Vol.4, Part? DECEMBER, 1922 , . Number 25 BULLETIN OF THE NATIONAL RESEARCH COUNCIL CELESTIAL MECHANICS A Survey of the Status of the Determination of the General Perturbations of the Minor Planets BY A. 0. LEUSCHNER PUBLISHED BY THE NATIONAL RESEARCH COUNCIL OP THE NATIONAL ACADEMY OP SCIENCES WASHINGTON, D. C, 1922 Announcement Concerning Publications of the National Research Council The Proceedings of the National Academy of Sciences is partly supported by the National Research Council which is represented officially on its Editorial Board and Executive Committee. It is open for the publication of papers to members of the National Research Council on the same terms as to members of the National Academy of Sciences. Subscription rate for the "Proceedings" is $5 per year. Business address: Home Secretary, National Academy of Sciences, Smithsonian Institution, Washington, D. 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December, 1922 Number 25 CELESTIAL MECHANICS A SURVEY OF THE STATUS OF THE DETERMINATION OF THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. Appendix to the Report of the Committee on Celestial Mechanics, National Research Council* BY A. 0. LEUSCHNER Professor of Astronomy, University of California CONTENTS PAGE Introduction 2 1 Ceres 11 2 Pallas 15 3 Juno , 20 4 Vesta 23 10 Hygiea 28 28 Bellona 31 93 Minerva 35 94 Aurora 38 127 Johanna 40 128 Nemesis 42 175 Andromache 44 433 Eros 46 447 Valentine 51 588 Achilles 54 617 Patroclus 57 624 Hector 60 659 Nestor 63 716 Berkeley 66 718 Erida 68 884 Priamus 70 911 Agamemnon 73 *This Committee of the Division of Physical Sciences of the National Re- search Council consists of the following members: E. W. Brown, Professor of Mathematics, Yale University, Chairman; G. D. Birkhoff, Professor of Mathe- matics, Harvard University ; A. O. Leuschner, Professor of Astronomy, University of California; H. N. Russell, Professor of Astronomy, Princeton University, 670869 A/3 { ' INTRODUCTION. With approximately one thousand asteroids discovered and believed to be sufficiently observed to permit of fairly reliable orbit determina- tions, as indicated by the permanent numbers assigned to them, the task of preserving these discoveries has grown so stupendous that the time seems to have arrived for an analysis of the present astronomical practice in providing the necessary additional observations and calculations. Hitherto, the burden of correcting orbit elements and computing ephemerides has rested principally on the Berlin Recheninstitut. In recent years the Marseilles Observatory has rendered notable service in contributing orbits and ephemerides. Observations, photographic and visual, are regularly made at a number of observatories. The Berlin Recheninstitut publishes ephemerides and other results in the Astronomische Nachrichten, and in the Ephemeriden der Kleinen Planeten. Up to 1918 these data appeared also in the Astronomisches Jahrbuch. The number of oppositions during which the minor planets have been observed, and the status of orbit determinations are an- nually summarized in the Viertelj ahrschrift der Astronomischen Gesell- schaft. In 1901 Bauschinger published the latest reliable elements, etc., with data concerning the perturbations for the then known 463 planets, "Tabellen zur Geschichte und Statistik der Kleinen Planeten." The latest available collection of elements is contained in the Berlin Jahrbuch for 1918. The adopted Jahrbuch elements serve the purpose of providing ephemerides from opposition to opposition. Their origin may be traced from the notes given from year to year in the Jahrbuch, ending with 1918, and in Kleine Planeten. Some of the elements include arbitrary corrections to the mean motion and to the mean anomaly for the purpose of representing late oppositions so as to serve for prediction of immediately following oppositions. In other cases, approximate or accurate perturbations are included, with or without correction of the elements by the usual least squares adjustment. For thirty-six planets the elements in the Jahrbuch of 1918 are mean or osculating elements, derived in connection with general perturbations which are approximately included in the pre- diction of ephemerides. Until similar fundamental data shall have become available for the remaining planets, the present practice of the Recheninstitut appears to furnish the only certain method for the preservation of planetary discoveries. CELESTIAL MECHANICS: LEUSCHNER 3 In addition to the general pertubations of the thirty-six planets which are being used by the Jahrbuch, the general pertubations of a number of other planets have been .derived on the basis of what, at the time, appeared to be reliable osculating elements. These pre- sumably valuable data have been replaced by later elements derived independently, more or less accurately, with or without general perturbations, or upon the basis of arbitrary corrections. Bausch- inger's Tabellen form a valuable key to some of these investigations, but even in the Tabellen the elements and perturbations cited do not, in all cases, represent the best elements and perturbations available, although perhaps in every case they are the most reliable for subse- quent oppositions. This arises from the fact that earlier investigations were abandoned by Bauschinger in favor of later ones. Preliminary calculations have shown, however, that some of the earlier elements and perturbations represent distant oppositions at a later date more satisfactorily than his adopted elements and perturbations represent earlier oppositions equally remote. Of great importance for the program of the Recheninstitut are the contributions of Brendel, who has developed methods for the approxi- mate determination of the perturbations for certain groups of planets. Perturbations greater than 3'.4 within fifty years are included, with the object of reproducing geocentric places within 20' for 100 years. So far the necessary data have been published by Brendel, Labitzke, and Boda for 230 planets, approximately 25 per cent of the total number of known minor planets. The advantage to be gained from BrendeFs contributions for these planets is that for the practical purpose of preserving these planets, by following their motion, it should become unnecessary as a rule to compute special perturbations for them, or even to apply corrections to the elements. Brendel plans to continue the work of supplying instantaneous elements and approximate perturbations for other groups of planets so that the program of the Recheninstitut, of the Marseilles Observatory, and of various investigators who, from time to time, publish improved elements and perturbations for ephemeris purposes, will become more and more simplified. The preservation of planetary discoveries by observation and pre- diction with the aid of approximate perturbations is not the ultimate aim of astronomical science, but a necessary and unavoidable means to the end. The ultimate aim rests on the determination of mean elements and general perturbations which hold for all time or at least for very long periods within the limits of accuracy set by observation. It is expected that the elements and perturbations determined under 4 CELESTIAL MECHANICS: LEUSCHNER the Newtonian law of gravitation may serve this purpose, provided that the mathematical difficulties will not prove insurmountable. It may be assumed that the rigid mathematical methods hitherto developed are satisfactory for planets with moderate eccentricity and inclination which are not in a very near commensurable ratio with any of the major planets, but it has not been established so far whether an accurate application of the Newtonian law would fully account for the motion of the minor planets even in the ordinary cases just referred to. Exhaustive researches are available only for a very limited number of planets. Among these are (4) Vesta, (13) Egeria and (447) Valen- tine. The researches on (4) Vesta are due to Leveau, whose extraordi- nary investigations extend approximately over a complete century of oppositions. In connection with his work on the motion of Vesta, Leveau has aimed at a determination of the masses of Jupiter and Mars. His final value is larger than the best available mass of Jupiter by approximately one one-thousandth. On account of the moderate perturbations, the motion of Vesta does not lend itself as well to a determination of the mass of Jupiter as the motion of other minor planets with very large perturba- tions. Any slight departure from the true mass of Jupiter, et cetera, can reveal itself through the motion of Vesta only in long intervals of time, which accounts for Leveau's gradual improvement of his adopted mass by successively including longer periods of observation. For the present his results may be considered funda- mental and final, so far as this planet is concerned. No other case has been studied so exhaustively. Later predictions are well within the errors of observation, and not the slightest departure from the New- tonian law is noticeable. It remains, however, to establish the same result for planets with large perturbations, particularly for such planets as have a mean motion commensurable with that of Jupiter. To avoid the necessity of gradually improving the Jupiter mass by means of subsequent observations of Vesta, it appears advisable to base further predictions on the best determined values of the masses of the major planets. Vesta also furnishes an example of the weight to be assigned to observations in the early part of the last century. Leveau's investigations furnish a striking example of the funda- mental researches necessary for the promotion of astronomical science as distinguished from the generally accepted program of observation and prediction for the preservation of discoveries. For the study and interpretation of planetary statistics, particularly with reference to the origin of minor planets, the explanation of the CELESTIAL MECHANICS: LEUSCHNER 5 gaps, the question of stability and ultimate destiny, or in general regarding their place in any hypothesis concerning the solar system, final mean elements derived on the basis of accurate developments of the perturbations are most essential. Fragments of fundamental investigations of perturbations are available for a number of minor planets. The value of some of these has been vitiated by corrections made in connection with the accepted program of approximate pre- diction, such as, for example, the correction of an accurate set of osculating elements derived by special or general perturbations, to represent later oppositions, either without perturbations or by taking account only of approximate or incomplete perturbations. Fundamental investigations here are understood to include the determination of osculating or mean elements from a limited number of oppositions with complete regard of the perturbations, either special or general, in so far as they may have been appreciable. In con- nection with the study of the data existing for a limited number of selected planets, it has been found that the failure of such elements and perturbations to represent future oppositions in some cases can be accounted for by the fact that the masses of the major planets were known at the time with insufficient accuracy. The mere cor- rection of the perturbations therefore, for the latest known values of the masses may render such elements and perturbations far more satisfactory than they appeared to be at the time when they were discarded in favor of new determinations of elements with or without perturbations. Freed from effects of changes which affect disadvantageously their permanent value the fragments of fundamental investigations referred to are of great importance as a basis for researches and their intel- ligent application will involve a vast saving in computational and theoretical work. At present it appears next to hopeless to the investigator to adopt a profitable form of attack in connection with any of the older minor planets without an enormous expenditure of time in searching astro- nomical records. This accounts for the many duplications of effort and for the disregard of previous valuable investigations. If systematically undertaken, the task of bringing to light the important data available for a final determination of the elements and general perturbations of the minor planets, does not appear insurmountable. Once available, such research surveys will be invaluable and should prove an encouragement to research, particularly to young investigators. The research surveys of the few planets which are given below are 6 CELESTIAL MECHANICS: LEUSCHNER intended to serve as illustrations of the data which should be made easily accessible. No claim is made for the absolute completeness of these data. The time for active work, with the aid of a few assistants, to prepare these preliminary surveys has extended only over a little more than a month. A great mass of material had to be consulted which was found to be of no importance to the purpose in hand.. This is being preserved on cards for easy reference, if required at any time. Thus care has been taken to eliminate elements which would not be considered as fairly accurate osculating elements particularly those which have resulted from corrections on the basis of subsequent oppositions purely for ephemeris purposes, without complete con- sideration of the perturbations and of the earlier oppositions in the final adjustment. This policy, however, has not been adhered to strictly, partly for historical and theoretical reasons with reference to preliminary elements, and partly for other reasons with reference to later elements. Whenever possible, the reasons for the abandonment of previous investigations are given, but in many cases no reasons could be found, at least not in the astronomical records available in the library of the University of California. Some of these reasons are probably to be found in the records available in the library of the Lick Observatory, but in the limited time it has not been possible to consult these or other additional records for this preliminary survey. An immense amount of fundamental work has been accomplished by the Berlin Recheninstitut, particularly in computing special perturbations and deriving osculating elements, but has been published only in part. The remainder reposes in the archives of the Recheninstitut. It may be assumed that the immense task of providing ephemerides has interfered with the publication of the accumulated material. With- out this material, research surveys such as those presented here are not complete. A simple way of accomplishing the introduction of the improved mass of disturbing planets referred to above, is to multiply the final sum of all the terms for each component of the perturbations by the ratio of the new to the old mass. Aside from the improvement which it may be possible to make to some of the older fundamental data, par- ticularly those which are no longer used for ephemeris purposes, by the introduction of the best determined values of the masses of the major planets, it is probably possible to enhance their values still further by correcting the elements on the basis of the existing develop- ments of the general perturbations, with the aid of subsequent oppositions and in case of appreciable changes in the elements, by CELESTIAL MECHANICS: LEUSCHNER 7 also correcting the numerical coefficients in the general developments by differential methods. After revision of independently determined elements and pertur- bations for separate series of fairly consecutive oppositions, discon- nected by a gap including a number of oppositions to which neither series was extended either backward or forward, the separate funda- mental investigations will, in some cases, probably be found to be entirely consistent and thus become of permanent value, such as Leveau's investigations on (4) Vesta, without involving extensive theoretical and numerical work. In other similar cases the correction of the elements and perturbations pertaining to fundamental investi- gations for groups of oppositions separated by considerable gaps, so as to represent the osculating data at a subsequent epoch, may establish satisfactorily the connection between one or more groups of oppositions for which elements and perturbations have been independently deter- mined with accuracy. The mode of attack will vary with the avail- able data for different planets, as indicated by the research surveys, which this discussion advocates. The resurrection of the classical contributions of the pioneer investigators of planetary perturbations on a permanent basis, should produce material of great value for the ultimate aims of astronomical science concerning planetary investigations. The proposed program of fundamental investigations cannot supersede the present astronomical practice in caring for the minor planets in the immediate future, but as stated above it will be of great assistance for the practical purposes of prediction, and should gradually solve the now stupendous task of preserving planetary discoveries, while furnishing at the same time the data for the more fundamental aims of astronomical science. For the majority of the minor planets, probably the application of four successive steps or processes will be necessary to preserve the discoveries until final elements and perturbations can be made avail- able. The first step or process represents the present practice principally conducted by the Berlin Recheninstitut. The second step is illustrated by Brendel's plan of supplying instantaneous elements and approximate perturbations. The third step corresponds to the determination of the elements and perturbations of the Watson asteroids undertaken by Leuschner, which are intended to provide fairly accurate but not final results. Hansen's and the Bohlin-v. Zeipel methods have been found most practical and accurate in this connection. The fourth and final step is demonstrated by the funda- mental work of Leveau on (4) Vesta. It is the object of this discus- 8 CELESTIAL MECHANICS: LEVSCHNER sion to encourage researches similar to Leveau's, and by supplying samples of research surveys for a limited number of planets to pave the way for a comprehensive international program in this connection. It cannot be too strongly emphasized that accurate osculating elements are absolutely essential for fundamental investigations of the perturbations. While this requirement is fully recognized, the prevailing practice of changing elements for immediate ephemeris purposes is apt to lead to erroneous interpretation of available elements. Mean elements, in general, can be determined only after osculating elements and perturbations shall have become available. Some investigators have adopted as approximate mean elements the average of elements published for more or less extensive series of oppositions, assuming that these elements represent fairly reliable osculating elements. Even if this were the case, it hardly ever occurs that a sufficiently large number of elements, uniformly distributed over the orbit, are available to guarantee that, in taking the average, the effect of the periodic terms is entirely eliminated. But, as previously stated, many of the apparently reliable sets of elements are not osculating, but inferior elements produced by arbitrary changes or with incomplete perturbations. Practically the only reliable method of arriving at accurate initial osculating elements consists in representing the observations of a limited number of oppositions by taking into account the special perturbations and in testing the validity of the resulting elements for one or more oppositions following. Osculating elements thus obtained will rarely require later changes which would affect the coefficients of the general perturbations. No correction of such elements should be attempted, except on the basis of the determination of complete special or general perturbations. As it was not considered necessary, at the time, to adhere strictly to the foregoing principle in Leuschner's program for the determination of the perturbations of Watson's asteroids, allowances for slight inaccuracies may later become necessary for some of the Watson planets. In particular, corrections to the larger coefficients of the perturbations may be necessary for the planets for which the initial adopted elements, con- sidered at the time as sufficiently accurate, were neither accurate mean elements nor accurate osculating elements. Attention has recently been called, in the Proceedings of the National Academy of Sciences of 1921, Vol. 8, No. 7, p. 170, and in the report of the Committee on Celestial Mechanics of the National Research Council, Bulletin of the National Research Council, Vol. 3, Part 4, No. 19, June, 1922, to the extremely satis- CELESTIAL MECHANICS: LEUSCHNER factory results obtained for the planets (10) Hygiea, and (175) Andromache, by the application of Leuschner's revision of von ZeipePs tables for the Hecuba group. Further reference to the great importance of "Gruppenweise Berechnung der Stoerungen," inaug- urated by Bohlin, may therefore be omitted here. The methods of Bohlin and his followers serve admirably in connection with the third of the four stages outlined above for the determination of funda- mental results. In certain cases of limited eccentricity and inclination, they will, no doubt, lead to final results. No claim is made that the planets for which research surveys are given below are the ones most in need of immediate attention. Further study of available data will be necessary to classify the planets with reference to the requirements of observation and com- putation, as outlined in the report of the American Committee on Comets and Asteroids, presented at the Brussels meeting of the International Astronomical Union in 1919; nor are the planets con- sidered below the most important for fundamental scientific purposes. The list, however, may be considered as fairly representative of the immediate research requirements. To some extent the selection has been accidental. Thus the computing section of the British Astro- nomical Society has undertaken the computation of the ephemerides of the first four planets. In this connection it sought advice regarding the best available data and methods of procedure. The research sur- veys of the first four planets were undertaken to aid the computing section in its undertaking. The importance of the Trojan group is too well known to be emphasized. For further investigations con- cerning the theories of the six planets belonging to this group the research surveys given will be of considerable value. It is of interest to note that Leuschner's orbit methods as applied by Einarsson, appear to be the most promising for the determination of preliminary osculating elements, while Wilkens' method deserves careful trial in deriving the perturbations. E. W. Brown's unpublished theory promises to be thoroughly fundamental. For the two planets of the Trojan group last discovered, more accurate preliminary osculating elements are immediately needed. For other planets, the list of research surveys themselves will reveal the most necessary work to be done. In general, reference to theoretical investigations is included only in connection with a simultaneous new determination of elements. Thus the numerous and important investigations on the theory of the Trojan group are not considered here, the chief object of the surveys of these planets being to furnish numerical data and encourage their improvement as a basis for such theories. 10 CELESTIAL MECHANICS: LEUSCHNER The form in which the research surveys are presented must be con- sidered experimental. That adopted is the outcome of several other attempts at presenting the material. It is hoped that this report will call forth helpful criticisms and suggestions which may ultimately lead to the adoption of some definite plan for international coopera- tion. Much material has been collected on planets not included in the list, which, it is hoped, may be printed later. It was found that the research surveys for the various planets could not be made so complete that the investigator may abstain from referring to the sources themselves. This applies also to the collection of elements. The elements are collected merely for purposes of com- parison and are not reproduced with uniform accuracy. For practically all the planets in the list, except the first four and several others, a fairly complete bibliography of observations has been prepared, but this bibliography is published here only for the last two of the Trojan group. Attention might well be called here to the need of curtailing indiscriminate observations. Even in recent years observations have been multiplied for planets for which two or three accurate observa- tions at each opposition would be sufficient for all scientific purposes. It is planned to formulate in the near future definite proposals for an international program of observations. The main purpose of this report is the encouragement of funda- mental researches essential to the ultimate aims of astronomical science, which, for their consummation, require the knowledge of accurate elements and perturbations of the minor planets. The surveys have been prepared in the main by Dr. W. F. Meyer, and by Dr. H. Thiele, assisted by several advanced students in astronomy, who have gathered the necessary references. For the Trojan group, unpublished data collected by Dr. Sturla Einarsson have been available. As a rule the abbreviations adopted for the references are those of the Astronomischer Jahresbericht. The usual notations of the elements are adhered to, both ^ and n being used for the mean daily motion. CELESTIAL MECHANICS: LEUSCHNER 11 (1) CERES. The first and largest of the minor planets was discovered 1801, January 1, by Piazzi in Palermo. 1 Piazzi assumed that the object was a comet, but several astronomers succeeded in proving from the 22 meridian observations near the stationary point over an heliocentric arc of 9 that it was a planet moving in a nearly circular orbit; thus Burckhardt 2 computed Elements A, Olbers 3 the circular Elements B, Piazzi* the circular Elements C. Only the computation by Gauss, 5 Elements D, was accurate enough, especially in the determination of perihelion and eccentricity, to indicate where the planet might be found the following year. Olbers found Ceres again 1802, January 1, % from the predicted place, near the place where, three months later, he discovered the second of the minor planets. The new observations naturally increased the accuracy of the elements notably; thus Gauss 6 computed Elements E, from observations in 1801, and January 1802; represen- tation in February 1802, +7" in a, 20" in 8. Burckhardt 7 including the perturbations larger than 30' found Elements F. For some years the orbit of Ceres was investigated by Oriani, Burckhardt, and Gauss by taking the perturbations into account, but the efforts of Gauss went farther than those of the others. Burck- hardt 8 started with the computation of perturbations at intervals of two days, and later computed tables founded upon them. Oriani 9 used Laplace's method, with which also Gauss started. Gauss developed the perturbations first in 1802, together with Elements VIII, G, and formed tables of perturbations 10 and later in 1805 11 when he used the same interpolatory development of the perturbative function as Hansen later used in 1830. The orbit computation was taken up later by Heiligenstein. 12 He derived Elements H from the oppositions 1818, 1820, 1821, 1822, 1825, 1826, 1827, with special perturbations of the elements by Jupiter, (mass 1/1053.924). Representation of the normal places 10" to +6" in mean longitude. Correction to the ephemeris for 1830 April, May, 6" in a, 10" in 8. Heiligenstein's ephemeris deviates 15' from the ephemeris in B. J. 1830, which was based on the elements of Gauss (XIII, 1809), using the tables of perturbations by Gauss and an empirical correction by Encke of 14' to the mean longitude determined from the last observations. 18 12 CELESTIAL MECHANICS: LEVSCHNER In B. J. for 1831 Encke 14 gives an ephemeris from new Elements I of his own based on the oppositions 1820, 1821, 1822, 1825. Jupiter mass 1/1053.924. Special perturbations by Jupiter only. Representation: 1820 1821 1822 1825 1827 1829 in a 6" +2" 4" 3" 2" 27" in 8 0" 0" +6" +1" 0" 11" In B. J. 1832 to 1836 the ephemerides by Heiligenstein were published. Later the computation by Encke and Wolfers was used to 1871. In the meantime Damoiseau 16 had given expressions for the per- turbations containing a large number of terms but the individual coefficients do not seem to be very exact, according to Hill. For the use of the American Ephemeris, E. Schubert 16 undertook to correct the elements by 250 observations in 14 oppositions, 1832- 1854, using special perturbations of the elements by Jupiter as com- puted by Encke and Wolfers but corrected for the secular variation of the obliquity. 17 Elements J. Residuals in a 22" to +21", in 8 8" to +8" ; corrected according to A. J., Vol. 5, p. 73. A further correction of the elements by Schubert 18 was based on only four normal places in 1853, 1854, 1855, 1857; he applied the special per- turbations of the elements by Jupiter and Saturn. Representation of the normals 0", "by which the correctness of the whole is proved." Elements K. Godward 19 repeats the process of Heiligenstein, Encke, Wolfers, Schubert. The errors for fifteen oppositions 1857 to 1876 of the ephemerides in Nautical Almanac which include the perturbations of Venus, the Earth, Mars, Jupiter, Saturn gave by a least squares solution the Elements L. Ephemerides by these elements were given in the Nautical Almanac to 1913. The corrections to Encke's ephemerides increased after 28 years to 3 8 in a 20" in 8. The corrections to Schubert's ephemerides increased after 23 years to +6 8 in a 40" in 8. The corrections to Godward's ephemerides increased after 36 years to +2 a in a 10" in 8. For the purpose of illustrating his modified form of computing absolute perturbations Hill 20 computed the first order perturbations of Ceres by Jupiter starting with the first elements by Schubert (un- corrected). It was found that the osculating mean motion differed CELESTIAL MECHANICS: LEUSCHNER 13 widely from the mean mean motion. An arbitrary value was substi- tuted. The Jupiter mass is taken to be 1/1047.355. The expressions for the periodic terms of the perturbations are given. In order to arrive at mean elements as well as to see how closely the perturba- tions represent the observations, ten normal places 1802, 1807, 1830, 1857, 1863, 1866, 1873, 1883, 1885, 1890, were formed. Secular per- turbations of Mars, Jupiter, and Saturn were computed by the method of Gauss. The periodic perturbations by Mars and Saturn were taken from the tables of Damoiseau. Preliminary elements and a least squares solution led to mean Elements M. The residuals are 40" to +40" in hel. longitude, 20" to +13" in geoc. latitude. Hill originally intended to enlarge and complete his theory of Ceres; for this purpose he collected the observations into 75 normals from 1801- 1897. 21 He published the positions because he did not expect to finish the work. The collection is not complete. As an extension of Hill's work Merfield 22 has given a computation of the secular perturbations of Ceres arising from the action of the eight Major Planets. From Hill's theory and his mean elements using the method of Gauss as set forth by Hill the numerical values of the action of the planets were derived. M. Wolf 23 has developed the expression (p) according to the theory of Gylden in the case of Ceres. Cf. Tisserand, Mecanique Celeste, Vol. 4. M. Viljev 24 has published tables of absolute perturbations of Ceres after the method of Hansen. REFERENCES 'BODE'S B. J. 1804, p. 249. B. J. "B. J. 1831, p. 277. A. N. vol. 27, 1805, p. 202. v. Zach, Monatliche p. 177. Correspondance. Not available here. 1 f ' T V o1 16 n ' 87 GAUSS. Werke Bd. 6, p. 207. J J; * 3/51 7 BODE'S B. J. 1805, p. 96. M fjf V l R7 P ' *ai 8 A J vol 16 p 57 ' N< vo ' 67 ' p> 551< 9 v.' Zach'. Monatliche Correspon- "WOLF. Bur les termes elemen- dance 1802, Dec. Not available here. taires dans I expression du rayon- 10 GAUSS. Werke Bd. 7, p. 375. vecteur. Stockholm, 1890. 11 GAUSS. Werke Bd. 7, p. 401. Publications de 1'Observatoire Cen- 13 A. N. vol. 7, p. 413. tral Nicolas, Poulkovo. Not available 18 B. J. 1830, p. 245. here. 14 CELESTIAL MECHANICS: LEUSCHNER TABLE I. Elements (1) Ceres Letter Date MT L IT Q i o / O t o t * e / ^^ A. . . . 1801 Jan. 1 . 3328 68 59 37 248 59 37 80 58 30 10 47 B 1801 Jan. 1 68 35 51.5 80 22 45 11 3 36 C. . . . 1801 68 46 41 80 46 48 10 51 12 D. . . 1801 Palermo .... 76 28 14 150 33 20 81 2 35 10 36 30 E 1801 Palermo .... 77 27 31 145 57 15 80 58 40 10 37 57 F 1802 155 32 35 146 44 37 81 5 35 10 36 52 G 1801 Seeberg 77 19 34.9 146 33 37 80 54 59 10 37 56.0 H.... 1818 Oct. 15.0.. G6ttingen. . 28 21 52.291 148 2 14.084 80 48 32.192 10 38 21.682 I 1822 Jan. 22.0. . Gottingen. . 127 36 44.2 147 36 57.6 80 41 55.0 10 38 7.7 M o / J 1854 Jan. Washington. 113 22 25.08 148 55 23.41 80 50 50.79 10 37 8.54 K.... 1854 Jan. Washington. 113 18 22.40 148 56 3.72 80 50 31.11 10 37 4.76 L 1854 Jan. Washington . 113 22 11.73 148 55 26.54 80 50 31.06 10 37 5.81 L o / M.... 1850 Jan. 0.0. . . Greenwich.. 309 30 32.4 148 28 32.5 80 48 5.6 10 37 6.2 Letter Date" MT r M Equinox Author 1801 Jan 1 3328 / 2 5 * 859 05 Burckhardt I B 1801 Jan 1 786 528 Olbers c 1801 795 937 Piazzi D 1801 Palermo .... 4 2 45 784.254 Gauss I E F . 1801 1802 Palermo.. . . 4 40 10 4 31 25 769.7925 771.363 Gauss VII Burckhardt II G. ... 1801 Seeberg . . 4 31 17.8 770.7951 Gauss VIII H I J K.... L M.... 1818 Oct. 15.0.. 1822 Jan. 22.0.. 1854 Jan. 1854 Jan. 1854 Jan. 1850 Jan. 0.0... Gottingen. . Gottingen. . Washington. Washington. Washington. Greenwich . . 4 31 5.183 4 .31 18.0 4 24 28.41 4 24 29.36 4 24 29.65 4 29 57.8 771.2273825 770.72468 769.63875 769.62476 769.64746 770.718276 1818.00 1810.00 1854.00 1854.00 1850.00 Heiligenstein Encke Schubert Schubert Godward Hill* *Mean elements. CELESTIAL MECHANICS: LEUSCHNER 15 (2) PALLAS Discovered by Gibers at Bremen 1802, March 28. 1 Gibers attempted to compute a circular and a parabolic orbit for the new planet, both of which failed. His computation showed the orbit had a large inclination and considerable eccentricity. From observations extending from April 1 to July 8, Gauss 2 com- puted Elements A (Gauss V). They are improvements on preliminary sets. With these elements an ephemeris for 1803 was computed. From observations extending from April 4 to May 20, Burkhardt 3 computed Elements B. With these elements Burkhardt computed the perturbations in longitude, latitude, and radius vector covering the period April 4 to May 20. The planet was reobserved by Harding 1803, Feb. 21st. The com- parison between Gauss' ephemeris and observations was as follows: 1803 Aa AS Feb. 21 +2' 02" 34" Feb. 23 +2 35 57 On the basis of these residuals, Gauss 4 improved Elements A Gauss (V). These new Elements C (Gauss VI) represent the obser- vations as follows: 1803 Aa AS Feb. 21 - 20"0 + 15"8 Feb. 23 + 7.8 7.7 From a set of elements, based on oppositions 1804, 1805, 1807, 1808, Gauss 5 derived an improved set of Elements D from a least squares solution. This solution includes also the oppositions 1803 and 1809 and forms the basis for the computation of perturbations as outlined below. Gauss 6 first attempted to construct tables of perturbations for the four known minor planets, but the large eccentricity and inclination forced him to formulate a theory for Pallas based on the variation of the elements expressed analytically and integrated by mechanical integration. From two successive calculations of the special pertur- bations, due to Jupiter, Gauss derived the improved Elements E, which represented the heliocentric longitudes for the first seven oppo- sitions within 8". In 1811 Gauss 6 began his first computation of general perturbations due to Jupiter. For this purpose he used Laplace's elements of Jupiter, 16 CELESTIAL MECHANICS: LEUSCHNER epoch 1805, and his own elements of Pallas for the same epoch. This computation led to a set of mean Elements (F). With these mean elements for epoch 1810 and similar elements for Jupiter (Laplace) a second computation of general perturbations due to Jupiter was undertaken. This computation led to the following results: The mean motion of Pallas oscillates between 18/7 of QJ. motion 0".2153, and 1894 revolutions of Pallas = 737 of Jupiter. A new value for Jupiter's mass = 1/1042.86. Then follows (1816-1817) the computation of perturbation tables due to Jupiter, Saturn and Mars. In this latter work, Gauss was assisted by Encke and Nicolai. In Astronomisches Jahrbuch 1816, page 234, Bode gives the best set of elements by Gauss up to that time (Elements G). About 1824, Encke 7 used Gauss' elements based on early oppositions and computed the perturbations due to Jupiter. He reports that Gauss' elements with Jupiter's perturbations represent the opposition of 1823 as follows: 1823 Aa AS Oct. 9 +13:2 +25^6 He then gives Elements (H) for the epoch 1826, and with these computes the next ephemeris. For the opposition in 1825, Encke 8 reports that the correction to the ephemeris is very large. But if the perturbations are included, the difference between observation and computation is as follows: 1825 Aa AS March 23 +4276 33"2 He then gives a set of elements for the epoch 1827, and computes an ephemeris for 1827. By 1834 Encke 9 reports a deviation of Pallas from computed places amounting to 5'. He states this may be due to use of Laplace's value for Jupiter's mass. It will be necessary to recompute elements covering all observations. In A. N. No. 636, Encke publishes osculating Elements I for each year from 1831 to 1838; his fundamental starting elements are for the epoch of 1810, January 0. In B. J. 1838, p. 286, Encke draws atten- tion to an error which he had committed in neglecting the corrections for the secular variation of the obliquity. In the British Nautical Almanac for 1837 Airy points out this error to which Encke refers. Galle 10 undertook the reinvestigation of the orbit based on opposi- tions 1816, 1821, 1827, 1830, 1834, 1836, making use of Airy's value for the mass of Jupiter 1/1048.69. The former perturbations were retained CELESTIAL MECHANICS: LEUSCHNER 17 except for the change due to Jupiter's mass. The resulting Elements J represent the heliocentric longitude and latitude as follows : 1816 1821 1827 1830 1834 1836 AL 19" +34" +4" 14" +5" 13" AB 1 +4 +6 10 +1 +4 Galle states these differences may be accounted for if the perturba- tions of Saturn and Mars were taken into account. In A. N. No. 636 osculating Elements K are published for each year from 1839 to 1850. These were computed by Galle. The start- ing elements are those for epoch 1810, January 0. In computing the special perturbations, Encke and Galle used mass of Jupiter 1/1053.924. From 1851 to 1870 Galle 11 continues the special perturbations by Jupiter and later with the elements of Giinther also those by Saturn. These were used in computing the ephemerides published in the Astronomiches Jahrbuch from 1862 to 1870. (See Elements L.) Beginning with the year 1871 and continuing to 1919, the Jahrbuch published and used Farley's 12 osculating elements for computing the ephemeris. (See Elements M and N). Farley's computation includes the perturbations by Venus, Earth, Mars, Jupiter and Saturn. His computations are also the basis for the ephemerides published in the British Nautical Almanac. With Farley's elements we have the fol- lowing comparisons: Corrections to Ephemerides. 1883 1892 1895 1906 1908 1914 Aa -1 B 4 -1?2 -1?0 -2?5 -5?4 -2 s ! A5 -J-2'7 +07 +0'8 +8'2 -14-0 +4'3 In Annales de 1'Observatoire de Paris, Vol. I, Le Verrier pub- lishes the results of his investigation on "Developpement de la fonction perturbatrice relative a Faction de Jupiter sur Pallas. Calcul du terme dont depend une inegalite a longue periode du mouvement de cette derniere planete." Le Verrier states that the aphelion of Pallas is 54 from the intersection of the orbit with Jupiter. Conse- niiently when Pallas is at aphelion the distance from Jupiter is in- creased on account of the great inclination of the orbit. This large inclination diminishes the effect due to the large eccentricity. Le Verrier gives the series for the reciprocal of the distance in a more convergent form and develops the equation in longitude depending on the argument 18 QJ. 7 Pallas. The maximum of the term is 895". In his report before the Paris Academy, 13 Cauchy compares his theory 18 CELESTIAL MECHANICS: LEVSCHNER with the results by Le Verrier; his value for the inequality is 906". Cauchy's investigation is more fully elaborated by M. Puiseux in An- nales de 1'Observatoire de Paris, Vol. VII. In Vol. VIII, ibid., Hoiiel has recomputed the inequality. The development of the reciprocal of the distance was later extended by Tisserand. 14 He shows that the development depending upon the inclination and eccentricity is divergent in some parts of the orbit of Pallas and proceeds to give the analytical development and to apply it to the case of Pallas. In Bulletin Astronomique Vol. XII, 1895, M. P. Bruck has pub- lished the results of his work on 'The secular variations of the ellip- tic elements of Pallas due to the action of Jupiter." He used the method developed by Gauss and extended by Hill and Callandreau. He utilizes elements by Farley for the epoch 1878. In 1910 Georg^ Struve 15 published his results on "Die Darstellung der Pallasbahn durch die Gauss'sche Theorie fur den Zeitraum 1803 bis 1910." The result of his work based on 63 normal places is a more accurate value for the mean motion of Pallas (769".1385). The new value for the annual motion of Pallas compared with Jupiter becomes 18n' 7n = 123". The deviation between observation and computation still amounts to 4' which is attributed to the second order perturbations. These residuals are somewhat reduced by empiri- cal terms. In A. N. No. 205, p. 225, M. Viljev has published his "Recherches sur le mouvement de Pallas." He attempts to reduce the residuals from Struve's work (4') by taking into account second order terms in the general perturbations, employing the method by Hill. He reports his results as negative. REFERENCES CODE'S A. J. 1805, p. 102. "British Nautical Almanac 1860 (not "BODE'S A. J. 1805, pp. 106, 111, 228. available here). "BoDE's A. J. 1805, pp. 181, 182. 13 Tisserand Merhaninup TpWp vol CODE'S A. J. 1806, pp. 179-180. ^c Mecnam 1910 > P- 188 - vol. 55, p. 194. B. A. 28, p. 184. CELESTIAL MECHANICS: LEUSCHNER 19 .S J u nts > a a S 6^ Sfl o a^ -t-> ^- O> ill l if !* l!i!l i-^j-" a ? " '-3 o s 03 fl O W5CM CO U5CMOCM OCO COCM CM lOrJfCOiO rH rH CM CM CM CM CM 3 8 OS OS OO O iS io :~0 O O OO PH % 3 ic S n o PQ PQ PQ o . It Letter Epoch M. T. P Equinox Authority A 1805 Gottingen Of f 14 48 11.5 813 8468 GaussVII B c 1811 1810 Gottingen Gottingen 14 44 IX 14 43 95 813.25748 812 7140 1811 1810 Wachter Mobius D.... E 1815 Dec. 31.0.. 1819 Gottingen Mannheim 14 43 28.84 14 53 17 44 812.9304 813 86981 1816.0 1819 Nicolai F G.... H 1820 May 11.... 1810 1826 Nov. 1. . . . Mannheim Gottingen Berlin 14 55 1.78 14 44 39.19 14 53 22 6 814.40238 813.4837354 813 88514 1820 May 11 1810 1810 Nicolai Nicolai Encke I 1861 Nov. 21.0.. Greenwich 14 47 14.1 813 34555 1861 Nov 21 Hind J. ... . 1861 Nov. 21.0.. Greenwich 14 47 13 81 813 35271 1861 Nov 21 Downing K 1900 Jan. 14 50 813.434 1900 Boda CELESTIAL MECHANICS: LEUSCHNER 23 (4) VESTA Vesta was discovered by Olbers 1 at Bremen on March 29, 1807. Preliminary elements were computed by Gauss 2 and also by Burk- hardt. 3 The third set of Elements A by Gauss is based on obser- vations from March 29 to July 11. They were used to compute the ephemeris for 1808-1809. The preliminary work of Gauss was continued by Gerling, 4 who supplied the ephemeris for a number of years. His last set of Ele- ments B, are based on the first six oppositions. Burkhardt's preliminary work was continued by Daussy 5 (refer- ence not available here). In his work he took into consideration the perturbations by Jupiter, Saturn, and Mars, and was able to represent the first seven oppositions satisfactorily. On account of the small eccentricity and inclination, the methods of La Place and Le Verrier were sufficient. On account of increased error in the ephemeris for 1818 based on Gerling's first elements, Encke computes a set of Elements C, based on oppositions 1812, 1815, 1816, and 1818. In Astronomisches Jahrbuch 1829, pp. 156-158, Encke gives the results of his work on Vesta based on fourteen oppositions by also using Nieolai's value of Jupiter's mass (1/1054), for the perturbations due to Jupiter. He also makes use of Daussy's tables for the per- turbations of Saturn and Mars. His new set of Elements D repre- sents the oppositions from 1807 to 1825 within 6". Elements D are then brought up to the epoch of 1827. Astronomisches Jahr- buch from 1830 to 1866, contains the elements and ephemerides com- puted by Encke. (See Elements E.) The method of computation 7 was that of the variation of the elements by special perturbations of Jupiter. In this work Encke was assisted by Bruhns and Schiaparelli. In A. N. 332, Encke comments on the poor results obtained for planets (1), (2) and (3), by using Laplace's value for the mass of Jupiter and also publishes a new value of Jupiter's mass (1/1050), obtained from the results of Vesta. The Berliner Jahrbuch for 1868 publishes mean Elements F by Briinnow. 8 They are also published in Watson's Theoretical Astron- omy. Briinnow completes the work of Wolfers and Galle 9 who de- veloped expressions for the perturbations in longitude and radius vector after Hansen's method. Bninnow's elements represent the oppositions from 1810 to 1851 within 6" to +10". For Jupiter's mass he used 1/1050. 24 CELESTIAL MECHANICS: LEVSCHNER From 1871 to 1910 the Berliner Jahrbuch publishes elements (see Elements G), by Farley. 10 His work is based on twelve oppositions, 1840 to 1855. This work forms the basis for later investigations of the general perturbations. Probably the most extensive work on a minor planet are' the tables of Vesta by Leveau published in Annales de FObservatoire de Paris Memoires, XV, XVII, XX, XXII, XXV. The method applied by Leveau is that of Hansen "Auseinandersetzung einer Zweckmassigen Methode zur Berechnung der Absoluten Storungen der Kleinen Planeten, I, II, III." The explanation for the choice of this method is that the application to Vesta is a preparatory study to the motion of Pallas, as Gauss' theory of Ceres was a preliminary study to his theory of Pallas. Memoir XV contains the perturbations of the first order of the masses of Venus, Earth, Mars, Jupiter, Saturn, Uranus, and Neptune. The memoir concludes with the determina- tion of the constants of integration and the expressions for nSz, v, u seci and a representation of an observation 1858, April, 23.5 as follows: Aa= s .! and AS= 0".4. Memoir XVII contains the terms depending upon the square of the mass of Jupiter in n8z and 2i/. The effect on u seci becomes noticeable after 100 years. Memoir XX contains the terms depending upon the product of the masses and concludes with a set of mean elements and corresponding expressions for nSz, v, and u seci. The mean elements given in memoir XX are slightly changed in memoir XXII (see Elements H), on ac- count of some perturbations of the second order that Farley had in- cluded and which form the basis of Leveau's work. Memoir XXII contains the comparisons with 215 normal places founded on 5000 observations extending from 1807 to 1889. The computation has been changed to conform to the solar tables by Newcomb. The correction to the mean motion is zero. The new Elements I represent the obser- vations in right ascension between 2" and +4" and in declination between V and +4". A new value of Jupiter 1/1046, and Mars 1/3648000. The perturbations are collected in tables introducing the mean anomaly and corrective terms for the eccentric anomaly. Mem- oir XXV contains some supplementary terms depending upon the product of the masses. One of the larger terms has a period of 3000 years. The effect of the critical terms, which are mentioned in the beginning of his work, shows itself by comparing these terms before and after integration. The coefficients of corresponding terms are ten or twenty times larger, whereas the other terms mostly decrease. Leveau has made a comparison between observations and calcu- lation of later positions 11 showing the residuals as obtained from CELESTIAL MECHANICS: LEUSCHNER 25 the British Nautical Almanac, and his own computations. The fol- lowing comparisons are illustrations of the results: 1890 1892 1894 1896 1898 Nautical Almanac Aa +l s .18 +1 8 .00 +1 8 .73 +l s .71 +2 8 91 AS +0".9 5".8 +8".9 2".0 + 16".3 Leveau ' Aa +0 3 .03 +0 S .01 +0 S .25 +0 S .06 +0 3 .19 A8+0".4 +0".5 +2".0 Further results of Leveau's theory are published in Comptes Rendus, T. 145, p. 903-906, "Determination des Elements Solaires et des Masses de Mars et de Jupiter par les Observations Meridiennes de Vesta." Extending the comparison with the meridian observations from 1807 to 1904 and taking into consideration the masses of Jupiter and Mars and also the solar elements, Leveau determines a new set of smaller corrections to the elements of Vesta and for Jupiter's mass, 1/1046, and mass of Mars 1/3601280. The tables of residuals shows the poor quality of the meridian observations before 1826. A period of 36 years, (three revolutions of Jupiter, or ten of Vesta) , points to the effect of the critical terms in the residuals ; the amplitude is about 1". The effect of the earlier observations on the value for the mean motion is also illustrated by the last residuals. In Annales de 1'Observatoire Astronomique de Toulouse, T. I., B. 1 to B. 90, M. J. Perrotin published his extensive investigation on the "Theorie de Vesta," applying the method of Le Verrier (Annales de 1'Obs. de Paris, T. X.). The method consists first of deriving mean elements from previous osculating elements by computing provisional periodic perturbations and applying these to the osculating elements for a first approximation. In order to avoid considerable labor, Per- rotin starts with a certain fixed major axis and develops corrective terms for the variation in the assumed value. The final mean motion is determined from two extreme groups of observations in 1807 and 1876 when the planet was near the same place in its orbit. The per- turbative function is developed by Le Verrier to the seventh degree in the inclination and eccentricity ; thus Perrotin includes terms of 10n' 3n. Venus, Earth, Mars, Jupiter, and Saturn are taken into account. Derived from the secular terms e is always smaller than 0.15, the mean motion of the perihelion is +38", that of the node 38", and the inclination remains less than 9. The terms of the second order are then considered. Those due to the square of Jupiter's mass of the second degree are small. Those depending on the product of the masses are more important, especially those depending upon 5n" 2n', 26 CELESTIAL MECHANICS: LEVSCHNER 2n"+4n' n, 2n"+9n' 3n. For getting these terms the develop- ment of the perturbative function is used to the seventh degree and for determining the terms of the eighth degree the method of Cauchy, as extended by Puiseux, is used. No comparison with observations is attempted. REFE.RENCES 1 BODE'S B. J. 1810, p. 194, 6 Connaissance des Temps, 1818, 1819, 'BODE'S B. J. 1810, pp. 198, 213; 1812, 1820. p.253; Gauss Werke. vol. vi. 3 BODE'S B. J. 1810, p. 199; Annales de 1'Observatoire Astronomique de Toulouse, vol.i, p. B4. 10 British N. A. 1860. 4 B. J. 1814, p. 253; B. J. 1817, p. 255; "Bull. Astr. vol. 19, p. 434; Comptes B. J. 1819, p. 224. Rendus, T. 135, p. 525. CELESTIAL MECHANICS: LEUSCHNER 27 3rd set of preliminary Based on first six op Al. A57 B. J. 1831, B. J. 1871. Mem. XX Mem. XX - OOCO in a and 1" in 8. This was the beginning of Zech's important work which made this planet suitable for testing theories in the case of near commensurability. In B, J. 1858, new Elements D, by Zech, 11 are published, based upon five oppositions taking the perturbations by Jupiter, Saturn, and Mars into account. These elements are very closely the same as those v. Zeipel selected from Zech's manuscript. In this the general perturba- tions by Jupiter, Saturn, and Mars were computed and partly tabu- lated. The basic elements for this computation were founded on eight oppositions. After Zech's death in 1864 his computations of the general perturbations of the planets (5) and (10) were discon- tinued by the Recheninstitut, whereas his elements and special per- turbations for (10) Hygiea were used in B. J. until 1875 by Powalky and Becker. In 1873 the correction to the ephemeris had increased to 4 s in a and 20" in 8. In B. J. 1876 E. Becker published a preliminary set of elements which finally was corrected to the Elements E, given by Bauschinger 12 based upon the oppositions 1868, 1869, 1871, 1873 and 1874. Special per- turbations by Jupiter and Saturn were included and brought forward to the osculation 1898, December 20. The elements given in Kleine Planeten for 1920 are Becker's brought forward by Strehlow with Jupiter's perturbations. The mean motion is corrected empirically by +0"-07 (since 1898, August 22), represent- ing the fourteen oppositions since 1900 within ztO^. Osculation 1920 January 0. Elements F. In the mean time v. Zeipel 13 had computed the tables for the general perturbations by Jupiter according to the method by Bohlin in the case CELESTIAL MECHANICS: LEUSCHNER 29 of near commensurability, Hecuba group. As a test he applied his tables to the orbit of (10) Hygiea and started with the Elements G (a) from Zech's manuscript. These were first transformed to mean ele- ments and then compared with nine oppositions from 18491884 with the aid of his tables. A least squares solution gave the corrected Elements H. The work of computing tables after Bohlin's method for the Hecuba group had also been undertaken by A. 0. Leuschner 14 and an appli- cation to (10) Hygiea was made by Miss E. Glancy and Miss S. H. Levy 14 . In order to compare the results with those of v. Zeipel, Miss Glancy 15 used the Berkeley tables and after a comparison with nine oppositions between 1849-1884 obtained the Elements I, starting with the mean Elements G(b). Further comparisons were made by repre- senting observations in 1910, 1914, 1917. The residuals before solution from Zech's elements with the Berkeley tables were 11' to +10' in the plane; from Zech's elements and v. ZeipePs tables 24' to +5'. After Miss Glancy 's solution the residuals are 8' to +7', and after v. Zeipel's solution 7' to +8'; but in 1917 they are +9' and +19' respectively, apparently in favor of the Elements I and the Berkeley tables. The residuals from Zech's elements and the Berkeley tables ( 11' to +10') show a decided periodicity of 30 years, thus pointing to the influence of Saturn. Later observations in 1917 and 1921 are repre- sented much better (0' and +10') by the Berkeley tables and Zech's elements than by any of the solutions. 16 The best future representa- tion may be expected from Zech's elements G(b) and the Berkeley tables. The residuals, probably chiefly due to Saturn, keep within fixed limits 10'. The most obvious next step would be to correct the residuals for some of the earlier oppositions by means of the per- turbations of Saturn and Mars, available in manuscript in the Rechen- institut. Until that shall have been done, no corrections should be applied to Zech's elements, which appear to be the best available. REFERENCES *A. N. vol. 28, p. 391. "H. v. Zeipel, Angenaherte Jupiter- 2 A. N. vol. 29, p. 15. storungen f iir die Hecuba-Gruppe. 3 A. N. vol. 29, p. 49. Memoires de TAcademie des Sciences *A. N. vol. 29, p. 81. de St. Petersburg, vol. 12, Nr. 11, 6 A. N. vol. 29, p. 81. iqno 6 A. N. vol. 29, p. 81, p. 126; vol. 30, ' .. . , a . p 320. "National Academy of Sciences. 7 A. N. vol. 30, p. 81, p. 82. Memoirs, vol. 14, third memoir. 8 A. N. vol. 30, p. 87. 15 A. J. vol. 32, p. 27. .f"2' V0 }' 81> P * 27 * " A> ' Leuschner Comparison of " A ' N 1 ^Q P ' -U7 theory with observation for the minor "VerSffentUchungen des astronomis- P lanets < 10 > H y iea and < 175 > Andr - chen Recheninstituts zu Berlin. Nr. 16, mache. Proc. N. A. S., Washington, p. 48. vol. 8, No. 7, p. 170. 30 CELESTIAL MECHANICS: LEUSCHNER TABLE 5. Elements (10) Hygiea Letter Date M.T. M V fl i A. . . 1849 Apr. 15 Berlin. . . 330 52 8.56 227 49 54.23 287 37 8.64 3 47 15 51 B 1851 Sept. 28.5 Berlin 128 44 20.7 228 2 32.1 287 39 8.3 3 47 8.3 C D 1851 Sept. 17.0. . 1851 Sept. 17.0.. Berlin Berlin 126 59 37.2 126 59 48.76 227 48 9.2 227 47 58.77 287 38 37.6 287 38 34.21 3 47 9.22 3 47 9.29 E 1874 Dec 26 Berlin 174 55 300 CO 312 40 30 5 285 18 57 5 3 47 43 2 F . 1925 Jan Greenwich 181 38 49 2 305 25 22 8 285 52 55 2 3 48 46 8 G(a).. G(b).. H 1851 Sept. 17.0. . 1851 Sept. 17.0. . 1851 Sept. 17.0 Berlin Berlin Berlin.... 126 59 48.6 121 51 58 121 35 27.6 TT 227 46 36.6 230 47 49.6 231 2 9 287 37 11.4 287 37 11.28 287 8 33 6 3 47 8.4 3 47 8.5 3 47 45 6 I 1851 Sept. 17.0 . Berlin 121 31 53.8 231 4 56.6 287 27 28 1 3 47 30.1 Letter Date M.T. V M Equinox Authority A 1849 Apr. 15.0 Berlin Of , 5 47 55.37 634.6406 1849.0 d' Arrest VI B 1851 Sept. 28.5 Berlin.... 5 46 34.7 634.83504 Chevallier C D E F G(a).. G(b).. H I 1851 Sept. 17.0 1851 Sept. 17.0 1874 Dec. 26.0..... 1925 Jan. 0.0 1851 Sept. 17.0 1851 Sept. 17.0 1851 Sept. 17.0 1851 Sept. 17.0 Berlin.... Berlin.... Berlin.... Greenwich Berlin.... Berlin.... Berlin Berlin 5 46 16.8 5 46 16.57 6 18 23.7 6 40 48.0 5 46 16.8 6 23 8.9 6 22 1.2 6 21 31.0 634.84564 634.84912 636.58673 638.517 634.850 636.8566 636 849 636 . 86105 1851 Sept. 17 1851 Sept. 17 1870.0 1925.0 1850.0 1850.0 1850.0 1850 Zech Zech E. Becker Strehlow Zech Zech v. Zeipel Glancy CELESTIAL MECHANICS: LEUSCHNER 31 (28) BtiLLONA Discovered by R. Luther 1 at Bilk near Diisseldorf, March 1, 1854. Preliminary elements were published by Bruhns, 2 3 4 Chevallier 5 and Ruemker, 5 Oudemans. 6 From 141 observations formed into five normal places Bruhns 7 8 derived, originally using four longitudes and two latitudes correspond- ing to the first, second, fourth and fifth normals, the first reliable Elements A. As Elements A differ considerably from his previous elements and those of Oudemans. He made a comparison of ephem- erides which satisfied him regarding the correctness of Elements A. This case is somewhat indeterminate and small errors of observation would produce considerable changes in resulting elements. From Elements A Bruhns 7 has published an ephemeris for 1855. Further ephemerides are published by Bruhns, among them for 1856, 9 1866, 10 (a star correction 11 November 29 by Engelmann), for 1867, 12 correction Aa 50 s , AS 3'.4 by Tietjen, 13 and for 1871-72. 14 Other elements by Bruhns are published in the B. J. from 1857 to 1860. The B. J. from 1861 to 1891 contains new elements and ephemerides by Bruhns originally based upon the observations of the first four oppositions with perturbations by Jupiter, Saturn, and Mars. Ele- ments C. 19 Whether these elements are merely brought forward by perturbations or contain corrections is difficult to determine. From 1892 the B. J. uses von der Groeben's elements instead of those by Bruhns. By a process of successive correction of osculating elements and special perturbations by Jupiter, Saturn and Mars (perturbations by the Earth and Venus were found negligible) , based on 16 normal places extending over a period of thirty-two years from 1854 to 1886, von der Groeben, 15 starting with a set of elements osculating for 1870, September 18, derived Elements Ba, Bb, Be, Bd osculating for differ- ent epochs from 1861 to 1882. These elements were brought forward with special perturbations of Jupiter, Saturn and Mars to the epoch 1886, February 26, Elements Be, and to the epoch 1889, October 28, Elements Bf. Elements Bf are adopted by the B. J. for 1892 16 and 1893. 17 Three observations by Ball at Luttich, October 31 to Novem- ber 15, 1889, are well represented by the ephemeris. Mean Aa + s .19, mean AS ".5. From seven observations at Washington and Tacubaya with star places newly determined by Bruns and four observations by himself at Diisseldorf, Luther 18 obtained a mean correction to the ephemeris 32 CELESTIAL MECHANICS: LEUSCHNER from von der Groeben's Elements Bf brought forward by special perturbations of Aa O s .42, AS +3".4 in 1894. The Elements Bg 20 to Bu given in the B. J. and in Kleine Planeten from 1894 to 1919 are von der Groeben's elements probably brought up to date in the same manner as before with the special perturbations by Jupiter, Saturn and Mars and without any other correction. The following is a partial list of published corrections to the B. J. ephemerides from von der Groeben's elements: 1902, August 7 + 8 8 +0' Luther 21 1903, October 26 +6 ra 38 +25' 2 Luther 22 1905, March 1 + 2.77 - 8'3 Iwanowski 28 1906, June 20 + 8.99 +11.3 Luther 24 1907, September 7 + 21.68 +1' 23'l Luther 26 1908, November 28 - 17.77 +0' 9'0 Luther 26 1916, August 8 +0 m 3 +1' Luther 27 1917, December 4 +l m 4 +7' Luther 28 1919, April 1 -I m 8 +6' Luther 29 In a dissertation on the Jupiter perturbations of the group of small planets whose mean daily motions are in the neighborhood of 750", D. T. Wilson 30 gives an application of the Hansen-Bohlin method to the Jupiter perturbations of this group. "The integration divisors for certain values of the integers n, r and s become as small as 0.2. These terms increase rapidly as the series advance. They were computed to the third power of the eccentricities and to the fourth power of w. It was found that all the terms of the third and fourth powers of w and some of those of the third power of the eccentricities are negligible when the eccentricity of the disturbed planet does not exceed 0.34 and when the mean daily motion lies within the limits 720" to 780". There- fore only those terms of the third power of the eccentricities which are appreciable within the above limits have been retained. All the secular terms have been computed to the fourth power of eccentricities." By means of these tables the Jupiter perturbations of Bellona were computed and compared with the results previously obtained by Hansen's method by Bohlin. 31 The mass of Jupiter is taken as 1 : 1048 in the tables by Wilson. Of the three applications of these tables by D. T. Wilson that of Bellona is by far the most interesting. This depends on the greater proximity to the commensurability (748") and the cross position of the line of apsides to that of Jupiter. The inequality 2g-5g' in rj8z is the largest of all and amounts to 40'. But the difference between the coefficients computed by Hansen's and Bohlin's methods is large (2') and the comparison with the observations ought to decide between the application of these methods to numerous planets in this group. CELESTIAL MECHANICS: LEUSCHNER 33 REFERENCES S A.N. vol. 38, pp. Ill, 143. 18 A. N. vol. 139, p. 302. 3 A. N. vol. 38, p. 155. M B. J. 1861, p. 506. 8 A. N. vol. 38, p. 218. " B. J. 1894, p. 394. 4 A. N. vol. 38, p. 351. 31 A. N. vol. 159, p. 295. 6 A.N. vol. 38, p. 158. M A. N. vol. 163, p. 383. 6 A.N. vol. 38, p. 158. "A. N. vol. 167, p. 319. T A.N. vol. 40, p. 203. *A. N. vol. 171, p. 349. 8 B. J. 1858, p. 407. a A. N. vol. 175, p. 401. 9 A. N. vol. 44, p. 235. M A. N. vol. 179, p. 243. 10 A. N. vol. 68, p. 125. a7 A. N. vol. 203, p. 164. 11 A. N. vol. 69, p. 104. M A. N. vol. 206, p. 16. " B. J. 1869. * A. N. vol. 208, p. 248. 13 A. N. vol. 65, p. 176. 80 Astrpnomiska lakttagelser och Un- 14 A. N. vol. 79, p. 5. dersokningar a Stockholms Observa- 15 A. N. vol. 123, p. 369. torium. vol. 10, No. 1. 16 B. J. 1892, p. 390. "Manuscript in the office of the 17 B. J. 1893, p. 390. Recheninstitut. 34 CELESTIAL MECHANICS: LEUSCHNER iij i ijijf jjifji j jjjjj 3 o o o o o a o o o o o a o o o o o o o o o o qqqqqqqqqqqqqqqqqqqqq q D 00 rH CO CO r-J OQ rH ,-H ^ t>- iO l2 S * i-tTjtOSOCOOi^OOcOCOiOOr-lO^iCINt^fNlO O O iH iOiH(NiO from 1887 to 1897 5 ), Bauschinger gives Leppig's Elements C in the "Tabellen" 6 for the epoch 1883, July 12.0. The various elements in the B. J. to 1915 are probably brought up from Leppig's Elements B, with special perturbations. The char- acter of the perturbations is not given. Nor is any reference made to arbitrary corrections. In Kleine Planeten, 1916, the elements are changed by estimating the perturbations 1883-1910 and roughly determining, M and /*,, Ele- ments D. In Kleine Planeten, 1921, the last elements are again corrected by the computation of Jupiter's perturbations and a representation of observed positions, 1884-1918, within l m .5. Osculating elements, 1921, April 24, not available. From 1884 to 1899 the planet was practically lost, mainly on ac- count of the misprint in to. Coddington 7 computed a place with the elements of B. J. 1901, and found the planet Aa +5 m .2, AS 20', 1899. For the computation of the perturbations and tables of the Watson asteroids, Leuschner made a collection of Leppig's elements in the B. J. including Elements C and derived average Elements E from those in B. J. 1871, 1873, 1874, 1875, 1877, 1878, 1881, 1898. From these, approximate mean elements were derived. The perturbations were developed and a preliminary correction of the mean motion and mean anomaly from observations in 1867 and 1899 was at- tempted. A mistake in one of the main terms of the perturbations was discovered and corrected. Nevertheless, large discrepancies be- tween observation and computation remained ( 5 in a, 1867). These differences were used for a preliminary correction of the mean motion and of the mean anomaly. With the new value of the mean motion the perturbations were corrected, the residuals re-determined, and further corrections made to the mean motion and to the mean anomaly. A least square solution of 12 places from 1867 to 1899 was then made including the corrected perturbations. The residuals CELESTIAL MECHANICS: LEUSCHNER 39 were thereby reduced from 2.2 to a maximum of dbO.14. Ele- ments F. Further correction of the perturbations by means of the new mean motion produced larger residuals. The best representation as above is obtained by the use of the adopted Elements F without further correcting the perturbations. From the experience of the Recheninstitut and of Leuschner with Leppig's elements, it is evident that before a satisfactory repre- sentation for all oppositions, without making arbitrary corrections to the elements, can be obtained, it will be necessary to derive an ac- curate set of osculating elements by connecting a limited number of oppositions with accurate determination of the perturbations. With an accurate set of osculating elements the perturbations may then be corrected, but further correction of the elements should be made only after higher order perturbations and perturbations by planets other than Jupiter shall have been considered. REFERENCES 'A. N. 2 A. N. 8 A. N. 4 B. J. vol. 70, p. 79. vol. 70, p. 219. vol. 71, p. 47. 1871. A. N. vol. 72, p. 331. 8 A. N. vol. 139, p. 63. *Tabellen zur Geschichte und Stat- istik der Kleinen Planeten. 7 A. N. vol. 153, p. 225. TABLE 8. Elements (94) Aurora Letter Epoch M.T. M 0> D i A. . . . 1867 Nov. 28 Berlin . . 340 30 39 5 40 50 23 2 4 32 9 3 8 5 27 B... . 1870 Jan. O.O.... Berlin 115 9 43.74 40 2 43.08 4 34 36.38 8 5 18.49 C... . 1883 July 12.0.. Berlin 256 3 4.3 45 22 31.8 4 25 0.9 8 4 14.0 D... . 1925 Jan. 0.5 Greenwich . 20 40 1.2 57 20 9.6 4 22 26.4 8 4 8.4 E... . 1875 Jan. 0.0... Greenwich . 73 37 45 41 51 21 4 28 46 8 4 54 F... . 1875 Jan. 0.0... Greenwich. 65 20 12 48 13 54 4 24 21 8 3 51 Letter Epoch M.T. 9 M Equinox Author A 1867 Nov 28 Berlin . . Of W 5 10 18 6 630 5129 1867 Tietjen B 1870 Jan Berlin .. . 5 6 8 13 631 5264 1870 Leppig c 1883 July 12 Berlin 4 44 18 3 630.6584 1900.0 Leppig D 1925 Jan. 05. Greenwich. . 5 4 58.8 631.800 1925.0 Berberich E 1875 Jan. 00 Greenwich. . 4 56 22 631.2196 1900.0 Leuschner F 1875 Jan. 0.0 Greenwich. . 5 17 16 631.9473 Leuschner 40 CELESTIAL MECHANICS: LEUSCHNER (127) JOHANNA Discovered by Prosper Henry 1 at Paris, 1872, November 5. In B. J. 1875 a set of Elements A by Baillaud is published, based on observations Nov. 9, 22, 28. Preliminary Elements B by Renan 2 are based on seven observations for the opposition 1872-73. An ephemeris for 1874 April and May is also published. An observation on April 17, 1874, shows corrections to the ephemeris Aa +2 m 43 s , AS 16'. An improvement on Elements B is made by Renan 3 on the basis of six normal places (1872-1874). The resulting Elements C represent the normal places within +7".8 and 9".l. Renan states these differ- ences are within the limits of error and Elements C may be considered definitive. An ephemeris is then computed for 1876 September. Elements D are published by Bauschinger. 4 They are by Maywald and are based on oppositions 1872, 1874, 1876, 1879. The special perturbations of Jupiter and Saturn to 1890 are included. Renan's elements are published and used by B. J. 1882 to 1887. Beginning with B. J. 1888, Maywald's elements are used. An empirical correction was applied to the mean anomalie in 1913 by Berberich. 5 A correction to the R. I. ephemeris 6 for 1918, November 23, is Aa 5 m .2, AS 20'. An improved ephemeris is published 7 for the period 1920, March 21 to April 21, from elements based on 6 oppositions, 1897-1908. Jupiter's perturbations are included. Representation m .3. Osculation 1921, July 13. The correction to this ephemeris 8 from an observation 8 February 28, 1920, is Aa ^-O'M, AS 6'. Another ephemeris 9 for 1920 by P. Maitre is based on elements published in Connaissance des Temps for 1915 with the mean anomaly corrected, on the basis of observations in 1917 and 1918, AM 1.305. General perturbations applying Bohlin's method for this planet have been published by D. T. Wilson 10 and similar perturbations with Hansen's method, by M. Viljev. 11 Olson 12 has published general perturbations of the first order by Jupiter. The basic elements are those of Maywald. Jupiter mass 1:1047.568 (Bessel-Schur) . Method that of Hansen. Terms of 6th to 8th order are below 1". No comparison with observations. REFERENCES 'A. N. vol. 80, p. 239. 7 B Z der A. N. No. 11, 1920. a A. N. vol. 83, p. 349: C. R. vol. 8 B Z der A. N. No. 14, 1920. 78, 1874, p. 1219. ' 8 C. O. M. No. 297. 8 C. R. vol. 83, 1876, p. 567. 10 Ast. lakttagelser och Undersoknin- 4 Veroffentlichungen R. I. No. 16. gar, Stockholm, vol. 10, No. 1. 6 B. J. 1915, p. 27. "Bull. Soc. Astr. Russio. No. 22. *A. N. vol. 208, p. 14. "Swed. Akad. Handl., Stockholm, 1895. CELESTIAL MECHANICS: LEUSCHNER TABLE 9. Elements (127) Johanna 41 Letter Epoch M.T. M <> fl i A 1872 Dec. 18 Berlin 293 6 46 90 25 17 31 40 11 8 19 42 B 1874 Apr. 17 . Berlin 35 3 42 91 13 48 31 41 41 8 17 28 C 1876 Sept 5 5 Berlin 223 47 46 90 50 37 31 46 38 8 16 40 D 1879 Apr. 4.0 Berlin 67 49 52 89 18 47 31 45 2 8 16 48 Letter Epoch M.T.

M Equinox Authority A 1873 Jan. 1.0... Greenwich O I It 7 21 59 // 776.86 1873.0 Bossert B C 1873 Feb. 25.5... 1873 Feb. 25.5... Berlin.... Berlin 7 13 5 7 12 51 778.030 778.1516 1873.0 1873.0 De Ball De Ball D 1872 Nov. 25 0. . Berlin 7 13 17 764.990 1872.0 Richter E F 1873 Feb. 25.5... 1875 Apr. 25 0... Berlin.... Berlin .... 7 11 59 7 13 20 778.0333 777.4729 1870.0 1875.0 DeBall DeBall G 1880 July 70... Berlin 7 21 52 777.4964 1880.0 Palisa H I 1892 Feb. 15.0... 1896 July 3.0... Berlin.... Berlin 7 10 53 7 16 50 777.6921 777.8761* 1890.0 1900.0 De Ball Leuschner *Mean Elements. 44 CELESTIAL MECHANICS: LEUSCHNER (175) ANDROMACHE. Discovered by J. Watson 1 1877 October 1, but observed only Oc- tober 5, 6, 16, 29. These were all the observations available until rediscovery May 19, 1893. By an unfortunate delay the news did not reach other interested observatories until two months later. The preliminary orbit, Ele- ments A, by Tietjen, 2 was erroneous, partly on account of errors in Watson's ringmicrometer observations. Watson 3 computed Elements B, and with them perturbations for three years until his death in 1880. Bidschof 4 made an attempt to improve the orbit, Elements C, but perceived the impossibility of this undertaking. 1893, May 19, Charlios 5 at Nice discovered by photography a planet, 1893 Z, and noticed the close similarity between its orbit and that of (175) Andromache. He referred the case to Berberich 6 who followed it out from a preliminary orbit to the best to be obtained from the data in 1893 May 19- August 1, Elements D, and compared his results with the observations in 1877 and with one in 1892 from a photographic plate taken at Heidelberg. Berberich 7 next computed the extremely large perturbations by Jupiter (in 1887) and Saturn, and improved the orbit by a solution from four normal places in 1877, 1892, 1893, with an ephemeris for the coming opposition, Elements E "this planet therefore deserves peculiar attention for it will furnish an excellent means for determin- ing an accurate value of the mass of the planet Jupiter." Cf. (3) Juno, (4) Vesta, (13) Egeria, (24) Themis, (33) Polyhymnia, (447) Valentine. Berberich 8 improved his elements by the following oppositions and brought them forward with the special perturbations of Jupiter and Saturn. The Elements F, G, H, illustrate the large changes in this case of near commensurability (Hecuba group). This case among others impelled A. 0. Leuschner to undertake the computation of the tables 9 for the Hecuba group after Bohlin's method. The application of these tables to the orbit of (175) Andromache was carried out by Miss S. H. Levy. Her unpublished computation contains the transformation of Berberich's elements to Mean Elements I, tables of the perturbations, determination of con- stants, the comparison with ten oppositions between 1893 and 1907, and at least squares solution which led to the Mean Elements J. The representation of observations in 1914 was O m .4 in a and + 1' in 8. The comparison between theory and observation has been discussed by A. 0. Leuschner. 10 CELESTIAL MECHANICS: LEUSCHNER 45 REFERENCES 1 C. R. vol.85, p. 1006. 91, p. 127. A. N. vol. Circular zum B. J. Nr. 81. 8 B. J. 1881, 1882, 1886. 4 Sitz-Ber. Akad. Wien. Bd. I 5 A. N. vol. 132, p. 367. e A. N. vol. 134, p. 143. 7 A. J. vol. 14, p. 36. 8 A. N. vol. 153, p. 59. 9 National Academy of Sciences, Me- moirs, vol. 14, third memoir. 10 A. O. LEUSCHNER, Comparison of theory with observation for the minor planets (10) Hygiea and (175) Andro- mache with respect to the perturba- tions by Jupiter. Proc. N. A. S. Wash- ington, vol. 8, No. 7, p. 170. TABLE 11. Elements (175) Andromache Letter Date M. T. M CO Q . i A . . 1877 Oct. 29.5 1878 Dec. 5.O.... 1889 Apr. 7.5 1893 Aug. 1.5.... 1894 Aug. 23.0.... 1877 Oct. 11.0.... 1900 Sept. 1.0... 1920 Jan. 20 1877 Oct. 11.0.... 1877 Oct. 11.0 Berlin 45 6 25.6 105 28 46.7 317 3 18.5 297 57 33.9 3 52 54.5 32 5 0.1 16 10 41.5 76 24 14 3 49 12 3 59 32 269 26 21.1 269 31 45.6 269 42 7.7 299 49 1.8 299 46 4.9 298 31 16.6 301 33 8.5 306 45 58 304 6 54 * 304 13 9 23 32 56.0 23 34 50.2 23 43 24.9 25 27 14.8 25 27 32.5 25 36 3.8 25 23 37.7 25 7 12 25 36 4 25 43 27 3 46 38.8 3 46 36.7 3 46 45.8 3 10 51.4 3 10 59.4 3 11 42.8 3 10 38.9 3 10 37 3 11 43 3 11 29 B . . Berlin C D E F G H Berlin Berlin Berlin Berlin Berlin 1925.0 Greenwich. . Berlin I J Berlin Letter Date M. T. M Equinox Authority A 1877 Oct 29 5 Berlin ... o / 20 26 45 7 * 542 173 1877 Tie t jen B 1878 Dec 5.O.. Berlin . 20 26 12 6 541 779908 1880 Watson c 1889 Apr. 7 5 Berlin. . 20 15 17 8 544 411 1890 Bidschof D 1893 Aug. 1 5 . Berlin... 11 39 45 8 614 943 1890 Berberich E 1894 Aug. 23.0 . Berlin .. . 11 36 51 4 614 63354 1890 Berberich F 1877 Oct 11.0 Berlin. . 12 8 54 1 617 7375 1890 Berberich * J^ G 1900 Sept. 1.0 Berlin 11 7 42 9 612 2868 1900 Berberich H 1920 Jan. 20 1877 Oct. 11.0 1925.0 Greenwich.. Berlin .. 10 42 11 12 21 26 607.899 619 5629 1925.0 1890 Berberich Berberich J 1877 Oct. 11.0 .. Berlin 12 19 34 619 025 1890 Miss Levy C7 46 CELESTIAL MECHANICS: LEUSCHNER (433) EROS, 1898 DQ. Discovered 1898, August 13, by Witt at Berlin and by Charlois at Nice. Fayet 1 computed Elements A from three observations on August 15, 26, and September 7. Later he 2 computed Elements B from normal places 1898 August 16.5(7 observations), September 17.5(6 observations), and October 22.4(2 observations). The residuals vary from s . 11 to 8 .36 in a, and +5".l to +8".2 in 8. Hussey 3 gives Elements C, computed from the mean of two Kiel observations made on the 15th of August, and observations made at Mt. Hamilton on September 6 and 27. Hussey 4 later computed Ele- ments D from observations at Mt. Hamilton on August 15, September 27, and November 11, 1898. The residuals for the observations in 1898 vary from +0 S .04 to 8 .18 in a, and +2".2 to +4".4 in 8 and observations to May 4, 1899, are closely represented. Chandler 5 computed Elements E from observations August 14 to November 16, 1898. Elements F 6 and G 7 are also due to Chandler. Elements F are from eight normal places from August 17.5 to No- vember 26.5. The representation is within the errors of observation. In a later article he gives the following residuals for earlier observa- tions found on plates taken in 1896 at Arequipa: 1896 Aa AS April 6 -O m 36?0 -4'9 June 5 -1 16 7 -5'8 Elements G 7 are the preceding ones with corrections applied so as to fit the observations made at Arequipa 1896. Representation 8 for 1893, 1894, 1896, gives maximum residual of +7 8 .6 in a and I'.O in 8. The observations from 1893 to February 16, 1894, were found by Pickering and Mrs. Fleming on plates taken at Cambridge. Per- turbations were not considered in applying the corrections to Ele- ments F to obtain Elements G. Elements H are due to Berberich 9 and are given under the title of "First elements," and are based upon observations made at Urania (Berlin) on the 14, 23, and 31 of August. Berberich gives forty-three sets of residuals covering the period August 13 to August 31, Aa be- ing greater than O s .40 but three times, and AS being larger than 10".0 but twice. Elements I 10 J 11 K 12 and L 13 are due to Russell. Elements I are computed from three normal places obtained by comparison of ob- servations in the A. N. and A. J. with places computed from Ber- CELESTIAL MECHANICS: LEUSCHNER 47 bench's Elements H. 9 (1898, August 18.5, 34 observations, August 26.5, 16 observations, and September 9.5, 26 observations, heliocentric arc about 8.) The set J 11 is based upon nine normal places, although Dr. Chandler's value of the mean motion was taken (Elements G) and the other elements determined by varying the ratio of the ex- treme geocentric distances. The normal places are well represented. Elements K, Russell, 12 are the same as J 11 except for changes in M, (o, and based upon observations in 1899.0 at the Chamberlin and Lick Observatories. The representation August 17, 1898, to May 20, 1899, is satisfactory except for the normal place of November 11.5 for which Aa s .28, AS 3". Elements L are merely the preceding ones brought up to the epoch 1900.0 and mean equinox of 1900.0. In his article 13 Russell develops the general perturbations of the major axis of Eros by the action of Mars. He does this by Le Verrier's method of interpolation. Rus- sell finds eight terms of the general perturbations of the mean longi- tude larger than 1".50. The largest is 35" with a period of about 1000 years. The greatest displacement due to the first 7 terms will be +38" in 1927 and 53" in 1959 in mean longitude, and "will eventually lead to a valuable determination of the mass of Mars." Russell then gives tables of the perturbative function, perturbations of log a and perturbations of the mean longitude. Elements M 14 were developed by Robbins from Elements G by applying special perturbations of Venus, Earth, Mars, Jupiter, and Saturn by the method of the variation of constants. (Nautical Almanac 1837, appendix.) Elements N 15 are due to H. Osten, who has computed eight normal places based upon Elements P, with special perturbations of Venus, Earth, Mars, Jupiter, and Saturn according to Encke's method. Millosevich has produced numerous sets of elements. Elements O 16 were computed from observations made during the interval August 14 to September 21, 1898. An observation by Millosevich October 8. gives Aa l s .93, AS +7".5. Elements P 17 were computed from a normal place of date August 14.5 and Millosevich's observations on September 21 and October 24, 1898. Millosevich states that Berberich's ephemeris requires a correc- tion of +131 8 in a and +5'.5 in 8 on December 23, 1898. Elements Q 18 are from photographic observations at Greenwich by the variation of the distances. Later 19 he stated that there is an error of about 2 s in his ephemeris after five months. Elements R 19 are based upon four normal places and are Elements P improved by the method of variation of the distances. They rep- resent 17 normal places from 1000 observations in 1898-1899 perfectly. 48 CELESTIAL MECHANICS: LEUSCHNER Elements S 20 are based upon 17 normal places made from 999 ob- servations in a and 992 observations in 8 in the years 1898-1899, and are derived from Elements R brought forward with the perturbations of Venus, Earth, Mars, Saturn and Jupiter for 20-day intervals. Elements T 21 U 22 V 23 W 24 X 25 Y 26 are improvements of preceding elements, as are Elements Z 27 AA 27 AB 27 AC 28 AD 29 and AE 30 . Elements AE are Elements AC with special perturbations for 20-day intervals for the period 1901, March 20, to 1903, June 8.0 applied. These perturbations were computed by Wedemeyer, those of Venus, Earth, Mars, Jupiter and Saturn being considered. Elements AF were found only in the B. J. for 1907. They are probably Elements AE with the epoch changed and perturbations ap- plied. Dziewulski has published "Sekulare Marsstorungen in der Bewegung des Eros.," Bull, de PAcad. des Science de Cracovie 1905. Not available here. Some mistakes are corrected in A. N. vol. 175, p. 171. In A. N. vol. 175, p. 17, Merfield gives the secular perturba- tion of Eros due to all major planets. Gauss' method by Hill. Elements from Hill's memoir, Ast. Papers of the Amer. Ephm. Vol. IV and Elements W. The secular perturbations by Jupiter, the Earth and Venus are the largest. Elements AG are by G. Witt. 31 They are based upon observations from 1893 to 1903, the perturbations of Venus, Earth, Mars, Jupiter and Saturn being included. The perturbations were calculated by the method of variation of constants. For Mars, Jupiter and Saturn the perturbations were calculated for 20-day intervals and for Venus and the Earth at 10-day intervals. With these elements he calculates the perturbations from 1903 to the beginning of 1908, and includes these in an ephemeris for 1905 for dates from July 17 to August 22 (B. J. 1907, p. 476.). Elements AH 32 AI 33 AJ 34 are preceding elements with change of osculation. Elements AI are also found in the Connaissance des Temps for 1915, but with the equinox changed in 1920.0, Greenwich M. T. In an article "Beitrage zur Theorie der Bewegung des Planeten 433 Eros," E. Noteboom 35 uses Elements AL to compute the general perturbations of Mercury, Uranus, and Nepture. A recurring run in the residuals is not due to these planets. Noteboom then uses the twenty normal places of Witt 36 (which lie between dates 1893 October 31 and 1907 October 8) and forms four more normal places, one in 1910, one in 1912, and two in 1914. He then gets Elements AM out of a least square solution by a correc- tion of Elements AL. He gives the mass of Earth and Moon as 1/328370102 and 7r=8".799. No authority is available for Elements AK. 87 CELESTIAL MECHANICS: LEUSCHNER 49 REFERENCES I A. N. vol. 147, p. 335. "A. N. vol. 153, p. 25. 3 A. N. vol. 148, p. 27. C. R. 127, 33 A. N. vol. 153, p. 25. A. J. B. p. 806. 1900, p. 158. 3 A. J. vol. 19, p. 120. M A. N. vol. 153, p. 217. 4 A. J. vol. 20, p. 61. A. N. vol. "A. N. vol. 153, p. 217. A. J. B. 148, p. 143. A. J. B. 1899, p. 132. 1900, p. 158. 6 A. J. vol. 19, p. 148. 35 A. N. vol. 154, p. 142. 6 A. J. vol. 19, p. 155. *A. N. vol. 154, p. 142. B. J. 1903. 7 A. J. vol. 19, p. 160. A. J. B. 1899, 3T A. N. vol. 156, p. 327. p. 132. * A. N. vol. 156, p. 328. B. J. 1904. 8 A. N. vol. 148, p. 189. A. J. B. 1901, p. 184. 9 A. N. vol. 147, p. 221. ""A. N. vol. 155, p. 25. A. J. B. "A. J. vol. 19, p. 147. 1901. II A. J. vol. 20, p. 8. A. J. B. 1899, 80 B. J. 1905, p. 534 and p. 428. p. 132. 81 G. WITT; Untersuchung iiber die "A. J. vol. 20, p. 134. A. J. B. 1899, Bewegung des Planeten (433) Eros, p. 132. Berlin, 1905, Druck der Norddeutschen 13 A. J. vol. 21, p. 25. Buchdruckerei. A. N. vol. 176, p. 211- 14 M. N. vol. 60, p. 614. A. J. B. 213. 1900, p. 158. 33 B. J. 1908, p. 496 15 A. N. vol. 150, p. 362. A. J. B. 83 B. J. 1909. 1899, p. 132. 34 B. J. 1916. "A. N. vol. 147, p. 363. 85 A. N. vol. 214, p. 153. "A. N. vol. 147, p. 397. B. J. 1901. 3*> ^ -3 T3 III"! H9q-q|l ^ _ . r^ r"\ r^t m 93 d d j O O O 5 3 oi QQ ad W rt g ^3J^J5^3^3X)J3^3X Ifewllllllllisllsiiilidbdd d w CD CD CO O KOOt>t^ -OOCN CS CD CO t> ^ I - 00 00 00 00 OOP OOOOOOO -OOP COCO COCOCOCOCN CO i-liCCOc6c5lNr-<00(NCNc5cMM^H?3 OS t^ TH CD CO 00 S^2^S? d S5SSSS (N CN CN OO ^ooooooooooooooooooooooooooooooo oooiC OS 00 OS 1C OS TH CO iH Tjl i-H 1C O CO O CD c q *$% O CO t^ * CO CO 00 CO t*- CN i-l CN s2 cSSco 1 00 >C N. CO 00 05 d co c>i q oo co CN CO CO CS OS CO OS O OSOOcOt-.-i -t i-H CO OS 1C CO 1C 00 t^ b- t CS CO CO CO CO >C 10 CO CN ^H 1-1 1C CNJ O CN CN CN O IN O I llll O O PQ W PQ I I J I I I I I J I J I I I J I II 1 I I .S .3 B w 6 v D 8 8 *9 V 9 v 8 V w V w v w 2 9 V pqp5pqpqpqpqpqpQpqwpqpqfflpqOPQ COCOCO CO CO CO COCO co rH CNCNI 00 00 X 00 GO o W M -j CELESTIAL MECHANICS: LEUSCHNER 51 (447) VALENTINE (1899 ES). Discovered by Wolf and Schwassmann 1 at Heidelberg 1899, Octo- ber 27. Preliminary elements were computed by Kreutz 2 based on observa- tions 1899, October 29, November 11 and December 3. Elements A. Corrections to the ephemendes based on improved elements in 1902 3 Aa +16 S , AS 2'; in 190* January 13, Aa +36 S , AS 0'.3. A more complete investigation of the orbit was undertaken by Hans Osten. 5 He received from Kreutz five sets of elements, B to F, which refer to different epochs in order to show the effects of the perturba- tions. Kreutz states that Elements B to E are comparable and are to be preferred to Elements F. Elements B to E were determined from five normal places (1899 to 1904). The residuals for the normal places are la Ib II III IV V AacosS +5'1 -f-8'7 +i:2 -3'2 -0'7 +35^5 A5 +14 +8-1 -1.1 +4-0 -0-2 - 3.0 For the purpose of investigating the general perturbations for this planet, Osten makes use of Kreutz' Elements B, Leverrier's elements for Jupiter and Saturn, except for the adoption ofTETilFs values of the mean motion for each, for Jupiter's mass Newcomb's value and for Saturn's mass Bessel's value. The perturbations of the first order of the masses are determined according to Hansen's method. As a test on his results, he compares observations with theory for seven normal places (1894 to 1906) with the following results: Aa cos 6 A6 1. 1894 -10!73* - 0;54* 2. 1899a +5.07 + 1.41 3. 1899b + 8.28 + 8.06 4. 1901 + 0.32 1.01 5. 1902 + 2.44 + 1.39 6. 1904 +33.26 +16.12 7. 1906 +32.06 2.52 On the basis of these residuals the elements were corrected, which resulted in Elements G. The comparison between observation and computation for the normal places gives: * Weight 1/9. Observation uncertain. 52 CELESTIAL MECHANICS: LEVSCHNER Aa cos 6 A5 1. 1894 +8!76t +7!22 2. 1899a 2.81 2.36 3. 1899b +1.57 +4.86 4. 1901 2.51 +0.73 5. 1902 +1.86 +2.00 6. 1904 0.68 0.31 7. 1906 +1.97 1.05 The approximate perturbations due to Mars were found to be in- effective since they remained less than 0".01. For the purpose of improving Elements G by means of additional observations, Osten 6 first computes special perturbations due to Jupiter and Saturn by the method of variation of elements, and determines new Mean Elements H from the corrections which the observations successively indicate. For the other six major planets the general perturbations are computed after the method in Tisserand, Vol. 1, Chap. 22. All the masses are taken from Bauschinger, "Tafeln z. Theor. Astr." From this combination of special and general pertur- bations a set of osculating elements for the ten oppositions 1899-1912 results, forming the basis for the formation of normal places. A de- tailed investigation of the comparison stars and the observations leads to normal places with relative weights and corrections for magnitude equation in a. The equations of conditions are solved under four different assumptions, thus Elements I without and Elements J with magnitude equation. No definite solution is accepted, but the impor- tance of using a observations free from magnitude equation is em- phasized. In the next work Osten 7 proceeds to determine the perturbations of the second order starting with Elements I. Hansen's method is followed throughout (correcting an error in "Auseinandersetzung II, page 98," which acts nearly as a change of the perturbing mass). A capital difficulty is encountered in the near commensurabilities (+2e 5e' +le") with a period of 8650 years. Osten proposes to treat such inequalities by expansion into power series in the time and then eliminate one of the anomalies. For (447) the term 3e 7e' is thereby much enlarged. A comparison is made between the special and general perturbations by Jupiter and Saturn. Some deviations of the order of V are attributed to the perturbations of the third order. Eight tables contain the perturbations. In the hope of obtaining as accurate results for (447) as for the major planets Osten 8 completes his former theory and gives the last t Uncertain. j Normal places from Kreutz. CELESTIAL MECHANICS: LEUSCHNER part of the perturbations of the second order and those of the third in longitude and radius vector. The accuracy of V in 100 years is extremely difficult to obtain. A first test is the comparison between special and general perturbations in the longitude and radius vector. He finds small deviations which probably are due to the computation of the special perturbations. As a further test Osten gives the comparison with 16 normal places 1894-1918. The Jupiter mass is also included as an unknown. The value 1 : 1047.49 is found. 1 : 1047.35 according to Newcomb is adopted. Thus Elements K are found. The representation of the normals from micrometer observations is V to +2" in the plane. X A. N. vol. 150, p. 431. 2 A. N. vol. 151, p. 159. 8 A. N. vol. 158, p. 271. 4 A. N. vol. 170, p. 195. REFERENCES 5 Astronomische Abhandlungen No. 15, 1908. A. N. vol. 194, No. 4639, p. 113. 7 A. N. vol. 199, p. 393. 8 A. N. vol. 210, p. 129. TABLE 13. Elements (447) Valentine (1899 ES) Letter Epoch M. T. M. M Q i A 1899 Dec. 3.5 Berlin 4 21 32 319 16 21 72 18 33 4 49 33 B c 1899 Dec. 5.5 Osc. Nov. 5.0 1901 Feb. 8.0 Berlin Berlin 4 40 42 87 2 32 319 15 3 318 53 31 72 24 5 72 24 4 49 4 4 49 4 D 1902 Apr 4 Berlin 167 42 14 318 29 20 72 23 32 4 49 4 E 1904 Oct. 10.0 Berlin 345 41 51 316 22 10 72 19 44 4 49 5 F 1906 Feb 2 Berlin 79 55 40 313 38 14 72 25 5 4 49 21 G 1899 Nov 5 Berlin 358 57 25 319 13 45 72 24 10 4 49 4 H 1899 Nov. 5 Berlin 2 42 27 315 27 43 72 25 38 4 49 9 1 1899 Nov. 5 Berlin 358 52 18 319 13 42 72 24 16 4 49 4 j 1899 Nov. 5 Berlin 358 52 21 319 13 42 72 24 11 4 49 3 K 1899 Nov. 5 Berlin 358 57 21 319 13 40 72 24 17 4 49 4 Letter Epoch M. T. *> ft Equinox Authority A 1899 Dec. 3.5 Berlin / * 2 36 37 * 687.012 1900.0 H. Kreutz B . ... 1899 Dec. 5.5 Berlin 2 34 33 687.5969 1900.0 H. Kreutz c Osc. Nov. 5.0 1901 Feb 8 Berlin 2 35 5 687.6846 1900.0 H. Kreutz D 1902 Apr 4 Berlin 2 36 17 688.1604 1900.0 H. Kreutz E 1904 Oct 10 Berlin 2 40 15 686.5435 1900.0 H. Kreutz F 1906 Feb 2 Berlin 2 38 34 687.9066 1910.0 H. Kreutz G . 1899 Nov. 5 Berlin 2 34 32 687.3550 1900.0 Osten* H 1899 Nov. 5 Berlin 2 25 42 687.3504 1900.0 Osten I 1899 Nov. 5 Berlin 2 34 32 687.6016 1900.0 Osten J 1899 Nov. 5.0 Berlin 2 34 32 687.6018 1900.0 Osten K 1899 Nov. 5.O.. .. Berlin 2 34 32 687.3884 1900.0 Osten Mean elements. 54 CELESTIAL MECHANICS: LEUSCHNER (588) ACHILLES, 1906 TG. Discovered by Wolff 1 at Heidelberg 1906, February 22. From observations of February 22, and March 5, Berberich 2 com- puted Elements A for a circular orbit (search ephemeris for April 1906). The first elliptic Elements B were published by Berberich. 3 They are based on observations 1906, February 22, March 23, and April 22. An ephemeris for May and June, 1906, is also included. He also points out that the aphelion point lies far beyond Jupiter's orbit, and that the present orbit has had its form and position for a long time. From Elements B, Charlier 4 finds that Achilles is approximately 55 ahead of Jupiter, consequently very close to one of Lagrange's libration points. He also points out that we may have here a case (as he has shown) 5 where the planet does not remain at the apex of the equilateral triangle, as suggested by Lagrange, but oscillates about it with a period of about 148 years. Comments on the character of the orbit of Achilles have been pub- lished by Berberich, 6 Crommelin, 7 Ristenpart, 8 Stroobant. 9 On the basis of observations extending from 1906, February 22, to May 19, Bidschof 10 has published a set of Elements C and an ephemeris for 1907. A continuation of the ephemeris with corrections to the same was published by Bidschof. 11 An observation 1907, February 12, gives Aa 51 s , and AS +6'.7. An ephemeris for 1909 based on Elements B. J. 1911 with special perturbations due to Jupiter was published by Franz. 12 For the following years, to 1919, the B. J. (Kleine Planeten) pub- lishes elements by Bidschof brought forward to a new epoch and mean equinox, 13 Elements D. As an application of Leuschner's 14 satellite method, Einarsson 15 computed a preliminary orbit, (Elements E), based on observations 1907, February 12, April 15, and June 2. Special perturbations, due to Jupiter, computed by Encke's method, were included for the period covering the observations. The maximum residuals, for nine observa- tions of 1907, were Aa cos 8 3".6, AS 0".7. Elements E were used, without perturbations, to represent an observation 1906, May 19, with the following results: Aa cos 8 +1'01".8, AS 39".2. The most recent work on Achilles was done by Julie M. Vinter- Hansen. 16 With Bidschof's Elements D, the special perturbations, due to Jupiter and Saturn, were computed from 1906 to 1914. All observations of 1906 and 1907, two of 1913, and one of 1914, were then represented and residuals determined. With these as a basis, CELESTIAL MECHANICS: LEUSCHNER 55 eleven normal places were formed, weighted according to number of observations in each. The residuals, (freed from perturbations), for the 1906 and 1907 normal places are small. For normal place X, (1913), AacosS +2781".2, AS + 1685".0; for normal place XI, (1914), Aa cos 8 +1664".!, A8+852".7. The resulting orbit improvement gave Ele- ments F. The residuals for the normal places from Elements F vary from + 19".8 to 40".9 in Aa cos 8, and 9".2 to +23"8 in A8. With Elements F the special perturbations were recomputed by J. Braae. With Elements F and the new perturbations, new residuals for the normal places were determined and the orbit improved on this basis. When the corrections to the elements were put back into the equations of condition, the residuals were of the same order as prior to the solution. Miss Hansen concludes a normal place must be incor- rect. The equations of condition were again solved, with the last normal place omitted. The resulting Elements G represent the ten normal places in Aa cos 8 1".5 to +5".8 and A8 1".3 to +2".3. The normal place XI gives residuals Aa cos 8 1817".2, A8 1172".0, which indicates that it does not belong to Achilles. The special perturbations due to Jupiter for 1915 to 1920 are pub- lished by Miss Hansen. 17 They were computed from Elements G. These elements are published and utilized in Kleine Planeten since 1920. In Ark. for Math. Bd. 4 Nr. 20 Linders developed the approxi- mate theory for planets near the libration points and gives the prin- cipal perturbations of the elements of 588. Cf. Heinrich V. J. S. 1913. REFERENCES 1 A. N. vol. 170, p. 353. 2 A. N. vol. 171, p. 11. 8 A. N. vol. 171, p. 127. 4 A. N. vol. 171, p. 213. 5 Meddelanden fran Lunds Observa- torium, No. 18. Nat. Rund. vol. 21, p. 485^86. 7 Observatory, vol. 29, p. 352-355. Pop. Ast. vol. 14, p. 472-475. 8 H. u. E. vol. 18, p. 517-521. Ciel et Terre. vol. 27, p. 161-164. 10 A. N. vol. 174, p. 45-48. 11 A. N. vol. 174, p. 175. "A. N. vol. 180, p. 295. 13 B. J. 1914, p. 30. 14 L. O. Pub. vol. vii, p. 455. 15 In Manuscript (Berkeley). 18 Pub. og mindre Meddelelser fra Kobenhavns Obs. No. 37. 17 A. N. vol. 208, p. 345. TABLE U. Elements (688) Achilles (1906 TG.) Letter Epoch M. T. M. Q i A 1906 Mar 5 5 Berlin (u) 186 15 '9 316 1/2 11 43/2 B 1906 Feb 22 5 Berlin / 48 57 24 o / 120 25 50 o / 315 34 7 Of 10 20 53 C 1906 Feb. 22.5 Berlin 43 45 37 (129 24 11 315 31 7 10 16 36 D 1907 Apr 15 5 Berlin . . 80 18 12 \129 24 10 125 37 50 315 31 58 315 36 2 10 16 36 10 18 25 E F G 1907 Apr. 15.66... 1907 May 28.0.... 1907 May 28.0 Greenwich. . Berlin Berlin 80 23 50 82 54 47 84 3 2 125 25 21 127 7 10 125 36 22 315 35 45 315 34 26 315 35 59 10 18 45 10 17 53 10 18 14 56 CELESTIAL MECHANICS: LEUSCHNER TABLE UEkments(588) Achilles (1906 TG.)- Continued Letter Epoch M. T. )/V W,63. A preliminary orbit, Elements A, was computed by Stromgren 2 based Afi on observations from February 10 to April 16. These elements gave the following residuals: 1907 AA A Feb. 10 1".0 +2".2 April 19 +2 .8 +8 .3 He reports that Hector is another planet with mean motion nearly equal to that of Jupiter. Since the perturbations of Jupiter are small and the perturbations of the other planets will be ineffective for a long time, this planet will remain in the neighbourhood of the libration point for a long time. An observation by Palisa February 29, 1908, gives a correction to f7, Jl3 the ephemeris 3 based on Elements A as follows: Aa= 37 s A8=+6'.3. r 0,317 An ephemeris is published by Stromgren 4 for the opposition of 1909 based on Elements A. In preparation for the ephemeris of 1911, Stromgren, 5 assisted by J. Fischer-Petersen, computes a new set of Elements B, based on nine normal places (oppositions 1907, 1908, 1909), taking into account the perturbations due to Jupiter and Saturn. These elements represent the normal places, AacosS between +0'.30 and .17, and AS between 0'.04 and +OM6. The planet was next observed in July 1911. The comparison be- tween observation and ephemeris (from Elements B) was unsatisfac- tory. Stromgren, assisted by Ruben Andersen, 6 again investigated the orbit based on observations from 1907 to 1911. For this purpose Elements A were used to compute the residuals for five observations, July 4 to 16, 1911. A tenth normal place was formed from these five places with the following residuals Aacos8 = 52' 23" .9 AS = 11' 36". 6. A weight of five (number of observations) was given to this normal place. Combining this normal place with the solution that led to Elements B, a new set of elements was derived, Elements C. With these elements the special perturbations, due to Jupiter and Saturn, were computed by the method of Encke, for 1906 to 1912. The observations for 1907 and 1908 were then computed without taking into account the perturbations (reported as being small) and the obser- vations of 1909 and 1911 were represented with perturbations. From these representations ten new normal places were formed and new Elements D were obtained from a least square solution. This set CELESTIAL MECHANICS: LEUSCHNER 61 represents the normal places, AaCOsS between 7".l and +3".8, and AS between 1".6 and +3".0. With Elements D and special perturbations, the maximum residuals for opposition of 1912 were Aacos8 = +2M2 and A8 = +26".6. Stromgren, assisted by Julie M. Vinter Hansen, 7 again investigated the orbit of Hektor on the basis of thirteen normal places (1907 to 1912). A least square solution led to Elements E. At the conclusion of this work, 7 osculating elements were computed for the epoch of each year, 1907 to 1913, based on Elements E and the special perturbations pub- lished in the same reference. The ephemeris for 1913 was published in A. N., vol. 198, p. 367. In A. N., vol. 200, pp. 79-82, Stromgren publishes the comparisons between the observations and the ephemeris, which was based on Elements E and the perturbations due to Jupiter and Saturn, for 1913 and 1914. The maximum residual for 1913 for Aacos8=+0 8 .53 and for A8=+7".l. For 1914 the maximum residual for AacosS=+0 3 .51 and AS=+H"-6- Stromgren regards these residuals as satisfactory. As an application of Leuschner's satellite method, S. Einarsson 3 computed a preliminary orbit of Hector based on observations 1907, February 10, March 11, and April 16, Elements F. These elements represented an observation of May 2, 1908, as follows: Aacos8 = 4' 32".4 A8=+3' 31".9. With Elements F, the special perturba- tions by Encke's method were computed for the 1907 opposition. New elements were then computed from observations 1907, February 10, March 11, and a normal place from observations April 12, 16, and 19, by a differential correction of Elements F. These new Elements G represented the opposition of 1908, May 2, as follows: Aa + 30", AS -^. 95", and for 1909, April 17, Aa + 6M; AS 6'.7. The application of Leuschner's method thus yielded far better results than the ordinary method followed by Stromgren: 1908, Aa= 9' 15" AS=+6' 18". No further work was done on this planet by Einarsson since Strom- gren and his colleagues had already made extensive investigations. An ephemeris for 1915 is published by J. Fischer-Petersen, 9 using AN. Elements E and taking into account perturbations due to Jupiter and Saturn. The elements and ephemeris for 1916 10 are based on Stromgren's KPJ; Elements E with special perturbations by Jupiter and Saturn brought forward. Elements and ephemerides are given in Kleine Planeten for each year to 1921. No further comment is published regarding the elements. An ephemeris is published by M. Henri Blondel 11 for the opposition M of 1921 which has been corrected on the basis of an observation at 62 CELESTIAL MECHANICS: LEUSCHNER / J7 / Algiers 12 on May 6, 1921. This observation gave a correction to the ephemeris published in Kleine Planeten of +7 m -0 in right ascension and 80' in declination. In A. N., vol. 215, p. 249, A. Wilkens has published an article, "Uber die Sakularen Veranderungen der Grossen Achsen der Bahnen der Planeten der Jupiter Gruppe." In A. N., vol. 175, p. 89, Charlier gives a brief discussion on the orbits of the Trojan group, regarding their motion about the libration points. In A. N., vol. 206, p. 235, A. Koref outlines his investigation regard- ing the motion of Hector. His preliminary work is based on eighteen normal places (1907 to 1914). The investigation will be completed when observations of 1918 and 1919 are available. REFERENCES 'A. N. vol. 174, p. 63. a A. N. vol. 175, p. 14. ' A. N. vol. 177, p. 123. 4 A. N. vol. 180, p. 327. 6 Publikationer og mindre Meddel- elser fra Kobenhavns Observatorium No. 6. A. N. vol. 188, p. 395. * Publikationer og mindre Meddel- elser fra Kobenhavns Observatorium No. 8. 7 Publikationer og mindre Meddel- elser fra Kobenhavns Observatorium No. 12. 8 In manuscript (Berkeley). 9 A. N. vol. 201, p. 335. B. A. J. 1917. 10 Eph. der Kleinen Planeten, 1916, p. 17, p. 77, p. 93, p. 97. "Cir. O. M. No. 480 (1921). "Cir. 0. M. No. 171, second series. TABLE 16. Elements (624) Hector (1907 XM) Letter Epoch M.T. M M i A 1907 Feb. 10.0 Berlin 335 47 12 183 51 52 341 58 25 18 7 17 B 1907 Feb. 10.0 Berlin 343 51 43 175 6 42 341 58 57 18 8 34 c 1907 Feb 10 Berlin 345 38 38 173 5 26 341 59 47 18 9 13 D E F 1907 Feb 10.0 1907 Feb. 10.0 1907 Mar. 11.36 Berlin Berlin Greenwich . . 343 40 12 343 48 55 341 56 43 175 19 175 9 30 179 46 25 341 59 18 341 59 15 341 57 32 18 8 50 18 8 45 18 8 05 G 1907 Mar. 11.36 Greenwich . . 348 27 23 172 44 30 341 57 14 18 8 19 Letter Epoch M.T.

u Equinox Authority t * , A B 1911 Aug. 17.5 1911 Aug 18 5 Berlin Berlin 5 28 46 5 5 17 753.940 754 565 1911.0 1911 Hopfner Stracke C 1911 Aug. 22.4 Berlin 5 23 38 753.7233 1911.0 Neubauer 68 CELESTIAL MECHANICS: LEUSCHNER (718) ERIDA, 1911 MS. Discovered by Palisa 1 at Vienna, 1911, September 29. Preliminary Elements A, and an ephemeris based on observations 1911, September 29, October 13, and October 28, are published by Cohn. 2 These elements were used by the B. J. 1915 and 1916. New Elements B were obtained by Strehlow 3 from observations 1914, February 28, March 18, and March 29. The method of the variation of the distances was utilized so that observations of 1911 and 1914 were well represented. A correction of +1".5 to /A was applied in order to represent observa- tions of 1904 for (1904, OD) which is supposed to be identical to (718) Erida. As a test of Leuschner's Short Method for computing orbits, 4 Mundt 5 has published two sets of elements C and D, which are based on the observations used by Cohn. 2 For the purpose of comparison, ephemeris places were computed from Cohn's Elements A, Strehlow's Elements B, and Mundt's Ele- ments C. 1917 G. M. T. a 8 Oct. 22.5 2 h 56 m 3 +17 09' Elements C. Oct. 22.5 255.9 +17 07 Elements B. Oct. 22.5 253.5 +1656 Elements A. Dec. 1.5 225.3 +15 39 Elements C. Dec. 1.5 225.1 +15 36 Elements B. Dec. 1.5 222.9 +15 24 Elements A. -Ne comparison with observations in 1917 has been made. Strehlow's Elements B have been published and utilized by B. J. (Kleine Planeten) since 1915. An observation by Palisa 6 on 1919, January 6, gave corrections to the ephemeris as follows: Aa l m .7 AS 3'. The object of Mundt 's work was to test the possibility of deriving by properly chosen methods as satisfactory elements from a single opposition as are ordinarily obtained from at least two oppositions. This result was realized in this case. His work was duplicated by Miss Easton 5 on a slightly different plan of removing the residuals. Elements D. REFERENCES 1 A. N. vol. 189, p. 295. 4 L. O. Pub. vol. vii. 3 A. N. vol. 192, p. 421-423. 6 L. O. Bull. No. 302. 'B. J. 1917, p. 36 and 106. 'E. Z. der A. N. 1919, No. 561. CELESTIAL MECHANICS: LEUSCHNER TABLE 19. Elements (718) Erida (1911 MS) Letter Epoch M.T. M H a i A 1911 Sept. 29.5 Berlin 149 40 169 56 47 39 22 47 7 3 55 B C 1914 Apr. 1.5 1911 Oct. 13.4 Berlin Berlin 320 18 15 156 34 10 168 8 30 166 36 12 39 44 16 39 42 41 6 58 13 6 59 5 D 1911 Oct. 13.4 Berlin 156 03 25 166 55 59 39 43 51 6 58 49 Letter Epoch M.T. V M Equinox Authority Of* r A 1911 Sept. 29.5 Berlin.... 12 5 35 664.65 1911.0 F. Cohn B 1914 Apr. 1.5 Berlin.... 11 28 39 664.412 1910.0 Strehlow C 1911 Oct. 13.4 Berlin.... 11 19 7 663.865 1911.0 C. Mundt D 1911 Oct. 13.4 Berlin.... 11 20 11 663.769 1911.0 Miss E. J. Easton 70 CELESTIAL MECHANICS: LEUSCHNER (884) PRIAMUS, 1917 CQ. Discovered by Wolf 1 at Heidelberg on September 22, 1917. Wilkens outlines 2 his preliminary investigation regarding the motion of this fifth member of the Trojan group. For this purpose he utilizes preliminary Elements A, by Berberich. The libration point is 60 behind Jupiter and the planet oscillates about this point in approximately 150 years. He states that /* varies between 294".27 and 303".99. He also points out that his succeeding work will show that Priamus and Patroclus are diametrically opposite in the small libration ellipse. Klose applies 3 Wilkens' method 4 for taking into account the prin- cipal perturbations of Jupiter, to Priamus. In this method the prin- cipal perturbations by Jupiter are accounted for by centering in the Sun the mass of the Sun-Jupiter system. For this study Klose utilizes Berberich 's Elements A, bringing them up to mean equinox 1925.0, Elements B, and compares the coordinates computed from Elements B plus special perturbations with those computed from Elements C, which were derived by Wilkens' method. The differ- ence in the representation by the two sets of elements for observa- tions from 1917 October 14 to 1918 December 28, did not exceed O s .5 in right ascension and 2" in declination. Klose concludes that Wil- kens' method is applicable beyond this short period for immediate Ephemeris purposes. As an example of his method 5 of integrating the differential equa- tions for the perturbations in the coordinates for planets of the Jupiter group, Wilkens 6 gives a numerical application for Priamus. His results are similar to those of Klose. Klose 7 publishes a further comparison between the usual method of special perturbations due to Jupiter and Wilkens' method. For this purpose he compares results gotten from Elements B and a new set of Elements D, derived by Wilkens' method. He then brings up Elements B with perturbations due to Jupiter up to epoch 1918 (Elements E), and Elements D are brought forward to epoch 1918 by Wilkens' method. An ephemeris is published for the opposition 1919 for which the per- turbations of Saturn are also taken into account. In an article on "Bemerkenswerte Eigenschaften der Bahnen der Planeten der Jupitergruppe," Wilkens 8 points out that the ascending nodes of the six Trojan planets lie with one exception, (Patroclus), in the same quadrant. He also forms a mean value of the ascending nodes from certain planets of the group and compares the individual values with these mean values. He also refers to his article 9 regarding the time of maximum and minimum for the mean motion of Priamus CELESTIAL MECHANICS: LEUSCHNER 71 and Patroclus. He finds the maximum of Priamus (1986.8) takes place when the planet is on a line with and between the point of oscillation and the Sun, and the minimum for Patroclus (1985.8) when the planet is on the line with and beyond the point of oscillation and the Sun. This verified his previous conclusion 2 that the two planets are diametrically opposite in their paths of oscillation. A set of elements and an ephemeris are published in Kleine Planeten for 1920. The elements are probably Elements B, brought up to epoch 1925. An ephemeris is published in Kleine Planeten for 1921. In A. N. Vol. 215, No. 5147, Wilkens outlines his investigation regarding the secular variations of the major axes of the orbits for the Trojan group. REFERENCES 'A. N. vol. 205, p. 141. M. N. 78, p. 289. 2 A. N. vol. 207, p. 9. 8 A. N. vol. 207, p. 183. 4 A. N. vol. 205, No. 4906. 'A. N. vol. 206, No. 4937. A. N. vol.208, No. 4984. 7 A. N. vol. 209, No. 5016. 8 A. N. vol. 215, No. 5147. 9 A. N. vol. 206, No. 4945. TABLE 20. Elements (884) Priamus (1917 CQ.) Letter Epoch M.T. M Q i A 1917 Sept. 24 . 5 Greenwich . . 83 18 55 329 32 38 300 41 27 8 51 26 B 1917 Sept. 24 5 Greenwich. . 83 18 55 329 32 18 300 48 28 8 51 28 C D 1917 Sept. 24.5 1917 Sept. 24.5 Greenwich.. Greenwich . . 83 46 41 82 22 47 329 4 44 329 51 19 300 48 28 300 49 27 8 51 28 8 51 24 E 1918 Oct. 29.5 Greenwich . . 115 11 59 329 45 49 300 49 27 8 51 24 F 1918 Oct. 29.5 Greenwich. . 115 33 19 329 24 42 300 49 27 8 51 24 Letter Epoch M.T. V ft Equinox Authority A 1917 Sept. 24.5 Greenwich . . o / 6 46 53 * 294.427 1917.0 Berberich B C 1917 Sept. 24.5 1917 Sept. 24.5 Greenwich . . Greenwich . . 6 46 53 6 46 54 294.427 294.989 1925.0 1925.0 Berberich Berberich-Klose D 1917 Sept. 24 5. . . . Greenwich . . 7 5 53 294.427 1925.0 Klose E 1918 Oct. 29.5 Greenwich . . 7 5 59 294 5850 1925.0 Klose F 1918 Oct. 29.5 Greenwich. . 7 7 34 295.0965 1925.0 Klose 72 CELESTIAL MECHANICS: LEUSCHNER TABLE 21. References for Observations of (884) Priamus (1917 CQ.) Date Place Reference 1917... September 22.. . Heidelberg f A.N. 205 p. 141, M.N. 78 p. 289* \E.Z. 1917 No. 535 September 23. . . Heidelberg A.N. 205 p. 141; E.Z. 1917 No. 535 September 24 ... Heidelberg A.N. 205 p. 141; E.Z. 1917 No. 535 September 25... Heidelberg A.N. 205 p. 141; E.Z 1917 No. 535 September 26. .. Heidelberg A.N. 205 p. 141; E.Z. 1917 No. 535 October 6... Wien A.N. 207 p. 149 October 11... Wien A.N. 207 p. 149 October 13 ... Wien A.N. 207 p. 149 October 16. .. Wien A.N. 207 p. 149 October 21 ... Heidelberg A.N. 205 p. 239 E.Z. 1917 No. 537 November 8 ... Heidelberg A.N. 205 p. 239 E.Z. 1918 No. 537 A.N. 206 p. 63 December 4 ... Heidelberg A.N. 205 p. 279 E.Z. 1918 No. 538 A.N. 206 p. 63 1917... December 4 ... Bergedorf . . . . A.N. 208 p. 39 E.Z. 1918 No. 559 1918... January 2 ... Bergedorf . . . . A N. 208 p. 39 E.Z. 1918 No. 559 January 3 ... Heidelberg /A.N. 206 p. 63 \A.N. 206 p. 15 January 14 ... Bergedorf. . . . A.N. 208 p. 39 E.Z. 1918 No. 559 October 5 ... Heidelberg A.N. 207 p. 239 E.Z. 1918 No. 555 October 30. .. Heidelberg. . . . A.N. 207 p. 283 E.Z. 1918 No. 557 November 23 ... Heidelberg f A.N. 208 p. 13 E.Z. 1918 No. 558 \A.N. 208 p. 167 1919... October 21 ... Heidelberg B.Z. 1919 No. 13 Vol. 1 1921... January 15... Heidelberg B.Z. 1921 No. 3 Vol. 3 'Discovery date. CELESTIAL MECHANICS: LEUSCHNER 73 (911) AGAMEMNON, 1919 FD. This planet was discovered by Reinmuth 1 at Heidelberg on March 19, 1919. The nature of its motion (Jupiter group) was noted about the same time by Palisa and Berberich. 2 The preliminary elements A by Berberich are published in A. N. Vol. 208, p. 332. A comparison between computation and observation 3 for 1919 May 20, gives Aa = 4 s AS = 2'. Elements A are brought forward to mean equinox of 1925.0 and with an ephemeris are published in Kleine Planeten for 1920, p. 25 and p. 47. An observation 4 on March 11, 1920, gives a correction to the ephemeris of +l m -5 in right ascension and 18' in declination. An ephemeris for 1921 is given in Kleine Planeten for 1921, p. 20. REFERENCES 1 A. N. vol. 208, p. 231. Royal A. S. 210, p. 241. vol. 80, p. 405, 406. 8 A. N. vol. 210, p. 244. 2 A. N. vol. 208, p. 331. A. N. vol. 4 B-Z der A. N. No. 13, 1920. Elements (911) Agamemnon (1919 FD.) Epoch M.T. M. cu i A 1919 Mar. 19.5 Grw. 8848'19" 7846'08" 33655'10" 2156'50" < /A Equinox Authority Remarks 455'43" 303".190 1919.0 Berberich. Preliminary orbit. TABLE 22. References for Observations of (911) Agamemnon (1919 FD) Date Place Reference 1919... March 19 April 2 Konigstuhl. . . Konigstuhl . . . A.N. 208 p. 231* A.N. 208 p. 231 May 20 Konigstuhl . . . A.N. 208 p. 347 April 5 Wien A.N. 211 p. 430 April 6 Wien A.N. 211 p. 430 April 19 . ... 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