30-, 34-, 37-; Mntp. Ac. Sc. Mm. 2 (1851-54) 1-, 315-. Sacchi, G. N. A. Mth. 2 (1863) 172-. Purser, F. QJ. Mth. 7 (1866) 129-. Aoust, (Vabbg) . C. E. 70 (1870) 978-. Habich, E. (xn) Mathesis 2 (1882) 145-. Gossot, (le capit.) F. Ev. Mar. et Col. 89 (1886) 319-. case. Lane, A. V. Am. J. Mth. 8 (1886) 132-. centres of curvature. Hadamard, . Bor- deaux S. Sc. Mm. 5 (1895) xxxvii-. for conic on other curves. Ekama, H. Arch. Mth. Ps. 8 (1890) 388-. Descartes's theorem. Frenet, F. N. A. Mth. 13 (1854) 299-. described by conic. Ekama, H. N. Arch. Wisk. 3 (1898) 175- ; Fschr. Mth. (1897) 525. generalisation of theorem of Delaunay. Saint- Germain, A. de. Bll. Sc. Mth. 17 (1893) 110-. generated by rolling circle, orthogonal trajec- tories. Breton, P. [1861] (xn) Isere S. Bll. 7 (Livr. 3 & 4) (1867) 346-. geometrical transformation applicable to. De- moulin, A. Brux. Mm. Cour. 8, 45 (1891) No. 3, 35 pp. nomenclature for. Willson, F. N. Am. As. P. (1887) 59-. and pedal curves. Catalan, E. C. Brux. Ac. Bll. 27 (1869) 144-. plane. Hennig, R. [1864] Crelle J. 65 (1866) 52-. and polar paths of plane kinematic systems. Sellentin, R. Z. Mth. Ps. 28 (1883) 116-. problems, inverse. Chrystal, G. Edinb. Mth. S. P. 5 (1887) 38-. quadrature. Orlov, T. E. [1883] (xn) Eec. Mth. (Moscou) 11 (1883-84) 457-; Fschr. Mth. (1884) 764. rectification. Steiner, J. Berl. Ab. (1838) 19-. and rolling surfaces. Pirondini, G. G. Mt. 26 (1888) 352-. skew. Laurent, H. Par. S. Mth. Bll. 2 (1874) theorem. Lamarle, E. Brux. Ac. Bll. 4 (1858) 239-. . Serret, P. N. A. Mth. 18 (1859) 341-. attributed to de La Hire. Le Paige, C. Bb. Mth. (1887) 109. . Curtze, M. Bb. Mth. (1888) 65-; (1895) 33-. theory. Maxwell, J. C. Edinb. E. S. T. 16 (1849) 519-. . Catalan, E. C. N. A. Mth. 15 (1856) 102-. . Mannheim, A. L'l. 26 (1858) 69. . Cesdro, E. N. A. Mth. 7 (1888) 209-. Euled mechanics. Saussure, R. de. C. E. 123 (1896) 796-. surface traced by line forming part of in- variable moving system. Rouquet, V. [1900] Toul. Ac. Sc. Mm. 1 (1901) 9-. Eunners, problem. Manderlier, . Quetelet Cor. Mth. 2 (1826) 199-. Skew curves, 3rd degree, analogue of helicoidal motion. Appell, P. C. E. 82 (1876) 70-; Par. EC. Norm. A. 5 (1876) 245-. Sphere sliding or rolling on horizontal plane. Resal, H. N. A. Mth. 11 (1872) 193-. Spherical cycloidal and trochoidal curves. Je/ery, H. M. [1882] QJ. Mth. 19 (1883) 44-. loop line (tortuous curve). Schiffner, F. Arch. Mth. Ps. 5 (1887) 160-. ( ). Janisch, E. Arch. Mth. Ps. 8 (1890) 184-, 334-; 9 (1890) vn ; 10 (1891) vn. 597 8420 Kinematic Geometry Straight line sliding against curves, locus of point in. Enneper, A. Z. Mth. Ps. 11 (1866) 434-. Surface, 4th degree, with generators capable of moving normally to their trajectories. Chasles, M. C. R. 52 (1861) 1094-. Surfaces capable of generating a Lame" system. Cosserat, E. G. R. 124 (1897) 1426-. generated by motion of lines, partial diffe- rential equations. BlaSek, G. [1864] Wien Sb. 51 (1865) (Ab. 2) 186-. moving curve. Pranghofer, J. Wien Sb. 53 (1866) (Ab. 2) 505-. plane curve when plane rolls on developable surface. Pirondini, G. G. Mt. 30 (1892) 188-. having generators of constant form, partial differential equation for. Gebbia, M. Pa- lermo Cir. Mt. Rd. 1 (1887) 74-. , helicoidal, construction of parallel projec- tion, and especially shadow. Burmester, L. Z. Mth. Ps. 18 (1873) 184-. related to curve (skew) rolling on another. Svechnicoff [SveSnikov], P. N. A. Mth. 14 (1895) 501- ; Kazan S. Ps.-Mth. Bll. 6 (1896) 16 pp. ; Fschr. Mth. (1896) 537. , screw, with circumscribed cylinders, tan- gent curves. Sobotka, J. Prag Sb. (1893) (Mth.-Nt.) No. 22, 38 pp. susceptible of generation by motion of plane curve. Mainardi, G. Mod. S. It. Mm. 20 (1828) 482-. System of 3 equal figures in same plane, properties. Stephanos, C. [1881] Par. S. Phlm. Bll. 6 (1882) 12-. 2 similar bodies. Chasles, M. Quetelet Cor. Mth. 7 (1832) 352-. Theorem. Reveille, J. Par. S. Mth. Bll. 16 (1888) 130- . on description of areas. Johnson, W. W. B. A. Rp. (1881) 536-. Theorems of Schooten and de La Hire. (1) Ellipse, kinematic description. (2) Hypo- cycloid degenerating to t straight line. La Gournerie, J. de. Par. EC. Pol. J. 36 e cah. (1856) 255-. Trajectory of point on straight line displaced under conditions. Mannheim, A. C. R. 70 (1870) 1215-. surfaces of figure of invariable form, normals, construction. Mannheim, A. As. Fr. C. R. (1878) 152-. , meet two straight lines. Mannheim, A. As.Fr.C.R.2(1873)95-. moving under 4 conditions. Mann- heim, A. (x) C. R. 77 (1873) 752-; Par. Mm. Sav. Etr. 22 (1876) No. 12, 19 pp. Transformation of demonstration in kinematic geometry. Mannheim, A. C. R. 112 (1891) 475-. , general, of group of Euclidean movements. Page, J. M. [1898] A. Mth. 12 (1898-99) 87-. method. Mannheim, A. C. R. 110 (1890) 220-, 270-. for ruled surfaces. Mannheim, A. C. R. 88 (1879) 1128-. Transformations. Mannheim, A. C. R. 110 (1890) 391-. Plane Curves 8430 Translation surface. Crelle, A. L. Crelle J. 51 (1856) 100-. surfaces. Sucliarda, A. Wien Ak. Sb. 92 (1886) (Ab. 2) 836-. . Pirondini, G. A. Mt. 17 (1889-90) 225-. . Sanctis, P. de. N. A. Mth. 9 (1890) 552-. . Lie, S. Leip. Mth. Ps. B. 44 (1892) 447-, 559-. , and Abel's theorem. Lie, S. C. R. 114 (1892) 277- ; Leip. Mth. Ps. B. 48 (1896) arising in several ways. Scheffers, G. C. R. 116 (1893) 1352-. , construction of tangent rays. Sucharda, A. Prag Ceske Ak. Fr. Jos. Rz. (TfUa 2) 6 (1897) Art. 24, 36 pp. ; Prag Fr. Jos. Ac. Sc. Bll. (Mth. Nt.) 4 (1897) (Pt. 2) 63-. , determination of all. Lie, M. S. Arch. Mth. Ntvd. 7 (1882) 155-. in general and biquadratic in particular. Sucharda, A. Casopis 28 (1899) 257-, 336- ; Fschr. Mth. (1899) 571-. "surfaces" and Abel's theorem. Lie, S. Leip. Mth. Ps. B. 49 (1897) 181-. Trochoids. Studnicka, F. J. Casopis 1 (*1872) 252- ; Bll. Sc. Mth. As. 6 (1874) 97. , curvature. Gogu, C. [1890] Bucarest Ac. Rom. A. 13 (1890-91) (Pt. Admin.) 45-. , double generation. Wiener, L. C. Z. Mth. Ps. 26 (1881) 257-. , evolutes. Wiener, L. C. Z. Mth. Ps. 27 (1882) 129-. , foliate curves. Hyde, E. W. Des Moines Anal. 2 (1875) 12-. , history. Hoza, F. Casopis 1 (*1872) 54-. of points on rolling conies. Ekama, H. N. Arch. Wisk. 16 (1889) 58-; Fschr. Mth. (1889) 719-. Volumes generated by closed contour in any motion. Koenigs, G. C. R. 106 (1888) 1512-. Watt's curve. Catalan, E. Mathesis 5 (1885) 154-, 222-. , theorem. Tchebitchef, P. L. Bll. Sc. Mth. As. 5 (1881) (Pt. 1) 216. 8430 Curvature of plane curves; other applications of the differen- tial calculus to plane curves. (See also 7210.) Aberration of curvature. Husquin de Rheville, . N. A. Mth. 9 (1890) 138-. . Servais, C. Mathesis 12 (1892) 129-. Antipedals, theorem. Catalan, E. C. Rm. At. N. Line. 25 (1872) 349-. Asymptotic chords. Hermann, O. Grunert Arch. 12 (1849) 323-. Barycentre of curvature. Steiner, J. Berl. Ab. (1838) 19-. of algebraic curves. Neumann, C. Z. Mth. Ps. 12 (1867) 172-, 425-. 598 8430 Applications of Calculus to Plane Curves 8430 Boundary relations, and formula for radius of curvature. Voller, . Grunert Arch. 32 (1859) 97- ; 33 (1859) 350-. Caustics, construction. Cornu, A. N. A. Mth. 2 (1863) 311-. identical with pedals. Weyr, E. Z. Mth. Ps. 14 (1869) 376-. , plane, application of coordinate system to. Habich, E. A. Mt. 2 (1868-69) 134-. Centre of curvature. Ocagne, M. d'. Par. S. Mth. fill. 19 (1891) 31-. , graphic construction. Anon, (vi 541) Gergonne A. Mth. 12 (1821-22) 135-. , . Gergonne, J. D. Gergonne A. Mth. 12 (1821-22) 137-. of locus of points, distances of which from 2 given curves are in constant ratio. Mannheim, A. Tortolini A. 1 (1858) 364-. and normal of polytropic curves. Kosch, E. Z. Mth. Ps. 45 (1900) 161-. with polar coordinates. Husquin de Rheville, . N. A. Mth. 10 (1891) 411-. . of certain spiral curves. Rulf, W. Mh. Mth. Ps. 3 (1892) 211-. Centres of curvature of integral curve. Ma- chovec, F. Casopis 19 (1890) 67- ; Fschr. Mth. (1890) 703. isogonal curves. Seiliger, D. N. N. Bs. S. Nt. Mm. (Mth.) 14 (1892) 155- ; Fschr. Mth. (1892) 582. , successive. Haton de la Goupilliere, J. N. C. B. 46 (1858) 930- ; Liouv. J. Mth. 4 (1859) 183-. of y n =a r x n ~ r . Machovec, F. Prag Sb. (1885) '(Mth.-Nt.) 386-. \x n + /j.y n = d n . Machovec, F. Prag Sb. (1891) (Mth.-Nt.) 83-. Circles of curvature. Schellbach, C. H. Crelle J. 45 (1853) 263-. . Tait, P. G. Edinb. Mth. S. P. 14 (1896) 26. and asymptotes. Laisant, C. A. Par. S. Mth. Bll. 23 (1895) 95-. complex quantities. Petersen, J. Sk. Nf. F. (1892) 354; Fschr. Mth. (1892) of conies. Weiler, A. Ziir. Vjschr. 33 (1888) 119-. ellipse. Cesdro, E. Mathesis 5 (1885) 7-. family of curves. Allegret, [A.] N. A. Mth. 9 (1870) 30-. nodal cubic. Janisch, E. Mh. Mth. Ps. 7 (1896) 142-. at point of curve of 4th degree with 3 double points, of which 2 are circular points at infinity. Saltel, L. Brux. Ac. Bll. 34 (1872) 51-. , radii. Pagani, G. M. Brux. Ac. Bll. 14 (1847) 185-. , relation to curve at point of contact. Mack, L. Arch. Mth. Ps. 64 (1879) 182-. having double contact with conic, point transformations in elliptic coordinates. Lovett, E. 0. Am. J. Mth. 20 (1898) 242-. Cochleoid. Falkenburg, C. Arch. Mth. Ps. 70 (1884) 259-. Cochleoid, properties. Benthem, A. N. Arch. Wisk. 10 (1884) 76-. Conies having higher order of contact with given curve. Ocagne, M. d'. N. A. Mth. 16 (1897) 252-. osculating curves of 3rd degree. Gent, E. Z. Mth. Ps. 17 (1872) 476-. , systems, differential equations. Roberts, R. A. [1895] N. Y. Am. Mth. S. Bll. 2 (1896) 11-. touching curves of any degree. Cayley, A. C. B. 59 (1864) 224-. Contact. Christie, A. S. Smiths. Misc. Col. 33 (1888) Art. 1, 157-. (Wash. Ph. S. Bll. 6 (1884).) with conies or curves of 3rd degree. Hal- phen, G. H. [1875] Par. S. Mth. Bll. 4 (1876) 59-. and curvature, theory. Grunert, J. A. Grunert Arch. 30 (1858) 361-. of 2 curves, ratio of curvatures at point of. Wolffing, E. Z. Mth. Ps. 38 (1893) 237-. lines, successive intersections. Desmarest, E. N. A. Mth. 3 (1844) 220-. of given order, of conic with curve, conditions. Painvin, L. C. B. 78 (1874) 55-. 5th order, of conic with curve, conditions. Painvin, L. C. B. 78 (1874) 436-, 835-. with parabola. Schell, W. Marb. Schr. 8 (1857) 135-. of plane cubic with curve of any degree, positions where closest. Disteli, M. Z. Mth. Ps. 38 (1893) 257-. , 6-point, of curves of 3rd degree with conic, 27 points of. Steiner, J. Berl. B. (1845 386-. points of parallel tangent lines or planes, centre of mean distances. Liouville, J. Liouv. J. Mth. 6 (1841) 345-; C. B. 13 (1841) 412-. , theory. Poselger, F. T. Berl. Ab. (1836) 55-. , , point in. Halphen, G. H. Par. S. Mth. Bll. 2 (1874) 94-. Coordinates in plane, Watson's. Hill. T. Am. As. P. 22 (1873) 27-. , , problems. Hill, T. Am. As. P. (1875) (Pt. 1) 41-. Cubic parabolas, property. Cellerier, G. Arch. Sc. Ps. Nt. 11 (1884) 408-. Curvature. Gergonne, J. D. Gergonne A. Mth. 21 (1830-31) 1-. . Zimmermann, H. E. M. 0. Z. Mth. Ps. 28 (1883) 115-. . Demoulin, A. Brux. Mm. Cour. 8, 44 (1891) No. 5, 48 pp. conies. Berg, F. J. van den. Amst. Ak. Vs. M. 9 (1892) 85-; Fschr. Mth. (1892) 654-. curve of basis point of pencil of curves of nth degree. Zahradnik, K. Prag Sb. (1878) 250-. at cusp. Anon, (xi 52) N. A. Mth. 10 (1871) 318-. double point. Cayley, A. QJ. Mth. 3 (1860) 322-. formula. Torry, A. F. Mess. Mth. 2 (1863) 81-. 599 Applications of Calculus to Plane Curves 8430 Curvature formulae in trilinear coordinates. Walton, W. QJ. Mth. 8 (1867) 39-. at multiple point. Painvin, L. C. B. 68 (1869) 131- ; A. Mt. 4 (1870-71) 215-. of 3rd order. Dupin, C. C. E. 26 (1848) 321-, 393-. properties of ellipse and hyperbola. Wright, C. Camb. (M.) Mth. M. 2 (1860) 198-. , question in. Cesdro, E. Mathesis 13 (1893) 177-. relations of pencil of curves at basis point. Weyr, E. Z. Mth. Ps. 15 (1870) 486-. Curves. (See also 7610, 7630.) algebraic, curvature. Servais, C. Brux. Ac. Bll. 21 (1891) 587-; 22 (1891) 512-. , , etc. Ocagne, M. d'. Mathesis 12 (1892) 100-. , equal to circular arcs [p=r (a-r)]. Fuss, N. [1823] St Pet. Ac. Sc. Mm. 11 (1830) 274-. , hyperbolic arcs. Fuss, N. [1798] St Pet. Ac. Sc. N. Acta 14 (1805J 111-. , intersecting orthogonally, and their con- focality. Rummer, E. E. N. A. Mth. 11 (1852) 426-. analogous, intrinsically, to catenary of equal resistance. Cifarelli, T. G. Mt. 36 (1898) 183-. angles with straight lines. Monteiro, A. S. G. Teix. J. Sc. 1 (1878) 81-. angularly related, theorems. Holditch, H, QJ. Mth. 3 (1860) 271-. apses. Longchamps, G. de. As. Fr. C. E. (1891) (Pt. 2) 23-. arcs, Graves' and Chasles's theorem, generali- sation. Humbert, G. C. E. 104 (1887) 1826- ; N. A. Mth. 7 (1888) 5-. with arcs between same radii vec tores equal. Fuss, N. [1801] St Pet. Ac. Sc. Mm. 2 (1810) 29-. asymptotes from polar equation. Nell, A. M. Grunert Arch. 15 (1850) 315-. case where curve is its own evolute. Binet, J. P. M. Liouv. J. Mth. 6 (1841) 61-. pedal. Haton de la Goupil- liere, J. N. Liouv. J. Mth. 11 (1866) 329-. s varies as tan 6. Fuss, N. [1818-26] St Pet. Ac. Sc. Mm. 9 (1824) 115- ; 11 (1830) 314-. cases where p varies as r 2 . Euler, L. St Pet. Ac. So. Mm. 9 (1824) 47-. normal. Farcy, A. N. A. Mth. 3 (1844) 528-. (1) y n ; (2) r; (3) p; (4) s\ Hill, T. Am. As. P. 22 (1873) 31-. r is proportional to projection of p upon r. Du Chatenet, M. N. A. Mth. 5 (1886) 233-. s varies as ylx. Velde, A. Arch. Mth. Ps. 14 (1896) 200-. characteristic equation 1 , interpretation, cases. Brocard, H. P. P. B. (xn) Mathesis 2 (1882) 25-, 49-. classes, two remarkable. Laisant, C. A. As. Fr. C. E. (1890) (Pt. 2) 74-. 8430 closed, belonging to same family, criterion. Crocchi, L. G. Mt. 10 (1872) 302-. , mean curvature, formulae. Lindelof, L. L. [1870] Helsingf. Acta 9 (1871) 361-. , parallels. Hoppe, R. Arch. Mth. Ps. 66 (1881) 46-. with constant polar tangent. Giard, . N. A. Mth. 1 (1862) 70-. continuous development, extension of theorem of Bernoulli. Mainardi, G. A. Sc. Lomb. Yen. 7 (1837) 29-. convexity and concavity, theorem. Gerono, G. C. N. A. Mth. 18 (1859) 397. cubic, curvature. Hoist, E. Arch. Mth. Ntvd. 7 (1882) 109-, 177-. with several cusps. Laurent, H. N. A. Mth. 19 (1860) 210-. cutting at given angle. Hoppe, JR. (vi Adds.) Berl. Pol. Gs. Vh. 20 (1859) 135-. given curve at angle which is given function of point. Franconi, E. G. Mt. 38 (1900) 232-. cyclic, curvature, and projective geometry. Schwarz, A. Mh. Mth. Ps. 10 (1899) 250- ; 11 (1900) 71-. , of direction. Humbert, G. Liouv. J. Mth. 4 (1888) 129-. denned by differential equation, 1st order, be- tween two variables, systems. Dyck, W. Munch. Ak. Sb. 21 (1892) 23-; 22(1893)101-. intrinsic relation. Pirondini, G. G. Mt. 30 (1892) 326-; 31 (1893) 51. law of rectification. Ocagne, M. d\ N. A. Mth. 10 (1891) 82-, 133. property of centre of gravity of area. Collignon, E. As. Fr. C. E. (1886) (Pt. 2) 53-. relation between ordinate area and sub- tangent. Schlomilch, O. Grunert Arch. 10 (1847) 215-. "derived." Druckenmiiller, N. Crelle J. 26 (1843) 1-. " ," algebraic and differential equations. Tortolini, B. A. Mt. 6 (1864) 88-. from two plane curves. Laisant, C. A. Par. S. Mth. Bll. 16 (1888) 172-. determination, application of method of para- meters. Moseley, H. Thomson A. Ph. 11 (1826) 321-. by tangential distances from given curves. Ocagne, M. d'. N. A. Mth. 17 (1898) 115-. of direction of 3rd class. Laguerre, E. N. A. Mth. 2 (1883) 97-. for which to - L Appell, P. N. A. Mth. 15 (1896) 491-. dividing circular arcs through 2 fixed points into n equal parts. Burali-Forti, C. G. Mt. 27 (1889) 153-. double tangents. Cayley, A. Phil. Trans. (1859) 193-. enveloped, theory. Heilermann, . Grunert Arch. 24 (1855) 438-. equidistant, properties. Mahistre, G. A. [1855] (xn) Lille S. Mm. 3 (1857) 1-. with evolutes and involutes equal inter se, inte- grals. Aoust, . C. E. 84 (1877) 385-; Par. S. Mth. Bll. 7 (1879) 143-. 600 8430 Applications of Calculus to Plane Curves 8430 and their evolutes, ratio of curvatures. Ruffini, F. P. Bologna Ac. Sc. Mm. 6 (1884) 71ft-, families, remarkable. Cesdro, E. N. A. Mth. 7 (1888) 171- ; 19 (1900) 489-. family denned by algebraic differential equation of 1st order. Fouret, G. C. B. 86 (1878) 586-. represented by differential equation. Amigues, E. As. Fr. C. B. 8 (1879) 117-. generalisation of Graves' theorem on confocal conies. Zehfuss, G. Schlomilch Z. 5 (1860) 210-. given by tangential equation, curvature. Pain- vin, L. Bll. Sc. Mth. As. 3 (1872) 174-. having given relation between arc and area in polar coordinates. Collignon, E. As. Fr. C. E. 11 (1882) 69-. at infinity, curvature. Torroja, E. Fschr. Mth. (1893-94) 1077-. intrinsic equation, and application. Whewell, W. Camb. Ph. S. T. 8 (1849) 659- ; 9 (1856) 150-. equations. Gergonne, J. D. Gergonne A. Mth. 4 (1813-14) 42-. Lame's, curvature. Fouret, G. C. B. 110 (1890) 843-. limiting, of given manifoldness of curves. Ascoli, G. Bm. E. Ac. Line. Mm. 18 (1883) 521-. maximum and minimum inclinations to radius vector. Exner, K. Wien Sb. 57 (1868) (Ab. 2) 75-. 2, meeting tangents of 3rd at related angles, curvature. Habich, E. Par. S. Mth. Bll. 13 (1885) 201-. neighbouring, 2, order of approach between finite limits. Leaute, H. C. E. 86 (1878) 1537-. orthogonal, curvature, theory. Eeecli, F. Par. EC. Pol. J. Cah. 41 (1865) 201-. orthotomic, of system of lines in plane. Cayley, A. Mess. Mth. 22 (1893) 45-. osculating. Wolstenholme, J. QJ. Mth. 11 (1871) 210-. . Pirondini, G. G. Mt. 26 (1888) 257-, 380. , applications of linear quasi-inversion to. Servais, C. Mathesis 8 (1888) 28-. , of triply infinite system. Krey, H. [1875] Mth. A. 10 (1876) 221-. parabolic and other, curvatures. Machovec, F. Prag Sb. (1892) (Mth.-Nt.) 77-. parallel. Peter sen, J. P. C. (xn) Ts. Mth. 1 (1871) 154-. , theorems embracing those relating to. Steiner, J. Crelle J. 32 (1846) 75-. , theory. Hoppe, E. Crelle J. 55 (1858) 95-. , . Cantor, M. Schlomilch Z. 5 (1860) 219-. ' ' periplegmatic ' ' , given by y ~-^ = - X. Gylden , H. Stockh. Ofv. (1892) 69-. polar reciprocal, and radius of curvature. Hir- man, G. N. Vars. S. Nt. Tr. (1891-92) (C. R., Ps. C.) No. 8, 6. of "raccordement," bicircular for 2 straight lines. Fontes, . Toul. Ac. Sc. Mm. 4 (1892) 375-. -- , with 3 centres. Fichera, F. Catania Ac. Gioen. At. 20 (1888) 323-. , several centres. Du Hays, . Liouv. J. Mth. 15 (1850) 241-. -- , --- . Revellat, J. P. C. B. 77 (1873) 434-. -- , cubic parabola for easing. Merfield, C. J. N. S. W. B. S. J. 29 (1895) 51-. -- , -- as transition curve. Merfield, C. J. N. S. W. E. S. J. 31 (1897) Ivi (bis)-. -- for junctions. Brianchon, C. J. Par. EC. Pol. J. 19" cah. (1813) 187-. -- , lemniscate for junctions. Oostinjer, H. Brux. A. Tr. Pbl. 4 (1899) 335-. -- , method of easing. Shellshear, W. N. S. W. B. S. J. 22 (1889) 89-. -- , parabolic. Verdugo, J. A. Santiago de Chile Un. A. 55 (*1879) 123-. -- , curves for junction of 2 circular arcs. Bernis, . As. Fr. C. B. (1892) (Pt. 2) 212-. -- , --- junctions. Mene trier, L. S. (vm) A. Cond. Pon. Chauss. 3 (1859) 14-. rectifiable, deduction of equation. Azzarelli, M. [1877] Em. N. Line. At. 31 (1878) 6-. with relation of 2nd degree between arc and coordinates. Hoppe, R. [1862] Crelle J. 62 (1863) 193-. representing integral of w 2 + ( ~ e J 2 = w 2 , property. Hirst, T. A. Liouv. J. Mth. 2 (1857) 392. rhizic (i.e. fulfilling condition Rankine, W. J. M. B. S. P. 15 (1867) 468-. . Walton, W. QJ. Mth. 11 (1871) 91-. , curvature at multiple points. Walton, W. QJ. Mth. 11 (1871) 274-. , spoke asymptotes. Walton, W. [1870] QJ. Mth. 11 (1871) 200-. satisfying certain condition of maximum or minimum, common properties. Vicaire, E. C. E. 106 (1888) 456-. scale increasing or diminishing indefinitely. Swellengrebel, J. G. H. Grunert Arch. 16 (1851) 419-. with similar evolutes. Ettingshausen, A. von. Baumgartner Z. 9 (1831) 178-. . Tucker, R. Mess. Mth. 3 (1866) 191-. and straight lines, agreement. Franci&re, . A. Cond. Pon. Chauss. 17 (1873) 137-. surfaces, curvature, and point-transforma- tions. Mehmke, R. Z. Mth. Ps. 36 (1891) 206-. symmetric, and images. Longchamps, G. de. N. A. Mth. 18 (1899) 373-. systems, differential equations appertaining to. Czuber, E. Wien Ak. Sb. 102 (1893) (Ab. 2a) 1141-. tangential, of conies. Hochheim, A. [1869] Z. Mth. Ps. 15 (1870) 377-. tangents, lines cutting at constant angle. Boklen, 0. Arch. Mth. Ps. 43 (1865) 14-. , theorems. Marchesano, S. Tortolini A. 4 (1853) 161-. theory. Jdinek, C. Wien SB. (1849) (Ab. 2) 19-. 601 8430 Curves Envelopes Evolutes 8430 triangular, and symmetric tetrahedral curves, curvature. Fouret, G. Par. S. Mth. Bll. 20 (1892) 60-. s =f ( ~ j , construction. Plana, G. Mod. Mm. S. It. 16 (1813) 361-. r =a m cosm0. Allegret, [A.~\ N. A. Mth. 11 (1872) 162-. r m cosm6 = a m . Bassani, A. G. Mt. 24 (1886) 23-. I [(pla) n - l]~*dp, and its evolute. Cesdro, E. Fschr. Mth. (1892) 708. y= derived from y=f(x). Azzarelli, M. Em. N. Line. At. 46 (1893) 186-. Curvilinear angles. Miguel, A. Liouv. J. Mth. 9 (1844) 20-; 10 (1845) 347-; 11 (1846) 65-. "Curvital" functions, sin (y, x)=y- I ydx* + I ydx 4 etc. Gavarni, . C. B. 60 (1865) 1079-. Diameters, tangents at points on. Serret, P. C. E. 82 (1876) 67-. Differential equation L (xdy - ydx) - Mdy + M2z = 0(p=l,r=l). Fouret, G. C. B. 78 (1874) 1693-. Direction of curvature in polar coordinates. Schulten, N. G. of. [1843] (vm) Helsingf. Acta 6 (1861) 43-. Dual curvature and duals of evolutes and in- volutes. Je/ery, H. M. [1869] QJ. Mth. 10 (1870) 321-. Envelopes. Fergola, E. Mod. S. It. Mm. 24 (1850) 290-. Cayley, A. Mess. Mth. 1 (1872) 3-. Veeehio, A. G. Mt. 11 (1873) 318-. Brunei, . Bordeaux S. Sc. Mm. 4 (1888) xi. Aoust's problem. Hoppe, JR. Arch. Mth. Ps. 66 (1881) 386-. of carried curves, Savary's theorem. Resal, H. A. N. A. Mth. 1 (1882) 7-. case. Walton, W. [1867] QJ. Mth. 9 (1868) 142-. . Tognoli, 0. G. Mt. 11 (1873) 376-. . Fais, A. G. Mt. 12 (1874) 150-. . Baehr,G.F.W. N. A. Mth. 20 (1881) 250- . , special. Gilbert, . Brux. S. Sc. A. 15 (1891) (Pt.,1) 37-. of circles. Eibaucour, A. (xi) Par. S. Phlm. Bll. 4 (1867) 88-; ,5 (1868) 30-. decomposition. Saltel, L. '[1881] Bordeaux S. Sc. Mm. 4 (1882) 443-. determination of class. Oiasles, M. C. B. 81 (1875) 757-. c-discriminant as. Macdonald, J. A. Edinb. B. S. T. 39 (1900) 27-. imaginary, of conjugates of plane locus. Marie, M. C. B. 75 (1872) 7-. Lagrange's methods. Flauti, V. [1848] Mod. S. It. Mm. 24 (1850) 251-. of line with ends on 2 straight lines, con- sequences. Terquem, O. N. A. Mth. 3 (1844) 182-. of line with 2 fixed lines cutting off constant intercept. Gomes Teixeira, F. Fschr. Mth. (1899) 530. normals to, and envelopes with doubly infinite set of straight lines as normals. Elliott, E. B. [1879] Mess. Mth. 9 (1880) 85-. osculating. Moller, J. Lund. Un. Acta 24 (1887-88) (Mth., No. 1) 4 pp. of pedal line of triangle. Ferrers, N. M. [1866] QJ. Mth. 8 (1867) 209- ; 9 (1868) 147-. . Besant, W. H. [1868] QJ. Mth. 10 (1870) 110-. plane series of curves, order. Zimmermann, O. Crelle J. Mth. 116 (1896) 10-. polars of curve. Hoppe, E. [1860] Crelle J. 58 (1861) 374-. singularities. Henrici, 0. L. Mth. S. P. 2 (1869) 177-. of straight line. Desmarest, E. N. A. Mth. 3 (1844) 154-. . SaeeJii, G. N. A. Mth. 2 (1863) 31-. . Genese, R. W. QJ. Mth. 13 (1875) 260-. , application to centre of curvature of conies. Mannheim, A. N. A. Mth. 16 (1857) 322-. joining ends of hands of watch. Gilbert, L. P. (xn) Brux. S. Sc. A. 4 (1880) (Pt. 2) 159-. making equal chords in 2 circles. Taylor, H. M. Mess. Mth. 4 (1875) 163. moving with ends on 2 other lines. Hyde, E. W. Des Moines Anal. 1 (1874) 60-. such that sum of squares of distances of number of points from it is constant. Sykora, A. Prag Sb. (1881) 23-. lines, cases. Ocagne, M. d'. N. A. Mth. 2 (1883) 252-. system of curves. Laisant, C. A. N. A. Mth. 13 (1874) 571-. . Bosi, L. G. Mt. 35 (1897) 120-, 353. tangent to 2 variable circles. Terre, E. N. A. Mth. 10 (1851) 340-. theorem. Laurent, H. N. A. Mth. 13 (1874) 273-. theorems. Magnus, L. J. Gergonne A. Mth. 16 (1825-26) 80-. theory. Petersen, J. P. C. (xn) Ts. Mth. 2 (1872). 81-. . Genese, R. W. Mess. Mth. 7 (1878) 61-. . Cayley, A. Mess. Mth. 7 (1878) 62-. , general, connexion with ^-discriminant of differential equation of 1st order. Chrystal, G. Edinb. B. S. T. 38 (1897) 803-. , problem. Wagner, C. Arch. Mth. Ps. 56 (1874) 1-. Evolutes. Wolfers, J. P. Grunert Arch. 4 (1844) 135-. Dunesme, M. C. B. 35 (1852) 518-. Enneper, A. Schlomilch Z. 7 (1862) 120-. Nicolaides, N. Les Mondes 8 (1865) 621-. Bassani, A. G. Mt. 24 (1886) 371-. Jamet, V. N. A. Mth. 9 (1890) 496-. 8430 Bvolutes Plane Curves (Calculus) 8430 of algebraic curves. Henneberg,L. Ziir. Vjschr. 21 (1876) 71-. analogues to. Halphen, G. H. Liouv. J. Mth. 2 (1876) 87-. centres of curvature. Mannheim, A. L. Mth. S. P. 20 (1889) 241-. . Ocagne, M. d\ Par. S. Mth. Bll. 20 (1892) 49-. and centres of curvature of conies. Machovec, F. Casopis 16 (1887) 235-; Fschr. Mth. (1887) 722-. of conies. Terquem, O. N. A. Mth. 6 (1847) 205-. , centres of curvature. Machovec, F. Casopis 20 (1891) 97-; Fschr. Mth. (1891) 672. cubic curves. Jeffery, H. M. [1870] QJ. Mth. 11 (1871) 78-, 145-. curves of 4th degree, equations. Tucci, F. P. Nap. Ed. 5 (1846) 34-. nth degree, determination of degree and class. MUinowski, . Z. Mth. Ps. 19 (1874) 182-. , identity. Pirondini, G. N. A. Mth. 5 (1886) 460-. ellipse. Gamma [7]. (vi Adds.) Camb. Mth. J. 1 (1859) 47-. . Lignieres, , & Trenquelleon, C. de. N. A. Mth. 2 1863) 85-. . Laguerre, E. C. E. 84 (1877) 224-. , equation. Taillier, L. N. A. Mth. 2 (1863) 143-. , theorems. Pitt, G. N. A. Mth. 11 (1852) 412-. elliptic. Amiot, B. C. E. 21 (1845) 348-. imperfect, conjugate. Aoust, L. C. E. 30 (1850) 523-. , elongated or contracted. Levi, D. Tor. At. Ac. Sc. 4 (1868-69) 576-. and involutes. Maggi, P. G. A. Sc. Lomb. Yen. 8 (1838) 161-, 249-; 9 (1839) 187-. . Puiseux, V. Liouv. J. Mth. 9 (1844) 377-. . Brioschi, F. Tortolini A. 4 j(1853) 50-. (nth). Holditch, H. QJ. Mth. 3 (I860) 236-. . Krantz, H. J. N. Arch. Wisk. 10 (1884) 97-. , theory. Beltrami, E. (vii) A. Mt. 4 (1861) 257-. from involutes. Hoppe, E. Grunert Arch. 25 (1855) 125-. and osculating curves and trajectories, reduced to a single problem. Toblini, G. Poligrafo 3 (1830) 14-, 265-. of reciprocal curves, relation between radii of curvature. Godefroy, E. Par. S. Mth. Bll. 19 (1891) 109-. referred to oblique axes. Bary, . Gergonne A. Mth. 21 (1830-31) 249-. successive. Sharp, W. J. C. Mess. Mth. 9 (1880) 95-. . Pirondini, G. Nap. Ed. 30 (1891) 139-. , centre of mean distance of centres of curvature. Haton de la Goupilliere, . C. E. 115 (1892) 856-. , . Fouret, G. C. E. 115 (1892) 1055-. , property. Anon, (vi 898) N. A. Mth. 2 (1863) 347-. symmetric bicircular quartic. Hart, H. QJ. Mth. 18 (1882) 382-. theory. Catalan, E. C. C. E. 45 (1857) 135-. . Cayley, A. Ph. Mg. 29 (1865) 344-. of a 2 x' i + b' 2 y 2 =(x 2 + y 2 )' 2 . Perrodil, de. N. A. Mth. 6 (1847) 275-. fan- Formula I pdu of Legendre, generalisation. CavallinC. B. S. (xn) Ts. Mth. 6 (1882) Hyperbolic, parabolic and elliptic curvature. Bergsten, N. J. Stockh. Ak. Hndl. 30 (1809) Implexes of surfaces, denned by two charac- teristics, properties. Fouret, G. C. E. 79 (1874) 689-. "Infinitesimal analysis of plane curves," by Aoust, and a priority claim. Habich, E. Les Mondes 50 (1879) 150-. geometry of curves. Sobotka, J. Prag Sb. (1898) (Mth.-Nt.) No. 26, 38 pp. , auxiliary curves. Ocagne, M. d\ Lisb. J. Sc. Mth. 12 (1888) 193- ; 2 (1892) 227-. integral curves. Sobotka, J. Wien Ak. Sb. 107 (1898) (Ab. 2a) 569-. Inflexions. Bordoni, A. Mil. G. I. Lomb. 3 (1842) 314-. Intrinsic equations. Wolffing, E. Bb. Mth. 1 (1900) 142-. , discontinuity. Walton, W. QJ. Mth. 5 (1862) 260-. and properties of Cassini's ovals and conies. Cesdro, E. Mathesis 11 (1891) 51-. of parabolas. Eecoq. A. N. A. Mth. 4 (1865) some curves. Cesdro, E. 360-. , equation. Turnbull, J. C. QJ. Mth. 5 (1862) 152. and parallel curves. Cayley, A. QJ. Mth. 11 (1871) 183-. radii of curvature. Dubois-Ayme, . [1807] Turin Mm. Ac. (1809-10) (pte. 2) , application of theory of limits. Azzarelli, M. Em. N. Line. At. 33 (1880) Mathesis 9 (1889) 209-. , use. Cesdro, E. N. A. Mth. 5 127-, 352. Invariants of n points. Lovett, E. O. [1897] N. Y. Am. Mth. S. Bll. 4 (1898) 58-. Inverse method of tangents. Fuss, N. [1799] St Pet. Ac. Se. Mm. 1 (1809) 88-, 119-. _ _ . Tortolini, B. G. Arcad. 79 (1839) 32-; 80 (1839) 124. , problem. Minich, S. E. A. Sc. Lomb. Yen. 7 (1837) 57-, 171-. 603 8430 Plane Curves (Calculus) Pedals 8430 Inverse method of tangents, problems. Fuss, P. H. [1822] St Pet. Ac. Sc. Mm. 10 (1826) 130-. , . Colligrwn, E. N. A. Mth. 18 (1899) 488-; 19 (1900) 11-. Involute of given curves. Rouquet, V. Toul. Fac. Sc. A. 1 (1899) 79-. Involutes. Dienger, J. Grunert Arch. 20 (1853) 69-. , oblique, any order, integrals. Aoust, . C. E. 85 (1877) 609-. of point. Cesdro, E. Mathesis 8 (1888) 36-. , problem. Pirondini, G. A. Mt. 17 (1889- 90) 65-. , successive. Anon, (vi 530) Gergonne A. Mth. 9 (1818-19) 73-. , theory. David, C. A. [1862] (xn) Lille S. Mm. 9 (1863) 499-. Loci, general equations and applications. Callegari, P. Em. At. 7 (1853-54) 179-, 261-. Macrograph. Morgan, W. B. Franklin I. J. 48 (1864) 16-. Maximum and minimum properties of curves. Sonine, N. J. Fschr. Mth. (1887) 359-. Method of investigating curves, and application to evolutes and caustics. Watson, W. Am. As. P. (1859) 151-. Multiple points. Droop, H. R. QJ. Mth. 2 (1858) 149-. Multipolar coordinates, relations. Ghysens, E. Brux. Ac. Bll. 45 (1878) 231-. Networks with equal invariants, and their asymptotic lines. Koenigs, G. C. E. 114 (1892) 55-, 728-. Normal and centre of curvature of "radial " of curve. Sucharda, A. Prag Ceske Ak. Fr. Jos. Ez. (Tfida 2) 6 (1897) Art. 25, 3 pp.; Prag Fr. Jos. Ac. Sc. Bll. (Mth.-Nt.) 4 (1897) (Pt. 2) 100-. of curvature. Whitlock, G. C. Silliman J. 16 (1853) 231. Osculating conies. Amiot, B. Par. S. Phlm. PV. (1846) 144-. . Lundberg, E. Ts. Mt. Fys. 4 (1871) 241-. . Enneper, A. Z. Mth. Ps. 19 (1874) 138-. . Dewulf, E. Mathesis 10 (1890) 55-. . Pirondini, G. G. Teix. J. Sc. 11 (1892) 9-. of cubics. Durege, H. Prag Sb. (1870) (pt. 2) 47-. .' Servais, C. Brux. Ac. Bll. 23 (1892) 522-. etc., theorem of Steiner. August, F. [1867] Crelle J. 68 (1868) 235-. and problem of Carnot's. Minich, S. R. Yen. Mm. I. 6 (1856) 111-. parabola. Hoppe, R. Arch. Mth. Ps. 12 (1894) 168-. , construction at any point of curve. Koenigs, G. N. A. Mth. 20 (1881) 11-. , given curve. Ocagne, M. d'. N. A. Mth. 11 (1892) 326-. parabolas and differential functions un- affected by change of axes. Ampere, A. M.. [1803] Par. 6c. Pol. J. 7 (1808) 159-. Osculation. Cesdro, E. N. A. Mth. 9 (1890) 143-. , formula. Busse, F. G. von. Gilbert A. 71 (1822) 91-. , 5-pointic. Cayley,A. Phil. Trans. (1859) 371-. Osculations of higher orders. Dupin, C. C. E. 32 (1851) 569-. Pedal coordinates. Purkiss, H. J. [1865] Mess. Mth. 3 (1866) 83-. Pedals. Drobisch, M. W. Leip. B. 9 (1857) 49-. Bacaloglo, E. Grunert Arch. 35 (1860) 41-. Cayley, A. (vii) Ph. Mg. 26 (1863) 20-. Albrich, (Prof.) C. Arch. Mth. Ps. 52 (1871) 56-. Juel, C. S. (xn) Ts. Mth. 3 (1873) 177- ; 4 (1874) 9-. Reuschle, C. (jun.) Z. Mth. Ps. 21 (1876) 139-. Hicks, W. M. Mess. Mth. 6 (1877) 94-. Johnson, W. W. Des Moines Anal. 4 (1877) 143-. Laisant, C. A. As. Fr. C. E. (1882) 104-. Weinmeister, J. P. Z. Mth. Ps. 28 (1883) 256. Ruffini, F. P. Bologna Ac. Sc. Mm. 2 (1891) 123-. Petrini, H. Stockh. Ofv. (1893) 613-. and anti-pedals. Azzarelli, M. Em. N. Line. At. 27 (1874) 127-. of circle. Richter, 0. Z. Mth. Ps. 34 (1889) 338-. conies. Wolf, R. Crelle J. 20 (1840) 88-. . Schiltte, . Grunert Arch. 20 (1853) 175-. . Roberts, S. [1870] L. Mth. S. P. 3 (1869-71) 88-. construction of circle of curvature. Weyr, E. Wien Sb. 59 (1869) (Ab. 2) 169-. and director curves. Raabe, J. L. Crelle J. 50 (1855) 189-. , relations to original curve and solu- tions by singular integrals. Raabe, J. L. Crelle J. 48 (1854) 105-. of ellipse. Jefabek, V. Mathesis 16 (1896) 15. , determination of real foci and of vector equation. Wolstenholme, (Rev.) J. L. Mth. S. P. 13 (1881-82) 70-. and hyperbola, curvature and evolute. Tortolini, B. Palomba Eac. 1 (1845) 278-. equations and properties. Tortolini, B. Pa- lomba Eac. 3 (1847) 305-. first, curvature, and polar reciprocal of curve. Servais, C. Mathesis 11 (1891) 84-. of hyperbola, construction of arcs with rectifi- able difference. Wiegers, C. Schlomilch Z. 3 (1858) 308-. inverse. Dieu, T. N. A. Mth. 13 (1854) 259-. law of formation, generalisation. Janisch, E. Arch. Mth. Ps. 8 (1890) 171- ; 9 (1890) vii. of logarithmic spiral. Amato, V. G. Mt. 38 (1900) 149-. negative. Tortolini, B. N. A. Mth. 5 (1846) 365-. 604 8430 Planes Curves (Calculus) negative, first, of central conic, properties, de- monstration. Malet, J. G. [1878] Ir. Ac. T. 26 (1879) 431-. , theory. Ameseder, A. Arch. Mth. Ps. 64 (1879) 164-. oblique. Pellet, E. Bll. Sc. Mth. As. 3 (1872) 278-. any order. Wetzig, F. Schlomilch Z. 4 (1859) 319- ; 5 (1860) 1-, 81-. and orthogonal trajectories. Hochheim, A. Arch. Mth. Ps. 47 (1867) 121-. pedal surfaces. Roberts, W. C. E. 49 (1859) 742-. . Bacaloglo, E. Grunert Arch. 35 (1860) 41-. (Bacaloglo). Magener, A. Grunert Arch. 36 (1861) 375-. property. Besant, W. H. [1870] QJ. Mth. 11 (1871) 38-. . MacMahon, P. A. Mess. Mth. 10 (1881) 190-. and reciprocal curves, curvature. Steede, B. H. [1898] Ir. Ac. P. 5 (1898-1900) 124-. successive. Barisien, E. N. A. Mth. 14 (1895) 89-, 157-, 207-, 233-, 463-. , radii of curvature. Niewenglowski, B. C. B. 84 (1877) 765-. , theory. Roberts, W. N. A. Mth. 3 (1864) 80-. of system of conies. Amstein, H. Laus. S. Yd. Bll. 22 (1887) 87-. tangent constructions for. Janisch, E. Arch. Mth. Ps. 9 (1890) 196-, viii. theorem. [Salvatore-~\Dino, N. G. Mt.. 1 (1863) 217-. theory. Tortolini, B. [1870] Bm. At. N. Line. 24 (1871) 63-. Plane sections of cones, formula relating to. Guimaraes, R. As. Fr. C. B. (1893) (Pt. 2) 93-. surfaces, curvature. Exner, K. Z. Mth. Ps. 17 (1872) 416-. Points of inflexion. Hesse, L. 0. Crelle J. 41 (1851) 272-. of cubics. Hesse, L. O. Crelle J. 28 (1844) 97- ; 38 (1849) 257-. curve corresponding to plane or skew curve, traced on developable surface. Olivier, T. Par. EC. Pol. J. cah. 22 (1833) 78-. and cusps. Jeffery, H. M. [1869] QJ. Mth. 10 (1870) 232-. in development of plane section of cone. Balitrand, F. N. A. Mth. 15 (1896) 65-. , infinitesimal space round. Long- champs, G. de. As. Fr. C. E. (1893) (Pt. 2) 277-. , symmetrical investigation. Walton, W. Camb. Mth. J. 4 (1845) 13-. , visible, and curvature at multiple points. Walton, W. QJ. Mth. 6 (1863) 204-. singular curvature. Walker, E. QJ. Mth. 2 (1858) 107-. Radii of Curvature 8430 Polar coordinates, formulae, f, , . d (p^sin (n+ 1) o)~l tan (u + na)=- r j t ~ l > ,( ( etc. L d (p n+l cos (ra + 1) a)J Lucas, F. Les Mondes 18 (1868) 611-. Polars of point of curve of nth degree, curva- ture. Servais, C. Brux. Ac. Bll. 21 (1891) 362-. Radii of curvature. Moore, W. Nicholson J. 21 (1808) 256-. Liouville, J. Liouv. J. Mth. 9 (1844) 435. Schellbach, C. H. Crelle J. 45 (1853) 265-. Schlnmilch, O. Schlomilch Z. 2 (1857) 273-. Schlechter, . Grunert Arch. 31 (1858) 327-. Mannheim, A. N. A. Mth. 1 (1862) 123-. Haag, . (x) Par. S. Phlm. Bll. 3 (1866) Soufflet, . Les Mondes 17 (1868) 110- ; 24 (1871) 337-. Laisant, C. A. N. A. Mth. 13 (1874) 367-. Wasteels, C. Mathesis 11 (1891) 268-. application of differential calculus. Le Barbier, . (vm) Gergonne A. Mth. 22 (1831) 31-. of caustics at cusps. Breton (de Champ), . N. A. Mth. 4 (1865) 25-. and centres of curvature. Lamarle, E. Brux. Ac. Bll. 2 (1857) 33-, 307-, 528- ; 3 (1857) 295-; 5 (1858) 5-; 6 (1859) 11-. circles of curvature. Schellbach, C. H. N. A. Mth. 12 (1853) 390-. of class of curves. Godefroy, R. N. A. Mth. 5 (1886) 272-. , geometrical determination. Transon, (Prof.) A. Liouv. J. Mth. 10 (1845) 147-. collinear curves, relation between. Geisen- heimer, L. Z. Mth. Ps. 25 (1880) 214-. , reciprocal, and inverse curves, relation between. Geisenheimer, L. Z. Mth. Ps. 25 (1880) 300-. construction for symmetric triangular curves, etc. Fouret, G. C. E. 110 (1890) 778-, 876. of curve and its polar reciprocal, relation. Chemin, J. N. A. Mth. 6 (1867) 49-. determination. Laguerre, E. (x) Par. S. Phlm. Bll. 4 (1867) 15-. in parallel point-coordinates. Ocagne, M. d\ Brux. Ac. Bll. 21 (1891) 220-. expression in dipolar coordinates. Frost, W. Mess. Mth. 10 (1881) 18-. polar coordinates. Smith, Arch. Camb. Mth. J. 2 (1841) 191-. , and curves, equation of which is r k =a k sink0. Unferdinger, F. Arch. Mth. Ps. 51 (1870) 72-. for, in Salmon's Higher Plane Curves, p. 83, reduction by determinants. Story, W. E. [1880] (xn) J. H. Un. Cir. [1] (1882) 35. expressions for. Wolfers, J. P. Grunert Arch. 9 (1847) 60-. in point- and tangential coordinates, relation between. Ocagne, M. d\ Par. S. Mth. Bll. 19 (1891) 26-. of Lame's curves. Godefroy, R. Par. S. Mth. Bll. 21 (1893) 20-. 605 8430 Plane Curves (Calculus) Trajectories 8430 of normal section of given surface determined on mechanical considerations. Kolacek, F. Casopis 24 (1895) 225- ; Fschr. Mth. (1895) 698. and normals (Chasles-Transon construction). Boklen, O. Schlomilch Z. 3 (1858) 252-. , and tangents of certain system of curves, construction. Stegmann, F. Grunert Arch. 7 (1846) 48-. , unknown curves, construction. Pressel, W. Grunert Arch. 4 (1844) 337-. of plane sections of anallagmatic surfaces. Laguerre, E. (x) Par. S. Phlm. Bll. 7 (1871) 241-. at point of inflexion, or at cusp. Breton [de Champ], P. N. A. Mth. 13 (1854) 127-. and polar sub-normals. Ghysens, E. Brux. Ac. Bll. 43 (1877) 544-. properties, transformation by reciprocal polars. Mannheim, A. Liouv. J. Mth. 11 (1866) 193-. of reciprocally polar curves, relation between. Ocagne, M. d\ Par. EC. Norm. A. 4 (1887) 313-. successive, of certain curves. Godefroy, E. C. B. 117 (1893) 1062- ; Par. EC. Pol. J. 2 (1897) 19-. , theory of curves deduced from. Timmer- mans, J. A. Lille Mm. S. (1827-28) 46-. of systems of curves, relation between. Chemin, J. N. A. Mth. 7 (1868) 120-. Badius vector and tangent, angle between, application to optics. Piani, D. [1850] Bologna Mm. Ac. Sc. 4 (1853) 309-. Semicurvilinear coordinates, system. Petro- vitch, M. Prag Sb. (1898) (Mth.-Nt.) No. 7, 21pp. Sextactic points. Cayley, A. Phil. Trans. 155 (1865) 545-. __ _. Spottiswoode, W. Phil. Trans. 155 (1865) 653-. . Battaglini, G. Bm. B. Ac. Line. Bd. 4 (1888) (Sem. 2) 238-. of algebraic curves. Gerbaldi, . Palermo Cir. Mt. Ed. 4 (1890) 65-. Singular point, motion. Jacobi, C. G. J, Crelle J. 24 (1842) 5-. points. Cremona, G. F. [1820] (vn) Mod. Mm. Ac. Sc. 1 (1833) (pte. 2) 79-. , form, etc. Matthiessen, L. Schlomilch Z. 8 (1863) 451-. Tac-loci. Workman, W. P. [1882] Mess. Mth. 12 (1883) 21-. Tangent and centre of curvature of certain curve, constructions. Sucharda, A. Prag C~eske Ak. Fr. Jos. Ez. (Trida 2) 8 (1899) Art. 40, 6 pp.; Fschr. Mth. (1899) 471. normal of curve, locus derived from. Bassani, A. G. Mt. 22 (1884) 211-. Tangential triangle, infinitesimal ratio of seg- ment. Voller, . Grunert Arch. 31 (1858) 449-; 33 (1859)433-. , . Weiler, A. Grunert Arch. 32 (1859) 418-. Tangents, normals, curvature. Gaudard, J. A. Gen. Civ. 2 (1873) 484-. , , etc., of curves in general, and of conies. Dostor, G. Arch. Mth. Ps. 51 (1870) 129-. Theorem. If Q = x cos + y sin 6-f (6), then 6' = is normal to envelope of = 0, 6" = is normal to evolute of envelope, etc. Tail, P. G. QJ. Mth. 3 (1860) 364-. Theorems of Fouret and Jamet. Cesdro, E. Mathesis 13 (1893) 217-. Theory of curvature. Tichomandrickij , M. A. Kharkov Mth. S. Com. (1886) 33-. and kinematics of rigid body, con- nexion between; Landsberg, G. Crelle J. Mth. 118 (1897) 163-. Trajectories. of confocal conies, plane and spherical. Roberts, W. A. Mt. 6 (1864) 28-. determination, so that the two radii of curvature at points of intersection are in given relation. Hansen, C. (xii) Ts. Mth. 2 (1866) 117-. isogonal, areas, etc., of systems of curves. Ekama, H. N. Arch. Wisk. 1 (1895) 55-; Fschr. Mth. (1893-94) 1079. oblique, of system of confocal ellipses. Mukhopadhyay, A. [1887] Beng. As. S. J. 56 (Ft. 2) (1888) 117-. orthogonal. Rummer, E. E. Crelle J. 35 (1847) 5-. . Kiepert, L. Z. Mth. Ps. 17 (1872) 420-. , in bipolar coordinates. Baur, C. W. Z. Mth. Ps. 12 (1867) 430-. , . Varies, G. N. A. Mth. 13 (1894) 283-. , composed solely of conies, families. Appell, P. [1878] Arch. Mth. Ps. 63 (1879) 50-. , example in Boole's "Differential equa- tions." Glaisher, J. [1879] Mess. Mth. 9 (1880) 46-. and homofocal. Legoux, A. N. A. Mth. 20 (1881) 406-. , of surface. Euler, L. [1782] St Pet. Ac. Sc. Mm. 7 (1820) 33-. of points and envelopes of movable lines in plane. Ocagne, M. d'. C. B. 109 (1889) 959-; 110 (1890) 60. of moving figure, centres of curvature. Marcolongo, JR. Palermo Cir. Mt. Ed. 7 (1893) 29-. rigid body in motion, curvature. Schonfties, A. Liege S. Sc. Mm. 11 (1885) #o. 6, 9 pp., No. 9, 15 pp. in solid subject to 4 conditions, radius of curvature. Reveille, J. C.E. 104 (1887)1827-. on straight line. Ocagne, M. d'. As. Fr. C. E. (1889) (Ft. 2) 228-. moving in space. Mannheim, A. C. B. 76 (1873) 551-, 635-; (x) Par. S. Mth. Bll. 1 (1873) 106-. Transformation, ortho tangential. Cesdro, E. N. A. Mth. 8 (1889) 116-. , y' = -*L of curves. Colli- , (x) ' y f (x) gnon, E. As. Fr. C. B. (1897) (Pt. 2) 7-. x'-- 606 8440 Skew Curves (Calculus) Curvature 8440 Transformations, isogonal, curvature in. Laisant, C. A. Par. S. Mth. Bll. 15 (1887) 39-. , laws of curvature in. Transon, A. N. A. Mth. 8 (1869) 114-. Vector pedal equation of curve. Nanson, E. J. Mess. Mth. 28 (1899) 80-. 8440 Curvature of skew curves; other applications of the dif- ferential calculus to skew curves. (See also 7240, 8420.) Centre of curvature. Hotter, J. Lund. Un. Acta 21 (1885-86) (Mth., No. 2) 18 pp. Characteristic differential equation. Hoppe, E. Arch. Mth. Ps. 15 (1897) 244-. Cone-loxodrome as osculator of curve. Tesar, J. Prag Sb. (1886) (Mth.-Nt.) 347-. Contact and osculation. Pirondini, G. G. Teix. J. Sc. 10 (1891) 113-. Contingence angle. Rodrigues, (Dr) . [1813] Par. EC. Pol. Cor. 3 (1814-16) 36-. . Terquem, O. N. A. Mth. 4 (1845) 266-. Coordinates in terms of arc. Mathews, G. B. QJ. Mth. 26 (1893) 27-. Correspondence. Daug, H. T. Ups. Arsk. (1861) (Mth.) 56 pp. Curvature. Chelini, D. Palomba Eac. 1 (1845) 105-, 129-, 140-, 156- ; Tortolini A. 4 (1853) 337-. Spottiswoode, W. Camb. and Dubl. Mth. J. 9 (1854) 234-. Boklen, O. N. A. Mth. 19 (1860) 136-. Greer, H. R. QJ. Mth. 4 (1861) 183-. Routh, E. J. QJ. Mth. 7 (1866) 37-. Constable, E. C. Mess. Mth. 1 (1872) 50. Hart, H. Mess. Mth. 4 (1875) 172-. Stolz, O. Mh. Mth. Ps. 1 (1890) 433-. Zur Hammer, A. Arch. Mth. Ps 15 (1897) 14-. absolute, radius. Smith, Arch. Camb. Mth. J. 1 (1839) 92-. , . Ferrers, N. M. QJ. Mth. 2 (1858) 356. of curve, and locus of centres of osculating spheres. Daug, H. T. Stockh. Ofv. 15 (1858) 1-. curves on spherical surface. Fuss, N. [1807] St Pet. Ac. Sc. Mm. 2 (1810) 73-. surface. Beltrami, E. N. A. Mth. 4 (1865) 258-. . Issoly, P. N. A. Mth. 3 (1884) 522-. and surfaces. Stuyvaert, M. Brux. Mm. Cour. 8, 55 (1896-98) No. 6, 19 pp. , elements. Mangeot, S. Par. EC. Norm. A. 10 (1893) 87-. and deformation. Olivier, T. Par. EC. Pol. J. cah. 24 (1835) 61-, 252-. of intersection of 2 given surfaces. Binet, J. M. P. C. B. 19 (1844) 210-. of intersection of 2 quadrics. Tortolini, B. (vm) A. Mt. 5 (1863) 305-. kinematic analogy. Hoppe, . D. Nf. Tbl. (*1873) 118. treatment. Szily, K. Mth. Termt. Ets. 7 (1889) 140- ; Mth. Nt. B. Ung. 7 (1890) 220-. law, demonstration. Grunert, J. A. Arch. Mth. Ps. 42 (1864) 467-. of orthogonal projections of curve. Catalan, E. C. Liege S. Sc. Mm. 15 (1888) No. 1, 208-. radii, of curve and of its projection, relation. Peaucellier, . N. A. Mth. 20 (1861) 427-. radius. Bammert, (Dr) . Z. Mth. Ps. 10 (1865) 163-. . Hermite, C. N. A. Mth. 5 (1866) 297-. , of helicoidal curves on developable surfaces. Resal, H. A. [1856] (xii) Doubs S. Mm. 8 (1857) 100-. , and osculating circle, determination. Fuss, N. [1810] St Pet. Ac. Sc. Mm. 6 (1818) 66-. , of projection of curve. Caron, J. N. A. Mth. 14 (1895) 138-. =f(s), resulting differential equation. Goursat, E. Toul. Fac. Sc. A. 1 (1887) C, 26pp. at singular points. Wolffing, E. Arch. Mth. Ps. 15 (1897) 146-. spherical. Sobotka, J. Wien Ak. Sb. 104 (1895) (Ab. 2a) 144-. , special kind. Grunert, J. A. Grunert Arch. 25 (1855) 301-. systems of curves. Lilienthal, R. von. Mth. A. 32 (1888) 545- ; 38 (1891) 429-. and torsion. Hoppe, R. Arch. Mth. Ps. 15 (1897) 124-. in collineation and reciprocity. Servais, C. Brux. Mm. Cour. 8, 58 (1898-99) No. 2, 48pp. , determination of curves by relation between. Hoppe, R. Arch. Mth. Ps. 65 (1880) 287-. , , Goursat' s reduction of problem. Hoppe, R. Arch. Mth. Ps. 9 (1890) 43-. , properties connected with. Ramorino, A. Tor. Ac. Sc. At. 32 (1896) 331- or 471-. , radii, and spherical indicatrix. Paulli, H. (xii) Ts. Mth. 5 (1881) 86-. 1 representation of curves by. Hoppe, R. Crelle J. 60 (1862) 182- ; 63 (1864) 122-. total, and torsion of two curves, relations between. Rouquet, . Toul. Ac. Sc. Bll. 1 (1898) 39-. Curves. Lancret, M. A. [1802] Par. Mm. Sav. Etr. 1 (1806) 416-. Barre de Sainj-Venant, . C. E. 19 (1844) 547-; Par. EC. Pol. J. 18 (1845) 1-. Trenet, F. [1847] Liouv. J. Mth. 17 (1852) 437-. Enneper, A. Gott. Nr. (1866) 134- ; Z. Mth. Ps. 12 (1867) 510-. Bianchi, L. G. Mt. 21 (1883) 222-. Pirondini, G. G. Mt. 26 (1888) 104-. 607 8440 Curves Skew Curves (Calculus) 8440 algebraic, as evolutes. Stdckel, P. Mth. A. 45 (1894) 341-. apparent singularities, when projected from arbitrary point. Kneser, A. Mth. A. 34 (1889) 204-. barycentric properties, use of intrinsic equation. Cesdro, E. Ev. Mt. 5 (1895) 90-. Bertrand's. Demartres, G. C. E. 106 (1888) 1065-. . Bioche,C. Par. S. Mth. Bll. 17 (1889) 109-. . Cesdro, E. Ev. Mt. 2 (1892) 153- ; Mathesis 14 (1894) 265-. , generalisation. Demoulin, A. C. E. 116 (1893) 246-. case with one asymptotic point, and its tangent surface. Schiffner, F. [1881] Arch. Mth. Ps. 67 (1882) 207-. given by relation between curvature and torsion angles. Hoppe, E. Arch. Mth. Ps. 2 (1885) 417-. with known properties as to p and j, p.-, are equal. Bruno, G. Tor. At. Ac. Sc. 6 (1870-71) 133-. Spherical lines on surfaces. Pirondini, G. G. Teix. J. Sc. 9 (1889) 65-. representation, Gauss's. Ruffini, F. P. Bologna Ac. Sc. Mn>. 8 (1887) 661-. Spheroids and surfaces generally, curvature. Naccari, G. Yen. Aten. 1 (1893) 237-; 2 (1893) 133-. Sub-normals, polar, and curvature of surfaces. Ghysens, E. Brux. Ac. Bll. 44 (1877) 220-. Surface of centres. Codazzi, D. Tortolini A. 8 (1857) 129-. Curtis, A. H. Liouv. J. Mth. 3 (1858) 79-. Darboux, G. C. E. 70 (1870) 1328- ; 71 (1870) 267-. Catalan, E. C. C. E. 71 (1870) 50-. Backlund, A. V. Stockh. Ofv. 29 (No. 8) (1872) 3-. Halphen, G. H. G. E. 80 (1875) 258-. Voss, A. E. Mth. A. 16 (1880) 560-. developable. Hoppe, R. Arch. Mth. Ps. 63 (1879) 205-. of ellipsoid. Darboux, G. Bll. Sc. Mth. As. 3 (1872) 122-. , equation. Salmon, G. QJ. Mth. 2 (1858) 217-. elliptic paraboloid. Caspary, F. [1875] Crelle J. Mth. 81 (1876) 143-. with lines of curvature plane. Bonnet, O. C. E. 36 (1853) 1046-. projective. Voss, A. Munch. Ak. Ab. 16 (1888) 243-. of quadric, bitangents to. Laguerre, E. C. E. 78 (1874) 556-. of constant curvature, application of calculus of variations to demonstration of sum of angles of geodesic triangles on. Sabinin, E. Eec. Mth. (Moscou) 21 (1901) 54-; Fschr. Mth. (1900) 476. , functions. Dobriner, H. Acta Mth. 9 (1887) 73-. , transformations. Bianchi, L. A. Mt. 3 (1899) 185-. mean curvature. Jellett, J. H. Liouv. J. Mth. 18 (1853) 163-. negative curvature. Bianchi, L. Mth. A. 16 (1880) 577-. positive curvature. Bianchi, L. G. Mt. 20 (1882) 287-. containing centres of one curvature. Binet, J. P. M. Liouv. J. Mth. 6 (1841) 61- ; C. E. 12 (1841) 435-. determination of form by local measurements. Christoffel, E. B. [1864] Crelle J. 64 (1865) 193-. envelopes of planes cutting off constant volume. Padula, F. Nap. Mm. Ac. Sc. 2 (1855-57) 308-. generation, cases. Curtis, A. H. (n) Mess. Mth. 2 (1864) 76-. with given surface of centres, and curvature relation. August, F. Arch. Mth. Ps. 68 (1882) 315-. inverse to Cayley's cylindroid. Janisch, E. Mh. Mth. Ps. 8 (1897) 278-. , relation between curvatures. Demoulin, A. Bll. Sc. Mth. 16 (1892) 268-. loci of centres of curvature, theory. Lilienthal, JR. von. Mth. A. 30 (1887) 1-. with multiple points. Skew curve of 3rd degree, osculating sphere. Saltel, L. Brux. Ac. Bll. 35 (1873) 539-. of negative curvature and their geodesies. Ha- damard, . Liouv. J. Mth. 4 (1898) 27-. non - spherical, application of spherometer. Larmor, J. Camb. Ph. S. P. 7 (1892) 327-. parabolic points. Salmon, G. Camb. and Dubl. Mth. J. 2 (1847) 74-. of positive curvature, minimum of angle be- tween two conjugate tangents on. Hoppe, R. Arch. Mth. Ps. 69 (1883) 19-. with radii of curvature equal and opposite. Padula, F. Nap. Ed. 1 (1852) 76-. ;__. Purser, F. QJ. Mth. 13 (1875) __ _ _'_ ___*_. Catalan, ~E. C. C. E. 338-. , expressed as contravariant of two quaternary forms. Bjorling, C. F. E. Stockh. Ofv. 34 (1877) No. 4, 3-. relation between curvature elements of two sheets. Mannheim, A. C. E. 74 (1872) 458-; 79 (1874) 1328-. Surfaces. of buoyancy. Saint Germain, A. de. Par. S. Mth. Bll. 3 (1875) 37-. canal. Bertrand, J. Liouv. J. Mth. 13 (1848) 73-. of constant curvature. Minding, E. F. A. Crelle J. 19 (1839) 370-. . Bukrejew, B. Fschr. Mth. (1897) 546. 41 (1855) 35-, 274-, 1019- ; Par. EC. Pol. J. 37 e cah. (1858) 129-. reciprocal, curvature. Franke, J. N. Liouv. J. Mth. 3 (1877) 415-. of revolution, construction from fragment. Barbarin, P. As. Fr. C. E. (1887) (Pt. 2) 123-. , curvature of geodesies. Resal, H. N. A. Mth. 6 (1887) 57-. ruled, theory. Bonnet, 0. C. E. 46 (1858) 906-. screw, constructions for indicatrix. Sobotka, J. Wien Ak. Sb. 102 (1893) (Ab. 2a) 1204-. sections, curvature. Grunert, J. A. Arch. Mth. Ps. 41 (1864) 241-. and skew curves, theory, historic development. Christensen, S. A. (xn) Ts. Mth. 1 (1883) 97-. 615 8450 Surfaces. Theorems. Curvilinear Coordinates 8450 skew, curvature. Weyr, E. Prag Sb. (1870) (pt. 2) 29-. , at points of generating straight line. Buka, F. Z. Mth. Ps. 26 (1881) 15-. , osculation hyperboloid, construction. Weyr, E, Prag Ceske Ak. Fr. Jos. Ez. (Tfida 2) 5 (1896) Art. 5, 6 pp. ; Fschr. Mth. (1896) 415. , , . Prochdzka, B. Prag CeskS Ak. Fr. Jos. Ez. (Tfida 2) 6 (1897) Art. 15, 38 pp. ; Prag Fr. Jos. Ac. Sc. Bll. (Mth. Nt.) 4 (1897) (Pt. 1) 14-. , with radii of curvature in certain relation. Beltrami, E. A. Mt. 7 (1865) 139-. theory, angle between adjacent normals. Lamarle, E. C. E. 60 (1865) 851-. umbilical, 2nd order, properties derived from sphere. Booth, J. B.A.Ep. (1858) (^.2)2-. Symmetrical investigation of curvature, etc. Johnson, A. [1882] (xn) Cn. E. S. P. & T. 1 (1883) (Sect. 3) 31-. Tangent-plane. Bedetti, G. Bologna N. Cm. 5 (1842) 485-. Theorems. Dim, U. [1869] Mod. S. It. Mm. 2 (1876) 1-. Dupin's, demonstration. Oergonne, J. D. Gergonne A. Mth. 4 (1813-14) 368-. , . Cayley, A. QJ. Mth. 12 (1873) 185-. , . Le Pont, H. G. Teix. J. Sc. 7 (1886) , . Lichtenfels, 0. von. Mb. Mth. Ps. 5 (1894) 380-. , and application. Biffignandi, A. G. Mt. 28 (1890) 202-. , , elementary. Thomson, (Sir) W. Camb. Mth. J. 4 (1845) 62-. , generalisation. Cremona, L. Tortolini A. 3 (1860) 325-. Gauss's. Chelini, D. G. Arcad. 115 (1848) 257-; 116 (1848) 3-. . Bertrand, J. C. E. 42 (1856) 1088-. , analogue to. Sturm, R. Mth. A. 21 (1883) 379-. , industrial application. Levy, M. C. E. 86 (1878) 111-. Meunier's, analogue to. Mannheim, A. [1870] (x) Par. S. Phlm. Bll. 7 (1871) 138-. and Bonnet's. Brisse, C. Par. S. Mth. Bll. 4 (1876) 96-. , curvature of skew curves. Besant, W. H. QJ. Mth. 6 <1863) 140-. , demonstration. Bonnet, 0. Par. S. Phlm. PV. (1848) 59-. , generalisation. Mannheim, A. C. E. 74 (1872) 372-. l + l = ^ + ^ + dn lications P! p 2 ax ay dz E. E. Mess. Mth. 9 (1880) 170-. Theory. Bonnet, O. Par. EC. Pol. J. 32" cah. (1848) 1-; C. E. 33 (1851) 89-; 37 (1853) . Grunert,J.A. Grunert Arch. 28 (1857)163-. . Lipschitz, E. O. S. Crelle J. Mth. 81 (1876) 230-. Theory. Salvert, (le vicomte) F. de. (xn) Brux. S. Sc. A. 5 (1881) (Pt. 1) 63-, (Pt. 2) 291-. . Lilienthal, E. von. Crelle J. Mth. 104 (1889) 341-. . Voss, A. Mth. A. 39 (1891) 179-. , analytic. Amiot, B. Liouv. J. Mth. 12 (1847) 129-. , and Dupin's cyclides. Holzmiiller, G. Z. Mth. Ps. 44 (1899) 194-. , fundamental formulae. Laguerre, . N. A. Mth. 11 (1872) 60-. , Gauss's. Cayley, A. L. Mth. S. P. 12 (1880-81) 187-. , , principle of least action in relation to its foundations. Beke, M. Mth. Tennt. fits. 2 (1884) 133- ; Mth. Nt. B. Ung. 2 (*1883-84) 282-. , general. Beltrami, E. Arch. Mth. Ps. 42 (1864) 117. , geometrical. Mannheim, A. C. E. 74 (1872) 598-. , interpretation of quantities of 2nd order. Hoppe, E. Arch. Mth. Ps. 60 (1877) 65-. , use of direction cosines of normal. Gilbert, P. Brux. S. Sc. A. 18 (1894) (Pt. 2) 1-. Total curvature, and application to double definite, integrals. Tortolini, B. Em. At. 4 (1850-51) 53-. of bounded portions of surfaces. Graves, C. Ir. Ac. P. 5 (1853) 339-. at edge of regression. Stouff, X. Par. EC. Norm. A. 9 (1892) 91-. Transformation to chief axes. Cayley, A. QJ. Mth. 12 (1873) 34-. Umbilic, conditions for. Stone, E. J. QJ. Mth. 3 (1860) 146-. , . Hall, A. A. Mth. 9 (1894-95) 42. , lines of curvature near. Frost, P. QJ. Mth. 10 (1870) 78-. , (Frost). Cayley, A. QJ. Mth. 10 (1870) 111-. , through. Bioche, C. Par. S. Mth. Bll. 18 (1890) 95-. Umbilics. Bonnet, 0. Par. EC. Pol. J. 30 e cah. (1845) 165- ; Liouv. J. Mth. 16 (1851) 191-. . Souillart, C. Crelle J. 65 (1866) 320-. . Catalan, E. Mathesis 5 (1885) 73-. and differential equations. Cayley, A. (vn) Ph. Mg. 26 (1863) 373-, 441-. , lines of curvature at. Hoppe, E. Arch. Mth. Ps. 70 (1884) 289-. of surface of order n, problem comprising determination of number. Berzolari, L. Tor. Ac. Sc. At. 30 (1895) 756-. tetrahedral surfaces. Lebon, E. Palermo Cir. Mt. Ed. 4 (1890) 115-. Variation in contact transformation of surface. Mehmke, E. Ev. Mt. 2 (1892) 159-. /tt s sin 2 ' + i 2 sin 2 i" = a 2 , demonstration. Chasles, M. C. E. 22 (1846) 517-. CUEVILINEAE COOEDINATES. Lame, G. C. E. 6 (1838) 43-; Liouv. J. Mth. 5 (1840) 313- ; 16 (1851) 171- ; (vra) C. E. 49 (1859) 341-. Aoust, L. C. E. 48 (1859) 842-; Crelle J. 58 (1861) 352-. 616 8450 Curvilinear Coordinates 8450 Boklen, O. Grunert Arch. 34 (1860) 26-. Aoust, L. G. E. 54 (1862) 461-. Combescure, E. G. E. 74 (1872) 1517-. Frattini, G. G. Mt. 10 (1872) 235. Roger, E. A. Mines 5 (1874) 110- ; 7 (1875) 92-. Darbmtx, G. [1877] Par. EC. Norm. A. 7 (1878) 101-, 227-, 275-. Mathieu, E. L. Liouv. J. Mth. 8 (1882) 5-. Cayley, A. QJ. Mth. 19 (1883) 1-. Catalan, E. C. Liege S. Sc. Mm. 12 (1885) No. 2, 234-, 403. Williamson, B. [1890] Ir. Ac. T. 29 (1887-92) 515-. Coordinates of point in relative motion, ex- pansion, and curvature of orthogonal lines. Balitrand, . Par. S. Mth. Bll. 23 (1895) 26-. reducing square of arc to form de t =(U+V)(du* + dv*). Bicci, G. Em. E. Ac. Line. Ed. 2 (1893) (Sem. 1) 73-. Curvature and geodesic torsion. Reina, V. Em. E. Ac. Line. Ed. 6 (1890) (Sem. 2) 103-, 176. of system of plane curves in curvilinear coordinates, extension to 3 dimensions. Harst, A. D. van der. N. Arch. Wisk. 4 (1899) 226-. Curves referred to any system of coordinates, analysis. Aoust, (Vabbe) . Par. EC. Norm. A. 6 (1869) 205-. Curvilinear coordinates and curves on surfaces. Aoust, (Vabbe) . C. E. 65 (1867) 814-. geodesic lines of isothermic surfaces. Salvert, (le vicomte) de. Brux. S. Sc. A. 10 (1886) (Pt. 2) 293-; 11 (1887) (Pt. 2) 1-. in space and on surfaces. Codazzi, D. A. Mt. 1 (1867-68) 293- ; 2 (1868-69) 101-, 269- ; 4 (1870-71) 10- ; 5 (1871-73) 20ft-. . Chelini, D. Bologna Ac. Sc. Mm. 8 (1868) 483-. and normal coordinates. Warren, J. W. [1876-77] Camb. Ph. S. T. 12 (1879) 455-, Development of surfaces, linear element of which is expressible by homogeneous func- tion. Levy, M. C. E. 87 (1878) 788-. Differential parameters. Somigliana, C. Mil. I. Lomb. Ed. 22 (1889) 275-. . Knoblauch, J. Crelle J. Mth. Ill (1893) 329-. . Pennacchietti, G. Catania Ac. Gioen. At. 9 (1896) Mem. 1, 11 pp. , application to theory of surfaces. Bendixson, I. Stockh. Ofv. (1897) 331-. , and curvature of surface expressed in curvilinear coordinates. Somoff, J. [1864] St Pet. Ac. Sc. Mm. 8 (1865) No. 16, 45 pp. ; St Pet. Ac. Sc. Mm. (Rs.) 8 (*1866) (Suppl.) No. 4, 66 pp. , theory. Beltrami, E. Bologna Ac. Sc. Mm. 8 (1868) 549-; Bologna Ed. (1869) 55-. Elliptic coordinates. Boklen, 0. Grunert Arch. 34 (1860) 308-. . Somov, O. I. St Pet. Ac. Sc. Mm. (Rs.) 9 (*1866) 295-. Elliptic coordinates. Darboux, G. Bll. Sc. Mth. As. 3 (1872) 122-. , application to determination of oblique trajectories. Mylord, H. (XH) Ts. Mth. 1 (1871) 33-, 65-. , development. Jacobi, C. G. J. As. Nr. 28 (1849) 257-. , evaluation of surface of ellipsoid in. Obenrauch, F. J. Arch. Mth. Ps. 12 (1894) 155-. in plane. Grunert, J. A. Grunert Arch. 39 (1862) 377-. space. Grunert, J. A. Grunert Arch. 39 (1862) 402-. of tangents common to two quadrics (Liouville's formula). Roberts, W. Ir. Ac. P. 4 (1850) 288-. , transformation of differentials by. Gundelfinger, S. [1877] Crelle J. Mth. 85 (1878) 80-. , rectangular rectilinear coordinates into. Somoff, J. St P6t. Ac. Sc. Bll. 10 (1866) 41-. Equations of motions of heat referred to curvilinear coordinates. Thomson, (Sir) W. Camb. Mth. J. 4 (1845) 33-. Formulae, analytical. Enneper,A. Schlomilch Z. 7 (1862) 75-, 313-, 365-; 8 (1863) 241-. on surface and volume integrals, in curvi- linear coordinates. Gilbert, P. Brux. S. Sc. A. 14 (1890) (Pt. 2) 1-. Geodesic coordinates, transformation. Ldska, V. Prag Sb. (1894) (Mth.-Nt.) No. 12, 6 pp. polar coordinate system, properties, demon- stration. Bonnet, G.O. C.E. 97 (1883)1422-. Isometric coordinates on sphere. Enneper, A. Z. Mth. Ps. 24 (1879) 256. Lemniscate coordinates. Lommel, E. Z. Mth. Ps. 12 (1867) 45-. , nth order. Holzmiiller, F. G. Crelle J. Mth. 83 (1877) 38-. geometry, etc., deduced by aid of function of complex argument. Holzmiiller, F. G. Z. Mth. Ps. 21 (1876) 325-. Lines of zero length on surface as curvilinear coordinates. Burnside, W. Mess. Mth. 19 (1890) 99-. Orthogonal coordinates. Darboux, G. C. E. 60 (1865) 560-. . Enneper, A. Gott. Nr. (1867) 232-, 277- ; (1868) 258-, 421- ; (1874) 125-. , application to curvature of curves on surfaces. Aleksyeev, N. N. (xn) Eec. Mth. (Moscou) 2 (1867) (Pt. 1) 79-. ( equation V'*=0 in. Petrini, H. Ups. Arsk. (1890) (Mt. Nt.) iv + 104 pp. , especially for conicoids. Levy, M. [1867] Par. EC. Pol. J. Cah. 43 (1870) 157-. 1 system belonging to differential equation V 2 $ + Wf(xyz)$ = 0. Petrini, H. Stockh. Ak. Hndl. Bh. 19 (Afd. 1) (1894) No. 4, 36 pp. ; , new, for problems in heat and electricity. Liouville, J. Liouv. J. Mth. 12 (1847) 265-. geodesic coordinates on ellipsoid. Ldska, V. Prag Ceske- Ak. Fr. Jos. Ez. (Tfida 2) 3 (1894) Art. 17, 8 pp. 617 8450 Curvilinear Coordinates. Surfaces 8450 Orthogonal hyperboloids, families. Van&dek, J. S. Prag Ceske" Ak. Fr. Jos. Rz. (Tflda 2) 4 (1895) Art. 28, 4 pp. projection on surface of revolution. Piron- dini, G. N. A. Mth. 17 (1898) 246-. surfaces. Frenet, F. Lyon Ac. Sc. Mm. 3 (1853) 5-. trajectories of polhodes. Catalan, E. C. Liege S. Sc. Mm. 15 (1888) No. 1, 57-. Parabolic coordinates, and application to para- boloids. Valson, C. A. N. A. Mth. 19 (1860) 298-. Quadric surfaces, rotational, contour lines. Langer, K. Wien Az. 36 (1899) 3. Eadii of curvature of sections. Brioschi, F. Tortolini A. 4 (1853) 232-. Surfaces referred to a conjugate azimuthal network. Eaffy, L. Par. S. Mth. Bll. 24 (1896) 51-. with 2 sets of conjugate geodesies. Razzaboni, A. Bologna Ed. (1888-89) 66. , and certain unicursal curves on them. Lelieuvre, . C. B. Ill (1890) 568-. System, particular, of curvilinear coordinates. Habich, E. N. A. Mth. 3 (1884) 353-. Theorem. Issaly, (I'abbe) . Bordeaux S. Sc. PV. (1898-99) 38-. of Stokes. Webster, A. G. N. Y. Am. Mth. S. Bll. 4 (1898) 438-. Theory. Codazzi, D. Tortolini A. 8 (1857) 129-. . Aoust, (I'abbe) . A. Mt. 6 (1864) 65- ; 2 (1868-69) 39-, 268 ; 3 (1869-70) 55- ; 5 (1871-73) 261-. . Brioschi, F. A. Mt. 1 (1867-68) 1-. . Padova, E. Em. E. Ac. Line. Ed. 4 (1888) (Sem. 2) 369-, 454-. , application to integral calculus. Frattini, G. G. Mt. 15 (1877) 1-. of surfaces, and partial differential equations. Enneper, A. Schlomilch Z. 7 (1862) 1-. Trajectories of generators on skew surface. Molins, H. Toul. Ac. Sc. Mm. 7 (1895) 421-. Transformation of class of differential equa- tions into curvilinear coordinates. Gundel- finger, S. Crelle J. Mth. 85 (1878) 295-. Triply conjugate systems. Tzitzeica, . Bll. Sc. Mth. 23 (1899) 330-. Use in evaluation of surfaces, masses, etc. Cauchy, A. L. C. E. 16 (1843) 413-. OTHEE APPLICATIONS OF THE DIFFERENTIAL CALCULUS TO SUEFACES. Analysis applied to geometry. Beltrami, E. G. Mt. 2 (1864) 267-, 297-, 331-, 355-; 3 (1865) 15-, 33-, 82-, 228-, 311-. Angle at which surfaces cut. Morin, P. C. E. 66 (1868) 741-. Asymptotic lines of surface XYZ = T 3 . Appell, P. Arch. Mth. Ps. 61 (1877) 144-. Conchoidal surfaces and conchoids. Minich, S. B. Padova Ev. Period. 10 (1862) 135-. Conformal representation in configurations of three dimensions. Cotton, E. C. E. 125 (1897) 225-. Contact of conies and surfaces. Darboux, G. C. E. 91 (1880) 969-. . Moutard, T. C. E. 91 (1880) 1055-. curves and surfaces. Collet, J. Liouv. J. Mth. 4 (1878) 315-. . Darboux, G. Bll. Sc. Mth. As. 4 (1880) 348-. , multiple, of surfaces. Spottiswoode, W. [1875] Phil. Trans. 166 (1877) 227-. , 3rd order, of two surfaces. Mannheim, A. C. E. 74 (1872) 856-, 928-. of quadrics with other surfaces. Spottis- woode, W. (xi) L. Mth. S. P. 5 (1873-74) 70-. surface of revolution and developable, construction of tangents to curve. Drasch, H. Wien Ak. Sb. 77 (1878) (Ab. 2) 174-. Contacts of line and surface, Schubert's method. Cayley, A. QJ. Mth. 17 (1881) 244-. Curve arising from illumination of revolution quadric. Enneper, A. Gott. Nr. (1868) 174-. of 4th degree, bicircular. Cayley, A. Phil. Trans. 167 (1877) 441-. Curves drawn from point on surface having contact of nth order, theorem. Mannheim, A. (x) Par. S. Phlm. Bll. 7 (1871) 228-. on surfaces, flexion. Demartres, . Par. S. Mth. Bll. 15 (1887) 129-. Cyclide, envelope of planes perpendicular to radii vectores. Roberts, W. C. E. 53 (1861) 1118-. Cylinder orthogonal to certain surfaces. Piron- dini, G. G. Teix. J. Sc. 13 (1897) 77-. Cylinders circumscribing surfaces of revolution, lines of contact and tracing. Nicoli, F. (x) Mod. Ac. Sc. Mm. 13 (Pt. 1) (1872) 39-. Developable circumscribed to quadric or cubic. Roberts, M. N. A. Mth. 14 (1855) 268-. , sextic, edge of regression. Sucharda, A. Prag Sb. (1891) (Mth.-Nt.) 369-. Developoids (envelopes of straight lines meeting any curve under a constant angle). Haton de la Goupilliere, J. N. Par. S. Mth. Bll. 5 (1877) 126- ; (xn) Brux. S. Sc. A. 2 (1878) (Pt. 2) 1-. (Haton de la Goupilliere). Aoust, . C. E. 85 (1877) 331-. Diagonal curves, crossing two families of curves. Breton, P. Isere S. Bll. 25 (1887) 28-. Duals of geodesies and lines of curvature on ellipsoid and on its pedal surfaces. Jeff'ery, H. M. QJ. Mth. 12 (1873) 322-. Envelope of quadric surface, case. Cayley, A. QJ. Mth. 11 (1871) 244-. Evolutes of surfaces, cases. Calf), B. A. Mt. 21 (1893) 195-. Generating characteristics of curved surfaces. Minich, S. R. Yen. I. Mm. 22 (1882) 169-. Geometry of surface of ellipsoid. Clifford, W. K. [1872] L. Mth. S. P. 4 (1871-73) 215-. Homography, case in geometry of surfaces. Somoff, J. [1862] St Pet. Ac. Sc. Mm. 5 (1863) No. 9, 22 pp. 618 8450 Applications of Calculus. Surfaces 8450 Hyperboloid osculating ruled surface along generator. Duporcq, E. N. A. Mth. 17 (1898) 10&-. skew surface along generator. Chosles, M. Brux. Ac. fill. 5 (1838) 643-. touching skew surface along generator. Bruno, G. [1871] Tor. At. Ac. Sc. 7 (1871- 72)235-. Hyperboloids of one sheet, construction of line of striction. Petersen, J. N. Ts. Mth. 10 (B) (1899) 39-; Fschr. Mth. (1900) 542. Infinitely near surfaces. Maggi, P. G. [1850] Yen. Mm. I. 5 (1855) 299-. Jacobian of three surfaces, applications. Arzela, C. G. Mt. 9 (1871) 32-. Normal curves and surfaces. Brenner, . Grunert Arch. 13 (1849) 225-. Normals. Bedetti, G. [1844] Bologna N. Cm. 10 (1849) 523-. . Bertrand, J. Liouv. J. Mth. 9 (1844) 133-. . Rothig, O. Crelle J. Mth. 85 (1878) 250-. , adjacent. Joachimsthal, F. Liouv. J. Mth. 13 (1848) 415-. and normal planes of certain curves and surfaces, construction. Schonemann, T. Berl. B. (1855) 255-. pedal curves and surfaces. Sturm, JR. [1872] Mth. A. 6 (1873) 241-. , proof of Bertrand' s relation. Mannheim, A. Liouv. J. Mth. 17 (1872) 403-. , theory. Bertrand, J. Liouv. J. Mth. 12 (1847) 343-. , 3rd variation. Mayer, A. C. G. Leip. Mth. Ps. B. 33 (1881) 28-. Orthogonal trajectories of systems of surfaces. Morera, G. Mil. I. Lomb. Ed. 19 (1886) 282-. Osculation problem. Clebsch, A. Liouv. J. Mth. 8 (1863) 297-. Parallel lines and surfaces. Bordmi, A. [1811] Mod. Mm. S. It. 16 (1813) 72-. , equations. Bordoni, A. Mil. At. I. Lomb. 1 (1858) 209-. Parallelism of curved lines and surfaces. Crelle, A. L. Gergonne A. Mth. 12 (1821- 22) 1-. Plane arid spherical envelopes of curves and torses. Pirondini, G. Bologna Ac. Sc. Mm. 9 (1888) 641-. Pseudo-surfaces, extension of Laguerre's formu- la to. Issaly, . Par. S. Mth. Bll. 25 (1897) 243-. , triple systems. Issaly, (I'abbe) . N. A. Mth. 9 (1890) 204-. Quadric cones osculating developable surfaces. Whittemore, J. K. [1900] N. Y. Am. Mth. S. Bll. 7 (1901) 203, 209. Quadrics, equation deduced from a polar re- lation. Jamet, V. Brux. Ac. Bll. 6 (1883) 885-. Radius of geodesic curvature of a curve S, of constant normal curvature, theorem. La- guerre, E. C. B. 80 (1875) 822- Reciprocal tangents. Dupin, C. Par. EC. Pol. Cor. 3 (1814-16) 138-. Ruled helicoid, the most general. Ocagne, M. d'. Par. S. Mth. Bll. 23 (1895) 114-. Solid angle subtended by quadric at point. Beltrami, E. G. Mt. 1 (1863) 68-. Spherical conic, and developable formed by its tangents. Huber, G. Z. Mth. Ps. 45 (1900) 86-. coordinates, transformation. Necker, K. As. Nr. 130 (1892) 49-. Sphere-conjugate tangents. Cremona, L. Tortolini A. 6 (1855) 382-. Surface envelope of planes cutting rational skew curve of 4th degree in 4 concyclic points. Sturm, E. [1869] A. Mt. 4 (1870- 71) 73-. j, A. Tortolini A. 2 (1859) 168-. ~ sphere. Monge, G. Par. EC. Pol. J. 6 (1806) 41-. envelopes of spheres. Eibaucour, A. (xi) Par. S. Phlm. Bll. 5 (1868) 30-. , . Lecornu, L. Par. Ee. Pol. J. Cah. 53 (1883) 135-. . Jamet, V. Mars. Fac. Sc. A. 10 (1900) Fasc. 2, 18 pp. - , property. Ribauc 67 (1868) 1334-. our, A. C. R. touching two circles in plane. Niemtschik, R. Wien Ak. Sb. 73 (1876) (Ab. 2) 7-. , theory, characteristics, etc. Raabe, J. L. Crelle J. 15 (1836) 125-. of variable quadric surface, edge of regression. Livet, . Par. EC. Pol. Cor. 1 (1804-08) 422-. Surfaces. admitting systems of equal and similar lines. Pirondini, G. A. Mt. 23 (1895) 93-. algebraic, theory, new invariant in, and re- duction of double integrals. Picard, E. C. R. 126 (1898) 297-. capable of generating a Lame! system. Darboux, G. C. R. 124 (1897) 1428. . by helicoidal movement, two classes. Bianchi, L. Toul. Fac. Sc. A. 11 (1897) H, 8 pp. screw displacement. Petot, A. C. R. 118 (1894) 1409-. 2 Lame systems by 2 different move- ments. Medolaghi, P. Rm. R. Ac. Line. Rd. 8 (1899) (Sem. 2) 304-. conical and cylindrical, property. Merrifield, C. W. Mess. Mth. 1 (1872) 81-. and curves. Scheffers, G. Leip. Mth. Ps. B. 52 (1900) 1-. , derived. Tortolini, B. Tortolini A. 2 (1859) 316-. , theory. Pellet, A. Par. EC. Norm. A. 14 (1897) 287-. thereon, general formulae and special cases. Enneper, A. Z. Mth. Ps. 9 (1864) 96-. cyclic. Enneper, A. Gott. Nr. (1866) 243-; Z. Mth. Ps. 14 (1869) 393-. 619 8450 Applications of Calculus. Surfaces 8450 described by points of straight line and re- ciprocal problem. Mannheim, A. As. Fr. C. E. 2 (1873) 98-. developable. Poisson, S. D. [1825] Baum- gartner Z. 1 (1826) 110-. . Beltrami,E. Yen. At. Aten. 5 (1868) 535-. . Voss, A. E. Mth. A. 13 (1878) 232-. . Prochdzka, F. Fschr. Mth. (1899) 492-. , characteristic property. Mangoldt, H. von. Mth. A. 18 (1881) 604-. , circumscribed to 2 conies. Regis, D. Tor. Ac. Sc. At. 11 (1875-76) 971-. , given surfaces. Enneper, A. Gott. Nr. (1866) 321-; Z. Mth. Ps. 13 (1868) 322-; 15 (1870) 283-. _, 2 quadrics. Painvin, L. F. [1868- 74] (xn) Lille S. Mm. 9 (1871) 97-; 11 (1873) 81-; 13 (1874) 329-. , . Saint- Germain, A. L. de, (xn) Caen Ac. Mm. (1883) 99-. , of conical screw (envelope of tangents). Schiffner, F. Arch. Mth. Ps. 69 (1883) 444-. , edge of regression. Combescure, E. Crelle J. 62 (1863) 174-. , . Lebon, E. [1879] Par. S. Mth. Bll. 8 (1880) 27-. , denned by tangential equations, determination. Painvin, L. C. B. 71 (1870) 217- ; Liouv. J. Mth. 17 (1872) 177-. , envelope of moving plane, curvature. Mannheim, A. C. E. 70 (1870) 1259-. , formed by refraction of pencil of parallel rays by a given curve. Levy, L. Par. S. Mth. Bll. 11 (1883) 186-. , formulae. Cayley, A. B. A. Ep. 41 (1871) (Sect.) 19-. , generated by straight lines drawn through points of conic at 45 to its plane. Fiedler, W. Ziir. Vjschr. 29 (1884) 348-. , isogonal trajectories of generators. Piron- dini, G. Crelle J. Mth. 118 (1897) 61-. , properties. Pirondini, G. Liouv. J. Mth. 3 (1897) 405-. , theorem. Catalan, E. C. C. E. 17 (1843) 738-. , . Saltel, L. [1881] Bordeaux S. Sc. Mm. 4 (1882) 443-. , theorems. Pirondini, G. A. Mt. 19 (1891- 92) 247-. , deduced by quaternions. Graefe, F. [1882] Arch. Mth. Ps. 69 (1883) 1-. divided into equal parallelograms by their isogonal trajectories. Fibbi, C. Em. E. Ac. Line. Ed. 4 (1895) (Sem. 1) 413-. of equal incidence. Lecornu, L. As. Fr. C. E. (1892) (Pt. 2) 172-. inclination. Regis, D. Tor. At. Ac. Sc. 6 (1870-71) 500-. generated by line, equations of which contain arbitrary parameter. Tortolini, B. G. Arcad. 57 (1832) 110-. motions of curve, determination of all. Lie, M. S. Arch. Mth. Ntvd. 7 (1882) 155-. normals. Cliasles, M. N. A. Mth. 18 (1859) 192-. principal normals of skew curve. Curtis, A. H. Liouv. J. Mth. 1 (1856) (1856) 223-. with given properties, deduction of equation. Schlomilch, O. X. Z. Mth. Ps. 21 (1876) 75-. having higher contact with surface along con- tinuous line, characteristic. Bordoni, A. (vi Adds.) Opusc. Mt. Fis. 1 (1832) 22-. infinitesimal geometry. Bukrejeiu, B. J. Fschr. Mth. (1900) 600. with infinitesimal and linear transformations. Lie, M. S. Arch. Mth. Ntvd. 7 (1882) 179-. intrinsic differential equations. Ricci, G. Yen. I. At. (1894-95) 445-. isostatic. Weingarten, J. L. G. J. Crelle J. Mth. 90 (1881) 18-. maximum and minimum inclinations to radii vectores. Exner, K. [1870] Wien Sb. 63 (1871) (Ab. 2) 149-. new system of coordinates for. Bonnet, 0. Liouv. J. Mth. 5 (1860) 153-. normal, of developable along intersection with surface. Peschka, G. A. von. Wien Ak. Sb. 83 (1881) (Ab. 2) 790-. to given surface. Colnet-d'Huart, . Lux. S. Sc. Nt. 5 (1857-62) 69-. , along plane sections. Peschka, G. A. von. Wien Ak. Sb. 81 (1880) (Ab. 2) 1163-. , of quadrics along plane sections. Peschka, G. A. von. Wien Ak. Sb. 85 (1882) (Ab. 2) 381-. , surface along intersection with another surface. Peschka, G. A. von. [1881] Wien Ak. Sb. 84 (1882) (Ab. 2) 30-. , theory. Peschka, G. A. von. Wien Ak. Sb. 81 (1880) (Ab. 2) 1128-. normals of which at point cut off spherical triangle of constant area. Aonst, (I'abbe) . C. E. 81 (1875) 963-. touch given surface. Braunmuhl, A. von. Munch. Ak. Ab. 14 (1883) (^46. 3) 91-. of oblique revolution. Walton, W. QJ. Mth. 6 (1863) 134-. parabolic lines, and singular points of curves. Segre, C. Em. E. Ac. Line. Ed. 6 (1897) (Sem. 2) 168-. parallel. Mannheim, A. L. Mth. S. P. 12 (1880-81) 177-. . Craig, T. Crelle J. Mth. 94 (1883) 162-. , of developables and skew curves. Roberts, S. (xi) L. Mth. S. P. 5 (1873-74) 90-. , extension of concept. Enneper, A. Gott. Nr. (1870) 70-. pedal. Bacaloglo, E. Grunert Arch. 35 (1860) 41- ; Schlomilch Z. 5 (1860) 67-. . Hirst, T. A. QJ. Mth. 3 (1860) 210-. . Enneper, A. Schlomilch Z. 8 (1863) 53-. with rational generating lines. Lelieuvre, . Par. EC. Norm. A. 12 (1895) 57-. reciprocal, a particular kind. Bonnet, O. C. E. 42 (1856) 485-. of revolution, construction of tangent to shadow. Mannheim, A. (x) Par. S. Phlm. Bll. 2 (1865) 9-. , quartic, geodesies of which are algebraic. Tannery, . Par. S. Phlm. Bll. 4 (1892) 85-. , tangents of boundary of shadow, deter- mination. Staudigl, R. Wien Sb. 68 (1873) (Ab. 2) 228-. 620 8450 Surfaces of revolution, tangents of boundary of shadow, determination. Pelz, C. Wien Ak. Sb. 79 (1879) (Ab. 2) 447-. ruled. Pirondini, G. G. Mt. 25 (1887) 25-, 115-. , 2nd asymptotic line. Hoppe, R. Arch. Mth. Ps. 60 (1877) 276-. , belonging to skew curve. Ciani, E. G. Mt. 27 (1889) 233-. , class. Rouquet, V. Toul. Ac. Sc. Mm. 7 (1895) 117-. , contact. Binet, J. P. M. Par. EC. Pol. Cor. 2 (1809-13) 331-. , and curves with constant curvature and torsion. Allen, A. J. C. [1882] Mess. Mth. 12 (1883) 26-. , expression for ds 2 on. Bioche, C. [1889] Par. S. Mth. Bll. 18 (1890) 91-. , geodesic and line of striction, theorem. Mannheim, A. [1877] Par. S. Mth. Bll. 6 (1878) 7-. , intrinsic equations. Cesdro, E. N. A. Mth. 9 (1890) 294-. , line of striction. Nicolaldes, N. Les Mondes 9 (1866) 146-. , given, construction. Hoppe, R. Arch. Mth. Ps. 11 (1892) 345-. , satisfying certain conditions. Amigues, E. N. A. Mth. 8 (1889) 77-. , orthogonal trajectories of generators. Fais, A. [1879] Bologna Ac. Sc. Mm. 1 (1880) 67-. , paraboloid of normals. Mannheim, A. Par. S. Mth. Bll. 5 (1877) 190-. , passing through curve. Bioche, C. N. A. Mth. 12 (1893) 412-. , and surface envelopes. Stammer, W, Liege Mm. S. Sc. 1 (1866) 137-. , tangential helicoid. Mannheim, A. Liouv. J. Mth. 4 (1878) 57-. singular points. Gregory, D. F. Camb. Mth. J. 2 (1841) 252-. . Amiot, B. Brux. Mm. Cour. 4, 21 (1846) 48 pp. singularities. Maur, A. Grunert Arch. 25 (1855) 335-. , pinch points. Cayley, A. QJ. Mth. 9 (1868) 332-. skew. Catalan, E. C. [1865] Brux. Mm. Cour. 8, 18 (1866) (No. 5) 80 pp. . Dewulf, E. Par. EC. Norm. A. 3 (1886) 189-. , asymptotic lines. Anon, (xi 19) Bll. Sc. Mth. As. 1 (1870) 228-. helicoidal, determination of contour. Pelz, C. Wien Ak. Sb. 87 (1883) (Ab. 2) 473-. , intrinsic equations. Cesdro, E. N. A. Mth. 8 (1889) 445-. , lines of striction, application to illumination. Prochdzka. B. Casopis 18 (1889) 1-; Fschr. Mth. (1889) 589-. , with plane line of striction. Molins, H. Toul. Ac. Sc. Mm. 9 (1887) 516-. , principal tangent curves. Voss, A. E. Mth. A. 12 (1877) 485-. , properties. Dini, U. G. Mt. 3 (1865) 281-. , property. Chome, F. (xn) Mathesis 2 (1882) 82-. Congruences 8455 skew, with same plane curve for line of stric- tion and of curvature. Amigues, E. N. A. Mth. 14 (1895) 491-. , theorems. Amigues, E. C. B. 104 (1887) 1092-. , theory. Catalan, E. C. Par. S. Phlm. PV. (1848) 67-. , . Liiroth, J. [1866] Crelle J. 67 (1867) 130-. , . Voss, A. E. Mth. A. 8 (1875) 54-. , . Korteweg, D. J. N. Arch. Wisk. 3 (*1877) 66-. theory. Pellet, A. C. E. 124 (1897) 451- ; Par. S. Mth. Bll. 26 (1898) 138-. , fundamental relations referred to movable trihedra. Issaly, . N. A. Mth. 19 (1900) 49-, 240. , general. Padova, E. Bologna Ac. Sc. Mm. 10 (1889) 745-. , . Pellet, A. C. B. 124 (1897) 739-. , infinitesimal, formulae. Frattini, G. G. Mt. 13 (1875) 161-. , new element (Christoffel). Beltrami, E. Mil. I. Lomb. Ed. 2 (1869) 853-. , solutions of problems depending on small quantities of 3rd order. Mannheim, A. C. E. 80 (1875) 541-, 619-. transformation method. Habich, E. N. A. Mth. 8 (1869) 253-. transformations. Enneper, A. Z. Mth. Ps. 9 (1864) 126-. triply asymptotic system, impossibility. Eisen- hart, L. P. [1900] N. Y. Am. Mth. S. Bll. 7 (1901) 184-. useful formulae for. Baehr, G. F. W. Grunert Arch. 32 (1859) 221-. vectorial geometry. Genty, . As. Fr. C. E. (1888) (Pt. 2) 95-. Tangent planes, parallel, centre of mean dis- tances of their points of contact. Liouville, J. Liouv. J. Mth. 6 (1841) 345-. Terrestrial ellipsoid, theorems. Angelitti, F. Nap. Ac. Pont. At. 28 (1898) No. 2, 124 pp. Theorem of Betti and Weingarten. Fieri, M. G. Mt. 24 (1886) 290-. Variation of straight lines in connexion with surface. Hoppe, R. Arch. Mth. Ps. 3 (1886) 290-. 8455 Differential geometry of congruences and other ap- plications of the differential calculus to elements of space. Congruences conjugate to systems C (K con- gruences). Guichard, C. C. E. 126 (1898) 718-. which are K congruences in several man- ners. Guichard, C. C. E. 126 (1898) 1011-. , intrinsic analysis. Cifarelli, T. G. Mt. 36 (1898) 145-. , geometry. Cesdro, E. Nap. Ed. 33 (1894) 141-. 621 8455 Congruences. Space Elements. Areas 8460 Congruences and nets of curves, spherical re- presentation. Guicliard, C. G. E. 116 (1893) 1238-. of normals. Transon, (Prof.) A. N. A. Mth. 2 (1863) 138-. special type for transformation. Brill, J. L. Mth. S. P. 20 (1889) 102-. Conjugate systems with equal invariants. Koenigs, G. C. E. 113 (1891) 1022- . Contact transformations and optics. Lovett, E. 0. Camb. Ph. S. T. 18 (1900) 256-. Curves, tangents of which are part of line complex of 1st order, theorem. Appell, P. Arch. Mth. Ps. 60 (1877) 274-. Cyclic system, in envelopes of spheres. Ribaucour, A. C. E. 92 (1881) 233-. systems. Eibaucour, A. C. E. 113 (1891) 304-, 324-. Differential equations. Lie, M. S. Christiania F. (1882) #o. 21, 12 pp., No. 22, 6 pp.; (1883) No. 10, 4 pp., No. 18, 5 pp. forms and systems of elements. Koenigs, G. Acta Mth. 10 (1887) 313-. Infinitesimal properties of circular complexes. Cosserat, E. C. E. 106 (1888) 1514-. complexes. Bourget, H. C. E. 104 . (1887) 1253-. ruled space. Koenigs, G. Par. EC. Norm. A. 11 (1882) 219-. Kinematic geometry (ruled space). Saussure, R. de. Am. J. Mth. 18 (1896) 304- ; 19 (1897) 329-. Line congruences, first differential projective invariant. Waelsch, E. C. E. 118 (1894) 736-. , and problem of Eibaucour. Cosserat, E. C. E. 118 (1894) 335-. surfaces, infinitesimal geometry. Waelsch, E. Wien Ak. Sb. 100 (1891) (Ab. 2a) 158-. , theorem of Darboux on orthogonal systems. Cosserat, E. Toul. Fac. Sc. A. 8 (1894) B, 9 pp. Normals to surface. Bonnet, 0. C. E. 52 (1861) 1081-. . Baehr, G. F. W. N. A. Mth. 2 (1863) 35-. Osculatory transformations of surfaces. Back- lund, A. V. [1875] Mth. A. 9 (1876) 297-. Plane curves with factorable parallels. Snyder, V. [1900] N. Y. Am. Mth. S. Bll. 7 (1901) 299-. Pseudospherical congruences. Candida, G. N. A. Mth. 17 (1898) 275-. Quadratic forms in geometry of complexes. Koenigs, G. Toul. Fac. Sc. A. 1 (1887) E, 13-. Euled surfaces. Bouquet, J. Liouv. J. Mth. 11 (1846) 125- ; Lyon Mm. Ac. 2 (1847) 385. Surfaces with circular generators. Demartres, G. Par. EC. Norm. A. 2 (1885) 123-. Translation surfaces. Lie, S. Leip. Mth. Ps. B. 44 (1892) 447-, 559-. and Abel's theorem. Lie, S. Leip. Mth. Ps. B. 48 (1896) 141-. , special class. Wiegner, G. Arch. Mth. Ntvd. 16 (1893) 325-. 8460 Rectification and quadrature of curves ; areas and volumes of surfaces. Other applications of the integral calculus to geometry. Arcs of algebraic curves, plane or skew, pro- perties. Humbert, G. Liouv. J. Mth. 1 (1895) 181-. conies, plane and spherical, and other curves, comparison. Maccullagh, J. Ir. Ac. P. 2 (1840-44) 507-. plane algebraic curves. Humbert, G. Par. EC. Pol. J. 57 (1887) 171-. curves, approximate calculation, application of theorem of Poncelet's. Resal, H. A. C. E. 94 (1882) 1375-. ,2, Bertrand's proposition on curve defined by sum or difference. Sucharda, A. Casopis 18 (1889) 49-; Fschr. Mth. (1889) 269. with rectifiable, and zones with quadrable difference. Lebesgue, V. A . Liouv. J. Mth. 11 (1846) 331-. Areas. Amsler's theorems, as regards related variable systems. Kleiber, J. Arch. Mth. Ps. 14 (1896) 405-. of anallagmatic curves. Liguine, V. As. Fr. C. E. (1881) 231-. approximation, by seven equidistant ordinates. Weddle, T. Camb. and Dubl. Mth. J. 9 (1854) 79-. calculation. Lorenz, L. Mth. Ts. 6 (1864) 33-. , formula, Poncelot's, error in. Chevilliet, . C. E. 80 (1875) 823-. , , Simpson's, degree of accuracy. Che- villiet, . C. E. 78 (1874) 1841-. , , , transformation. Saigey, J. F. N. A. Mth. 15 (1856) 291-. , Simpson's two rules, comparison, and Woolley's rule. Merrifield, C. W. Nv. Archt. T. 6 (1865) 40-. , , , . Purkiss, H. J. Nv. Archt. T. 6 (1865) 48-. of circular ring, subdivision. Davidson, G. Calif. Ac. P. 4 (1868-72) 278-, 290-. closed skew curve. Laisant, C. A. As. Fr. C. E. (1899) (Pt. 2) 135-. cones and cylinders. Azzarelli, M. Tor- tolini A. 8 (1857) 441-. conic, from trilinear equation. Ferrers, N. M. QJ. Mth. 2 (1858) 247-. , (Ferrers). Cayley, A. QJ. Mth. 2 (1858) 248-. conical, calculation, cases. Guimaraes, R. As. Fr. C. E. (1892) (Pt. 2) 166-. of conical wedge-surfaces. Schlomilch, 0. Schlomilch Z. 8 (1863) 142-. convex contours. Petrini, H. Stockh. Ak. Hndl. Bh. 19 (Afd. 1) (1894) No. 5, 58 pp. ; Fschr. Mth. (1893-94) 1079. 8460 Rectification and Quadrature. Areas 8460 of corresponding surfaces. Craig, T. (xn) J. H. Un. Cir. [1] (1882) 209-. curve and its in- and circum-scribed n-gons, Voller's theorem, and application to skew curves. Matthiessen, L. Schlomilch Z. 5 (1860) 146-. between curve and rectilinear base, calculation, formula. Como, G. da. (xn) Brescia At. Cm. (1878) 131-, 279; (1880) 152-. of curve T = -. Bender, C. Arch. Mth. 1 + tan a Ps. 47 (1867) 45-. curved surfaces. Borletti, F. G. Mt. 22 (1884) 207-. . Maggi, G. A. Km. K. Ac. Line. Ed. 5 (1896) (Sem. 2) 440-. cut off by spiral on hemisphere (theorem of Pappus). Resal, H. A. N. A. Mth. 20 (1881) 433-. of 'cycloid. Walton, W, Camb. and Dubl. Mth. J. 9 (1854) 263-. cylindrical figures. Wasteels, C. E. Mathesis 19 (1899) 9-. described by line moving in plane under given conditions. Bardelli, G. Mil. I. Lomb. Ed. 12 (1879) 290-. division, historico-critical notice. Favaro, A. Ven. I. Mm. 22 (1882) 129-. of ellipse. Mossbrugger, L. Grunert Arch. 6 (1845) 19-. and hyperbola. Grunert, J. A. Grunert Arch. 23 (1854) 385-, 477-. , problems. Matzka, W. Arch. Mth. Ps. 46 (1866) 300-. ellipsoid. Lobatto, R. Liouv. J. Mth. 5 (1840) 115-. . Catalan, E. C. Brux. Ac. Bll. 30 (1870) 97-. . Zurria, G. Catania. At. Ac. Gioen. 5 (1871) 97-. . Stickelberger, . D. Nf. B. (*1883) 56. , approximate values. Peano, G. Em. E. Ac. Line. Ed. 6 (1890) (Sem. 2) 317-. , and contour of ellipse, approximate expressions for. Boussinesq, J. C. E. 108 (1889) 695-. , elongated. Peyronny,A. N. A. Mth. 3 (1844) 466-. ellipsoidal zone. Wolfers, J. P. Grunert Arch. 22 (1854) 473-. zones, cases. Humbert, G. C. E. 109 (1889) 611-, 734-. ellipsoids of revolution. Grillo, S. N. A. Mth. 17 (1858) 272-. elliptic cone. Dienger, J. Tortolini A. 2 (1851) 119-. paraboloid. Tortolini, B. (vra) A. Mt. 4 (1861) 293-. on elliptic paraboloid, expression of. Humbert, G. Par. S. Mth. Bll. 21 (1893) 13-. , new. Humbert, G. Par. S. Mth. Bll. 21 (1893) 17-. of elliptic sectors. Grunert, J. A. Grunert Arch. 30 (1858) 472-. extension of term. De Morgan, A. Camb. and Dubl. Mth. J. 5 (1850) 139-. extension of term, first, to autotomic plane circuit. Muir, T. Ph. Mg. 45 (1873) 450-. of figure on sphere formed by small circle arcs. Wasteels, C. E. Mathesis 12 (1892) 105-. formed by arc of curve and two consecutive normals. Dahlander, G. R. Stockh. 6fv. 14 (1857) 269-. of lemniscate. Dippe, M. C. Crelle J. 16 (1837) 65-. between 2 loxodromic curves and meridian of sphere. Berg, F. J. van den. Amst. Arch. Wisk. Gn. 2 (1860-63) 1-, 10-, 23-. of loxodromic triangles on oblate spheroid. Grunert, J. A . Grunert Arch. 27 (1856) 143-. mean, of prismoid, and associated theorems. Echols, W. H. [1894] A. Mth. 9 (1894-95) of oblique cone. Schonbichler, K. Wien SB. 16 (1855) 447-. parabola, geometric determination. Mathiot, 0. L. Des Moines Anal. 9 (1882) 106-. parabolas of higher order. Schoute, P. H. [1896] C. E. 122 (1896) 1113- ; Amst. Ak. Vs. 5 (1897) 35. , theorem of Schoute. Korteweg, D. J. C. K. 122 (1896) 1399. , . Mannoury, G. C. E. 122 (1896) 1399-. pedal curves. Lecornu, L. As. Fr. C. E. (1894) (Pt. 2) 276-. and perimeter of 6-faced surface formed by orthogonal quadrics. Hoppe, R. Arch. Mth. Ps. 56 (1874) 354-. , to 3rd or 4th order, of geodesic circle. Bonnet, G. O. C. E. 97 (1883) 1360-. plane, approximate calculation. Mansion, P. (xn) Brux. S. Sc. A. 5 (1881) (Pt. 2) 231-. , . Harnack,A. Giving. 28 (1882) 257-. , . Petit Bois, G. Mathesis 5 (1885) 5-, 27-. , . Mansion, P. Mathesis 7 (1887) 77-. curvilinear, calculation. Leclert, E. A. Ge"n. Civ. 7 (1868) 630-. of 3 plane sections of surface. Olivier, T. Fe~russac Bll. Sc. Mth. 15 (1831) 312- . plane, of Simpsonian curves, approximate calculation. Petit Bois, G. Liege S. Sc. Mm. 18 (1895) No. 6, 19 pp. , theorem, extension to curved surfaces. Elliott, E. B. Mess. Mth. 10 (1881) 156-. of portions of ellipsoid. Ghysens, E. (xn) Brux. S. Sc. A. 2 (1878) (Pt. 2) 89-. projection of lemniscate on sphere. De- laire, . N. A. Mth. 15 (1856) 53-. quadric surfaces, central. Schlomilch, O. Leip. B. 14 (1862) 23-. right cone cut obliquely. Horn, W. Crelle J. 2 (1827) 364-. segment of conic. Cremona, L. G. Mt. 1 (1863) 360-. sinusoids, and Wallis's formula for. Man- sion, P. Mathesis 20 (1900) 183-. spherical surface between great and small circles which intersect. Edelmann, . Ger- gonne A. Mth. 3 (1812-13) 141-. zone. Escher, P. (vi Adds.) D. Nf. Vsm. B. 34 (1858) 143-. 623 Areas. Applications of Integral Calculus 8460 of spheroidal triangle, sides of which are lines of shortest distance. Puissant, L. Par. Mm. Ac. Sc. 10 (1831) 530-. star-polygons. Jacobi, C. G. J. Crelle J. 65 (1866) 173-. surface of elasticity. Roberts, W. Liouv. J. Mth. 11 (1846) 81-. --- ellipsoid, and surface of ellipsoidal vault on rectangular base. Plana, G. Crelle J. 17 (1837) 345-. -- , expression for. Strebor [Roberts, W.]. N. A. Mth. 9 (1850) 310. -- , -- . Faure, H. N. A. Mth. 11 (1852) 393-. -- , from perimeter, application to volumes. Linotte, L. Bologna Opusc. Sc. 2 (1818) 108-. surfaces. Essen, E. Grunert Arch. 22 (1854) 56-. -- , analogies to Viviani's theorem. Crocchi, L. G. Mt. 13 (1875) 170-. -- , by coordinates determining direction of normal to an element. Schlomilch, 0. Z. Mth. Ps. 11 (1866) 505-. , and length of curves. Goedseels, . Brux. S. Sc. A. 16 (1892) (Pt. 1) 79-. in polar coordinates. Bing, F. (xn) Ts. Mth. 4 (1874) 164-. -- , various. Schlomilch, 0. Leip. B. 18 (1866) 38-. theorem including Holditch's, and analogue in 3 dimensions. Elliott, E. B. Mess. Mth. 7 (1878) 150- . of trajectories described in plane motion of figure of invariable form. Liguine, V. Bll. Sc. Mth. As. 2 (1878) (Pt. 1) 306-. 2 triangles, problems in maximum. Dippe, M. C. Crelle J. 16 (1837) 65-. and volumes. Terquem, 0. N. A. Mth. 5 (1846) 232-. -- (shallow surfaces) , approximate formulae for. Geisenheimer, L. Z. Mth. Ps. 30 (1885) 325-. volumes, centroids, etc., application of approxi- mation formula for definite integrals. Li- gowski, W. Ev. Mar. et Col. 51 (1876) 159-. and volumes of conical wedge. Monteiro, A. S. G. Teix. J. Sc. 2 (1880) 68-, 81-, 110-. cylindrical ungula. Schell, W. Grunert Arch. 19 (1852) 70-. -- , determination. Juel, C. N. Ts. Mth. 8 (B) (1897) 49-; Fschr. Mth. (1897) 497. -- of ellipsoid, reduction of integrals for. Lobatto, R. ' Amst. Vs. Ak. 13 (1862) 193-. -- , fundamental problem. Rausenberger, 0. Mth. A. 43 (1893) 601-. generated by motion of curves and surfaces. NehU, C. Arch. Mth. Ps. 13 (1895) 225-, 337-. -- and lengths related to regular polygons. Gelin, E. Mathesis 11 (1891) 160-. -- , measurement. Sang, J. [1836] Edinb. N. Ph. J. 22 (1837) 231-. -- of surfaces of equation 8460 and volumes of surfaces generated by regular polygon rotating about a diameter. Sohncke, L. A. Arch. Mth. Ps. 48 (1868) 457-. tore (anchor ring). Gelin, E. Mathesis 10 (1890) 190- ; 11 (1891) 66-. in trilinear and quadriplanar coordinates. Slesser, G. M. QJ. Mth. 2 (1858) 357-. zone, etc. Stein, J. P. Gergonne A. Mth. 16 (1825-26) 257-. of zoneof pedal surfacesof quadrics. Schlomilch, O. Leip. B. 14 (1862) 51-. Argument - of plane curve F (x, y) = 0, Spitzer, S. D. Nf. B. (*1877) 111-. properties. Roberts, R. A. L. Mth. S. P. 15 (1883-84) 4-; 16 (1884-85) 238-. Boundary integral. Prym, F. E. [1869] Crelle J. 71 (1870) 305-. Closed plane curves, theorem. Risteen, A. D. A. Mth. 3 (1887) 104. Conical surfaces, integrals analogous to angle and its bisector. Humbert, G. C. B. 105 (1887) 739-. Conies, pedals and inverses of division of arcs. Strebor [Roberts, W.]. N. A. Mth. 11 (1852 182-. Cono-cuneus of Wallis, problem. Berg, F. J. van den. Amst. Arch. Wisk. Gn. 2 (1860-63) 1-, 10-, 23-. Curves, arc of which is elliptic or hyperelliptic integral of 1st kind. Schwering, K. Z. Mth. Ps. 25 (1880) 234-. , integral of 1st kind. Kiepert, L. [1874-76] Crelle J. Mth. 79 (1875) 304- ; Freiburg B. 7 (1880) 1-. , equals elliptic arc. Euler, L. [1781] St Pet. Ac. Sc. Mm. 11 (1830) 95-. , parabolic arc. Euler, L. [1781] St Pet. Ac. Sc. Mm. 11 (1830) 100-. , arcs of which can be expressed by circular arcs. Euler, L. [1781] St Pet. Ac. Sc. Mm. 11 (1830) 114-. , . Serret, J. A. [1852] Par. EC. Pol. J. 33 e cah. (1853) 69-. , represent elliptic functions. Serret, J. A. Liouv. J. Mth. 10 (1845) 421-. , Legendre's elliptic functions of 1st kind. Talbot, W. H. F. Gergonne A. Mth. 14 (1823) 380. conjugate hyperbolas, perimeters, two theorems. Roberts, W. Liouv. J. Mth. 13 (1848) 179-. on ellipsoid. Dienger, J. Tortolini A. 2 (1851) 104-. with equal arcs. Leaute, H. Toul. Ac. Sc. Mm. 6 (1874) 419-. quadrable by elliptic or circular functions, Clebsch's two theorems. Marie, M. C. E. 84 (1877) 227-. Cycloidal arcs. Tortolini, B. A. Mt. 7 (1865) 211-. Cylinder and sphere, problems. Berg, F. J. van den. Amst. Arch. Wisk. Gn. 2 (1860- 63) 1-, 10-, 23-. 624 8460 Application of Integral Calculus 8460 Division of surface of right cone on circular base. Lampe, E. Fschr. Mth. (1895) 736. Ellipse, circumference. Brinkley, J. [1802] Ir. Ac. T. 9 (1803) 145. , . Bakewell, T. W. Franklin I. J. 29 (1855) 369-. , , approximate calculation. Schldmilch, 0. Giving. 12 (1866) 31-. , , values. Cavallin, C. B. S. (xn) Ts. Mth. 1 (1883) 33-. , , expression by series. Hoppe, R. Grunert Arch. 3 (1843) 265-. , quadrant. Schl&milch, O. Z. Mth. Ps. 13 (1868) 530. Elliptic arc. Morin, A. N. A. Mth. 2 (1843) 493. arcs, calculation. Seewald, E. [1877] Wien Ak. Sb. 76 (1878) (Ab. 2) 365-. , comparison. Guimardes, R. G. Teix. J. Sc. 7 (1886) 111-. , property. Buzengeiger, K. H. J. Lindenau Z. 2 (1817) 98-. , Schmidt's tables. Verdam, G. J. Amst. I. (1842) 57-. functions,- application to problems of mean values. Mukhopadhyay, A. [1888] Beng. As. S. J. 58 (Pt. 2) (1890) 199-, 213-. Equal surfaces and lines cut off by common radii, systems. Hoppe, R. [1871] Ups. N. Acta S. Sc. 8 (1873) No. 4, 18 pp. Extension of Woolley's rule to integration of moments, proof. Merrifield, C. W. Nv. Archt. T. 8 (1867) 218-. Frustum of pyramid, divided in given ratio. Grunert, J. A. Grunert Arch. 36 (1861) 503-. Geodesic circles, Bertrand's theorems, demon- strations. Bonnet, G. O. Par. EC. Pol. J. 56 (1886) 143-. Hyperbola, arc and asymptote. Roberts, M. Ir. Ac. P. 3 (1847) 385. , rectangular, property of arcs. Talbot, W. H. F. E. S. P. 3 (1834) 258. Hyperbolic arc, Landen's theorem. Plana, G. Palomba Eac. 2 (1846) 233-. , . Griffiths, J. L. Mth. S. P. 11 (1879-80) 87-. , . Cayley, A. [1881] L. Mth. S. P. 13 (1881-82) 47-. , (Cayley). Griffiths, J. [1881] L. Mth. S. P. 13 (1881-82) 49-. arcs, comparison. Merrifield, C. W. Phil. Trans. (1859) 171-. Integration bv approximation. Martini, E. G. Mt. 3 (1865) 91-. of dz 2 -t- di/ 2 + dz 2 = ds 2 . Serret, J. A . Liouv. J. Mth. 13 (1848) 353-. = . Darboux, G. Liouv. J. Mth. 3 (1887) 305-. dx' 2 + dy 2 =ds* and analogous equations. Darboux, G. Liouv. J. Mth. 18 (1873) 236-. d 2 = Edu 2 + 2Fdu dv + Gdv*. Legoux , A . Toul. Fac. Sc. A. 3 (1889) F, 2 pp. Inverse of ellipse from centre, division of arcs. Strebor [Roberts, W.}. N. A. Mth. 11 (1852) 182-. Length of arc and of asymptote, difference. Fuss, P. H. St Pet. Ac. Sc. Mm. 9 (1824) 175-. on sphere and spheroid, difference, approximate formula. Bruns, H. As. Nr. 97 (1880) 73-. Cartesian oval arc, expression by elliptic functions. Roberts, S. [1873-75] (xi) L. Mth. S. P. 5 (1873-74) 6-; 6 (1874-75) 200. curve, first determination. Christensen, S. A. Ts. Mth. 5 (1887) 121- ; Bb. Mth. (1887) 76-. elliptic arc, geometrical researches. Camus, G. Yen. I. At. 1 (1883) 997-. helix on cone. Dienger, J. Grunert Arch. 16 (1851) 454-. line. Goedseels, (le It.) E. Brux. Ac. Bll. 16 (1888) 86-, [800]. Lengths of arcs. Ribaucour, A. (xi) Par. S. Phlm. Bll. 6 (1869) 12-. of curves, geometrical comparison. Mannheim, A. Par. EC. Pol. J. 40 e cah. (1863) 205-. Mean radius vector of curve, surface or solid. Drobisch, M. W. Leip. B. 10 (1858) 124-. Measurement of arcs of curves, etc. Tarry, G. As. Fr. C. E. (1893) (Pt. 2) 217-. convex surface of prism or truncated cylinder. Breton [de Champ], P. Liouv. J. Mth. 2 (1837) 133-. curved surfaces by ordinates. Merrifield, C. W. Nv. Archt. T. 6 (1865) 64-. prisms. Dieu, T. N. A. Mth. 16 (1857) 143-. Perimeter of ellipse. Le Francois, (Ing.) . A. Gen. Civ. 1 (1862) 206-. . Brocard,H.P.P.B. (xn) Mathesis 2 (1882) 180-. . Mansion, P. (xn) Mathesis 2 (1882) 211-. . Muir, T. Mess. Mth. 12 (1883) 149-. , and approximation of definite in- tegrals. Mansion, P. [1883] (xn) Brux. S. Sc. A. 8 (1884) (Pt. 1) 51-; (Pt. 2) 11-. , certain geometric means. Cavallin, C. B. S. [1882-83] (xn) Mathesis 3 (1883) 56-. , limits. Peyronny, A. N. A. Mth. 3 (1844) 232-. , volume, centroid, etc. of certain spaces, relations. Dainelli, U. G. Mt. 16 (1878) 279-. Perimeters of curves derived from conjugate hyperbolas, two theorems. Roberts, W. Liouv. J. Mth. 13 (1848) 179-. Planimeter, Amsler's, theory. Thire, A. N. A. Mth. 5 (1886) 353-. , polar, theory. Grunert, J. A. Arch. Mth. Ps. 51 (1870) 385-. Polyhedra of maximum volume for given surface. Lindelof, L. L. [1869] St Pet. Ac. Sc. Bll. 14 (1870) 257-. Prismatoid, Wittstein's. Bretschneider, C. A. Grunert Arch. 36 (1861) 18-. Quadrable cones. Sucksdorff, C. G. [1855] Helsingf. Acta 4 (1856) 617-. VOL. I. 625 RB 8460 Integral Calculus. Quadrature 8460 Quadrable figures on cylinders. Schlomilch, 0. Grunert Arch. 9 (1847) 149-. lanes. Clausen, T. Crelle J. 21 (1840) 375-; N. A. Mth. 8 (1849) 395-. Quadratrix of curve of with degree, conditions ior disappearance of periods. Marie, M. C. B. 76 (1873) 757-. , cyclic periods. Marie, M. C. B. 80 (1875) 872-; 84 (1877) 27-, 120-. , generalisation. Fouret, G. N. A. Mth. 5 (1886) 39-. , tangent, construction. Resal, H. A. N. A. Mth. 15 (1876) 337-. Quadrature. Servois, J. F. Gergonne A. Mth. 8 (1817-18) 73-. Menabrea, L. F. [1844] Tor. Mm. Ac. 8 (1846) 195-. Hill, C. J. D. Stockh. Ofv. 13 (1856) 15-. Tardy, P. [1865] Mod. Mm. S. It. 2 (1866) 133-. Oppermann, L. H. F. (xn) Ts. Mth. 1 (1871) 11-. TcJiebichejf, P. de. As. Fr. C. B. 2 (1873) 69-. Posse, K.A. N. A. Mth. 14 (1875) 49-. of algebraic curves, classification of integrals. Marie, M. C. B. 76 (1873) 692-, 943-; Par. EC. Pol. J. Cah. 44 (1874) 109-. algebraic and logarithmic. Raffy, L. Par. EC. Norm. A. 2 (1885) 185-. approximate. Halt, A. Des Moines Anal. 3 (1876) 1-. . Liventsov, A. I. [1878] (xn) Bee. Mth. (Moscou) 9 (1878-81) (Pt. 1) 569-. . Jadanza, N. Bv. Mt. 3 (1893) 15-. . Bardelli, G. Bv. Mt. 3 (1893) 16-. . Peano, G. Bv. Mt. 3 (1893) 17-. by approximation. Dupain, J. C. N. A. Mth. 17 (1858) 288-. . Bierens de Haan, D. Amst. Vs. Ak. 6 (1872) 185- ; Arch. Neerl. 8 (1873) 113-. and areas and volumes, simple formulae for. Francke, A. Hann. Archt.-Vr. Z. 21 (1875) 177-. of conies. Bary, . Gergonne A. Mth. 19 (1828-29) 245-. , elementary. Azzarelli, M. Em. N. Line. At. 32 (1879) 331-. , series 'for. Wallace, (Prof.) W. [1808] Edinb. B. S. T. 6 (1812) 269-. curves. Bgrard, . Gergonne A. Mth. 7 (1816-17) 101-. (Berard). Amptre, A. M. Gergonne A. Mth. 8 (1817-18) 117-. . Cost Jordens, G. H. Leijd. A. Ac. (1821-22) 64 pp. , method, new, any function. Berard, . Gergonne A. Mth. 7 (1816-17) 101-. , , unpublished, of Newton. De Morgan, A. Camb. and Dubl. Mth. J. 8 (1853) 93-. by polar coordinates. Lindman, C. F. Grunert Arch. 26 (1856) 471-. , principle. Steiner, J. Berl. B. (1840) by elementary methods. Gretschel, H. Arch. Mth. Ps. 42 (1864) 424-. of ellipsoid. Tortolini, B. G. Arcad. 78 (1839) '-. Carda, K. Mh. Mth. Ps. 7 (1896) 129-. of revolution. Tortolini, B. (vra) A. Mt. 4 (1861) 170-. elliptic paraboloid. Laciani, F. Tortolini A. 6 (1855) 406-. sectors. Grunert, J. A. Grunert Arch. 17 (1851) 313- ; 20 (1853) 217-. envelope of negative pedal. Tortolini, B. Palomba Bac. 2 (1846) 339-. exact, of certain curves. Breton [de Champ}, P. N. A. Mth. 2 (1843) 351-. , surfaces. Borchardt, C. W. Liouv. J. Mth. 19 (1854) 369-. of geodesic triangle, small, Gauss's theorem. Jacobi, C. G. J. Crelle J. 16 (1837) 344-. geometric. Hudson, C. I. Bv. 2 (1838) 206-. of hyperbola. Nizze, E. Grunert Arch. 26 (1856) 111-. . Essen, E. Grunert Arch. 27 (1856) 40-. , and hyperbolic natural* logarithms. Grunert, J. A. Grunert Arch. 25 (1855) 82-. , by method of limits. Prudot, H. N. A. Mth. 3 (1844) 565-. hyperbolic paraboloid. Tucci, F. P. Nap. At. Ac. 6 (1851) 69-. hypocycloids. Baur, C. W. Schlomilch Z. 4 (1859) 311-. from integrals of differentials in two variables, properties. Andreiewsky, . Mth. A. 4 (1871) 391-. of logarithmic spiral, y = a x . Rispal, A. N. A. Mth. 4 (1845) 117-. mechanical, theory. Stieltjes, T. J. Par. EC. Norm. A. 1 (1884) 409-. method. Catalan, E. C. Par. S. Phlm. PV. (1851) 16-. . Saviotti,C. CuyperBv.Un. 14(1883) 584-. , Gauss's. Callandreau, 0. C. B. 84 (1877) 1225- ; 90 (1880) 1067-. , . Radau, R. C. B. 90 (1880) 913-. , . Mansion, . Brux. S. Sc. A. 15 (1891) (Pt. 1) 57-. , . Raffy, L. N. A. Mth. 15 (1896) 249-. , , extension. Bourget, H. C. B. 126 (1898) 634-. , , generalisation. Christoffel, E. B. Crelle J. 55 (1858) 61-. , , remainder. Mansion, P. C. B. 102 (1886) 412- ; Brux. Ac. Bll. 11 (1886) 293-. , , and solution of Kepler's problem. Strehlke, F. Grunert Arch. 32 (1859) 433-. methods. Gaudard, J. A. Gen. Civ. 2 (1873) 480-. , comparison and new formulae. Parmentier, T. N. A. Mth. 14 (1855) 370- ; As. Fr. C. B. (1875) 179- ; (1882) 64-. , formula of Parmentier. Anon, (vi 892) N. A. Mth. 16 (1857) 11-. , formulas with equal coefficients. Radau, R. C. B. 90 (1880) 520-. , , practical. Piobert, . N. A. Mth. 13 (1854) 323-. 626 8460 Quadrature methods, Simpson's formula, elementary de- monstration. Steen, A. N. A. Mth. 10 (1871) 301-. , generalisation. Peano, G. Tor. Ac. Sc. At. 27 (1892) 608-. , Stirling's and Wallis's formulae, demonstra- tion. Mansion, . Brux. S. Sc. A. 23 (1899) (Pt. 1) 1-. of parabola. Voller, . Grunert Arch. 32 (1859) 420-. , cycloid, and logarithmic spiral. Stain- ville, de. Par. EC. Pol. Cor. 2 (1809-13) 434-. parabolas of 3rd degree. Parmentier, T. As. Fr. C. R. (1879) 150-. parabolic curves. Catalan, E. C. [1880] Brux. Ac. Sc. Mm. 43 (1882) (No. 8) 9 pp. segments cut off by focal chords. Grunert, J. A. Grunert Arch. 26 (1856) 351-. parallel surfaces. Roberts, W. Tortolini A. 1 (1850) 143-. plane curves. Carvallo, J. N. A. Mth. 1 (1842) 370-. polygons. Tessari, D. A. G^n. Civ. 3 (1864) 239-. problems reducible to, approximate solution. Monteiro da Rocha, J. Lisb. Ac. Sc. Mm. 1 (1797) 218-. of quadrics. Jellett, J. H. Camb. and Dubl. Mth. J. 1 (1846) 57-; Liouv. J. Mth. 12 (1847) 92-. . Azzarelli, M. Rm. N. Line. At. 29 (1876) 337-. , central. Schlomilch, 0. Liouv. J. Mth. 8 (1863) 89-. and rectification. Carmichael, R. Camb. and Dubl. Mth. J. 9 (1854) 266-. , reduction to "cubature" and quadra- ture respectively. Denzler, W. Ziir. Mt. 3 (1853-55) 75-. reduction to rectification. Plana, G. Palomba Rac. 2 (1846) 276-. of simple curves, synthetic demonstration of rule. Young, M. [1786] Ir. Ac. T. 1 (1787) 31. spherical curves, application of elliptic functions. Tortolini, B. [1849] Mod. S. It. Mm. 24 (1850) 337-. surface of cone. Sucksdorff, C. G. Helsingf. Acta 4 (1856) 121-. , ellipsoidal, and volume of surface of 8th degree, similarity of expressions for. Torto- lini, B. Tortolini A. 3 (1852) 530-. , reduction to rectification of curve, Euler's and Legendre's formulae compared. Plana, G. Palomba Rac. 2 (1846) 169-, 254-. surfaces. Schubert, F. T. St Pe"t. Ac. Sc. Mm. 9 (1824) 277-. . Dirksen, E. H. Berl. Ab. (1833) 123-. . Bedetti, G. Bologna N. Cm. 6 (1844) 341-. . Carmichael, R. Camb. and Dubl. Mth. J. 9 (1854) 266-. . Stem, A. (xn) Ts. Mth. 1 (1865) 38-. , formula. Roberts, W. Tortolini A. 1 (1850) 318-. Rectification 8460 of surfaces parallel to ellipsoid. Tortolini, B. G. Arcad. 119 (1849-50) 3-. surface of elasticity. Tortolini, B. Tortolini A. 7 (1856) 373-. resulting from intersection of cylinders. Lanciani, F. A. Mt. 7 (1865) 169-. , ruled, cases, and volumes of solids en- closed. Saladini, G. [1804] Bologna Mm. I. It. 1 (1806) 69-. , theorem. Jellett, J. H. Camb and Dubl. Mth. J. 5 (1850) 176-. , and volumes of solids. Tortolini, B. Palomba Rac. 1 (1845) 96-; Crelle J. 31 (1846) 12-; 34 (1847) 101-. vaults and lunes. Riccati, G. Verona S. It. Mm. 5 (1790) 48-. and volumes. Carvallo, J. N. A. Mth. 1 (1842) 370-. , approximate. Mansion, P. C. R. 95 (1882) 384-. , equations being given implicitly in Cartesian coordinates. Azzarelli, M. Rm. N. Line. At. 28 (1875) 134-. by infinite series. Trowbridge, D. Des Moines Anal. 7 (1880) 47-. of surface of elasticity. Roberts, W. Liouv. J. Mth. 11 (1846) 124. of y* = s - x*. Bonnet, O. N. A. Mth. 3 (1844) 75. /*\ 2 - - + \a/ Mth. Ps. 6 :1. Spitzer, S. Arch. 61 (1877) 329-. Ratio of triangle to sector of conic. Fabritius, W. As. Nr. 129 (1892) 49-. . Ball, L. de. As. Nr. 129 (1892) 283-. Rectifiable algebraic curves. Humbert, G. C. R. 104 (1887) 1051-. arcs of ellipse. Ocagne, M. d'. G. Teix. J. Sc. 7 (1886) 178-. curves. Schlomilch, 0. Z. Mth. Ps. 15 (1870) 124-. . Koenigsberger, .L. Mth. A. 32 (1887) 589-. , finding. Doppler, C. Grunert Arch. 19 (1852) 236-. on surfaces of revolution. Fuss, N. [1818] St Pdt. Ac. Sc. Mm. 8 (1822) 198-. plane algebraic curves. Humbert, G. Liouv. J. Mth. 4 (1888) 133-. Rectification. of algebraic curves. Kobb, G. Stockh. 6fv. (1887) 713- ; Fschr. Mth. (1887) 504. approximate. Pellet, A. E. Par. S. Mth. Bll. 19 (1891) 5-. , of circular arcs. Rankine, W. J. M. Ph. Mg. 34 (1867) 381-. , , case. Berg, F. J. van den. N. Arch. Wisk. 4 (*1878) 200- ; 10 (1884) 186-. , circumference of circle. Barrachina, E. S. Fschr. Mth. (1897) 447. , curves. Somoff, J. [5omov, O. I.] [1870] St Pet. Ac. Sc. Bll. 15 (1871) 257-; St Pet. Ac. Sc. Mm. (Rs.) 18 (*1871) 25-. 627 KR2 8460 Integral Calculus. Rectification 8460 approximate, of ellipse. Schlomilch, 0. Z.Mth. Ps. 10 (1865) 501-. , . Peano, G. C. B. 109 (1889) 960-. _, and trochoid. Rankine, W. J. M. [1864] Ph. Mg. 29 (1865) 22-. of arc of meridian, means of shortening calcu- lation. Puissant. L. C. E. 13 (1841) 53-. Cartesian ovals. Roberts, W. Liouv. J. Mth. 15 (1850) 194-. . Genocchi, A. C. B. 80 (1875) 112-. . Darboux, G. C. B. 87 (1878) 595-, 741. and other curves. Genocchi, A. Mathesis 4 (1884) 49-. cassinoid, n foci. Roberts, W. Liouv. J. Mth. 13 (1848) 38-. circular unicursal cubics. Longchamps, G. de. C. B. 104 (1887) 964-. class of curves of 4th degree. Darboux, G. C. B. 87 (1878) 692-. plane curves by elliptic functions. Roberts, W. Liouv. J. Mth. 10 (1845) 177-. conies. Hellins, J. Phil. Trans. (1802) 448-. , geometrical theorems. Maccullagh, J. [1830] Ir. Ac. T. 16 (1831) (pt. 2) 79-. convex curve, Legendre's formula, general- isation. Cavallin, C. B. 8. Ts. Mth. 2 (1884) 147- ; Fschr. Mth. (1884) 623. curve cutting generators of cone at constant angle. Turquan, . N. A. Mth. 4 (1845) 659-. curves. Beverley, T. Tilloch Ph. Mg. 67 (1826) 393-. . Gill, C. Ph. Mg. 9 (1831) 250-. . Dieu, T. N. A. Mth. 12 (1853) 210-. . Booth, J. A. Mt. 2 (1868-69) 81-. . Schlomilch, 0. Z. Mth. Ps. 15 (1870) 215. . Scheeffer, L. Acta Mth. 5 (1884) 49-. , cases. Roberts, W. Liouv. J. Mth. 12 (1847) 445-. , . Roberts, R. A. [1886] L. Mth. S. P. 18 (1886-87) 97-. , , remarkable. Longchamps, G. de. Mathesis 7 (1887) 127-, 170-. , 4th degree, singular case. Sang, E. Edinb. B. S. P. 7 (1872) 613-. , difficulty. Poisson, S. D. Par. EC. Pol. Cor. 3 (1814-16) 23-. , using projective measurement. Wirtinger, W. Mh. Mth. Ps. 5 (1894) 92-. cycloid. Peyronny, A. N. A. Mth. 3 (1844) 232-. . Dainelli, U. (xn) Bv. Sc.-Ind. 13 (1881) 369-. ellipse. Ivory, J. [1796] Edinb. B. S. T. 4 (1798) 177-. . Wilder, C. Silliman J. 18 (1830) 38-. . Verdam, G. J. [1851] Amst. Ts. Ws. Nt. Wet. 5 (1852) 42-. . Strehlke, F. Grunert Arch. 22 (1854) 444-. . Grunert, J. A. Grunert Arch. 27 (1856) 99-; 30 (1858) 213-. . Schlomilch, 0. Schlomilch Z. 6 (1861) 330-. of ellipse. Gaudard, J. A. Gen. Civ. 2 (1873) 483-. . Williot, V. A. Pon. Chauss. 12 (1896) 22-. , application of Taylor's theorem. Benz, 0. Arch. Mth. Ps. 8 (1890) 378-. , calculation, mode. Vaudoncourt, C. F. G. de. (vm) A. Cond. Pon. Chauss. 5 (1861)167-. , given, table for finding. Vaudoncourt, C. F. G. de. (vm) A. Cond. Pon. Chauss. 3 (1859) 252-. and hyperbola. Bangma, 0. S. Amst. N. Vh. 1 (1827) 55-. . Lobatto, R. Amst. N. Vh. 4 (1833) 115-. , resultant integrals. Fergola, N. [1828] Nap. At. Ac. 4 (1839) 13-. elliptic arc. Wallace, (Prof.) W. [1802] Edinb. B. S. T. 5 (1805) 253-. by series. Stainville, de. Par. EC. Pol. Cor. 2 (1809-13) 429-. arcs. Nieuport, C. F. F. de. Amst. Vh. 4 (1818) 171-. . Guimaraes, R. G. Teix. J. Sc. 8 (1887) 30-; Lisb. J. Sc. Mth. 12 (1888) 133-. by elliptic functions. Ramus, C. Kiob. Ov. (1843) 83-; Kiob. Dn. Vd. Selsk. Afh. 12 (1846) 95-. Fagnanian arcs of ellipse, and 2 corresponding arcs of hyperbola. Griffiths, J. (x) L. Mth. S. P. 5 (1873-74) 95-. , curves, arcs of which possess same property as. Griffiths, J. L. Mth. S. P. 11 (1879-80) 87-. Fagnano's point. Andrien, . Mathesis 20 (1900) .105-. theorem. Grunert, J. A. Grunert Arch. 26 (1856) 198-. . Kupper,C. Schlomilch Z. 1 (1856) 363-. . Talbot, W. H. F. Edinb. B. S. T. 23 (1861) 285-. . Malmsten, C. J. Schlomilch Z. 8 (1863) 306-. , analogous theorem on associated ellip- soids. Besge, V. A. Bordeaux Mm. S. Sc. 2 (1861-63) 247-. . Arcs of conies with rectifiable difference. Ghasles, M. C. B. 17 (1843) 838-. . . Serret, J. A. N. A. Mth. 3 (1844) 425-. , and associated points of ellipse. Lebesgue, V. A. N. A. Mth. 4 (1845) 573-. for any conic. Azzarelli, M. Bm. At. N. Line. 24 (1871) 336-; 25 (1872) 427-. , proof. Walker, J. J. L. Mth. S. P. 13 (1881-82) 172-. on sphere. Enneper, A. Schlomilch Z. 8 (1863) 231-. , stereometric analogues. Schlomilch, 0. Leip. B. 13 (1871) 13-. ; and surface of oblique cylinder, etc. Brinkley, J. [1802] Ir. Ac. T. 9 (1803) 145. first, and Torricelli. Loria, G. Bm. B. Ac. Line. Bd. 6 (1897) (Sem. 2) 318-. and Fontaine's theorem on triangle. Lecointe, (Vabbe) . N. A. Mth. 10 (1851) 196-. of geodetic curve. Young, (Dr) T. QJ. Sc. 21 (1826) 153-. 628 8460 Rectification graphic, of ellipse. Somoff, J. [1850] St Pet. Ac. Sc. Bll. 9 (1851) 97-. of hyperbola. Hellins, J. Phil. Trans. (1811) 110-. . Kiipper, C. Crelle J. 55 (1858) 89-. lemniscates and other curves. Eoberts, W. Ir. Ac. P. 2 (1844) 180-. lines of curvature on cylinders. Ahrendt, A. Arch. Mth. Ps. 8 (1890) 442-. ellipsoid. Roberts, M. [1871] A. Mt. 5 (1871-73) 17-. on surfaces. Boklen, O. Grunert Arch . 36 (1861) 32-. plane curves. Prouhet, E. N. A. Mth. 3 (1864) 403-. , cases. Tortolini, B. G. Arcad. 105 (1845) 193-. unicursal cubics. Eoffy, L. C. B. 107 (1888) 944-, 1030; Par. EC. Norm. A. 6 (1889) 103-. curves. Raffy, L. C. E. 104 (1887) 892-. and property of secondary caustics. Genocchi, A. A. Mt. 6 (1864) 97-. quadrature. Cauchy, A. L. [1832-41] Par. Mm. Ac. Sc. 22 (1850) 3-; C. E. 13 (1841) 1060-. . Grunert, J. A. Grunert Arch. 26 (1856) 48-. of conies. Azzarelli, M. Em. N. Line. At. 28 (1875) 287-. intersections of quadrics. Azzarelli, M. Em. N. Line. At. 29 (1876) 337-. involute of circle and of developable helicoid. Pressel, W. Grunert Arch. 4 (1844) 337-. spherical ellipse. Booth, J. E. S. P. 4 (1842) 387-; Ph. Mg. 25 (1844) 18-. toroid. Dienger, J. Grunert Arch. ,9 (1847) 438-. and volume. Dirksen, E. H. Berl. Ab. (1833) 123-. with reference to complex roots of algebraic equations. Lebesgue, V. A. N. A. Mth. 3 (1844) 145-. of roulettes. Steiner, J. Berl. Ab. (1838) 19-. same, pairs of curves having. Euler, L. [1781] St Pet. Ac. Sc. Mm. 11 (1830) 102-. of spherical curves. Tortolini, B. Tortolini A. 5 (1854) 71-. . ellipse and division of arcs. Tortolini, B. G. Arcad. 109 (1846) 231-. sphero-conics. Strebor [Eoberts, W.~\. N. A. Mth. 9 (1850) 141-. trisectrix of Maclaurin by elliptic integrals. Longchamps, G. de. C. E. 104 (1887) 676-. Euled surfaces, theorem. Thue, A. Thrond. Skr. (1896) No. 6, 6 pp. Sectors in space on same base. Zeuthen, H. G. (xn) Ts. Mth. 2 (1866) 60-. , theorems. Oppermann, L. As. Nr. 66 (1866) 129-. Segment of circle and Simpson's formula. Wittstein, T. Grunert Arch. 39 (1862) 12-. Skew curves rectifiable in algebraic terms. Stackel, P. Mth. A. 43 (1893) 171-. Volumes 8460 Skew curves, theorems, general. Hurwitz, A. Mth. A. 25 (1885) 287- ; 27 (1886) 162. Solid bounded by sphere and lemniscate cone. Azzarelli, M. Tortolini A. 8 (1857) 372-. Spherical curves analogous to Serret's. Eoberts, W. Liouv. J. Mth. 12 (1847) 479-. . Serret, J. A. Liouv. J. Mth. 12 (1847) 480-. Surface of body in terms of a volume-in- tegral. Tait, P. G. Edinb. E. S. P. 9 (1878) 541-. conoid with elliptic base. Korteweg, D. J. N. Arch. Wisk. 3 (*1877) 60-. , 4th degree, o?z = x 2 i/ 2 , cubature, etc. Schlomilch, 0. Grunert Arch. 12 (1849) 193-. of ellipsoid, integrations over. Tallqvist, H. Helsingf. Acta 18 (1891) 329-. in terms of Weierstrass's elliptic functions. Boyd, J. H. [1892] A. Mth. 7 (1892-93) 1-. integrals reducible to contour, theory. Ee-thy, M. (xn) Mag. Tud. Ak. Etk. (Mth.) 3 (1875) (No. 7) 20 pp. of oblique cyclinder, Brinkley's theorem. Grunert, J. A. Grunert Arch. 10 (1847) 222-. Tetrahedrometry, differential formula. Milller, J. H. T. Schlomilch Z. 5 (1860) 49-. Theorem. Mendeleef, . C. E. 121 (1895) 421-. of Holditch, extension. Echols, W. H. A. Mth. 4 (1888) 47-. Pappus. Henck, J. B. Camb. (M.) Mth. M. 1 (1859) 200-. Viviani, generalisation. Weinberg, W. P. Fschr. Mth. (1900) 624-. Vaulted surfaces (generated by semicircles on radii vectores of plane curve). Schlomilch, 0. Schlomilch Z. 8 (1863) 121-. Volumes. application of Simpson's rule. Isely, (Prof.) . [1869] Neuch. Bll. 8 (1870) 262-. . to cylindrical wedge. Graeber, . Arch. Mth. Ps. 17 (1900) 401-. and areas, determination. Ghysens, E. Brux. Ac. Bll. 44 (1877) 532-. bounded by ruled surface and parallel ends. Chelini, D. G. Arcad. 96 (1843) 3-. calculation. Gaudard, J. A. Ge"n. Civ. 2 (1873) 483. ; Maleyx, L. N. A. Mth. 19 (1880) 529-. , approximate. Daug, H. T. [1865] Stockh. Ofv. 22 (1866) 415-. , . Mansion, P. (xn) Brux. S. Sc. A. 6 (1882) (Pt. 2) 228-. , formulae, application to ships. Grunert, J. A. Arch. Mth. Ps. 47 (1867) 176-. t formula. Latars, G. (xn) Isere S. Bll. 11 (1882) 341- ; 13 (1884) 18. for pedal surface of ellipsoid. Magener, A. Grunert Arch. 34 (1860) 450-. solids. Dirksen, E. H. Berl. Ab. (1833) 123-. , cases. Finck, B. N. A. Mth. 7 (1848) 241-. 8460 Integral Calculus. Volumes 8460 calculation for solids derived from ellipsoid. Dienger, J. Grunert Arch. 12 (1849) 81-. of revolution. Collignon, E. As. Fr. C. E. 10 (1881) 196-. wave surface. Roberts, W. (vm) A. Mt. 4 (1861) 345-; (v) C. B. 55 (1862) 503. and centroid of class of solids, formula for. Lobatto, R. (vm) Amst. N. Ws. Ntk. Vh. 2 (1854) 121-. of class of solids. Grunert, J. A. Grunert Arch. 31 (1858) 481-. conical and cylindrical ungulse. Timmer- mans, J. A. Quetelet Cor. Mth. 2 (1826) 149-. heaps of ore. Heyse, G. Karsten Arch. 5 (1832) 511-. wedge. Hube, K. (vi Adds.) Krk. Boczn. Uniwers. 8 (1823) 115-. cylinder cut by plane oblique to base. Eilles, J. Arch. Mth. Ps. 42 (1864) 186-. (Eilles). Lobatto, R. Arch. Mth. Ps. 43 (1865) 235-. determination by help of centroid. Blazek, G. Wien SB. 46 (Ab. 2) (1863) 342-. , in Loba6evskij's geometry. Simon, M. Mth. A. 42 (1893) 471-. of ellipsoid, geometrical determination. Ma- thiot, 0. L. Des Moines Anal. 8 (1881) 124-. equal, division of ellipsoid into. Weddle, T. Mathematician 2 (1847) 302-. generated by closed contour in any motion. Koenigs, G. Liouv. J. Mth. 5 (1889) 321-. turning about all straight lines of space, distribution. Koenigs, G. C. B. 106 (1888) 927-. contour rigidly connected with trihedron of curve. Koenigs, G. C. B. 107 (1888) 474-. revolving arc of circle. Noel, J. N. Quetelet Cor. Mth. 6 (1830) 61-. Guldin's rule. Grunert, J. A. Grunert Arch. 32 (1859) 348-. , extension. Richter, P. B. Z. Mth. Ps. 37 (1892) 172-. theorem. Bordoni, A. [1826] Mod. Mm. S. It. 20 (1828) 86-. , extensions. Jung, G. Bm. B. Ac. Line. T. 7 (1883) 97-. , generalisation. Fabry, . [1887-90] Mntp. Ac. Mm. 11 (1892) 257-. , . Kuscow, . N. A. Mth. 17 (1898) 209-. integrals, classification. Geometrical definition of surfaces capable of cubature. Marie, M. C. B. 80 (1875) 757-. of loci of connected points. Elliott, E. B. [1882] L. Mth. S. P. 14 (1883) 62-. paraboloid, geometrical determination. Mathiot, 0. L. Des Moines Anal. 10 (1883) 46. of revolution, extension of theorem of Archimedes to. Haedicke, H. Dingier 237 (1880) 165, 328. pedal surfaces. Hirst, T. A. C. B. 55 (1862) 572- ; Phil. Trans. (1863) 13-. of portion of elliptic cone cut off by two planes. Unferdinger, F. Arch. Mth. Ps. 41 (1864) 178-. right circular cone between a circular and a hyperbolic segment. Zambelli, A. Yen. Aten. At. 12 (1875) 267-. prismatoid, formula. Becker, J. C. [1877] Z. Mth. Ps. 23 (1878) 412-. between ruled surface and two parallel planes. Terrier, L. [1874] Neuch. S. Sc. Bll. 10 (1876) (App.) 6-. of ruled surfaces. Haillecourt, A. Bordeaux Ac. Act. 39-40 (1877-78) 155-. segment of surface, 2nd degree, bounded by parallel planes. Booth, J. Ph. Mg. 20 (1842) 472-. segments, and frusta of quadrics. Un- ferdinger, F. [1869] Wien Sb. 60 (1870) (Ab. 2) 631-. ship-shaped bodies. Grunert, J. A. Grunert Arch. 13 (1849) 443-. skew surface. Boulanger, A. N. A. Mth. 16 (1897) 171-. solid bounded by hyperbolic paraboloid. Kawalki, W. [1899] Hamb. Mth. Gs. Mt. 3 (1900) 400-. , formula of " three levels" for. Goulard, . Mathesis 17 (1897) 105-. generated by motion of plane in space. Christiansen, C. (xn) Ts. Mth. 1 (1865) 163 -. solids of revolution. Sinram, T. Hamb. Mth. Gs. Mt. 1 (1889) 20-. , J. Wilson's method. Clark, T. Glasg. Ph. S. P. 2 (1844-48) 161-. spherical segment. Les Enfants, J. Les Mondes 3 (1882) 173-. surface, 4th degree, pedal of hyperboloid. Tortolini, B. [1847] Mod. S. It. Mm. 24 (1848) 378-. of revolution given in polar coordinates. Boije af Gennas, . Ts. Mt. Fys. 3 (1870) 193-. surfaces and lengths of arcs of figures described by straight line. Schumann, A. Z. Mth. Ps. 25 (1880) 87-. of revolution generated by conies, and areas of conies. Gaboardi, S. Brugnatelli G. 10 (1827) 266-. , sections of which parallel to plane are quadratic functions of distance. Wein- meister, . Arch. Mth. Ps. 17 (1900) 190-. and surfaces of 2 solids of revolution. Dostor, G. Arch. Mth. Ps. 68 (1882) 421-. of truncated cone by indeterminate coefficients. Ivon, L. N. A. Mth. 2 (1843) 23-. ungulse. Lehrnus, (Dr) . (vin) Z. Bauw. 4 (1854) 573-. various bodies. Stamkart, F. J. (vm) Amst. N. Ws. Ntk. Vh. 2 (1854) 187-. zones of equal height of solids of revolution, formulas. Winkhaus, W. Crelle J. 44 (1852) 375. 630 8470 Special Transcendental Curves 8470 8470 Special transcendental curves. (For cycloids, epicycloids and tro- choids see 8420.) Catenary. Hill, T. Gould As. J. 2 (1852) 10-. , areas and volumes related to. Wasteels, C. E. Mathesis 16 (1896) 241-. .centrifrugal. Webb,J.B. Am. As. P. (1889) 76. , constants. Loffler, A. Grunert Arch. 36 (1861) 323-. , construction, graphic method. Palmer, W. K. Kan. Un. Q. 7 (1898) 211-. of equal strength, involute. Azzarelli, M. Palomba Eac. 4 (1848) 9-, 17-, 35-, 57-. -, , identity of tractrix with. Calegari, P. (vn) Bologna Opusc. Sc. N. Col. (1824) 233-. and other transcendental curves, properties. Masetti, G. Bologna Opusc. Sc. 4 (1823) 179-. , evolute (Huygens' problem). Egger, E. A. Mt. 6 (1864) 19-. , mechanical description. Robillier, . Angers Mm. S. Ag. 1 (1831) 41-. , by parabolic trigonometry. Booth, J. R. S. P. 8 (1856-57) 443-. problem, Huygens' solution. Korteweg, D. J. Bb. Mth. 1 (1900) 97-. , spherical. Greenhill, A. G. [1894] L. Mth. S. P. 27 (1896) 123-. on surface of revolution. Molins, H. Toul. Mm. Ac. 5 (1849) 261-. , vertical force on elements and distance of elements from horizontal plane. Molins, H. Toul. Mm. Ac. 4 (1848) 221-. Closed class of transcendental curves. Schicer- ing, K. Z. Mth. Ps. 20 (1875) 457-. Curve of elliptic functions of 1st kind. Serdobinskij, V. E. [1893] Rec. Mth. (Moscou) 17 (1895) 337-; Fschr. Mth. (1893- 94) 1158. logarithmic sines. Barbour, L. G. Des Moines Anal. 4 (1877) 114-. , evolute. Barbour, L. G. Des Moines Anal. 4 (1877) 169-. second sines and its variations. Sang, E. [1874] Edinb. R. S. P. 8 (1875) 356-. r = - . Fontana, G. Verona S. It. Mm. 2 (1784) 123-. Differential curve of parabola, foci. Hochheim, A. [1875] Arch. Mth. Ps. 58 (1876) 278-. Elastic curve. Halphen, G. Par. EC. Pol. J. 54 (1884) 183-. . Ise, E. Nap. Ac. Pont. At. 18 (1888) 203-. . Tedone, 0. Rm. R. Ac. Line. Rd. 3 (1894) (Sem. 1) 265-. , intrinsic equation. Greenhill, A. G. Mess. Mth. 8 (1879) 82. Evolutes of circular arcs. Poinsot, L. Par. EC. Pol. Cor. 1 (1804-08) 245-. Exponential curve. Church, I. P. Des Moines Anal. 1 (1874) 84-. Helices. Tissot, A. N. A. Mth. 11 (1852) 454-. , characteristic property. Appell, P. Arch. Mth. Ps. 64 (1879) 19-. and helicoidal surfaces. Richardson, J. M. Franklin I. J. 33 (1857) 231-. , problem. Cesaro, E. G. Mt. 24 (1886) 46-. , theory. Tos y Feitto, J. de. [1898] Barcel. Ac. Mm. 2 (1892-1900) 307-. , , general. Schoelcher, (le col.) . As. Fr. C. R. (1891) (Pt. 1) 160. , of Torricelli. Ferroni, P. Mod. S. It. Mm. 15 (1811) 60-. Helix, only curve with ratio of curvatures con- stant. Bertrand, J. Liouv. J. Mth. 13 (1848) 423-. , cylindrical, lines connected with. Piron- dini, G. Crelle J. Mth. 121 (1900) 245-. projected into cycloid. Olivier, T. Par. S. Phlm. PV. (1847) 106-. , property. Reye, T. Z. Mth. Ps. 15 (1870) , sense. Anon, (vi 889) N. A. Mth. 14 (1855) 175-. Hirundo. Hill, (Rev.) T. Am. As. P. (1886) 67. Hypergeometric curves. Multedo, . Genova Mm. Ac. 3 (1814) 328-. Involute of circle, Bakewell, T. W. Franklin I. J. 7 (1831) 99-. . Crofton, M. W. [1864] Mess. Mth. 3 (1866) 93-. . D'Auria, L. Franklin I. J. 80 (1880) 45-. . Skinner, J. J. Franklin I. J. 80 (1880) 105-. Involutes of circle. Hill, T. Am. As. P. 19 (1870) 21-. depending on Diophantine equation. Sylvester, J. J. L. Mth. S. P. 2 (1869) 137-. and their relation to analytic functions. Thoman, F. C. R. 26 (1848) 498-. , successive. Sylvester, J. J. Ph. Mg. 36 (1868) 295-, 459-. Kumaids. (Plane curves becoming ellipses on cylinder.) Vieth, G. U. A. Gilbert A. 54 (1816) 311-. Logarithmic curve', researches of Torricelli. Loria, G. Bb. Mth. 1 (1900) 75-. exponential curves. Vincent, A. J. H. Gergonne A. Mth. 15 (1824-25) 1-; 16 (1825- 26) 92-. parabola. Booth, J. Camb. and Dubl. Mth. J. 7 (1852) 35-. Logocyclic curve. Booth, J. R. S. P. 9 (1857- 59) 256-; QJ. Mth. 3 (1860) 38-, 127-. . or strophoid. Gilnther, A. W. S. (xn) Mathesis 1 (1881) 81-. Loxodromes of cone. Piccioli, H. N. A. Mth. 18 (1899) 508-. Mechanical properties of some curves. Heulen, J. van. N. Arch. Wisk. 7 (*1881) 33-. Normals to several mechanical curves, con- struction. Chasles, M. [1829] (ix) Par. S. Mth. Bll. 6 (1878) 208-. Plane curves, Roberval's method. Geer, P. van. N. Arch. Wisk. 11 (1884) 28-; Fschr. Ps. (1884) (Ab. 1) 217-. 631 8470 Spirals Plane curves, theory. Enneper, A. Gott. Nr. (1884) 364-. transcendental curve, arc, representation of elliptic function, 1st species, by. Torto- lini, B. Tortolini A. 4 (1853) 485-. curves belong to system (/*, v). Fouret, G. Par. S. Mth. Bll. 2 (1874) 96-. " Points de dedoublement. " Mansion, P. Bll. Sc. Mth. As. 3 (1879) (Pt. 1) 514-. Singular curves. Peano, G. Tor. Ac. Sc. At. 26 (1891) 299-. Sinusoidal curves. Cesdro, E. N. A. Mth. 13 (1894) 102-. Spirals. Pagani, G. M. Brux. Mm. Cour. 5 58 pp. Olivier, T. Par. S. Phlm. N. Bll. (1833) 21-. Hill, T. Am. As. P. (1859) 158. Schou, T. V. Mth. Ts. 2 (1860) 6-. La Gournerie, J. de. G. B. 66 (1868) 283-, 832-; Liouv. J. Mth. 14 (1869) 9-, 103-. approximate construction. Thallmayer, V. Dingier 226 (1877) 156-. of Archimedes, minimal property. Janisch, E. Arch. Mth. Ps. 9 (1890) 445-. class, properties. Laquiere, E. M. N. A. Mth. 2 (1883) 118-. conical, graphic method for tangents. Gar- binski, H. Gergonne A. Mth. 16 (1825-26) 167-. equiangular, geometrical proofs. Whitworth, W. A. Mess. Mth. 1 (1862) 5-, 138-. harmonic. Eaffy, L. Par. EC. Norm. A. 12 (1895) 145-. hyperbolic or inverse. Palmer, W. K. Kan. Un. Q. 7 (1898) 155-. , tangents. Schiffner, F. Arch. Mth. Ps. 66 (1881) 334-. logarithmic. Turquan, . N. A. Mth. 5 (1846) 88-. . Pirondini, G. Mathesis 19 (1899) 153-. , construction. Wiebe, F. K. H. Berl. Pol. Gs. Vh. 23 (1862) 106-. , history of applications. Favaro, A. Bb. Mth. (1891) 23-. , and involute of circle. Orlando, L. Ma- thesis 20 (1900) 110-. , new. Binet, J. P. M. Par. S. Phlm. PV. (1841) 15. new. Drach, S. M. Mathematician 3 (1850) 34-. peculiar. Wolfers, J. P. Grunert Arch. 28 (1857) 114-. properties. Laguerre, E. (x) Par. S. Phlm. Bll. 6 (1869) 39-. system, intrinsic equations. Onnen, H. Arch. Neerl. 10 (1875) 361-. r=a^-. Stockly, L. Arch. Mth. Ps. 48 (1868) 109-. where p=r. Schlomilch, 0. Z. Mth. Ps. 14 (1869) 162-. p n = Asinnu. Haton de la Goupilliere, J. N. N. A. Mth. 15 (1876) 97-. r0=asin0. Jung, G. Bm. B. Ac. Line. T. 7 (1883) 97-. Transcendental Surfaces 8480 p m - p n =(a m -a n ) -, properties. Cabreira, A. Lisb. J. Sc. Mth. 5 (1898) 1-; Fschr. Mth. (1897) 536. p=Rsio.fj.u, algebraic if /u. is integral, trans- cendental otherwise. Loria, G. Fschr. Mth. (1899) 515. Strophoidals. Longchamps, G. de. Mathesis 14 (1894) 138-. Tractrix. Gesdro, E. (xn) Mathesis 2 (1882) 217-. of curve with planimetric application. Leh- mer, D. N. A. Mth. 1 (1900) 14-. , polar. Webb, J. B. Am. As. P. (1889) , problem. Dubuat, . Gergonne A. Mth. 4 (1813-14) 332-; 5 (1814-15) 210-. Trajectories, isogonal, of co-axal circles. Lodge, A. Mth. Gz. 1 (1900) 213-. , , . Macaulay, F. S. Mth. Gz. 1 (1900) 214. Transcendental curve, properties. Fuss, N. [1817] St Pet. Ac. Sc. Mm. 8 (1822) 147-. , . Azzarelli,M. (vn) A. Mt. 5 (1863) 72-. , . Egger, E. A. Mt. 6 (1864) 21-. curves, properties of two. Collins, E. [1812] St Pet. Ac. Sc. Mm. 6 (1813-14) 86-. Trigonometric curves, circular and hyperbolic analogies. Anon, (vi 888) N. A. Mth. 14 (1855) 151-. y=tfc. Milller, Hub. Arch. Mth. Ps. 44 (1865) 128. sin y sin my = a sin x sin nx + b. Newton, H.A., & Phillips, A. W. [1875] Conn. Ac. T. 3 (1874-78) 97-. 8480 Special transcendental surfaces. Curves and surfaces. Pirondini, G. G. Mt. 26 (1888) 352-. Envelope of sphere, centre of which describes cycloid, edge of regression. Livet, . Par. EC. Pol. Cor. 1 (1804-08) 195. Helical surfaces, theory. Stiickel, P. Leip. Mth. Ps. B. 50 (1898) (Mth.) 3-. Helicoid. Schiffner, F. Arch. Mth. Ps. 7 (1889) 54-. , general. Issaly, . N. A. Mth. 19 (1900) 499-. , principal radii of curvature. Paignon, F. M. N. A. Mth. 4 (1845) 413-. of revolution, properties of lines of shadow. Negri, C. Tor. Ac. Sc. At. 14 (1878) 116-. Helicoids. Enneper, A. Gott. Nr. (1870) 335-. . Bianchi, L. [1878] G. Mt. 17 (1879) 9-. . Fouret, G. As. Fr. C. B. 7 (1878) 173-. . Schiffner, F. Arch. Mth. Ps. 68 (1882) 72-. . Machovec, F. Casopis 13 (1884) 111-; Fschr. Mth. (1884) 519-. 632 8490 Transcendental Surfaces Helicoids. Pirondini, G. N. A. Mth. 6 (1887) 87- ; A. Mt. 16 (1888-89) 137-. , curvature and indicatrix. Sobotka, J. Wien Ak. Sb. 101 (1892) (Ab. 2a) 899-. with director plane. Catalan, E. C. Par. S. Phlm. PV. (1843) 113-. , sections. Pecharman, P. C. B. 103 (1886) 987-. , orthogonal and tangential. Pirondini, G. Ev. Mt. 2 (1892) 127-. Skew helicoids. Bruno, G. Tor. At. Ac. Sc. 2 (1866-67) 601- ; 3 (1867-68) 194-. , contact-curves and enveloping torses of. Schmid, T. Wien Ak. Sb. 99 (1891) (Ab. 2a) 952-. , developable. Lamarle, E. Brux. Ac. Bll. 11 (1861) 321-. , surfaces connected with. Sobotka, J. Mh. Mth. Ps. 4 (1893) 59-. surfaces, asymptotic lines. Bioche, C. Toul. Fac. Sc. A. 2 (1888) A, 7 pp. Spiral surfaces, characteristic property, and theorem of Bibaucour. Demoulin, A. Par. S. Mth. Bll. 23 (1895) 198-. , linear element. Demoulin, A. C. B. 118 (1894) 337-. with parallel lines of curvature, linear element. Eaffy, L. Par. S. Mth. Bll. 20 (1892) 22-. , theory. Ebner, . Arch. Mth. Ps. 14 (1896) 241-. Surface, equation of which is e r = ' , Padu- la's work. Eubini, E. N. A. Mth. 15 (1856) 183-. generated by logarithmic curve. Bruno, G. [1879] Tor. Ac. Sc. At. 14 (1878) 735-. Surfaces associated with Peterson's. Mlodze- evskij, B. K. Bee. Mth. (Moscou) 21 (1901) 450- ; Fschr. Mth. (1900) 613. Transcendental surfaces including cyclide as case. Holzmilller, F. G. [1882] Crelle J. Mth. 94 (1883) 239-. 8490 Hypergeometric configura- tions and higher elements of Curvature of ' ' curves " in n dimensions. Lands- berg, G. Crelle J. Mth. 114 (1895) 338-. higher varieties. Voss, A. E. Mth. A. 16 (1880) 129-. , measure. Beez, E. (vn) Mth. A. 7 (1874) 387-; (ix) Z. Mth. Ps. 21 (1876) 373-; 24 (1879) 1-, 65-. , measure, Gauss's, theory. Alle, M. [1876] Wien Ak. Sb. 74 (1877) (Ab. 2) 9-. , -, Biemann's, theory. Voss, A. E. Mth. A. 16 (1880) 571-. of varieties. Berzolari, L. Tor. Ac. Sc. At. 33 (1897) 462-, 527- or 692-, 759-. Curves of triple curvature and their parallels. Hoppe, E. Arch. Mth. Ps. 65 (1880) 373-. 4th dimension, uniform positive curvature in. Newcomb, S. Crelle J. Mth. 83 (1877) 293-. Hypergeometry 8490 3-dimensional varieties. Cotton, E. Toul. Fac. Sc. A. 1 (1899) 385-. with one zero and two equal curva- tures. Banal, E. A. Mt. 24 (1896) 213-. 3- and higher dimensional space, curvature. Hadamard, . Bordeaux S. Sc. PV. (1897- 98) 85-. 4-dimensional geometry. Lovett, E. 0. [1900] N. Y. Am. Mth. S. Bll. 7 (1901) 88-. n-dimensional curves, intrinsic treatment. Cesdro, E. Em. B. Ac. Line. Ed. 5 (1889) (Sem. 2) 165-. Euclidean space, expansion of coordinates in powers of arc. Eichmond, H. W. QJ. Mth. 31 (1900) 315-. rotation ellipsoid, geodesies on, application of calculus of variations. Eudio, F. Ziir. Vjschr. 43 (1898) 340-. space. Keyser, C. J. N. Y. Am. Mth. S. Bll. 4 (1898) 177, 181-. , differential geometry. Lovett, E. O. [1900] N. Y. Am. Mth. S. Bll. 7 (1901) 203, 210. , linear, metrical properties of skew curves. Brunei, G. E. A. [1881] Mth. A. 19 (1882) 37-. spheres, volumes and surfaces of. Story, W. E. [1880] (xn) J. H. Un. Cir. [1] (1882) 48. theory of curves. Hoppe, E. Arch. Mth. Ps. 6 (1888) 168-. Elliptic space, helicoidal and ruled surfaces. Gigli, D. Mil. I. Lomb. Ed. 33 (1900) 717-. Generalisation to p-dimensional space of trans- formation of line into surface integrals. Franchis, M. de. Palermo Cir. Mt. Ed. 12 (1898) 163-. Hyperspace, Codazzi's formulae in. Cesdro, E. Nap. Ed. 33 (1894) 87-. , complex variables in. Volterra, V. Em. B. Ac. Line. Ed. 5 (1889) (Sem. 1) 291- ; 6 (1890) (Sem. 2) 241-. , extension of Euler's and Meunier's theo- rems to. Berzolari, L. Em. E. Ac. Line. Ed. 6 (1897) (Sem. 2) 283-; 7 (1898) (Sem. 1) 4-. , functions. Fabri , C. Yen. I. At. (1892-93) 283-. , , and their differential parameters. Vol- terra, V. Em. E. Ac. Line. Ed. 5 (1889) (Sem. 1) 630-. , theory. Eicci, G. Em. B. Ac. Line. Ed. 4 (1895) (Sem. 2) 232-. Hyperspheric lines of constant curvature. Piccioli, H. N. A. Mth. 19 (1900) 385-. Knotted curve, equation, and solution in 4th dimension. Hoppe, E. Arch. Mth. Ps. 64 (1879) 224. , Hoppe' s. Durege, H. [1880] Wien Ak. Sb. 82 (1881) (Ab. 2) 135-. , solution in 4th dimension. Hoppe, E. Arch. Mth. Ps. 65 (1880) 423-. Lobacevskij's planimetry, concrete interpreta- tion. Lachtin, L. K. Eec. Mth. (Moscou) 17 (1895) 767-; Fschr. Mth. (1895) 534-. Non-Euclidean space, geodesic geometry of surfaces. Whitehead, A. N. L. Mth. S. P. 29 (1898) 275-. 633 8800 Differential Geometry Osculating sphere of curve and analogues in n dimensions. Hoppe, B. Arch. Mth. Ps. 12 (1894) 96-. Parallel varieties, invariants. Forsyth, A. R. B. A. Bp. (1899) 640-. Pseudospherical surfaces. Bianchi, L. A. Mt. 24 (1896) 347-. varieties of three dimensions. Brill, A. Mth. A. 26 (1886) 300-. Pseudo-surfaces, geometrical study of curva- ture. Issaly, . Par. S. Mth. Bll. 17 (1889) 84-. Space of constant negative curvature, funda- mental equations in geometry. Kagan, H. N. A. Mth. 14 (1895) 20-. Spaces of constant curvature. Banal, R. Em. E. Ac. Line. Ed. 7 (1898) (Sem. 1) 7-. , connexion with projective spaces. Schur, F. Mth. A. 27 (1886) 537-. , metrics. Giudice, F. Tor. Ac. Sc. At. 35 (1900) 119- or 193-. Differential geometry; applications of differential equations to geo- metry. 8800 General. Automorphic curves and surfaces, family. Klein, F., & Lie, S. C. E. 70 (1870) 1222-, 1275-. Curves represented by Mth. 10 (1881) 21-. Developable surfaces. Kobb, G. Par. S. Mth. Bll. 19 (1891) 1-. Differential equation of conies. Halphen, G. H. Par. S. Mth. Bll. 7 (1879) 83-. , 1st order, curves denned by. Picard, E. C. E. 120 (1895) 522-. , , , asymptotes. Zeuthen, H. G. N. Ts. Mth. 4 (B) (1893) 73-. of all parabolas. Mukhopadhyay, A. [1888] Beng. As. S. J. 57 (Ft. 2) (1890) 316-. for points of curve or surface. Halphen, G. H. C. E. 81 (1875) 1053-. equations, curves denned by. Poincare', H. C. E. 93 (1881) 951- ; Liouv. J. Mth. 7 (1881) 375-; 8 (1882) 251- ; C. E. 98 (1884) 287-; Liouv. J. Mth. 1 (1885) 167- ; 2 (1886) 151-. representable by rational curves. Fano, G. Em. E. Ac. Line. Ed. 4 (1895) (Sem. 1) 51-. of ruled surfaces and surfaces of revo- lution. Dolezal, E. Arch. Mth. Ps. 14 (1896) 1-. trajectories. Mukhopadhyay, A . [1887] Beng. As. S. J. 57 (Pt. 2) (1890) 72-. geometry, question in. Piccioli, H. N. A. Mth. 18 (1899) 454-. ,2 theorems in. Lamioni, C. N. A. Mth. 19 (1900) 557-. Curves on Surfaces 8810 Differential parameters in theory of surfaces. Frobenius, G. Crelle J. Mth. 110 (1892) 1-. Envelopes. Lecornu, L. Caen Ac. Mm. (1890) (Pt. 1) 12-. Fundamental magnitudes in theory of surfaces. Knoblauch, J. Crelle J. Mth. 103 (1888) 25-. Geometry in 3-dimensional space. Calinon, A . Nancy S. Sc. Bll. (1891) 35-. on surfaces of constant curvature. Carda, K. Wien Ak. Sb. 107 (1898) (Ab. 2a) 44-. Infinitesimal geometry of surfaces. Arvay, W. Prace Mt.-Fiz. 9 (1898) 164-. Isometric curves. Ocagne, M. d'. Brux. S. Sc. A. 9 (1885) (Pt. 1) 56-. Linear partial differential equations belonging to theory of surfaces. Craig, P. C. E. 123 (1896) 634-. Partial differential equations derived from Co- dazzi's, applications. Craig, T. Crelle J. Mth. 120 (1899) 165-. in geometry, and projectivity. Sanio, T. Arch. Mth. Ps. 1 (1884) 225-. Points on curve satisfying algebraic differential equation, space analogue. Halphen, G. H. Liouv. J. Mth. 2 (1876) 257-, 371-. Shortest line on sphere. Hoiiel, J. N. A. Mth. 7 (1868) 73-. Surfaces cutting lines at given angle. B&re- nius, H. G. [1838-39] St Pet. Mm. Sav. Etr. 4 (1845) 255-; Helsingf. Acta 1 (1840) 381-. on which point can move according to certain law. Saint Germain, A. de. Liouv. J. Mth. 2 (1876) 325-. of revolution with minimum line in given plane. Molins, H. Toul. Ac. Sc. Mm. 2 (1890) 360-. Transformations, class. Clairin, J. C. E. 130 (1900) 309-. Unique ellipsoid. Adan, E. [1878] Brux. Mm. Cour. 8, 29 (1880) (No. 4) 10 pp. 8810 Determination of curves on surfaces. Algebraic curves of constant torsion, and alge- braic minimal surfaces inscribed in sphere. Cosserat, E. C. E. 120 (1895) 1252-. Arc element of surfaces, geodesic circles of which admit of contact transformation. Lie, S. Arch. Mth. Ntvd. 9 (1884) 40-. Asymptotic lines and conjugate systems on surface. Lelieuvre, . C. E. 109 (1889) 792-. of Kummer surface. Hutchinson, J. I. N. Y. Am. Mth. S. Bll. 5 (1899) 465-. on ruled surfaces. Snyder, V. N. Y. Am. Mth. S. Bll. 5 (1899) 343-. surf ace, condition that family of plane curves should be projection of. Koenigs, G. Par. S. Phlm. Bll. 4 (1892) 94. tore. Maschke, H. [1895] N. Y. Am. Mth. S. Bll. 2 (1896) 19-. Bisectors of network of lines on any surface. Aoust, . Liouv. J. Mth. 9 (1883) 43-. 634 8810 Determination of Curves on Surfaces 8810 Characteristic lines and characteristic angle of surface. Pucci, E. Em. B. Ac. Line. Ed. 5 (1889) (Sem. 1) 501-. , properties. Reina, V. Em. E. Ac. Line. Ed. 5 (1889) (Sem. 1) 881-. Conjugate lines on surfaces, functions of angle between. Massimi, P. Em. N. Line. At. 52 (1899) 126-. systems, one of which is geodesic. Guichard, C. C. E. 128 (1899) 599-. Contour and slope lines. Cayley, A. Ph. Mg. 18 (1859) 264-. Contours on surfaces. Jordan, C. Liouv. J. Mth. 11 (1866) 110-. Curvature of isogonals. Seiliger, D. N. Fschr. Mth. (1895) 725-. Curves on surfaces. Spitzer, S. Wien SB. (1853) 435-. Mainardi, G. Mil. G. I. Lomb. 9 (1856) 385-. Bdklen, 0. Grunert Arch. 39 (1862) 204-. Peterson, K. M. [1866] (xn) Eec. Mth. (Moscou) 2 (1867) (Ft. 1) 17-. Gilbert, P. [1868] Brux. Ac. Sc. Mm. 37 (1869) 47 pp. (Gilbert.) Aoust, (I'abbe) . Brux. Ac. Bll. 26 (1868) 471-. (Aoust.) Gilbert, P. Brux. Ac. Bll. 26 (1868) 480-. Aoust, (Vabbe) . C. E. 68 (1869) 526-. Pirondini, G. Nap. Ed. 28 (1889) 87-. on algebraic surfaces. Darboux, G. Bll. Sc. Mth. As. 2 (1871) 23-, 184-, 221-, 314- ; 3 (1872) 221-, 251-, 281-. , "normal segments." Laguerre, E. [1870] (x) Par. S. Phlm. Bll. 7 (1871) 69-. that are always geodesies of surfaces they generate. Hazzidakis, J. N. [1882] Crelle J. Mth. 95 (1883) 120-. on cone or cylinder, developed on plane, equa- tion. Sornin, J. Toul. Mm. Ac. 5 (1855) 301-. of constant curvature on sphere. Zeuthen, H. G. N. Ts. Mth. 1 (B) (1890) 67-; Fschr. Mth. (1890) 774. contact of conic and surface. Hachette, J. N. P. Par. EC. Pol. Cor. 1 (1804-08) 188-. cylinder circumscribed to helicoid with director plane. Verstraeten, T. (xn) Mathesis 1 (1881) 49-. i . Mister, J. (xn) Mathesis 1 (1881) 137-. skew surface and its director surface, construction of tangents. Drasch, H. [1880] Wien Ak. Sb. 82 (1881) (Ab. 2) 1140-. surface of revolution and circum-cone or cylinder, construction of tangents. Seidelin, C. J. L. (xn) Ts. Mth. 6 (1882) 33-. contained in algebraic surfaces, linear differ- ential equations denning. Fano, G. Em. E. Ac. Line. Ed. 4 (1895) (Sem. 1) 232-, 292-, 322-. curvature. Beltrami, E. N. A. Mth. 4 (1865) 258-. curvature (Beltrami). Bonnet, 0. N. A. Mth. 4 (1865) 267-. , geodesic. Cayley, A. L. Mth. S. P. 12 (1880-81) 110-. and torsion. Pellet, A. E. As. Fr. C. E. (1890) (Pt. 2) 200-. and curvilinear coordinates. Aoust, (I'abbe) . C. E. 65 (1867) 814-. of 4th degree on sphere, and their osculating planes at point. Laguerre, E. (x) Par. S. Mth. Bll. 1 (1873) 101-. determination. Enneper, A. Gott. Nr. (1873) 785. on developables. Codazzi, D. Tortolini A. 8 (1857) 165-. . Michel, C. Bll. Sc. Mth. 24 (1900) 157-. cut by osculating planes at constant angle. Molins, H. Liouv. J. Mth. 12 (1847) 394-. , properties. Olivier, T. Par. S. Phlm. N. Bll. (1832) 166-. development. Minding, E. F. A. Crelle J. 6 (1830) 159-. differential equation of class. Enneper, A. Gott. Nr. (1871) 577-. equations. Eicci, G. Yen. I. At. (1892- 93) 1336-. on ellipsoid, geodesic radius of curvature. Liouville, J. Liouv. J. Mth. 19 (1854) 368. , Poinsot's poloids. Allman, G. J. Camb. and Dubl. Mth. J. 3 (1848) 287-. of equal illumination on algebraic ruled surface. Burali-Forti, C. Palermo Cir. Mt. Ed. 4 (1890) 57-. sphere. Chretien, L. V. A. Cond. Pon. Chauss. 16 (1872) 45-. . surfaces of revolution, construction by tangent cones. Matzek, F. Wien Sb. 58 (1868) (Ab. 2) 49-. equidistant, on sphere. Gudermann, C. Crelle J. 25 (1843) 119-. _, . Eoberts, W. [1869] A. Mt. 4 (1870-71) 207-. geodesic curvature. Pirondini, G. A. Mt. 21 (1893) 33-. and normal curvature, loci of centres. Gattorno, G. G. Mt. 37 (1899) 41-. torsion. Burgatti, P. Palermo Cir. Mt. Ed. 10 (1896) 229-. of given curvature on given surface. Hoppe, E. Arch. Mth. Ps. 11 (1892) 193-. on given surfaces. Graves, C. Ir. Ac. P. 2 (1844) 576-. having same polar surface, integrals of differential equations. Aoust, (I'abbe) . C. E. 78 (1874) 1290- ; A. Mt. 7 (1876) 1-. , (Aoust). Serret, J. A. C. E. 78 (1874) 1329-. , (Serret) . Aoust, (I'abbt) . C. E. 78 (1874) 1481-. integral. Delassus, E. Bll. Sc. Mth. 22 (1898) 318-. networks of, and congruences connected with them. Guichard, C. C. E. 128 (1899) 1149-. plane, on conical surfaces. San Bertolo, N. Em. At. 12 (1858-59) 173-. 635 8810 Curves on Surfaces plane, on spheres, shadows. Jeffery, H. M. [1875] QJ. Mth. 14 (1877) 53-. of principal tangents of skew surfaces. Clebsch, R.F.A. [1867] Crelle J. 68 (1868) 151-. properties. Raabe, J. L. Crelle J. 2 (1827) 368-. . Brioschi, F. Tortolini A. 5 (1854) 232-. . Gilbert, P. Brux. Ac. Bll. 9 (1860) 46-. . Ribaucour, A. C. B. 80 (1875) 642-. (Eibaucour). Mannheim, A. C. E. 80 (1875) 725-. property. Laguerre, E. [1870] (x) Par. S. Phlm. Bll. 7 (1871) 49-. on quadrics, formula. Laguerre, . N. A. Mth. 9 (1870) 5-. radii of curvature, demonstration of Bonnet's formulas. Beltrami, E. G. Mt. 4 (1866) 123-. on right cylinder with circular base. Az- zarelli, M. [1876] Em. N. Line. At. 30 (1877) 1-. scroll, number of intersections. Story, W. E. (xn) J. H. Un. Cir. [2] (1883) 143. skew, on sphere, absolute radius of osculating circle, and evolutes. Plana, G. Mod. S. It. Mm. 24 (1848) 343-. on sphere. Bacaloglo, E. Grunert Arch. 35 (1860) 57-. . Gilnther, S. Arch. Mth. Ps. 56 (1874) 267-. ,arcs. Newengloivski,(Prof.). Par. EC. Norm. A. 2 (1873) 137-. , class. Roberts, W. Ir. Ac. P. 2 (1844) 194-. , evolutes. Mehler, F. G. [1863] (vm) Crelle J. 64 (1865) 183-. and plane, transmutation method. Roberts, W. Liouv. J. Mth. 13 (1848) 209-. surface of right cone, involutes. Valles, F. Gergonne A. Mth. 17 (1826-27) 349-. surfaces of revolution. Biehringer, (Dr) . Z. Mth. Ps. 18 (1873) 552-; 21 (1876) 229-; 22 (1877) 151- ; 28 (1883) 157-. , transformation. Pirondini, G. Nap. Ed. 29 (1890) 155-. touching one another, relation due to Laguerre. Mannheim, A. C. E. 82 (1876) 554-. theory. Cosserat, E. Toul. Ac. Sc. Mm. 7 (1895) 366-. torsal, on ruled surface, determination of number. Schubert, H. C. H. Mth. A. 17 (1880) 575-. transformation. Faure, H. Mntp. Ac. Sc. Mm. 2 (1851-54) 463-. Darboux or D lines. (Curves on surface, with osculating sphere touching surface at every point.) Darboux, G. C. E. 73 (1871) 732-. . Cosserat, E. C. E. 121 (1895) " . Hardy, J. G. Am. J. Mth. 20 (1898) 283-. on quadrics. Pell, A. N. Y. Am. Mth. S. T. 1 (1900) 315-. Geodesies 8810 Differential parameters, application in theory of surfaces. Bendixson, I. Stockh. Ofv. (1897) 331-. Expression for ds* in three variables, new. Levi-Civita, . C. E. 124 (1897) 1434-. Extension to any surface of Euler's problem on ds 2 =dx* + dy*. Koenigs, G. C. E. 108 (1889) 221-. Fundamental equations, geometrical signi- ficance. Knoblauch, J. Acta Mth. 15 (1891) 249-. Geodesic arcs on ellipsoid whose difference is algebraic. Staude, E. 0. Mth. A. 20 (1882) 185-. circles. Ribaucour, A. [1870] (xi) Par. S. Phlm. Bll. 7 (1871) 22-, 110-, 124-. , contact transformations. Oseen, C. W. Stockh. Ofv. (1900) 957-. and parallel curves. Betti, E. Tortolini A. 3 (1860) 336-. curvature. Bonnet, O. C. E. 42 (1856) 1137-. . Schlesinger, L. Orv.-Termt. Ets. (Termt. Szak) (1891) 267-. . Lilienthal, R. von. Mth. A. 42 (1893) 505-. . Staude, O. Dorpat Sb. 11 (1896) 72-. . Vostokov, I. A. Vars. S. Nt. Tr. (1895-96) (C.R., Ps. C.) No. 8, 5 pp. , centres. Caronnet, T. C. E. 115 (1892) 589-. orthogonals on sphere and spheroid. Berardinis, G. de. G. Mt. 27 (1889) 127-, 318- ; 28 (1890) 53-, 138-. polygons on quadrics. Staude, E. O. Mth. A. 21 (1883) 219-. representation, most general, of geodesic circles of surface. Lie, S. Arch. Mth. Ntvd. 9 (1884) 62-. torsion. Mathews, G. B. QJ. Mth. 27 (1895) 145-. angles, of intersection of two surfaces. Mannheim, A. Mess. Mth. 8 (1879) 122. triangle on spheroid. Puissant, L. Par. Mm. Ac. Sc. 10 (1831) 530-. surface and corresponding spherical triangle, ratio of areas. Clausen, T. As. Nr. 20 (1843) 13-; 21 (1844) 145-. Geodesies. Minding, E. F. A. Crelle J. 20 (1840) 323-. Spitzer, S. Grunert Arch. 23 (1854) 125-. Bonnet, 0. C. E. 40 (1855) 1311- ; 41 (1855) 32-. Grunert, J. A. Grunert Arch. 37 (1861) 264-. Anon, (xi 41) Mess. Mth. 5 (1871) 87-. Enneper, A. Z. Mth. Ps. 18 (1873) 613-. Hoppe, R. Arch. Mth. Ps. 64 (1879) 60-. Boklen, O. Z. Mth. Ps. 26 (1881) 264-. Lie, M. S. Mth. A. 20 (1882) 357-. Pirondini, G. G. Mt. 23 (1885) 288-. Ermakov, V. P. Eec. Mth. (Moscou) 15 (1891) 576-; Fschr. Mth. (1890) 760. Lechalas, G. N. A. Mth. 10 (1891) 527-. Koenigs, G. [1892] Par. Mm. Sav. Etr. 31 (1894) No. 6, 318 pp. 636 8810 Curves on Surfaces. Geodesies 8810 Sochocki, J. Prace Mt.-Fiz. 3 (1892) 82-; Fschr. Mth. (1892) 732. (Koenigs.) Eicci, G. Rm. R. Ac. Line. Ed. 2 (1893) (Sem. 2) 146-, 338-. (Ricci.) Koenigs, G. Rm. R. Ac. Line. Rd. 2 (1893) (Sem. 2) 336-. Hadamard, . Bordeaux S. Sc. PV. (1896- 97) 115- , 131. application of theta functions. Hoppe, R. Arch. Mth. Ps. 3 (1886) 75-. Dximate calculation. Mobile, A. Nap. 1. 34 (1895) 139-. cases. Albeggiani, M. L. Palermo Cir. Mt. Rd. 3 (1889) 80-. . Waelsch, E. C. R. 125 (1897) 521-. complex of principal normals to. Gebbia, M. Palermo Cir. Mt. Rd. 1 (1887) 25-. on cone. Antomari, X. Par. S. Mth. Bll. 17 (1889) 118-. . Piccioli,H. N. A. Mth. 17 (1898) 207-. conjugate system , surfaces having. Eazzaboni , A. Bologna Ac. Sc. Mm. 9 (1888) 765-. of constant curvature on pseudospherical surfaces. Reina, V. Rm. R. Ac. Line. Rd. 5 (1889) (Sem. 1) 448-. and other curves on surfaces. Kummell, C. H. Smiths. Misc. Col. 33 (1888) Art. 1, 123-. (Wash. Ph. S. Bll. 6 (1884).) departure from normal sections of surface. Berardinis, G. de. Tor. Ac. Sc. Mm. 36 (1885) 159-. on developable surface. Molins, H. Toul. Mm. Ac. 5 (1861) 401-. differential equation. Issaly, . N. A. Mth. 19 (1900) 392-. and elliptic functions. Braunmiihl, A. von. Mth. A. 26 (1886) 151-. envelopes. Braunmiihl, A. von. [1878] Mth. A. 14 (1879) 557-. equation, and applications. Azzarelli, M. Rm. N. Line. At. 31 (1878) 327-. , general, discovery. Enestrom, G. Bb. Mth. (1899) 19-. , integration. Brioschi, F. Tortolini A. 4 (1853) 133-. generalisation of problem. Schlomilch, 0. Z. Mth. Ps. 13 (1868) 156-. and geodesic triangle. Brill, A. Munch. Ak. Ab. 14 (1883) (Ab. 2) 109-. geometrical demonstrations. Hart, A. S. Camb. and Dubl. Mth. J. 4 (1849) 80-. at infinity, and geodesies of ruled quadrics. Hadamard, . Par. S. Mth. Bll. 26 (1898) 195-. integrable, surfaces having. Laguerre, E. (x) Par. S. Mth. Bll. 1 (1873) 281. Joachimsthal's theorem, geometrical proof. Graves, C. [1850] Crelle J. 42 (1852) 279. length. Ivory, J. QJ. Sc. 21 (1826) 361-. letters to Johann and Daniel Bernoulli. Clairaut, A. C. Rm. N. Line. At. 45 (1892) 245-. lines analogous to. Darboux, G. Par. EC. Norm. A. 7 (1870) 175-. and lines of curvature. Joyce, M. J. QJ. Mth. 5 (1862) 265-. , general equations. Dickson, B. Camb. and Dubl. Mth. J. 5 (1850) 166-. and Liouville's isothermic systems. Ricci, G. Yen. I. At. (1893-94) 643-. normal plane sections at two adjacent points, divergence between. Berg, F. J. van den. Amst. Ak. Vs. M. 10 (1876) 1-; Arch. Neerl. 12 (1877) 353-. partial differential equation of 3rd order con- nected with. Levy, M. C. R. 85 (1877) 904-. 4th order connected with. Levy, H. C. R. 85 (1877) 938-, 1009-. problem, and equations with mixed functions. Koenigs, G. C. R. 117 (1893) 683-. , intermediary integrals of equation. Levy, M. C. R. 85 (1877) 1150-. , rational integrals. Levy, M. C. R. 85 (1877) 1065-. projection. Schols, C. M. Delft EC. Pol. A. 5 (1889) 133-. , curvature. Schols, C. M. Delft EC. Pol. A. 2 (1886) 179-. properties. Young, (Dr) T. QJ. Sc. 21 (1826) 136-. property, fundamental. Ferrers, N. M. QJ. Mth. 1 (1857) 245. with quadratic integrals. Koenigs, G. Par. S. Phlm. Bll. 5 (1893) 26-. representing motion with 2 degrees of freedom. Kelvin, (Lord). B. A. Rp. (1892) 652-. on ruled surfaces. Astor, A. As. Fr. C. R. (1887) (Pt. 2) 1-. special, on harmonic surfaces. Raffy, L. Par. S. Mth. Bll. 22 (1894) 8-. on special surfaces. Guldberg, A. N. Ts. Mth. 6 (B) (1895) 1-; Fschr. Mth. (1895) 704-. and spheroidal trigonometry. Grunert, J. A. Grunert Arch. 22 (1853) 64-; 40 (1863) 33-. on spiral surfaces, and their differential equa- tions. Hadamard, J. Bordeaux S. Sc. PV. (1895-96) 55-. any surface. Euler, L. [1779] St Pet. Ac. Sc. N. Acta 15 (1806) 44-. surface of centres. Hoppe, R. Arch. Mth. Ps. 63 (1879) 81-. surfaces of constant curvature. Liouville, R. Am. J. Mth. 10 (1888) 283-. anticlastic curvature. Hadamard, . Bordeaux S. Sc. PV. (1896-97) 60-; C. R. 124 (1897) 1503-. revolution. Jamet, . Brux. Ac. Bll. 13 (1887) 421-. . Catalan, E. C. Brux. Ac. Bll. 13 (1887) 425-; Liege S. Sc. Mm. 15 (1888) #0. 1, 213-. , Clairaut's equation. Lorenzoni, G. Ven. I. At. 2 (1875-76) 591-. , theorem. Emilio, R. d\ Ven. I. At. (1892-93) 1191-. between 2 points of generator. Castet, . Bordeaux S. Sc. Mm. 2 (1878) 185-. , theorem. Lommel, E. Grunert Arch. 38 (1862) 201-. system, characteristic of differential equation. Liouville, R. C. R. 108 (1889) 495-. contained in cyclic systems. Guichard, C. C. R. 128 (1899) 1308-. 637 8810 Geodesies. Geodesies on Quadrics 8810 theorem. Dini, U. G. Mt. 3 (1865) 65-. , fundamental. Fact de Bruno, F. Les Mondes 8 (1865) 739-. theorems of Jacobi, two. Para/, . C. B. 106 (1888) 1139-. theory. Boklen, 0. Grunert Arch. 39 (1862) 189-. . Beltrami, E. Mil. I. Lomb. Kd. 1 (1868) 708-. . Jamet, V. Mars. Fac. Sc. A. 8 (1898) 117-. . Stdckel, P. D. Nf. Vh. (1900) (Th. 2, Hdlfte 1) 5-. Geodesies on quadrics. (See also Geography 70.) Hart, A. S. Camb. and Dubl. Mth. J. 4 (1849) 192-. Graves, C. Ir. Ac. P. 4 (1850) 283-. Cayley, A. [1872] L. Mth. S. P. 4 (1871-73) 191-, 368-. Laguerre, E. N. A. Mth. 15 (1876) 10-. on confoqals, and lines of curvature. Chasles, M. Liouv. J. Mth. 11 (1846) 105-. ellipsoid. Jacobi, C. G. J. Crelle J. 19 (1839) 309-. . Liouville, J. Liouv. J. Mth. 9 (1844) 401-. . Jacobi, C. G. J. Berl. B. (1846) 351-. . Roberts, M. Liouv. J. Mth. 12 (1847) 491-. . Schlfifli, L. Bern Mt. (1847) 96-. . Besge, V. A. Liouv. J. Mth. 14 (1849) 247-. . Roberts, W. Liouv. J. Mth. 2 (1857) 213-. . Aoust, L. C. B. 50 (1860) 484-. . Boklen, 0. Grunert Arch. 35 (1860) 101-. . Weierstrass, C. Berl. Mb. (1861) 986-. . Cayley, A. [1871] Ph. Mg. 41 (1871) 534-; As. S. Mm. 39 (1872) 31-. , application of elliptic functions. Roberts, M. Liouv. J. Mth. 15 (1850) 275-. , centres of maximum and minimum curvature, etc. Schldfli,L. Bern Mt. (1846) 145-. , equation, geometrical demonstration. Liouville, J. C. E. 22 (1846) 111-. , Jacobi's method. Tortolini, B. Km. At. 4 (1850-51) 287-. , and lines of curvature. Roberts, M. Liouv. J. Mth. 11 (1846) 1-; 13 (1848) 1-. , , two theorems. Liouville, J. C. B. 22 (1846) 893-. , radii of two curvatures. Tortolini, B. Tortolini A. 2 (1851) 345-. through umbilic. Hart, A, S. Ir. Ac. P. 4 (1850) 274. and their envelopes. Braunmilhl, A. von. Mth. A. 20 (1882) 557-. geometrical demonstration. Liouville, J. Liouv. J. Mth. 11 (1846) 21-. and lines of curvature. Joachimsthal, F. Crelle J. 26 (1843) 155-. and lines of curvature. Chasles, M. Liouv. J. Mth. 11 (1846) 5-, 105-. (Chasles). Liouville, J. Liouv. J. Mth. 12 (1847) 255. loci. Rutledge, J. Y. Camb. and Dubl. Mth. J. 6 (1851) 21-. on non-central quadrics. Marcolongo, R. Nap. Ed. 30 (1891) 32-. paraboloids. Marcolongo, R. Bm. E. Ac. Line. Ed. 6 (1890) (Sem. 1) 392-; G. Teix. J. Sc. 11 (1892) 145-. , and lines of curvature. Chelini, D. G. Arcad. 106 (1846) 1(51-. quadrics not of revolution. Forsyth, A. R. L. Mth. S. P. 27 (1896) 250-. right cone. V6sz, J. A. [1868] (xn) Mag. Tud. Ak. Etk. (Mth.) [1] (No. 5) (1869) 19pp. . Czuber [Cuber], E. Arch. Mth. Ps. 69 (1883) 125-. spheroid. Soldner, J. Zach M. Cor. 11 (1805) 7-. . Baeyer, J. J. As. Nr. 71 (1868) 289-. . Ciscato, G. Ven. I. At. (1891-92) 1087- , 1333-. , approximate calculation. Pizzetti, P. Nap. Ed. 35 (1896) 75-. , arcs. Halphen, G. H. C. E. 105 (1887) 583-. , Dalby's method of finding difference of longitude of 2 points. Tiarks, J. L. Ph. Mg. 4 (1828) 364-. , new geometrical representation. Schwering, K. Z. Mth. Ps. 24 (1879) 405-. , oblate. Cayley, A. Ph. Mg. 40 (1870) 329-. , . Fergola, E. Nap. Ed. 33 (1894) , . Pizzetti, P. Tor. Ac. Sc. At. 30 (1895) 217-. , . Forsyth, A. R. Mess. Mth. 25 (1896) 81-. , , conjugate points. Forsyth, A. R. Mess. Mth. 25 (1896) 161-. , , properties. Ivory, J. Tilloch Ph. Mg. 67 (1826) 241-, 340-. , between 2 points, latitude and longitude of which are given. Puissant, L. Par. S. Gg. Bll. 9 (1828) 184-. , prolate. Halphen, G. H. C. E. 105 (1887) 535-. torsion at a point. Laguerre, E. Par. S. Mth. Bll. 4 (1876) 160-. Geodesy, solution of fundamental question in. Chandrikov, M. T. Eec. Mth. (Moscou) 17 (1895) 575-. Helix on cone. Junge, A. Giving. 4 (1858) 172-. Integrals of dynamics. Painleve, P. C. B. 114 (1892) 1168-. Isoperimetric figures on any surface. Bordoni, A. (vi Adds.) Opusc. Mt. Fis. 1 (1832) 1-. Latitude, longitude and azimuth. Andrae, . Kjob. Ov. (1858) 230- ; As. Nr. 50 (1859) 161-. 638 8810 Curves on Surfaces. Loxodromes 8810 Latitude and longitude, formulae. Oriani, B. Mil. Effem. As. (1808) 3-. Line-element of surface, certain forms. Massimi, P. Em. N. Line. At. 52 (1899) 93-. Lines of equal slope on given surfaces. Schlamilch, 0. Schlomilch Z. 1 (1856) 250-. greatest slope. Stem, A. (xn) Ts. Mth. 1 (1865) 123-. . Breton (de Champ), . C. B. 70 (1870) 982-. , differential equation. Breton (de Champ), . C. B. 64 (1867) 407-. , ridge lines and channels. Boussinesq, J. C. B. 73 (1871) 1368- ; 75 (1872) 198-, 835-. shadow for some surfaces. Pirondini, G. G. Mt. 29 (1891) 356-. striction. Larmor, J. QJ. Mth. 19 (1883) 381-. of hyperboloid. Monteiro, A. S. G. Teix. J. Sc. 3 (1881) 131-. as curves generated by higher configurations. Schmid, T. [1881] Wien Ak. Sb. 84 (1882) (Ab. 2) 908-. rational skew curves of 4th order. Migotti, A. [1879] Wien Ak. Sb. 80 (1880) (Ab. 2) 1023-. of one sheet. Bobek, K. Par. S. Mth. Bll. 11 (1883) 125-. and of hyperbolic paraboloids. Bauer, M. Z. Mth. Ps. 28 (1883) 274-. . ruled surfaces. Nicolaides, N. Les Mondes 9 (1866) 146-. of 2nd and 3rd degrees. Adler, A. Wien Ak. Sb. 85 (1882) (Ab. 2) , etc. of skew surfaces. Enneper, A. Gott. Nr. (1867) 454-. Loxodromes. on conical surfaces. Enneper, A. Gott. Nr. (1869) 459-. cylinder and cone. Grebe, E. W. Grunert Arch. 2 (1842) 127-. ellipsoid and sphere. Plagemann, W. Grunert Arch. 32 (1859) 1-. equation and properties. Fournier- Vannson, . N. A. Mth. 20 (1861) 31-, 225-. and geodesies on tore. Puchta, A. Mh. Mth. Ps. 1 (1890) 443-. loxodromic figures. Friesach, K. Steierm. Mt. (1878) 78-. on paraboloid of revolution. Boyman, J. B. Grunert Arch. 13 (1849) 375-. projecting into logarithmic spirals, surfaces with. Molins, H. Toul. Ac. Sc. Mm. 7 (Son. 2) (1885) 293-; 8 (1886) 426-. properties. Arrest, H. d'. As. Nr. 36 (1853) 351-. on quadrics. Boyman, J. E. Grunert Arch. 7 (1846) 337-. and shortest paths on spheroid. Caluso, (I'abbe) T. V. de. Turin Mm. Ac. 4 (1788-89) 325-; 5 (1790-91)100-. spherical catenary. Gudermann, C. Crelle J. 11 (1834) 394-. on any surface. Legoux, A. Toul. Ac. Sc. Mm. 1 (1889) 352-. surfaces. Dina, C. G. Mt. 19 (1881) 298-. of revolution. Schubert, F. T. [1803] St Pet. Ac. Sc. N. Acta 15 (1806) 202-. . Gergonne, J. D. Gergonne A. Mth. 8 (1817-18) 125-. . Aoust, L. Liouv. J. Mth. 11 (1846) 184-. . Junge, A. Schlomilch Z. 5 (1860) 296-. . Laisant, C. A. N. A. Mth. 13 (1874) 573-. , equations. Grunert, J. A. Grunert Arch. 21 (1853) 304-. Loxodromic arc between two points on sphere. Friesach, K. Steierm. Mt. (1880) 3-. and great circles on sphere, relation. Friesach, K. Steierm. Mt. (1876) 97-. triangles. Grunert, J. A. Grunert Arch. 16 (1851) 23-. Maximum area on surface with given perimeter. Aelt, . N. A. Mth. 19 (1860) 100-. Meridian curve of surface of revolution. Eesal, H.A. C. B. 85 (1877) 5-. Motion of point on surface. Enneper, A. Gott. Nr. (1869) 62-. Number of conditions for existence of an r-fold straight line on surface of nth order is Machovec, F. Casopis 16 (1887) 230-; Fschr. Mth. (1887) 780. Orthodromic sailing. Glotin, P. [1876-79] Bordeaux S. Sc. Mm. 1 (1876) xlviii-; 2 (1878) 189- ; 3 (1880) 377-. Orthogonal lines on surface, and linear partial differential equations of 2nd order (elliptic type). Burgatti,P. A. Mt. 23 (1895) 225-. Points on surfaces such that geodesies through them never cease to be shortest lines. Man- goldt, H. von. [1880] Crelle J. Mth. 91 (1881) 23-. Projection, quasi- stereographic, due to Gauss. Forsyth, A. E. QJ. Mth. 21 (1886) 376-. Pseudo-spheres, Viviani's theorem. Torelli, G. G. Mt. 10 (1872) 128-. Quadrable figures on surface of right cone. Magistrini, D. [1845] Bologna N. Cm. 8 (1846) 427-. Bibaucour's problem. GuicJuird, C. C. B. 125 (1897) 1013-. Bidge lines and channels. Barre" de Saint- Venant, . Par. S. Phlm. PV. (1852) 24-. _ --- . Jordan, C. C. B. 74 (1872) 1457- ; 75 (1872) 625-, 1023-. ---- . Breton (de Champ), P. Liouv. J. Mth. 3 (1877) 99-. ____ , geometrical characters. Breton [de Champ], P. C. B. 53 (1861) 808-. --- lines connected with them. McCowan, J. Ph. Mg. 37 (1894) 227-. Spherical catenaries. Gudermann, C. Crelle J. 33 (1846) 189-, 281-. class-cubics with 3 single foci. Jeffery, H. M. B. A. Bp. (1878) 490-. 639 Minimal Surfaces 8820 Sphero-conics, confocal, differential equation. Burnside, W. Mess. Mth. 20 (1891) 60-. Spheroidal trigonometry. Puissant, L. [1813- 29] Par. S. Phlm. Bll. 3 (1813) 273-; Par. Mm. Ac. Sc. 10 (1831) 457-. , especially geodesies. Gudermann, C. [1847] Crelle J. 43 (1852) 294-. Spiral, conical, development. Jube, E. N. A. Mth. 4 (1845) 454-. on tore. Kinealy, J. H. St Louis Ac. T. 5 (1892) 1-. Surfaces, reciprocally polar, relations between infinitesimal elements. Demoulin, A. C. B. 114 (1892) 1102-. of revolution cut by sphere in geodesies. Molins, H. Toul. Ac. Sc. Mm. 3 (1891) 296-. System of variables, new, of Bonnet. Tortolini, B. [1868] Em. At. N. Line. 22 (1869) 172-. Trajectories cutting at given angle tangents to any curve. Molins, H. Toul. Mm. Ac. 2 (1846) 65-. skew curve. Molins, H. Toul. Mm. Ac. 6 (1839-41) 33-. , on given surface or in space, rendering minimum I (v) ds, where v is the velocity. Roger, E. C. E. 40 (1355) 1176-. surfaces. Gilbert, P. Brux. Ac. Bll. 25 (1868) 288-. Triangle on spheroid, reduction of angle to angle of plane or spherical triangle. Weingarten, J. As. Nr. 75 (1870) 91-. Triangles formed by lines of system (a) or by geodesies, sum of angles. Boklen, O. Arch. Mth. Ps. 43 (1865) 18-. on spheroid. Legendre, A. M. Par. Mm. de PI. 6 (1806) (Sem. 1) 130-. , problems. Proteus, P. Nicholson J. 8 (1804) 151-. , solution. Baeyer, (Gen.-Lt.) J. J. As. Nr. 61 (1864) 225-; 71 (1868) 289-. 8820 Minimal surfaces. Gergonne, J. D. Gergonne A. Mth. 7 (1816-17) 143-, 283-. Bonnet, O. C. E. 40 (1855) 1107- ; 41 (1855) 1057- ; 42 (1856) 532-. Lamarle, E. Brux. Ac. Bll. 6 (1859) 329-. Christoffel, E. B. Crelle J. 67 (1867) 218-. Roger, E. C. E. 71 (1870) 351-. Schwarz ,C.H.A. Ziir. Vj schr. 19 (1874) 243-. Kiepert, L. [1875-77] Crelle J. Mth. 81 (1876) 337-; 85 (1878) 171-. Cayley, A. QJ. Mth. 14 (1877) 190-. Lie, M. S. Arch. Mth. Ntvd. 4 (1879) 477-. Schwarz, H. A. Helsingf. Acta 15 (1888) 315-. Vivanti, J. Z. Mth. Ps. 33 (1888) 137-. Cayley, A. C. E. Ill (1890) 953-. Carvallo, E. Bll. Sc. Mth. 18 (1894) 12-. Pad, P. Tor. Ac. Sc. At. 29 (1894) 446-. Stiickel, P. Leip. Mth. Ps. B. 48 (1896) 478-. Guichard, C. C. E. 126 (1898) 1487-. Bromwich,T.J.I'A. L. Mth. S. P. 30 (1899) 276-. Richmond, H. W. Camb. Ph. S. T. 18 (1900) 324-. Algebraic surfaces. Geiser, C. F. [1870] Mth. A. 3 (1871) 530-. Algebraic surfaces, determination of lowest class number. Henneberg, L. [1877] A. Mt. 9 (1878-79) 54-. inscribed in algebraic developable. Lie, M. S. Arch. Mth. Ntvd. 3 (1878) 340-. sphere, and algebraic curves of constant torsion. Cosserat, E. G. E. 120 (1895) 1252- . inscriptible in cone, determination of all. Lie, M. S. Arch. Mth. Ntvd. 3 (1878) 224-. , projective and metrical researches. Lie, M. S. Mth. A. 14 (1879) 331- ; 15 (1879) 465-. Apsidal or conjugate surfaces. Gilbert, P. Brux. Ac. Bll. 28 (1869) 31-. Area, second variation, especially for helicoids. Schwarz, H. A. Berl. Mb. (1872) 718-. Case. Catalan, E. C. Par. S. Phlm. PV. (1841) 59-. . Salvatore-Dino, N. Nap. Ed. 19 (1880) 148-. . Zeeman, P. Haarl. Ms. Teyl. Arch. 5 (1898) 299-. , construction of model. Tallqvist, H. Hel- singf. Ofv. 31 (1889) 32-. Cases, determination. Herzog,A. Ziir.Vjschr. 20 (1875) 217-. Catenoid as minimal surface, limits. Lindelof, L. L. [1870] Helsingf. Acta 9 (1871) 353-. Class. Weierstrass, C. Berl. Mb. (1867) 511-. . Stouff, X. Toul. Fac. Sc. A. 6 (1892) A, 8pp. Continuous surface bounded by skew quadri- lateral. Tucci, F. P. [1852] Nap. Mm. Ac. Sc. 1 (1852-54) 49-. Curve, rotation of which generates minimal surface. Saalschiitz, L. Arch. Mth. Ps. 5 (1887) 131-, vn. Cyclic surface, determination of constants. Juga, G. Mth. A. 52 (1899) 167-. Developable surface with known edge of re- gression. Monge, G. Par. EC. Pol. Cor. 1 (1804-08) 209-. passing through curve which developes into arc of circle of given radius. Molins, H. Liouv. J. Mth. 1 (1856) 265-. Division of space with minimum partitional area. Thomson, (Sir) W. Ph. Mg. 24 (1887) 503-. Doubly connected surface. Kluyver, J. C. Amst. Ak. Vh. (Sect. 1) 3 (1896) No. 9, 42pp. ; Fschr. Mth. (1895) 746-. Expression of line element. Soler, E. Palermo Ac. At. 4 (1897) (Sc. Nt.) 10 pp. Family. Weingarten, J. Gott. Nr. (1887) 272-. . Rouquet, V. Toul. Ac. Sc. Mm. 2 (1890) 127-. of spheres connected with minimal surfaces. Rouquet, V. Toul. Ac. Sc. Mm. 1 (1889) 56-. Focal conies of confocals, construction of sur- faces, with centres of curvature on. Roberts, W. C. E. 53 (1861) 799-. General equation. Weiler, A. Grunert Arch. 38 (1862) 356-. Generation. Mathet, G. Liouv. J. Mth. 8 (1863) 323-. Geodesies. Lie, M. S. Arch. Mth. Ntvd. 6 (1882) 490-. 640 8820 Minimal Surfaces 8820 Geometrical researches. Sturm, R. Crelle J. Mth. 105 (1889) 101-. Helicoids. Falchi, M. G. Mt. 34 (1896) 89-. Joachimsthal's theorem. Cayley, A. C. B. 106 (1888) 995-. Minimal curves on surfaces. Minding, E. F. A. Crelle J. 5 (1829) 297-. . Berner, T. [1865] Z. Mth. Ps. 11 (1866) 81-. . Minding, E. F. A. [1878-79] St Pet. Ac. Sc. Bll. 25 (1879) 190- ; Crelle J. Mth. 86 (1879) 279-. of revolution. Minding, E. F. A. [1875] St Pet. Ac. Sc. Bll. 21 (1876) 252-. Minimum areas (alveole). Jacob, . N. A. Mth. 2 (1843) 160-. circumscribed and maximum inscribed poly- gons to or in ellipse, and analogous polyhedra to or in ellipsoid. Mainardi, G. Em. At. N. Line. 26 (1873) 86. property of square and cube. Bouvier, L. C. Gergonne A. Mth. 15 (1824-25) 115-. . (Criticism of Bouvier's proof.) Gergonne, J. D. Gergonne A. Mth. 15 (1824-25) 265-. Non-algebraic surfaces, containing system of algebraic curves. Schwarz, C. H. A. Crelle J. Mth. 87 (1879) 146-. Non-Euclidean surfaces. Guichard, . Par. EC. Norm. A. 13 (1896) 401-. Periodic surfaces with octahedral symmetry, two, equations. Schoenfties, A. C. B. 112 (1891) 515-. Problem. Darboux, G. C. E. 104 (1887) 728-. Problems. Pincherle, S. Mil. I. Lomb. Ed. 9 (1876) 444-; G. Mt. 14 (1876) 75-, 192. Properties. Mathet, G. C. E. 57 (1863) 868-. . Beltrami, E. Bologna Ac. Sc. Mm. 7 (1867) 411-. Property. Berdelle, . Par. S. Mth. Bll. 17 (1889) 102-. , characteristic. Bianchi, L. G. Mt. 22 (1884) 374-. Pseudo-spherical surfaces, applications of hyperbolic functions. Forti, A. (xn) Ev. Sc.-Ind. 15 (1883) 221-, 250-. Eeal algebraic surfaces. Lie, M. S. Arch. Mth. Ntvd. 2 (1877) 157-. Euled surface. Bonnet, O. Bll. Sc. Mth. 9 (1885) 14-. surfaces. Catalan, E. C. Liouv. J. Mth. 7 (1842) 203-. , and surfaces with plane lines of curvature. Demoidin, A. Brux. Mm. Cour. 8, 58 (1898- 99) No. 7, 38 pp. Schwarz's formulae, generalisation. Montcheuil, de. Par. S. Mth. Bll. 28 (1900) 268-. Singularities. Neovius, E. E. Helsingf. Acta 16 (1888) 529-. Special surfaces. Schroarz, H. A. Berl. Mb. (1872) 3-. Surface bounded by skew quadrilateral formed by 4 edges of regular tetrahedron. Schwarz, H. A. Berl. Mb. (1865) 149-. of 5th class. Schilling, C. Bll. Sc. Mth. As. 4 (1880) (Ft. 1) 395-. VOL. i. 641 Surface generated by straight line. Lewanen, S. Z. Mth. Ps. 18 (1873) 423-. with given boundary. Riemann, [G. F.] B. Gott. Ab. 13 (1868) (Mth.) 1-. between given limits. Poisson, S. D. Crelle J. 8 (1832) 361-. . Scherk, H. F. Crelle J. 13 (1835) 185-. having Neil's parabola for plane geodesic. Henneberg, L. Ziir.Vjschr. 21 (1876) 66-. of revolution. La Vallee-Poussin, C. J. de. Brux. S. Sc. A. 24 (1900) (Pt. 1) 49-. between 2 straight lines in space. Serret, J. A. C. E. 40 (1855) 1078-. Surfaces applicable to surfaces of minimum area. Dini, U. C. E. 60 (1865) 340-. revolution or spiral surfaces. Demoulin, A. Bll. Sc. Mth. 21 (1897) 244-. themselves. Sinigallia, L. G. Mt. 36 (1898) 172- ; 37 (1899) 171-. bounded by 4 edges of skew quadrilateral. Schoenflies, . C. E. 112 (1891) 478-, 548. 3 straight lines. Neovius, E. R. Helsingf. Acta 16 (1888) 573-; 19 (1893) No. 4, 37 pp. straight lines, theory. Schwarz, H.A. Berl. Ak. Sb. (1894) 1237-. , 2-branched, of less area than others with same limits. Schwarz, H. A. Gott. Ab. 34 (1887) (Mth.) 48 pp. of 3 first classes of polyhedra. Hoppe, R. Arch. Mth. Ps. 58 (1876) 327-. enveloped by system of cones of 2nd degree. Schwarz, C.H.A. Crelle J. Mth. 80 (1875) 301-. of given contour, Eiemann's method for determination. Niewengloicski, B. Par. EC. Norm. A. 9 (1880) 227-. with given plane or spherical curve as curvature line. Tallqvist, H. Helsingf. Acta 17 (1891) 473-. principal curvatures equal and opposite. Roberts, M. Liouv. J. Mth. 11 (1846) 300-. . Simon, 0. E. Grunert Arch. 27 (1856) 322-. , case. Bjorling, E. G. Grunert Arch. 4 (1844) 290-. , _. Brunei, G. Bor- deaux S. Sc. PV. (1896-97) 62-. , . Hadamard, . Bordeaux S. Sc. PV. (1896-97) 163-. , pair. Roberts, M. Liouv. J. Mth. 15 (1850) 323-. , ruled, case. Serret, J. A. Liouv. J. Mth. 11 (1846) 451-. related to minimal surfaces. Appell, P. Am. J. Mth. 10 (1888) 175-. representable by elliptic integrals. Lilienthal, von. Crelle J. Mth. 99 (1886) 179-. of revolution, to which helicoid can be ap- plied. Lamarle, E. Brux. Ac. Bll. 19 (1865) 537-. in spaces of constant curvature. Bianchi, L. Em. E. Ac. Line. Mm. 4 (1887) 503-. with system of lines of spherical curvature. Dobriner, H. Acta Mth. 10 (1887) 145-. of zero mean curvature. Catalan, E. C. Liege S. Sc. Mm. 12 (1885) No. 2, 254-; 13 (1886) 180- ; 15 (1888) 259. 8830 Surfaces defined by Differential Properties Theorems. Lie, M. S. Arch. Mth. Ntvd. 3 (1878) 166-. Theory. Enneper, A. Gott. Nr. (1882) 34-, 89-. . Darboiix, G. C. E. 102 (1886) 151 3-. . Goursat, E. C. B. 105 (1887) 743-. , extension. Lipschitz, R. O. S. [1872-73] (x) Crelle J. Mth. 78 (1874) 1-. Transcendental surface. Lichtenfels, 0. von. Wien Ak. Sb. 94 (1887) (Ab. 2) 41-. Transformation. Goursat, E. Acta Mth. 11 (1887-88) 135-, 257-. Transformations. Bianchi, L. Km. B. Ac. Line. Ed. 8 (1899) (Sem. 2) 151-. 8830 Surfaces determined by re- lations of curvature and by other differential properties. Condition that lines can be cut orthogonally. Cayley, A. [1876] L. Mth. S. P. 8 (1877) 53-. Corresponding surfaces, two classes. Mangeot, S. Mathesis 14 (1894) 34-. Determination of all algebraic integral surfaces of the differential equation s = inscribed in an algebraic developable. Lie, M. S. Arch. Mth. Ntvd. 4 (1879) 334-. surfaces admitting continuous system of projective transformations. Lie, S. Leip. Mth. Ps. B. 47 (1895) 209-. ovaloids by measure of curvature for each normal section. Liebmann, H. Gott. Nr. (1899) 134-. spiral surfaces by line element. Raffy, L. C. E. 112 (1891) 1421-. surface by its 2 fundamental quadratic forms. Raffy, L. C. E. 126 (1898) 1852-. one of its two surfaces of centres. Hoppe, R. Arch. Mth. Ps. 68 (1882) 256-. surfaces by curvature relations. Lipschitz , R. 0. S. Berl. Ak. Sb. (1882) 1077- ; (1883) 169-, 541-. with prescribed expression for line element. Lipschitz, R. Berl. Ak. Sb. (1884) 649-. system of equal lines of curvature. Bricard, R. C. E. 130 (1900) 475-. Developable, surface of centres. Petersson, V. Fschr. Mth. (1886) 748-. formed by tangent planes along curve on surface. Enneper, A. Gott. Nr. (1869) 207-. through 2 given curves, generation. Taylor, H. M. Mess. Mth. 5 (1876) 1-. Developables. Rehofovsky, V. Casopis 9 (*1880) 31-, 60-, 161-, 223-; Fschr. Mth. (*1880) 569-. , representation of geodesies on. Puchta, A. Wien Ak. Sb. 97 (1889) (Ab. 2a) 1269-. Differential equations of some families of sur- faces. Tanner, H.W. L. [1876-77] Mess. Mth. 6 (1877) 120-. geometry, theorem. Padova, E. Mil. I. Lomb. Ed. 23 (1890) 840-. Envelope of a surface. Enneper, A. Mth. A. 5 (1872) 304- ; Gott. Nr. (1873) 217-. 8830 Enveloping surfaces. Garbieri, G. Ven. I. At. (*1883-84) 1201-. Fuchsian surfaces. Bianchi, L. Em. B. Ac. Line. Ed. 4 (1888) (Sem. 2) 161-. Geodesic representations of surfaces. Liouville, R. C. E. 108 (1889) 335-. Geometrical applications of Picard's method of successive approximations. Bianchi, L. Em. E. Ac. Line. Ed. 3 (1894) (Sem. 1) 143-. Harmonic ruled surfaces. Raffy, L. C. E. 110 (1890) 223-. spirals. Raffy, L. C. E. 112 (1891) 518-. surfaces. Raffy, L. Par. S. Mth. Bll. 22 (1894) 63-, 84-. , class. Raffy, L. C. E. 112 (1891) 424-. surfaces, properties. Raffy, L. Toul. Fac. Se. A. 9 (1895) C, 44 pp. Isometric surfaces. Voss, A. Mth. A. 46 (1895) 97-. . Pellet, A. C. E. 124 (1897) 1337-. Isometry of lines of curvature. Knoblauch, J. Crelle J. Mth. 103 (1888) 40-. Kinematic classification of surfaces according to geodesic triangles. Mangoldt, H. von. [1882] Freiburg B. 8 (*1882-85) 73-; (x) Crelle J. Mth. 94 (1883) 21-. . Weingarten, J. L. G. J. Berl. Ak. Sb. (1882) 453-. Liouville's surfaces, characteristic property. Stackel, P. Mth. A. 35 (1890) 91-. Locus of centres of curvature. Arndt, F. Grunert Arch. 9 (1847) 68-. Orthogonal surfaces. Cayley, A, C. E. 75 (1872) 177-, 246-, 324-, 381-^1800-. . Vanecek, J. S. Prag Ceske Ak. Fr. Jos. Ez. (Tfida 2) 4 (1895) Art. 16, 4 pp. Parallel surfaces. Steiner, J. Berl. B. (1840) 114-. . Roberts, W. C. E. 54 (1862) 797-. . Roberts, S. L. Mth. S. P. 4 (1871-73) 218-. , etc. Combescure, E. Par. EC. Norm. A. 7 (1878) 409-. , and application to ellipsoid. Tortolini, B. Tortolini A. 1 (1850) 5-. , theorem, applications. Roberts, W. Ph. Mg. 24 (1862) 39-. Partial differential equation, Cayley's, in theory of surfaces. Bianchi, L. Em. E. Ac. Line. Ed. 4 (1888) (Sem. 1) 442-. of surfaces of constant curvature. Darboux, G. C. E. 97 (1883) 946-. equations connected with theory of surfaces. Craig, T. Am. J. Mth. 19 (1897) 94-. of cylinders. Tanner, H. W. L. [1876] Mess. Mth. 6 (1877) 113. and representation of surfaces. Dar- boux, G. C. E. 96 (1883) 766-. theory of surfaces, relation between. Stouff, X. Par. EC. Norm. A. 13 (1896) 9-. Euled surface, dependence of characters on characters of guiding curves. Rupp, O. Mth. A. 18 (1881) 366-. surfaces. Hachette, J. N. P. Crelle J. 8 (1832) 358-. 642 8830 Surfaces of Constant Curvature 8830 Euled surfaces, differential equation. Sincov, D. [1894-95] Kazan S. Ps.-Mth. Bll. 4 (1895) 198; N. A. Mth. 14 (1895) 58-. , line element of which is reducible to Liouville's form. Demartres, . C. E. 110 (1890) 329-. , partial differential equation. Mansion, P. [1876] Mess. Mth. 6 (1877) 45-. Skew helicoids developable on one another. Lamarle, E. Brux. Ac. Bll. 19 (1865) 407-. surface, locus of principal normals of 2 curves. Mannheim, A. Liouv. J. Mth. 17 (1872) 406-. surfaces, applicable on surfaces of revolution, and skew surfaces formed by principal normals of curve. Dini, U. G. Mt. 4 (1866) 298-. , one radius of principal curvature being a function of the other. Dini, U. A. Mt. 7 (1865) 205- . , satisfying partial differential equations of 2nd order. Dini, U. G. Mt. 3 (1865) 321- ; 4 (1866) 305- ; C. E. 61 (1865) 1001-. Surface of centres of curvature of surface of wth degree. Marcks, L. Mth. A. 5 (1872) 27-. 4th class, lines of curvature of which can be determined. Laguerre, E. Liouv. J. Mth. 2 (1876) 145-. least resistance. Armanini, E. A. Mt. 4 (1900) 131-. , tangent plane of which makes with radius vector an angle which is given function of radius vector. Turazza, D. Tortolini A. 2 (1851) 55-. Surfaces admitting lines of spherical curvature and having same spherical representation for their lines of curvature. Blutel, . C. E. 116 (1893) 249-. , asymptotic lines of which in one system are curves of constant torsion. Bianchi, L. A. Mt. 18 (1890) 301-. with circular generator. Boulanger, A. N. A. Mth. 11 (1892) 159-. , class related to pseudo-spherical surfaces. Bianchi, L. Em. E. Ac. Line. Ed. 5 (1896) (Sem. 1) 131-. with common evolute. Sanctis, P. de. Em. N. Line. At. 46 (1893) 90- confocal quadrics as surfaces of centres. Rudio, F. Crelle J. Mth. 95 (1883) 240-. 2 or more conies, traced on surface, passing through each point. Koenigs, G. C. E. 105 (1887) 407-. conjugate geodesies. Guichard, C. C. E. 110 (1890) 995-. Surfaces of constant curvature. Dini, U. G. Mt. 3 (1865) 241- ; C. E. 60 (1865) 340-. Nicolaldes, N. Bll. Sc. Mth. As. 9 (1875) 142-. Enneper, A. Gott. Nr. (1876) 597-. Lie, M. S. Arch. Mth. Ntvd. 4 (1879) 345-, 355-; 5 (1881) 282-, 328-, 518-. Darboux, G. C. E. 97 (1883) 892-. Kuen, T. [1884] Munch. Ak. Sb. 14 (1885) 193-. Waelsch, E. Wien Ak. Sb. 102 (1893) (Ab. 2a) 1317-. Beliankin, J. J. Fschr. Mth. (1898) 523. Bianchi, L. Em. E. Ac. Line. Ed. 8 (Sem. 1) 223-, 371-, 484-. Darboux, G. C. E. 128 (1899) 1018-. mean. Folie, F. Brux. Ac. Bll. 30 (1870) 15-. . Tilly, J. M. de. Brux. Ac. Bll. 30 (1870) 28-. . Chini, M. G. Mt. 27 (1889) 107-. . Vivanti, G. Mil. I. Lomb. Ed. 28 (1895) 353-. negative. Nannei, E. G. Mt. 26 (1888) 201-. . Kahan, V. Kazan S. Ps.-Mth. Bll. 5 (1896) 111-, 145- ; Fschr. Mth. (1895) 532, 533. , case. Sabinin, E. T. Eec. Mth. (Moscou) 18 (1896) 507- ; Fschr. Mth. (1896) 537. , geometry on. Escherich, G. von. Wien Ak. Sb. 69 (1874) (Ab. 2) 497-. properties. Hazzidakis, J. N. [1878] Crelle J. Mth. 88 (1880) 68-. of line element. Weingarten, J. L. G. J. Crelle J. Mth. 94 (1883) 181- ; 95 (1883) 325-. and surfaces with Liouville's line element. Waelsch, E. C. E. 116 (1893) 1435-. total. Darboux, G. C. E. 97 (1883) 848-. . Genty,E. Par. S. Mth. Bll. 22 (1894) 106-. . Guichard, G. C. E. 126 (1898) 1556-, 1616-. . Tzitzeica \Titeica], . C. E. 128 (1899) 288. , and congruences formed of optical axes. Cosserat, E. Toul. Fac. Sc. A. 8 (1894) C, 3 pp. , related surfaces. Guichard, C. Par. EC. Norm. A. 7 (1890) 233-. , transformations. Darboux, G. C. E. 128 (1899) 953-. Surfaces containing one set of curvature centres. Binet, J. P. M. C. E. 12 (1841) 435-. corresponding to given surfaces of centres. Enneper, A. Gott. Nr. (1871) 210- ; (1872) 577-. defined by property of straight lines or spheres touching them. Koenigs, G. C. E. 100 (1885) 847-. R + R' = F (u) + F^ ( Wl ). MontcJieuil, de. Par. S. Mth. Bll. 26 (1898) 103-. t 3rd degree, with parabolic umbilical line. Saint-Germain, A. de. Bll. Sc. Mth. 12 (1888) 177-. w ith difference of principal curvatures constant. Lipschitz, R. C. E. 104 (1887) 418, 534. divisible by their lines of curvature into squares. Cayley, A. [1872] L. Mth. S. P. 4 (1871-73) 8-, 120- ; C. E. 74 (1872) 1445-. , differential equation. Weingarten, J. L. G. J. Berl. Ak. Sb. (1883) 1163-. generated by motion of lines, partial differ- ential equations. Blazek, G. [1864] Wien Sb. 51 (1865) (Ab. 2) 186-. generation, analytic expression. Monge, G. Turin Mm. Ac. 1 (1784-85) 19-. 643 88 2 8830 Surfaces defined by Differential Properties 8830 Surfaces of given curvature. Weingarten, J. Schlomilch Z. 3 (1858) 43-. with line element of form du 2 + K 2 cos 2 u . dv 2 . Soler, E. Palermo Ac. At. 4 (1897) (Se. Nt.) 7pp. reducible to form f(V+V) (du 2 + dvZ). Petot, A. C. B. 110 (1890) 330-. reducible to Liouville's form in several ways. Koenigs, G. C. B. 109 (1889) 565-, 639-. lines of curvature equal in 1 system. Demoulin, A. C. B. 130 (1900) 823-. forming network with equal tangential invariants. Thybaut, A. C. B. 121 (1895) 519-. helices on any cylinders. Petot , A. C. E. 106 (1888) 1517-. isometric. Craig, T. C. E. 123 (1896) 794-. plane. Dini, U. [1867] A. Mt. 1 (1867-68) 146-. . Razzaboni, A. Bologna Ac. Sc. Mm. 10 (1879) 529-. , all. Bonnet, 0. C. E. 36 (1853) 81-, 219- ; 42 (1856) 1067-. , . Rouquet, V. Toul. Ac. Sc. Mm. 9 (1887) 233- ; 10 (1888) 161-. , . Burnside, W. Mess. Mth. 20 (1891) 49-, 148. and equal in 1 system. Caronnet, T. C. E. 117 (1893) 842-. or spherical. Enneper, A. Gott. Ab. 23 (1878) 95 pp. ; 26 (1880) 139 pp. . Cayley, A. Am. J. Mth. 11 (1889) 71-, 293-. . Waelsch, E. C. E. 128 (1899) 920-. , all. Serret, J. A. Liouv. J. Mth. 18 (1853) 113- ; C. B. 36 (1853) 432-. inl system. Serret, J. A. C. B. 42 (1856) 194. . Joachimsthal, F. Crelle J. 54 (1857) 181-. . Combescure, E. [1883] Mntp. Ac. Mm. 10 (1884) 401-. . Pirondini, G. G. Mt. 22 (1884) 118-. and isothermic. Darboux, G. C. E. 96 (1883) 1202-, 1294- ; Bll. Sc. Mth. As. 7 (1883) 257- spherical. Blutel, E. C. E. 122 (1896) 301-. inl system. Serret, J. A. C. E. 42 (1856) 109-, 190-. . Dint, 17. [1869] Mod. S. It. Mm. 2 (1876) 135-. . Enneper, A, Gott. Nr. (1872) 17-, 80-. . Dobriner, H. [1882] Crelle J. Mth. 94 (1883) 116- Surfaces with normals tangent to developable surface, case. Monge, G. Par. EC. Pol. J. 5 (1806) 1-. given surface. Weingarten, J. [1861] Crelle J. 62 (1863) 61-. sphere, all. Vdlyi, J. Arch. Mth. Ps. 68 (1882) 217-. , case. Monge, G. Par. EC. Pol. J. 4 (1801-02) 28-. parallel sections cutting lines of curva- ture under a constant angle. Bianchi, L. Em. E. Ac. Line. Ed. 7 (1891) (Sem. 1) 4-. every portion subtending solid angles in given ratio from two fixed points. Weyr, E. [1877] Wien Ak. Sb. 76 (1878) (Ab. 2) 859-. principal planes equidistant from fixed point. Guichard, . C. B. 116 (1893) 487-. having ratio of distances from fixed point constant. Bianchi, L. Bm. B. Ac. Line. Ed. 3 (1894) (Sem. 2) 77-. rational generators. Lelieuvre, . C. B. 113 (1891) 635-. of revolution of constant mean curvature. Delaunay, C. Liouv. J. Mth. 6 (1841) 309-. . Sturm, J. C. F. Liouv. J. Mth. 6 (1841) 315-. . Lindelof, L. L. [1862] (vm) Helsingf. Acta 7 (1863) 345-. cut by sphere in geodesic line. Molins, H. Toul. Ac. Sc. Mm. 3 (1891) 296-. , loxodromes of which (or their analogues) project into logarithmic spirals. Molins, H. Toul. Ac. Sc. Mm. 7 (Sem. 2) (1885) 293- ; 8 (1886) 426-. with loxodromic relation. August, F. Z. Mth. Ps. 33 (1888) 154-. minimum line in given plane. Molins, H. Toul. Ac. Sc. Mm. 2 (1890) 360-. , theory. Jung, V. Casopis 9 (1880) 216- ; Fschr. Mth. (*1880) 569. with RiR 2 constant. Metzler, G. F. Am. J. Mth. 20 (1898) 76-. with singular meridian curves. Enneper, A . Gott. Ab. 29 (1882) 87 pp. system of umbilical lines, arising from system of envelopes. Gremigni, M. [1882] Em. B. Ac. Line. Mm. 15 (1883) 3-. , 2 systems of geodesies on which form a conjugate system. Voss, A. Munch. Ak. Sb. 18 (1889) 95-. of uniform slope, and proportional networks. Lecornu, L. Caen Ac. Mm. (1884) 65-. system of curves on them. Lecornu, L. C. B. 98 (1884) 972-. zero mean curvature. Weierstrass, C. Berl. Mb. (1866) 612-, 855-. . Ribaucour, A. [1880] Brux. Mm. Cour. 4, 44 (1882) No. 1, vi + 236 pp. (Eibaucour). Calalan, E. C. Brux. Ac. Bll. 50 (1880) 333-. Liouville's element of arc. Waelsch, E. , case. Mensbrugghe, G. van der. Wien Ak. Sb. 106 (1897) (Ab. 2a) 323-. Brux. Ac. Bll. 21 (1866) 552-. normals tangent to conic surface, case. ' of which 4 straight lines enclose Monge, G. Par. EC. Pol. J. 4 (1801-02) finite area. Schwarz, C. H. A. C. E. 96 59-. (1883) 1011-. 644 8840 Eepresentation of Surfaces on Others 8840 Bonnet, O. cases. Hentschel, O. Z. Mth. Ps. 17 (1872) 39-. Lomb. Ed. 18 (1885) Surfaces with Bi + R<> constant. C. B. 42 (1856) 110-. . Dini, U. A. Mt. 7 (1865) 5-. proportional to distance of /tangent plane from fixed point. Goursat, E. Am. J. Mth. 10 (1888) 187-. . . Baroni,E. G. Mt. 28 (1890) 349-. j?j R O constant. Codazzi, D. Tortolini A. 8 (1857) 346-. R 1= f(R z ). Weingarten, J. [1862] Crelle J. 62 (1863) 160-. = _. Halphen, G. H. Par. S. Mth. Bll. 4 (1876) 94-. = . Mannheim, A. C. B. 84 (1877) 932-; Par. S. Mth. Bll. 5 (1877) 163-. = . Lie, M. S. Arch. Mth. Ntvd. 4 (1879) 507-. = _ . Weingarten, J. Crelle J. Mth. 103 (1888) 184-. = , cases. Ea/y, L. Par. S. Mth. Bll. 19 (1891) 158-. = , theory. Eaffy, L. Par. S. Mth. Bll. 25 (1897) 147-. R l = E 2 . Forsyth, A. R. Mess. Mth. 27 (1898) 129-. Uniformly illuminated lines of any surface. Fieri, M. Tor. Ac. Sc. At. 27 (1892) 347-. Voss's surfaces. Guichard, C. C. R. 129 (1899) 23-, 188. Weingarten's surfaces. Pellet, A. C. E. 125 (1897) 601-. , two surfaces that may be associated with. Demmdin, A. C. E. 131 (1900) 330-. Pdx + Ody + Rdz - 0, geometrical interpretation. Voss, A.E. Mth. A. 16 (1880) 556-. 8840 Conforinal and other repre- sentations of surfaces on others. (See also 3600, 6840 and Geo- graphy 7095.) Complex variables on any surface. Beltrami, E. [1867] A. Mt. 1 (1867-68) 329-. Conformal relation of parallelism between two surfaces. Mlodzeevskij, B. K. [1898] Eec. Mth. (Moscou) 20 (1899) 620- ; Fschr. Mth. (1899) 606. CONFOBMAL BEPBESENTATION. Eisenlohr, F. Crelle J. 72 (1870) 143-. Neumann, C. Leip. B. 22 (1870) 49-, 264-. Klein, C. FI [1881] Mth. A. 19 (1882) 159-. Phragmtn, E. Acta Mth. 14 (1890-91) 225-. Escher, R. J. N. Arch. Wisk. 20 (1893) 111-. Stcickel, P. Mth. A. 44 (1894) 553-. Lindemann, F. Konigsb. Schr. 35 (1895) [27]-. Voss, A. Mth. A. 46 (1895) 133-. Naetsch, E. Dresden Isis Sb. (1896) 33. branch points of polynomial. Stolp, C. N. Arch. Wisk. 17 (1890) 235-; Fschr. Mth. (1890) 600. case. Amstein, H. [1877] Laus. S. Vd. Bll. 15 (1878) 175-. . Ascoli, G. Mil. I. 349-, 390-, 440-. , and their application to hydrodynamics. Amstein, H. [1879] Laus. S. Vd. Bll. 16 (1880) 249-. by central projection. Weyr, E. Prag Sb. (1877) 273-. of circle on interior of epicycloid. Huber, G. Bern Mt. (1891) 42-. configurations in three dimensions. Cotton, E. C. B. 127 (1898) 349-. with conservation of area. Razzaboni, A. [1889] Bologna Ed. (1889-90) 21-. of cyclide. Holzmiiller, F. G. Crelle J. Mth. 94 (1883) 342-. on rectangle and unlimited plane. Holzmiiller, F. G. [1882] Crelle J. Mth. 94 (1883) 237-. depending on zeta-function. Kluyver, J. C. Z. Mth. Ps. 40 (1895) 129-. of Earth by epicycloidal projection. August, F. Berl. Z. Erdk. 9 (1874) 1-. by electricity. Kelvin, (Lord). Nt. 46 (1892) 490-, 541-. of ellipsoid on plane. Jacobi, C. G. J. Crelle J. 59 (1861) 74-. elliptic area on circular area. Schwarz ,H.A. A. Mt. 3 (1869-70) 166- . extension to 3-dimensional space. Landsberg, G. D. Nf. Vh. (1896) (Th. 2, Halfte 1) 7. of half plane on polygon approaching limiting circle. Pick, G. Wien Ak. Sb. 100 (1891) (Ab. 2a) 1387-. simply connected surface bounded by algebraic curve. Lindemann, F. Munch. Ak. Sb. 24 (1895) 403-. surface bounded by circular cubic or bicircular quartic. Goettler, J. [1900] Munch. Ak. Sb. 30 (1901) 165-. inner surface of circle on inner surface of regular polygon. Bigler, U. Arch. Mth. Ps. 14 (1896) 360-. multiply connected plane surfaces. Schottky, F. Crelle J. Mth. 83 (1877) 300-. paraboloid on plane. Koch, H. von. Stockh. Ofv. (1889) 181- ; Fschr. Mth. (1889) 834. pentagon on half plane. Bar well, (Miss) M.E. L. Mth. S. P. 29 (1898) 695-. one plane on another. Wangerin, A. Arch. Mth. Ps. 55 (1873) 5-. plane area on circular area. Neumann, C. G. Leip. Mth. Ps. B. 29 (1877) 154-. areas. Tallqvist, H. Helsingf . Acta 21 (1896) No. 3, 29 pp. figure bounded by straight lines in half plane, general possibility. Schlcifli,L. Crelle J. Mth. 78 (1874) 63-. on prism. Cassel, G. Stockh. Ak. Hndl. Bh. 16 (Afd. 1) (1891) No. 3, 11 pp. ; Fschr. Mth. (1891) 893-. on planes, theorem. Hoppe, R. Arch. Mth. Ps. 59 (1876) 59-. of polygons. Painleve, P. C. B. 106 (1888) 473-. problem. Cassel, G. Acta Mth. 15 (1891) 33-. 645 8840 Conformal Eepresentation. Projection 8840 problem. Burnside, W. L. Mth. S. P. 24 (1893) 187-. . Zorawski, K. Krk. Ak. (Mt.-Prz.) Bz. 19 (1902) 218- ; Crc. Ac. Sc. Bll. (1900) 325-. of pseudosphere, and applications. Marco- longo, R. Nap. Ed. 27 (1888) 111-. quadrics. Hoppe, R. [1869] Mth. A. 2 (1870) 504-. by rational function. Fuchs, L, Crelle J. Mth. 108 (1891) 181-. of spherical triangles by algebraic functions. Fischer, O. Leip. Mth. Ps. B. 36 (1884) 17-. one surface on another. Gauss, C. F. Schumacher As. Ab. 3 (1825) 1-. . Souslow [Suslov], G. K. C. E. 126 (1898) 30-; Mosc. S. Sc. Bll. 93 (No. 2) (1898) 12-. surface of rectangle on hemisphere. Schicarz , C.H.A. Gcitt. Nr. (1883) 51-. tetrahedron on sphere. Schwarz ,H.A. Crelle J. 70 (1869) 121-. surfaces of revolution on plane. Btichwaldt, F. Is. Mth. 6 (1889) 73-; Fschr. Mth. (1888) 869-. theorems of Poincare. Brun, F. de. Stockh. Ofv. (1889) 677-. theory. Painleve, P. C. E. 112 (1891) 653-. of Poincare. Brun, F. de. Stockh. Ofv. (1890) 135- ; Fschr. Mth. (1890) 410. of trochoids. Amstein, H. Laus. S. Vd. Bll. 28 (1892) 67-. Conformal transformation. Nanson, E. J. [1876] Mess. Mth. 6 (1877) 69-. , application of theory. Emmanuel, D. [Bucarest S. Sc. Bl. 7 (1898)] 14-. transformations in space, limitation. Capelli, A. A. Mt. 14 (1886-87) 227-. Conformation of areas, special. Forsyth,A. R. QJ. Mth. 26 (1893) 145-. Correlation between two plane figures, special case. Brill, J. Mess. Mth. 19 (1890) 57-, 151-. Correspondence of surfaces, general theory. Egorov, D. T. Eec. Mth. (Moscou) 18 (1896) 86-; Fschr. Mth. (1896) 500. , geodesies. Busse, F. Berl. Ak. Sb. (1896) 651-. Corresponding surface elements. Neumann, C. G. Leip. Mth. Ps. B. 28 (1876) 253-. Differential equation of second order with tran- scendental coefficients. Cassel, G. Stockh. Ofv. (1890) 381-. Flamsteed's modified development, change t of angles and distances. Tissot, A. Par. EC. Pol. J. cah. 37 (1858) 217-. Form of surface elements. Exner, K. Arch. Mth. Ps. 51 (1870) 7-. Geodesy. Pizzetti, P. Mex. S. " Alzate " Mm. 4 (1890) 257-. , fundamental equations. Ldska, W. Prag Sb. (1897) (Mth.-Nt.) No. 36, 13 pp. Great circle on Mercator chart. Chauvenet, W. Camb. (M.) Mth. M. 1 (1859) 135. sailing. Janse, L. N. Arch. Wisk. 7 (*1881) 91-. Isogonal relation of two plane spaces. Weyr, E. Casopis 3 (*1874) 1- ; Fschr. Mth. (*1874) 532. transformation, use of supplementary curves. Harris, R. A. Am. J. Mth. 14 (1892) 291-. Isothermic reflexion. Holzmilller, F. G. [1880] Crelle J. Mth. 94 (1883) 179-. Locus of centres of curvature and involutes. Fuss, N. [1804] St Pet. Ac. Sc. Mm. 3 (1811)91-. Maps, construction, Cebysev's theorem. Grave, D. A. As. Fr. C. E. (1894) (Pt. 2) 196-. , , development method. Obrecht, A. [1893] Santiago de Chile Un. A. 82 (1892) 871-. , on rotation ellipsoid. Vostokov,I.A. Vars. S. Nt. Tr. (1894-95) (C. R.) No. 11, 3-. , geographical. Korkine, A. Mth. A. 35 (1890) 588-. , , construction. Grave, D. A. Liouv. J. Mth. 2 (1896) 317-. , theory. Heger, R. Giving. 36 (1890) 47-. Measurement on plane map of lines on terres- trial sphere. Collignon, E. As. Fr. C. E. (1884) (Pt. 2) 15-. Normal curve of genus p=3. K^pinski, S. Prace Mt.-Fiz. 11 (1900) 1-; Fschr. Mth. (1900) 455. Orthogonality of elements, correspondence by. Demoulin, A. C. E. 116 (1893) 682-. Orthomorphosis. Cayley, A. QJ. Mth. 25 (1891) 203-. of circle into parabola. Cayley, A. QJ. Mth. 20 (1885) 213-. , problems. Cayley, A. Crelle J. Mth. 107 (1891) 262-. , theory. Cayley, A. QJ. Mth. 26 (1893) 282-. Point transformations leaving areas of all sur- faces invariant. Carda, K. Wien Ak. Sb. 105 (1896) (Ab. 2a) 787-. Projection. adopted for "Carte de France," properties. Puissant, L. Par. S. Phlm. Bll. (1821) 17-. central. Mogni, A. G. Mt. 13 (1875) 186-. conformal. Veltmann, W. As. Nr. 83 (1874) 225-. of curved surfaces. Dienger, J. N. A. Mth. 11 (1852) 252-. equivalent, with minimum deviation for limited circular area. Schols, C. M. Amst. Ak. Vs. M. 2 (1886) 130-; Arch. Neerl. 20 (1886) 388-. of figure of revolution. Codazzi, D. Tortolini A. 1 (1858) 89-. gnomonic, notions of infinity derived from. Purkiss, H. J. [1865] Mess. Mth. 3 (1866) horizontal, stereographic, of terrestrial hemi- sphere. Castrilli, C. G. Mt. 30 (1892) 31-, 105. map. Bremiker, C. As. Nr. 41 (1855) 353-. . Collignon, E. Par. EC. Pol. J. Cah. 41 (1865) 73-. . Markov, A. A. Kharkov Mth. S. Com. 1 (1889) 113- ; Fschr. Mth. (1888) 870-. 646 8840 Surfaces. Projection map, conformal. Wittstein, T. As. Nr. 71 II , . Grunert, J. A. Arch. Mth. Ps. 50 (1869) 176-. , graduation of instruments for. Driencourt, . A. Hydrog. 20 (1899) 78-. , for oblique cylinder and cone. Heger, R. Giving. 41 (1895) 401-. , problems. Rollet de L'Isle, . A. Hydrog. 15 (1893) 58-. , with tables. Schols,C.M. N. Arch. Wisk. 8 (*1882) 113-. Mercator's. Timerding, H. E. Z. Mth>Ps. 43 (1898) 320-. , of skew hyperboloid. Cayley, A. Mess. Mth. 4 (1875) 17-. , for triangulation near equator. Schols, C. M. Delft EC. Pol. A. 1 (1885) 1-. methods, merits. Chelini, D. G. Arcad. 74 (1838) 47-. oblique, axonometric. Kinkelin, H. Ziir. Vjschr. 6 (1861) 358-. orthographic. Avillez, J. F. de. Lisb. J. Sc. Mth. 3 (1895) 76-; Fschr. Mth. (1893-94) 1888. orthomorphic, of ellipsoid on sphere. Craig, T. Am. J. Mth. 3 (1880) 114-. plagiographic. Anger, C. T. Grunert Arch. 8 (1846) 235-. problem. Cayley, A. QJ. Mth. 13 (1875) 19-. properties. Poisson, S. D. Par. EC. Pol. Cor. 1 (1804-08) 389-. quincuncial, of sphere. Pierpont, J. Am. J. Mth. 18 (1896) 145-. of sphere, new system. Cornu, A. (ix) Par. S. Phlm. Bll. 3 (1866) 111-. , . Guyou, . C. R. 102 (1886) 308- ; Rv. Mar. et Col. 94 (1887) 228-. stereographic. Delambre,J.B.J. [1799] Par. Mm. de PI. (1804) 393-. . Chasles, M. Gergonne A. Mth. 19 (1828- 29) 157-. . Catalan, E. C. Liouv. J. Mth. 19 (1854) 132-. , analytic development. Grunert, J. A. Grunert Arch. 32 (1859) 250- . , application. Zeuthen, H. G. (xii) Ts. Mth. 2 (1866) 97-. , of theory of caustics. Quetelet, L. A. J. [1825] Brux. Ac. Sc. Mm. 4 (1827) 81-. , construction of plane zonal arcs in. Websky, M. Berl. Ak. Sb. (1886) 33-. , history. Gilnther, . D. Nf. Tbl. (1887) 227-. , property. Chasles, M. Liouv. J. Mth. 7 (1842) 272. , of spherical conic. Cayley, A. Ph. Mg. 25 (1863) 350-. , and theorems of Quetelet and Dandelin. Pelz, C. Prag Sb. (1898) (Mth.-Nt.) No. 31, 4pp. of straight line on spheroidal surface. Grunert, J. A. Grunert Arch. 6 (1845) 293-. on surface. Schoute, P. H. As. Fr. C. R. (1879) 155-. of surface of revolution on plane. Grave, D. A. St Pet. Ac. Sc. Bll. 1 (1894) 73-. Representation 8840 Projective relation between infinitesimal elements. Oliver, J. E. [1884] A. Mth. 1 (1884-85) 40-. Reduction of altitude curves. Reveille, J. A. Hydrog. 15 (1893) 147-. Relation between great circle and loxodrome joining two points on the Earth. Berg, F. J. van den. N. Arch. Wisk. 9 (*1882) 15-. Representation. by algebraic functions of whole plane on circle. Fuchs, L. Crelle J. Mth. 77 (1874) 339-; 78 (1874) 338-. of algebraic surfaces. Cremona, L. Gott. Nr. (1871) 129-. . Darboux, G. Bll. Sc. Mth. As. 2 (1871) 40-, 155-. , especially those of 4th and 5th degrees. Clebsch, R. F. A. [1868] Mth. A. 1 (1869) 253-. on plane. Clebsch, R. F. A. Mil. I. Lomb. Rd. 1 (1868) 794-. on circle, of plane areas with proportionality of small areas. Schellhammer, F. Z. Mth. Ps. 23 (1878) 69-. , single sheeted simply connected plane surface. Christqffel, E. B. Gott. Nr. (1870) 283-. , n-sheeted simply connected plane sur- face. Christo/el, E. B. Gott. Nr. (1870) 359-. of circular ring on rectangle, geometrical con- struction. Study, E. Crelle J. Mth. 97 (1884) 13-. class of skew surfaces on plane, and de- termination of their asymptotic curves. Cremona, L. A. Mt. 1 (1867-68) 248-. surfaces of 5th degree. Clebsch, R.F.A. Gott. Nr. (1869) 486-; Gott. Ab. 15 (1870) 64pp. on conic, of curve of 4th degree with cusp. Weyr, E. J. [1878] Wien Ak. Sb. 78 (1879) (Ab. 2) 396-. with conservation of area. Wierzbicki, E. Prace Mt.-Fiz. 10 (1899-1900) 129-. continuous, of line on surface. Hilbert, D. [1890] Mth. A. 38 (1891) 459-. of continuous linear configuration on discon- tinuous one. Harnack, A. Mth. A. 23 (1884) 285-. curved surfaces, new principle. Voss, A. E. Mth. A. 19 (1882) 1-. curves on surface and application to maps. Du Chatenet, M. N. A. Mth. 5 (1886) 142-. equivalent, of rotation ellipsoid on sphere. Klingatsch, A. Mh. Mth. Ps. 8 (1897) 175-. of exterior of polygon of circular arcs on circular surface. Sanio, T. Arch. Mth. Ps. 3 (1886) 1-. geodesic. LUroth, J. Mth. A. 51 (1899) 161-. of geodesic circles on surface by tangential transformations. Oseen, C. W. Stockh. Ofv. (1900) 957-. half plane on polygon bounded by arcs of confocal conies. Lindeinann, F. Munch. Ak. Sb. 25 (1896) 219-. 647 8840 Representation homocyclic, of sphere on plane. Andreev, K. A. Kharkov Mth. S. Com. 3 (1893) 35-; Fschr. Mth. (1891) 894. logarithmic, and orthogonal curve systems arising therefrom. Holzmuller, G. Z. Mth. Ps. 16 (1871) 269-. of lunes. Igel, B. Z. Mth. Ps. 17 (1872) 251-. orthogonal, representation of collinear projec- tions in space in. Schlesinger, J. Wien Sb. 59 (1869) (Ab. 2) 636-. , of surfaces of revolution, relation between meridian and contour curve. Mutter, R. Z. Mth. Ps. 21 (1876) 265-. of small part of ellipsoidal surface. Hammer, E. Ac. Nt. C. N. Acta 71 (1898) 447-. parts of curved surface on plane, principle. Weber, H. Crelle J. 67 (1867) 229-. plane, of ruled surfaces. Mannheim, A. C. K. 85 (1877) 788-, 847-, 941-. , solid. Cayley, A. Ph. Mg. 41 (1871) 286-. of polygons of circular arcs. Pochliammer, L. Crelle J. 76 (1873) 170-. problem. Weber, H. Mth. A. 2 (1870) 140-. problems. Schwarz, H. A. Crelle J. 70 (1869) 105. on sphere. Dini, U. A. Mt. 4 (1870-71) 175-. of sphere on plane. Bianchi, L. Rm. E. Ac. Line. Ed. 6 (1890) (Sem. 1) 226-. . Mlodzeevskij, B. K. Mosc. S. Sc. Bll. 65 (No. 1) (1890) 83-. rotation-cone. Klingatsch, A. Mh. Mth. Ps. 10 (1899) 75-. spherical. Darboux, G. C. E. 68 (1869) 253-. . Ribaucour, A. C. B. 75 (1872) 533-. . Darboux, G. C. E. 94 (1882) 120-, 158-, 1290-, 1343- ; 96 (1883) 366-; Par. EC. Norm. A. 5 (1888) 79-. . Adler, A. Wien Az. 37 (1900) 222-. of Steiner's surfaces and of skew surfaces of 3rd degree on plane. Cremona, L. Mil. I. Lomb. Ed. 4 (1867) 15-. straight lines of space on plane. Scheffers, G. Leip. Mth. Ps. B. 47 (1895) 201-. one surface on another. Dini, U. A. Mt. 3 (1869-70) 269-. (Dini). Darboux, G. Bll. Sc. Mth. As. 1 (1870) 383-. . Weyr, E. Casopis 9 (*1880) 201- ; Fschr. Mth. (*1880) 628. -r. Somov, P. O. Vars. S. Nt. Tr. (1894-95) (C. R.) No. 11, 14-. in Gauss's manner. Razzaboni, A. G. Mt. 27 (1889) 274-. surface on plane so that geodesies are repre- sented by straight lines. Beltrami, E. A. Mt. 7 (1865) 185-. of revolution. Saija, G. Catania Ac. Gioen. At. 10 (1897) Mem. 17, 10 pp. surfaces of revolution with constant negative curvature on one another. Wangerin, A. D. Mth. Vr. Jbr. 1 (1892) 71-. x + yi = ?/X+Yi. Holzmuller, F. G. Crelle J. Mth. 83 (1877) 38-. Deformation of Surfaces 8850 Schwarz's triangles. Kapteijn, W. N. Arch. Wisk. 1 (1895) 185- ; Fschr. Mth. (1893-94) 698. " with simple nodal point. Schilling, F. [1896] D. Mth. Vr. Jbr. 5 (1901) (Heft 1) 73-. Skew surfaces and their application on one another. Enneper, A. Z. Mth. Ps. 9 (1864) 377-. Spheroidal trigonometry. Oriani, B. Zach M. Cor. 10 (1804) 244-; 11 (1805) 551- ; Bologna Mm. I. It. 1 (1806) 119- ; 2 (1808) 1-; (pte. 2) (1810) 1- ; Mil. Ephem. As. (1827) 3-; (1828) 3-; (1829) 3-; Mil. Mm. I. Lomb. Yen. 4 (1832) 325-. , formulae. Bonsdarff, A. [1885] St Pet. Ac. Sc. Bll. 30 (1886) 205-. Stereogramsof surf aces, construction. Maxwell, J. C. [1868] L. Mth. S. P. 2 (1869) 57-. Sumner lines on Mercator chart. Greenhill, A. G. Mess. Mth. 16 (1887) 162-. and stereographic chart. Greenhill, A. G. Mess. Mth. 20 (1891) 4-. Surface of revolution, tangents to boundary of shadow. Staudigl, R. Wien Sb. 68 (1873) (Ab. 2) 228-. Surfaces of first 7 degrees developable on plane. Schwarz, H. A. [1864] Crelle J. 64 (1865) 1-. , development, case. Minding, E. F. A. Crelle J. 20 (1840) 171-. divided into squares by system of circles and their orthogonal trajectories. Demartres, G. Par. EC. Norm. A. 4 (1887) 145-. representable on plane. Bonnet, 0. [1875] A. Mt. 7 (1876) 61-. point for point. Guccia, J. As. Fr. C. E. 9 (1880) 191-. of revolution, applicable to sphere. Ocagne, M. d'. As. Fr. C. E. (1894) (Pt. 2) 11-. , homolographic development. La- marle, E. Brux. Ac. Bll. 10 (1860) 530- ; 11 (1861) 42-. in orthogonal and perspective re- presentations, construction of contours. Niemtschik, R. Wien Sb. 52 (1866) (Ab. 2) 573-. with same spherical representation. Pellet, A. C. B. 124 (1897) 1291-. Symmetric S (Schwarzian) functions. Schilling, F. Ac. Nt. C. N. Acta 71 (1898) 207-. Systems of maps with all circles of sphere represented by circles. Du Chatenet, M. N. A. Mth. 5 (1886) 168-. 8850 Deformation of surfaces. Minding, E. F. A. Crelle J. 18 (1838) 365-. Maxwell, J. C. [1854] Camb. Ph. S. T. 9 (1856) 445-. Bour, E. [J. E. E.] Par. Ee. Pol. J. cah. 39 (1862) 1-. Jordan, C. Liouv. J. Mth. 11 (1866) 105-. Moutard, T. (x) Par. S. Phlm. Bll. 6 (1869) 45-. Ribaucour, A. (xi) Par. S. Phlm. Bll. 6 (1869) 51-; (vm) C. E. 70 (1870) 330-. 648 8850 Surfaces. Deformation 8850 Daug, H. T. Stockh. Ofv. 33 (1876) 3-. Laisant, C. A. As. Fr. C. E. (1878) 180-. Weingarten, J. Berl. Ak. Sb. (1886) 83-. Mlodzeevskij, B. K. Mosc. Un. Mm. (Ps.-Mth.) 7 (1887) 141 pp. ; Bll. Sc. Mth. 15 (1891) 97- Bianchi, L. Em. E. Ac. Line. Ed. 1 (1892) (Sem. 2) 41-. Goursat, E. Am. J. Mth. 14 (1892) 1-. Cosserat, E. Toul. Fac. Sc. A. 8 (1894) E, 46pp. Adam, P. Par. S. Mth. Bll. 23 (1895) 106-, 219-; C. E. 121 (1895) 551-. Guichard, . Liouv. J. Mth. 2 (1896) 123-. Staeckel, P. C. E. 123 (1896) 677-. Weingarten, J. Acta Mth. 20 (1897) 159-. Applicable figures, construction. Cayley, A. Am. J. Mth. 2 (1879) 186. , theorem. Cayley, A. L. Mth. S. P. 10 (1878-79) 143-. Application of invariants of linear and quad- ratic differential forms. Hessenberg, G. Acta Mth. 23 (1900) 121-. Cases. Minding, E. F. A. Crelle J. 18 (1838) 297-. . Enneper,A. Gott. Nr. (1875) 129-. Contact transformations. Lovett, E. O. N. Y. Am. Mth. S. Bll. 5 (1899) 113, 118-. Covariants. Knoblauch, J. Crelle J. Mth. Ill (1893) 277-. and differential parameters. Stiickel, P. Crelle J. Mth. 113 (1894) 58-. DEFOEMATION. of class of surfaces. Bianchi, L. G. Mt. 16 (1878) 267-. closed surfaces of positive curvature. Lieb- mann, H. Mth. A. 53 (1900) 81-. and conjugate systems. Cosserat, E. C. E. 113 (1891) 460-. . Stdckel, P. Mth. A. 49 (1897) 255-. with conservation of principal curvatures. Hazzidakis, J. N. Crelle J. Mth. 117 (1897) 42-. continuous. Mair, D. B. QJ. Mth. 27 (1895) 1-. and curvature and representation of surfaces. Dienger, J. Grunert Arch. 19 (1852) 306- . of curved surfaces under certain conditions. Stdckel, P. D. Mth. Vr. Jbr. 1 (1892) 70. and cyclic systems. Cosserat, E. C. E. 113 (1891) 498-. equilibrium of skew surface. Cayley, A. L. Mth. S. P. 12 (1880-81) 103-. of evolute of catenoid. Bazzaboni, A. G. Mt. 28 (1890) 154-. extensible surfaces, mechanical theory. Finsterwalder, S. D. Mth. Vr. Jbr. 6 (1899) (Heft 2) 43-. flexible and inextensible surfaces. Volterra, V. Em. E. Ac. Line. Ed. 1 (1885) 274-. geodesies. Allegret, [_A.~\ C. E. 66 (1868) 342-. inextensible surfaces. Weingarten, J. Crelle J. Mth. 100 (1887) 296-. of inextensible surfaces, Gauss's theorem, simple demonstration. Boussinesq, V. J. (xn) Lille S. Mm. 8 (1880) 381-. infinitesimal. Genty, E. Par. S. Mth. Bll. 22 (1894) 221-; Toul. Fac. Sc. A. 9 (1895) E, . Voss, A. [1895-97] D. Mth. Vr. Jbr. 4 (1897) 132- ; Munch. Ak. Sb. 27 (1898) 229-. . Pszeborski.A.P. Fschr. Mth. (1900) 610-. , and Bianchi's associated surfaces. Cos- serat, E. C. E. 115 (1892) 1252-. , of curves. Perna, A. G. Mt. 36 (1898) 286-. of infinitesimal element. Desmousseaux de Givre, F. X. E. Par. Mm. Ing. Civ. (1864) intrinsic treatment. Cesdro, E. Nap. Ed. 33 (1894) 149-. of model of hyperboloid. Cayley, A. Mess. Mth. 8 (1879) 51-. . Lucas, E. N. A. Mth. 20 (1881) 9-. paraboloid. Thybaut, A. Par. EC. Norm. A. 14 (1897) 45-. , cases, and Weingarten's theorem. Cos- serat, E. C. E. 124 (1897) 741-. of rotation in spaces of constant curva- ture. Bianchi, L. A. Mt. 4 (1900) 1-. plane closed polygon. Macaulay, F. S. L. Mth. S. P. 28 (1897) 442-. quadrics. Peterson, K. M. (xn) Rec. Mth. (Moscou) 10 (1882-83) (Pt. 1) 476-. . Guicliard, C. C. E. 125 (1897) 596-. . Darboux, G. C. E. 128 (1899) 760-, 854-, 1264-. , isothermal surfaces connected with. Darboux, G. C. E. 128 (1899) 1483-. of revolution. Guichard, C. C. E. 128 (1899) 232-. . Tzitzeica [Titeica], . Bll. Sc. Mth. 23 (1899) 153-. ruled surfaces. Beltrami, E. A. Mt. 7 (1865) 105-. . Daug, H. T. Ups. S. Sc. N. Acta Vol. extra ord. (1877) (No. 5) 18 pp. . Chini, M. Tor. Ac. Sc. At. 26 (1891) 20-. skew surfaces. Bonnet, 0. C. E. 57 (1863) 805-. small, of curves and surfaces (helix and circular ring). Michell, J. H. Mess. Mth. 19 (1890) 68-. of sphere. Soler, E. Palermo Cir. Mt. Ed. 8 (1894) 209- ; Palermo Ac. At. 5 (1900) (Sc. M.) 35 + 23 pp. spherical surface. Cayley, A. Mess. Mth. 6 (1877) 88-. spiral surfaces. Raffy, L. C. E. 112 (1891) 850-, 976 ; Par. S. Mth. Bll. 19 (1891) 65- ; Par. EC. Norm. A. 9 (1892) 145-. and strain of surfaces. Gosiewski, W. Par. T. Nauk Sc. Pam. 1 (*1871) 57-, 183-. of one surface of revolution into another. Bro- de-n, T. Fschr. Mth. (1886) 731-. surfaces connected with quadrics. Tzitzeica [Titeica], . C. R. 128 (1899) 1276- . 649 8850 Deformation of Surfaces 8850 of surfaces of revolution. Rayleigh, J. W. Strutt (Lord). [1881] L. Mth. S. P. 13 (1881-82) 4-. . Ocagne, M. d' . Par. S. Mth. Bll. 21 (1893) 85-. . Bianchi, L. Em. E. Ac. Line. Ed. 8 (1899) (Sem. 1) 141-. , case. Demoulin, A. Brux. Ac. Bll. 30 (1895) 61-. , and isometric networks. Picard, E. (xi) Par. S. Phlm. Bll. 4 (1867) 73-. translation. Adam, P. Par. S. Mth. Bll. 23 (1895) 204-. Dilatations of elements of small portion of elastic membrane. Boussinesq, V. J. (xn) Lille S. Mm. 8 (1880) 381-. Gauss's theorem, product of radii constant. Bertrand, J. Liouv. J. Mth. 13 (1848) 80-. Invariants. Zorawski, K. Krk. Ak. (1ft.- Prz.) Bz. 3 (1891) 225-; Fschr. Mth. (1891) 805- ; Krk. Ak. (Mt.-Prz.) Ez. 8 (1895) 1- ; Crc. Ac. Sc. Bll. (1895) 91-. , and fundamental equations in theory of surfaces. Voss,A. Munch. Ak. Sb. 22 (1893) 247-. Ivory's theorem and confocal quadrics. Mann- heim, A. C. E. 102 (1886) 310-. Partial differential equations. Serret, J. A. Liouv. J. Mth. 13 (1848) 361-. Problem. Adam, P. Par. S. Mth. Bll. 24 (1896) 28-. , general. Eaffy, L. C. E. 114 (1892) 1407- ; Par. S. Mth. Bll. 22 (1894) 119-. , , fundamental equations. Aoust, (Vabbe) . C. E. 68 (1869) 1095-. Surfaces. applicability. Calo, B. A. Mt. 4 (1900) 123-. on surfaces of revolution, conditions for. Levy, M. C. E. 86 (1878) 947-. applicable. Nicolaides, N. Les Mondes 8 (1865) 710-. . Amigues, E. C. E. 104 (1887) 564-. . Chini, M. G. Mt. 27 (1889) 265-. , class, new. Weingarten, J. Gott. Nr. (1887) 28-. , and curvilinear coordinates.. Hoppe, R. Arch. Mth. Ps. 59 (1876) 225-; 68 (1882) 439-. , with distance of corresponding points con- stant. Demoidin, A. Par. S. Mth. Bll. 23 (1895) 71-. on given surface. Dini, V. G. Mt. 2 (1864) 282-. . V . Weingarten, J. C. E. 112 (1891) 706-. , theory. Bonnet, O. Par. EC. Pol. J. Cah. 41 (1865) 209- ; Cah. 42 (1867) 1-. one another, class. Weingarten, J. Crelle J. 59 (1861) 382-. , criterion. Minding, E. F. A. Crelle J. 19 (1839) 370-. , pairs. Caronntt, T. Par S. Mth. Bll. 21 134-. applicable, pairs. Genty, E. Par. S. Mth. Bll. 22 (1894) 36-. , , and conjugate systems. Petot, A. C. E. 115 (1892) 1250-. , problem. Farkas, G. Orv.-Termt. Ets. (Termt. SzaJc) (1889) 260-. , . Raffy, L. C. E. 108 (1889) 493-. , ruled, with parallelism of generators. An- tomari, X. Par. S. Mth. Bll. 22 (1894) 58-. , skew. Razzaboni, A. G. Mt. 21 (1883) 92-. on surface of revolution. Pellet, A. C. E. 125 (1897) 1159- ; 126 (1898) 392-. surfaces. of minimum area. Dini, U. C. E. 60 (1865) 340-. revolution. Picart, A. N. A. Mth. 20 (1881) 113-. . Raffy, L. Bll. Sc. Mth. 13 (1889) 161-. , and theorem of Weingarten. Goursat, E. Toul. Fac. Sc. A. 5 (1891) E, 34 pp. , theory. Dini, U. A. Mt. 7 (1865) 25-. __, _. Weingarten, J. C. E. 112 (1891) 607-. , . Goursat, E. C. E. 112 (1891) 707-. application. Combescure, E. C. E. 105 (1887) 434-. . Ricci, G. Yen. I. At. (1896-97) 1230-. on one another. Codazzi, D. Par. Mm. Sav. Etr. 27 (1883) ,Yo. 6, 45 pp. , formulae of Codazzi and Wein- garten. Cayley, A. L. Mth. S. P. 24 (1893) 210-. , theory. Ribaucour, A. (xi) Par. S. Phlm. Bll. 6 (1869) 37-. bent (umgestiilpt). Deschwanden, J. W. von. Ziir. Vjschr. 9 (1864) 1-. with circles for isothermal lines. Demartres, . C. E. 104 (1887) 217-. conservation of lines of curvature after de- formation. Delfino,C. Tortolini A. 7 (1856) 410-. deformable, applicable on surfaces of revolu- tion, determination. Raffy, . Par. S. Mth. Bll. 19 (1891) 34-. , with lines of equal curvature parallel. Raffy, . Par. S. Mth. Bll. 19 (1891) 54-. of 3-dimensions, deformability. Banal, R. Em. E. Ac. Line. Ed. 8 (1899) (Sem. 2) 13-. finite simply connected, of negative curvature, theorem. Liebmann, H. Leip. Mth. Ps. B. 52 (1900) 28-. inextensible. Jellett, J. H. [1853] Ir. Ac. T. 22 (1855) 343-. with measure of curvature invariable. Brios- chi, F. Tortolini A. 3 (1852) 293-; 4 (1853) 232-. of revolution, applicable, families. Razzaboni, A. [1887] Bologna Ed. (1887-88) 19-. , on given surface of revolution. Adam, A. N. A. Mth. 10 (1891) 18-. which by deformation are superposable in all their parts. Reech, F. C. E. 80 (1875) 1388-, 1442-. t to which helicoid can be applied. Lamarle, E. Brux. Ac. Bll. 19 (1865) 537-. 650 Orthogonal and Isothermic Surfaces 8860 ruled, applicable on surface of revolution. Pellet, A. C. E. 106 (1888) 654. , . Bioche, C. C. E. 106 (1888) 829-. skew, applicable on given skew surface. Cay- ley, A. L. Mth. S. P. 23 (1892) 217-. susceptible of special infinitesimal deformations. Padova, E. Mil. I. Lomb. Bd. 24 (1891) 821-. Theory. Weingarten, J. Acta Mth. 22 (1899) 193-. , continuous group of transformations with 28 parameters in. Staeckel, P. C. E. 121 (1895) 396-. , general differential equation in. Craig, T. Franklin I. J. 78 (1879) 289-. 8860 Orthogonal and isothermic surfaces. (See also 5630.) Lame, G. (vm) C. E. 17 (1843) 338-. Thomson, (Sir) W. Camb. Mth. J. 3 (1843) 286-; 4 (1845) 179-. Tail, P. G. [1872] Edinb. E. S. T. 27 (1873) 105-. Pellet, A. C. E. 124 (1897) 552-. Bonnet's problem. Guickard, C. C. E. 125 (1897) 643-. Condition that family of surfaces should belong to orthogonal system. Weingarten, J. L. G. J. Crelle J. Mth. 83 (1877) 1-. . Lilienthal, E. von. Mth. A. 44 (1894) 449-. for isogonal transformation of curvilinear coordinates in 3 dimensions. Maxwell, J. C. [1872] L. Mth. S. P. 4 (1871-73) 117-. Confocals of surfaces cut them orthogonally. Ltvy, M. C. E. 74 (1872) 176-. Conjugate functions and equipotential curves. Brill, J. Camb. Ph. S. P. 6 (1889) 187-. Curves and surfaces normal to one another, theory. David, C. A. [1864] (xn) Lille S. Mm. 1 (1865) 151-. Cyclidal surfaces, application of physical methods to study. Darboux, G. C. E. 83 (1876) 1037-, 1099-. Dirichlet's problem of stationary tempera- tures, and representation of given surface. Christoffel, E. B. A. Mt. 1 (1867-68) 89-; 4 (1870-71) 1-. Equations of motions of heat referred to cur- vilinear coordinates. Thomson, (Sir) W. Camb. Mth. J. 4 (1845) 33-. Equipotential cylindrical surfaces, class. Pomey, J. B. A. Tel. 21 (1894) 460- ; 24 (1898) 150-. surfaces, class. Peirce, B. O. Am. J. Mth. 18 (1896) 130-. , plane. Burnside, W. Mess. Mth. 27 (1898) 138-. system of curves, singular points. Brill, J. Camb. Ph. S. P. 6 (1889) 313-. , (Brill). Burnside, W. Camb. Ph. S. P. 7 (1892) 126-. 8860 Isothermic coordinate systems. Kotteritzsch, T. Z. Mth. Ps. 19 (1874) 265-. curves, plane, property. Burnside, W. Mess. Mth. 20 (1891) 64-. ISOTHEEMIC SUEFACES. Pellet, A. C. E. 125 (1897) 291-. Darboux, G. C. E. 128 (1899) 1299-, 1538. Guichard, C. C. E. 130 (1900) 159-. Thybaut, A. C. E. 131 (1900) 932-; Par. EC. Norm. A. 17 (1900) 541-. application of vectorial geometry. Genty, . As. Fr. C. E. (1889) (Pt. 2) 53-. class. Thybaut, A. C. E. 123 (1896) 295-. and deformation of paraboloid. Thybaut, A. C. E. 128 (1899) 1274-. "doubly eylindric" surfaces. Raffy, L. C. E. 128 (1899) 285-. geodesic lines, use of curvilinear coordinates. Salvert, (le vicomte) de. Brux. S. Sc. A. 10 (1886) (Pt. 2) 293-; 11 (1887) (Pt. 2) and harmonic equations. Thybaut, A . C. E. in heterotropic media. Moutier, J. Par. S. Phlm. Bll. 8 (1884) 184-. homogeneous solids in temperature equi- librium. Poisson, S. D. Liouv. J. Mth. 2 (1837) 147-. paraboloidal. Lame,G. [1843 or 1844] Liouv. J. Mth. 19 (1874) 307-. with plane lines of curvature in one or both systems. Adam, P. C. E. 116 (1893) 1036- ; Par. EC. Norm. A. 10 (1893) 319-. systems, algebraic. Haton de la Goupilliere, J. N. C. E. 50 (1860) 307-. , Liouville's, and geodesies. Eicci, G. Ven. I. At. (1893-94) 643-. transformation. Guichard, C. C. E. 130 (1900) 477-. triple systems. Betti, E. A. Mt. 8 (1877) 138-. Isotherms, plane algebraic. Schwarz, C.H.A. Crelle J. Mth. 77 (1874) 38-. Laplace's equation, transformations leading to new solutions. Forsyth, A. E. L. Mth. S. P. 29 (1898) 165-. Non-ruled surfaces applicable on plane. Lebesgue, H. C. E. 128 (1899) 1502-. Orthogonal and confocal curves, system. Legoux, A. Toul. Fac. Sc. A. 4 (1890) D, 7 pp. curves, two systems. Jamet, V. C. E. 106 (1888) 830-. Orthogonal surfaces. Bouquet, J. Liouv. J. Mth. 11 (1846) 446-. Serret, J. A. Liouv. J. Mth. 12 (1847) 241-. Lebesgue, V. A. N. A. Mth. 10 (1851) 265-. Bonnet, O. C. E. 54 (1862) 554-, 655-. , Darboux, G. C. E. 59 (1864) 240- ; Par. EC. Norm. A. 3 (1866) 97-; (rx) Par. S. Phlm. Bll. 3 (1866) 16-. Catalan, E. C. Brux. Ac. Bll. 26 (1868) 180-. 651 8860 Orthogonal Surfaces 8860 Darboux, G. C. E. 67 (1868) 1101-. Ribaucour, A. (xi) Par. S. Phlm. Bll. 6 (1869) 1-, 26-. Tisserand, F. C. B. 72 (1871) 734-. Enneper, A. Gott. Nr. (1872) 226-; (1873) 423-. Darboux, G. C. B. 76 (1873) 41-, 83-, 160-. Catalan, E. C. C. E. 79 (1874) 28-. Tilly, J. M. de. C. E. 87 (1878) 361-. application of elliptic coordinates. Roberts, W. C. E. 53 (1861) 546-; Crelle J. 62 (1863) 50-. Cayley's differential equation for. Johnson, A. R. Mess. Mth. 16 (1887) 27-, 111. , extension. Johnson, A. R. QJ. Mth. 22 (1887) 81-. and confocal surfaces. Legoux, A. QJ. Mth. 20 (1885) 1-. curvature. Cayley, A. Phil. Trans. 163 (1873) 229-. Dupin's theorem and its converse, geometrical proof. Sommerfeld, A. D. Mth. Vr. Jbr. 6 (1899) (Heft 1) 123-. and lines, formulae. Beltrami, E. Mil. I. Lomb. Ed. 5 (1872) 474-. represented by a partial differential equation of 1st order. Legoux, A. (xn) Isere S. Bll. 10 (1880) 125, 132-. system, elements, expressed by sigma functions, with application to rotation of connected solids. Jahnke, E. Liouv. J. Mth. 5 (1899) 155-. formed by most general system of quadrics and two other systems. Schlafli, L. [1870] Crelle J. 76 (1873) 126-. systems. Puiseux, V. Liouv. J. Mth. 8 (1863) 335-. . Enneper, A. (vn) Mth. A. 7 (1874) . Darboux, G. [1877] Par. EC. Norm. A. 7 (1878) 101-, 227-, 275-. . Adam, P. C. E. 121 (1895) 812-. . Goursat, E. C. E. 121 (1895) 883-. . Weyr, E. Casopis 25 (1896) 42-; Fschr. Mth. (1896) 506. . Tzitzeica [Titeica], . C. E. 127 (1898) 856. . Pellet, A. C. E. 128 (1899) 284-. . Servant, . C. E. 130 (1900) 28-. admitting continuous group of Combescure transformations. Egorov, D. T. C. E. 131 (1900) 668-. . Fouche, M. C. E. 131 (1900) 873-. , algebraic. Darboux, G. C. E. 69 (1869) 392-. , associated. Guichard, C. C. E. 125 (1897) 929-. , class including isothermic systems as particular case. Darboux, G. C. E. 84 (1877) 298-. and congruences. Guichard, . C. E. 125 (1897) 564-. cyclic systems. Guichard, C. C. E. 125 (1897) 519- ; Par. EC. Norm. A. 14 (1897) 467-; 15 (1898) 179-. including family of quadric surfaces. Darboux, G. C. E. 84 (1877) 336-. systems obtained by substitution with 4 para- meters. Mehler, F. G. Crelle J. Mth. 84 (1878) 219-. , and orthogonal systems of curves. Brill, J. Camb. Ph. S. P. 6 (1889) 230-. , partial differential equation of 3rd order for. Darboux, G. Acta Mth. 4 (1884) 93-. and partial differential equations. Bdcklund, A. V. Mth. A. 40 (1892) 194-. theorem. Adam, P. C. E. 103 (1886) 996-. theory, definite integrals deduced from. Ostrogradsky, . (vm) St Pet. Ac. Sc. Bll. 7 (1840) 362-. , theorems demonstrated by. Lame, G. C. E. 21 (1845) 112-. triple system. Combescure, E. [1862] (vn) A. Mt. 5 (1863) 39-. . Hoppe, R. Arch. Mth. Ps. 55 (1873) 362-; 56 (1874) 153-, 250- ; 57 (1875) 89-, 255-, 366-; 58 (1876) 37-. , condition that family of surfaces should belong to. Hoppe, R. Arch. Mth. Ps. 63 (1879) 285-. of developable. Molins, H. Toul. Ac. Sc. Mm. 1 (1879) (Sem. 1) 81-. , elements of volume between. Chelini, D. Palomba Eac. 2 (1846) 109-, 126-. formed from surfaces of 4th degree. Wangerin, F. H. A. Crelle J. Mth. 82 (1877) 145-, 348. , isothermic, theory. Salvert, (le vicomte) de. Brux. S. Sc. A. 13 (1889) (Pt. 2) 117- ; 14 (1890) (Pt. 2) 71-; 15 (1891) (Pt. 2) 201- ; 16 (1892) (Pt. 2) 273-; 17 (1893) (Pt. 2) 103- ; 18 (1894) (Pt. 2) 61-. , new. Puchta, A. Wien Ak. Sb. 102 (1893) (Ab. 2a) 1197-. systems. Betti, E. A. Mt. 8 (1877) 138-. . Kotteritzsch, T. Z. Mth. Ps. 23 (1878) 158-. . Levy, L. Liouv. J. Mth. 8 (1892) 351- ; C. E. 117 (1893) 477-; Brux. Mm. Cour. 4", 54 (1896) No. 9, 92 pp. . Geiser, C. F. Zur. Vjschr. 43 (1898) 317-. of algebraic. Roberts, W. C. E. 58 (1864) 291-. , class. Morin, P. C. E. 67 (1868) 788-. , . Bianchi, L. [1882] Em. E. Ac. Line. T. 7 (1883) 46-. , including system of helicoids. Bianchi, L. A. Mt. 13 (1885) 39-. , , new. Bianchi, L. A. Mt. 18 (1890) 301-. , cyclic. Bianchi, L. G. Mt. 21 (1883) 275-; 22 (1884) 333-. , cyclic systems. Ribaucour, A. C. E. 76 (1873) 478-. , of Eibaucour. Backlund, A. V. Stockh. Ak. Hndl. Bh. 16 (Afd. 1) (1891) No. 1, 32 pp., No. 8, 41 pp.; 17 (Afd. 1) (1892) No. 2, 41 pp.; Fschr. Mth. (1891) 395. , Dupin's theorem. Hamilton, (Sir) W. R. Ir. Ac. P. 6 (1853-54) 86-. , equation of condition for parameters. Ricci, G. Em. E. Ac. Line. Ed. 3 (1894) (Sem. 2) 93-. 652 8870 Configurations and Higher Elements of Hyperspace 8870 triple systems, one family with same spherical representation for its lines of curvature. Fouche, M. C. B. 126 (1898) 210-. including series of surfaces with one system of lines of curvature plane. Bianchi, L. A. Mt. 19 (1891-92) 177-. system of pseudospherical surfaces. Bianchi, L. Km. B. Ac. Line. Kd. 2 (1886) (Sem. 1) 19-. and lines of curvature. Stahl, H. [1870] Mth. A. 3 (1871) 488-. , property. Catalan, E. C. Liege S. Sc. Mm. 15 (1888) No. 1, 90-. , pseudospherical, Backhand's transfor- mation for. Bianchi, L. Bin. B. Ac. Line. Bd. 1 (1892) (Sem. 2) 156-. , theorem. Bertrand, J. C. B. 121 (1895) 921-. , theory. Geiser, C. F. Arch. Sc. Ps. Nt. 6 (1898) 363-. of Weingarten. Bianchi, L. Em. B. Ac. Line. Bd. 1 (1885) 163-, 243-; A. Mt. 13 (1885) 177- ; 14 (1886-87) 115- ; Palermo Cir. Mt. Bd. 8 (1894) 25-. Surfaces cut orthogonally by spheres. Bonnet, 0. C. B. 36 (1853) 1133-. with lines of curvature isothermic. Burgatti, P. Bm. B. Ac. Line. Bd. 9 (1900) (Sem. 2) 352-. and plane in both systems. Caronnet, T. C. B. 116 (1893) 1240-. orthogonal to xyz = c 3 , integration of equa- tion. Catalan, E. C. Liege S. Sc. Mm. 13 (1886) 73-. Systems of plane curves, orthogonals of which form a similar system. Tait, . Edinb. B. S. P. 20 (1895) 497-. spherical coordinates, and corresponding triply orthogonal systems. Petot, A. C. B. 112 (1891) 1426-. Theorem in theory of functions, and applica- tion to isothermic systems of curves. Holzmiiller, . Z. Mth. Ps.42 (1897) 217-. 8870 Hypergeometric configura- tions and higher elements of hyperspace. Applicability of 2 spaces with same Biemann's curvature constant. Bianchi, L. Bm. B. Ac. Line. Bd. 7 (1898) (Sevi. 2) 147-. surfaces of spaces of constant curvature. Bianchi, L. Bm. B. Ac. Line. Mm. 2 (1878) 479-. Conditions that displacement of invariable system is possible in one or several directions. Levy, M. C. B. 86 (1878) 875-. Cubes of different dimensions, representation one on another. Schoenfties, A. Gott. Nr. (1896) 255-. Curvature in hyperspace. Beez, E. Z. Mth. Ps. 21 (1876) 373-. Deformation of 3-dimensional space in plane 4-dimensional space. Schur, F. Mth. A. 28 (1887) 343-. spaces. Monro, C. J. L. Mth. S. P. 9 (1877-78) 171-. with Biemann's curvature constant. Schur, F. Mth. A. 27 (1886) 163-. Deformations of figures in n dimensions. Stdckel, P. Crelle J. Mth. 113 (1894) 102-. Infinitesimal deformations of hyperspaces. Cesdro, E. Nap. Bd. 34 (1895) 47-. - surfaces in spaces of constant curvature. Fubini, G. Bm. B. Ac. Line. Bd. 8 (1899) (Sem. 1) 246-. Loci of centres of hyper-spherical curvature of w-dimensional curves. Schoute, P. H. Amst. Ak. Vs. 8 (1900) 622-; Amst. Ak. P. 2 (1900) 527-. Meunier's theorem in higher dimensional space. Brunei, . Bordeaux S. Sc. Mm. 2 (1891) xv-. Non-Euclidean billiards. Hadamard, J. Bor- deaux S. Sc. PV. (1897-98) 147-. surfaces of constant mean curvature. Servant, . C. B. 131 (1900) 827-. Orthogonal systems in any space. Ricci, G. C. B. 125 (1897) 810-. tt-dimensional space, and reduction of differential systems. Drach, J. C. B. 125 (1897) 598-. n variables. Pellet, A. Toul. Fac. Sc. A. 2 (1900) 137-. Quadric in 5-dimensional space, superlines. Cayley, A. QJ. Mth. 12 (1873) 176-. ' n-dimensional space. Snyder, V. [1897] N. Y. Am. Mth. S. fill. 4 (1898) 68-. Beal representation of imaginary configurations, and application to theory of numbers. Busche, E. Crelle J. Mth. 122 (1900) 227-. Bepresentation of n-dimensional surface in (n - 1) -dimensional space. Vaes, F. J. Fschr. Mth. (1900) 642-. Biemann's measure of curvature in hyperspace. Beez, E. Z. Mth. Ps. 24 (1879) 1-, 65-. Spaces of constant curvature. Banal, R. Bm. B. Ac. Line. Bd. 6 (1897) (Sem. 2) 357-. . Tedone, O. Mil. I. Lomb. Bd. 32 (1899) 592-. Surface images in 4-dimensional Mannoury, G. N. Arch. Wisk. 4 112-. Total curvature in hyperspace. Banal, R. Yen. I. At. (1894-95) 998-. Weingarten's systems in spaces of constant curvature. Bianchi, L. Bm. B. Ac. Line. Mm. 4 (1887) 221-. 653 APPENDIX. 0000 Limits of our knowledge respecting theory of parallels. Kelland, P. Edinb. E. S. T. 23 . (1864) 433-. Metaphysics of geometry. Dandelin, G. P. [1842] Brux. Mm. Ac. Sc. 17 (1844) 44 pp. 0010 HISTOEY. Algebraic surfaces. Bertrand, J. J. Sav. (1867) 644-. Arabian Mathematics. Construction for 2 mean proportionals. Carra de Vaux, . Bb. Mth. (1898) 3-. Curvature of surfaces and skew curves, develop- ment of theory. Christensen, S. A. (xn) Ts. Mth. 1 (1883) 97-. Desargues (17th century), Dutch edition of his method of perspective. Le Paige, C. Bb. Mth. (1888) 10-. Duality. Milller, J. H. T. Grunert Arch. 34 (I860) 1-. Geometrical methods. Chasles, M. Brux. Mm. Cour. 11 (1837) 571 pp. Geometry, priority claim for various theorems. Mainardi, G. Em. At. N. Line. 26 (1873) 88-. of triangle. Vigarie, E. As. Fr. C. E. (1887) (Pt. 2) 87-; (1889) (Pt. 2) 117-. . Fuhrmann, . Konigsb. Schr. 40 (1899) [37]-. Greek Mathematics. Archimedes, "Arenarius." Eigaud, S. P. Ashmol. S. T. 1 (1835-37) No. 9, 32 pp. Mathematics in Bohemia, 1848-98. Studnicka, F. J. Prag Ceske Ak. Fr. Jos. Pam. (1898) (II a) 3-. Mensuration in Norway in 14th century. Enestrim, G. Bb. Mth. (1898) 19-. Plane curves, anonymous treatise. Loria, G. Bb. Mth. (1899) 10-. Quadrature of circle. Studnicka, F. J. Casopis 1 (*1872) 35-; Fschr. Mth. (*1872) 254. . Eudio, F. Ziir. Vjschr. 35 (1890) Skew curves, 3rd degree. Eeye, T. Hamb. Mth. Gs. Mt. 2 (1890) (Festschr. , Tl. 2) 43-. Various letters. Clairaut, A. C. Em. N. Line. At. 45 (1892) 233-. BIOGEAPHY. Bacaloglu, Emmanuel. [Bucarest S. Sc. Bl. 2 (1893)] 101-; [3 (1894) (Supl.)] 51-; 4 (1895) (Supl.) 97-. Broch, Ole Jacob. Par. Poids et Mes. PV. (1889) 3-. Claret y Parera, Felipe. Barcel. Ac. Bl. 1 (1892-1900) 28-. Davidov, Avgust Julievic. Mosc. S. Sc. Bll. 51 (App.) (1887) 55 pp. Henoch, Max. Fschr. Mth. (1888) 6 pp. Hessel, Johann Friedrich Christian. N. Jb. Mn. (1896) (Bd. 2) 107-. Kambly, Ludwig. Bresl. SI. Gs. Jbr. (1887) 403-. Kirchhoff, Gustav Eobert. Bresl. SI. Gs. Jbr. (1887) 172-. Kletke, Caesar Albano. Bresl. SI. Gs. Jbr. (1893) (Nek.) 6-. Le Fe"bure de Fourcy, Luis. Santiago de Chile Un. A. 33 (*1869) 134-. Lie, Marius Sophus. Christiania F. (1899) (Ov.) 4-. Ostrogradskij, Michail Vasiljevic. St Pet. Ac. Sc. Mm. (Es.) 3 (*1863) 1-. Eaymond, Georges Marie. Sav. Ac. Mm. (1891) 19-. Sadebeck, Benjamin Adolph Moritz. Bresl. SI. Gs. Jbr. (1885) 432-. Saint-Martin, Michel. Sav. Ac. Mm. (1891) 46-. Somov, Osip Ivanovic. St Pet. Ac. Sc. Mm. (Es.) 31 (*1877-78) 248-. Wittiber, Gustav. Bresl. SI. Gs. Jbr. (1886) 317-. Zurria, Giuseppe. Catania Ac. Gioen. Bll. 46 (1897) 31-. 0032 Cartesian ovals. Liguine, V. Bll. Sc. Mth. As. 6 (1882) (Pt. 1) 40-. Theory of parallels. Stdckel, P. Bb. Mth. (1899) 47-. 0035 Trigonometry, tables. Lindman,C.F. Stockh. Ofv. 11 (1854) 183- ; Grunert Arch. 25 (1855) 284-. 0050 Analysis, course applied to mechanics. Prom/, E. de. Par. EC. Pol. J. l cr cah. (1794) 92-; 2 cah. (1795) 1-; 3 e cah. (1796) 209- ; 4" cah. (1796) 459-. 654 0050 Appendix 0410 Mathematical study at Cambridge. Gautier, A. Bb. Un. 26 (1824) 161-, 243-. 0080 (For integrators and planimeters see also Mechanics 0120.) INSTRUMENTS. Compasses, homographic. Koenigs, G. C. R. 131 (1900) 1179-. , universal. Weber, L. Carl Bpm. 16 (1880) 388. Curve-tracing, use of flexible metallic strip in. Resal, H. A. C. R. 78 (1874) 709- ; Par. EC. Norm. A. 4 (1875) 191-. Curves of 2nd degree of a continuous motion, apparatus for tracing. Uchard, A. Rv. Artl. 15 (1879) 496-. Division of circle, apparatus for. Anon. Gilbert A. 65 (*1820) 329-. , (Anon). Liebherr, J. Gilbert A. 67 (1821) 109-. , (Liebherr). Reichenbach, G. R. Gilbert A. 68 (1821) 33-. , - (Reichenbach). Liebherr, J. Gilbert A. 69 (1821) 320-. Drawing instrument for circles and other curves. Cerebotani, L. Rm. N. Line. At. 46 (1893) 11-, 86-. envelopes, and application to teeth of wheels, etc. Hele-Shaw, H. S. B. A. Rp. (1898) 619-. parabolic curves. Inwards, R. L. Ps. S. P. 11 (1892) 337-; Ph. Mg. 34 (1892) 57-. Graphometer for interpolating values in curves. Hensen, V. Z. Instk. 20 (1900) 357-. Instrument for describing spiral of Archimedes. Quinby, A. B. Silliman J. 9 (1825) 316-. Integrators. Abdank Abakanowicz, B. Lum. Elect. 27 (1888) 23-. and integral curve. Aubert, . A. Cond. Pon. Chauss. 31 (1887) (Pt. 1) 165-. Line divider. Marks, (Miss) S. L. Ps. S. P. 7 (1886) 1-; Ph. Mg. 19 (1885) 280-. Oval tracer. Rebicek, G. Z. Instk. 17 (1897) 289-. Pantographs. Fischer, (Prof.) E. Carl Rpm. 1 (1866) 257-. , graduation. Woollgar, J. W. Tilloch Ph. Mg. 59 (1822) 401-. Planimeter, Amsler's, "no-rolling" curves. Morley, F. A. Mth. 1 (1900) 21-. , new. Fessenden, C. Cn. I. P. 5 (1888) 27-. , polar. Barbarin, P. N. A. Mth. 19 (1880) 212- ; 20 (1881) 266-. , , problem. Echols, W. H. [1889] A. Mth. 5 (1889-90) 9-. for rectangular coordinates. Schmidt, J. Giving. 26 (1880) 437-. Reduction-compasses. Schiereck, P. F. Ding- ier 82 (1841) 265-. , improvement. Prony, R. de. Par. A. Pon. Chauss. 9 (1835) 81-. Registering instruments. Gassier, J. F. Rouen Tr. Ac. (1833) 74-. Scale, Marquoi's, improved. Marks, W. D. Franklin I. J. 75 (1878) 418-. Spherical integrator, modifications of a form of. Ventosa, V. Ph. Mg. 23 (1887) 381. CALCULATING MACHINES. Buniakovskij, V. Ja. St Pet. Ac. Sc. Mm (Rs.) 27 (*1876) (App.) No. 4, 28 pp. Algebraic machines. Torres, L. C. R. 121 (1895) 245-. Difference-calculating engine of Scheutz. Bab- bage, H. P. (vi Adds.) CE. I. P. 15 (1855- 56) 497-. Principle of moments. Auchincloss, W. S. Am. .S. CE. T. 10 (1881) 135-. Sonne's calculating machine. Hoyer, (Inspect.) . A. G6n. Civ. 5 (1866) 111-. MODELS. Cubic surface with 27 real lines. Frost, P. QJ. Mth. 18 (1882) 89-. Equatorial surfaces, Pliicker's models. Cayley, A. L. Mth. S. P. 3 (1869-71) 281-. Peaucellier cell. Peaucellier, A. N. A. Mth. 12 (1873) 71- . Lemoine, E. As. Fr. C. R. (1874) 122- ; Par. Ing. Civ. Mm. (1875) 247-. and circular inversion. Baker, A. L. [1894] Rochester (N. Y.) Ac. Sc. P. 2 (1895) 256-. Riemann surfaces. Durege, H. Prag Sb. (1864) (Pt. 2) 112-. Skew curves of 4th degree. Rohn, C. D. Mth. Vr. Jbr. 1 (1892) 43-. surfaces, construction of paper models. Schonemann, P. Z. Mth. Ps. 28 (1883) 243-. Vernier, model. Mannheim, A. J. de Ps. 2 (1873) 392-. 0090 Abacus, logarithmic, for cable calculations. Bois, J. Lum. Elect. 39 (1891) 375-. Graphic methods of solving electrical problems. Guerout, A. Lum. Elect. 9 (*1883) 113-. Nomographic method. Ocagne, M. d'. C. R. 126 (1898) 397-. , application to equations with variables up to 10. Ocagne, M. d\ C. R. 117 (1893) 216-, 277-. Nomography. Ocagne, M. d'. Par. EC. Pol. J. 4 (1898) 205-. Periodic curves, analysis. Boucherot, P. Lum. Elect. 49 (1893) 251-. functions, instruments for analysis. Hess, A. Lum. Elect. 52 (1894) 551-. Plane representation of equations with four variables. Ocagne, M. d'. C. R. 112 (1891) 421-, 504. 0400 Negative and imaginary quantities. Vidal, S. E. [1887] I. Egypt. Bll. 8 (1888) 31-. 0410 Weights problem. Clausen, T. Crelle J. 7 (1831) 30-. 655 0420 Appendix 1210 0420 Cube, duplication. Mansfield, J. Conn. Mm. Ac. 1 (1810) lift-. , . Fontes, . Toul. Ac. Sc. Bll. 1 (1898) 129-. , , ancient methods. Mackay, J. S. [1885] Edinb. Mth. S. P. 4 (1886) 2-. , , approximate constructions. Mayor, . Gergonne A. Mth. 15 (1824-25) 90-. , , and circle, quadrature, approximate solutions. Hayden, W. E. S. P. 20 (1872) 525-. , multisection. Deladereere, A. N. A. Mth. 3 (1844) 231-. Fractions approximating to square root of a variable. Tchebychef, P. L. Par. EC. Norm. A. 15 (1898) 463-. 0430 Continuous representation of line on surface. Hilbert, D. [1890] Mth. A. 38 (1891) 459- Curve covering plane area. Peano, G. Mth. A. 36 (1890) 157-. Function, derivative of which is not integrable near certain points. Osgood, W. F. N. Y. Am. Mth. S. Bll. 5 (1899) 326, 328-. Functions having no differential coefficients. Fambri, P. Ven. I. At. (1884-85) 823-. Point aggregates. Veltmann, W. Z. Mth. Ps. 27 (1882) 313-. in continuous space of N dimensions. Cantor, G. Acta Mth. 2 (1883) 409-. , infinite linear. Cantor, G. Mth. A. 23 (1884) 453-. , theorem of Bendixson, proof. Phmgmen, E. Acta Mth. 5 (1884) 47-. 0830 Physical quantities, classification. Maxwell, J. C. L. Mth. S. P. 3 (1869-71) 224-. Quaternion functions and astatics. Joly, C. J. [1899] Ir. Ac. P. 5 (1898-1900) 366-. operator u + Va ( ) as motor, use in statics and kinematics. Everett, J. D. Mess. Mth. 4 (1875) 36-, 135- ; 5 (1876) 72-. Quaternions, application to skew curves, 3rd degree. Hamilton, (Sir) W. E. [1863] Ir. Ac. P. 8 (1864) 331-. Stereometric multiplication. Grassmann, H. [1852] Crelle J. 49 (1855) 10-. 0840 Couples, normal, and quadratic functions of 2 variables, connexion. Graves, J. T. Ph. Mg. 26 (1845) 315-. Directive calculus. Berdelle, C. As. Fr. C. E. (1886) (Pt. 2) 103- ; (1891) (Pt. 2) 181-. and calculus of position. Hyde, E. W. Am. J. Mth. 6 (1884) 1-. Vector analysis of bicircular quartics and circles. Cox, H. QJ. Mth. 19 (1883) 74-. field, theory. Vaschy, A. A. Tel. 21 (1894) 97-. 656 Vectors, aconic function of six. Hamilton, (Sir) W. E. Ir. Ac. P. 5 (1853) 177-. , projective theory. Kotelinikov, A. Kazan S. Ps.-Mth. Bll. 8 (1899) (App.); 9 (1900) (App.) 317 + 40 pp.; Fschr. Mth. (1899) 501-. 0850 (See also 2010.) Algebraic differential equations. Torelli, G. G. Mt. 24 (1886) 280-. Differential systems, existence of integrals in. Eiquier, C. C. E. 125 (1897) 933-. , most general. Eiquier, C. Par. EC. Norm. A. 14 (1897) 99-. , transformations. Delassus, E. C. E. 123 (1896) 1246-. Linear systems. Laguerre, . Par. EC. Pol. J. Cah. 42 (1867) 215-. Polar umbrse. Sylvester, J. J. E. S. P. 12 (1862-63) 563-. Systems of differential equations and their connexion with partial linear equations of 1st order. Jacobi, C. G. J. Crelle J. 23 (1842) 1-. , theory. Stodolkievitz , A. J. C. E. 120 (1895) 36-, 595-. infinite order of differential equations. Koch, H. von. Stockh. Ofv. (1899) 395-. n 2 integral elements, reduction. Kro- necker, L. Crelle J. Mth. 107 (1891) 135-. 0870 Syllogism No. Ill, and logic in general. De Morgan, A. [1858] (vn) Camb. Ph. S. T. 10 (1864) 173-. IV, and logic of relations. De Morgan, A. [1860] (vn) Camb. Ph. S. T. 10 (1864) 331-. V, and points of onymatic system. De Morgan, A. [1863] (vn) Camb. Ph. S. T. 10 (1864) 428-. 1200 Finite screw motions, composition. Wiener, H. Leip. Mth. Ps. B. 42 (1890) 13-. Geometrical analysis of motions. Wiener, H. Leip. Mth. Ps. B. 42 (1890) 245- ; 43 (1891) 424-. Groups of interchangeable two-reflexion rela- tions. Wiener, H. Leip. Mth. Ps. B. 45 (1893) 555-. motions. Schonflies, A. Mth. A. 28 (1887) 319- ; 29 (1887) 50-. , proof of a theorem. Schonflies, A. Gott. Nr. (1886) 497-. Inversions, theory. Wiener, H. Leip. Mth. Ps. B. 42 (1890) 71-. Eelations compounded of two reflexions. Wiener, H. Leip. Mth. Ps. B. 43 (1891) 644-. 1210 Cross-ratio group of 120 Cremona quadratic transformations. Slaught, H. E. Science 8 (1898) 121- ; Am. J. Mth. 22 (1900) 343-. 1210 Appendix 1630 Cubic surfaces, polar hexahedra. Cremona, L. Mth. A. 13 (1878) 301-. Cyclic groups, number in Cremona transforma- tion. Kantor, S. [1880] A. Mt. 10 (1880- 82) 71-. , quadratic plane transformation. Kantor, S. [1880] A. Mt. 10 (1880-82) 64-. . , space transformation. Kantor, S. C. E. 90 (1880) 1156-. Equations of geometry. Jordan, C. C. E. 68 (1869) 656-. Finite groups of plane collineations with irre- ducible quartic invariant. Ciani, E. Mil. I. Lomb. Ed. 33 (1900) 1170-. Function, five times transitive, of twenty-four quantities. Mathieu, E. Liouv. J. Mth. 18 (1873) 25-. Galois group of bitangents of curves of 4th degree. Weber, H. Mth. A. 23 (1884) 489-. Groups of class n-u and degree n, at least (n-1) times transitive. Maillet, E. C. E. 128 (1899) 277. transformation in 3-dimensional space, class. Blichfeldt, H. E. Am. J. Mth. 22 (1900) 113-. (PJsflO' (njseo arising from 6 line complexes mutually in involution. Veronese, G. A. Mt. 11 (1882-83) 284-. Hamiltonian groups. Miller, G. A. C. E. 126 (1898) 1406-. Inflexion points of 3rd degree and stationary tangents of curves of 3rd class. Hesse, L. O. Crelle J. 38 (1849) 257- Rummer surface, equation, 16th degree, con- nected with. Jordan, C. Crelle J. 70 (1869) 182-. Linear substitutions, new theory. Laurent, H. Liouv. J. Mth. 4 (1898) 75-. Number of values assumable by function of n letters, theorem of Bertrand. Cartan, . C. E. 119 (1894) 902. Octahedral and ikosahedral groups, two skew cubics automorphic for. Kohn, G. Wien Ak. Sb. 108 (1899) (Ab. 2a) 58-. Octahedron, equation. Brioschi, Fr. Em. E. Ac. Line. T. 3 (1879) 233-. and equation of 4th order. Puchta, A. Wien Ak. D. 41 (1879) (Ab. 2) 57-. Substitution group connected with 16 straight lines of quartic surface with double conic. Pereno, I. A. Mt. 21 (1893) 57-. groups connected with 27 straight lines of cubic surface. Pascal, E. A. Mt. 20 (1892- 93) 269-; 21 (1893) 85-. , Napier's rules in spherical trigono- metry. Fund, 0. [1897] Hamb. Mth. Gs. Mt. 3 (1900) 290-. 1230 Automorphic curves and surfaces, family. Klein, F., & Lie, S. C. E. 70 (1870) 1222-, 1275-. Continuous system of transformations, deter- mination of all surfaces admitting. Lie, S. Leip. Mth. Ps. B. 47 (1895) 209-. VOL. I. 657 Discontinuity in certain homographies. Fricke, R. Mth. A. 47 (1896) 557-. Infinitesimal transformations, geometry. Zo- rawski, K. Krk. Ak. (Mt.-Prz.)Bz. 14 (1899) 218-; 17(1900) 154-; Crc. Ac. Sc. Bll. (1897) 365-; (1899)81-. Invariants of n points. Lovett, E. O. [1897] N. Y. Am. Mth. S. Bll. 4 (1898) 58-. Transformations, geometric, of line systems. Lie, S. [1870] C. E. 71 (1870) 579-; Christiania F. 13 (1871) 506-. 1600 Algebraic continued fractions, generalisation. Fade, . C. E. 118 (1894) 848-. Equalities in two degrees 2a r =SJ r , 2a r 2 = SV- Frolov, M. C. E. 107 (1888) 831-. 1620 Chess. Knight's move, problem. Cliambure, de. [1861] 1. Egypt. Mm. 1 (*1862) 67-. COMBINATIONS. Lock, Bramah's, improved. Renter, J. Wien Jb. Pol. I. 10 (1827) 32-. with 10,000 combinations. Niciiolson, W. Nicholson J. 13 (1806) 158-. combinations, insecurity. Crivelli, A. Wien Jb. Pol. I. 5 (1824) 22-. , Egyptian. Crivelli, A. Wien Jb. Pol. I. 6 (1825) 1-. Partition of numbers. theorem. Clausen, T. As. Nr. 21 (1844) 367-. PEEMUTATIONS. automatic tit-tat-to machine. Freeland, F. T. [1878] Franklin I. J. 77 (1879) 1-. circular, sequences. Andre, D. C. E. 120 (1895) 714-. of first 71 numbers, distribution into 4 groups. Andre, D. C. E. 115 (1892) 872-. 1625 Figurate numbers. polygonal numbers. Marchand, D. Les Mondes 42 (1877) 164-. in n-dimensional space. Hannuth, T. [1882] Arch. Mth. Ps. 69 (1883) 90-. 1630 Curves of errors. Thiele,T.N. (xn) Ts.Mth. 2 (1878) 54-. Ellipses of equal probability. Breger, P. Ev. Artl. 10 (1877) 82-. Error triangles. Grnnert, J. A. Grunert Arch. 4 (1844) 348-. 1630 Appendix 2060 Errors of observation arising from precon- ception of result. Dufour, C. Laus. Bll. S. Vd. 5 (1856) 99-. 1635 Annuities, theory. Hinard, C. A. Cond. Pon. Chauss. 27 (1883) (Pt. 1) 250-. 2000 (See also 8080 Connexes.) Linear substitutions, extension of calculus. Stephanos, C. C. R. 128 (1899) 593-. 2010 Determinant of Smith and Mansion. Cesdro, E. N. A. Mth. 5 (1886) 44-. Determinants, theorems. Anglin, A. H. Par. S. Mth. Bll. 15 (1887) 120-. , . Studnicka, F. J. Fschr. Mth. (1898) 121, 122, 123; (1899)150-. 2020 Resultants in spherical trigonometry, concept of. Meyer, F. Crelle J. Mth. 15 (1895) 209-. 2030 Determination of points and planes after rota- tion about an axis. Hudson, E. W. H. T. Mn. Mg. 12 (1900) 343-. Orthogonal coefficients, relation between sys- tems. Hoppe,E. Arch. Mth. Ps. 61 (1877) 111-. Rotatory transformations of space. Study, E. Leip. Mth. Ps. B. 42 (1890) 341-. 2040 Algebraic forms, Hilbert's theory. Schonflies, A. Gott. Nr. (1891) 339-. Concomitants of systems of conies and quadrics. Johnson, A. E. L. Mth. S. P. 21 (1891) 432-. Configuration corresponding to connexes, 2nd order and 1st class. Krause, E. Mth. A. 14 (1879) 294-. Invariants of conies and quadrics given by motion and similitude groups. Lovett, E. O. A. Mth. 11 (1896-97) 33-. Polar constructions. Morley, F. Mth. A. 51 (1899) 410-. operations, numerical formulae. Capelli, A. G. Mt. 21 (1883) 343-. rectilinear figures of quadric and cubic. Johnson, A. E. Mess. Mth. 16 (1887) 63-. Quadratic equation, discriminant, conditions for circular sections. Dingeldey, F. Crelle J. Mth. 122 (1900) 186-. form, homographic transformation into itself. Cayley, A. Ph. Mg. 6 (1853) 326-; 7 (1854) 208-. functions, simultaneous transformation of two. Campbell, J.E. Mess. Mth. 21 (1892)78-. Quaternary connexes (1, 1), linear systems. Lazzeri, G. Rm. R. Ac. Line. Mm. 4 (1887) 259-. Reduction of general equation of 2nd degree. Boole, J. Camb. Mth. J. 2 (1841) 64-. 2050 Binary cubic and quadratic forms, application to skew cubic curves. Pittarelli, G. G. Mt. 17 (1879) 260-. and conies. Pittarelli, G. G. Mt. 21 (1883) 19-. forms, representation on skew cubic curves. Sturm, E. Crelle J. Mth. 86 (1879) 116-. Covariant geometry of triangle. Morley, F. QJ. Mth. 25 (1891) 186-. Discriminant surface, singularities. Hilbert, D. Mth. A. 30 (1887) 437-. Equation, 4th order. Eussell, E. [1887] L. Mth. S. P. 19 (1889) 56-. Invariants and covariants of function trans- formed by quadratic substitution. Spottis- woode, W. Rm. R. Ac. Line. T. 7 (1883) 218-. (Spottiswoode). Pittarelli, G. Rm. R. Ac. Line. Rd. 1 (1885) 327-, 374-. Octahedron, equation. Brioschi, Fr. Rm. R. Ac. Line. T. 3 (1879) 233-.__ Poncelet's theorem on ^/a + bx + cx*. Syl- vester, J. J. Ph. Mg. 20 (1860) 307-. Solution of differential equation of sources. Hammond, J. Am. J. Mth. 5 (1882) 218-. Symbolic notation of binary forms, study of skew cubic curves by. Ovidio, E. d'. [1879] Tor. Ac. Sc. Mm. 32 (1880) 1-. 2060 Analytical study of quartic curves. Brioschi, Fr. A. Mt. 7 (1876) 202-. Conjugate connexes, theory. Stephanos, C. Bll. Sc. Mth. As. 4 (1880) (Pt. 1) 318-. Connex, new element of plane geometry. Clebsch, E. F. A. Gott. Nr. (1872) 429-. Connexes (1, 2) and (2, 2), construction. Peano, G. Tor. Ac. Sc. At. 16 (1880) 497-. Contravariant (Cayleyan), etc. of cubic curves, general properties. Cayley, A. [1856] Phil. Trans. (1857) 415-. Covariant curves, Steiner's, Cayley's, etc., behaviour at singular points of original curve. Wolffing,E. Z. Mth. Ps. 40(1895)31-. theory of quartic curves, X 2 X 3 + X 3 3 X l + X l 3 X 2 ' Brioschi, F, Mil. I. Lomb. Rd. 17 (1884) 401- ; Rm. R. Ac. Line. T. 8 (1884) 164-. Covariants, 2, applied to construction of certain curves. Mangeot,S. N.A.Mth.l6(1897) 76-. of ternary forms. Gundelfinger, S. A. Mt. 5 (1871-73) 223-. Curves, decomposition into straight lines. Gordan, P. Mth. A. 45 (1894) 410-. Invariants of two conies. Kluyver, J. C. As. Fr. C. R. (1887) (Pt. 2) 132-. Tactinvariant of conic and cubic. Lindemann, F. B. A. Rp. (1884) 647-. Ternary connexes, 1st order and 1st class. Battaglini, G. [1880] Nap. Ac. At. 9 (1882) No. 4, 16 pp. 658 2060 Appendix 2830 Ternary connexes, 1st order and 1st class, geo- metrical significance of invariants. Pannelli, M. G. Mt. 36 (1898) 81-. , 2nd order and 2nd class, in double involution. Pannelli, M. G. Mt. 26 (1888) 1-. forms of 4th, 5th, 6th, 7th orders, equa- tions arising from certain. Briosc'hi, F. I'm. B. Ac. Line. T. 8 (1884) 164-. 2070 Application of symbolic calculus to quadrics. Garbieri, G. Yen. I. At. (1885-86) 943-. Quadratic form in m variables, decomposition into sum of m-n squares. Benmt, . C. E. 101 (1885) 869-. forms, 2, and skew quartic curve. Westphal, G. Mth. A. 13 (1878) 1-. 2400 Algebraic functions, representation, theorem of Abel's. KiJnigsberger, . [1868] Mth. A. 1 (1869) 168-. Besolution of functions into conjugate factors. Young, J. E. Ph. Mg. 34 (1849) 278-. 2420 Sturm's functions. Vidal, . [1886] I. Egypt. Bll. 7 (1887) 400-. Sylvester's theorem, new version. Horner, J. [1867] QJ. Mth. 9 (1868) 221-. 2430 Equations, 3rd order. geometry. Bussell, R. [1893] Ir. Ac. P. 3 (1893-96) 170-. Polygonal and polyhedric coordinates, etc. Sautreaux, F. Brux. Mm. Cour. 8, 36 (1884) No. 3, 14 pp. 2440 Fluctuating functions, numerical calculation of roots. Rayleigh, John William Strutt (Lord), (xi) L. Mth. S. P. 5 (1873-74) 119-. Numerical equations, separation of roots. Vidal, . [1882] I. Egypt. Bll. 3 (*1883) 91-. .solution. Vidal, . [1880] I.Egypt. Bll. 1 (*1882) 3-. 2450 Equations, 5th order. Guldberg, A. S. [1869] Christiania F. 12 (1870) 308-. Solution, equation of icosahedron in. Klein, C. F. Mil. I. Lomb. Ed. 10 (1877) 253-. Icosahedron. Klein, C. F. Erlang. Ps. Md. S. Sb. 9 (1877) 16-, 70-, 179- ; Mth. A. 12 (1877) 503-. Irreducibility of equation in division of circle. Arndt, F. Crelle J. 56 (1859) 178-. 2460 Equations representable by systems of three points on curve. Ocagne, M. d\ Acta Mth. 21 (1897) 301-. 2800 Bardwell, F. W. Kan. Ac. Sc. T. 3 (1874) (Re- print, 1896) 94-. Application of theorem in analysis situs. Busche, E. [1898] Hamb. Mth. Gs. Mt. 3 (1900) 333-. Function f(s) = Sn~ 8 , zeros. Piltz, . D. Nf. Vh. (1891) (Th. 2) 15-. Functional theorem of Lagrange, extension. Degen, C. F. (vi Adds.) Kiob. Ov. (1818- 19) 3-. Indeterminate equations and n-dimensional geometry. Brunei, G. Bordeaux S. Sc. Mm. 2 (1886) 129-. Laws of reciprocity, and sub-groups of arith- metical group. Stouff, X. C. E. 116 (1893) 308-. Products of numbers, properties. Teixeira, J. P. G. Teix. J. Sc. 14 (1900) 179-. 2810 Congruence x l = 1 (mod. M), roots, etc. Arndt, F. Crelle J. 31 (1846) 259-. nx=n + I (mod. p). Graves, J. T. B. A. Ep. (1856) (pt. 2) 1-. Congruences, linear, solution by binomial fac- torial. Bouniakowsky, V. [1842] St Pet. Ac. Sc. Mm. 5 (1844) 287-. , , of systems. Demeczky de Gyergyo- szentmiklos, D. C. E. 88 (1879) 1311-. DIVISIBILITY. of numbers of form of 2 2m + l, new case (Pervusin). Buniakovskij , V. Ja. St Pet. Ac. Sc. Mm. (Rs.) 31 (*1878) 223-. Function entering into convergents of con- tinued fractions, and roots of congruences of first degree. Porcelli, 0. G. Mt. 10 (1872) 37-. 2815 INDETEEMINATE EQUATIONS. 2 or 3 variables, having only finite number of integral solutions. Maillet, E. C. E. 128 1383-. 2830 Congruences, 2nd degree. Bouniakowsky, V. [1830] St Pet. Ac. Sc. Mm. 1 (1831) 563-. _, , solution. Tonelli, A. Gott. Nr. (1891) 344-. 659 2830 Appendix Quadratic binary forms, system of three, rela- tion to spherical trigonometry. Stephanos, C. Par. S. Mth. Bll. 10 (1882) 134-. 2840 Congruences, 2nd degree. Jordan, G. C. E. 62 (1866) 687-. , -- , canonical form and number of solu- tions. Jordan, C. Liouv. J. Mth. 17 (1872) 368-. , -- , reduced forms. Jordan, C. C. E. 74 (1872) 1093-. , -- , with several unknowns, case. Le Besgue, V. A. C. E. 62 (1866) 868-. Decomposition of numbers into sums of squares. 4 squares, Jacobi's theorem. Weill, M. C. E. 99 (1884) 859-. Quadratic forms, application of non-Euclidean geometry to. Poincare, H. As. Fr. C. E. (1881) 132-. 2850 Binomial congruences, classification of roots, application to Canon Arithmeticus. Le Besgue, V. A. C. E. 63 (1866) 1100-. , Daniel's formula. Horta, F. Lisb. A. 1 (1857) 705-. with prime modulus, theorem, and appli- cation to Canon Arithmeticus. Le Besgue, V.A. C. E. 61 (1865) 1041-. Congruences, formula. Bellavitis, G. Ven. At. (1859-60) 821-. , general theorems, and applications. Scho- nemann, T. Crelle J. 19 (1839) 231-, 289-. , higher, with real prime modulus. Dede- kind, R. Crelle J. 54 (1857) 1-. , , theory. Eados, G. Crelle J. Mth. 99 (1886) 258-. 2880 Cyclotomy. Eisenstein, G. Crelle J. 27 (1844) 269-. Gauss's series, Dirichlet's method for finding coefficients. Kronecker, L. Hamb. Mth. Gs. Mt. 2 (1890) (Festschr., Tl. 2) 32-. 2890 Automorphic functions, theory, and theory of numbers, relations between. Fricke, E. D. Mth. Vr. Jbr. 6 (1899) (Heft 1) 94-. Fuchsian functions and quadratic forms. Picard, E. Am. J. Mth. 11 (1889) 187-. . Stouff,X. Toul. Fac. Sc. A. 8 (1894) D, 20 pp. , indefinite ternary. Poincare, H. C. E. 102 (1886) 735-. groups and quadratic forms. Stouff, X. Toul. Fac. Sc. A. 3 (1889) B, 28 pp. Gamma function, applications to theory of numbers. Berger, A. [1880] Ups. S. Sc. N. Acta 11 (1883) (No. 1) 87 pp. 3250 Hyperfuchsian functions and ternary quadratic forms. Picard, E. Acta Mth. 5 (1884) 121-. Koch, H. von. 2900 C. E. 130 (1900) 1243-. 2920 converging series for. Ferrel, Smiths. Ct. 18 (1873) (Art. 5) 3200 W. [1870] 6 pp. mg Fambri, P. Ven. I. At. (1884-85) 823-. , monomial, 2, property of primitives and derivatives of product. Sang, E. Ph. Mg. 6 (1829) 262-. 3220 INFINITE SEEIES. Integral series. Hadamard, . C. E. 124 (1897) 492; Bordeaux S. Sc. PV. (1896-97) 110-. 3230 Analysis, variable parameters in. Cauchy, A.L. C. E. 40 (1855) 261-. 3240 Expansion of functions satisfying an algebraic equation. Hurwitz, A. Par. EC. Norm. A. 6 (1889) 327-. Lagrange's Series. and application to Kepler's equation. Weiss, E. Wien Ak. D. 49 (1885) (Ab. 1) 133-. (Weiss). Stole, 0. Wien Ak. Sb. 95 (1887) (Ab. 2) 199-. 3250 INTEGEATION. graphical methods. Massau, J. Ev. Un. Mines 16 (1884) 243-, 579-; 17 (1885) 1-; 20(1886)1-, 475-; 21 (1887) 85-, 308-, 572-; 22 (1887) 1-. A. Mth. 15 Quadrature, Cebysev's formula. Biles, J. H. [1898] Glasg. I. Eng. T. 42 (1899) 176-. , Gauss's method. Mansion, . Brux. S. Sc. A. 15 (1891) (Pt. 1) 57-. , Gauss's method. Raffij, L. N. . (1896) 249-. , , extension. Bourget, H. C. E. 126 (1898) 634-. , , generalisation. Christoffel, E. B. Crelle J. 55 (1858) 61-. , , remainder. Mansion, P. C. E. 102 (1886) 412- ; Brux. Ac. Bll. 11 (1886) 293-. , , and solution of Kepler's problem. Strehlke, F. Grunert Arch. 32 (1859) 433-. 660 3250 Appendix 4060 Theorem of Cotes, new demonstration. Littrow, J. J. von. Zach Cor. 3 (1819) 22-. Transcendental integrals of form Thomson, (Sir) W. QJ. Mth. 1 (1857) 316-. 3260 [1839] St Pet. Ac. Sc. Ostrogradsky, M. A. Mm. 4 (1841) 309-. Poisson's integral. Mukhopadhyay, A. [1888] Beng. As. S. J. 57 (Pt. 2) (1890) 100-. I [/ (sin a*, cos ax) - /(sin /So:, cos /3x)] ~ Schlffinilch, 0. Grunert Arch. 5 (1844) 152-. Zehfuss, G. Grunert Arch. 30 (1858) 465-. 7T /o~ "(2 cosx) a + 6 cos (a - b)xdx. Lipschitz,R. 0. S. Bll. Sc. Mth. As. 5 (1881) (Pt. 1) 387-. 3270 Curvilinear coordinates, Stokes's theorem. Webster, A. G. N. Y. Am. Mth. S. Bll. 4 (1898) 438-. Double integrals. of 2nd kind in theory of algebraic surfaces. Picard, E. C. R. 125 (1897) 909- ; 127 (1898) 579-; 129 (1899) 539-. Integrals deduced from theory of orthogonal surfaces. Ostrogradnky, . (vm) St Pet. Ac. Sc. Bll. 7 (1840) 362-. 3280 Saint- (1886) Cuttings and embankments, problem. Germain, A. de. Caen Ac. Mm. (Pt. 1) 23-. Maximum or minimum, properties common to all curves satisfying certain condition of. Vicaire, E. C. E. 106 (1888) 456-. Trajectories, on given surface or in space, rendering minimum I (v) ds. Roger, E. C.R. 40(1855) 1176-. 3600 (See also 8800-8840.) Analytic forces. Lecornu, L. Par. EC. Pol. J. 55 (1885) 253-. Conformal representation. Neumann, C. Leip. B. 22 (1870) 49-, 264-. Curves along which the absolute value of a function is constant, use. Schlafli, L. [1873] (xi) A. Mt. 6 (1873-75) 1-. Singular points, essential, theory. Guichard, C. Par. EC. Norm. A. 12 (1883) 301-. Transformation of polygons in z plane. Lecornu, L. C. R. 92 (1881) 695-; Par. EC. Pol. J. Cah. 49 (1881) 205-. 4010 Cubic corpus of functions, complete system of differentials of 1st class in. Baur, L. Mth. A. 46 (1895) 31-, 160, 608. , functions, theory. Baur, L. Mth. A 43 (1893) 505-. Fundamental system for given class domain, determination. Mertens, F. Prace Mt.-Fiz. 6 (1895) 106-. Period systems of functions of real variables. Kronecker, L. Berl. Ak. Sb. (1884) 1071-. 4040 Approximate equation F 1 (c) = log - . Verhulst, P. F. Brux. Ac. Bll. 10 (1843) (pte. 2) 378. Curves represented by _ Tr _*L__ * 77 __^__ r _ = o, V Turner, H. H. Mess. Dixon, A. C. ivelopes. Mth. 10 (1881) 21-. Deformable frameworks. Mth. 29 (1900) 1-. Doubly periodic functions. class of two. Farkas, J. C. R. 90 (1880) 1269-. of 3rd kind, application of Mittag-Leffler's theorem. Appell, P. Par. EC. Norm. A. 2 (1885) 67- ELLIPTIC FUNCTIONS. applications to flexible cords. Marcolongo, R. Nap. Ed. 31 (1892) 71-, 89-. of 1st kind, representation by arc of plane transcendental curve. Tortolini, B. Tor- tolini A. 4 (1853) 485-. 4050 Curves, 4th degree, x^x^ + x^Xi+x^x.,, co- variant theory. Brioschi, F. Mil. I. Lomb. Ed. 17 (1884) 401- ; Em. E. Ac. Line. T. 8 (1884) 164-. DIVISION. of lemniscate arcs. Tortolini, B. Palomba Eac. 1 (1845) 17-; Mod. S. It. Mm. 1 (1862) 91-. 4060 Abelian functions. in connexion with 2-dimensional fluid motion. Baker, H. F. Camb. Ph. S. P. 9 (1898) 381-. 661 TT 3 4060 Appendix ABELIAN INTEGRALS, inversion, AppeU, P. C. R. 99 (1884) 1010- 5230 Solution. Algebraic differential equations, algebraic in- tegrals. Jahnke, E. Z. Mth. Ps. 35 (1890) 148-. , irreducibility. Koenigsberger, L. Berl. Ak. Sb. (1899) 783-. differentials, algebraic integrals. Humbert, G. Acta Mth. 10 (1887) 281-. , . Ptaszycki, J. Acta Mth. 11 (1887-88) 395-; Prace Mt.-Fiz. 1 (1888) 81-; Kharkov Mth. S. Com. 1 (1889) 61-; Fsehr. Mth. (1888) 29&-. Conditionally periodic functions in mechanics. Staude, O. Crelle J. Mth. 105 (1889) 298-. 4420 Differential equations of tetrahedron, octa- hedron and icosahedron. Brioschi, F. A. Mt. 10 (1880-82) 104-. Greatest height of mast or tree consistent with stability. Greenhill, A. G. [1881] Camb. Ph. S. P. 4 (1883) 65-. HYPEKGEOMETRIC FUNCTIONS, analogues. Goursat, E. C. R. 102 (1886) 204-. Kepler's problem and function 1 (2 J fc h = | cos 7te . cos (k sin e) de. Anger, C. T. Danzig N. Schr. 5 (1855) (Hej't 2) vi + 29 pp. 4440 Conformal representation of half plane on polygon approaching limiting circle. Pick, G. Wien Ak. Sb. 100 (1891) (Ab. 2a) 1387-. spherical triangles by algebraic func- tions. Fischer, 0. Leip. Mth. Ps. B. 36 (1884) 17-. Fuchsian surfaces. Bianchi, L. Rm. R. Ac. Line. Rd. 4 (1888) (Sem. 2) 161-. Surfaces with curve-symmetry of systems of planes. Mangeot, S. C. R. 112 (1891) 1497-. Symmetric S-functions with singularity. Schilling, F. Ac. Nt. C. N. Acta 71 (1898) 207-. 4800 Integration, graphic methods. Massau, J. Rv. Un. Mines 16 (1884) 243-, 579-; 17 (1885) 1-; 20 (1886) 1-, 475-; 21 (1887) 85-, 308-, 572-; 22 (1887) 1-. 4830 Equations. Lie, M. S. Christiania F. (1882) No. 21, 12 pp., No. 22, 6 pp.; (1883) No. 10, 4 pp., No. 18, 5 pp. of problem of 3 bodies, report on progress. Whittaker, E. T. B. A. Rp. (1899) 121-. Surfaces of 4th degree admitting an integral of total differential of 1st kind. Berry, A. C. R. 129 (1899) 449-. System of surfaces defined by linear equation, 1st order, fundamental points. Fouret, G. C. R. 86 (1878) 654-. 4840 Connexes of line pairs. Weber, E. von. Gb'tt. Nr. (1896) 282-. Integrals. intermediate, of equations, 2nd order. Goursat, E. C. R. 112 (1891) 1117-. Integration. case. Zaremba, . C. R. 110 (1890) 127-. Transformations, class. Clairin, J. C. R. 130 (1900) 309-. 4850 Equations. of tetrahedron, octahedron, and icosahedron. Brioschi, F. A. Mt. 10 (1880-82) 104-. 4870 Algebraic equation, family of plane curves de- nned by. Fouret, G. C. R. 86 (1878) 586-. Connex C (x, x, 0; M, , 0), principal coinci- dence. Moller, J. Lund. Un. Acta 16 (1879-81) (Mth.) (No. 1) 16 pp. Equation between two variables, systems of curves defined by. Dyck, W. Munch. Ak. Sb. 21 (1892) 23-; 22 (1893) 101-. 5200 Differential forms, and systems of elements. Koenigs, G. C. R. 104 (1887) 673-, 842-; Acta Mth. 10 (1887) 313-. 5210 case. Miil- (1883) 113-. Equation of connex (1, n), s lendorff, E. Arch. Mth. I Partial differential equations. Duport, H. C. R. 130 (1900) 232-. P dx + Q d 11 + R dz = 0, geometrical interpreta- tion. Voss, A. E. Mth. A. 16 (1880) 556-. 5230 TRANSFORMATION. infinitesimal, trajectories of motions of systems admitting. Painleve, P. C. R. 116 (1893) 21-. 662 5240 Appendix 6810 5240 Covariants of skew curves. Gicyther, R. F. B. S. P. 56 (1894) 272-. 5600 Curves fulfilling the condition ~ + -r = Q dx 2 dyz (rhizic curves), property. Rankine, W. J. M. B. S. P. 15 (1867) 468-. Differential equation V 2 * + ft 2 / (.ryz) * = 0, system of orthogonal coordinates belonging to. Petrini, H. Stockh. Ak. Hndl. Bh. 19 (Afd. 1) (1894) No. 4, 36 pp. 5610 Harmonic analysis by polar planimeter. Fin- sterwalder, S. Z. Mth. Ps. 43 (1898) 85-. Bate of solution of solids. Bottomley, J. Manch. Lt. Ph. S. Mm. & P. 3 (1890) 113-. Surface generated by solution of isotropic solid. Bottomley, J. Manch. Lt. Ph. S. Mm. & P. 2 (1889) 154-, viii. 5620 Shells bounded by confocal quadrics. Klein, C. F. Mth. A. 18 (1881) 410-. 5630 1 d 2 r 2 -m- d ^ = m^+(Xx+Yy + Zz). Mantel, W. N. Arch. Wisk. 18 (1891) 127-. , 6 2 't> 9 2 v 9 2 w A 'W = W + 'W Class f m ^ ois ' Levi - Civita, T. N. Cim. 6 (1897) 204-. 5660 Dirichlet's problem. ZaremlM, S. C. B. 124 (1897) 940-. -- of stationary temperatures, and represen- tation of given surface. Christoffel, E. B. A. Mt. 1 (1867-68) 89-; 4 (1870-71) 1-. Neumann's method and Dirichlet's problem. Poincare, H. C. B. 120 (1895) 347-. 6000 Calculus of functions. Spottisicoode, W. B. S. P. 11 (1860-62) 447-. 6400 Tracing of line by point, paradox. Zehfust, G. Grunert Arch. 30 (1858) 229. 6410 Absolute geometry, trigonometry in. Simon, M. Crelle J. Mth. 109 (1892) 187-. Area of triangle, imaginary geometry. 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Singularities. Zeuthen, H. G. C. R. 67 (1868) 225-. . Fine, H. B. Am. J. Mth. 8 (1886) 156-. of curve and developable surface. Zeuthen, H. G. A. Mt. 3 (1869-70) 175-. of intersection of 2 surfaces having certain number of multiple points in com- mon. Saltel, L. C. R. 80 (1875) 1285-. curves and characteristics of polar developables. Bjorling, C. F. E. Stockh. Ofv. 38 (1881) No. 4, 3-. 8010 Connexes, 2nd order and 2nd class, generation. Armenante, A. Rm. R. Ac. Line. At. 3 (1876) (Pte. 2) 123-. Linear connexes. Voss, A. E. Mth. A. 15 (1879) 355-. 8020 Characteristic combinations in transformation of singular point of curve. Noether, M. Palermo Cir. Mt. Rd. 4 (1890) 89-, 299-. Substitutions, instability. Levi-Civita, . C. R. 131 (1900) 103-. 8075 SPECIAL CONFIGURATIONS OF POINTS, LINES, PLANES OR OTHER ELE- MENTS. 9 point configuration. Clausen, T. Crelle J. 7 (1831) 36-. 8080 Systems of forces (Dynamen) , geometry . Study , E. Leip. Mth. Ps. B. 51 (1899) (Mth.) 29-. 8090 SYSTEMS OF CURVES. Differential equations. Czuber, E. Wien Ak. Sb. 102 (1893) (Ab. 2a) 1141-. 8420 Deformable frameworks. Mth. 29 (1900) 1-. Dixon,A.C. Mess. 8430 Curves. of "raccordement." Allaume, . A. Cond. Pon. Chauss. 34 (1890) 76-. , parabolic. Fepoux, . A. Cond. Pon. Chauss. 31 (1887) 233-. 8460 Areas. of hemisphere, calculation. Dumas, J. B. A. Cond. Pon. Chauss. 37 (1893) 505-. Rectification. of ellipse. Bronwin, B. Ph. Mg. 2 (1827) 378-. . Moreau, A. A. Cond. Pon. Chauss. 38 (1894) 563-. , extension of J. Bernoulli's rule. Magis- trini, G. B. Bologna Opusc. Sc. 1 (1817) 244-. 8820 Class of fluid motions connected with minimal surfaces, and particular integrals of V 2 V0. Weingarten, J. Gott. Nr. (1890) 313-. CAMBRIDGE : PRINTED BY JOHN CLAY, MJ AT THE UNIVERSITY PRESS. I/./ University of California SOUTHERN REGIONAL LIBRARY FACILITY 405 Hllgard Avenue, Los Angeles, CA 90024-1388 Return this material to the library from which It was borrowed. S. 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