THE SCIENTIFIC PAPEKS OF JOHN COUCH ADAMS. Hon&on: C. J. CLAY AND SONS, CAMBRIDGE UNIVERSITY PRESS WAREHOUSE, AVE MAEIA LANE, lasgoto : 263, ARGYLE STEEET. F. A. BROCKHAUS. JletD gorfe: THE MACMILLAN CO. THE SCIENTIFIC PAPEES OF JOHN COUCH ADAMS, M.A., Sc.D., D.C.L., LL.D., F.R.S., LATE LOWNDEAN PROFESSOR OF ASTRONOMY AND GEOMETRY IN THE UNIVERSITY OF CAMBRIDGE. VOL. I. EDITED BY WILLIAM GKYLLS ADAMS, Sc.D., F.R.S. WITH A MEMOIR BY J. W. L. GLAISHER, Sc.D., F.R.S. CAMBRIDGE : AT THE UNIVERSITY PRESS. 1896 [All Bights reserved.] CAMBRIDGE : PRINTED BY J. AND C. F. CLAY, AT THE UNIVERSITY PRESS. PKEFACE TO VOLUME I. THE present volume of the Collected Works of the late Professor JOHN COUCH ADAMS contains all the original papers which were published by him during his lifetime, extending from 1844 (when he was 25 years of age) to 1890. They consist of about 50 Astronomical Papers which were for the most part printed in the Memoirs or Monthly Notices of the Royal Astronomical Society and 11 Papers on Pure Mathematics. Besides these there are many papers on various branches of Astronomy which were left in an incomplete state among Professor Adams' manuscripts. These are being prepared for publication by Professor Sampson. There is also a great quantity of unpublished work in an incomplete state on Legendre's and Laplace's Coefficients and on Terrestrial Magnetism which was taken up from time to time extending over a period of 40 years, but no part of which has been published except a short paper (No. 60) on Legendre's Coefficients. It is hoped that a considerable portion of this unpublished work may shortly be brought into shape for publication, and that it will form the continuation of these Collected Works. Since the Appendix to Paper 19 (p. 124 of this volume) was printed, more exact expressions of the coefficients for Jupiter's Satellites II, III and IV have been found among Professor Adams' unpublished papers. Thus in forming the Tables for Satellite II, in addition to the terms 2 s '5 sin (II A n ) 1 s "5 sin (II A m ) given on p. 118 of this volume, another term + S- 127 sin (II A IV ) was employed in the calculation for the vi PREFACE TO VOLUME I. period 1890 1900. In place of the expressions given on p. 124 for this period, 1890 1900, the more exact values of the coefficients are For Satellite IT +0 8 756 sin (5ii-2u - 177), Satellite III + 2'233 sin (5u-2u - 177), Satellite IV +12'33 sin (5u-2u - 177). The full paper on the attraction of an indefinitely thin ellipsoidal shell on an external point, which was given before the Cambridge Philo- sophical Society, has been reproduced (see p. 414 of this volume) by the aid of the notes taken by Professor Greenhill at Professor Adams' lectures on the Figure of the Earth. In 1876 a translation of the paper on the discovery of the planet Neptune was published in Liouville's Journal de Mathematiques with the addition of an Appendix by Professor Adams which forms the seventh paper of this Volume. In March 1867, a paper " Sur les e"toiles filantes de Novembre " was published in the Paris Acad. Sci. Compt. Rend., LXIV. which was also communicated to, but not published by, the Cambridge Philosophical Society. A paper on the lunar inequalities due to the ellipticity of the Earth was overlooked when the papers on Astronomy were being printed : these papers are printed at the end of this Volume. The biographical notice prefixed to this volume has been written by Dr J. W. L. Glaisher. My thanks are due to Mr W. H. Wesley, the Assistant Secretary of the Royal Astronomical Society, for kind help which he has given me. W. GRYLLS ADAMS. KINO'S COLLEGE, LONDON. Oct. 8th, 1896. CONTENTS. 1. Results of calculations of the elements of an exterior planet, which mil account for the observed irregularities in the motion of Uranus .......... 1 Monthly Notices of the Royal Astronomical Society, Vol. VII. (1846), pp. 149152 2. An explanation on the observed irregularities in- the motion of Uranus, on the hypothesis of disturbances caused by a more distant planet; with a determination of the mass, orbit, and position of the disturbing body ...... 6 Memoirs of the Royal Astronomical Society, Vol. xvi. (1847), pp. 427 460. Appendix to Nautical Almanac for 1851, pp. 265293 3. Corrected elements of Neptune . . . . . . . 54 Monthly Notices of the Royal Astronomical Society (1847), Vol. vn., pp. 244245 4. New elements of Neptune . . . . . . . . 57 Monthly Notices of the Royal Astronomical Society (May, 1847), Vol. vn., pp. 268269 5. Ephemeris of Neptune and meridian observations . . . 58 Astronomische Nachrichten. xxvi. (1847). No. 604, pp. 51, 52. No. 616, pp. 241244 6. The mass of Uranus ......... 62 Monthly Notices of the Royal Astronomical Society, Vol. ix. (1849), pp. 159160 A. b viii CONTENTS. PAGE 7. Appendix on the discovery of Neptune . . . 63 Liouville's Journal de Mathdmatiques, New Series, Tome II. (1876), pp. 8386 8. Elements of the comet of Faye .... 66 Monthly Notices of the Royal Astronomical Society, Vol. vi. (1844), pp. 2021 9. The orbit of the new comet .... 67 Times, October 15, 1844 10. The relative position of the two heads of Biela's comet 68 Monthly Notices of the Royal Astronomical Society (March 14, 1846), p. 83 11. On the application of graphical methods to the solution of certain astronomical problems, and in particular to the determination of the perturbations of planets and comets . 70 Report of the British Association for 1849, (pt. 2), p. 1 12. Elements of Comet II. 1854 71 Monthly Notices of the Royal Astronomical Society, Vol. xiv. (1854), pp. 251252 13. Observations of Comet II. 1861 .... 72 Monthly Notices of the Royal Astronomical Society, Vol. xxn. (1862) and Astronomische Nachrichten, LVII. (1862), col. 235 240 14. On the orbit of y Virginis ...... 78 JEdes Hartwellianse, letter to Admiral Smythe, June, 1851, pp. 340 342 15. On the total eclipse of the Sun, 28 July 1851, as seen at Frederiksvaern ........ 80 Memoirs of the Royal Astronomical Society, Vol. xxi. (1852), pp. 101107 16. On an important error in Bouvard's tables of Saturn . . 87 Memoirs of the Royal Astronomical Society (1849), Vol. xvn., pp. 1 2, and Monthly Notices of the Royal Astronomical Society (1847), Vol. vii., pp. 251252 CONTENTS. PAliK 17. On new tables of the Moon's parallax . . . . . . 89 Monthly Notices of the Royal Astronomical Society (1853), Vol. MIL, pp. 175 180, and Nautical Almanac for 1856, App. pp. 35 43 18. On the corrections to be applied to Burckhardt's and Plana's parallax of the Moon, expressed in terms of the mean arguments .......... 108 Monthly Notices of the Royal Astronomical Society, Vol. xin. (1853), pp. 226264 19. Continuation of tables I. and III. of Damoiseau's tables of Jupiter's satellites ........... 113 Appendix to the Nautical Almanac for 1881, pp. 15 23 20. On Professor Challis's new theorems relating to the Moon's orbit 129 Philosophical Magazine, Vol. vin. (1854), pp. 2736 21. On the secular variation of the Moon's mean motion . . .140 Philosophical Transactions of the Royal Society, Vol. cxmi. (1853), pp. 397 406. Abstract of same, Proceedings of the Royal Society, June 16, 1853, and Monthly Notices of the Royal Astronomical Society, Vol. xiv. (1853), pp. 5962 22. On the secular variation of the eccentricity and inclination of the Moon's orbit . . . . . . , . .158 Monthly Notices of the Royal Astronomical Society (1859), Vol. xix., pp. 206 208 23. Reply to various objections against the theory of the secular acceleration of the Moon's mean motion (with postscript) . 160 Monthly Notices of the Royal Astronomical Society (1860), Vol. xx., pp. 225240 and pp. 279280 24. On the motion of the Moon's node in the case when the orbits of the Sun and Moon are supposed to have no eccentricities, and when their mutual inclination is supposed to be in- definitely small ......... 181 Monthly Notices of the Royal Astronomical Society, Vol. xxxvin. (1877), pp. 4349 62 X CONTENTS. PAGE 25. Note on a remarkable property of the analytical expression for the constant term in the reciprocal of the Moon's radius vector ...... Monthly Notices of the Royal Astronomical Society, Vol. xxxvni. (1878), pp. 460472 26. Note on Sir George Airys investigation of the theoretical value of the acceleration of the Moons mean motion 205 Monthly Notices of the Royal Astronomical Society (1880), Vol. XL., pp. 411415 27. Investigation of the secular acceleration of the Moon's mean motion, caused by the secular change in the eccentricity of the Earth's orbit .... . 211 Monthly Notices of the Royal Astronomical Society, Vol. XL. (1880), pp. 472482 28. Note on the constant of lunar parallax Monthly Notices of the Royal Astronomical Society, Vol. XL. (1880), pp. 482488 29. Note on the inequality in the Moons latitude which is due to the secular change of the plane of the ecliptic Monthly Notices of the Royal Astronomical Society, Vol. XLI. (1881), pp. 385403 30. Note on Delaunays expression for the Moon's parallax 253 Monthly Notices of the Royal Astronomical Society, Vol. XLIII. (1883), pp. 397401 31. Remarks on Mr Stone's explanation of the large and increasing errors of Hansens lunar tables by means of a supposed change in the unit of mean solar time . . . . 259 Monthly Notices of the Royal Astronomical Society, Vol. XLIV. (1883), pp. 4347 32. Remarks on Sir George Airys numerical lunar theory . 264 Monthly Notices of the Royal Astronomical Society, Vol. XLVIII. (1888), pp. 319322 33. On the meteoric shower of November, 1866 . . . . 268 Proceedings of the Cambridge Philosophical Society, Vol. n., p. 60 CONTENTS. xi PAGE 34. On the orbit of the November meteors . . . . . '. 269 Monthly Notices of the Royal Astronomical Society, Vol. xxvu. (1867), pp. 247252 35. Note on the ellipticity of Mars, and its effect on the motion of the satellites . . . . . . . . .275 Monthly Notices of the Royal Astronomical Society, Vol. XL. (1879), pp. 1013 36. Note on William Ball's observations of Saturn . . . . 279 Monthly Notices of the Royal Astronomical Society, Vol. XLIII. (1883), pp. 9397 37. On the change in the adopted unit of time ..... 286 Monthly Notices of the Royal Astronomical Society, Vol. XLIV. (1884), pp. 8284 38. On Neivton's solution of Kepler's problem ..... 289 Monthly Notices of the Royal Astronomical Society, Vol. XLIII. (1882), pp. 43 49 39. Note on Dr Morrison's paper (on Kepler's problem) . . . 297 Monthly Notices of the Royal Astronomical Society, Vol. XLIII. (1883), pp. 365368 40. On Newton's Theory of astronomical refraction, and on his explanation of the motion of the Moon's apogee . . . 302 British Association Report for 1884, p. 645 41. On the general values of the obliquity of the ecliptic, and of the precession and inclination of the equator to the invariable plane, taking into account terms of the second order . . 303 The Observatory, No. 109 (1886), pp. 150154 42. Address on presenting the gold medal of the Royal Astronomical Society to M. Peters 309 Memoirs of the Royal Astronomical Society, Vol. xxi. (1852), pp. 211223 43. Address on presenting the gold medal of the Royal Astronomical Society to Mr Hind . . . . . . .321 Memoirs of the Royal Astronomical Society, Vol. xxn. (1853), pp. 254261 xii CONTENTS. PAGE 44. Address on presenting the gold medal of the Royal Astronomical Society to M. Charles Delaunay . . . . . .328 Monthly Notices of the Royal Astronomical Society, Vol. xxx. (1870), pp. 122132 45. Address on presenting the gold medal of the Royal Astronomical Society to Professor H. D' Arrest . . . . . .341 Monthly Notices of the Royal Astronomical Society, Vol. xxxv. (1875), pp. 265275 46. Address on presenting the gold medal of the Royal Astronomical Society to M. le Verrier . . . . . . .355 Monthly Notices of the Royal Astronomical Society, Vol. xxxvi. (1876), pp. 232246 47. Astronomical observations made at the Observatory of Cambridge, under the superintendence of Professor Adams . . .374 Introduction to Vol. xxi. (18611865) 48. On the mean places of 84 fundamental stars, as derived from the places given in the Greenwich catalogues for 1840 and 1845, when compared with those resulting from Bradley' s observations .......... 386 Appendix to Astronomical Observations made at the Cambridge Observatory, Vol. xxn. (1866 1869), pp. xxxiii Ixxxiv 49. Account of some trigonometrical operations to ascertain the difference of geographical position between the Observatory of St John's College and the Cambridge Observatory . . 403 Cambridge Philosophical Society's Proceedings, Vol. I. (1852), pp. 119120 50. Proof of the principle of Amsler's planimeter .... 405 Cambridge Philosophical Society's Proceedings, Vol. I. (1857), p. 192 51. Note on the resolution of x n + n - 2 cos no. into factors . . 407 X. Cambridge Philosophical Society's Transactions, Vol. XL, Part 2 (1868), pp. 444 -446 CONTENTS. xiii PAGE 52. On a simple proof of Lambert's theorem . . . . .410 Report of the British Association for 1877, 2nd pt., pp. 15 18 53. On the attraction of an indefinitely thin shell bounded by two similar and similarly situated concentric ellipsoids on an external point .......... 414 Cambridge Philosophical Society's Proceedings, Vol. n. (1871), pp. 213215 54. On the calculation of the Bernoullian numbers from B K to S m . 426 Appendix I. to the Cambridge Observations, Vol. xxn. (1866 1869). App. I., pp. i xxxii 55. On some properties of Bernoulli's numbers . . . . .454 Communicated to the Cambridge Philosophical Society (1872). Vol. II., pp. 269270 On the calculation of Bernoulli's numbers . . . . .454 Crelle's Journal, Vol. 85, pp. 269272. Report of the British As- sociation for 1877, (Sect.), pp. 814 56. Note on the value of Euler's constant; likewise on the values of the Napierian logarithms of 2, 3, 5, 7, and 10, and of the modulus of common logarithms, all carried to 260 places of decimals .......... 459 Proceedings of the Royal Society, Vol. xxvn. (1878), pp. 8894 57. Supplementary note on the values of the Napierian logarithms of 2, 3, 5, 7, and 10, and of the modulus of common logarithms . 467 Proceedings of the Royal Society, Vol. XLII. (1886), pp. 2225 58. Note on Sir William Thomson's correction of the ordinary equilibrium theory of the Tides . . . . . .471 Report of the British Association for 1886, p. 541 59. On certain approximate formulce for calculating the trajectories of shot 473 Proceedings of the Royal Society, Vol. xxvi. (1877) and Nature, Vol. XLI. (1890), pp. 258262 xiv CONTENTS. PAGE 60. On the expression of the product of any two Legendre's coefficients by means of a semes of Legendre's coefficients 487 Proceedings of the Royal Society, No. 185 (1878), pp. 6371 61. Sur les etoiles filantes de Novembre . 497 Paris Acad. Sci. Compt. Rend. LXIV. (1867), pp. 651652 62. Lunar Inequalities due to the Ellipticity of the Earth 499 The Observatory, No. 108, (1886), pp. 118120 BIOGKAPHICAL NOTICE. JOHN COUCH ADAMS was born on June 5, 1819, at the farmhouse of Lidcot, seven miles from Launceston in Cornwall. His father, Thomas Adams, was a tenant farmer, and his ancestors for at least four generations had been tenant farmers in or near Laneast. His mother, whose maiden name was Tabitha Knill Grylls, possessed a small estate which was bequeathed to her by her aunt, Grace Couch. She had also inherited her uncle's library, and these books, which included some on astronomy, were Adams's early companions. He was the eldest of seven children. His brother Thomas, born April 28, 1821, was a missionary in Tonga and completed the translation of the Bible into the Tongan language : he died in 1885. His brother George, born November 5, 1823, assisted his father at Lidcot and became a farmer. His youngest brother, William Grylls Adams, born February 16, 1836, is the editor of this volume. He had three sisters who all died before him. From his mother, who belonged to a musical family, he inherited a correct ear and a love of music. At a village school in Laneast he made rapid progress, and with the schoolmaster, Mr R. C. Sleep, as his fellow student he was learning algebra before he was ten years old. At the age of twelve he went to a private school at Devonport, kept by the Rev. John Couch Grylls, a first cousin of his mother. He remained under Mr Grylls's tuition for several years, first at Devonport and afterwards at Saltash and Landulph, and received the usual school training in classics and mathematics. Astronomy had been his passion from very early boyhood, and at fourteen years of age he made copious notes and drew tiny maps of the constellations. He read with avidity all the astronomical books to which he could obtain access, and in particular he studied the astronomical articles in Rees's Cyclo- pcedia, which he met with in the library of the Devonport Mechanics' Institute, where he used to spend his spare time in reading astronomy and mathematics. In the same library he came across a copy of Vince's Fluxions, which was his first introduction to the higher mathematics. The intense interest which as a boy he felt in all astronomical questions is shown by the number of carefully written out manuscripts, belonging to this period, which exist among his papers, as well as by his letters to his parents and brothers. Some A. C xvi BIOGRAPHICAL NOTICE. of the manuscripts are copies from books, others contain calculations of his own. On October 17, 1835, he wrote from Landulph to his parents telling them that he had watched for the comet three weeks before without success, and that at last he had seen it: "you may conceive with what pleasure I viewed this, the first comet I had ever had a sight of, which at its visit 380 years ago threw all Europe into consternation, but which now affords the highest pleasure to astronomers by proving the accuracy of their calculations and predictions." The annular eclipse of the sun of May 15, 1836, interested him greatly and on May 13 he wrote from Stoke a long letter to his brother Thomas at Lidcot in order to give him "a brief description of the large eclipse of the sun which will take place next Sunday." He proceeds "As the almanacs only give the time &c. to this eclipse for London and some other remarkable places, I have taken some pains to calculate it, and I herewith send you, what I believe has not been done for some time, a calculation of this eclipse for the meridian and latitude of Litcott." He finds that it will begin at 1 h. 28 m. p.m., that the greatest eclipse will be at 3 h. m. and that it will end at 4 h. 22 m., the digits eclipsed being 10. He also gives a diagram showing the eclipse as it will appear from Lidcot. At the conclusion of the letter, he adds " There will also happen next Thursday evening between 6 and 7 o'clock a remarkable conjunction of the Moon and the planets Jupiter and Venus, which I wish you would observe. These planets are now approaching each other and will then be very near, as also will the moon." This early calculation of an eclipse (the manuscript of which still exists) is especially interesting in connexion with the remarkable theoretical calculations which he was to undertake and carry out so successfully only a few years later. On April 24, 1837, he wrote from Stoke "I observed the eclipse last Thursday with a small spy- glass which I borrowed : the moon looked most delightful after the end of the eclipse. At the request of Mr Bate, a young man of my acquaintance, who reports for the Telegraph, I wrote next morning a few lines on the eclipse, which were inserted in the paper the following day.... Mr Richards, the editor of the Telegraph, tells me that my article on the eclipse has been copied into several of the London papers." He was also interested in practical astronomy, and there was long preserved in the home at Lidcot a simple instrument constructed by him, when very young, in order to determine the elevation of the sun. It consisted of a vertical circular card with graduated edge, from the centre of which a plumb bob was suspended. Two small square pieces of card, with a pin-hole in each, projected from the circular disc at right angles to its face at opposite ends of a diameter. The card was to be so placed that the sun shone through the pin-holes, and the elevation was read off on the circle. It is also re- membered that on the window sill at Lidcot he had made lines or notches to mark the positions of shadows at noon. He showed such signs of mathematical power that in 1837 the idea of his going to Cambridge was entertained. He accordingly entered St John's College, Cambridge, in October, 1839. During his undergraduate career he was invariably the first man of his year in the college examinations, and in 1843 he graduated as Senior Wrangler, being also first Smith's Prizeman. In the same year he was elected Fellow of his college. His attention was drawn to the irregularities in the motion of Uranus by reading Airy's report upon recent progress in astronomy in the Report of the British Asso- BIOGRAPHICAL NOTICE. xvii ciation for 1831-32 1 , and on July 3, 1841, he made the following memorandum: "Formed a design at the beginning of this week of investigating, as soon as possible after taking my degree, the irregularities in the motion of Uranus which are yet unaccounted for, in order to find whether they may be attributed to the action of an undiscovered planet beyond it ; and, if possible, thence to determine the elements of its orbit &c. approximately, which would probably lead to its discovery." This memorandum was made at the beginning of his second long vacation, when he had just entered upon his twenty- third year 2 . In 1843, the year in which he took his B.A. degree, he attempted a first rough solution of the problem on the assumption that the orbit was a circle with a radius equal to twice the mean distance of Uranus from the Sun. The result showed that a good general agreement between theory and observation might be obtained In order to make the data employed more complete, application was made through Professor Challis, to Mr Airy, the Astronomer Royal, in February 1844, for the errors of the tabular geocentric longitudes of Uranus for 1818 1826, with the factors for reducing them to errors of heliocentric longitude. The Astronomer Royal at once supplied all the results of the Greenwich observations of Uranus from 1754 to 1830. Adams now undertook a new solution of the problem, taking into account the most important terms depending on the first power of the eccentricity of the orbit of the supposed disturbing planet, but retaining the same assumption as before with respect to the mean distance. In September, 1845, he gave to Professor Challis a paper containing numerical values of the mean longitude at a given epoch, longitude of perihelion, eccentricity of orbit, mass, and geocentric longi- tude for September 30, of the assumed planet. On September 22, 1845, Challis wrote a letter of introduction to the Astronomer Royal beginning, "My friend Mr Adams, who will probably deliver this note to you, has completed his calculations respecting the perturbation of the orbit of Uranus by a supposed ulterior planet, and has arrived at results which he would be glad to communicate to you, if you could spare him a few moments of your valuable time." Adams called at the Royal Observatory, Greenwich, in September, but the Astronomer Royal was absent in France. In the following month, on October 21, 1845, Adams called again at the Royal Observatory, and not being suc- cessful in seeing the Astronomer Royal, left a paper giving the following values of the mass and orbit of the new planet: Mean distance (assumed nearly in accordance with Bode's law) 38'4 Mean sidereal motion in 365'25 days 1 30' 9" Mean longitude, 1st October, 1845 323 34' Longitude of perihelion 315 55' Eccentricity 0'1610 Mass (that of the Sun being unity) 0'0001656 The paper which he left on this occasion also contained a list of the residual 1 This report does not contain any reference to the elliptic orbit, and that Bouvard was therefore obliged possibility of the irregularities being due to an undis- to reject the ancient observations entirely (Report, p. 154). covered exterior planet. It is merely mentioned that it a The original memorandum, written by itself on a seems impossible to unite all the observations in one slip of paper, is reproduced in facsimile facing p. liv. c2 xvili BIOGRAPHICAL NOTICE. errors of the mean longitude of Uranus, after taking account of the disturbing effect of the new planet, the errors being small except in the case of Flamsteed's observation of 1690 '. On November 10, 1845, Le Verrier presented to the French Academy an elaborate investigation of the perturbations of Uranus produced by Jupiter and Saturn, in which he pointed out several small inequalities which had previously been neglected. After taking these into account he still found that the theory was quite incapable of explaining the observed irregularities of the motion of Uranus. On June 1, 1846, Le Verrier presented to the French Academy his second memoir on the theory of Uranus. After reducing afresh nearly all the existing observations, he came to the conclusion that there was no other possible explanation of the discordances except thab of a disturbing planet exterior to Uranus. He investigated the elements of the orbit of such a planet, and assuming its mean distance to be double that of Uranus, and its orbit to be in the plane of the ecliptic, he gave as the most probable result that the value of the true longitude of the disturbing body for January 1, 1847 was about 325, and that it was not likely that this place was in error by so much as 10. Neither the elements of the orbit nor the mass of the planet were given. The position thus assigned by Le Verrier to the disturbing planet differed by only 1 from that given by Adams in the paper which he had left at the Royal Observatory more than seven months before. As will be mentioned subsequently, Le Verrier's third memoir, containing the elements of the orbit, was communicated to the French Academy on August 31, 1846. On July 9, 1846 the Astronomer Royal, who was then staying with Dean Peacock at Ely, wrote a letter to Challis suggesting that search should be made for the new planet with the Northumberland Equatorial at Cambridge, and offering to supply him with an assistant if he were unable himself to make the examination ; and on July 13 he transmitted to Challis a paper of suggestions with respect to the proposed sweep for the planet, which was to extend over a part of the heavens 30 long in the direction of the ecliptic, and 10 broad, having the theoretical place of the planet as its centre. On July 18, Challis, who had been absent from Cambridge, replied to these communications, stating that he had determined to sweep for the hypothetical planet himself, and that he should therefore not require the services of an assistant. The actual search for the planet was commenced by Challis with the Northumberland telescope on July 29, 1846, three weeks before the planet was in opposition, and the observations were continued steadily until September 29. The plan adopted was to make three sweeps over the whole zone, completing one sweep before commencing the next, and mapping the positions of the stars. When the observations were completed, a planet could be at once detected by its motion in the interval. For the first few nights the telescope was directed to the part of the zone in the immediate neighbourhood of the place indicated for the planet by theory. On September 2, in a letter to the Astronomer Royal, Challis said that he had lost no opportunity of searching for the planet, and that the nights being pretty good he had 1 A facsimile of this paper is given after p. liv. BIOGRAPHICAL NOTICE. taken a considerable number of observations, but that his progress was slow as he thought it right to include all stars to the 10-11 magnitude. He found that to scrutinise thoroughly, according to his plan, the proposed part of the heavens would require more observations than he could take in the year. On the same day Adams wrote to the Astronomer Royal a letter, the opening paragraphs of which are as follows: "In the investigation, the results of which I communicated to you last October, the mean distance of the supposed disturbing planet is assumed to be twice that of Uranus. Some assumption is necessary in the first instance, and Bode's law renders it probable that the above dis- tance is not very remote from the truth : but the investigation could scarcely be considered satisfactory while based on anything arbitrary ; and I therefore determined to repeat the calculation, making a different hypothesis as to the mean distance. The eccentricity also resulting from my former calculations was far too large to be probable; and I found that although the agreement between theory and observation continued very satisfactory down to 1840, the difference in subsequent years was becoming very sensible, and I hoped that these errors as well as the eccentricity might be diminished by taking a different mean distance. Not to make too violent a change, I assumed this distance to be less than the former value by about ^th part of the whole. The result is very satisfactory, and appears to show that, by still further diminishing the distance, the agreement be- tween the theory and the later observations may be rendered complete, and the eccentricity reduced at the same time to a very small quantity. The mass and the elements of the orbit of the supposed planet, which result from the two hypotheses, are as follows : Hypothesis I. Hypothesis II. Mean Longitude of Planet, 1st October, 1846 ... 325 8' 323 2' Longitude of Perihelion ............ 315 57' 299 11' Eccentricity ............... 0'16103 012062 Mass (that of Sun being 1) ......... 000016563 0'00015003." Adams also gave the errors of mean longitude, exhibiting the difference between theory and observation on the two hypotheses, and, after pointing out that the errors given by the Greenwich Observations of 1843 are very sensible on both hypotheses, he proceeds : " By comparing these errors it may be inferred that the agreement of theory and observation would be rendered very close by assuming =0'57, and the corresponding mean longitude on October 1, 1846, would be about 315 20', which I am inclined to think is not far from the truth. It is plain, also, that the eccentricity corresponding to this value of , would be very small." In consequence of the divergence of the results of the two hypotheses, Adams asked for two normal places near the oppositions of 1844 and 1845. In the Astronomer Royal's absence on the Continent, these were sent by Mr Main ; and on September 7 Adams wrote : " I hope by to-morrow to have obtained approximate values of the inclination and longitude of the node." Two days earlier, on August 31, 1846, Le Verrier had presented to the French XX BIOGRAPHICAL NOTICE. Academy his third paper on the motion of Uranus, in which he gave the following elements of the disturbing planet : Semi-axis Major ...... 36154 or = 0'531 V a, Periodic Time ..................... 217'387 Eccentricity ..................... 010761 Longitude of Perihelion ... ... ... ... ... 284 45' Mean Longitude, 1st January, 1847 ...... ... 318 47' Mass ... = * = 0-0001075 True Heliocentric Longitude, 1st January, 1847 ... ... 326 32' Distance from the Sun ...... ... ...... 33'06 and also comparisons between theory and observation. The paper also contained a detailed investigation, the object of which was to restrict as far as possible the limits within which the planet should be sought. Le Verrier concluded that it would have a visible disc and sufficient light to make it conspicuous in ordinary telescopes. The number of the Comptes Rendus containing this paper could not reach this country until the third or fourth week in September. Le Verrier communicated his principal conclusions to Dr Galle, of the Berlin Observatory, in a letter which was received by him on Sep- tember 23, 1846. The same evening Dr Galle examined the heavens, comparing the stars with Bremiker's map (Hora xxi of the Berlin Academy's star maps). He soon found a star of about the eighth magnitude, nearly in the place pointed out by Le Verrier, which did not exist on the map. There could be little doubt that this was the new planet, and the observations made on the following day showed that its motion was nearly the same as that of the predicted planet. The discovery of the planet was due, not to its disc, but to its absence as a star on Bremiker's map. The existence of this map, which had been but lately published, was unknown to the English astronomers. On October 1 Challis heard of the discovery of the planet at Berlin. He then found that he had actually observed it on August 4 and August 12, the third and fourth nights of his search, so that if the observations had been compared with each other as the work proceeded, the planet might have been discovered by him before the middle of August. When the search was discontinued, on October 1, Challis had recorded 3150 positions of stars and was making preparations for mapping them 1 . Adams's researches, therefore, preceded Le Verrier's by a considerable interval ; and, in spite of the delay in commencing the search, it had been carried on at Cambridge 1 Even as it was, the planet was nearly discovered by of July 30 included all those of August 12. After the the middle of August. Challis used two methods of discovery of the planet, Challis, continuing this corn- observation, one with telescope fixed and the other with parison, found that No. 49, a star of the 8th magnitude telescope moving. On July 30, the second day of the in the series of August 12, was wanting in the series of search, he observed by the second of these methods, and July 30. This was the planet, which had entered the on August 12, the fourth day of the search, he observed zone between July 30 and August 12. The former com- the same zone by the first method. Shortly afterwards parisou had not been continued beyond No. 39 "probably he compared the observations of these days, in order to from the accidental circumstance that a line was there verify the adequacy of his course of procedure, and as far drawn in the memorandum-book in consequence of the as the comparison was carried, he found that the positions interruption of the observations by a cloud." BIOGRAPHICAL NOTICE. xxi for eight weeks before the planet was found at Berlin. Adams's first complete in- vestigation may be regarded as having been finished on October 21, 1845, when he left his paper at the Royal Observatory. This was three weeks before Le Verrier presented to the French Academy his first memoir, in which it was shown that the irregularities in the motion of Uranus could not be attributed to the known planets, and seven months before the date of presentation of his second memoir in which he first investigated the orbit of the supposed disturbing planet. As we know, Adams had resolved to undertake the work in 1841, and his first rough solution was effected, as soon as he had leisure, in 1843. We may presume that Le Verrier did not attempt to determine the position or orbit of the disturbing planet until after the completion of his memoir of November 10, 1845. The discovery of the actual planet by Dr Galle, in consequence of Le Terrier's pre- diction, was received with the greatest enthusiasm by astronomers of all countries, and the planet was at once called " Le Verrier's Planet." Adams's work was only known to the Astronomer Royal, Challis, and a few other persons, chiefly private friends. The first public mention of Adams's name occurred in a letter to the Athmceum from Sir J. Herschel, which appeared under the heading "Le Verrier's Planet" in the number for October 8, 1846. In this letter, which is dated October 1, Herschel refers to the address he had delivered on September 10, on the occasion of resigning the Presidential Chair of the British Association at Southampton, in which, after referring to the astronomical events of the year, which included the discovery of a new minor planet, he added : "It has done more. It has given us the probable prospect of the discovery of another. We see it as Columbus saw America from the shores of Spain. Its movements have been felt, trembling along the far-reaching line of our analysis, with a certainty hardly inferior to that of ocular demonstration." To justify the confidence which these words express, Herschel first describes a conversation with Bessel in 1842, in which the latter had said that it was highly probable that the deviations of Uranus might be due to an unknown planet (being systematic, and such as an exterior planet would produce), and then proceeds: "The remarkable calculations of M. Le Verrier, which have pointed out, as now appears, nearly the true situation of the new planet by resolving the inverse problem of the perturbations if uncorroborated by repetition of the numerical calculations by another hand, or by independent investigation from another quarter would hardly justify so strong an assurance as that conveyed by my expressions above alluded to. But it was known to me at that time (I will take the liberty to cite the Astronomer Royal as my authority) that a similar investigation had been independently entered into, and a conclusion as to the situation of the new planet very nearly coincident with M. Le Verrier's arrived at (in entire ignorance of his conclusions) by a young Cambridge mathe- matician, Mr Adams, who will, I hope, pardon this mention of his name (the matter being one of great historical moment), and who will doubtless in his own good time and manner, place his calculations before the public." This passage seems to have passed almost unnoticed by astronomers, in the excitement produced by Le Verrier's discovery, and it was not till October 17, when a letter from Challis appeared in the Athenceum, giving an account of the proceed- ings at Cambridge in connexion with the new planet, that general attention was directed to Adams's calculations. It was then known for the first time that his xxii BIOGRAPHICAL NOTICE. conclusions had been in the hands of the Astronomer Royal and Challis since 1845, and that the latter had actually been engaged in searching for the planet. There was naturally a disinclination to give full credit to facts thus suddenly brought to light at such a time. It was startling to realise that the Astronomer Royal had had in his possession the data which would have enabled the planet to have been discovered nearly a year before. On the other hand, it seemed extraordinary that a competent mathema- tician, who had determined the orbit of the disturbing planet, should have been content to refrain for so long from making public his results. No time was now lost in bringing the evidence before the world. On November 13, 1846, the Astronomer Royal com- municated to the Royal Astronomical Society an "Account of some Circumstances historically connected with the Discovery of the Planet exterior to Uranus " ; and Challis also described the observations which he had undertaken in search of the planet. At the same meeting Adams communicated a memoir containing an account of his mathematical investigations in connexion with the determination of the mass, orbit, and position of the new planet, by which he had obtained the elements communicated to the Astronomer Royal on October 21, 1845, and September 2, 1846. All of these papers are published in Vol. xvi. of the Memoirs of the Society; but as it was felt that the immediate publication of Adams's memoir was a matter of national interest, it was at once printed separately by Lieut. Stratford, superintendent of the Nautical Almanac Office, as a special appendix to the Nautical Almanac for 1851, and widely circulated at the beginning of 1847. This appendix was also issued as a supplement to No. 593 (March 2, 1847) of the Astronomische Nachrichten. Having thus given in chronological order an outline of the main facts relating to the discovery of the new planet, it remains to describe in more detail some of the incidents which, apart from their historical interest, are of importance in connexion with the discussions which have taken place on the subject. At the time of Adams's first visit to the Royal Observatory, in September, 1845, the Astronomer Royal was abroad. On the occasion of the second visit, on October 21, 1845, he was engaged, and was unable to see Adams, who therefore left at the Observatory the paper containing the elements of the planet. Fifteen days afterwards, on November 5, 1845, the Astronomer Royal wrote to Adams, " I am very much obliged by the paper of results which you left here a few days since, showing the perturbations on the place of Uranus produced by a planet with certain assumed elements. The latter numbers are all extremely satisfactory: I am not enough acquainted with Flamsteed's observations about 1690 to say whether they bear such an error, but I think it extremely probable. But I should be very glad to know whether this assumed perturbation will explain the error of the radius vector of Uranus. This error is now very considerable, as you will be able to ascertain by comparing the normal equations, given in the Greenwich observations for each year, for the times before opposition with the times after opposition." Un- fortunately Adams did not reply to this enquiry or communicate again with the Astronomer Royal until September 2, 1846, when he forwarded to him the results of his second investigation. Le Verrier's memoir of June 1, 1846, reached the Astronomer Royal about the 23rd or 24th of June, and on June 26th the latter addressed to Le Verrier the following letter, containing the same question with respect to the radius vector which he had previously BIOGRAPHICAL NOTICE. xxiii put to Adams : " I have read with very great interest the account of your investigation on the probable place of a planet disturbing the motions of Uranus, which is contained in the Compte Rendu de I' Academic of June 1 ; and I now beg leave to trouble you with the following question. It appears, from all the later observations of Uranus made at Greenwich (which are most completely reduced in the Greenwich observations of each year so as to exhibit the effect of an error either in the tabular heliocentric longitude, or the tabular radius vector), that the tabular radius vector is considerably too small. And I wish to inquire of you whether this would be a consequence of the disturbance produced by an exterior planet, now in the position which you have indicated ? I imagine that it would not be so, because the principal term of the inequality would probably be analogous to the moon's variation, or would depend on sin 2 (v v') ; and in that case the perturbation in radius vector would have the sign for the present relative position of the planet and Uranus. But this analogy is worth little until it is supported by proper symbolical computations." Le Verrier replied to the Astronomer Royal's enquiry on June 28. In this letter he says, "Je compte avoir termini la rectification des elements de la planete troublante avant 1'opposition qui va arriver; et parvenir a connaitre ainsi les positions du nouvel astre avec une grande precision. Si je pouvais espeYer que vous aurez assez de confiance dans mon travail pour chercher cette planete dans le ciel je m'empresserais, Monsieur, de vous envoyer sa position exacte, des que je 1'aurai obtenue." He then explains that the errors in radius vector are well accounted for by the disturbing planet. On June 29, before Le Verrier's reply had been received, a meeting of the Board of Visitors of the Royal Observatory took place, at which Sir J. Herschel and Challis, among others, were present. In the course of a discussion the Astronomer Royal referred to the probability of shortly discovering a new planet, giving as his reason the very close coincidence between the results of Adams's and Le Verrier's positions of the supposed disturbing planet. It was in consequence of this opinion that Herschel felt justified in speaking so confidently of the approaching discovery in his address at Southampton on September 10. When the planet was discovered at Berlin, the Astronomer Royal was on the con- tinent, and on his return to Greenwich he wrote to Le Verrier, on October 14, 1846 : " I was in Germany at the latter part of the month of September, when I received the intelligence of the actual discovery of the new planet whose place had been so clearly pointed out by you. And I beg you to accept my sincere congratulations on this suc- cessful termination to your vast and skilfully directed labours. Not many days past, I was in company with Professor Schumacher of Altona, and there I had the pleasure of reading the manuscript paper which you have transmitted to him. I was exceedingly struck with the completeness of your investigations. May you enjoy the honours which await you ! and may you undertake other work with the same skill and the same success, and receive from all the enjoyment which you merit ! I do not know whether you are aware that collateral researches had been going on in England, and that they had led to precisely the same result as yours. I think it probable that I shall be called on to give an account of these. If in this I shall give praise to others, I beg that you will not consider it as at all interfering with my acknowledgment of your claims. You are to be recognised beyond doubt as the real predicter of the planet's place. I may add that the A. d xxiv BIOGRAPHICAL NOTICE. English investigations, as I believe, were not quite so extensive as yours. They were known to me earlier than yours." The rest of the letter relates to the name proposed for the new planet. Le Verrier's reply, of October 16, was written under a sense of injustice and irritation produced by Herschel's letter in the Athenceum, which he considers " bien mauvaise et bien injuste pour moi." He feels very much hurt that Herschel should have said that he should not have felt justified in expressing himself so confidently at Southampton if his results had not been independently corroborated by Adams's work. He gives a succinct account in historical order of his own publications on the subject, and, in con- nexion with the paper of June 1, 1846, refers to Airy's letter of June 26, 1846, which he says shows that at that time Airy had no precise information with respect to the position of the planet, and that he was even surprised that he (Le Verrier) had placed it where he had, " parce qu'ainsi situ^e elle ne lui paraissait pas rendre compte des inexactitudes du rayon vecteur." With reference to Adams he writes, " Pourquoi Mr Adams aurait-il garde le silence depuis quatre mois ? Pourquoi n'aurait-il parle des le mois de juin s'il eut eu de bonnes raisons a donner ? Pourquoi attend-on que 1'astre ait e"te vu dans les lunettes ? " He appeals to Airy to defend his rights, and states that he has documents to prove that on September 28 and 29 Challis was still searching for the planet " sur mes indications." The Astronomer Royal's reply to this letter contained a statement of the facts with regard to Adams's work and the search for the planet. The French astronomers were at first very unwilling to admit that Adams had any rights whatever in connexion with the planet, either as an independent discoverer or otherwise : and Arago, the secretary of the Academy, was especially violent in his de- nunciations. Le Verrier, who had at first inclined to the name of Neptune for the planet, delegated the right to name it to Arago, who insisted that it should be called Le Verrier. It is unnecessary to enter further into the discussions which took place on this subject: a very fair view of the whole matter was taken by Biot, and ultimately the name of Neptune was adopted by general consent. Strange as it may seem, the course of events in this country was somewhat similar, it being contended by some English astronomers that the fact that Adams's results had not been publicly announced deprived him of all claims in relation to the discovery. The recognition of the merit of Adams's researches was mainly due to the warm and generous advocacy of two Cambridge men, Sedgwick and Sheepshanks. Adams's determination of the orbit of the new planet was completed by October 1845, and by this date his results were in the possession of Challis and the Astronomer Royal, and yet no announcement whatever was made with respect to them until October 3, 1846. It is a most striking fact in the history of science that researches of such novelty and importance could have been known to two official astronomers besides their author for nearly a year without any steps being taken to make them public. The causes which produced this result are necessarily peculiar, and require to be examined in some detail. Adams, having completed his determination, took the results in person to the Royal Observatory, in the hope that steps would forthwith be taken to find the planet. He was disappointed at not seeing the Astronomer Royal, and probably had expected more encouragement than the letter he received a fortnight afterwards with the enquiry relative to the radius vector. Regarding this as a matter of trifling importance, he delayed to BIOGRAPHICAL NOTICE. xxv reply to it, and applied himself to his second calculation with a different mean distance. With respect to Challis, he has explained in his report to the Cambridge Observatory Syndicate 1 that it might reasonably be supposed that the position of the planet was only roughly determined, and that a search for it must necessarily be long and laborious. In 1845, when Adams had completed his calculations, the planet was considerably past opposition, and Challis had no thought of commencing the search then. The succeeding interval until June 1846 was occupied with observations of the planet Astrsea, Biela's double comet, and several other cornets, and during this period he had little communication with Adams respecting the new planet. Attention was again called to the matter by Le Verrier's paper of June 1, and, as has been stated, the search was commenced on July 29. From the Astronomer Royal's "Account &c." we learn that he attached great importance to the explanation of the error in radius vector. After giving the letter which he addressed to Adams on this subject he states that he considered the establishment of the error of the radius vector of Uranus to be a very important determination and proceeds, " I therefore considered that the trial, whether the error of radius vector would be explained by the same theory which explained the error of longitude, would be truly an experimentum crucis. And I waited with much anxiety for Mr Adams's answer to my query. Had it been in the affirmative I should have exerted all the influence which I might possess, either directly, or indirectly through my friend Professor Challis, to procure the publication of Mr Adams's theory. From some cause with which I am unacquainted, probably an accidental one, I received no immediate answer to this enquiry. I regret this deeply for many reasons. While I was expecting more complete information on Mr Adams's theory, the results of a new and most important investigation reached me from another quarter." This refers to Le Verrier's paper of June 1, 1846, after giving an account of which, the Astronomer Royal proceeds : " This memoir reached me about the 23rd or 24th of June. I cannot sufficiently express the feeling of delight and satisfaction which I received from it. The place which it assigned to the disturbing planet was the same, to one degree, as that given by Mr Adams's calculations which I had perused seven months earlier. To this time I had considered that there was still room for doubt of the accuracy of Mr Adams's investigations... But now I felt no doubt of the accuracy of both calculations, as applied to the perturbation in longitude. I was however still desirous, as before, of learning whether the perturbation in radius vector was fully explained." Le Verrier replied to this enquiry in a letter from which some passages have already been quoted. With reference to Le Verrier's explanations regarding the error of radius vector the Astronomer Royal writes : " It is impossible, I think, to read this letter without being struck with its clearness of explanation, with the writer's extraordinary command, not only of the physical theories of perturbation, but also of the geometrical theories of the deduction of orbits from observation, and with his perception that his theory ought to explain all the phenomena, and his firm belief that it had done so. I had no longer any doubt upon the reality and general exactness of the prediction of the planet's place." After describing the contents of Le Verrier's third paper, of August 31, 1846, the Astronomer Royal proceeds : " My analysis of this paper has necessarily been exceedingly imperfect, as regards the astronomical and mathematical parts of it ; but I am sensible that in regard to another part it fails totally. I cannot attempt to convey to you the 1 This report, on account of its importance, is reprinted in extemo on pp. xlix liv. d2 xxvi BIOGRAPHICAL NOTICE. impression which was made on me by the author's undoubting confidence in the general truth of his theory, by the calmness and clearness with which he limited the field of observation, and by the firmness with which he proclaimed to observing astronomers, 'Look in the place which I have indicated, and you will see the planet well.'... It is here, if I mistake not, that we see a character far superior to that of the able, or enterprising, or industrious mathematician : it is here that we see the philosopher." Adams was not fortunate in the two astronomers to whom he communicated his results: neither of them gave to a young and retiring man the kind of help or advice that he should have received. Challis, a most conscientious and painstaking astronomer, had obtained for him the places of Uranus that he required, and written him a letter of introduction to the Astronomer Royal. Although quite appreciative of Adams's calculations, he was occupied with his own observatory work, and seems to have left the matter in the hands of Airy. He undertook the search for the planet when it was suggested to him by Airy, after the publication of Le Verrier's paper, and carried it out methodically and with scrupulous care, as was his practice in everything; and in course of time the planet would have been discovered : but he does not seem to have been alive to the importance of making known in a more public way than by communi- cation to the Astronomer Royal the results which Adams had obtained. As professor in the University he should not have allowed a young Senior Wrangler, through modesty or diffidence or inexperience, to do such injustice to himself. It is evident that even if the planet had been discovered at Cambridge, the same difficulty would have had to be encountered as that which actually occurred in bringing Adams's claims before the world, as Le Verrier's work had been already published and his indications had been used in the search. Airy states that he regarded the question of the radius vector as an experimentum crucis, and waited with much anxiety for Adams's reply to his query. When he found that Le Verrier assigned nearly the same position to the planet as Adams, and when Le Verrier had explained to him that the error in radius vector was corrected, any doubt with respect to the quality of Adams's work, which the absence of a reply to his enquiry may have caused, must have been removed, and the time had clearly come to take some notice of the paper which had been in his possession for seven months. But though he mentioned the matter at the meeting of the Board of Visitors on June 29 and suggested the search to Challis on July 9, he took no steps, either directly or through Challis, to bring about the public announcement of Adams's results. Of course Airy knew that Adams had Challis and possibly other Cambridge men to advise him with respect to publication. Challis was a man of gentle and kindly nature, but slow in action and wanting in initiative : Airy, however, was a man of vigorous character, and it seems unaccountable that he should have taken no steps to secure the publication of Adams's results, even after his correspondence with Le Verrier in June 1846 '. The fact that no reply had been received to the radius vector question affords no adequate explanation; he could have written to Adams again or applied to Challis, if he still considered an answer essential. It is easy to understand the " delight and satisfaction " which Airy as a mathe- matician may have received from Le Verrier's paper confirming Adams's place of the 1 Sedgwick's letter, from which the interview with out without more delay. Was Adams ever so much as Adams is quoted on the next page, contains the following told that Le Verrier was at his heels? Our astronomers passage: "When it was found that Adams was confirmed ought to have got up a flare in an instant." by the fortunate Frenchman the facts ought to have been BIOGRAPHICAL NOTICE. xxvn planet, but one would have thought that at the same time he would have felt some regret that Adams's paper had remained so long untouched in his keeping, thus depriving this country and his own University of the merit of the first announcement. It is impossible not to contrast the admiration with which he received Le Verrier's published writings with the indifference shown towards Adams's still unpublished work. Adams was certainly as clearly convinced of the reality of the planet as Le Verrier, and what- ever claims the latter has to the name of philosopher rather than mathematician apply equally to the former. It is difficult also to see how Airy could have felt justified in writing to Le Verrier, after the discovery of the planet, the words, "you are to be recognised beyond doubt as the real predicter of the planet's place." It has been said, and truly, that it was no part of the Astronomer Royal's duty to search for a new planet, and that he had . no telescope available for the purpose even if he had desired to do so: but Adams (who possibly acted on Challis's advice) cannot be much blamed for taking his paper to Greenwich, in hopes that the planet might be found in this country. Adams himself seems to have been content to leave the matter in the hands of the Astronomer Royal, and it is to be remarked that at that time he was not only the official head of Astronomy, but was much looked up to by Cambridge men as one who had recently given a great impulse to astronomical studies in the University, as professor and director of the Observatory 1 . When it became known in Cambridge that Airy and Challis had been in possession of results which would have enabled the planet to be discovered in 1845 a good deal of indignation was naturally felt at the apathy and incredulity with which Adams's work had been received. This led Sedgwick, an intimate friend of Airy, to write two letters on the subject, which are now in the archives of the Royal Observatory at Greenwich. The second of these letters, dated December 6, 1846, contains the following interesting passages. "Adams, though a great philosopher in his way, has shown no worldly wisdom, indeed has acted like a bashful boy rather than like a man who had made a great discovery. " Again, he was certainly wrong in not answering Airy's letter. How strange and how unfortunate ! Surely he must have been ill advised on this point ; but I will try to learn this from himself. "Just as I had written so far, in came Adams, to return my call, and five minutes after in came Sheepshanks, who, after chatting for half an hour with his surplice on, went to drink tea at the Lodge. Adams remained and drank tea with me, and we have had a very long chat " (1) He called at the Observatory soon after his calculations were finished the Astronomer Royal away bad luck, but no blame anywhere this was September 1845. (2) Called again (October, the same Autumn) and the Astronomer out left his card heard that Airy would return soon, and therefore left word that he would call again. (3) Did call again (I think in a little more than an hour) and was told that the 1 Adams did at last contemplate publication, for he British Association," and in his letter of November 18, concludes his letter of September 2, 1846 to the Astronomer 1846 (p. xxviii) he states that he drew up such a paper but Koyal with the words, " I have been thinking of drawing arrived at the meeting too late to present it. up a brief account of my investigation to present to the xxviii BIOGRAPHICAL NOTICE. Astronomer was at dinner ; had no message, and therefore went away. But he added that he did not call by appointment. He only took his chance on his way back from Devonshire to Cambridge, &c. &c. I collected that he had been mortified (I am not using his own words) at receiving no message on the second call in October. 'I thought' (said he) 'that though he had been at dinner he would have sent me a message, or perhaps spoken a word or two to me : but I am now convinced that in fact he never knew of my second call that the servant had not delivered my message along with my card.' These were mainly his words. I asked him whether the circumstances just mentioned had any influence in preventing his reply to Professor Airy's note. He said in answer, that had these not happened he possibly might have replied more readily; but assuredly had he considered the question about the radius vector as of great import- ance (' as an experimentum crucis ') he should have answered the note instantly. ' But,' said he, 'I could not look on the corrections of the radius vector as an experimentum crucis; because any hypothesis (however wrong) which gave a correction in longitude must give a correction in the radius vector of the same kind as the correction deduced from the perturbations of the new planet' (I think I state this correctly). 'Again,' said he, 'I wanted to send my papers in good order to the Astronomer Royal. I went over all my calculations three times. I added a few terms, without changing my results. I was much interrupted, so it was my vacation before I could finish my last revision,' &c. &c. ' I lament very much that I did not immediately answer the first note. I ought to have answered it,' &c. &c. 'But,' he added, 'I did think that the Astronomer Royal would have communicated my results among his correspondents. I took all that for granted, and I thought it a publication,' &c. &c. He is anxious to have no misunderstanding with Airy. He spoke very earnestly on this subject, and expressed himself grieved at the ill-natured things that had been said." The following letter from Adams to Airy was written five days after the meeting of the Royal Astronomical Society at which Airy's 'Account &c.' was read. ' ' ST JOHN'S COLLEGE, 18 November, 1846. "DEAR SIR, "Allow me to thank you for your able, interesting, and impartial account of circumstances connected with the discovery of the new planet. I need scarcely say how deeply I regret the neglect of which I was guilty in delaying to reply to the question respecting the radius vector of Uranus, in your note of Nov. 5th, 1845. " In palliation, though not in excuse of this neglect, I may say that I was not aware of the importance which you attached to my answer on this point, and I had not the smallest notion that you felt any difficulty on it, such as you subsequently mentioned to M. Le Vender. "For several years past, the observed place of Uranus has been falling rapidly more and more behind its tabular place. In other words, the real angular motion of Uranus is considerably slower than that given by the tables. This appeared to me to show clearly that the tabular radius vector would be considerably increased by any theory which represented the motion in longitudes, for the variation in the second member of the equation r 2 -j- = Jpa (1 e 2 ) is very small. at BIOGRAPHICAL NOTICE. xxix " Accordingly, I found that if I simply corrected the elliptic elements, so as to satisfy the modern observations as nearly as possible, without taking into account any additional perturbations, the corresponding increase in the radius vector would not be very different from that given by my actual theory. Hence it was that I was led to defer writing to you till I could find time to draw up an account of the method employed to obtain the results which I had communicated to you. More than once I commenced writing with this object, but unfortunately did not persevere. I was also much pained at not having been able to see you when I called at the Royal Observatory the second time, as I felt that the whole matter might be better explained by half-an-hour's conversation than by several letters, in writing which I have always experienced a strange difficulty. " I entertained, from the first, the strongest conviction that the observed anomalies were due to the action of an exterior planet; no other hypothesis appeared to me to possess the slightest claims to attention. " Of the accuracy of my calculations I was quite sure, from the care with which they were made, and the number of times I had examined them. The only point which appeared to admit of any doubt was the assumption as to the mean distance, and this I soon proceeded to correct. The work however went on very slowly throughout, as I had scarcely any time to give to these investigations except during the vacations. "I could not expect however that practical astronomers, who were already fully occupied with important labours, would feel as much confidence in the results of my investigation as I myself did ; and I therefore had our instruments put in order, with the express purpose, if no one else took up the subject, of undertaking the search for the planet myself, with the small means afforded by our Observatory at St John's. "I remain, dear Sir, " Yours very respectfully, "J. C. ADAMS. "I drew up a paper for the meeting of the British Association at Southampton, but did not arrive there in sufficient time to present it, as Section A closed its sittings one day earlier than I expected." In connexion with Adams's researches on the new planet, and his omission to reply to Airy's enquiry 1 , the following interesting extracts from a letter from Challis to Airy, of December 19, 1846, should also find a place here. "In the Athencewn of Dec. 5 there was an article on the new planet, ably and fairly written in general, but so unjust with respect to Mr Adams's scientific merits, that I wrote a letter to the Editor, which is in the Athenceum of to-day. . .There is one point in the story which is in an unsatisfactory state. Why did not Adams answer your question ? I know that he is extremely tardy about writing, and that he pleads guilty to this fault. 1 In 1883, when the present writer was preparing the the Astronomer Boyal's letter about the radius vector, obituary notice of Challis for the Royal Astronomical Adams said, "I should have done so: but the enquiry Society, in reply to a question why he had not answered seemed to me trivial." XXX BIOGRAPHICAL NOTICE. He experiences also a difficulty, which all young writers feel more or less, in putting into shape and order what he has done, and well done, so as to convey an adequate idea of it to others by writing. After receiving your questions it occurred to him that it would be well for him to send you a full account of his methods of calculation, and that he might send the answer at the same time. I believe that nothing but procrasti- nation in fulfilling this intention was the reason of his not sending an answer at all. I have always found him more ready to communicate orally than by writing. It will hardly be believed that before I began iny observations 1 had seen nothing of his in writing respecting the new planet, except the elements which he gave me in September written on a small piece of paper without date. " I first got an idea of the nature and value of his researches by an abstract which he drew up to produce at the meeting of the British Association at Southampton. The public would hardly take such a reason as that I have mentioned to be the true reason for his not answering your question, and I fear therefore a hiatus must remain in the history." As the Astronomer Royal laid so much stress upon the explanation of the error of radius vector, regarding it as an experimentum crucis with respect to the value of Adams's calculations, and as his views upon the matter have been much criticised, it seems proper to quote the following explanatory passages which were written by him after he had received Adams's letter of November 18, and when the matter was attracting general attention. Writing to Sheepshanks on December 17, 1846, he says: "Concerning the radius vector of Uranus, the error was certain as to sign. It was determined with reasonable accuracy as to magnitude (perhaps the probable error might be or ^ of the whole). Now, suppose that Adams's elements which gave longitude-corrections had given a wrong sign for the correction of the radius vector, what would his theory have been worth ? The alternation of signs of errors H in longitude does not exclude any other hypothesis than that of an exterior planet. If the law of force differed slightly from that of inverse square of the distance (of which two years ago there was great probability) and if tables were calculated strictly on the law of inverse square of distance (as was done in existing tables), then the discordances in longitude would have the alternate signs + . Le Verrier evidently attached great importance to the radius vector... The radius vector, as you say, was to be used as an indirect verification, but its error de- manded explanation quite as imperatively as the other." And writing to Challis, December 21, 1846, he says: " I am sure that you cannot have a higher opinion of Adams's ability in the scientific parts of this matter than I have.... But with regard to one part of your own published letter in the last Athenceum, I must make one remark 1 . There were two things to be explained, which might have existed each independently of the other, and of which one could be ascertained independently of the other : viz. error of longitude and error of radius vector. And there is no a priori reason for thinking that a hypothesis 1 Challis had written: "Again, as to the error of the of the other. Mr Adams actually employed a method of radius vector : it is quite impossible that its longitude calculation which required him to compute the coefficients could be corrected during a period of at least 130 years of the expression for error of radius vector, before corn- independently of correction of the radius vector.. ..The puting the coefficients of the expression for error of investigation of one correction necessarily involves that longitude." (Athenieum, December 19, 1846.) BIOGRAPHICAL NOTICE. xxxi which will explain the error of longitude will also explain the error of radius vector. If, after Adams had satisfactorily explained the error of longitude, he had (with the numerical values of the elements of the two planets so found) converted his formulae for pertur- bation of radius vector into numbers, and if these numbers had been discordant with the observed numbers of discordance of radius vector, then the theory would have been false, not from any error of Adams's, but from a failure in the law of gravitation. On this question therefore turned the continuance or fall of the law of gravitation. This, it appears to me, has been totally overlooked in your letter. It was a question of vast importance. " The progress of science almost always depends on questions of this kind. Thus, in Chemistry, the phlogistic theory explained the concurring facts of oxidation of metals and vitiation of air, or gaseous formation in water. But did it also account for the increased weight of the metal ? No. Then it was false. Laplace's notion of forces gave an explanation of the course of extraordinary pencils of light. But did it or could it give an explanation also of the separation of pencils and of their polarisation ? No. Then it was false. "The theory of gravitation might have been in the same predicament with regard to Uranus. Adams's answer would have made this satisfactory. . . . What could be the reason of Adams's silence, I could not guess. It was so far unfortunate that it inter- posed an effectual barrier to all further communication. It was clearly impossible for me to write to him again." Looking back now upon Adams's achievement, which, as has been truly said, belongs at once to the science and to the romance of astronomy, there are several points that stand out as very remarkable : his extreme youth when he attacked, unaided, so difficult a problem, and steadily carried it through to success; his complete faith in the Newtonian law and in the results of his own mathematics; and his extreme modesty. As soon as he took his degree in 1843 he devoted his whole leisure, in term time at Cambridge, and in vacations in Cornwall, to the new planet's orbit, without assistance or encouragement from anyone. How quietly and unassumingly he pursued his investigations is shown by the fact that at the time of the finding of the planet his name was only known to Airy, Challis, Herschel, Earnshaw, and a few intimate university friends of his own standing. He was perfectly convinced of the reality of the planet from the first, and of the approximate accuracy of the place he had assigned to it ; and in the paper which he placed in the hands of Challis in September, 1845, he used the words "the new planet." Although containing no new facts it may be well to conclude the account of Adams's researches on the new planet with the following extract from a letter written by him at the time (November 26, 1846) to Professor James Thomson: " On considering the subject it appeared to me that by far the most probable hypothesis that could be formed to account for these irregularities was that of the existence of an exterior undiscovered planet whose action on Uranus produced the disturbances in question. None of the other hypotheses that had been thrown out seemed to possess the slightest claims to attention, as they were all improbable in themselves, and incapable of being tested by any exact calculation. Some had even supposed that, at the great distance of Uranus from the Sun, the law of attraction became different from that of the inverse square of the distance, but the law of gravitation was too firmly established for this to A. e xxxn BIOGRAPHICAL NOTICE. be admitted till every other hypothesis had failed to account for the observed irregularities ; and I felt convinced that in this, as in all previous instances of the kind, the discrepancies which had for a time thrown doubts on the truth of the law would eventually afford it the most striking confirmation. In contrast with all these vague hypotheses, the sup- position that the irregularities were caused by the action of an unknown planet appeared to be thoroughly in accordance with the present state of our knowledge, could be tested by calculation, and would probably lead to important practical results viz. the approxi- mate determination of the position of the disturbing body." After quoting the memorandum of July 3, 1841, he proceeds: "Accordingly, in 1843, I commenced my calculations, and in the course of that year I arrived at a first solution of the problem, which, though incomplete in itself, fully convinced me that the hypothesis which I had formed was quite adequate to account for the observed irregularities, and that the place of the disturbing body might be very approximately determined by a more extended investigation. Having received from the Astronomer Royal, in February 1844, the whole of the Greenwich observations of Uranus, I accordingly attacked the problem afresh, and in a much more complete manner than before, and, after obtaining several solutions, differing little from each other, by gradually taking into account more and more terms in the series expressing the perturbations, I communicated my final results to Professor Challis in September 1845, and the same, slightly corrected, to the Astronomer Royal in the following month. The near agreement of the several solutions which I had obtained gave me great confi- dence in my results, which included a determination of the mass, position and elements of the orbit of the supposed planet." Adams took no part whatever in the controversies or discussions which arose with regard to the discovery of the planet, either publicly or privately, and at no time in his life did he ever criticise the conduct of anyone, or say an unkind word in connexion with the matter. Fortunately all the facts relating to the calculations of Adams and Le Verrier and the discovery of the planet are undisputed, and any discussions that may take place in the future can have reference only to the conclusions to be drawn from them 1 . On the discovery of the planet the Royal Society at once awarded their highest honour, the Copley Medal, to Le Verrier (1846), and it was not till two years afterwards that it was awarded to Adams. The Royal Astronomical Society was saved from expressing a similar preference by the by-law requiring that the award of the medal should be confirmed by a majority of three-quarters of the Council. A sufficient minority were of opinion that " an award to M. Le Verrier, unaccompanied by another to Mr Adams, would be drawing a greater distinction between the two than fairly represents the proper inference from facts, and would be an injustice to the latter 2 ." 1 The principal contemporary publications relating these documents in writing the account in the text, to the new planet are to be found in Vol. xvi. of the Challis's report to the Observatory Syndicate at Cam- Memoirs of tlw Royal Astronomical Society, in the bridge, which contains an account of his own proceedings Comptes Ilendus, in the Athenceum, in the Astrona- relative to the new planet, is added as an appendix to this miiche Nachrichten, and in Vol. vn. (1847) of the North notice (pp. xlix liv). Eeferences to the discovery of the British Re-view, which contains an article by Brewster. planet occur in the Life and Letters of Adam Sedgwick, A number of letters bearing upon the subject are con- by Clark and Hughes, 1890, Vol. n. pp. 107 and 287. tained in the Archives of the Royal Observatory, and 2 In an interesting letter to Schumacher, in the pos- Sheepshanks's correspondence is in the possession of the session of the Royal Astronomical Society, Sheepshanks Royal Astronomical Society. Free use has been made of wrote as follows, under date April 7, 1847: "You will be BIOGRAPHICAL NOTICE. xxxm The honours so freely and deservedly bestowed upon Le Verrier in France and other countries form a striking contrast to the general want of appreciation with which Adams's work was at first received. But there were conspicuous exceptions. In 1847, on the occasion of the Queen's visit to Cambridge, the honour of knighthood was offered to Adams, but this offer he felt obliged to decline. The members of St John's College, also, were not slow in showing their sense of the honour he had conferred upon his college and the University, for in a very short time a fund, producing about 80 per annum, was raised for establishing a prize to be connected with his name. This fund was offered to the University, and accepted on April 7, 1848. The Adams Prize, which . is biennial, is awarded for the best essay on some subject of pure mathematics, astronomy, or other branch of natural philosophy. A French translation of Adams's memoir on the motion of Uranus was published in Liouville's Journal de Matheinatiques pures et appliquees for 1875. The editor, M. Resal, stated that he had been led to undertake this republication by the pressing solicitations of several eminent mathematicians. In introducing the memoir he writes : '' Le probleme fut resolu simultane'ment, en Angleterre par M. Adams, et en France par M. Leverrier, qui, ainsi que le reconnait M. Adams, a publie le premier les resultats de ses recherches. ...II est impossible de rencontre r, dans 1'histoire des sciences, une decouverte qui fasse plus d'honneur au genie humain. Les lois de Newton recevaient ainsi la plus eclatante des confirmations, et 1'Astronomie, de"sormais indiscutable dans ses principes, etait arrived a I'e'tat de science parfaite. Le Me"moire de M. Adams a valu, a juste titre, a son auteur la plus glorieuse ce'le'brite' : il est digne, en effet, de figurer a cote 1 des plus beaux me'moires de Laplace et Lagrange." This republication of the memoir, after an interval of thirty years, in a purely mathematical journal, derives additional interest from the fact that Adams added a few notes at the end, some of which relate to the objections made by Professor Benjamin Peirce to the legitimacy of the methods pursued by himself and Le Verrier. In Peirce's paper, which was published in 1847, it was contended that the period of Neptune differed so considerably from that of the hypothetical planet that the modes of procedure adopted were unreliable, so that the finding of the planet was partly due to a happy accident. In reply to this, Adams points out that the objection would be valid if the object in view had been to represent the perturbations of Uranus during surprised when I tell you that the strongest opponents to this was nothing at all, simply because the over-modest Mr Adama's claims to consideration are to be found in man communicated his results to Airy and Challis, that England, of course with the exception of France. All the planet might be looked for, instead of bringing his acknowledge M. Le Verrier's merits, and all admit his investigation before the world as he ought to have done. iinili/iilited claim to independent discovery. All are agreed, Surely it is a greater honour to science that two men too, that in making public his results and investigations in should independently have come to the same conclusion the masterly and confident way he did, he deserves the f r0 m the same data than that one should have hit on it, highest praise. As to national feeling (which, by the as it were, accidentally. Thanks to Struve and Biot, &c. way, is too often national injustice) there is absolutely our anti-Adamites are calmer, and as there never was none whatever, so far as I know, or among astronomers. any opposition to Le Verrier, we are quite satisfied at In England at present the current runs the other way, present, and so I hope are the two discoverers. I think and though I very much prefer this failing of the two, there is a hope that Mr Adams will continue his astro- yet it is provoking too. I assure you that it was with nomical researches. In any other country there could be difficulty that one could get a hearing, while pointing out no doubt of it, but in England there is no carriere for the fact that Mr Adams had deduced the elements and men of science. The Law or the Church seizes on all place of the planet in October, 1845. I have been told talent which is not independently rich or careless about repeatedly by those who should have known better that wealth." xxxiv BIOGRAPHICAL NOTICE. two or three synodic periods, but that the case is different when, as in this instance, it was only required to represent the perturbations for a fraction of a synodic period. Before leaving the subject of Neptune, it should be stated that Adams always ex- pressed the warmest appreciation of Le Verrier's work. It was a great pleasure to him when they met at Oxford in 1847. In the same year Le Verrier visited Adams at Cambridge. The honorary degree of LL.D. was conferred upon Le Verrier in 1874 by the University of Cambridge, and it cannot be doubted that this was owing to the action of Adams. In 1876, when Adams was President of the Royal Astronomical Society for the second time, the gold medal was awarded to Le Verrier for his planetary researches. In delivering the medal Adams spoke of " the admiration we feel for the skill and perseverance by which he has succeeded in binding all the principal planets of our system from Mercury to Neptune in the chains of his Analysis." In 1847 Adams communicated to the Royal Astronomical Society a paper on an im- portant error in Bouvard's tables of Saturn. Having been engaged upon a comparison of the theory of Saturn with the Greenwich observations, he was struck with the magnitude of the tabular errors in heliocentric latitude, which could not be attributed to imperfections in the theory. He found that the error was one of computation, two terms of different arguments having been, in effect, united into one. In 1848 he was occupied with the determination of the constants in Gauss's theory of terrestrial magnetism. This investigation he afterwards resumed, and the calculations connected with it, upon which he was engaged in the later years of his life, were left unfinished at the time of his death. When failing health prevented him from any longer giving his personal attention to the work, he placed the manuscripts in the hands of his brother, Professor W. G. Adams, for completion. In 1851 he was elected President of the Royal Astronomical Society, and held the office for the usual term of two years. As president he delivered the addresses on the presentation of the medal to Peters and to Hind. In 1852 he communicated to the Society new tables of the Moon's parallax, to be substituted for those of Burckhardt. Henderson had compared the parallaxes deduced from observation with those derived by calculation from the tables both of Damoiseau and of Burckhardt, finding a difference of no less than 1"'3, according as one set of tables or the other was employed. The parallax in JDamoiseau's tables is given at once in the form in which it is furnished by theory, but that in Burckhardt's tables is adapted to his peculiar form of the arguments, and requires transformation in order to be compared with the former. When this was done, Adams found that several of the minor equations of parallax deduced from Burckhardt differed completely from their theoretical values as given by Damoiseau. He discovered that these errors were due to Burckhardt's transformations of Laplace's formula, and he succeeded in tracing them to their sources. He also examined carefully the theories of Damoiseau, Plana, and Pontecoulant, with respect to the same subject, and supplied a number of defects and omissions. Burckhardt's value of the parallax having been em- ployed in the Nautical Almanac, Adams gave, in addition to the new tables, a table of corrections to be applied to the values in the Nautical Almanac for every day of the year from 1840 to 1855 inclusive. This contribution to astronomy is very characteristic of its author. It contains the results of a great amount of intricate and elaborate mathe- matical investigation, carried out with great skill and accuracy in all its details, both analytical and numerical, but no part of the work itself is given. The method of pro- BIOGRAPHICAL NOTICE. xxxv cedure is briefly sketched, and the final conclusions are stated in the fewest words and simplest manner possible. No one unacquainted with the subject would imagine how much careful research was represented by these few pages of results. The tables were printed as a supplement to the Nautical Almanac for 1856. As Adams had not taken holy orders, his Fellowship at St John's College came to an end in 1852, but he continued to reside in the college until February 1853, when he was elected to a Fellowship at Pembroke College, which he retained till his death. In the autumn of 1858 he was appointed Professor of Mathematics in the University of St Andrews, and shortly afterwards, in the same year, he was elected Lowndean Professor of Astronomy and Geometry at Cambridge, in succession to Peacock. He continued his lectures at St Andrews, however, until the end of the session in May 1859. In 1861 he succeeded Challis as Director of the Cambridge Observatory. In 1863 he married Eliza, daughter of Haliday Bruce, Esq., of Dublin, who survives him. In 1853 Adams communicated to the Royal Society his celebrated memoir on the secular acceleration of the Moon's mean motion. Halley was the first to detect this acceleration by comparing the Babylonian observations of eclipses with those of Albategnius and of modern times, and Newton referred to his discovery in the second edition of the Prindpia. The first numerical determination of the value of the acceleration is due to Dunthorne, who found it to be about 10" in a century. Tobias Mayer obtained the value 6"'7, which he afterwards increased to 9". Lalande's value was nearly 10". The discrepancies were due to the eclipses selected, the results derived from the different eclipses being in- consistent with one another. The history of the theoretical investigations relating to the acceleration may be summed up as follows: In 1762 the French Academy proposed as the subject of their prize the influence of a resisting medium upon the movements of the planets. The prize was won by Bossut, who showed that the principal effect of such a medium would be an acceleration in their motions, which would be much more sensible in the case of the Moon than in that of the planets. In 1770 the question proposed was whether the theory of gravitation could alone explain the acceleration. Euler obtained the prize, but he was unable to discover any term of a secular character, and concluded that the force of gravitation would not account for this inequality. The subject was proposed again in 1772, Euler and Lagrange sharing the prize between them. The former came to the same conclusion as before, attributing the acceleration to a resisting medium ; the latter did not carry the application of his formulae so far as to complete the investigation. The prize was again offered for the same subject in 1774, the competitors being required to examine whether the fact that the Moon appeared to have a secular acceleration, while there was no sensible effect of this kind in the case of the Earth, could be ex- plained by the theory of gravitation alone, taking into account not only the action of the Sun and the Earth upon the Moon, but also the action of the other planets, and even the non-spherical figure of the Moon and Earth. The prize was awarded to Lagrange, who, after showing that none of the causes proposed would suffice to explain the secular variation of the Moon, concluded that, if this variation is real, it must be produced in some other manner, such as by a resisting medium. But as the existence of such a medium was not confirmed by the motions of the other planets, and was even contradicted by the motion of Saturn, which seemed to show a retardation, Lagrange expressed doubts with respect to the reality of the lunar acceleration, resting as it does on observations of eclipses in Xxxvi BIOGRAPHICAL NOTICE. very remote ages. The next investigation relating to the subject is by Laplace, who showed that the acceleration could be accounted for by supposing that the transmission of the force of gravitation was not instantaneous, but that the rate of propagation was about eight million times that of light. Some years later, however, Laplace unexpectedly dis- covered the true gravitational cause of the acceleration. While working at the theory of Jupiter's satellites, he remarked that the secular variation of the eccentricity of Jupiter's orbit produced secular terms in their mean motions. Applying this result to the Moon, he found that the secular variation of the eccentricity of the Earth's orbit produced on the Moon's motion a secular term which agreed very well with the value assigned to it by observation ; he found also that the same cause produced secular terms in the motion of the Moon's node and perigee. This result was communicated to the French Academy in November, 1787, and the memoir containing the details of the calculation was published in the following year. The Stockholm Academy of Sciences had already proposed in 1787 the secular variations of the Moon, Jupiter and Saturn as the prize subject for 1791, but no essays being sent in, the prize was adjudged to Laplace for his memoir published in 1788. Laplace's discovery was received with general satisfaction, and the complete ex- planation of so intractable a variation by means of the Newtonian principles, after so many years of fruitless attempt, was an important event in the history of astronomy. The honour of the discovery might very easily have belonged to Lagrange, for the formulae given by him in a memoir published in 1783 would at once, if applied to the Moon, have produced Laplace's result. But Lagrange had found that, in the case of Jupiter and Saturn, these formulae gave nearly insensible values, so that he did not extend the investigation to the other planets, or to the Moon, although the latter application would only have involved easy numerical substitutions, much simpler than those required for the principal planets. In 1820, at the instigation of Laplace, the lunar theory was taken in hand afresh by Plana and Damoiseau, the approximations being carried to an immense extent, especially by the former. Damoiseau calculated the acceleration numerically, and found it to be 10"'72. Plana's process was algebraical, and he carried the series, of which Laplace had only calculated the first term, as far as to quantities of the seventh order. By reducing to numbers the twenty-eight terms of this series he found 10"'58 as the complete value of the acceleration, the first term, which alone had been included by Laplace, giving 10"'18. Subsequently Hansen gave the values 11"'93 (1842), ll'H7 (1847); and in his tables published in 1857 he used the value 12"'18. It does not seem clear, however, to what extent these values are to be regarded as theoretical determinations. Thus when Adams published his memoir in the Philosophical Transactions for 1853 no suspicion had arisen that Laplace's discovery was not absolutely complete, and that the question of the acceleration had not been finally set at rest. In this short paper of only ten pages Adams showed that the condition of variability of the solar eccentricity introduces into the solution of the differential equations a system of additional terms which affect the value of the acceleration. He found that the second term of the series on which the acceleration depends was really equal to ^p- m 4 , instead of ^V 8 g- m*, as found by Plana. The former is more than three times as great as the latter, and the amount of the acceleration is greatly decreased by the correction of this error. For some time BIOGRAPHICAL NOTICE. xxxvn the paper seems to have attracted no attention, but it then became the object of a long and bitter controversy. Plana, who was the person most concerned in the matter, published, in 1856, a memoir in which he admitted that his own theory was wrong upon this point, and he deduced Adarns's result from his own equations. But shortly after- wards he retracted his admission, and, rejecting some of the new terms which he had obtained, arrived at a result which differed both from his original value and from Adams's. The question was in this state when Delaunay, by employing his own special method of treating the Lunar Theory and extending the investigation only to the fourth order, had the satisfaction of obtaining Adams's coefficient ^l 1 , a result which he brought before the French Academy in January, 1859. This caused Adams to communicate to the Academy, in the same month, the values which he had obtained some time before for the terms in m 6 , m 6 , and m? ; and he pointed out at the same time that, when these terms were included, the value of the acceleration was reduced to 5"'78, and, inferring that the remainder of the series would be nearly equal to 0"'08, he concluded that the total value of the acceleration was about 5"70. Soon afterwards Delaunay carried his approximation as far as terms of the eighth order, and by reducing the forty-two terms in the ana- lytical expression to numbers he obtained the value 6"'ll. Delaunay's result, which was communicated to the Academy in April, 1859, confirmed the accuracy of Adams's values of the terms in m", m", and m 7 , and also those of ?n, 2 e 2 , and m 2 7 2 , which Adams had communicated to him privately. A month after the publication of this paper Pontecoulant made a vigorous attack on the new terms introduced by Adams, which he said had been rightly ignored by Laplace, Damoiseau, Plana, and himself, as they had no real existence. He also objected that if the result of Adams were admitted, it would " call in question what was regarded as settled, and would throw doubt on the merit of one of the most beautiful discoveries of the illustrious author of the Mecanique Celeste." Shortly after- wards he communicated a paper to the Monthly Notices of the Royal Astronomical Society on " the new terms introduced by Mr Adams into the expression for the co- efficient of the secular equation of the Moon," in which he characterised the mathematical process by which these terms had been obtained as " une veYi table supercherie analytique 1 ." It would appear that Le Verrier did not accept Adams's value, for in presenting a note by Hansen to the Academy in 1860 he states that Hansen's tables afford an irrefragable proof of the accuracy of the value 12" which is there attributed to the acceleration. Referring then to the fact that according to Delaunay the secular acceleration should be reduced to 6" he proceeds : " Pour un astronome, la premiere condition est que ses theories satisfassent aux observations. Or la theorie de M. Hansen les repre"sente toutes, et Ton prouve a M. Delaunay qu'avec ses formules on ne saurait y parvenir. Nous con- servons done des doutes et plus que des doutes sur les formules de M. Delaunay. Tres certainement la veYite est du cot6 de M. Hansen 1 ." 1 Hansen stated in 1866 (Monthly Notices, xxvi. p. m, and found the result to be 5"-70. Hansen says that 187) that he had never disputed the correctness of Adams's Adams's theory appeared too late to permit of his using theory, but that he was not satisfied with " the develop- it ; " and it was well that it so happened, for I had already ment of the divisors into series." If this refers to the found by my own theory a coefficient which represents expansion of the acceleration-coefficient in powers of m, ancient eclipses as well as could be desired." It is there- it should be noticed that Adams stated (Vol. xxi. p. 15) fore to be inferred that in this theory the new terms were that he had calculated the value of the acceleration by omitted by Hansen, as they had been by Flana and a method that did not require any expansion in powers of Damoiseau. xxxvni BIOGRAPHICAL NOTICE. In the Monthly Notices for April, I860, Adams replied to his objectors, pointing out simply and clearly the errors into which they had fallen. He mentions that before publishing his memoir of 1853 he had obtained his result by two different methods, and that he had subsequently confirmed and extended it by a third. In a series of letters addressed to Lubbock in June, 1860, Plana began by objecting to Adams's value of the term in m*, but he soon admitted its accuracy. Lubbock also was led to apply his own formulae to the question, and he too arrived at Adams's result. Another calculation was made by Cayley, who, by an entirely different method, also obtained the same result. As Ponte"coulant still continued his reiterated attacks upon the accuracy of the new terms, Cayley's calculation was printed in extenso in the Monthly Notices, where it occupies fifty-six pages. Delaunay had also made another calculation, in which, by following the method indicated by Poisson in 1833, he was led to the same value. The coefficient of in* had also been verified in 1861 by Donkin, who used Delaunay's method of the variation of the elements. Thus Adams's value of the term in m* was obtained by himself in three ways, by Delaunay in two ways, and by Lubbock, Plana, Donkin, and Cayley. Pontecoulant continued his attacks with no abatement of violence in the Comptes Rendus. Ultimately he abandoned Plana's value and obtained one of his own, which differed both from Adams's and Plana's. The whole controversy forms a very extraordinary episode in the history of physical astronomy; the indifference with which the memoir of 1853 was at first received, in spite of the interest and importance of the subject, being followed by the violent controversy which resulted in so many independent investigations by which Adams's result was confirmed. It is not known why Laplace did not carry the calculation beyond the term in m 2 ; but it may be supposed that he regarded the subsequent terms as not likely to modify the value of the first term to any considerable extent. Damoiseau's and Plana's theories passed under the review of Laplace, and may be regarded as having received his sanction. Thus Adams's result not only unsettled a matter which after years of difficulty and struggling had apparently received its full and final explanation, but it detracted from the completeness of a discovery which had long been regarded as one of the greatest triumphs of Laplace's genius. Although the point in dispute relates entirely to the mathematical solution of differential equations, in which observation in no way entered, there can be no doubt that the fact that Plana's result agreed with observation, while Adams's did not, created in the minds of many a presumption against the accuracy of the latter. This view was certainly taken by Le Verrier in the passage quoted above, and it seems also to have influenced Hansen. It is curious that it should have been possible for so much dif- ference of opinion to exist upon a matter relating only to pure mathematics, and with which all the combatants were fully qualified to deal, as is clearly shown by their previous publications. The whole controversy illustrates the peculiar nature of the lunar problem, and of the analysis by means of which the results are reached. The com- plete solution being unattainable by any of the methods which have as yet been applied, the skill of the mathematician is shown in selecting from a vast number of terms those which will produce a sensible influence in that particular portion of the complete solution which is under consideration. A most admirable account of the whole discussion was given by Delaunay in the BIOGRAPHICAL NOTICE. xxxix Additions to the Connaissance des Temps for 1864, in which the place occupied by Adams's memoir in the history of gravitational astronomy is so well summed up that it may be permissible to quote the passage in its entirety : " L'apparition du me"moire de M. Adams a etc" un veritable e"ve"nement: c'e"tait toute une revolution qu'il ope"rait dans cette partie de 1'astronomie theorique. Aussi le rdsultat qu'il renfermait fut-il vivement attaque 1 ; on ne voulait pas 1'admettre, et on ne manquait pas de raisons a donner pour cela. II est, disait-on, en disaccord complet avec les ob- servations ; il ne tend a rien moins qu'a enlever a Laplace 1'honneur d'une de ses plus belles de"couvertes ; il est base 1 d'ailleurs sur une analyse fautive et errone'e. Mais parmi toutes ces raisons il n'y en avait pas une bonne; et la persistance avec laquelle elles ont Ae" pre'sente'es et soutenues a produit un effet diame'tralement oppos^ a celui qu'on en attendait: les confirmations de ce re"sultat tant conteste" se sont accumule'es a un tel point, qu'il serait difficile de trouver dans les sciences une ve'rite' mieux e'tablie que ne 1'est maintenant celle que M. Adams a mise en avant le premier dans son me'moire de 1853. Toutes les objections qui avaient e'te' formule'es sont tombe'es d'elles-mmes. L'analyse declare'e fautive et errone'e a e'te' reconnue exacte. L'accord ou le disaccord du re*sultat thdorique avec les indications fournies par les observations n'a plus e'te' regard^ comme un moyen de contr61er 1'exactitude de ce resultat th4orique. Si le desaccord annonce" existe bien re"ellement, on en conclut simplement que la cause assignee par Laplace a 1'accele'ration seculaire du moyen mouvement de la Lune ne produit pas seule la totalitd du phenomene et on ne trouve dans ce disaccord rien qui soit de nature a amoindrir la de'couverte de 1'illustre ge"ometre franais." These sentences derive additional interest from the fact that they were written by one who was himself the author of the most comprehensive and elegant method by which the lunar problem has ever been treated, and who was the first to recognise the accuracy of Adams's result. In 1866 the Gold Medal of the Society was awarded to Adams for his con- tributions to the development of the Lunar Theory, the address on the occasion being delivered by Mr De la Rue. In the preparation of this very able address, which contains an excellent- history of the problem of the secular acceleration, Mr De la Rue had the invaluable assistance of Delaunay. To complete the account of Adams's connexion with the secular acceleration, it should be stated that in 1880, thirty-seven years after Adams's memoir, Airy com- municated to the Society a paper on the theoretical value of the acceleration (Monthly Notices, vol. xl. p. 368), in which he obtained the value of 10"1477. At the next meeting of the Society Adams pointed out that in Airy's method of treatment certain terms were omitted, the effect being that the expression for the coefficient was reduced to its first term, so that the result necessarily agreed with Laplace's. Subsequently, taking into account these terms, Airy obtained the value 5"'4773. Adams took the occasion of the matter being thus again raised to communicate to the Society the investigation of the acceleration which he had been in the habit of giving in his lectures. In the Monthly Notices for April 1867 Adams published an account of the results he had obtained with respect to the orbit of the November meteors. Professor H. A. Newton had concluded that these meteors belong to a system of small bodies describing an elliptic orbit about the Sun, and extending in the form of a stream along an arc of that orbit of such a length that the whole stream occupies about one-tenth or one-fifteenth of the periodic time in passing any particular point. He showed that the A. / xl BIOGRAPHICAL NOTICE. periodic time of this group must be either ISO'O days, 185 4 days, 354'6 days, 376'6 days, or 33'25 years, and that the node of the orbit must have a mean motion of 52" - 4 with respect to the fixed stars. Soon after the remarkable display of the November meteors in 1866 Adams undertook the examiuation of this question. From the position of the radiant-point observed by himself he calculated the elements of the orbit of the meteors, starting with the supposition that the periodic time was 354'6 days, the value which Professor Newton considered to be the most probable one. The orbit which corresponds to this period is very nearly circular, and he found that the action of Venus would produce an annual increase of about 5" in the longitude of the node, that of Jupiter about 6", and that of the Earth about 10". Thus the three planets, which alone could sensibly affect the motion of the node, would produce an increase of about 12' in 33'25 years. The observed motion of the node is about 29' in 33'2o years, which is therefore inconsistent with a periodic time of the meteors about the Sun of 354 - 6 days. If the periodic time were supposed to be about 377 days, the calculated motion of the node would differ very little from that in the case already considered, while if the periodic time were a little greater or a little less than half a year, the calculated motion of the node would be still smaller. Hence, of the five possible periods indicated by Professor Newton, four were incompatible with the observed motion of the node, and it only remained to examine whether the fifth period of 33'25 years would give a motion in accordance with obser- vation. In order to determine the secular motion of the node in this orbit the method given by Gauss in his memoir Determinatio Attractionis &c. was employed. By dividing the orbit of the meteors into a number of small portions, and summing up the changes corresponding to these portions, the total secular changes of the elements produced in a complete period of the meteors was determined, the result being that during a period of 33'25 years, the longitude of the node is increased by 20' by the action of Jupiter, nearly 7' by the action of Saturn, and about 1' by that of Uranus. The other planets were found to produce scarcely any sensible effects, so that the entire calculated increase of the longitude of the node is about 28', agreeing very closely with the observed amount of 29', and leaving no doubt as to the correctness of the period of 33'25 years. In order to obtain a sufficient degree of approximation it was requisite to break up the orbit of the meteors into a considerable number of portions, for each of which the attractions of the elliptic rings corresponding to the several disturbing planets had to be determined. These calculations were therefore of necessity very long, although a modification of Gauss's formula was devised which greatly facilitated its application to the actual problem. Subsequently certain parts of the orbit of the meteors were subdivided into still smaller portions, with the view of obtaining a closer approximation. Unfortunately the mathematical investigations which Adams carried out on this subject have not been published. They exist among his papers, together with a great amount of numerical work connected with the calculations. In 1877 Mr G. W. Hill published a memoir on the motion of the Moon's perigee, in which he calculated that part of c which depends only upon m to fifteen places of decimals by a new method in which the expansion in powers of m was avoided, the numerical value of c being obtained by means of an infinite determinant. The publication of this memoir led Adams to communicate to the Royal Astronomical Society in November 1877 a brief notice of his own work in the same field, in which, after con- BIOGRAPHICAL NOTICE. xli gratulating Mr Hill upon his investigation, he mentions that his own researches had followed in some respects a parallel course. In particular he remarks that the differential equation for z, the Moon's coordinate perpendicular to the ecliptic, presents itself naturally in the same form as that to which Mr Hill had so skilfully reduced his differential equations. In solving this equation, which was therefore of Mr Hill's standard form, he fell upon the same infinite determinant as that considered by Mr Hill, and developed it in a similar manner in a series of powers and products of small quantities, the coefficient of each such term being given in a finite form. This development was continued as far as the terms of the fourth order in 1868; and in 1875, when he resumed the subject, the approximation was extended to terms of the twelfth order, which is the same degree of accuracy as that to which Mr Hill had carried his researches. On making the reductions requisite in order to render the two results comparable, he found that they were in agreement with the exception of one of the terms of the twelfth order, and that this discrepancy was due to a simple error of transcription. He states that the calculations by which he had found the value of the determinant were very different in detail from those required by Mr Hill's method, but that he had not had time to copy them out from his old papers and put them in order. In this communication, therefore, he confined himself to making known the result which he had obtained for the motion of the Moon's node. After giving an outline of the method pursued, including the equation derived from the infinite determinant, he arrives at the formulae by means of which the value of g, as dependent only upon m, was obtained to fifteen places of decimals. It is difficult to appreciate too highly the mathematical ability shown by Adams and Hill in devising methods which did not require expansion in powers of m, and which yielded with such wonderful accuracy these values of g and c. Apart, however, from the mathematical and astronomical interest of the researches themselves, the co- incidence of methods and ideas is very striking. But for the publication of Hill's memoir it is probable that no account of these results of Adams's would have been published in his lifetime, and it is not unlikely that he would never have put into writing his views on the mathematical treatment of the lunar problem which give additional interest to this short paper. As far back as 1853, in his memoir upon the secular acceleration, he mentioned that the new terms in the expression of the Moon's coordinates occurred to him some time before, when he was engaged in thinking over a new method of treating the lunar theory, and it is well known that the theory itself, or problems connected with it, constantly occupied his attention. In this paper of 1877 he states that he had long been convinced that the most advantageous mode of treatment is by first determining with all possible accuracy the inequalities which are independent of e, e, and 7, and then in succession finding the inequalities which are of one dimension, two dimensions, and so on with respect to these quantities. Thus, the coefficient of any inequality in the Moon's coordinates would be represented by a series arranged in powers and products of e, e, and 7 ; and each term in this series would involve a numerical coefficient which is a function of m alone, and which admits of calculation for any given value of m without the necessity of developing it in powers of m. This method is particularly advantageous when the results are to be compared with those of an analytical lunar theory such as Delaunay's, in which the eccentricities and the inclination are left indeterminate, since each numerical coefficient admits of a separate comparison with its xlii BIOGRAPHICAL NOTICE. analytical development in powers of m. He mentions also that, many years before, he had obtained the values of the inequalities independent of the eccentricities and inclination to a great degree of approximation, the coefficients of the longitude and those of the reciprocal of the radius vector, or of the logarithm of the radius vector, being found to ten or eleven places of decimals. Adams always preferred to treat the lunar theory as far as possible by means of its special problems; and this was also the method which he followed in his Cambridge lectures. In 1878 he published a short paper on a property of the analytical expression for the constant term in the reciprocal of the Moon's radius vector. Plana had found that the coefficients of e 2 and 7" in this term vanished when account was taken of terms in- volving m 2 and m 3 , and Pontdcoulant, who carried the development further, had found that this destruction of the terms in the coefficients still continued when the terms involving m* and m 6 were included. Thinking it probable that these cases in which the coefficient had been observed to vanish were merely particular cases of some more general property, Adams was led to consider the subject from a new point of view, and, so far back as 1859, he succeeded in proving that not only did these coefficients necessarily vanish identically, but that the same held good also for coefficients which were much more general, so that the coefficients of e 2 e' 2 , e 2 e' 4 , &c. y"e'' 2 , 7 2 e' 4 , &c. were also identically equal to zero. Further reflection on the subject led him in 1868 to obtain a simpler and more elegant proof of the property in question. He also obtained subse- quently, in 1877, some very simple relations connecting the coefficients of e 4 , &f, and 7*. Of this theorem he says himself that it " is remarkable for a degree of simplicity and generality of which the lunar theory affords very few examples." We thus see that a striking result and one moreover which admitted of being isolated from the rest of the lunar theory was obtained in 1859, but was not published till nearly twenty years afterwards, although in the meantime he had obtained another and more satisfactory proof. This illustrates the disinclination that Adams seems always to have felt to prepare his work for publication; a disinclination which was mainly due to his desire to obtain a still higher degree of simplification or perfection. The discovery of the additional relations in 1877 shows that his attention was at that time still occupied with the theorem of 1859. It may be remarked that Adams's shorter papers deserve more attention than their mere length might seem to entitle them to, not only because they frequently consist wholly of results derived from laborious researches, but also because they afford glimpses of the nature and extent of the work with which he was occupied. For forty-five years his mind was constantly directed to mathematical research relating principally to astronomy ; and it is evident that what he had accomplished is very inadequately represented by what has been published. It is also noticeable that so few of his papers should have appeared quite spontaneously: it frequently happened that he was incited to give an account of something which he had done himself probably years before by the publication of a paper in which the same ground was partially covered by another investigator, and in several cases he was called upon to correct misapprehensions which were leading others astray. As already stated, there can be no doubt that he constantly allowed himself to postpone the immediate publication of his researches, with the intention of effecting BIOGRAPHICAL NOTICE. xliii improvements in the processes and mode of representing the subject, or of attaining to an even more accurate result. A striking instance of this innate craving for per- fection is afforded, even as early as 1845, by his calculation of the second orbit of the new planet. No able mathematician who is engaged upon a fruitful research can continually defer publication with impunity : the subject opens before him ; his views expand ; the earlier results, so interesting at the moment of discovery, lose their charm in comparison with the problems still unsolved and the novel vistas of thought opened out by them ; and the rearrangement and rewriting of the old work always an irksome task become intolerable when later and still unfinished developments on the same subject are exciting the mind. In Adams's case the difficulty of satisfying himself, and reaching his own standard of completeness, also contributed to his apparent reluctance to publish his work. Those who knew him will remember his words when pressed, "I have still some finishing touches to put to it." It was well known that he made important researches upon the motion of Jupiter's satellites, and their pub- lication was anxiously awaited. It does not appear that he ever made any serious attempt to put his longer investigations in order for the press, though occasionally, as his manuscripts on the different subjects increased in bulk, the feeling would come over him strongly that it was time for him to do so. Although there is no similarity between the simple and easy style of Adams's writings and the cold severity of Gauss's, there is a certain resemblance in their mode of work. Each had the same dislike to early or incomplete publication, and " Pauca sed matura " might have been the motto of both. In beginning a new research, Adams rarely put pen to paper until he had carefully thought out the subject, and when he proceeded to write out the investigation he developed it rapidly and without interruption. His accviracy and power of mind enabled him to map out the course of the work beforehand in his head, and his mathematical instinct, combined with perfect familiarity with astronomical ideas and methods, guided him with ease and safety through the intricacies and dangers of the analytical treatment 1 . He scarcely ever destroyed anything he wrote, or per- formed rough calculations; and the manuscripts which he has left are written so carefully and clearly that it is difficult to believe that they are not finished work which has been copied out fairly. The sheets are generally dated, and during many years he kept a diary of the work he had done each day. His contributions to pure mathematics show the same power and excellence, and, as the subject affords greater opportunities for the display of elegance and style, they indicate even more plainly the attention he bestowed upon the form of his results, as well as upon the substance. A paper communicated to the Royal Society in 1878 may be specially noticed, in which an expression is given for the product of two Legendrian coefficients, and for the integral of the product of three. The extent of his mathematical interests is perhaps best seen by looking over the series of papers which he set in the Smith Prize Examination. These questions, which cover a wide 1 This method of working characterised him from the wrote out rapidly the problems he had already solved ' in first, for in his Tripos Examination it was noticed that his head'." It may be mentioned here that in this exami- " in the problem papers, when everyone was writing hard, nation he received more than double the marks of the Adams spent the first hour in looking over the questions, Second Wrangler. This affords striking evidence of scarcely putting pen to paper the while. After that he Adams's mental powers, for he was not a rapid writer. xliv BIOGRAPHICAL NOTICE. field of mathematics, clearly indicate the bent of his mind and his favourite subjects of study: they are also noticeable for a high degree of finish, which is very unusual in examination questions. Like Euler and Gauss, he took very great pleasure in the numerical calculation of exact mathematical constants. We owe to him the calculation of thirty-one Bernoullian numbers, in addition to the first thirty-one which were previously known. The first fifteen were calculated by Euler, and the next sixteen by Rothe, the whole thirty-one being given in vol. xx. of Crelles Journal. Making use of Staudt's very curious theorem with respect to the fractional part of a Bernoullian number, Adams calculated all the numbers from B& to B w . The results were communicated to the British Association at the Plymouth meeting in 1877, and were also published in vol. Ixxxv. of Crelle's Journal. A much fuller account of the work, which was very considerable in amount, appeared in an appendix to vol. xxil. of the Cambridge Observations, where the process of calculation of the first, B 32 , and of the last, B m , is given in detail. Adams proved that if n be a prime number other than 2 or 3, then the numerator of the nth Bernoullian number is divisible by n. This afforded a good test of the accuracy of the work. Having thus at his command the values of sixty-two Bernoullian numbers, he was tempted to apply them to the calculation of Euler's constant. For this purpose, not only the Bernoullian numbers, but also the values of certain logarithms and sums of reciprocals were required. He accordingly calculated the values of the logarithms of 2, 3, 5, and 7 to 263 (afterwards extended to 273) decimal places, and by their means obtained the value of Euler's constant to 263 places. He also calculated the value of the modulus of the common logarithms to 273 places. The papers containing these results appeared in the Proceedings of the Royal Society for 1878 and 1887. Anyone who has had experience of calculations extending to a great many decimal places is aware of the difficulty of manipulating with absolute accuracy the long lines of figures ; but this was an enjoyment to Adams, and the work, as carried out with consummate care and neatness, in his beautiful figures, is an interesting memorial of the patience and skill that he devoted to any work upon which he was engaged. Some may think that the portion of his own time occupied by these calculations might have been more advantageously spent : but there is a charm of its own in carrying still further the determination of the historic constants of mathematics, which has exercised its attraction over the greatest minds. Those who feel the least possible interest in calculation for its own sake, and even dislike ordinary arithmetical com- putations, have been unable to resist the fascination of doing their share towards the calculation of the absolute numerical magnitudes which are so intimately connected with the foundations of the sciences dealing with abstract quantity. There is a special pleasure also in applying the resources of modern mathematics to obtain the values of these incommensurable constants to such an incredible degree of accuracy, and in verifying the distant figures by methods depending upon subtle principles and com- plicated symbolic processes, of the absolute truth of which we thus obtain so striking an assurance. Adams had the greatest possible admiration for Newton, and perhaps no one has ever devoted more careful and critical attention to Newton's mathematical writings, BIOGRAPHICAL NOTICE. xlv especially the Principia. When Lord Portsmouth presented to the University, in 1872, the large mass of scientific papers which Newton left at his death, the arrangement and cataloguing of the mathematical portion of the collection was willingly undertaken by Adams. It was a difficult and laborious task, extending over years, but one which intensely interested him, and upon which he spared no pains. He found that these papers threw light upon the remarkable extent to which Newton had carried the lunar theory, the method by which he had obtained his table of refractions (showing that the formula known as Bradley's was really due to Newton), and the manner in which he had determined the form of the solid of least resistance. In several instances he succeeded in tracing the methods that Newton must have used in order to obtain the numerical results which occurred in the papers. The solution of the enigmas presented by these numbers written on stray papers, without any clue to the source from which they were derived, was the kind of work in which all Adams's skill, patience, and industry found full scope, and his enthusiasm for Newton was so great that he had no thought of time when so employed. His mind bore naturally a great resemblance to Newton's in many marked respects, and he was so penetrated with Newton's style of thought that he was peculiarly fitted to be his interpreter. Only a few intimate friends were aware of the immense amount of time he devoted to these manuscripts or the pleasure he derived from them. In 1888 the Cambridge University Press published a catalogue of the papers, the mathematical portion of which was wholly written by Adams 1 . In 1887, on the occasion of the bicentenary of the publication of the Principia, he was asked by Trinity College to deliver a commemorative address. Unfortunately the state of his health prevented him from undertaking a task which he alone could have adequately performed ; but, with the kindness which all who sought his help invariably received, he most freely placed all the stores of his knowledge at the disposal of the present writer, who was appointed in his stead. He was frequently asked to undertake calculations in connexion with eclipses or other astronomical phenomena, and he never hesitated to lay aside his own work in order to comply with such requests. Mr Downing has written : " His readiness to help, and his magnificent ability to help, will long be remembered at the Nautical Almanac Office," and similar words might be used with reference to the invaluable assistance which he so willingly gave in other quarters. For more than forty years he rendered constant 1 After proving a general proposition from which it this point, and he referred to the matter in a eommunica- follows that the disturbing action of the Sun necessarily tion on the lunar theory which he made to the Plymouth produces a continual advance of the Moon's perigee, meeting of the British Association in 1877. His remarks Newton gave a numerical example which has been on the subject were not put into writing by himself, generally regarded as his calculation of the theoretical but a verbatim report appeared in the Athenaum for amount of this advance in the case of the Moon (Lib. I. August 25, 1877. He also referred to Newton's ex- Sect, ix. Prop. xlv. Cor. 2). The concluding worda " Apsis planation of the motion of the perigee, and to his lunae est duplo velocior circiter," which have been quoted theory of astronomical refraction, in a communication to in support of the view that the motion of the lunar the Montreal meeting in 1884. The catalogue referred apsides is the question considered in the corollary, were to in the text, which was published subsequently to the however intended to have exactly the opposite meaning, dates of these communications, contains a brief statement as can be shown by comparing the three editions of the of all the principal results which he derived from the Principia. Adams found that some of the papers in the examination of the manuscripts. Portsmouth Collection afforded further confirmation on xlvi BIOGRAPHICAL NOTICE. service to the Royal Astronomical Society, both as a referee and as a contributor to the annual reports. These references and notices often cost him much time and thought. He was President of the Royal Astronomical Society for the second time in 1874-76, when the medal was awarded to D'Arrest and to Le Verrier. In 1870, as Vice-President, he delivered the address on the presentation of the medal to Delaunay, of whose general method of treating the lunar theory he had the greatest possible admiration. In 1881 he was offered the position of Astronomer Royal, which he declined. In 1884 he was one of the delegates for Great Britain to the International Prime Meridian Conference at Washington. He was also present at the meetings of the British Association at Montreal and of the American Association at Philadelphia in the same year. This visit to America afforded him great enjoyment and gratification. He received the honorary degree of D.C.L. from Oxford, of LL.D. from Dublin and Edinburgh, and of Doctor in Science from Bologna and from his own university. He was a correspondent of the French Academy, of the Academy of Sciences of St Petersburg, and of numerous other societies. As Lowndean Professor he lectured during one term in each year, generally on the lunar theory, but sometimes on the theory of Jupiter's satellites, or the figure of the Earth. His lectures on these subjects have been prepared for press by Professor Sampson, who has also examined Adams's other mathematical manuscripts and arranged for publication those which were sufficiently complete. During Adams's tenure of the directorship of the Cambridge Observatory in 1870 a fine transit circle by Simms was added to its equipment. This instrument has been employed in observing one of the zones of the " Astronomische Gesellschaft " programme. The zone assigned to the observatory was that lying between 25 and 30 of north declination. Adams was a man of learning as well as a man of science, and his thoughts and interests were far from being restricted to astronomy and mathematics. He was an omnivorous reader, and his memory being exact and retentive, there were few subjects upon which he was not possessed of accurate information. Botany, geology, history, and divinity, all had their share of his eager attention. He derived great enjoyment also from novels, and when engaged in severe mental work always had one on hand. Among his more marked tastes may be mentioned his love of early printed books. His collection, containing about eight hundred volumes, eighty of which belong to the fifteenth century, was bequeathed by him to the University Library. The works relate principally to mathematics or astronomy, theology, medicine, and the occult sciences; but he seems always to have bought any fine old book that took his fancy. He was so little given to talk about himself or his pursuits that probably but few of his friends were aware of his affection for black-letter books. It may be mentioned that his other mathe- matical books were bequeathed to the Libraries of St John's College and Pembroke College. No one who knew him superficially, or who judged only by his quiet manner, could have imagined how deeply he was affected by great political questions or passing events. In times of public excitement (such as during the Franco-German war) his interest was so intense that he could scarcely work or sleep. His love of nature in all its forms was a source of never-failing delight to him, and he was never happier than when wandering BIOGRAPHICAL NOTICE. xlvii over the cliffs and moors of his native county. Strangers who first met him were in- variably struck by his simple and unaffected manner. He was a delightful companion, always cheerful and genial, showing in society but few traces of his really shy and retiring disposition. His nature was sympathetic and generous, and in few men have the moral and intellectual qualities been more perfectly balanced. An attempt to sketch his character cannot be more fitly closed than in the words of Dr Donald MacAlister, who knew him well, and attended him in his last illness : " His earnest devotion to duty, his simplicity, his perfect self-lessness, were to all who knew his life at Cambridge a perpetual lesson, more eloquent than speech. From the time of his first great dis- covery scientific honours were showered upon him, but they left him as they found him modest, gentle, and sincere. Controversies raged for a time around his name, national and scientific rivalries were stirred up concerning his work and its reception, but he took no part in them, and would generously have yielded to others' claims more than his greatest contemporaries would allow to be just. With a single mind for pure knowledge he pursued his studies, here bringing a whole chaos into cosmic order, there vindicating the supremacy of a natural law beyond the imagined limits of its operation ; now tracing and abolishing errors that had crept into the calculations of the acknowledged masters of his craft, and now giving time and strength to resolving the self-made difficulties of a mere beginner, and all the while with so little thought of winning recognition or applause that much of his most perfect work remained for long, or still remains, unpublished." He was suddenly attacked by severe illness at the end of October 1889, but he recovered sufficiently to resume his mathematical work in the usual way for several months. In June of the following year he was again attacked by an illness from which he never completely recovered, and he passed away on the early morning of January 21, 1892, after being confined to his bed for ten weeks. The funeral service took place in Pembroke College Chapel, and he was interred in St Giles's Cemetery, on the Huntingdon Road. There were many who thought that his resting-place should have been in Westminster Abbey, and a royal wish was expressed to this effect; but it is perhaps more fitting that he should lie in this quiet graveyard close to the Observatory where he passed so many happy and peaceful years. On February 20, 1892, a public meeting was held at St John's College, with the view of taking steps to place a bust or other memorial of him in Westminster Abbey. The proceedings on this representative occasion bore eloquent testimony to the admiration and affection in which he was held by his friends, and to the widespread wish throughout the country for such a memorial to one who was not only a great but a good man 1 . No suitable site for a bust could be found in the Abbey, but a medallion has been placed in an admirable position close to the grave of Newton. This medallion, executed by Mr Bruce Joy, was unveiled on May 9, 1895, after a ceremony in the Jerusalem Chamber, at which addresses were delivered by leading members of the University and others. A bust, also executed by Mr Bruce Joy, which represents Adams in the later years of his life, was presented to St John's College by Mrs Adams in the same 1 A report of this meeting was published in a special number of the Cambridge University Reporter, March 10, 1892, p. 607. A. 9 xlviii BIOGRAPHICAL NOTICE. year. In 1888 an excellent portrait was painted by Herkomer, which is now in the Combination Room of Pembroke College; a replica is in the possession of Mrs Adams. The portrait in the Combination Room of St John's College was painted by Mogford in 1850 51. The Royal Astronomical Society also possesses a bust of Adams which was executed when he was a young man. J. W. L. G. PROFESSOR CHALLIS'S REPORT TO THE OBSERVATORY SYNDICATE, xlix PROFESSOR CHALLIS'S FIRST REPORT TO THE CAMBRIDGE OBSERVATORY SYNDICATE UPON THE NEW PLANET 1 . AT a meeting of the Observatory Syndicate, held at the Observatory on December 4, for the despatch of ordinary business, a strong desire having been expressed by the Vice-Chancellor and the members of the Syndicate generally, to receive from me a Special Report of Observatory proceedings relating to the newly-discovered Planet, drawn up in such a manner, and in such detail, as would enable them to lay complete information on the subject before the members of the Senate, I considered it to be my duty at once to comply with this request. A new body of the solar system has been discovered, by means depending on the farthest advances hitherto made in theoretical and practical astronomy, and confirming, in a most remarkable manner, the theory of universal gravitation. It is, therefore, on every account desirable that the members of the Senate should be made fully acquainted with the part which has been taken by the Cambridge Observatory, relatively to this important extension of astronomical science. The obser- vations I shall have to speak of, and the reasons for undertaking them, are so closely connected with theoretical calculations performed by a member of this University, to account for anomalies in the motion of the planet Uranus, that the history of the former necessarily involves that of the latter. I hope that for this reason, and because of the peculiar nature of the circumstances, I may be allowed to make a communication less formal and restricted in its character, than a mere Report of Observatory proceedings. The tables with which the observations of the planet Uranus have been uniformly compared, were published by A. Bouvard in 1821. They are founded on a continued series of observations extending from 1781, the year of its discovery, to 1821. Previous to 1781, it had been accidentally observed seventeen times as a fixed star, the earliest observation of this kind being one by Flamsteed in 1690. Bouvard met with a difficulty in forming his Tables. On an attempt to found them upon the ancient, as well as the modern, observations, it appeared that the theoretical did not agree with the observed course of the planet. He thought this might be attributed to the imperfection of the ancient observations, and consequently rejected all previous to 1781, in the formation of the Tables finally published. These Tables represent well enough the observations in the forty years from 1781 to 1821 ; but very soon after the latter year, new errors began to show themselves, which have gone on increasing to the present time. It 1 Thia report, which is headed ' Special Eeport of view to the discovery of the new planet." This preamble Proceedings in the Observatory relative to the new Planet,' is signed by the syndics, H. Philpott (Vice- Chancellor), is signed by Challis and dated December 12, 1846. It is John Graham, B. Chapman, W. Whewell, Joshua King, preceded by the following introductory remarks. "The Geo. Peacock, James Cartmell, Chas. W. Goodwin, syndicate appointed to visit the Observatory, conceiving W. C. Mathison, G. G. Stokes. Professor Challis issued the subject at the present time to possess peculiar interest, a second report to the Syndicate, dated March 22, 1847, beg leave to submit to the Senate the following statement relating to the subsequent observations of the new of Professor Challis, describing the course of observations, planet. This second report was reprinted in the Ai- founded on the theoretical calculations of Mr Adams, of tronomiache Nachrichten (Vol. xxv. col. 309). St John's College, and made at the Observatory with a 1 PROFESSOR CHALLIS'S REPORT was now evident that the ancient observations had been rejected on insufficient grounds, and that from some unknown cause the theory was in fault. Were the Tables cal- culated inaccurately ? The difference between observation and theory (amounting in 1841 to 96" of geocentric longitude) was too great, and Bouvard's calculations were made with too much care to allow of this explanation. The effect of small terms neglected in the calculation of the perturbations caused by Jupiter and Saturn, could not be supposed to bear any considerable proportion to the observed amount of error. This state of the theory suggested to several astronomers the idea of disturbances, caused by an undiscovered planet more distant than Uranus. But there is no evidence of this hypothesis having been put to the test of calculation previous to 1843. The usual problem of perturbations is to find the disturbing action of one body on another, by knowing the positions of both. Here an inverse problem, hitherto untried, was to be solved; viz. from known disturbances of a planet in known positions, to find the place of the disturbing body at a given time. Mr Adams, Fellow of St John's College, showed me a memorandum made in 1841, recording his intention of attempting to solve this problem as soon as he had taken his degree of B.A. Accordingly, after graduating in January 1843, he obtained an approximate solution by supposing the disturbing body to move in a circle at twice the distance of Uranus from the Sun. The result so far satisfied the observed anomalies in the motion of Uranus, as to induce him to enter upon an exact solution. For this purpose he required reduced observations made in the years 1818 1826, and requested my intervention to obtain them from Greenwich. The Astronomer Royal, on my application, immediately supplied (February 15, 1844) all the heliocentric errors of Uranus in longitude and latitude, from 1754 to 1830, completely reduced. Mr Adams was now furnished with ample data from observation, and his next care was to ascertain whether Bouvard's theoretical calculations were correct enough for his purpose. He tested the accuracy of the principal terms of the perturbations caused by Jupiter and Saturn, and concluded that the small terms which Bouvard had not taken into account would not sensibly affect the final results, the chief of them being either of long period or of a period nearly equal to that of Uranus. Besides which he introduced into the theory several corrections which had been derived from observation and calculation by different astronomers since 1821. The calculations were completed in 1845. In September of that year, Mr Adams placed in my hands a paper containing numerical values of the mean longitude at a given epoch, longitude of perihelion, eccentricity of orbit, mass, and geocentric longitude, September 30, of the supposed disturbing planet, which he calls by anticipation "The New Planet," evidently showing the conviction in his own mind of the reality of its existence. Towards the end of the next month, a communication of results slightly different was made to the Astronomer Royal, with the addition of what was far more important, viz. a list of the residual errors of the mean longitude of Uranus, for a period extending from 1690 to 1840, after taking account of the disturbing effect of the supposed planet. This comparison of observation with the theory implied the determination of all the unknown quantities of the problem, both the corrections of the elements of Uranus and the elements of the disturbing body. The smallness of the residual errors proved that the new theory was adequate to the expla- nation of the observed anomalies in the motion of Uranus, and that as the error of longitude was corrected for a period of at least 130 years, the error of radius vector was TO THE OBSERVATORY SYNDICATE. li also corrected. As the calculations rested on an assumption, made according to Bode's law, that the mean distance of the disturbing planet was double that of Uranus, without the above-mentioned numerical verification, no proof was given that the problem was solved or that the elements of the supposed planet were not mere speculative results. The earliest evidence of the complete solution of an inverse problem of perturbations is to be dated from October 1845. Although the comparison of the theory with observation proved synthetically that the assumed mean distance was not very far from the truth, it was yet desirable to try the effect of an alteration of the mean distance. Mr Adams accordingly went through the same calculations as before, assuming a mean distance something less than the double of that of Uranus, and obtained results which indicated a better accordance of the theory with observation, and led him to the conclusion, which has since been confirmed by observation, that the mean distance should be still farther diminished. This second solution taken in conjunction with the first may be considered to relieve the question of every kind of assumption. The new elements of the disturbing body, and the results of comparing the observed with the theoretical mean longitudes of Uranus, were communi- cated to the Astronomer Royal at the beginning of September 1846. These were accompanied by numerical values of errors of the radius vector, the Astronomer Royal having inquired, after the reception of the first solution, whether the error of radius vector, known to exist from observation, was explained by this theory. It would be wrong to infer that Mr Adams was not prepared to answer this question till he had gone through the second solution. Errors of radius vector were as readily deducible from the first solution as from the other. The preceding details are intended to point out the circumstances which led astronomers to suspect the existence of an additional body of the solar system, and the theoretical reasons there were for undertaking to search for it. No one could have anticipated that the place of the unknown body was indicated with any degree of exact- ness by a theory of this kind. It might reasonably be supposed, without at all mistrusting the evidence which the theory gave of the existence of the planet, that its position was determined but roughly, and that a search for it must necessarily be long and laborious. This was the view I took, and consequently I had no thought of commencing the search in 1845, the planet being considerably past opposition at the time Mr Adams completed his calculations. The succeeding interval to midsummer of 1846 was a period of great astronomical activity, the planet Astrsea, Biela's double comet, and several other comets, successively demanding attention. During this time I had little communication with Mr Adams respecting the new planet. Attention was again called to the subject by the publication of M. Le Vender's first researches in the Gomptes Rendus for June 1, 1846. At a meeting of the Greenwich Board of Visitors held on June 29, at which I was present, Mr Airy announced that M. Le Verrier had obtained very nearly the same longitude of the supposed planet as that given by Mr Adams. On July 9 I received a letter from Mr Airy, in which he suggested employing the Northumberland Telescope in a systematic search for the planet, offering at the same time to send an assistant from Greenwich, in case I declined undertaking the observations. This letter was followed by another dated July 13, containing suggestions respecting the mode of conducting the observations, and an estimation of the amount of work they might be expected to require. lii PROFESSOR CHALLIS'S REPORT In my answer, dated July 18, I signified the determination I had come to of undertaking the search. Various reasons led me to this conclusion. I had already, as Mr Adams can testify, entertained the idea of making these observations; the most convenient time for commencing them was now approaching; and the confirmation of Mr Adams's theoretical position by the calculations of M. Le Verrier appeared to add very greatly to the pro- bability of success. I had no answer to make to Mr Airy's offer of sending an assistant, as I understood the acceptance of it to imply the relinquishing on my part of the undertaking. I have now to speak of the observations. The plan of operations was formed mainly on the suggestions contained in Mr Airy's note of July 13. It was recommended to sweep over, three times at least, a zodiacal belt 30 long and 10 broad, having the theoretical place of the planet at its centre ; to complete one sweep before commencing the next ; and to map the positions of the stars. The three sweeps, it was calculated, would take 300 hours of observing. This extent of work, which will serve to show the idea entertained of the difficulty of the undertaking before the planet was discovered, did not appear to me greater than the case required. It will be seen that the plan did not contemplate the use of hour xxi. of the Berlin Star Maps, the publication of which was equally unknown at that time to Mr Airy and myself. It may be proper here to explain that the construction of a good star-map requires a great amount of time and labour both in observing and calculating, and that precisely this sort of labour must be gone through to conduct a search of the kind I had undertaken. The stars must first be mapped before the search can properly be said to begin. With a map ready made, the detection of a moving body, as it happened in this instance, might be effected on a com- parison of the heavens with the map by mere inspection. Not having the advantage of such a map, I proceeded as follows. I noted down very approximately the positions of all the stars to the llth magnitude that could be conveniently taken as they passed through the field of view of the telescope, the breadth of the field with a magnifying power of 166 being 9', and the telescope being in a fixed position. When the stars came thickly, some were necessarily allowed to pass without recording- their places. Wishing to include all stars of the llth magnitude, I proposed, in going over the same region a second time, to avail myself of an arrangement peculiar to the Northumberland Equatorial, the merit of inventing which is due to Mr Airy. The Hour-circle, Telescope, and Polar Frame are movable by clockwork, which may be regulated to sidereal time nearly. While this motion is going on, the Telescope and Polar Frame are movable relatively to the Hour-circle, by a tangent-screw apparatus, and a handle extending to the observer's seat. This contrivance enables the observer to measure at his leisure differences of Right Ascension however small, and therefore meets the case of stars coming in groups. The observations made by this method might include all the stars it was thought desirable to take, and therefore might include all the stars taken in the first sweep. The discovery of the planet would result from finding that any star in the first sweep was not in its position in the second sweep. If two sweeps failed in detecting the planet among the stars of the first sweep, it might be among the stars of the second, which would be decided by taking a third sweep of the same kind as the second. It will appear that this plan carried out would not only detect the planet if it were in the region explored, but would also, in case of failure, enable the observer to pronounce that it was not in TO THE OBSERVATORY SYNDICATE. liii that region. The second mode of observing required the aid of my two assistants, Mr Morgan and Mr Breen, in reading off and recording the observations. I commenced observing July 29, employing on that day the first method, with telescope fixed. The next day I observed according to the second method, with telescope moving. On August 4, the telescope was fixed as to Right Ascension, but was moved in Declination in a zone of about 70' breadth, the intention of the observations of that day being to record points of reference for the zones of 9' breadth. On August 12, the fourth day of observing, I went over the same zone, telescope fixed, as on July 30 with telescope moving. Soon after August 12, I compared, to a certain extent, the observations of that day, with the observations of July 30, taken with telescope moving; and finding, as far as I carried the comparison, that the positions of July 30 included all those of August 12, I felt convinced of the adequacy of the method of search I had adopted. The observations were continued with diligence to September 29, chiefly with telescope fixed, and were made early in Right Ascension for the purpose of exploring as large a space as possible before I should be compelled to desist by the approach of daylight. On October 1, I heard that the planet was discovered by Dr Galle, at Berlin, on September 23. I had then recorded 3150 positions of stars, and was making pre- parations for mapping them. The following results were obtained by a discussion of the observations after the announcement of the discovery. On continuing the comparison of the observations of July 30 and August 12, I found that No. 49, a star of the 8th magnitude in the series of August 12, was wanting in the series of July 30. According to the principle of the search, this was the planet. It had wandered into the zone in the interval between July 30 and August 12. I had not continued the former comparison beyond No. 39, probably from the accidental circumstance that a line was there drawn in the memorandum-book in consequence of the interruption of the observations by a cloud. After ascertaining the place of the planet on August 12, I readily inferred that it was also among the reference stars taken on August 4. Thus, after four days of observing, two positions of the planet were obtained. This is entirely to be attributed to my having, on those days, directed the telescope towards the planet's theoretical place, according to instructions given in a paper Mr Adams had the kindness to draw up for me. I would also beg to call attention to the fact that, after August 12, the planet was discoverable by a closet-comparison of the observations, a method of observing, depending on novel and ingenious mechanism, having been adopted by which I could say of each star, to No. 48, " This is not a planet," and of No. 49, " This is a planet." I lost the opportunity of announcing the discovery by deferring the discussion of the observations, being much occupied with reductions of comet observations, and little suspecting that the indications of theory were accurate enough to give a chance of discovery in so short a time. On September 29, I saw, for the first time, the com- munication presented by M. Le Verrier to the Paris Academy on August 31. I was much struck with the manner in which the author limits the field of observation; and with his recommending the endeavour to detect the planet by its disk. Mr Adams had already told me that, according to his estimation, the planet would not be less bright than a star of the ninth magnitude. On the same evening I swept a considerable breadth in Declination, between the limits of Right Ascension marked out by M. Le Verrier, and I paid particular attention to the physical appearance of the brighter stars. Out of liv PROFESSOR CHALLIS'S REPORT TO THE OBSERVATORY SYNDICATE. 300 stars, whose positions I recorded that night, I fixed on one which appeared to have a disk, and which proved to be the planet. This was the third time it was observed before the announcement of the discovery reached me. This last observation may be regarded as a discovery of the planet, due to the good definition of the noble instru- ment which we owe to the munificence of our Chancellor. From the reduced places of the planet, on August 4 and August 12, and from observations since its discovery extending to October 13, Mr Adams calculated, at my request, values of its heliocentric longitude at a given epoch, its actual distance from the Sun, longitude of the node, and inclination of the orbit, which were published as early as October 17. I am now diligently observing the planet with the meridian instru- ments, and when daylight prevents its being seen on the meridian, I propose carrying on the observations as long as possible with the Northumberland Equatorial, for the purpose of obtaining data for a further approximation to the elements of the orbit. My report of proceedings relating to the planet here terminates. I beg permis- sion to add a few remarks, which the facts I have stated seem to call for. It will appear by the above account, that my success might have been complete, if I had trusted more implicitly to the indications of the theory. It must, however, be remembered, that I was in quite a novel position : the history of astronomy does not afford a parallel instance of observations undertaken entirely in reliance upon deductions from theoretical calcula- tions, and those too of a kind before untried. As the case stands, a very prominent part has been taken in the University of Cambridge, with reference to this extension of the boundaries of astronomical science. We may certainly assert to be facts, for which there is documentary evidence, that the problem of determining, from perturbations, the unknown place of the disturbing body, was first solved here ; that the planet was here first sought for ; that places of it were here first recorded ; and that approximate elements of its orbit were here first deduced from observation. And that all this may be said, is entirely due to the talents and labours of one individual among us, who has at once done honour to the University, and maintained the scientific reputation of the country. It is to be re- gretted that Mr Adams was more intent upon bringing his calculations to perfection, than on establishing his claims to priority by early publication. Some may be of opinion, that in placing before the first astronomer of the kingdom results which showed that he had completed the solution of the problem, and by which he was, in a manner, pledged to the production of his calculations, there was as much publication as was justifiable on the part of a mathematician whose name was not yet before the world, the theory being one by which it was possible the practical astronomer might be misled. Now that success has attended a different course, this will probably not be the general opinion. I should consider myself to be hardly doing justice to Mr Adams, if I did not take this opportunity of stating, from the means I have had of judging, that it was impossible for any one to have comprehended more fully and clearly all the parts of this intricate problem ; that he carefully considered all that was necessary for its exact solution ; and that he had a firm conviction, from the results of his calculations, that a planet was to be found. ^^^ s -, ^f *& *r?^ " V; "-s/~ >*r ^^f^^y * J&e ^xi*C^ . J&*^r. <<* --- ***~ *"^>. ,^ ^'. <^ ^^ f <**' *^*- **Z=~. ' fZr^-^r^~ ffsr + 22 18'8} - 8-35 sin {2(k-6'-l)n' i(i \ (z-w) 2 z(z 3 ({-IW 4 -l^'-lU 2 1 n 3 2tn' + j \ > n / __\a ^ _ -2n) 2z(z-2n) 2z 2 -w 2 z(z-n)(z-2n)J ' da z(z-r^)(z-2n) da 2 f3 (i- ON THE PERTURBATIONS OF URANUS. 13 2 (-)*( -Sn) 2 z(z-2) 3 __ 2z(z-2) z (z - n) (z - 2n) z(z-n)(z-2n) da? 12. Now, if we assume , or a = sin 30 = 0*5, the values of the funda- a mental quantities b, a , , a 2 -j a , will be log& = 0-33170 log a ~ = 9-53765 loga 2 rf - =977848 log 6, =974497 loga 1 =9-83868 loga 3 -^ =970857 aa da. log & 2 = 9-32425 log a -j- =9-68012 loga 2 ^ =9'87776 log 6 3 = 8-94670 log a '- = 9'463 15 loga 2 -,-j = 9'86253 da. da~ Hence the principal inequalities of mean longitude, produced by the / action of a planet whose mass is , that of the Sun being unity, and ouuu , e ' the eccentricity of whose orbit is will be the following: - 36-99 m' sin {nt - n't + e - e'} + 58-97 TO' sin 2 {nt - n't + e - e'} + 5 '80 TO' sin 3 {nt - n't + e - e'} + 2 '06m' sin {n't + e' - CT} 4'30 me' sin {n't + ^ - nr'} + 31'25m' sin {nt 2n't + e 2e / + cr} 12-14 m'e' sin {nt 2n't + e 2e' + CT'} + 48 '5 5m' sin {2nt 3n't + 2e 3ef + ar} -93-01 ?n'e'sin {2<-3n'< + 2e-3e / + cr'}. 14 ON THE PERTURBATIONS OP URANUS. [2 To these may be added the following, which are of two dimensions in terms of the eccentricities : + 6'-57 m' sin 3 {nt -n't + e- e'} - 1 '08 m'e' sin {3 (nt - n't + e - e') - nr + ra-'}. These expressions may be put under the following form : ^, cos (n n')t + h^ cos 2 (n n')t + h 3 cos 3 ( n n'}t + &] sin (n n')t + & 2 sin 2 (n n')t + k 3 sin 3 ( n n')t +p l cos n't +p. 2 cos (n 2n') t +p 3 cos (2n 3n') t + q l sin n't + q,, sin (n 2n') t + q 3 sin (2n 3n') t. 13. Let the time of the mean opposition in 1810 be taken as the epoch from which t is reckoned ; this date, expressed in decimal parts of a year, will be 1810'328. Also, let 3 synodic periods of Uranus, = 3'0362 years, be taken for the unit of time ; then the change of the mean anomaly in an unit of time will be 13 0''5 ; also n=\3 0'-6, n' = 4 36''0 .-. n-w'=8 24'-6, n-2n' = 3 48'"6, 2n-3n'= 12 13''2. Hence the equations of condition given by the modern observations will be of the form c= 8c+ &e,coe{l3 0-5} t + Bx, cos {26 ro}t + t Sn + 8y, sin {13 0-5}* + 8y 1 sin{26 1'0} + h l coa{ 82'6}t+ /i 2 cos{!6 49*-2} + A, cos {25 13'8} t + k 1 sin{ 824'6}+ jfe, sin{!6 49'2}t + k,am{25 13'8} + >,cos{ 4 36-0} t+ p 3 cos{ 348-6}t+p 3 cos{l2I3-2}t + 5jSin{ 436'0}t+ ^ 2 sin{ 3 48'6} t + q 3 s\n{!2 13'2} in which t assumes all integral values from 10 to +10 in succession, and the several values of c" are contained in the table given in Article 9. 14. The final equations for the corrections of the elliptic elements will be found by multiplying each equation successively by the coefficients of Se, &n, &EI, and Sy,, which occur in it, and adding the several results. Let the equations be treated in a similar manner with reference to the quantities h lt k lt h. 2> k t> h. it k a , p^, q z , p 3 3 , 2] ON THE PERTURBATIONS OF URANUS. 15 It will be seen that, in consequence of the arrangement which has been given to the equations of condition, the equations thus formed naturally separate themselves into two groups, one of which involves only 8e, Sx lt Sx.,, with the quantities h and p, while the other involves 8n, Sy lt 8y. 2 , with the quantities k and q. Also the coefficients in these equations are easily calculated by the following formulae, putting t10 in their right-hand members: sin m (t + i) S2cosm = \ -- *1 sin fin (t + l) smmt t sin m(t+ 1) sin mt = . .,, 2 sur^m 1 fsin(m n) (t + ^) sin (m + n) (t + i)} t = - ->--.-, ;v \*' -\ --- -> , \ \ 2 [ sin%(m n) am % (m + n) j . , . 1 (avn.(m-n)(t + ) am (m + n) (t 22 sin mt sin n = - -4- -,-7- - 7 -^ <-- L --- : . , ; v . 2 (^ sin ^ (m n) sin (m + n) } , 1 lsinm(2 + l) 22 cos-m< = i + - + - - . v 2 2 sin m 1 lsuim(2+l) S2sm 2 m< =( + --- ^ -L. 2 2 sin m 15. By performing the calculations, the equations of the first group are found to be the following : (e) 15l'-48= 21-00003e+ 6-06708^- 4 "4358 Sx, + 13-6320 A, + 0-4043 A,- 4'5608 h, + 18-6046^ + 19-3384 p,+ 7'3721 j> 8 (x) 246-48- 6-0670 8e+ 8-2821&B.+ 4-17628a? a + 7-4041 A! + 8-2523 ^, 2 + 4'6963 h t + 6-5389^+ 6-3978 p z + 8'1831 p a 20974= 13-63208e+ 7*4041 So;,- 0'2337 8x 2 + 10-7022 h, + 4-5356 A a - O'OOIS A 3 + 127013^+12-9883 p,+ 8'0038 jp 3 242-68= 0-4043 8e+ 8-25238:*;,+ 7'56508a; 2 + 4-5356 A, + 10-2960 h,+ 8'1944 h 3 + 1-7866^+ 1-3667 #,+ 7'6671 p s 16 ON THE PERTURBATIONS OF URANUS. [2 (h 3 ) 86-67 = - 4-5608 Se + 4'6963 8^+10-5023 8z 2 0-0018 A, + 8-1944 /;,+ 10-7071 7i 3 3-0812^- 3-5347 > 2 + 3'8855 p., (p 2 ) 165-99= 19'3384Se + 6-3978SZ!- 3-49488ce 2 + 12-98837^ + 1-3667 h,- 3'5347 h t + 17-2795^+17-9106 p.+ 7'5423 p t (p 3 ) 242-56= 7-3721 Se + 8-1831&C.+ 3'4071 8x, + 8-0038 h, + 7-6671 h,+ 3'8855 h 3 + 7-6127^+ 7-5423 p 2 + 8"2019 p t . 16. By means of (e) eliminate 8e from each of the other equations, and these latter become (x) (h,) (h,) (h 3 ) 20272= 6-5294 8^ + 5-4577 8x 2 + 3-4658 /^ + 8'1355/t, + 6-0139 7i,+ 1-1640 jpj + 0'8109^ + 6'0533j> 3 111-41= 3-4658 8*, + 2-6458 Sx., + 1-8531 ^ + 4-2731 A, 0-4349 p, + 3'2183^ t 4'2731 h, +10'2882 h., 0-9944^,+ 7'5252p 3 -9588 A-, + 8*28224, -6652jp., + 5'4866p 3 0-4349^+ O'9944/i, 0-1024^,,+ 0-7535jt> 3 (p 3 ) 189-38= 6-05338^ + 4-96438^ + 3-2183^+ 7'5252/i, + 5-4866 A 3 +1'0815 ^ + 07535^+ 5'6139^ 3 . 17. Again, by means of (x) eliminate S.TJ from each of the other equations, and we find + 2-9588 h, + -6243 23976= S'1355 8oj, + 7'6504 + 8-2822 /i 3 +l-4284 119-57= 6-0139 8x t + 9-5389 + 9-7166 /t 3 + 0-9593 p, 26-50= 0-81098^ + 0-59008 + 0-6652 A 3 + 0'1470 ^ (h,) 3-807= -0-2512 7^-0-0452 7i 2 -0-2334// :( + 0-0065 p, + 0-0045 p, + 0'0052^ :i (7i 2 ) -12-821= 0-8502 8x,-0'0452 7^ + 0-1515 7^ + 0-7890 7; :j -0-0219^-0-0160^-0-0171^, 2] ON THE PERTURBATIONS OF URANUS. 17 -6l49= 4-5 1 20 Sx,- 0'2334 h, + 07890 h, + 4'1775 h 3 - O'l I28p, - 0'0817_p 2 - 0-0888 p., (p t ) 1-327= -0-0878Sa; 2 + 0-0045/i 1 -0-0160/i 2 -0-0817A 3 (p,) 1-448= - 0-0955 8a; 2 + 0-0052 ^-0-0171/^-0-0888713 + 0-0024^, + 0-001 8 2 + 0-0020p 8 18. Similarly, the equations of the second group are found to be (n) -171-27= 77-0000 Sn+ 9-39388^- I'21838y 2 + 8-8463 ,+ 7-3034 k,- 0'5927 k, + 57519 ?1 + 4-8755 q,+ 9'5583 q 3 (y) -166-33= 93-9380 Sn+ 1271798^+ 1'8907 Sy a + 11-2022 &,+ 11-0848 k,+ 2'6731 k, + 7-0956 ^+ 5-9913 , + 0-6670p 2 -0-9023^ 3 + 0-6574 4-0-8493 g,- 07451 g. 2 -0'4310 q, ON THE PERTURBATIONS OF URANUS. 21 24. From each of these equations eliminate Se, Sn, 8x lt and S?/ J; by means of the equations (c), ()i), (x), and (?/), before found, and we have the following : -14,2-0= 17265 8as, + 0-8412 /t,+ 1-9521 h,+ 1-3230 h a - 11-3691 8y 2 + 3'6001 4~2'8793 4- 10'9578 4 -1 -6779^-1-6400^+ 0'2249p 3 + 2-68 15 0, + 1-8369 & + 0'2995 ft -105'2=- 0-4681 Sa- 2 -0-73ll 4- 1'2776 h a - 0'0609 ^ 9-62498^ + 3-7087^-2-19264- 9"5426 k, -1-7765^-1-4924^+ 0'2786p 3 + 1-6997^+1-1014^+ 07934(7, -126-1=- 0-20358^-0-9653/^-1-47304+ 0"1937 h, 977198^ + 3-5895 ^-2-5827 4- 9-5123&, -17649^,- 1-4598 p 3 - 0'2133_p s + 1-5629^ + 1-0070^+ 0-8437?, -199-1=- 0-1917 8x,- 1-3218 ^+1-5284 h,+ 0-0260 4 9-8232 Sy., + 0'8943 4-3-4359 4- 9 '9270 k t -07901jp 1 -0-5885jp a - 0'3497j> :) + 0-2540^ + 0-1607^+ 0'4028 q 3 -1747= 0-2985 Sa;,,- 1-1595 ^,+ 1-6072 /!.,+ 0'5979 h, - 9-5788 Sy,+ 0-7062 4-2-9425 4- 9-5877 A, -0-6712^-0-4970^- 0'3251j9 3 + 0-1946^ + 0-1238^+ 0'3277 g 3 -166-7= 0-81718^-1-0088 ^, + 1-6018 4+ 1'1442&, 9-1122 8y 2 + 0-5586 4-2-48904- 9*02584 -0-5688^-0-4203^- 0'2956j> 3 + 0-1498^ + 0-0958^+ 0'2658 g 3 -114-2= 2-0482 8cc 2 - 0-6027 /^ + 1-2894 A 2 + 2'2661 h 3 6-6781 Sy., + 0-2576 4-1-3421 4- 6-4080 A, -0-3256^-0-2384^- 0'1971^ 3 + 0-0628^ + 0-0419^+ 0'1298 q 3 22 ON THE PERTURBATIONS OP URANUS. [2 72*4= 2-2815 8x 2 - 0-3786 , + 0-9257^ + 2-3601 7t, - 4-4181 8y, + 0-1283 &,- 07339 L- 4'1495 k 3 -0'1957p 1 -0-1428p 2 - 0'1286^ 3 + 0-0283 (7. + 0-0198 fc + 0'0671 q t - 42-0= 2-1139 Sax, -0-2652 /i., + 0'6985 h,+ 2'1241 A 3 3-1027 8y, + 0-0772 fc,- 0-4646 A,- 2'8790 , - 0-1348 JJ,- 0-0984 p.,- 0'0924^ 3 + 0-0154^ + 0-0114(7,+ 0-0412 q 3 25. The largest terms depending on the eccentricity of the disturbing planet occur in p 3 , q 3 ; it will be proper, therefore, to combine the above equations in such a manner that these quantities may acquire the largest coefficients possible. This will be done by multiplying each equation by a quantity nearly proportional to the coefficient of each of the unknown quan- tities p 3 and q 3 , and adding together the several results. It was thought unsafe to employ the first of the above equations, since it is derived from the single observation of Flamsteed made in 1690, twenty-two years anterior to any other observation. Hence the equation for finding p 3 may be formed by multiplying the above equations, taken in order, by -0-8, -0-6, +1-0, +1-0, +0-9, +0-6, +0'4, +0-3, beginning with the second ; and the equation for q 3 by multiplying the same equations by 1-0, 1-0, 0-5, 0-4, 0-3, 0-2, O'l, 0-1. Hence we obtain _474'l= 4-114 8x,- 2-817 A, + 7-837 A, + 4-528 A, -20745 8y 3 - 2789 ,-6-551 &.,-20'666 & ;) + 0-193^ + 0-377^- I'489p 3 -1-660^-1-078^- 0-054^ -485-0= 0-446 Sx,- 3'308 ^ -0'442 A, + 1'629A 3 -32-961 8y s + 8-267 A.-8-805 & 2 -32'546 k a -4-473^-3-643^.,+ Q-Q37 p, + 3-530 g, + 2-27 8 &+ 2'086 q 3 2] ON THE PERTURBATIONS OF URANUS. 23 26. Eliminate Sx, and 8y., from these equations by means of (x) and (y) and they become (3) -4767= -2-930A + 7'572h,+ 4'332A 3 - 2751 A;, - 6-348 & 2 - 20'350 k 3 + 0-155^ + 0-350^- l-686p 3 -1-653^-1-074(7,+ 0-047 & (4) -485-9= -3-463A,- 0-805 /* 2 + 1'360A 3 + 8-345 k, - 8-39 Ik,- 31 '900 3 -4-525^-3-679^- 0'233j9 3 + 3-545^ + 2-286^+ 2'292g 3 These equations, with (l) and (2) of Article 22, suffice for the solution of our problem. 27. Eliminate the left-hand members from equations (2), (3), (4), by means of equation (l), and we have 0= 0-4819^-0-5950^- 5'0570 /* 3 + 0-2063p 1 + 0-1475^ 2 + 0'4300j? 3 -0-6812^ + 3-2982 & 2 + 29'5618 ^-07804^-0-2375 g- 2 -0'4789 q a 0= -1-2005 ^ + 2-44667^ -24-0122 ^ + 0-9735^ + 0-9412^-0-8575^ -2-6633^- 5-8825 k,- 19'6219 & 3 - 1'6362 ^-1-0648 g. 2 + 0= - 17003 /t 1 -6-0294/i 2 -27'52954 1 -3-6908^ 1 -3-0772j9 2 + -, + 2'2954 -,, + 28. If now we put e e' = and e w = JS, it is easily seen that \=- 36-99 sin 6 m! -,= -36-99 cos 6 ,= 5 8 '97 sin 20 ' m m m m -, = 58-97 5-80 sin 30 +0-007460^ + 0-008974% m m 5-80 cos 30 -0-008974^ + 0-007460% m m 0-18 sin (0-p) - 0-046247 M cos 20 - J sin 20 24 ON THE PERTURBATIONS OF URANUS. [2 % = - 0'18cos(0-/3) +0-046247 j ^ sin 20 + % cos 201 m |m m J ^,= 24-91 sin (20-fl) + 0-13055 {% cos 6- % sin 01 m (m m J % = 24-91 cos (20-0) + 0-13055 M sin 0+ % cos 01 m [m m J 29. Substituting these expressions in the equations of Article 27, and putting for fi its value 50 15 /- 8, we obtain, after a slight reduction, 0= - (1-24782) sin + (1-40248) cos 0- (1-57155) sin 20 + (2-27388) cos 20 - (1 -46746) sin 30 + (2'23430) cos 30 + (9'10380) P -, - (9'48254) ^> + (8-28455) {& cos - -% sin 01 - (8'49138) fe sin 0+ -% cos 01 1 \m in' } v ' \m m - (7-97958){% cos 20--% sin 201 -(8-55742) fe sin 20 + -% cos 201 ; ^m' m' j ' \m m = (1-65083) sin + (1 -99378) cos + (2'14259) sin 20- (2-58192) cos 20 -(2-14400) sin 30-(2-05631) cos 30- (9-93475) ^-(8-91803)^ + (9-08947) j% cos 0-%sin 01 -(9-14306) fcsin + ^ cos 01 ' \m! m J v ' \m, m' j - (8-65341) -fe cos 20 - % sin 201 - (8'87892) -fe sin 20 + -% cos 20} v ' (m mf J ; [m m' J = (1 -79213) sin - (2'49403) cos - (2'55700) sin 20 - (2'56972) cos 20 -(2-20337) sin 30-(2'25714) cos 30 + (9-83632)^+(0-3115G)' /:! / m m - (9-60395 jjr| cos - % sin 01 + (9'47665) fe sin + Jj cos + (9-23220) fe! cos 20-% sin 201 + (9'21679) \ P ] sin 20 + % cos 20 ' [m' m' j ' [m m } where the numbers enclosed within parentheses denote the logarithms of the corresponding coefficients. 2] ON THE PERTURBATIONS OF URANUS. 25 30. These equations may be rapidly solved by approximation. The coefficients of ! and , in the first equation being small, we may find m m J from it an approximate value of 6, the substitution of which in the second and third equations will give approximate values of ^7 and , . By means of these a more accurate value of 6 may be found from the first equation, and the process being repeated, will enable us to satisfy all the equations as nearly as we please. Thus we find 0=-51 30', ^ : ; = 27l"-57, -%= -207"'24. m' m' Now e is known and =217 55' .'. e / = 269 25' the mean longitude of the disturbing planet at the epoch 1810'328. The sidereal motion in 36 synodic periods of Uranus = 5 5 12', precession = 30', .'. mean longitude at the time 1846762, or October 6, 1846, =325 7'. Also, the analytical expressions for *-l and -4 are * m m -^ = 48"-55 sin (30 -/3)- 93-01 e' sin (30-/3') -2* = 48 -55 cos (30 -)- 93 '01 e' cos (30 -ft') m' where e ix' = (3'. Equating these to the values given above, we find e' = 3-2206, ' = 262 28', and .'. cr' = 315 27'. Hence longitude of perihelion in 1846 = 315 57'. Lastly, substituting the values just obtained in equation (1), we find m' = 0-82816. 31. Hence the values of the mass and elements of the orbit of the disturbing planet, resulting from the first hypothesis as to the mean distance, are the following : 5-0-5 OK Mean Long, of the Planet, October 6, 1846 ' 325 7 o / Longitude of the Perihelion 31557 Eccentricity of the Orbit 0'16103 Mass (that of the Sun being 1) 0'0001656 These are the results which I communicated to the Astronomer Royal in October, 1845. A. 4 26 ON THE PERTURBATIONS OF URANUS. [2 32. I next entered upon a similar investigation, founded on the as- sumption that the mean distance was about -g^th part less than before, so that or a = sin 31 = 0'515. The method employed was, in principle, Cv exactly the same as that given before ; but the numerical calculations were somewhat shortened by a few alterations in the process, which had been suggested by my previous solution. 33. Assuming then that a = sin 31, the values of the quantities b, db 2 d' 2 b <*~j-> a j^> da da? log& = 0'33385 loga^ = 9'57333 loga 2 ~ log& 1 = 976106 log a ^ = 9-86149 log a 2 ^ 976573 2 = 9-35361 log a ^ = 971359 log a 2 ^ = 9 '92466 da. da. = 8-98918 Hence, by means of the formulae given before, the principal inequalities of the mean longitude of Uranus, produced by the action of a planet whose mass is ^-- , that of the sun being unity, and the eccentricity of whose ouuu orbit is , may be found to be the following : 2i\) - 4 2'-3 3 m' sin {nt-n't+e e'} + 76'55 m' sin 2 {nt - n't + e - e'} + 7'25m' sin 3 {nt n't + e-e'} + 2'34 m' sin {'< + e / -ro-} - 474 m'e' sin {n't + d - n/} + 4172 m' sin {nt - 2n't + c - 2e' + CT} -16-47 m'e' sin {nt - 2n't + e - 2e' + w'} + 33-93 m' sin {2nt - 3n't + 2e - 3e' + m} - 63-41 m'e' sin {2nt - 3n't + 2e - 3e' + */}. 2] ON THE PERTURBATIONS OF URANUS. 27 To these we may add the following, which are of two dimensions in terms of the eccentricities : + 6'-40 m' sin 3 {nt n't + e- e'} - 0'74 mV sin {3 (nt - n't + e - e') - OT + ra-'}. 34. Now, on our present assumption, *i=130'-6, ?i' = 448'-5, n-n' = 8l2'-l, n-2n' = 3 23''6, 2n-3n'= 11 35'7. Hence the equations of condition given by the modern observations will be of the form c= Se + 8*, cos {13 6-5}< + 8* 2 cos{26 1-0} + t Sri + 8?/, sin {13 0'5}< + 8y 2 sin {26 I'O}^ + hi cos { 8 12-1} t+ A 2 cos{l624-2}< + A 8 cos{2436-3}< + yt,sin{ 8 12'1}+ 4 sin{!624-2}i + A; 8 sin{2436-3} + Pi cos { 448'5}+ p., cos { 3 23-Q}t+p 3 cos {11 357} + g,sin{ 448'5}+ g 2 sin{ 3 23-6} + g- s sin {11 357} . 35. Treating these equations of condition in the same manner as before, the equations in the first group, derived from them, are found to be the following : (e) 151-48= 21-0000 Se + 6-0670803,- 4'4358 8x, + 13-9515 A,+ 0-9471 h a - 4'5965 h t + 18-3916^ + 19-6752 p,+ 8'4184 p, (x) 246-48= 6'0670Se+ 8-28218^+ 4'17628a; 2 + 7-3540 A,+ 8-3027 A 2 + 5'0961 h 3 + 6-5793 p,+ 6-3319 p,+ 8'0850 p 3 207-58= 13-9515 8e + 7-35408*,- 0'4177 Sx 2 + 10-9735 A,+ 4-6775 A 2 - O'OOOS h 3 + 12-8697 p,+ 13-4050 p,+ 8 '4781 p t 245-17- 0-9471 Se + 8-30278*,+ 7-23628*,, + 4-6775 /i,+ 10-0259 A 2 + 8'3220 h t + 2-3661 pi+ 1-6727 p,+ 7'3073 p 3 (h t ) 103-48= - 4-5965 Se + 5'0961 S*,+ 10'5558 Sa;, - 0-0005 A,+ 8-3220 /i, 2 +10'9749 A, - 2-8935 p,- 37316 ^) 2 + 3'5852 p 3 . 42 28 ON THE PERTURBATIONS OP URANUS. [2 36. Similarly the equations in the second group are (w) -171-27= 77-0000 8n+ 9-3938S?/,- 1-2183%, + 87355 k,+ 7-6213 k,- 0'0590 k 3 + 5-9764 2V + 4-3875 q,+ 9 '61 52 ,- 15-4639 p,+ 2'4144 p 3 194-94= -11-7320 8n+ 3'6520 %,+ 15'0680 Sy a + 0-1951 k,+ 7-3294 ^+147009 k 3 - 0-5785 q,- 0'5496 q,+ 2'3663 q 3 . ON THE PERTURBATIONS OF URANUS. 29 38. By means of (e) and (n) of Articles 35 and 36, eliminate Se and from (x) and (y), and also from the equations just found, and we have (x) 20272= 6-5294 8*,+ 5'4577 Sa;,, + 3-3234 A,+ 8'029l h, + 6-4240 A,+ 1-2659 ^ + 0-6477^,+ 5'6529p 3 (y) 42-61= l-25788y,+ 3'3771 8y, + 0'3822 k,+ 2-0739 A, + 3-3916 k a + 0-0836 ^ + 0*0298^+ 0-9513^ 268-02= 11-1297 8.^+ 14-4919 8a; 2 + 5-3967 A, + 14-4642 h, + 15-4449 A :t + 2-0175 # + 1-0266^ + 9'4702^ 3 168-85= 5-0833 8^+14-8824 Sy, + 1-5261 k,+ 8'4906 k, + 14-6979 k,+ 0-3320 7, + 0-1189 ,+ 3-8313^. 39. Substituting for 8x 3 , 8y.,, their values in terms of 8x,, By lt we find 6-52948^+ 5-45778*,= 6'5700 S^ + 0'0490 8^ 1-25788^+ 3-37718^=- 0'0303 8aj,+ T2829 Sy, 11-1 297 8*, + 14-49 198*,= 11-2378 Sx, + 0-1300 8y, 5-08338^+14-88248^= - 0-13358^ + 5-19438^. Hence, if we add to the two latter equations -1-7106 (x)- 0-03607 (y) and 0-00165 (*)- 4-0487 (y) respectively, 8x, and 8_y, will be eliminated, and we shall obtain the following equations : (1) 80 /r 28= 0-2883 ^-07295 A, -4-4559 A, + 0-0138 ^ + 0-0748 ^ + 0'1223 k 3 + 0-1479^ + 0-0813^3,, + 0-1997 p 3 + 0-0030 ^ + 0-0011 q, + 0'0343 q 3 (2) 3-34= -0-0055 A, -0-0132 A,-0-0106 A, + 0-0212 A,- 0-0939 ^ 2 -0'9662 k 3 -0-0021^-0-0011^-0-0093^3 + 0-0066 7, + 0-0017 g 2 + 0-0203 &. 30 ON THE PERTURBATIONS OF URANUS. [2 40. Again, the equations of condition given by the ancient observations are 62-6= Se -0-8776 8x,+ 0'5402 &c a + 07923 h, + 0'2554 h, - 39-31 8-0'47958y 1 + 0-84158y s + 0-6101 , + 0-9668 k, -0-3875 A 3 - 0-9877 ^,-0-6870^-0-1009^, + 0-9219 , + 0-1566 - 07267 3 2-8583 & 3 + 4-6536 q,+ 1'9595 (/ + 1 17796 q 3 . 43. Substituting for 8x.,, By.,, their values in terms of Bx,, Sy,, we find -4-1831 &c,+ 7-1533 St/, + 0-6179 S 2 - 1'5388 8y,= -4'16478.T 1 + 7-14738^ -0-7686 8^+10-6926 Syi-0'1963 8*.,-4-2508 Sy.,= -07319 S^ + 10'6591 By,. Hence, if to the equations just found we add + 0-60808 (x)- 5-5942 (T/) and + 0-07 306 (a;)- 8 '31 10 (y) respectively, Bsc t and By, will be eliminated, and we shall obtain the following equations : (3) -476-84= -27630 A, + 6-9793 A, + 4'6473 A, -2-8290^-5-1777 k t -20'22^2k t + 0-0698^ + 0-3785^,- 2-5884p 3 - 1 7748 q, - 0-8036 q, - 0'2693 q, (4) - 486-08- -37091 A, -0-9682 A, + 2-2600 A, + 8-3364*,- 7-5348 ,-31 '0457 k, -4-6988^-3-1454^- 0'3772^ 3 + 3-9584^+17118 q,+ 3'8734 q t . 44. Eliminate the left-hand members from equations (2), (3), and (4) r of Articles 39 and 43, by means of equation (l), and we have 0= 0-4200 A, -0-4114 A,- 4'2014 -0-4964 &, + 2-3306 A; 2 + 23-3213 ^,-0-1567 ^-0-0409 2 2 -0-4531 q, 0= -1-0507 7^ + 2-6465 A 2 -21-8182A 3 + 0-9482^ 1 + 0-8614^ 2 - 1-4023^ 3 -27471 ^-47334^-19-4976^-17569 ^-0'7972 ?,- 0*0655 ? ( 0= -1-9638^-5-3845 A 2 -247155A 3 -3-8034p 1 -2-6532_p 2 + 0-8317p :t + 8-4199 A-,- 7-0819 ^-30'3051 ^ 3 +3'9767 g,+ l-7183 g., + 4-0811 q 3 . 2] ON THE PERTURBATIONS OF URANUS. 33 45. If, as before, we put e e' = 0, and e w = /3, it may be seen that , = - 42-'33 sin -> = 76"'55 sin 20 TO' TO' 7 7i % = -42-33 cos ^7 = 76-55 cos 20 TO TO % = 7-25 sin 30 + 0-007460 ^+0'008.974 % TO TO TO ^L = 7-25 cos 30 -0-008974 ^ + 0-007460% m TO m ^,= 0-20 sin ( - fi) - 0-074738 -^" cos 20-% sin 5 m [-HI m ^= - 0-20 cos ( 0-/3) + 0-074738 j-" sin 20 + % cos 201 m [TO fn } ^.= 32-91 sin (20- fi) + 0-259765 \^~ cos 0-%sin 01 m [TO TO J %= 32-91 cos (20-/3) + 0-259765 |^ sin + ^cos 01. tit ^ j 46. Substituting these expressions in the above equations, and putting for its value, 50 15' 8, we obtain 0= -(1-24872) sin + (1-32231) cos 0- (1-481 10) sin 20 + (2-24265) cos 20 -(1-48373) sin 30 + (2-22809) cos 30 + (9'26254) ^-(9-50079)^ + (8-44376) {g cos 0-| sin 0} -(8'02630) {^ sin + |, cos 0} - (8-17031) -fe cos 20-J^ sin 201 -(8'06861) fe sin 20 + J^, cos 20J . = (1-65190) sin + (2'06584) cos + (2'30220) sin 20 - (2'60306) cos 20 -(2-19916) sin 30-(2'15032) cos 30-(0'14305)|^- (9-60933) |> + (9-34981) {fcos0-| / sin0}-(9-31615){f;sin0+|cos0} - (8-85046) fe cos 20 -^ sin 201 - (9'1 1828) j|, 3 sin 20 + 1> cos 20| . 34 ON THE PERTURBATIONS OF URANUS. [2 = (1-91407) sin 6- (2-55189) cos 0- (2*62790) sin 20- (2*64230) cos 20 -(2-25331) sin 30- (2*34185) cos 30 + (9*96344) ^+ (0*56029)% TO' ' TO' - (9-83835) -fe cos 6 - % sin 4 + (9'64968) fe sin + % cos 01 ' [m' m j ' \m! m 1 J + (9-45371) I^COS 20- % sin 201 +(9'47306) -fe sin 20 + % cos 201 , 7 [TO m J ' [m m J where the numbers enclosed within parentheses denote the logarithms of the corresponding coefficients, as before. 47. From these equations we find, by the same method as before, 0=-4655' ^=138"*92 %=-109"*83 TO m Hence, since e = 21755', e' = 26450', the mean longitude of the disturbing planet at the epoch 1810*328. The sidereal motion in 36 synodic periods of Uramis = 57 42', Precession = 30'. .'. mean longitude at the time 1846762, or October 6, 1846, =323 2'. Also, the expressions for . and . are . TO TO - t = 33"*93 sin (30 -ft) - 63 /: 41 e' sin (30 -/3') m -^ = 33-93 cos (30 -ft)- 63-41 e' cos (30 -ft') ; TO' where c ny' ft'. Equating these to the values given above, we find e' 2*4123, ft' 279 14', and .'. w' = 2984l'. Hence longitude of the perihelion in 1846 =299 11'. Lastly, substituting the values just obtained in equation (1) of Article 39, we find m' = 075017. 48. Hence the values of the mass and elements of the orbit of the disturbing planet, resulting from the second hypothesis as to the mean distance, are the following : -, = 0*515 a ON THE PERTURBATIONS OF URANUS. 35 Mean longitude of the Planet, October 6, 1846... 323 2 Longitude of the Perihelion 299 11 Eccentricity of the Orbit 0'120615 Mass (that of the Sun being 1) 0'00015003. nr\ fj 49. From the values of m', 6, ,, and --. , found above, the values m m of the quantities h, k, p, and q, corresponding to each hypothesis, are im- mediately determined. Thus we find, IST HYPOTHESIS. a' h,= 23-98 ,= - 19"-07 h,= - 47-58 k,= - ll'OO *,= - 7-64 <7,= - 8-31 h t =- 1-93 9-93 = - 8-54 = 224-90 q. 2 = - 55-36 < 3 =- 171-63 2ND HYPOTHESIS. ", = 0-515 a h,= 23-19 A, = -21-69 A 2 = - 57'30 k. 2 = - 3-83 4,= - 3-40 k,= - 5-76 p,= 6-52 ^=-7-34 p,= - 11-62 q,= - 54-39 p 3 = 104-21 q a = - 82-39 50. And by substituting these values in the equations (e), (n), (x), and (y), we obtain IST HYPOTHESIS. a Se = - 4977 8n = - 0702 80;,= -130-69 817,= 222-38 8x 2 = 1-02 8y 3 = 2-83 2ND HYPOTHESIS. "=0-515 a 8e = -43-23 8n = - 0'5417 8^= 177 8^= 123-98 8x.,= 1-13 8w,= 0-91 and the corresponding corrections of the elliptic elements will be = 0-00000999 Sa - = 0-00000771 a 8e= 20-83 8e- 40-31 = 47-10 52 36 ON THE PERTURBATIONS OF URANUS. [2 It will be seen that the corrections of the eccentricity and longitude of perihelion vary very rapidly with a change in the assumed mean distance. 51. If these quantities be substituted in the expressions before given, we obtain the following theoretical corrections of the mean longitude, each of these corrections being divided into two parts, of which the first is due to the changes in the elements of the orbit of Uranus, and the second to the action of the disturbing planet. HYPOTHESIS I. Ancient Observations. Year. 1712 -288-0 + 365-8= +77'8 1715 -283-1+357-1= +74-0 1750 +210-5-260-7= -50-2 1753 +218-1-267-0= -48-9 1 756 +214-0 - 260-0 = - 46-0 1764 +154-0-1867= -32-7 1769 + 79-6-1007= -21-1 1771 + 27-6- 41-8= -14-2 Modern Observations. Year. ,, Year. 1780 -126-12+129-27=+ 3'15 1813 - 125'59 + 14772= +22'13 1783 -180-28 + 188-70=+ 8'42 1816 - 68'21 + 91'02=+22'81 1786 -227-66 + 240-36= +12-70 1819 10'40+ 33'18=+2278 1789 -265-70 + 281-63= +15-93 1822 + 44'84- 23'64= +21-20 1792 -292-25 + 310-38= +18-13 1825 + 94'69- 77'64=+17'05 1795 -305-84 + 325-27= +19-43 1828 +13673-127-48=+ 9'25 1798 -305-67 + 325-72= +20-05 1831 + 168-94- 172-17 =- 3'23 1801 -29177 + 312-05= +20-28 1834 + 189'85 -21T04 = -2T19 1804 -264-95 + 285-38= +20-43 1837 + 198'51 -243'59= -45'08 1807 -226-78 + 247-51= +2073 1840 + 194'54-269'36 = -74'82 1810 -179-43 + 20076= +21-33 2] ON THE PERTURBATIONS OF URANUS. 37 HYPOTHESIS II. Ancient Observations. 1712 -1337 + 211-9= +78-2 1715 -1177 + 191-5= +73-8 1750 + 85-2-134-4= -49-2 1753 + 73-8-122-2= -48-4 1756 + 59-1-105-2= -46-1 1764 + 2-7- 36-4= -33-7 1769 43-1+ 20-8= -22-3 1771 - 69-9+ 54-7= -15-2 Modern Observations. Year. 1780 -133-10+135-98=+ 2-88 1783 -149-47 + 157-87=+ 8'40 1786 -160-15 + 172-99 =+12-84 1 789 - 164-52 + 180-64 = + 16'12 1792 -162-30+180-58= +18-28 1795 -153-59 + 173-07= +19-48 1798 -138-87 + 158-86= +19-99 1801 -118-95 + 139-08= +20-13 1804 - 94-96 + 115-21= +20-25 1807 - 68-25+ 88-85= +20-60 1810 - 40-33+ 61-61= +21-28 Year. 1813 -1272+ 34-91 =+22-19 1816 +13-08+ 9-88= +22-96 1819 +35-71- 1274= +22-97 1822 +54-04- 32-68= +21-36 1825 +67-18- 50-08= +17-10 1828 +74-52- 65-37=+ 9"15 1831 +7574- 79-21=- 3'47 1834 +70-85- 92-31 =-21-46 1837 +60-08 -105-25 =-45-17 1840 +43-98-118-38= -74-40 52. Comparing these with the corrections of mean longitude derived from observation, we find the remaining differences to be the following : 38 ON THE PERTURBATIONS OF URANUS. ANCIENT OBSERVATIONS. Observation - Theory. Year. Hypoth. I. Hypoth. II. 1712 + 67 + 6-3 1715 - 6-8 - 6'6 1750-1-6 - 2'6 1753 + 57 + 5-2 1756 - 4-1 4-0 1764 - 5-1 - 4-1 1769 + 0-6 + 1'8 1771 +11-8 +12-8 MODERN OBSERVATIONS. Observation - Theory. Year. Hypoth. I. Hypoth. II. 1780 +0-27 + o'-54 1783 -0-23 -0-21 1786 -0-96 -1-10 1789 +1-82 +1-63 1792 -0-91 -1-06 1795 +0'09 +0'04 1798 -0-99 -0-93 1801 -0-04 +0-11 1804 +176 +1-94 1807 -0'21 -0-08 1810 +0-56 +0-61 Observation - Theory. Year. Hypoth. I. Hy'poth. II. 1813 -0'94 1816 -0-31 1819 -2-00 1822 +0-30 1825 +1-92 1828 +2'25 1831 -1-06 1834 -1-44 1837 -1-62 1840 +173 -1-00 -0-46 -2-19 + 0-14 + 1-87 + 2-35 -0-82 -1-17 -1-53 + 1-31 The largest difference in the above table, viz. that for 1771, is deduced from a single observation ; whereas the difference immediately preceding it, which is deduced from the mean of several, is very small. 53. The results of the two theories agree very closely with each other, and with observation, till we come to the later years of the series ; and it is to be observed, that the difference between the theories becomes sensible at precisely the point where they both shew symptoms of diverging from the observations, the errors of the second hypothesis, however, being less than those of the other. Recent observations shew that the errors of the theory soon become very sensible, though decidedly less for the second hypothesis than for the first. The following are the differences of mean longitude, as deduced from theory and observation, for the oppositions of 1843, 1844, and 1845 : Year. 1843 1844 1845 Observation - Theory. Hypoth. I. Hypoth. II. + 7-11 + 879 + 12-40 + 577 + 7-05 + 10-18 2] ON THE PERTURBATIONS OF URANUS. 39 For the observations of the last two years, I am indebted to the kindness of the Astronomer Royal. The three years nearly agree in shewing that the errors of the first hypothesis are to those of the second in the ratio of 5 to 4, from which I inferred, in a letter to the Astronomer Royal, dated September 2, 1846, that the assumption of , =sin 35 = 0'574, would ct probably satisfy all the observations very nearly. 54. The results which I have deduced from Professor Challis's obser- vations of the planet, strongly confirm the inference that the mean distance should be considerably diminished. It is of course impossible to determine precisely, without actual calculation, the alteration in longitude which would be produced by such a diminution in the distance. By comparing the values of 6 given by the two hypotheses, it may be seen, however, that if we took successively smaller and smaller values for the mean distance, the values found for the mean longitude in 1810 would probably go on diminishing, while at the same time the mean motion from 1810 to 1846 would rapidly increase, so that the corresponding values of the mean longitude at the present time would probably soon arrive at a minimum, and afterwards begin again to increase. This I believe to be the reason why the longitude found on the supposition of too large a value for the mean distance agrees so nearly with observation. In consequence of not making sufficient allowance for the increase in the mean motion, I hastily inferred, in my letter to the Astronomer Royal mentioned above, that the effect of a diminution in the mean distance would be to diminish the mean longitude. 55. I have already mentioned, that I thought it unsafe to employ Flamsteed's observation of 1690 in forming the equations of condition, as the interval between it and all the others is so large. The difference between it and the theory appears to be very considerable, and greater for the second hypothesis than for the first, the errors being +44 //< 5 and + 50"'0 respec- tively. These errors would probably be increased by diminishing the mean distance. It would be desirable that Flamsteed's manuscripts should be examined with reference to this point. "56. The corrections of the tabular radius vector of Uranus may be easily deduced from those of the mean longitude by means of the following formula : 40 ON THE PERTURBATIONS OP URANUS. [2 ._ __ _ 2 o ~"SB 4a 2 1 -e 2 6 n '' da m' + - a cos {i (nt n't + e e')nt e + rs-} i cos {i (nt -rit + e- e') - nt - e + B/} where 8 denotes the whole correction of the mean longitude at the time t, I dr . , , 3e 3 . - T~ = e sin {wi + e TO-} + sin 2 |i + e w| nearly, *-/.' ^~ i assuming all integral values positive and negative not including zero. 57. By substituting in this formula the values of m', Sa, 8e, &c., already obtained, and putting a=19'191, we find the following results corresponding to the two assumed values of the mean distance. HYPOTHESIS I. -- r r df. 2 ndt + 0*000069 cos {nt-n't + e-^} + 0-000259 cos 2 {nt - n't + e - e'} + 0-000109 cos 3 {nt-rit + e-e'} + 0-000016 cos {n't + ^-nr} -0-000168 cos + 0-000078 cos {nt - Zn't + e - 2^ + -0-000049 cos + 0-000209 cos 2] ON THE PERTURBATIONS OF URANUS. 41 HYPOTHESIS II. a ~ adr- adS - 8r --rOL, - -,- 0-000144 r r c/e 2 wcu + 0-000073 cos {nt - n't + e - e'} + Q'000266 cos 2 {n - n' + c - e'} + O'OOOl 1 5 cos 3 {nt -n't + e- e'} + 0-0000 16 cos {n't + f?-vr} -0-0001 88 cos + 0-000068 cos -0-000053 cos + 0-0001 65 cos 7S / 58. The values of 8 and y. for some recent years are the following: HYPOTHESIS I. Year. // // 1834 - 21-19 -20-93 1840 74-82 -32-34 1846 -148-65 -39-94 HYPOTHESIS II. 1834 21-46 -2o'-85 1840 74-40 -31-62 1846 -145-91 -38-30 Hence, by means of the above formulae, we find the corrections of the tabular radius vector to be Year. Hypothesis I. Hypothesis II. 1834 +0-00505 +0-00492 1840 +0-00722 +0-00696 1846 +0-00868 +0-00825 A. 6 42 ON THE PERTURBATIONS OF URANUS. [2 59. By far the most important part of these corrections arises from 1 7SJ X the term ~^ ) r ~r f ' an( i ma j therefore be immediately deduced from a com- parison of the observed angular motion of Uranus with that given by the tables. In fact, the corrections given by this term alone for the epochs above mentioned are Year. Hypothesis I. Hypothesis II. 1834 +0-00447 +0-00445 1840 +0-00694 +0-00678 1846 +0-00853 +0-00818 which, as we see, differ very little from the complete values just found. The correction for 1834 very nearly agrees with that which Mr Airy has deduced from observation in the Astronomische Nachrichten (No. 349). The corrections for subsequent years are rather larger than those given by the Greenwich Observations, the results of the second hypothesis, as in the case of the longitude, being nearer the truth than those of the first. 60. I made some attempts, by discussing the observations of latitude, to find approximate values of the longitude of the node and inclination of the orbit of the disturbing planet, but the results were not satisfactory. The perturbations of the latitude are, in fact, exceedingly small, and during the comparatively short period of three-fourths of a revolution are nearly con- founded with the effects of a constant alteration in the inclination and the position of the node of Uranus, so that very small errors in the observations may entirely vitiate the result. 61. The perturbations of Saturn produced by the new planet, though small, will still be sensible, and it would be interesting to enquire whether, if they were taken into account, the values of the masses of Jupiter and Uranus found from their action on Saturn would be more consistent with those determined by other means than they appear to be at present. The reduction of the Greenwich planetary observations renders such an inquiry comparatively easy, and it is to be hoped .that English astronomers will not be the last to avail themselves of the treasures of observation thus laid open to the world. THE SEARCH FOR THE PLANET NEPTUNE BY PROFESSOR CHALLIS. [From the Astronomische Nachrichten. No. 583 (1846). Pp. 101106.] CAMBRIDGE OBSERVATORY, October 21, 1846. My more immediate purpose in writing to you at present, is to give some account of observations which I undertook this summer in search of the recently-discovered planet. Mr Adams, a young Cambridge mathematician, had for a long time turned his attention to the perturbations of Uranus, and in the autumn of last year communicated to me and to Mr Airy, the Astronomer Royal, values which he had obtained of the heliocentric longitude, mass, eccentricity of orbit, and longitude of perihelion of a sup- posed disturbing planet, revolving at a mean distance from the Sun about double that of Uranus. These results were deduced entirely from a con- sideration of perturbations of Uranus not otherwise accounted for. M. Le Verrier, by an investigation published in June last, obtained almost precisely the same heliocentric longitude which Mr Adams had arrived at. This coincidence from two independent sources very naturally inspired confidence in the theoretical deductions, and accordingly Mr Airy shortly after suggested to me the employing of the Northumberland telescope of this Observatory in a systematic search after the planet. I commenced observing July 29. 62 44 THE SEARCH FOR THE PLANET NEPTUNE. [2 Unfortunately I was not then aware of the publication of hour XXI of the Berlin star-maps, and consequently had to proceed on the principle of comparison of observations made at intervals. On July 30 I recorded the approximate places of stars in a zone 9' in breadth, in such a manner as to be sure that none brighter than the llth magnitude escaped me, which a psculiar arrangement in the construction of the Northumberland Equatorial enabled me to do. On August 4 I took the places of the brighter stars in a zone 80' broad, and among these recorded a place of the planet. My next observations were on August 12, on which day I met with a star of the 8th magnitude in the zone which I had taken on July 30, which did not then contain this star. This again was the planet. So exactly had theory indicated the proper place for making the search, that in four days only of observing I had recorded two positions of the planet. Also according to the principle of search I had adopted, the observations of two of those days (July 30 and August 12) were sufficient to discover it. My time, however, was so occupied with comet reductions, and so little expec- tation had I of discovering the planet by a brief search, that I was only just preparing to map the places of the stars to see what success I had had, when the announcement of the discovery reached me. My observations after August 12 were purposely made early in Right Ascension for the sake of being able to carry them on during a longer portion of the year. Accordingly I did not again meet with the planet till September 29, on which day I saw for the first time the results of M. Le Verrier's last investigations. By these I was induced to return again to the theoretical position of the planet, and to endeavour to detect it by the appearance of a disk. In fact on the night of September 29, out of a very large number of stars whose approximate places I recorded. I fixed upon one which appeared to me to have a. disk, and which proved to be the planet. On October 1 I had intelligence of Dr Galle's discovery. The foregoing account, while it shews that I cannot lay claim to any discovery, may perhaps be regarded with some degree of interest. In particular, the places which I have obtained for the planet on August 4 and August 12, though they cannot pretend to great accuracy, for the present possess a value which they will lose when accurate observations have been continued for a longer period. I have, therefore, thought it worth while to send them to you, and to describe in detail the manner in which they have been deduced, that an opportunity may be given of judging of the degree of confidence they deserve. 2] THE SEARCH FOR THE PLANET NEPTUNE. 45 My observations were all made with the large Northumberland Hefractor, and with a magnifying power of 170. On August 4, the Hour Circle being fixed, the telescope was moved in declination, and the transits were all taken at the same part of the field, at the toothed edge of the comb of a micrometer eye-piece. Differences of declination were measured by means of a graduated sector-arc, which was read off by a microscope-micrometer, one revolution of which is 10". The stars were accurately bisected by a fixed wire equatorially adjusted, but to gain time the micrometer was read off to integral revolutions, and by estimation to a fourth part of a revo- lution. The error of reading off in this way could hardly be more than 3", and the error of comparison with a single star might possibly amount to 6". On August 12, the telescope was absolutely fixed, and the zone, which was 9' in breadth, was limited by the field of view. The transits were taken at the toothed edge of the comb carefully adjusted, and the differences of declination were measured by revolutions of the eye-piece micrometer, read off in integral revolutions, and by estimation to a fourth part of a revolution, by means of the teeth of the comb. Occasionally, as it happened in the instance of the planet, the tenth part of a revolution was estimated. The value of one revolution of this micrometer is 17", and I should therefore estimate the error of comparison with a single star, so far as it depended on error of reading off, to be at most 8". I now give the places of the planet resulting from a comparison with every known star that was taken in the same series on each of the two days. August 4 Star of Comparison and authority R; h Asc f plane( . Decl of planet for its place. * / s. . . *> v , ( ' h. m. s. ai a . (British Association Catalogue... 21 58 14'13 -12 57 18'4l * 50 Canricorni \ r ITU>ceAl 7 197 91 h ^7 m 1^ 8 1^-9 1 91-71 iJcootJl Zj. \-j-it L Of J.O ltfrJ.***a which agrees very closely with that found from Mr Lassell's observations. Although, therefore, more numerous observations will be requisite in order to obtain a mass which may be used with confidence in the theory of Neptune, I have no doubt that the value ai ^ 00 is much nearer the truth than either of those which have been previously given, and I shall ac- cordingly employ it in my subsequent calculations respecting the orbit of Neptune. The most probable values of the periods of the second and fourth satellites, given by the combination of the observations of Sir Win. Herschel, Sir J. Herschel, Lament, and Mr Lassell, are 8 d 7058435 and 13 d "463139 respectively ; but the remaining errors of the epochs are greater than can with probability be ascribed to mere errors of observation, and seem to indicate the existence of considerable perturbations. 7. APPENDIX ON THE DISCOVERY OF NEPTUNE. [From Liouville's Journal de Mathdmatiques, New Series, Tome II. (1876).] BESSEL a inse"r) au no. 48 des Astronomische Nachrichten, t. n., p. 441, une Lettre qui est accompagne'e d'une note explicative se rapportant k ses Tables d'Uranus et e"manant de Bouvard lui-meme. II rdsulte eVidemment des remarques I, II, III de M. Le Verrier, aux pages 92 94 de son Mdmoire sur les perturbations d'Uranus, qu'il n'avait pas connaissance de ces Lettres de Bessel et Bouvard ; car elles auraient fait disparaitre la plupart des doutes qu'il y exprime relativement aux Tables de ce dernier. II aurait vu, par exemple, que la correction 28e, qu'il suppose pouvoir s'elever k 100 secondes sexagesimals, n'e"tait reelle- ment que d'environ 10 secondes cente"simales. Au haut de la page 90 de son Mdmoire, M. Le Verrier remarque, avec beaucoup de justesse, qu'une erreur dans I'ine'galite' d'une longue periode n'a pas d'importance pour 1'objet en vue ; mais il aurait du aussi remarquer qu'une erreur dans une ine'galite', dont la pe"riode 6ta.it presque e"gale k celle d'Uranus, serait pareillement presque insignifiante, puisque 1'effet de cette erreur, durant le temps pendant lequel Uranus a e"te" observ^, serait, k peu de chose pres, repr^sent^ par une correction constante applique"e & 1'excentricit^ et k la longitude du pe"rihe"lie, comme je 1'ai dit k la fin du no. 7 de mon Me"moire. J'attache une tres-grande importance k la remarque faite au no. 9, relativement k 1'avantage d' employer la correction de la longitude moyenne au lieu de celle de la longitude vraie. M. Hansen a fortement insiste" sur ce point dans sa Theorie de la Lune et dans ses autres ouvrages. 64 APPENDIX ON THE DISCOVERY OF NEPTUNE. [7 Par suite de cela, les termes qui sont ndcessairement omis dans une premiere approximation sont plus faibles que si Ton avait employe les perturbations de la longitude vraie. Je vais maintenant faire un petit nombre de remarques, en re"ponse aux objections de M. le professeur Pierce, centre la legitimite' du procdde suivi, tant par M. Le Verrier que par moi-mdme, pour la solution de notre probleme. Le professeur Pierce prdtend que la pdriode de notre planete hypothe'tique differe si conside"rablement de celle de Neptune, que Ton pourrait indiquer quelques periodes interme'diaires, lesquelles seraient exactement commensurables avec la peViode d'Uranus, et qu'il y aurait une solution de continuity dans les perturbations d'Uranus, causee par deux planetes hypothe'tiques, dont 1'une aurait une plus grande pe"riode et 1'autre une pe"riode plus petite que la peViode commensurable dont il vient d'etre question. De plus, la periode de Neptune lui-m^me est, a tres-peu de chose pres, double de celle d'Uranus, et cette circonstance donne naissance a des perturbations re"ciproques tres-conside"rables, d'un caractere tout a fait different de celles qui seraient causees par nos planetes hypothe'tiques. Pen de mots, a mon avis, suffiront pour aplanir cette difficult^. II est vrai que, si nous voulions reprdsenter les perturbations d'Uranus causees par une planete superieure, pendant deux ou plusieurs periodes synodiques, cela ne pourrait se faire qu'en adoptant une peYiode approximativement vraie pour la planete perturbatrice ; mais le cas est different lorsque, comme ici, nous n'avons a repre"senter que les perturbations produites durant une fraction d'une pe'riode synodique. Dans ce cas, si nous prenions pour quantite*s inconnues, non les cor- rections applicables aux e'le'ments moyens de 1'orbite d'Uranus, mais celles qui seraient applicables aux e'le'ments adoptes pour I'e'poque de 1810, par exemple, alors toutes les considerations relatives a une commensurabilite approximative dans les deux periodes, deviendraient dtrangeres k la question, et les perturbations pour 1'intervalle limite" requis pourraient dtre representees approximativement, pourvu que les forces perturbatrices de la planete reelle et de la planete presumed fussent approximativement les memes en grandeur et en direction, durant le temps oil ces forces perturbatrices agiraient avec la plus grande intensite", c'est-a-dire lorsque les planetes ne seraient pas fort eloigne"es de leur conjonction. Sir John Herschel a montre dans ses Outlines of Astronomy que ces conditions sont remplies d'une maniere satis- faisante par les planetes hypothetiques de M. Le Verrier et de moi-meme, quand leur action est compared a celle de Neptune. 7] APPENDIX ON THE DISCOVERY OF NEPTUNE. 65 On ne devait attacher aucune valeur a la forte excentricite' ni a la longitude de 1'apside de 1'orbite de la planete pre'sume'e, si ce n'est en tant qu'elles fournissaient les moyens d'approcher de plus pres de la distance actuelle et du mouvement angulaire du corps perturbateur, dans 1'intervalle ou 1'action perturbatrice se faisait le plus sentir. Ainsi done, de la circonstance que le peYihelie de la planete pre'sume'e sortit du premier calcul, non loin de la ligne de conjonction, on aurait pu raisonnablement conclure, ce qu'a donne" en effet le second calcul, que 1'hypothese d'une plus faible valeur de la distance moyenne conduirait a une valeur plus faible de I'excentricite". On fera bien aussi de remarquer que les grands changements dans les valeurs de Se et eSw, qui se trouvent dans le no. 50, resultant de la transition de ma premiere a ma seconde hypothese, sont des changements dans les valeurs des elements moyens de 1'orbite d'Uranus, lesquels sont grandement affected par 1' megalith de la longitude moyenne avec les co- efficients p 3 et q 3 , dont la periode ne differe pas beaucoup de celle d'Uranus, particulierement pour le cas de la premiere hypothese. On verra que 8x l +p 3 et 8y 1 + q 3 varient bien moins en passant d'une hypothese a 1'autre que Sx^ et 8yi. Nous avons done : Premiere hypothese. Seconde hypothese. Sz, +p 3 = 94~2 1 8^ +p 3 = 105','98 = 50,75 %, + & = 41,59 Et les corrections des elements adopted, a 1'epoque de 1810, seront approximativement de"duites de ces quantit^s, absolument comme 8e et eSsr ont e'te' forme's de Sa^ et S,. L'observation de Flamsteed, en 1690, remonte a une e"poque trop pour qu'elle puisse Stre bien repr^sent^e par les formules dont les re"sultats s'accordent assez bien avec ceux des observations plus re"centes. Ma seconde hypothese a donn^ une erreur plus forte que la premiere. C'est done probablement pour avoir eu trop de confiance dans la possibilite d'appliquer ses formules k cette observation ancienne, que M. Le Verrier s'est trouve amene a fixer une limite infe"rieure a la distance moyenne de sa planete perturbatrice, laquelle ne concorde pas avec la distance moyenne de Neptune, telle qu'elle a 6te" observed. A. 9 rreenwich Mean Solar Time. Apparent B. A. of Comet. Apparent N. P. D. of Comet. h. m. s. 11 12 23 h. m. s. 5 21 37'5 o / // 84 24 55 9 59 18 5 17 287 85 47 53 11 55 45 5 13 33'0 86 35 55 8. ELEMENTS OF THE COMET OF FAYE. [From the Monthly Notices of the Royal Astronomical Society, Vol. vi. (1844).] THE observations used were made with the Northumberland telescope of the Cambridge Observatory ; and the deduced places are as follows : 1843 Nov. 29 Dec. 8 16 At first I computed the orbit by the method of Olbers, on the sup- position of its being a parabola, but found that the middle observation was so badly represented, that this hypothesis could not be correct. I then proceeded to determine the elements without making any hypothesis as to the conic section, and the resulting elements are as follows : Perihelion passage, 1843, October 2G d '33 Greenwich mean time. o / Longitude of Perihelion on the Orbit 54 27 '81 From the equinox Longitude of ascending Node 20738 'OJ of Dec. 5 Inclination to the Ecliptic 10 48'9 Perihelion Distance 1'687 Semi-axis Major 3'444 Eccentricity 0'510 Periodic Time 6'39 Sidereal years. Motion direct. I would suggest that the comet may not have been moving long in its present orbit, and that, as in the case of the comet of 1770, we are indebted to the action of Jupiter for its present apparition. In fact, supposing the above elements to be correct, the aphelion distance is very nearly equal to the distance of Jupiter from the Sun : also the time of the comet's being in aphelion was 1843'8 3'2 = 1840*6, at which time its heliocentric longitude was 234 '5 nearly, and the longitude of Jupiter was 231 '5 ; and, therefore, since the inclination to the plane of Jupiter's orbit is also small, the comet must have been very near Jupiter when in aphelion, and must have suffered very great perturbations, which may have materially changed the nature of its orbit. 9. THE ORBIT OF THE NEW COMET. [From the Times, October 15, 1844.] HAVING obtained some results of an interesting nature respecting the new comet, I am induced to communicate them to the world through the medium of your widely-spread journal. My first investigations were founded on three observations made by Prof. Challis with the Northumberland equatorial on the 15th, 20th and 25th of September, and the orbit found from them appeared to be an ellipse of moderate eccentricity and short period. To test the accuracy of this result, Prof. Challis kindly favoured me with some more recent observations, which were made on the meridian, and therefore entitled to more confidence. Availing myself of the extension thus given to the arc described by the comet, I have re-calculated the orbit from the observations on the 15th and 25th of September and the 5th of October. The following are the results which I have obtained : Perihelion passage, Sept. 2 '4 159 mean time at Greenwich. o t // Longitude of perihelion of the orbit... 342 28 25] From the mean equinox Longitude of ascending node 63 47 7J of Sept. 25 Inclination to the Ecliptic 2 56 13 Log. (f axis major) 0'500660 Eccentricity =sin 3840'22" Longitude perihelion distance '074841 Period in sidereal years 5'636 Motion direct. These elements compared with observations give the following errors : Date Error in Long. Error in S. Lat. Sept. 15 6 Sept. 25 +1-0 + 3-5 Oct. 2 +6'1 -28'9 (merid. obs.) Oct. 5 +0'0 O'O (merid. obs.) Though the period found may require considerable correction, I think there can be no doubt that the orbit is really elliptic. If this be the case, it is a remarkable fact that this is the second comet whose periodicity has been discovered during the present year. 92 10. THE RELATIVE POSITION OF THE TWO HEADS OF BIELA'S COMET. [Communicated to the Royal Astronomical Society (March 14, 1846).] THE diagram shows the relative position of the two heads of Biela's Comet on Jan. 26'5, Feb. 11 '5 and Feb. 27'5 mean Greenwich time, projected on a plane parallel to the equator. The rectangular coordinates of the smaller head, referred to the larger as origin, are as follows rx* 1] *Z Jan. 26-5 504-06 2574 85'06 Feb. 11-5 481-99 154'95 107'12 27-5 404-65 270-21 118'26 The unit of measure is a line subtending an angle of l" at the mean distance of the Earth from the Sun ; the plane parallel to the equator is the plane of xy ; and the axis of x is a line drawn in the direction of the first point of Aries. 10] THE TWO HEADS OF BIELA'S COMET. 69 The relative velocities on Feb. 11*5, in the directions of the axes are as follows ~= -3-2647, = 8-1047, ^ = 1'1415; dt dt dt the linear unit being the same as before, and the unit of time a mean solar day. \ V6^* t \ ^ ?i 4 %> &!> Larger j^>- head I n.'" 5 __i ^^m. T From these results it will be easy to deduce the differences of the elements of the orbits of the two heads. According to my calculations the periodic time of the smaller head is 8 '4 8 days longer than the periodic tune of the larger. 11. ON THE APPLICATION OF GRAPHICAL METHODS TO THE SOLUTION OF CERTAIN ASTRONOMICAL PROBLEMS, AND IN PARTICULAR TO THE DETERMINATION OF THE PERTURBATIONS OF PLANETS AND COMETS. [From the Report of the British Association (1849).] AFTER briefly pointing out the advantages of graphical methods, the author proceeded to give some instances of their practical application. It was shewn that the solutions of the transcendental equation which expresses the relation between the mean and eccentric anomalies in an elliptic orbit is obtained in the most simple manner by the intersection of a straight line with the curve of sines. Attention was directed to Mr Waterston's graphical method of finding the distance of a comet from the Earth, and an analogous method was given for determining the distance of a planet, on the supposition that the orbit is a circle in the plane of the ecliptic. The author then passed on to the more immediate object of his com- munication, the graphical treatment of the problem of perturbations of planets and comets. He first shewed how to obtain geometrical represen- tations of the disturbing forces, and then gave simple constructions for determining the changes produced by these forces in each of the elements of the orbit, in a given small interval of time. Having obtained the total changes of the elements in any number of such intervals, it was shewn in the last place how to find their effect on the longitude, radius vector and latitude of the disturbed body, and thus to effect the complete solution of the problem of perturbations without calculation. 12. ELEMENTS OF COMET II. 1854. [From the Monthly Notices of the Royal Astronomical Society, Vol. xiv. (1854).] PROBABLY you will have plenty of elements of the comet which is now starring it, nevertheless I may mention the following, which I deduced from Professor Challis's observations on March 30, April 1, 3. A comparison of these elements with an observation on April 7, gave an error of only 10" in longitude, and nothing in latitude, so that they are probably not far from the truth. Perihelion Passage, March 24-01221, G. M. T. Longitude of Perihelion 213 51 32 Longitude of the Ascending Node 315 29 52 Inclination 82 34 28 Log. Perihelion Distance 9 '4426 170 Motion retrograde. 13. OBSERVATIONS OF COMET II. 1861. [From the Monthly Notices of the Royal Astronomical Society, Vol. XXH. (1862) and Astronomische Nachrichten, LVII. (1862).] G. M. S. T. 1861 Observed K. A. Parallax xA. Observed N. P. D. Parallax xA. a. June 30 h. m. s. 11 6 7'4 11 19 51-1 h. m. s. 6 40 14-94 6 40 40-50 + 0-127 + 0-099 43 25 37-1 43 19 35-0 -8-343 -8-391 July 2 10 41 46-6 10 57 47-4 8 30 28-47 + 0-541 27 36 40-5 -6-571 3 9 57 55-6 11 4 52-4 9 39 53-92 9 43 15-42 + 0-822 + 0-726 24 10 11-5 24 4 38-6 -4-128 -5-330 5 8 10 29 33-1 9 54 51-8 11 44 52-88 13 17 34-82 + 0-868 + 0-624 23 38 32-8 27 54 50-1 -2-301 -0-573 10 53 7-9 13 18 22-15 + 0706 27 58 30-1 -1-627 9 11 4 31-9 11 56 10-5 13 35 4-31 13 35 36-11 + 0-673 + 0709 29 23 457 29 26 47-9 -1767 -2768 10 11 7 17 13 47 55-32 + 0-641 30 40 45-8 -1-800 13 11 22 27-6 14 13 11-05 + 0-588 33 47 49-3 -2-208 23 26 10 32 49-0 10 32 17-3 14 46 40-09 14 51 52-27 + 0-469 + 0-466 39 26 26-8 40 27 2-2 -2-151 -2-369 27 10 33 40-2 10 26 32-3 14 53 23-97 14 58 53-10 + 0-469 + 0-462 40 44 52-1 41 47 58-5 -2-456 -2-630 Aug. i 2 10 35 45-8 1032 1-3 15 870 15 1 2177 + 0-473 + 0-469 42 2 10-6 42 15 28-0 -2-835 -2-850 6 10 6 13-3 15 5 58-48 + 0-448 43 3 51-5 -2718 8 11 36 5-3 15 8 15-87 + 0-508 43 26 28-1 -4-273 13 H 15 10 50 16-2 10 5 46-4 10 17 49-8 15 13 38-37 15 14 40-53 15 15 45-60 + 0-489 + 0-459 + 0-470 44 14 56-1 44 23 31-9 44 32 18-2 -3-837 -3-202 -3-444 13] OBSERVATIONS OF COMET II. 1801. 73 G.M. S.T. Observed Parallax Observed Parallax 1861 K. A. xA. N.P.D. XA. a. h. m. s. h. m. . O / // Aug. 1 6 10 9 30-0 15 16 49-44 + 0-404 44 40 41-4 -3-375 19 10 29 28'4 15 20 3-57 + 0-480 45 4 48-1 -3-860 20 10 17 53-8 15 21 7'81 + 0-474 45 12 18-1 -3735 21 9 22 2-6 15 22 10-59 + 0-429 45 19 26-2 -2-960 23 9 45 40-3 15 24 22-39 + 0-454 45 33 50-1 -3-414 2 4 9 31 49-0 15 25 27-54 + 0-443 45 40 36-9 -3-265 27 10 12 6-3 15 28 50-08 + 0-475 46 31-3 -4-034 28 10 12 2'4 15 29 57-85 + 0-476 46 6 457 -4-088 30 9 22 47 15 32 10-76 + 0-445 46 18 33-4 -3-437 Sept. 3 9 57 56-4 15 36 50-93 + 0-472 46 40 58-0 -4-183 6 8 47 24-9 15 40 21-31 + 0-427 46 55 58-1 -3-285 7 9 17 36-4 15 41 35-06 + 0-453 47 49-0 -3-770 9 8 45 16-6 15 43 59-40 + 0-431 47 10 1-5 -3-394 10 9 44 31-5 15 45 16-09 + 0-470 47 14 35-8 -4-323 1 1 9 19 54-9 15 46 29-07 + 0-460 47 18 47-9 -3-996 12 10 24 59-8 15 47 46-77 + 0-477 47 23 9-1 -5-047 13 10 37 13'6 15 49 3-41 + 0-475 47 27 11-9 -5-284 14 9 45 6'2 15 50 16-27 + 0-472 47 30 53-2 -4-521 23 10 12 4-0 16 2 1-87 + 0-470 47 59 51-8 -5-342 Oct. 9 9 30 36-3 16 24 25-11 + 0-467 48 25 87 -5-358 1 1 8 57 45-5 16 27 20-63 + 0-468 48 25 45-7 -4-935 12 10 38 557 16 28 56-26 + 0-429 48 25 51-1 -3-473 H 9 55 27-1 16 31 52-46 + 0-452 48 26 0-1 -5-918 15 9 14 45'4 16 33 20-43 + 0-466 48 25 40-7 -5-342 16 8 32 50-6 16 34 47-90 + 0-467 48 25 22-8 -4-740 23 8 21 25-1 16 45 33-56 + 0-469 48 18 53-2 -4-813 28 7 35 56-1 16 53 23-50 + 0-460 48 10 7-4 -4-290 Nov. i 7 36 39-0 16 59 49-13 + 0-465 48 34-7 -4-426 2 8 57 36-6 17 1 31-62 + 0-464 47 57 36-8 -5-698 9 1 53-1 17 1 32-41 + 0-462 47 57 37-9 -5761 5 8 3 14-1 17 6 21-53 + 0-473 47 48 43-1 -4-959 6 7 52 51-5 17 7 59-25 + 0-473 47 45 28-2 -4-830 7 8 2 21-5 17 9 3879 + 0-474 47 41 56-1 -5-008 9 8 43 42-4 17 13 1-20 + 0-466 47 34 15-6 -5721 1 1 8 38 55-1 17 16 20-95 + 0-465 47 26 31-6 -5-689 20 6 53 55-6 17 31 30-01 + 0-476 46 43 49-4 -4-315 2 3 7 49 1-8 17 36 44-66 + 0-482 46 26 28-5 -5-327 A. 10 74 OBSERVATIONS OF COMET II. 1861. [13 G . M.S.T, , Observea Parallax Observea Parallax 1861 B. A. xA. N. P . D. xA. a. h. m. 8. h. m. 8. / Nov. 27 6 55 5 o 17 43 39-16 + 0-485 46 2 18-8 -4-507 28 6 56 14 3 17 45 24-37 + 0-486 45 55 48-9 -4-554 30 7 58 3 9 17 48 59-05 + 0-479 45 42 56-9 -5-575 Dec. 3 7 27 26 7 17 54 16-40 + 0-490 45 21 10-9 -5-193 4 7 55 49 1 17 56 5-36 + 0-480 45 13 45-6 -5-653 5 8 7 48 6 17 57 53-10 + 0-472 45 6 8-5 -5-882 The foregoing values were deduced as follows : E. A. No. of N. P. D. No. of Comet - Star. Comp. Comet - Star. Comp. Star. a. m. 8. // June 30 - 6 1-85 1 - 7 37-2 1 a -11 50-63 1 -31 27-0 1 b July 2 1 -11 - 8 26-54 10-07 3 ... ... {d f+ 5 23-9 \- 2 54-2 2 fe 3 -28 2-06 1 - 1 49-4 1 e + 3 46-57 3 + 18 55-2 3 f 5 - 4 25-66 6 -20 23-8 6 9 8 -27 41-47 1 + 5 53-8 1 h + 3 2.19 5 + 6 137 5 i 9 -29 25-99 1 -36 22-5 1 k - 3 33-03 2 + 17 31-6 2 I 10 + 2 11-99 6 - 5 287 6 m 13 - 6 13-30 4 + 24-9 4 n 23 - 5 8-51 4 -21 37-8 4 26 - 5 2-31 7 + 11 44-6 7 P 27 + 5 42-19 4 - 32-1 4 q 31 - 21-65 11 - 9-5 11 r Aug. i + 53-98 6 + 14 2-6 6 r 2 + 5 25-25 6 + 5 11-4 6 s 6 + 1 54-96 8 + 4 32-4 8 t 8 - 5 18-81 3 -25 49-5 3 u 13 - 5 4778 6 + 51-3 6 V H - 4 45-59 2 + 9 27-1 2 V 15 + 1 48-85 8 + 3 47-8 8 w 16 + 2 5272 6 + 12 10-9 6 w 13] OBSERVATIONS OF COMET II. 1861. 75 E. A. No. of N.P.D. No. of Comet - Star. Comp. Comet - Star. Comp. Star. d. m. s. Aug. 19 - 1 12-06 8 / // - 7 43-3 8 X 20 + 59-93 8 + 3 45-8 8 y 21 + 55-02 8 + 6 54-8 8 X 23 + 2 42-01 8 + 3 307 8 z 2 4 + 3 47-18 8 + 10 17-3 8 z 27 - 4 51-96 6 + 4 7-9 6 aa 28 - 3 44-16 6 + 10 22-2 6 act 30 + 1 44-81 8 - 3 36-4 8 bb Sept. 3 + 2 3-92 6 -10 3'6 6 cc 6 - 1 9'31 6 - 9 53-5 6 dd 7 - 4 41-08 6 - 3'6 6 ee 9 - 2 16-69 6 + 9 87 6 ee 10 - 2 37-66 6 + 5 12-8 6 ff 1 1 - 1 24-66 6 + 9 24-8 6 ff 12 - 6-93 8 + 13 45-9 8 ff 13 + 1 973 6 + 17 48-5 6 ff H + 2 22-62 6 + 21 297 6 ff 23 - 2 36-92 4 -32 34-3 4 99 Oct. 9 + 1 46-87 8 - 1 15-5 8 hh ii - 4 43-97 6 + 6 7-8 6 ii 12 - 3 8-32 2 + 6 13'0 2 ii H - 5 44-14 4 - 6 20-8 5 kk 15 - 4 16-15 6 - 6 40-4 6 kk 16 - 2 48-66 8 - 6 58-4 8 kk 23 + 1 0-70 6 + 13 15'6 6 11 28 - 2 2076 8 -10 55-2 8 mm Nov. i + 57-26 8 + 6 18-9 8 nn 2 - 34-14 7 - 6-2 7 00 + 2 40-55 2 + 3 21-9 2 nn 5 - 2 47-38 8 + 4 58-1 8 PP 6 - 1 9'64 7 + 1 42'9 7 PP 7 + 29-91 9 - 1 49-5 9 PP 9 + 2 3971 4 - 3 46-3 3 qq 1 1 - 48-10 9 - 9 12'8 9 rr 20 + 1 1-34 8 + 13 17'5 8 ss 23 + 19-00 6 - 59-9 5 tt 27 - 15-83 8 + 11 34-8 6 uu 102 76 OBSERVATIONS OF COMET II. 1861. [13 B. A Comet - Star. No. of Comp. N.P.D. Comet - Star. No. of Comp. Star. d. m . 8. Nov. 28 + 1 29 36 8 + 5 4 7 6 UU 30 - 51 62 8 - 5 34 2 6 vv Dec. 3 - 1 9 17 8 + 4 43 9 6 ww 4 + o 39 79 8 - 2 41 7 6 ww 5 + 2 27 53 8 -10 19 1 6 ww The determinations of N. P. D. from July 2 to July 9, inclusive, are liable to some uncertainty, in consequence of the defective state of the clamp by which the declination-rod was attached to the polar frame. The determinations of R. A., however, are trustworthy. The R. A. and N. P. D. for July 2 are obtained by taking a mean between the results of the comparisons with (c) and (d). It is probable that in the observation of Nov. 30 the recorded micro- meter-reading was too great by 5 revolutions, and that the N. P. D. should consequently be diminished by 5 r = 43 //- 2. Assumed Mean Places of the Stars of Comparison for 1861'0. Star. B.A. 1861-0. N.P.D. 1861-0. Authority. a h. m. B. 6 46 15-02 43 33' 14'-32 Johnson 1841 b c 6 52 29-36 8 41 53-41 43 51 2-00 27 31 18-42 Arg. Johnson 7473 2212 d e 8 38 36-47 10 7 54-05 27 39 33-30 24 12 1-81 Arg. Johnson 9299 2464 f 9 h 9 39 26-98 11 49 16-42 13 45 13-86 23 45 44-11 23 58 58-28 27 48 5873 M Arg. Johnson 2396 12183-84 3103 i k 13 15 17-63 14 4 27-81 27 52 18-45 30 10-61 Arg. Johnson 13563 3147 I 13 39 6-75 29 9 18-43 3084 m 13 45 40-92 30 46 16-61 3104 n 14 19 21-88 14 51 46-18 33 47 26-83 39 48 7-45 Arg. Johnson 14545 3293 P 14 56 52-21 14 47 39-45 40 15 2075 40 45 27-01 Arg. 15039 14924-5 and 6 r 14 59 12-46 41 48 11-30 Johnson 3318 s 14 55 54-28 42 10 19-81 3306 13] OBSERVATIONS OF COMET II. 1861. 77 Star. R. A. 1861-0. N. P. D. 1861-0. Authority. b. m. s. O / // t 15 4 1-33 42 59 22-55 Arg. 15138, 39 & U 15 13 32-48 43 52 21-45 15266 V 15 19 24-05 44 14 8-99 15347 w 15 13 5471 44 28 34-20 15272 X 15 21 13-64 45 12 35-38 Johnson 3385 y 15 20 5-93 45 8 36-32 Arg. 15355 z 15 21 38-49 45 30 23-41 Johnson 3387 aa 15 33 40-19 45 56 27-99 3423 bb 15 30 24-17 46 22 13-89 3413 cc 15 34 45-30 46 51 5-65 3431 dd 15 41 28-95 47 5 56-02 3448 ee 15 46 14-48 47 57-43 3462 ff 15 47 52-16 47 9 27-67 3464 99 16 4 37-39 48 32 30-79 H. C. 29530 hh 16 22 37-14 48 26 28-41 30042 ii 16 32 3-53 48 19 43-04 Eq. Comparison. kk 16 37 35-56 48 32 26-10 H. C. 30489 11 16 44 32-00 48 5 41-91 30687 mm 16 55 43-43 48 21 7-36 31031 nn 16 58 51-13 47 54 20-14 B. Z. 426 16 h 57 m 41 8 00 17 2 5-02 47 57 47-50 Eq. Comparison. PP 17 9 8-21 47 43 49-86 H. C. 31417 <1<1 17 10 20-85 47 38 5-93 31456 rr 17 17 8-42 47 35 49-01 31697 ss 17 30 28-18 46 30 36-13 32154 and 5 tt 17 36 25-19 46 27 32-75 Johnson 3741 uu 17 43 54-58 45 50 48-38 3763 vv 17 49 50-27 45 48 35-65 B. Z. 478. 17 h 47 m 53' WIV 17 55 25-32 45 16 31-59 Eq. Comparison. The place assumed for the star (ii) is derived from equatorial com- parisons made on Oct. 15 with H. C. 30489. The place of (oo) is derived from equatorial comparisons made on Nov. 20 with B. Z. 426. 16 d 57 m 41, and the place of (ww) from equatorial comparisons with Johnson 3795 made on Feb. 20, 1862. The observations up to July 13 were made by Professor Challis, and the subsequent ones by Mr Bowden, the senior Assistant at this Observatory. 14. ON THE ORBIT OF 7 VIRGINIS. [From jEdes Hartwelliance, Letter to Admiral Smythe, June, 1851.] I HAVE great pleasure in sending you the results which I have obtained respecting the orbit of y Virginia, and I feel the more indebted to you for having called my attention to the subject, inasmuch as the problem of determining the orbits of double stars is one with which I had previously only a theoretical acquaintance. The orbit, given by Sir John Herschel in the Results of his Cape Observations, was taken as the basis of the cal- culations, and equations of condition for the correction of the elements were formed by comparing certain selected angles of position deduced from ob- servation with the values calculated by means of Sir John Herschel's elements. The positions employed are those given by Bradley's observation in 1718, Sir William Herschel's observations in 1781 and 1803, a normal position for 1825 deduced from the observations of 1822, 1825, and 1828, one for 1833 from the observations of 1832, 1833, and 1834, another for 1839 from the observations of 1838, 1839, and 1840, and, lastly, a normal position for 1848 from the observations of 1846, 1847, 1848, 1849, and 1850. The number of these positions being greater by one than that absolutely necessary for the determination of the elements, I at first omitted the equation of con- dition for 1718 and solved the remaining ones in such a manner as to shew the effect which would be produced in each of the elements by a small given change in any one of the observed angles of position. The result proved that the elements would be greatly affected by small errors in the observed positions for 1781 and 1803, and I therefore called in the observation of 1718 to the rescue, and solved the equations anew, supposing the positions for 1825, 1833, 1839, and 1848 to be correct, and distributing the errors among the other three, according to the rules supplied by the method of least squares, giving double weight to the observations of 1781 and 1803. 14] ON THE ORBIT OF y VIRGINIS. 79 The following are the resulting elements : Inclination of the orbit to the plane of projection 25 27 Position of the node 34 45 Distance of perihelion from the node 284 53 Angle of eccentricity 61 36 Eccentricity 0'87964 Perihelion passage 1836'34 Period 174-137 yrs. The following table shews the differences between the observed positions and those calculated from the above elements : Observed Calculated Epoch. position. position. Differences. / / / 1718-22 150 52 151 3 -11 1781-89 130 44 130 29 + 15 1803-20 120 15 120 43 -28 1825-32 97 46 97 43 + 3 1833-27 61 16 61 11 + 5 1839-36 215 51 216 2 -11 1848-37 180 6 180 6 0. A better agreement could scarcely be desired. The observations made about the time of perihelion passage are liable to great errors in conse- quence of the excessive closeness of the stars, and therefore I did not take them into account in forming the equations of condition. Sir John Herschel was obliged to admit large differences between these observations and the results of his theory, and these differences are con- siderably increased by using my elements. I am inclined to think that these observations cannot be satisfied without materially increasing the errors on both sides of the perihelion passage. My elements agree very well with the latest observations which have come to my knowledge, as is shewn by the following comparison : Observed Calculated Observer. Epoch. position. position. Differences. Lord Wrottesley, 1851-172 175 55 175 52 +3 Mr Dawes, 1851-217 17635 17549 +46 Mr Fletcher, 1851-401 175 58 175 34 +24 15. ON THE TOTAL ECLIPSE OF THE SUN, 28 JULY 1851, AS SEEN AT FREDERIKSVAERN. Latitude, 58 59' 33"-9 N. Longitude, 40 m 15"-5 East. [From the Memoirs of the Royal Astronomical Society. Vol. xxi. (1852).] THE approach of the total eclipse of July 28, 1851, produced in me a strong desire to witness so rare and striking a phenomenon. Not that I had much hope of being able to add anything of scientific importance to the accounts of the many experienced astronomers who were preparing to observe it ; for I was not unaware of the difficulty which one not much accustomed to astronomical observation would have in preserving the requisite coolness and command of the attention amid circumstances so novel, where the points of interest are so numerous, and the time allowed for observation is so short. Certainly my experience has now shewn that I did not ex- aggerate these difficulties ; but I have at least the satisfaction of having formed a far more vivid idea of the phenomenon than I could have ob- tained from any description ; and I think that if I should ever have another opportunity of observing a total eclipse, I should be prepared to give a much better account of it than I can of the present. I left Hull, by steamer, on the evening of Saturday, July 19, together with a large party of astronomers bound on the same errand with myself. In the afternoon of Tuesday the 22nd, we arrived at Christ iania, where I landed with several other passengers, the remainder of the party going on to Gottenburg. We had no trouble in getting our instruments on shore ; 15] ON THE TOTAL ECLIPSE OF THE SUN, JULY, 1851. 81 the Norwegian Government having, in the most liberal and enlightened spirit, ordered the custom-house officers to allow them to pass without examination. This favour, I afterwards found, we owed to the kind offices of Professor Hansteen, whose acquaintance, as well as that of several other eminent Professors of the University, I had the happiness of making during my short stay at Christiania. On Thursday the 24th, in company with my friend Mr Liveing, of St John's College, Cambridge, I proceeded by steamer to Frederiksvsern, the point selected for making the observation, as being one easily accessible, and situated almost exactly on the central line of the path of the Moon's shadow. Here is one of the royal dockyards, containing a small observatory for giving time to the shipping. The officers of the dockyard shewed us much attention, and were anxious to render us every assistance in preparing for the observation. To Lieutenant Riis, in particular, we are under the deepest obligations. On Friday the 25th we inspected the Observatory, and examined the neighbourhood with the view of selecting a favourable spot for the observation. It rained heavily during the whole of Saturday, so that our prospects were not very encouraging, but on Sunday the weather im- proved, and on the morning of the eventful day, Monday the 28th, the sky was bright and clear, with the exception of a few light clouds, which, however, became more numerous as the day advanced, and at length over- spread the heavens, as fresh vapour was brought up by the wind, which blew quite a gale from the south-west. I had intended to observe the eclipse from the summit of a rocky island lying just off the dockyard, and commanding an extensive prospect over the sea, though the view on the land side is cut off by a lofty ridge of rocks rising behind the town. The violence of the wind, however, made it necessary to choose some sheltered position for the instrument, and I fixed upon one in an angle within the ramparts of the dockyard. The telescope which I employed was one of Dollond's, which was kindly lent me by the Master and Fellows of St John's College. The aperture of the object-glass is 2f inches, and its focal length 42 inches. The astronomical eye-pieces belonging to the instrument giving too small a field of view, I employed a terrestrial eye-piece, with a mag- nifying power of about 20. The field was limited by a diaphragm having small teeth of different sizes arranged at intervals of 45 around its cir- cumference, in order to enable me to estimate the position and magnitude of any small object that might be seen. As the eastern limb of the Moon advanced over the Sun, I observed A. 11 82 ON THE TOTAL ECLIPSE OF THE SUN, JULY, 1851. [15 that it appeared uneven in several places, and two mountains were parti- cularly noticed on the edge, about 5 apart and near the eastern extremity of the Moon's horizontal diameter. The cusps, too, as they were approaching each other, occasionally appeared to be somewhat blunted. I could see no trace of the Moon's limb extending beyond the Sun's disc. As the crescent became very narrow, it seemed to be in a state of violent agitation, and at last, just before the totality, it broke up into several parts. These, how- ever, were not like the " beads " described by Mr Baily, but were quite irregular, being evidently occasioned by the inequalities on the Moon's limb. As the totality approached, the gloom rapidly increased ; still, enough light remained up to the moment of total obscuration to render the change which then took place very marked and startling. For a few moments I felt somewhat confused, and did not immediately remove the dark glass. I then applied my eye to the finder, and saw the corona surrounding the dark body of the Moon. The light of the corona was pale, not sensibly coloured, and gradually faded away in receding from the Moon's edge. Its average breadth was perhaps about a third of the Moon's diameter, but it extended considerably farther in some directions than in others, its boundary being very irregular. It did not appear to consist of rays, and there was no marked annularity of structure, so that I could not decide whether it was concentric with the Sun or the Moon. I now quitted the telescope and looked first at the Moon and then around on the sky. The appearance of the corona, shining with a cold unearthly light, made an impression on my mind which can never be effaced, and an involuntary feeling of loneliness and disquietude came upon me. I had previously ascertained the position of the principal stars and planets, but none of them could be seen on account of the clouds. I did not notice any peculiarity in the colours of surrounding objects. The light remaining was only just sufficient to enable me to read off the face of a box chronometer which I had with me. A party of haymakers, who had been laughing and chatting merrily at their work during the early part of the eclipse, were now seated on the ground, in a group near the telescope, watching what was taking place with the greatest interest, and preserving a profound silence. About forty or fifty seconds after the commencement of the totality, I returned to the telescope, and cast my eye round the disc of the Moon. The light of the corona did not seem to be uniformly diffused round it, there being a patch brighter than the rest near the point where the Sun's 15] ON THE TOTAL ECLIPSE OF THE SUN, JULY, 1851. 83 last rays had disappeared. At the point nearly opposite, or about 105 from the upper point of the Moon, measured towards the west, I noticed a rosy-coloured prominence, about one minute in altitude. The upper or northern boundary of this was well denned, and had nearly the form of a quadrantal arc of a circle meeting the Moon's limb perpendicularly, the con- cavity being turned downwards ; the southern boundary was also somewhat concave downwards, but the illumination near it was less, and diminished gradually, so that it was difficult to ascertain its exact form. The appearance was somewhat like the enlightened portion of a hemispherical mountain standing on the Moon's limb and illuminated on its northern side, whilst more than half the hemisphere on the opposite side was invisible. After watching this for a short time, I observed that its altitude was gradually increasing, and my attention became in consequence entirely engrossed by it. The southern boundary of this prominence soon became better defined than at first, while the northern boundary remained perfectly even and well defined throughout. The altitude continued to increase till the moment of the Sun's reappearance, when it amounted to nearly three minutes. The form of the prominence now resembled that of a sickle, and it projected nearly perpendicularly from the Moon's limb, the part nearest the Moon being nearly straight, but the curvature gradually increasing in approaching the point, which was sharp and turned downwards. The breadth at the base was, perhaps, two-thirds of a minute. There was no sensible, or at any rate, no marked change of form in the several parts after they had once been seen, but only a gradual lengthening by additions at the base, of such a kind as would have been occasioned by the motion of the Moon if the prominence had really belonged to the Sun 1 . My impression, how- ever, is, that the increase of length was greater than can be accounted for by the Moon's motion, and that it proceeded more rapidly towards the end of the totality than at first, but I cannot feel certain on this point. A little before the end of the totality, the corona seemed to become brighter in the neighbourhood of the prominence, which was close to the point 1 "While the Sun is totally covered by the Moon, the latter appears surrounded by a luminous ring, with rays proceeding from it, something in the manner of the glory which is placed by painters round the heads of saints. The most extraordinary appearances how- ever were certain rosy-coloured flame-like projections from the limb of the Moon, one, which I noticed particularly, was very large. This was at the point of the limb at which the Sun reappeared, and it appeared gradually to lengthen out as the Sun's limb was approaching the Moon's, as if it had really been connected with the Sun and moved with it If these rosy flames really belong to the Sun, they must be of enormous magnitude, the one I noticed could not have been less than 50,000 miles in length." From Letter written Aug. 9, 1851. 112 84 ON THE TOTAL ECLIPSE OF THE SUN, JULY, 1851. [15 where the Sun was about to reappear. On account of the clouds, I felt no inconvenience in observing the reappearance without the intervention of a dark glass. As the first ray of the Sun appeared the corona vanished, and at the same moment the prominence seemed suddenly to contract and change its form, the point of it disappearing and the remaining part be- coming detached from the limb of the Moon. In about a second more the whole had vanished. I did not notice any interruption to the con- tinuity of the Sun's limb in its reappearance, like that with which I had been struck when it disappeared, the Moon's western limb being apparently much more regular than the eastern. The clouds now grew rapidly thicker, and completely hid the Sun from view before the end of the eclipse. At the small observatory the eclipse was observed by Lieutenants Smith and Hjorth, two officers of the Norwegian Royal Navy, and also by the well-known French traveller, M. D'Abbadie. Lieut. Smith, who was specially charged by Professor Hansteen with the determination of the time, found the following results : h. m. s. Beginning of the Eclipse ... 2 41 40 '3 Mean Time at the Observatory. Beginning of the Totality... 3 44 52'3 ,, ,, End of Totality 3 48 17'8 The end of the eclipse could not be observed. According to Professor Hansteen, the longitude of the Observatory is 2 m 39 fl< 3 west of Christiania, or 40 m 15 8- 5 east of Greenwich, and its latitude 5859'33".9 north. Lieut. Hjorth compares the appearance of the prominence to that of the flame of a candle acted on by the blowpipe. Besides this prominence, which was the only one seen by me, Lieut. Hjorth observed two much smaller ones to spring up a little before the end of the totality, on the same side of the Moon as the former, one being above and the other below it. Mr Liveing, who observed the eclipse from the same spot with myself, has kindly communicated the following observations, taken with the naked eye. 15] ON THE TOTAL ECLIPSE OF THE SUN, JULY, 1851. 85 " The first appearance I noted was the formation of a halo round the Sun soon after the eclipse commenced ; light clouds were at the same time flitting across the sky. When the totality approached, the passage of the shadow was not so rapid but that I could see the clouds to the north- west grow dark before the last direct beam of the Sun was extinguished. And at the reappearance of the Sun it was still more remarkable ; the clouds to the north-west lightened up, making it much lighter where I stood ; and I had time to exclaim that the Sun was going to appear, and to turn my eyes towards him, an appreciable interval before he actually shewed himself. The first appearance was a single point of light, like a very bright star, increasing in size, of course, very rapidly. " I did not observe that the landscape was peculiarly livid ; it had a cold appearance, but much such as it often has after sunset ; and the only clear part of the sky, towards the south-east horizon, had quite an orange hue, also such as is not unusual after sunset ; and it remained nearly the same colour the whole time of darkness. " I looked for colour in the corona, but could see none ; neither did it appear to me divided by a dark ring, or to be regular or well-defined on the outside ; in four points it certainly appeared to project to a greater distance than at the intermediate points, and these four points were at unequal intervals ; but I did not watch it long enough to observe how far this might be due to the clouds which covered it, and which had now become much thicker than at first. As I did not expect to be able to observe it, I had no means of exactly measuring the intensity of the light ; but I could not distinguish the features of people about four yards from me ; and a candle at about the same distance threw a well-defined shadow. " A crow was the only animal near me ; it seemed quite bewildered, croaking and flying backwards and forwards near the ground in an uncertain manner." I have also been favoured with the following interesting account by another friend, who observed the eclipse in company with several other persons, from an elevated point about thirty-three miles west of Christiania, which commands an extensive view of the surrounding country. "We observed the eclipse from the Skuderud SaBters, about nine miles north-east of Fossum, and nearly on the same parallel as Christiania. We had smoked glasses, and also a small telescope smoked. The eclipse appeared 86 ON THE TOTAL ECLIPSE OF THE SUN, JULY, 1851. [15 to begin about 2 h 45 m . As the shadow increased the change in the appearance of the country was most curious. The light became pale ; our shadows were sharply cut, as by moonlight, but the light was more yellow. A deep gray twilight seemed to come on. Perhaps two minutes before the totality a dark, thick shade appeared over the west and north-west mountains, which drew nearer, till, when the eclipse became total, it entirely surrounded us, though it was paler or less dense towards the east. But on the instant that we were in complete shade, a bright orange streak of light appeared on the horizon to the north-west, spreading west and south. The corona was orange. Bright, pale, and very irregular yellow rays streamed round like the glories round the heads of saints. Many stars were visible, but Venus was the only planet pointed out to me. The totality lasted 2 m 50 s to the best of our reckoning ; but before the Sun reappeared the clouds thickened rapidly, and afterwards we only caught stray glimpses. For a minute after the totality was passed the dark shade lingered over the south and south-east. " The following remarks are numbered with reference to the Suggestions drawn up by a Committee of the British Association. "16. We noticed no variation of colour in the sky. "18. The corona appeared to be formed instantaneously all round; equally broad ; not divided into rings. " 22. The corona cast no shadow. I read the word ' Observation ' at three yards, the remainder of the title at two, the interior print at the usual distance in my hand. I read the same at the same distances at 10 h 30 m the following evening, the book facing west; and at six, four, and two yards distance by sunlight. " 24. The outline of all the mountains was perfectly distinct." I cannot close this account without expressing my sense of the kind hospitality which I met with during a subsequent tour of six weeks in Norway. To Mr Crowe, Her Majesty's Consul-general at Christiania, whose kindness is so well known to all English travellers in that country, I feel particularly bound to return my warmest thanks. 16. ON AN IMPORTANT ERROR IN BOUVARD'S TABLES OF SATURN. [From the Memoirs of the Royal Astronomical Society (1849), Vol. xvn., and Monthly Notices of the Royal Astronomical Society (1847), Vol. vu.]. HAVING lately entered upon a comparison of the theory of Saturn with the Greenwich observations, I was immediately struck with the magnitude of the tabular errors in heliocentric latitude, and the more so, since the whole perturbation in latitude is so small, that it could not be imagined that these errors arose from any imperfection in the theory. In order to examine the nature of the errors, I treated them by the method of curves, taking the times of observation as abscissse, and the corresponding tabular errors as ordinates. After eliminating, by a graphical process, the effects of a change in the node and inclination, a well-defined inequality became apparent, the period of which was nearly twice that of Saturn. One of the principal terms of the perturbation in latitude (viz. that depending on the mean longitude of Jupiter minus twice that of Saturn] having nearly the same period, I was next led to examine whether this term had been correctly tabulated by Bouvard. The formula in the introdviction ap- peared to be accurate; but on inspecting the Table XLIL, which professes to be constructed by means of this formula, I was surprised to find that there was not the smallest correspondence between the numbers given by the formula and those contained in the table, the latter following the simple progression of sines, while the formula contained two terms. The origin of this mistake is rather curious. Bouvard's formula for the terms in question is 9"'67 sin {< - 2# - 60 "29} + 28"'19 sin {2< - 4<' + 66'12} but in tabulating the last term he appears to have taken the simple argument <-2<' instead of 2^-4^.', so that the two parts may be united 88 ON AN IMPORTANT ERROR IN BOUVARD'S TABLES OF SATURN. [16 into a single term, 25"'85 sm{<-2<' + 43-88} which I find very closely to represent Bouvard's Table XLII. After correcting the above error, and making a proper alteration in the inclination and place of the node, the remaining errors of latitude are in general very small. I subjoin a correct table, to be used instead of Bouvard's. The constant added being 36"'0 instead of 26"'0, it will be necessary to subtract 10 //- from the final result. TABLE XLII. Argument III. de la Longitude. Argument. Equation. Argument. Equation. Argument. Equation. Argument. Equation. 52-4 2500 17-4 5000 68-1 7500 6-1 100 54-4 2600 16-2 5100 69-4 7600 4-0 200 56-0 2700 15-5 5200 70-2 7700 2-3 300 57-2 2800 15-2 5300 70-5 7800 1-1 400 58-0 2900 15-2 5400 70-4 7900 0-4 500 58-3 3000 157 5500 69-8 8000 o-i 600 58-3 3100 16-6 5600 687 8100 0-4 700 57-8 3200 17-9 5700 67'2 8200 I'O 800 56-9 3300 19-6 5800 65-3 8300 2-2 900 557 3400 21-7 5900 62-9 8400 3-7 1000 54-1 3500 24-1 6000 60-1 8500 5-7 1100 52-2 3600 267 6100 57-1 8600 8-0 X200 50-0 3700 297 6200 537 8700 107 1300 47-5 3800 32-8 6300 50-0 8800 137 1400 44-9 3900 36-2 6400 46-2 8900 16-8 1500 42-1 4000 39-6 6500 42-1 9000 20-2 1600 39-2 4100 43-1 6600 38-0 9100 237 1700 36-2 4200 46-5 6700 33-9 9200 27-3 1800 33-3 4300 50-0 6800 29'8 9300 31-0 1900 30-4 4400 53-3 6900 257 9400 34-5 2000 277 4500 56-5 7000 21-8 9500 38-0 2100 25-1 4600 59-4 7100 18-1 9600 41-4 2200 22-8 4700 62-1 7200 14-6 9700 44-6 2300 20-6 4800 64-5 7300 11-4 9800 47-5 2400 18-8 4900 66-5 7400 8'5 9900 50-1 2500 17-4 5000 68-1 7500 6.1 10000 52-4 Constante ajoute"e 36"'0. 17. ON NEW TABLES OF THE MOON'S PARALLAX. [From the Monthly Notices of the Royal Astronomical Society (1853), Vol. xm., and Nautical Almanac for 1856.] THE importance of an accurate knowledge of the Moon's Parallax is very evident. No observation of the Moon's place can be compared with the Tables, or turned to any practical use, without undergoing a preliminary reduction of which the amount of the Parallax is the most important element. Now the same theory by which the angular motion of the Moon round the Earth is determined gives likewise the form of the orbit, and therefore the proportion between the Parallaxes at different times ; hence, as the theory is sufficiently perfect to represent the place of the Moon within 10", it cannot be doubted that it would be competent to give the variations of the Parallax within a small fraction of a second, provided the mean Parallax were known. To determine this, however, by theory, it is necessary to know, in addition to the elements furnished by observations of the Moon's motion, the ratio of the Moon's mass to that of the Earth. Hence, conversely, if the mean value of the Parallax be deduced from corresponding observations of the Moon's declination, made at distant points on the Earth's surface, one means is afforded of finding the ratio of the masses. The most recent determination of the Parallax by means of observations of this kind is contained in a paper by Mr Henderson in the tenth volume of the Memoirs of the Royal Astronomical Society, and is founded on his own observations made at the Cape of Good Hope, combined with cor- A. 12 90 ON NEW TABLES OF THE MOON'S PARALLAX. [17 responding observations at Greenwich and Cambridge. In this paper Mr Henderson compares the Parallaxes deduced from observation with those calculated by means of the Tables both of Burckhardt and Damoiseau. It is remarkable that he finds a difference of 1"'3 in the value of the mean Parallax, according as one set of Tables or the other is employed in the comparison, and not knowing which value to prefer, he adopts the mean of the two for his final result. If we consider, however, that the only part of this process which depends on the Tables consists in the reduction of the actual Parallaxes at the times of observation to the mean value, it is plain that so large a difference in the mean of thirty-four observations can only arise from intolerable errors in the periodic terms of Parallax given by one of the two sets of Tables. The Parallax in Damoiseau's Tables is given at once in the form in which it is furnished by theory, but that in Burckhardt's Tables is adapted to his peculiar form of the arguments, and requires transformation in order to be compared with the former. When this was done, I found that several of the minor equations of Parallax deduced from Burckhardt differed completely from their theoretical values given by Damoiseau. On further inquiry, I discovered that the difference between Burckhardt's equations of Parallax and those of Burg and Damoiseau had been long since remarked by Clausen in a comparative analysis of the three sets of Lunar Tables given in the seventeenth volume of the Astronomische Nachrichten, but no notice appears to have been taken of this remark. With regard to the Parallax, Burckhardt professes to have followed the theory of Laplace, but this agrees very closely with that of Damoiseau, so that errors have evidently been committed by him in the transformation of Laplace's formula. These appear to have originated in the following manner : In the formation of Burckhardt's Arguments of Evection and Variation, the mean longitude of the Sun is employed. Now four of the errors in the coefficients of the minor equations may be accounted for, by supposing him to have erroneously employed the true instead of the mean longitude of the Sun in forming the above-mentioned arguments. In another of these equations, the coefficient is taken with a wrong sign, and in another a wrong argument is employed. 17] ON NEW TABLES OF THE MOON'S PARALLAX. 91 A strange fatality seems to have attended all Burckhardt's calculations respecting the Moon's Parallax. In the Connaissance des Temps for the year xv of the Republic, he gives a comparison between the values furnished by Mayer's and Laplace's theories, and he concludes that the error of the former may sometimes amount to 7". But this difference is caused almost wholly by an error in his own transformation of Laplace's expression. In the formation of Mayer's Argu- ments of Evection and Variation, the true longitude of the Sun is employed, but Burckhardt appears to have inadvertently used the mean longitude instead of it, an error which is the exact converse of the one above noticed with respect to his own Tables. After examining Burckhardt's Table of Parallax, I was naturally led to scrutinize more closely the results of the theories of Damoiseau, Plana, and Ponte"coulant, with respect to the same subject. Although the differences between these were very trifling when compared with the errors of Burck- hardt, still they were greater than we had a right to expect, considering the close agreement which existed with respect to the equations of longitude. In the theories of Damoiseau and Plana, the expression for the projection of the Moon's radius vector on the Ecliptic in terms of her true longitude is required in order to find the relation between that longitude and the time, and therefore no pains have been spared to obtain it with accuracy ; but in the subsequent operations and transformations necessary in order to deduce the expression for the Parallax in terms of the time, the same care has not been employed. In Ponte"coulant's theory the time is taken as the independent variable, and consequently the analytical expression for the Parallax in the form required is obtained immediately, and is developed to as great an extent as the corresponding expression for the longitude, yet in the conversion of his formula into numbers he neglects all the terms beyond the fifth order, so that several of the resulting coefficients are sensibly in error. I have endeavoured to supply these defects and omissions. In the seventeenth volume of the Astronomische Nachrichten, M. Hansen gives the expression which he has obtained for the logarithm of the sine of the horizontal Parallax, by means of his new method of treating the Lunar Theory. I have transformed this expression with the care which its great value deserves, so as to compare it with the results of the former theories. 122 92 ON NEW TABLES OF THE MOON'S PARALLAX. [17 The agreement thus found between the several theories is most satis- factory, the difference of the separate values of each coefficient and the general mean rarely amounting to a hundredth of a second. There are only two instances in which this amount is much exceeded. One of these relates to the constant of Parallax, the value of which, given by M. Hansen's method, is //- 06 less than the corresponding value found from the same fundamental data by the other methods, and the second relates to the term whose argument in Damoiseau's notation is t + z, the coefficient being 0"'146 according to Damoiseau and Plana, 0" - 140 according to Pontecoulant, and 0"'181 according to Hansen. The values of the constant of Parallax which I have deduced from the theories of Damoiseau, Plana, and Pontecoulant agree perfectly with one another, and from the particular examination which I have given to this subject, I am induced to place considerable reliance on the result. It is possible that M. Hansen's definitive value of the constant may differ slightly from that which he has given in the paper above referred to. From the value of the constant of Nutation found by M. Peters, it follows that the ratio of the Moon's mass to that of the Earth is as 1 to 8T5 nearly. Employing this ratio, together with the dimensions of the Earth according to Bessel, and the length of the seconds' pendulum in latitude 35^, deduced from Mr Baily's Report on Foster's Pendulum experi- ments, I find the value of the constant of Parallax to be 3422 //- 325. Now Henderson, in the paper cited above, has found the value of the constant, by comparison with Damoiseau's Tables, to be 3422"'46. It should, however, be remarked that what the Table calls the Parallax is more strictly the sine of the Parallax converted into seconds of arc. In Henderson's calculations he has taken the tabular quantity to denote the Parallax itself, so that the value found must be diminished by 0"'15 in order to obtain the constant of the sine of the Parallax. Thus the value deduced in this manner is 3422 // '31, a result admirably agreeing with that just derived from theory. I have carefully transformed the expression for the Parallax given by theory, so as to make it depend on Burckhardt's Arguments of Longitude, and from the resulting formula Mr Farley has calculated the Tables which are appended to this paper. Constants are added to the several equations so as to render them always positive. 17] ON NEW TABLES OE THE MOON'S PARALLAX. 93 The Minor Equations of Equatorial Horizontal Parallax are comprised in Table I. Table II. contains the Equation depending on the Argument of Evection; Table III. that depending on the Argument of Variation ; and Table IV. that depending on the Argument of Anomaly. The formulae employed in their construction are the following, in which E denotes Burckhardt's argument of Evection ; V that of Variation ; and A that of Anomaly ; and the Arguments of the Minor Equations are denoted by their numbers as in Burckhardt. (Arg. 1) (Arg. 2) (Arg. 4) (Arg. 5) (Arg. 6) (Arg. 7) (Arg. 8) (Arg. 9) + 1"'81 cos 2 (Arg. 9) (Arg. 12) (Arg. 13) (Arg. 16) (Arg. 23) (Arg. 25) E+ 0"'41 cos 2E V+ 26"'34 cos 2 F+ 0"'16 cos 4 F A + 10"'27 cos 2A + 0"'64 cos 3A + 0"'04 cos 4A In this formula, a few terms have been neglected, the largest of the co- efficients of which does not exceed 0"'08. The sum of the constants in this formula is 3422"'29, slightly differing from what is called the constant of Parallax, in consequence of the change in the form of developement. 0-34- 0-34 cos T73 + 17 3 cos 1-46 + 1-46 cos 0-87 + 0'87 cos 071- 071 cos O'll- O'll cos 0-62- 0-62 cos 1-81- 0'05 cos 0-21- 0-21 cos 0-16- 0'16 cos 0-14 + 0'14 cos 0-12 + 0-12 cos 0-10 + O'lO cos 36-81 + 37 '22 cos 26-18- 0'94 cos 55' 50-92 + 187-1 4 cos 94 ON NEW TABLES OF THE MOON'S PARALLAX. [17 For the sake of comparison I will here give the formula on which Burckhardt's own Tables are constructed, which is as follows : 0-4- 0-4 cos (Arg. 1) 0'8+ 0'8 cos (Arg. 2) 0'3+ 0'3 cos (Arg. 4) 0'8+ 0'8 cos (Arg. 5) 1-1+ 0'8 cos (Arg. 6) 0-6- 0'6 cos (Arg. 8) 1-8+ 1-8 cos 2 (Arg. 9) 07+ 07 cos (Arg. 12) 1-0+ 1-0 cos (Arg. 13) 43-0+ 37-4008^ + 0"'4cos2J 30-0 - 1 -0 cos V+ 26"'3 cos 2 V+ 0"'3 cos 3 V 55' 40'0 + 187-0 cos A + 10"'2 cos 2A + 0"'3 cos 3A The sum of the constants in this formula is 3420"'5. The errors of the coefficients of Equations 2 and 12 arise from the mistake respecting the formation of the Argument of Variation before ex- plained, and those of the coefficients of Equations 4 and 13 from the similar mistake respecting the Argument of Evection. Equation 6 is taken with a wrong sign, and in the Variation Equation 3 V appears to be wrongly substituted for 4 V, though I find that the corresponding term, when reduced to Burckhardt's form, has a smaller co- efficient. In consequence of the way in which most of these errors originate, their amount will be generally greatest in March and September, and least about the beginning of January and July, when the Sun's mean and true places coincide. The total error of Burckhardt's Tables may amount to nearly 6", in- dependently of the change in the value of the constant. Looking at the accuracy of modern observations, it is easy to imagine to what an extent the value of comparisons between observed and tabular places may be diminished by their being liable to an error of this kind. In determining diiferences of longitude by means of occultations, it is 17] ON NEW TABLES OF THE MOON'S PARALLAX. 95 plain that the results may be considerably affected by such an error in the Parallax. It has often been remarked that differences of longitude obtained by means of different occultations are not so consistent with each other as might be expected from the precise character of the observation, and I have no doubt that a great part of the discrepancy is to be at- tributed to the use of an erroneous Parallax. Mr Maclear's observations at the Cape, combined with European obser- vations, would doubtless furnish most valuable materials for a new deter- mination of the constant of Parallax, care being of course taken to employ correct Tables in the reductions ; and such a work would be a useful contribution to Astronomy. In order to facilitate these and similar objects, Mr Stratford has calculated the Parallaxes from my Tables for each Greenwich mean noon in the years 1840 1855, and has thus obtained the corrections to be applied to the corresponding quantities given in the Nautical Almanac. These corrections are embodied in Tables which are appended to the present paper. Subsequently to 1855, the Moon's Parallax given in the Nautical Almanac is calculated from my Tables. 96 ON NEW TABLES OF THE MOON'S PARALLAX. [17 TABLE I. OF THE MOON'S EQUATORIAL HORIZONTAL PARALLAX. ARGUMENT : Arg 8 1,2,4, & c - from calculations of the Moon's Place by Burckhardt. Arg. Arg. i 2 4 5 6 7 8 9 12 13 16 23 25 // /, II l // tl // // it It tl ooo O'OO 3'46 2-92 74 o-oo o-oo o-oo 3'57 o-oo O'OO 0-28 0-24 O'2O IOOO OIO o'oo 3'46 2-92 74 o-oo O'OO O'OO 3-56 O'OO O'OO 0-28 0-24 O-20 990 O2O O'OO 3'45 2-91 73 o-oi O'OO O'OI 3'5 O'OO O'OO 0-28 0-24 O-2O 980 030 o-oi 3'43 2-89 72 o-oi o-oo o-oi 3'44 o-oo o-oo 0-28 O'24 O'2O 970 040 O'OI 3'4i 2-87 71 0-O2 o-oo O-O2 3'35 O'OI o-oi 0-27 0-24 0-20 960 050 O'O2 3-38 2-85 70 0-03 o-oo 0-03 3-23 O'OI O'OI 0-27 0-23 O'2O 95 060 O-O2 3'34 2-82 68 0-05 o-oi 0-05 3-08 O'O2 O'OI 0-27 0-23 0-19 940 070 0'O3 3'30 2-78 66 0-07 O'OI 0-06 2-92 O-02 O'O2 0-27 0-23 0-19 93 080 0-04 3-25 2-74 63 0-09 O'OI 0-08 274 0-03 O'O2 0-26 0-23 0-19 920 090 O'O5 3-19 2-69 60 o-n O'O2 o-io 2-54 0-03 0-03 0-26 O-22 0-18 910 IOO O'O6 3 -I 3 2-64 57 0-13 O-O2 O-I2 2'33 0-04 0-03 0-25 0'22 0-18 900 no 0-08 3-o6 2-58 53 0-16 0-03 0-14 2-n O-05 0-04 0-25 O'2I 0-18 890 1 20 0-09 2-99 2-52 50 0-19 0-03 O-I? 89 0-06 o'O4 0-24 O'2I 0-17 880 130 O'll 2-91 2-46 46 O-22 0-03 O-2O 66 0-07 o'05 0-24 0'2O 0-17 870 140 O'I2 2-83 2-39 42 0-26 0-04 0-23 "44 0-08 O'o6 0-23 O'20 0-16 860 150 O-I4 2-75 2-32 38 0-29 0-04 O-26 22 0-09 0-07 O-22 0-19 O'i6 850 160 0-16 2-66 2-24 '34 0-33 0-05 O-29 01 o-io o'07 0'2I 0-18 o'i6 840 170 0-18 2-56 2-16 29 o-37 0-06 0-32 0-82 o-ii 0-08 0'2I 0-18 0-15 830 i So 0-2O 2-47 2-08 24 0-41 0-06 0-36 0-63 O'I2 o'og O'20 0-17 0-14 820 190 O'22 2'37 2'OO 19 0-45 0-07 0-39 0'47 0-13 O'lO o'ig 0-16 0-14 810 200 0-24 2-27 91 4 0-49 0-07 0-43 o-33 0-14 o-ii 0-18 0-16 0-13 800 2IO O-26 2-16 82 09 0-53 0-08 0-47 O'2I 0-16 O'I2 0-17 0-15 0-13 790 220 0-28 2-05 73 03 0-58 0-08 0-50 0-12 0-17 0-13 0-17 0-14 O'I2 780 230 0-30 95 64 0-98 0-62 0-09 'S4 O-O5 0-18 0-14 o - i6 0-14 O'll 770 24O 0-32 84 55 0-92 0-67 O'lO 0-58 O'OI O-2O 0-15 0-15 0-13 O'll 760 250 0'34 73 46 0-87 0-71 O'll 0-62 o-oo O'2I o'i6 0-14 O'I2 O'lO 75 260 0-36 62 37 0-82 075 O'I2 0-66 O'O2 O'22 0-17 0-13 o-n 0-09 740 27O 0-38 'Si 28 0-76 0-80 0-12 0-70 O'O6 O-24 o-i8 O'I2 o-ii 0-09 730 280 0-40 41 19 0-71 084 0-13 0-74 O-I4 0-25 0-19 O'll O'lO 0-08 720 29O 0-42 30 TO 0-65 0-89 0-14 0-77 0-24 O-26 O'2O O'lO 0-09 0-07 710 300 0-45 19 01 0-60 0-93 0-14 0-8 1 0-36 0-28 0'2I O'lO 0-08 0-07 700 3 0-47 09 0-92 O'SS 0-97 0-15 0-85 0-51 0-29 0'22 0-09 0-08 0-06 690 320 0-48 0-99 0-84 0-50 01 0-16 0-88 0-68 0-30 0-23 0-08 0-07 0-06 680 33 0-50 0-90 0-76 0-45 05 0-16 0-92 0-87 0-31 O'24 0-07 0-06 0-05 670 34 0-52 0'8o 0-68 0-41 09 0-17 0-95 07 0-32 0-25 0-07 0-06 0-05 660 350 0-54 071 o'6o 0-36 13 0-18 0-98 28 0'33 0-25 O'o6 0-05 0-04 650 360 0-56 0-63 '53 0-32 16 0-18 01 50 0'34 O'26 o'os 0-04 0-04 640 37o 0-57 O'SS 0-46 0-27 19 0-19 04 73 0'35 0-27 o'O4 0-04 0-03 630 380 0-59 0-47 0-40 0-24 23 O'I9 07 96 0-36 0-28 o'O4 0-03 0-03 620 390 0-60 0-40 0-34 O'20 26 O'i9 10 2-19 0'37 0-28 0-03 0-03 O'02 610 400 0-62 - 33 0-28 0-17 29 O'2O 12 2-41 0-38 0-29 0-03 O'O2 O'O2 600 410 0-63 0-27 0-23 O-I4 31 O'2O 14 2-62 0'39 0-29 O'O2 O'O2 O-02 590 420 0-64 O'2I 0-18 o-n 33 O'2I 16 2-82 0'39 0-30 O'O2 O'OI O'OI 580 430 0-65 0-16 0-14 0-08 35 O-2I 18 3-01 0-40 0-31 o-oi O'OI O'OI 570 440 0-66 O'I2 O'lO 0-06 37 0-21 20 3'iS 0-40 0-31 O'OI O'OI O'OI 560 45 0-66 0-08 0-07 0-04 39 O'2I 21 3'32 0-41 0-31 o-oi O'OI O'OO 55 460 0-67 o"5 0-05 0-03 40 O-22 22 3'44 0-41 0-31 o-oo O'OO O'OO 54 470 0-67 0-03 0-03 O'O2 41 O'22 23 3'54 0-42 0-32 O'OO O'OO O'OO 53 480 0-68 o-oi O'OI o-oi 41 0-22 23 3'6i 0-42 0-32 O'OO O'OO O'OO 520 490 0-68 o-oo O'OO o-oo 42 O-22 24 3-65 0-42 0-32 o-oo O'OO o-oo 510 500 0-68 o-oo o-oo O'OO 42 O'22 24 3-67 O'42 0-32 O'OO O'OO o-oo 500 i 2 4 5 6 7 8 9 12 13 16 23 25 To be substituted for Burckhardt's Table XXVIII. 17] ON NEW TABLES OF THE MOON'S PARALLAX. 97 TABLE II. OF THE MOON'S EQUATORIAL HORIZONTAL PARALLAX. ARGUMENT : The Ai-gument of Evection from calculations of the Moon's Place by Burckhardt. o i 2 3 4 1 8 9 10 ii 12 13 14 11 17 18 '9 20 21 22 23 24 3 11 29 30 s diff. '4-44 i 1 I0 '5 2 10-22 9-90 9-58 a - 12 54 11-61 "36 t - XI 8 diff. II 6-24 5-82 5-39 4-96 , ' 37 '8 ' 43 45 5-02 57-47 56-92 56-36 55-79 55-22 S 5 g y*l II s diff. I n o 55-22 .. o 54;o5 o-ls o 52-88 5 9 o 52-28 ;5 _ 4 S 5 39-65 o 39-00 038-35 037-70 o 37-05 o 36-40 IX- IIP diff. o 36-40 o 35-75 035--0 o 22-57 02.-98 o 21-40 o 20-82 20-24 s irs . 6 4 o 26-20 2 5'59 024-98 2 4'37 0-60 VIII 8 IV diff. n O iS'OO T 4S 0-54 '*' 91 0-54 o 16-37 :.?! 1532 014-81 s o 13-30 S Ifli o 9-62 o 9-i9 o 8-78 g -37 o 7-97 r58 7 S r 46 o 6-ii o 5-76 C ^lo o 478 ' 4 3 ! 43 4 ' '33 VII s V s 478 1 diff. 0'3' 0-30 0-28 0-28 2-83 3 i '44 Ii s ?: ;; o-oo o . -3 *, 0-02 I ' o-oo o o-oo VI s 30 29 28 27 26 25 24 23 22 21 2O 19 18 i? 16 IS H 13 12 II IO 9 8 6 5 4 3 2 I O To be substituted for Burckhardt's Table XXIX. 13 98 ON NEW TABLES OF THE MOON'S PARALLAX. [17 TABLE III. OF THE MOON'S EQUATORIAL HORIZONTAL PARALLAX. ARGUMENT: The Argument of Variation from calculations of the Moon's Place by Burckhardt. O I 2 3 4 6 8 9 10 II 12 13 14 IS 16 17 18 '9 20 21 22 23 24 25 26 27 28 29 30 0" diff. 51-7 50-41 3 ^J 50-13 49-80 49-43 49-04 48-61 48-16 47-68 47-18 45-50 44-89 44-26 -6i 42-93 42-23 4i'5 4078 40-02 39-25 38-46 ~ O'3O > 4 XI 9 diff. o 36-83 o 36-00 035-15 34-30 o 28-08 5 53 o 25-35 3 JJ 24 ' 4 5 o 23-54 o 22-64 02174 020-85 '9-97 o 19-09 l8 '" o 14-04 '9 '9' O 'Q O o-o i 9' 9 : 3 H S! 0-86 : * ' IP o 12-46 o 11-70 o 10-95 O IO'22 o III o 8-16 diff. 0-76 0-75 0-7? 5-16 4-63 ft' 2-40 2-O-? 1-69 - 8 '9 0-03 0-06 o-oo * 'SS 0-37 " '34 -5 0-06 IX' IIP IV diff. 0-19 0-32 ' 88 1-71 2 '5 2-42 2-82 i 4-68 5-21 5-77 o-o, : 6 OI ;;3 "9 '3 diff. o 7-59 o ,S o 11-08 o "-S3 o 12-61 o 13-40 + '53 -' '74 .ll ? 8 79 VHP o 17-59 o 18-46 '9-35 o 20-24 o 2i-,l o 22-06 o 22-98 023-90 o 24-83 o 2575 o 26-68 o 27-62 o 28-55 o 29$ o 30-40 31-32 3 2 ' 2 J o 33-16 34-06 VIP diff. o 40-08 "9' '93 '9 2 0- 93 '94 ' 93 - 9 3 9 9 * 9 y 042-45 o 43-20 o 43-94 44-65 o 45-34 046-55 46-65 048-43 48-97 o 49H9 049-98 o 50-44 o 50-87 o 51-27 o 1-64 051-98 o 52-29 052-56 o 52-81 o 53-03 o g# 3-47 53-55 o 53-60 053-62 /;) ' 2 5 VI- 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 IS 14 '3 12 II IO 9 8 7 6 5 4 3 2 I O To be substituted for Burckhardt's Table XXX. 17] ON NEW TABLES OF THE MOON'S PARALLAX. 99 TABLE IV. OF THE MOON'S EQUATORIAL HORIZONTAL PARALLAX. ARGUMENT: The Argument of Anomaly from calculations of the Moon's Place by Burckhardt. O 9 i 8 IP III 8 IV 8 V s diff. diff. diff. diff. diff. diff. o / ft / // / // o M It to // O 59 9"oi 58 38-10 a 57 .8-70 55 40-69 54 12-84 ,.,. 53 13-97 30 I 59 8-98 . 3 58 36*10 . ..- 57 15-55 35 55 37-46 33 54 10-33 Lf: 53 12-64 .33 29 2 59 8-87 g 58 34-04 *. 57 12-38 3 |7 55 34-25 L!l 54 7-86 ^ 4 7 53 ii-35 .% 28 3 59 8-69 58 3.'92 2 . 17 57 9-20 3 ' 55 31-05 ^., 8 54 5-42 44 53 10-10 27 4 59 8-44 58 29-75 57 6-00 3 55 27-87 3 54 3-02 ^ 4 53 8-90 :* 26 6 7 59 8-. . 33 59 7-72 39 59 7-26 . 4 " 58 27-53 HI 58 25-26 ' 58 22-94 ,.3 fi 57 2-78 3" 56 59-55 f *3 56 56-31 3.H 55 24-70 37 55 21-55 3 5 55 '8-42 33 54 0-66 30 53 58-33 I'.** 53 56-03 ^.3 53 7-74 . 53 6-63 " 53 5-56 .11 25 24 23 8 59 6-73 ?3 58 20-58 t . 3 56 53-05 Hi 55 5-30 *" 53 53-77 ,. " 22 9 CQ ^* | ? 58 18-16 *. 4 3 564979 3 55 .2-20 3 5351-55 HI 53 3-56 9 21 10 59 5'45 . 7 . 58 15-70 . 4 56 46-51 a . n 55 9-12 3' 53 49-36 *: 53 2 -62 :g 4 2O ii 59 4-7I Q.oT 58 3;2o J.JJ 56 43-23 I* 55 6-06 3 53 47-22 I, 4 -- _._- O oQ JJ * /O o'Re 19 12 13 59 3-90 . 88 59 3-02 . 58 's-06 2 'l 9 56 39-94 3.^9 56 36-64 33 ss 3-03 3;3 55 0-02 **" 53 45-11 . A 53 43-03 l'. 53 o'88 53 o-os : 17 .4 59 2-07 .95 58 5-43 H? 56 33-33 ^., 54 57-03 I 99 . 53 4i-oo 3 52 59-33 " ,, 16 IS 16 ^ '.'I S : < 3; 56 30-03 33 56 26-72 33 S S- :: 53 39-oi n 53 37-05 l? 52 58-62 7 1 O'O7 52 57-95 "'67 IS H .7 58 58-82 .' 5 C*7 C*7'9O fi 57 57 ^9 ,..o 56 23-41 33 54 48-20 | 53 35-13 l.g. 52 57-33 .<7 13 18 58 57-6o " --, C4'CI 7 56 20-10 .' 54 45'3 ,. 8 g 53 33-26 |2 52 56-76 " 12 .9 58 56-31 .\ 9 . 57 5i"69 .0 56 16-79 33 54 42-45 ..a. 53 31-42 ,. 4 52 56-23 " II 20 58 54-97 .ft 57 48-84 56 I3-48 33 54 39-61 ^| 4 53 29-63 J4J 5 2 5575 f IO 21 58 53-56 4 57 45'95 . 56 10-17 "I 54 36-80 53 27-88 75 52 55-31 .tn 9 22 23 58 52-08 4S 58 50-55 ?3 57 40-o8 *;9| 56 6-87 33 56 3-57 3]3 54 34-02 "2 54 31-26 Jig 5326-16 .7 53 24-49 .. 6 ( 52 54-92 r.f 52 54-58 4 4 8 7 24 58 48-95 6 6 57 37-1 y 56 0-27 33 54 28-54 * 53 22-86 J2 52 54-28 3 6 25 26 58 47-29 . 58 45-57 .12 57 34-.0 3 57 31-07 3 3 55 56-99 3 8 55 53-71 3 _ 54 25-84 ^.i? 54 23-18 1 53 21-27 J.S 53 1973 ,.?; 152 U-0-? ' 25 52 53-82 ' 21 5 4 27 58 43-79 .0, 57 28-01 $* 55 50-44 {.A 54 20-54 g 4 53 18-22 5 i 52 53-66 . 3 28 58 41-95 .OQ 57 24-93 3 55 47-i8 1" 54 17-94 ,." 53 16-76 ['. 4 s 2 53-54 !" 2 29 30 5 8 4 o-o6 3 58 38-10 90 57 2.-S2 3" 57 18-70 3 ' 55 43-93 3.^5 55 40-69 54 iS'37 !" 54 12-84 53 53 I5-34 , 4 53 I3-97 37 52 53-47 to 52 53-45 I O XP X" IX 8 VHP VJP VI 8 To be substituted for Burckhardt's Table XXXI. 132 100 ON NEW TABLES OF THE MOON'S PARALLAX. [17 s PH oa O ij as 2 H II S & 5 w MM S a 5 & H co -^ w S3 bd K w co 8 w m S w ^ ^ 3 o ^7 3 si i H 1-1 ^ O S a & H -1 3 a PH H PH -d B 3 CO O g S 5 1 1 i + + ^ f}00 ^"OQ in ON O *** ^" ON M *d" N O >0 "^y= O l^vo CO 1-1 Ix 1-1 t10\"T-00 * o Z i i + + 1 1 M I~ WiOO Tj- r^ ON N ^- r*0 TJ- T}- N O N oo -3- "to aoo"HOcoT3\ M'O n + 1 1 + + 1 1 rf N 00 - NO "-1NO inNO ir> N in ^-NO ON N r^N r^rJ- ior^O N MimOOO ro 1 co N ~ ~ - -l---"l-W NN-OOOO-.~-iNNNNN-ii-. si> Tj- f> f} ON 10 i- N N l- r^\o NO r* OONt^r^txNNOOsN rOOO OO N VO Ix 1~, s + + 1 1 + + >, WOO <* -3- fO N 00 00 >O I-NO NO * Ot~-NO\-fr~f~OOOO"1~NIx f^OO * O I 1 3 'd- " + + 1 1 + + 00 ro ONVO -Tfi^NNNnNMO O N O OO O O O O N OO OOO *O Ix O !** ~ 1 00-NNNNNNN1SNN i-i>H-iO l -ii- i *- IB "*-i-iOOOOO BNNN >> N O 1^00 N vO O^ *^O 00 CM" i- -ONOrxO>"i-i-*tx"^ *oo oo f - m a a 1 1 + + 1 1 j O r^.\o O ^ ^ r^i TJ- o ^"QO M 10 mt^-oo 11 *) 'l-l^N ronono *- * 1 1 1 + + 1 1 N ^ in " t^ fl^O ON O '"'I O ^" O >ovO 00 O "-> O 00 ro 1- Ovoo ro w ul O N m 1 * b^Mf^NN | -'bbb'-''-'N i i + + 1 . rn*O tN. O ** * ^"O 00 m t^. O N N rn\o O*O M O ^1- * N O tx^O ro 1 11 + + N tn - - Tj-Tf^J-O M*-*rO rO-O iO N 1-1 ft CO ^J- ON ^C50 *3-00 Q\ ON tx N ^J-^o Ix I y NN -0000 0-,----- N N^t^ m r- 5 rorr, 1 N-0 am jo X*a T**"*".8-$as t-^2 J^S ff8 5 8 ff^ff'S ffS S-^P, . M t^ Tl- M O 00 O\O O O O O *0 30 tn ** f*i noo o\ O ^^ ^oo " r^. O ^o r** vn I + + 1 i + + vn H * UT* fO O. ^VO OO vO Tt- ~ N O NO ** \O f*l ^~ ^t" t^ 1 *^ inO *O O NO r'lCG o - W'if^ W-* Vj-fn - o b + + i 1 + + . O\ir>B- ior--i-i OOOvO "- N N t^ N N 00 GO ^O ^ t~^OO ON^O ^ ON 1^1 ^ t*> N t^> + + 1 i + + , xn f'l N f)ifi O N 1000 xo r^ t^ inO "^ GO O N O* fO f> O in ON ^*vO ONOO l^ 11+ +1 OOOOCNirom^-Tj-Tj-rocNiNOOO 1 + + . fl I^^O OOOOOOvO ^-O^O O -<*-f^ H oo oo -< r->. r^ ONOO m rnoo O 't'O int^ N N 4 1 1 + -OOH-HHn^inin^^rON000 t, ^OO*OTfi/TCO ONOO ON ** t^ in N O 00 00 O Tf 1 inO ^O M O | - __,oooooooo~ + +1 1 + + 00 GO ONOO ON in ON ** O ^" ro O N r. N ^""O ^-O *n t^ ON inoo "" O " ONGO in m I 1 g 1-5 rnrOfOfOro NI-H O O O O O " NNN " M " NNN + 1 * j^-bAJo^co-Mbb ON (^)00 OOvOvD O WOO f^Tj- CO N M xn LOOO ONOO 1^ rx ^J- N fO ro PH N^-invOr^^t^vO^N-00 OOOOOOOOOOOOOOO-N J N M N rOf">O int>Ooo NvO f) ^- r^\o WON *}"NO rx ON ^i n O ON 10 N Tf xrj a - i . r) ro ^- in^O *O 'O ^ ^ fl NH O bbbbobbbbb"-~i-ibbbb- . ON ^ fl O f^OO ON ^^ fl ^tvO - m looo ONONtxT)ro"---o-N * | + + i + + rOfOf^t^O MvO O M 'tNO \O Tj- fn N vo t^NO -^- ONNO N N tN OI^tx^t-NvO 0000 r.N NN NMNNMNN- OOOOO qjaOH N ro -^- IOVO 1^00 & O ^ N f> 2" ^"S rl2 2* 8 N S S" ^ S?^ R-'g S" ;3> S> am jo jCd 17] ON NEW TABLES OF THE MOON'S PARALLAX. 101 d oj O 1 1 + + 11+ + > O NO N CO O N flNO I~ ON PO O f^*O f^OO 00 N fO - t^flt^N xr>N N O-O o 1 1 + + 1 1 . ~ NO >- vn ON ro ro <( Tf m iJ-NO * ro 10 vn ro t^oo N "J-'+O^W-frON'-'OO*!- 1 1 1 + + 1 1 4 fXJO N vo Tj-NO to - vr> fO - Tt-00 ON O O* "10O "1 ONONONI^^J-^^-ONONVO i + + 1 1 N f m ty, n m UMJO 1^ mO NvO flfXjMnN "-VO o * ov n- st-oo N i-i m m ~ + + I 1 >, O\ v/^\o f^ fO ON to O\OO O O "1 O ^"1 O f*^ ^ ^ "^00 Tj-t^O ONONioN N i-.unt~ 3 1 * NrOfOfOr'}NNn-NNwN6b6bbbb + +11 + + 00 2 Tt^O VO moo Tft^OOOOi-MOcXlrrNfOrO--" tO\D VO ONroI^~ rOfOt^ "- 1 + + 1 + + ^, mvo vO >O -*o nfO t^oo ^- t" t-^. *^>O O ^ O O N S + +11 + + oooONONrofJONf>>-i "^NO N ui N moo oo r~ m O O " rooo VO 00 x vO O P* + +1 ! + + . M o O O O O f*^oO "- 1*1 **1 O "^ OO !* ^ N ""1 1*1 10 M vnsO t^vo VO 00 f) "^ N vr> j " bbbbbbbb->--'-b'-if^ "ii -bio i i + kO^^tj-MN^OOOOO + . NONrOTj-i*lON-Ov Tfoo "io VON -o-Tl-roi^ N rovomf^ioo * ~ PI oooobb---oooN^-^NOvONOvo^ vo ^ fo N ^^ o O O O^o 'OO vovoLno vnt^u-ivno ONOOO Tj-r^"- t*N tf C4 -O O fOOO ^ ON NO ^ + +11 + + ^"la -,* W o^ooo>o = 2 . ?S -. ? o. r o 2 , 8 sas?^^s- < 8s > ftfo . r-- ON rlO ONto-*ONO rIO * TJ- Tf - "1OO 1^ t- N 00 NO fioo OO vo-O N ON wi Q + +11 + + o * vb'in^-SN^'irSb'b'bbb b"b "fa b^ b" - 1 + + . CO O N co O OO t^^O xrt\O 10 "^ N N *H N co ON * - OO fO N 00 t-O r^ O 1^ 1 - vOvO^ro-'-OOOOOOOOOOOOO -00.-N-*VOO^>0 J O ON ""> t^ N N "^ f*5 N ^O ^^ fO ^ ^VO 1*1 floO 10 i-c IOVON vot^N Tj-VN 1 + + bbbbb fo'i- iovb 11+ + . T. N >*1NO COJ^ro^-00 ON N i-c ON -NO O- * O * - r^OO f> ON 9 J -.--00000000--NNNrocON N-00000-N ro ^ ^-^.-mcoT)-'*mo>n-- * r~ o NO ON t^ CONO oo m fo ON N (S) 3 -H- ^ + + 1 I + + 00 s rf Cl t^ Tj- r*5 r*o N u-ioO i-i \O rOVO ON t-^ fO ONOO O ""> ON J-ND vn ON ro ON r^ <( " 1 --OOOOOOO- MfOt5"-OOO g> t^ CO O **! ONOO r^ ON ON N M \O t-^vo ^- ^J-O P* ^}" f^tnioNoo O ONVO TJ- o oo S + + ' 1 + + 2 ON t^ fO tx 00 ONOO O O ON i*NOO O "- ONOO -lONNO ON ON 1 + + 1 1 + + ij S + + 1 cococON O O "-" coco^-^- i + + LO ON O ON ON ^ (S O OO t^^O NO N I s * co ^CC I~>.\O " ON - t^OO > O >*1N 1 +__ + ' 1 + + d >-3 O OO vo ON "1OO O " ^J" co N O ONNO iri N ON co^O ^" + +i D vO ^ f> co ^~ *^ ^o tv.00 O OC riMnn co i + + qinow - rorf^-o ^00 ONO - ro^vo j^oo O- O saffjjrs^ffas, 102 ON NEW TABLES OF THE MOON'S PARALLAX. [17 _ mr- * ^~O t^ ON t^ O OO O " O CO CO N fj ro N r^ O " O 00 * ^ ^"O OO Q + + 114 + 3 U-.TJ- C N N -" O ""1 *"" *O O O^ ^^OO ^OO OO W O fO *^1 "^ ro PO fO N O 00 t^ ^ o - - - 6 1 1 + + . O 10NO rOU1~ inxi N O* *O -!)-00 ^ ~ MOO i^i^it^oo - O\>OO * O 1- O 1 i + + i 7 I 1 i + + i 7 ec OO NO N MmoOt^rO~t^On Tl-00 ooOO>mOCT>O~OOOOOcO".^o I "bob + + 11+ +11 >, NO N Q> 1^ T^ i-< t^^O I^vO <-> <-> f^^O ^ M O ^" f^OO fl CTi r>.00 O f} 1^1 M ^ *J"l "^ ! i v MM 00 00 1-- * M 00 00 *00 00"1>-fO^-OM-iO~^O ONOO 00 O mo *O "1 Tl-00 M 1 + + 1 1 + + (J> O "> * ^ t^ t^OO t^t^~OOmMO"^--t^t^ "TOO O\00 - * "IvO ro ~ >O N a + + 1 1 + + t* f^O ^" N ON if) ON fO ^J* fOOO N !"> N OO N O ^" r^>OO O CO f*^ *"> *-OO "^ *-O ON ft + 1 OOO > ~** - NWN"** ip ro M trioo OO ON Tf O (^ "^ 1 + 0.. NM mmmMM-bboNmmmmrorOM... u-100 ON OOM'*"-OOO>ON>-O>nNVOmOMr~'?^"S S"B 2>8 5 8 !? JS"8 {f8 S^^m O N M N O OM^ -^1 m^O rONNVOmCT-m moo OO - I^OO N CMfl - ^C 10 Q 7 N N -00000- -NMNNMrONNN- -OO > o + + 1 1 + + . , 9 r * Ov M ~ OM-" 00 I^^O t^O *00 rr> ro Tf-OO ON O OO *O f^O -< VO - 00 O + + 1 1 + + _J ^" ^" ON N t-^ O\00 "1>O J^TfOCTM^O OvM N ^tOO *O O ^O i O\OO OO - *O N y> m N N M + 1 1 + + O ON O u-,O-*OS->-">r~C\Ni-'*NOO-t^mO\ 'J-OO N O> ONOO "1 t-. a + + 11+ + >, *~ N mON*r^ Mir>OSO>- -00 flONTl-N *inONOvONOOOOO NOO r^vn ^^p A - o o o 00 o> o^ o^co - N 00 O -i * -*O O\ - U1O vo r^ O O * m O>*Q - N O f> p + + 1 + + ;> MOO - t-l O M - CO * ON t-- -4-00 Nr^CT,N^-m-OON-^)-N Ti-NO - NO - MNO u^Nt^ON-r^f-OxO &4 * M M* (- + 1 1 -. CM^ ONNO OO t^ *^)00 OO OO t^ 'J' ^1 HM N O ^"^ N NO W10O OO OO w> ON x'l "-i C4 !* a v N fo + + 11+ + ,0 C*i t^ xn O O ON \f> O n r ONOO fO "1NO fl-OO *N t^mt^OsO ONt^r^I^ " N f^ Th + + 1 1 + + mo "^ - t^ ON t^ * - t^oo vn-NO ONN nu-iu->m- N txON^-"i^ i~-oo oo m - + + 1 1 + + w ow M ro * uvo i^oo ON o - N 2" ^"2 ^2 2 1 8 N a f? S" "8 fr'S ? m m aqi}' > X*(I 17] ON NEW TABLES OF THE MOON'S PARALLAX. 103 O CO O>O ^" ** N GO I*** ^OVO CO O N CO ^OO ON NM ^ LT> f 1 tprOMNN--bobOO------NNNN N N N rommrp 6 R 5 ;T i?!^ J???^ 2 n n :f2 fT'S ? m s s > i? - J^ ET^S ?T _^ LT> C< ON ON *^" ON O ON I s * ^ LO COO OO CO r O * N * ON ^OO um N o Ovro^ttx i i + + 1 1 + -j N O * CO O "1>O ^O t^T)--*'S-r^-OOvO I^O O VO f O N 00 t^ Tt- ^f 1 1 1 + + 1 1 ti -t O "-. m O\O >r> o * ui >^vO "ir-o N f^mmi^O 't 0000-rOMrt-O 1 1 1 + + 1 1 !, - O> O <^ 0^ ONOO VO t^OO ro O. N <*O 00 O * VO VO ro O *o M oo N m m m\o f: 3 1-5 + + 1 1 00 o> ON O HH O OO c*}\o HH 3 4- +1 1 + + k, ^O rO\O \D 00 CO r^. r O N N CO ON t^ O CO^O "1 N ^ ^* ^* f j-oo M a- 'too o O Ot**COt^. C^ 1 O tot^l^O + N ^^*^^ N O O O - . a. o\ t^^o oo - \ri~ wvo a- ON d a> o f>*o t~oo o N t^oo OvN " u-,mm\r>O 1-5 + + 1 1 + + oqTj 1 ow 3 AB(J N rO <* uivO t^OO O O <-" N *^ ^- "^O 1^00 CT* O ^ N r^ rt mvO t^OO O O "-" J ^NVOO-*N-"^->O rlX! O M O - -J3 1^ -f in- M i- CO 1^ O "1* T)-VO [^ Q - - - r. M ro N N 00000 5; >^t^vO>0 * * + + I 1 + + oo "loo ioc ooo t^m^-o UIONCON u->ma.a*ocso N i^vn-o-Noo T)-~ o i + +| i + + +j t^\O co ^ N ^ ^ O N ^" ^O M *O O OO ^ "^ BN N ^J"\O ON ^ O O O OO OO vo C\ i + +i i + + to mo i^^o ao i^ir^^r^oo NOO r^mo O O n * m r o N O\vO >O U-ITJ-X o -3 + +1 1 + 10 + _>, foo N - * oo man r^Ti-t^vo * i^ * m m o t~>o i~-^o t^ Os OssO O CN N ^_ 3 1-5 OOOxfO* 00 0) MO- t^l^O m ~ O O * 1^00 O >)-O"->I^Cs<100 Tl-rOTf N N N n >n > - i^>o * - rn * 10 O uir-lN N " O ** ^ + + 1 1 + +1 1 i-t\oo f 'i | - t Tj-'-ivor^oot^ONc>ovco l oWf^oo -00 * >^r^vO 1 + + 1 I + +1 1 + + b - oo - ro fovo ro N oo ui T)- - Tho -to*o O m&mm "-1U1NM3 -SO * 1 * * f> ^ N - 6 - "> M M fl N - O O O 0000--C.N _^ 10 Q\ to ^J- uloO rt f^^O 00 N fO^-fO^-"-00 r^ulf^ui moO "^ rO ON ^- r^ 2 N- N N CO CO Cs*O t^NOOWvO^OM"-OO^Cior')OWO'- ( M\O" p lON r^ ro N O <* t^-O "i N + + 1 + + sqi jc JUa N fO ^ UTO t^OO Qs O N r O ^" "1VO t->.OO ON O ^ N rO ^ w^o I^-OO ON O hH 104 ON NEW TABLES OF THE MOON'S PARALLAX. [17 ON 00 M n t^" f** N 30 OO in O NO f*^ ON N N CO ^NO O N r^. ro^O ON O r^ O OO O * ON ^-vO + + i i + + 1 ** HH tx, O NH NO O NO N OO ON N N O *"* f** '* s> ON to O *n ON Ln*o N O CO "O ro^o + + i i + + o O NO CO O *"" N OO tO t~^> ON^O N NO ^O -n *" ^O ON I^IFONVO O MI- ^00 f ^000 - + + i i + + "3. CO TfOsro.-.OsOI^Nas>-~sot^O~ N NO ^* N ^-NO O ^^O ON O N O - - - m + t^m + mmt. -00000 NNN ei i 4- * O "" "10O ro + +1 1 + + B 1-5 i-c - o CSI "1 >^00 t^OO O "1~ OS * ro CS O *O I^NO (S "100 O "1 r-~ rOOO ONOO Os N CO t~ NNNNNMrororONN -- O O-OO---r)Ncorr,ro aqi jo SB.OO Cv O "- 1 N r^ ^J" 1^>VC t^OO ON O ^ N <"O ^- u^vO r^OO ON O -" 00 00 HH a o t^ rooo N tn t^ f} f^oo ^o i^* G^ ^ ""^ ro ^ivo SO ^-OO ulTfro^uiOs-- uioo - + + K ^ O ^O O W^OO OOO'- l '- t f^^'NCO""fOO ^f OS t^OO t-*. COOO fO fO M N so N ^ i-iNNroc^rncOThTfVtfOMNNObb^ + + B ** t^ C\ ro Tj-\o u^root^Tj-r^N M, N o - - so roo ro Os ro r^ rooo r~00 + + 1 1 + 4- 1 *00 OS^O ^O iT*N ONiorOTJ-^^l-O tOOO OS "100 O OvOO CvOO "100 O ~ 1^ + " +1 : + + 60 a N Osr^fOOMOTj-O I^N ~ O i^lOv -^l-OO N + - - - - + ( 7 + " + lovnt^- fOO\>O r^OOOOOOO t^uT" Tj-OO m-NflfX^O-NOOOO -* + 1 1 + + a B i-i -iOO^OfOONOON^-OOOOOON"-O "1OO *J- ro rosO M 1^00 00 00 OS M - fo^-^J-Vi-fofoN>H>-NNN + + 1 i-c u-it^t^t^N rOfOI^O fOOO "I^O OOOOVO"-lO>O'-' "100 - O VO > CO 1 + + 1 " " 1 + i M in ON\O oo O t* ON r^ ro xt* ^^oo ^o *t t**oo r>. *^- O "1OO MOO "IN t^OsOO N i i + + 1 1 1 i noo to i-noo " * O to N in^o GO NO t** > Tf TJ- rf r^ " so fOOO T}- *H so 00 vo i i + + 1 1 OO>-"NOsNOONT}-OOiOOO>T*O>r<" ~ TfTtTOOst-T)-Os"1i- rooo + + 1 1 1 00-^--iOOr^Ni-.ONMNrou1NOON Tj-vc - 00 ** l~- OSOO 00 "1"1uirOOsO + fcii-tWMNNWNNNNNN-n + qsnow 9qi JO A-Bd CO .*"VO I>.00 ONO - rOTt^O - 00 Os O N ro 'J- "IsO t^OO Os O "-" 17] ON NEW TABLES OP THE MOON'S PARALLAX. 105 10 00 1 z t a am jo I I + OOOOOOOOOO^-H-OO** *"O *n insO ^O *n 10 ^~ ro N N "* "- O O + + i i + + p"? ~ ' i + ** - b + ioO O "-OOOO OCOOO T}- r^CX> +11 ^p + ^ O "* PO ^" * O + I I ?* P r 1 ^'f 1 T 4 ^y-i^ooop^p ^- ^t^r 1 "" P Mf^fOfoww^-^H^^bbbbbbbbb + i i l tf + fab---ob-. + i i + p --op o + O O " +1 "- O "- 1 1 + i N O CO O OO ^"OO H- O *n f* POOO O + I I + 1+ 00 1 . O O\cc vO 1M N POMNOO ^t + ! I + O "O N ONCp PONQO O PON O IN.N i-bbb^-NMi- bwpo + i i + + + 1 ** ro o T^ PO PO in ^ + + i + i hN y^Oscp + + N N N OO " - ^n ON < ON *n^O *n ON ^ * t*-* *" t** N r^ *- O OO PO PO PO NNHnbbO^^^-^-OOOOH^H-NM PO PO N + +i i + +1 t^ N t w M .'- i^OO >- N st^>p m ~00 ~ O *< Ov ~ N + 1 I + + O N U1 "H '-ON + I i, b b b - - i + + N- O - +1 ** O O O 1 + + p> < + O^CO T}- O 'O fO l-00 vO fOiniOCOO M ON O"OCO O N f*1 O CO \O 14 106 ON NEW TABLES OF THE MOON'S PARALLAX. [17 5 OO M ^-u^NCO f} r*>. O f*} W N "1O u"l (SO NO OS OS Os Cs 1-1 Os ONOO 00 1 1 + + 1 1 . 00 OS U1O "1 *s. CO m CO N w> N Os fi CO "^ O fl *1- rO CO N r1 *t O Tj-t^-N b fe 1 1 + + 1 1 . "1O O O\ * f>00 -O-Of-NfO-'l-flt^NOr's u-lsO O N OS fOOO t^OO < 1 1 + +11 4 OOcj^-o^"^" co *nvO oo * r^. ^00 ON r- N ^ ^* O N r~* HH p^ CTN O N 1 1 1 + +1 . N O ^ ON >n COOO ON r^ cOvO O OO ^ CO N H^ hn Q vncO O C4 O ^ "" ^~NO t- *-A N . + i i + + : i + + i !>, O rt'-iN "1>O ON OX O O >>O ^->O 00 - MO *NI^~ONr'5N CO a 1-5 1 + + 1 1 + + 00 OO O *t N f^) I'lOO Os I-^sO Tl-NOOvOTj-O(SNOOOr^ r-.OO OO "^ O V0 OO SO 1 1 a 1-5 1 1 + + 1 1 ^ rf inoo t^ f^OO -*O N ^"N -O ulm^-osN r~-0s~ -*N *-OO *'t>t^ a 1 1 + + 1 1 . r~I^iO- ^TOOs-^OsN Ost^x^U^O r^sO tOTfniow-r^NHOOOsO"' . 1 1 + + ij "ISO t~ O O t~.SO MI^O ^-OSTl-OQOr^O T)-sO 00 r^r>.. O OS"M~->n~ -4- <* 1 + 1 1 + + . -i OsoO ^ *^> ^ Os M f "^ OO fO O OS O O "* 00 ^so "^ "^ VO t~* *SI O *O V M "* i-i M M ro Tf ^ ^ fl f) NN'^^*"^O'^ B- '^OOO" H NNNNNi^ 1 1 + + . * m *oo rooo MOO o ooo inMoo "ro^-fooso -oo -*oo\vo so - "i 1-5 1 1 + + 1 HJt ow N fO ^- u^so r^oO Os O M f) 'i- i^ivo 1^00 Os O N ro Tj- mso t^20 OS O am ) 3X8(1 . so O m O "IsO u^ Qs * T}- vo u-i - tl-O LOr^--N-*OsNO^5-fOrO OSOO O 1 + +11+ + O + +1 1 + + fsrf O '-'OOf) Tj-OO OS OsvO TJ-T}- SO 00 Tj-I^Os^J-ui ^-vO OO 3 O + +1 1 + + 'S O OsoO M-fOOsoOsrxO-roNsO-Tl--oOM-sOM-l^O-Os>orO + +11+ + . *O ONN Nt^t-.MSOSO"1N O Nxonn rovo O ro * Os N >^100 Os w so 1 + + 1 1 + + ^ vO ^"OO co O ^~ co >-ooo N N <} *nO *n invO ON ^ ON ^^ N O t**"O NO t^. in O 10 i-s + + i i + + 00 0} g ; M^- : l-p rr OpOpsOp r )-^sOMpCppvpr-pp^-ps p pos r )pvp.~- S + + 1 0-^MrofoM M N --000-- N <,^ MMN c, N --00- : i/IOOO N rOt^OsOso >sO sol^Os"imt^-r^l^-Ot^ "^OO N U1O * 1 1 + +11 ^ f!O Tj-OsO ^J-sO OsO Os"^ t-^OO OO t^sO OOr^s OOO^Tj-ThONOO fOSO *<3- I 11+ +11 . t^ N t^ t-i M-s) POO OO OO Tj- m O ^ Tj- -^-OO O r^ f^r^Tt-Os MO iS 1 1 -1- ' +1 1 . ^ n ^ *N vo ^ ON * ^st~oo ^^ n N ^ oo NO N O GO >n co co ^ in O ON 1 11+ +11 q?n 8iIU< - N n * vno ^ ^2 = 2 PJ^S E"5 ?8 5 8 ff JS"8 S"S ^^P> 17] ON NEW TABLES OF THE MOON'S PARALLAX. 107 O N OO 00 !* O ^ r^ ON O\OO 00 ON O PO "^ fl ^^ ^ Q*> N p - --bbbbbbbb I i + + i i o i i + +i i NO f>NO O ""> t-~ ^ ro ONOO t~NO rOrON ONN r^ON ^-NO rO ON ~ Tt-O ro 1 1 + + ! 1 .j MO^OtiNi^t^roN mO t^O *NO NONO"1ONl-c--ONONU1~~Oi-lt^ 1 1 1 + +1 1 roONCNl^to~ONO ON ON ONOO in~uM/1fNOOO~OOOOOONOT!-~O 1 1 1 + +11 >, 10* O C1 NO t~ ON > OOO ^-rON 00 rOMON - M NO ONOO "1 - ONOO NOOOOOO--"ONT)-l^r-.tN 1 O a + + 1 n mm " i + + . -OON^^-t^O"^"^ ro ON N ON f^OO -*OO * U100 Nt^O~ON D. 1 + +1 1 M ^ O r~ * *NO oo m i-- O TfOO -r^~Mt^i-i^-t^--iONt^m t^NO M J_ 1 1 + + 1 1 "j ONOON'*-*"^ t^OO ^T-Nt-t>...ONNiONi-. mNO m o NO m m 1 i + +i i i- r^OO i^> t^ o M N 00 NO *~ONONNON- t^OO ON ONNO "1NON1^OMM d 1 1 + +1 1 mnow ts ro ^ IONO t^oo ON O * M f^ ^- ui\o r^OQ ON O *-" f m ^ t^iNO t^oo ON O ^ sin jo xua . 00 - ON r*>NO <^> N VO 00 r-. NO M 00 "Tt "> "> N ON ui ro O "i ro IN - - ^- I^iO O 2 + + 1 1 + + 1 1 . NO ONONNO ^irtl^r^Tj-^ ^- ON t-. (N.OO r^NO ON fT* m ON N "1OO " * t-- $5 + + 1 1 + + NOnr^OND>-OOt^ t^NO N -f m ~ - t^NO (NINO N r^~ o -^ ma-. r~ ro * ON 1 + + 1 1 + + ^ O ^O N ^ ^"00 ** O^ N N ON N O * tNi r^NO OO^J-'-'toOMONroOf^ r*5NO I + + i 1 + + 00 N M "1O r^to- t^'* MOO t^NO rON~Tl-^-.fOr^- ITN^OO TfOO I^NO OO 11+ +1 1 + ^ rr-u->ioirN-3-O t~m "It^- ONO mO " " " -*00 1-1 -^-ON"- ^--"l-^ONtt- s 3 1 1 + + 1 1 00 g f> t^ ro ON "1 J -*NO NO NO ^ N O f>NO ON N N O-OO UT- -*^ MI^O 1-1 HH 1 1 + +1 1 o,. 00 ro't-N.^roO n to ro r) N N O OO N li- t^OO f>00 ro (N) PO - 48 respectively. The corresponding corrections to Damoiseau's times of mean conjunction in 1880 and 1890 will be as follows: Sat. I. Sat. II. Sat. III. Sat. IV. 888 8 1880 -2-42 -272 -9'82 -17'89 1890 -2-42 -279 -9-80 -18'03 The mean anomaly of Jupiter, which forms Arg 1 1 for each Satellite, has been found from Le Verrier's Tables of the planet. Corrections have been applied to Damoiseau's values of the other arguments so as to make them consistent with the data in p. iii of the Introduction. 152 116 CONTINUATION OF DAMOISEAU'S TABLES OF JUPITER'S SATELLITES. [19 These corrections for 1880 and 1890, expressed in decimals of a degree, are given in the following Table : SAT. I. Arg' 1 456789 III. 1880 -'0011 -'007 -'002 -'224 '005 -'027 -'027 '003 1890 -'0034 -'008 -'001 -'241 '005 -'029 -'029 '003 SAT. II. Arg' 12 345678 1880 -'0011 -001 -001 '002 '003 '005 -'027 -'025 1890 -'0033 -001 -001 '002 '003 '006 -'029 -'027 Arg' I. II. UI. IV. 1880 -005 '023 -002 -003 1890 '005 -023 -002 "004 SAT. III. Arg' 14589 I. IV. 1880 -'0010 -007 -'002 -'031 -'026 '112 '132 1890 -'0032 '009 -'002 -'033 -'028 '120 '142 SAT. IV. Arg 1 123 45 6 7 1880 --0011 -'003 --002 -'002 '007 '003 -'059 1890 -'0034 --003 -'003 -'002 '007 '003 -'064 Arg' I. II. III. IV. 1880 -'058 -'066 -'059 '003 1890 -'063 -'071 -'064 '002 Corrections of the Mean Arguments on account of the Perturbations of Jupiter, J, which is the correction to be applied to Arg* 1, is the great in- equality of Jupiter, and is given in Table IX. of Le Verrier's Tables, where it is called 8L. The perturbations of longitude and of radius vector, which Damoiseau calls < and 1} are to be found in the following manner : 19] CONTINUATION OF DAMOISEAU'S TABLES OF JUPITER'S SATELLITES. 117 Let v a denote the longitude and r the radius vector, calculated from the mean longitude of Jupiter corrected by the secular term in Le Verrier's Table V., and the term 8L in Table IX., and the longitude of the Peri- helion corrected only by the secular term in Table V., employing the constant eccentricity e = 0-0480767, log 6 = 8-6819346, and the constant value of the mean distance a = 5'2025605, loga = 07162171. Also ' = 9916 // -53, log "=3-9963597, - = 0-0208955. These constant logarithms may be used when i\ is found by passing through the eccentric anomaly. If we employ series and call A the mean anomaly we shall have v = L + SL + 19827"'3 sin A + 595"'4 sin 2 A + 24"'8 sin 3 A + l"'2 sin I A, and then r c = - r, l+e cos (v t ro-) where loga(l -e 2 ) = 07152121. Next, let v denote the longitude in the orbit and r the radius vector, as calculated from Le Verrier's Tables, and we shall have = v - v , The value thus found for is to be used instead of , or < multiplied by a constant, occurs in Damoiseau's formula, J+ must be substituted instead of . 118 CONTINUATION OF DAMOISEAU'S TABLES OF JUPITER'S SATELLITES. [19 The following corrections are special to each Satellite : SATELLITE I. Add to the formula for Table III. - 4 9 '2 sin (n - A,,) + 8> 5 sin (II - A m ). SATELLITE II. Instead of the term -9 8 731 sin (n-A n ) in Table III., Substitute the terms - 2 S '5 sin (H - A H ) - 1 8 '5 sin (U - A m ). SATELLITE III. Instead of the term 5 8 775 sin (II A UI ) in Table III, Substitute the terms - 0"'4 sin (n - A n ) - 5 S 7 sin (n - A m ) + 8> 5 sin (U - A IT ). SATELLITE IV. In Table III. instead of the term 16 9 '694 sin (IT- A IV ), Substitute the terms 2 8> sin (n - A m ) + 16 8 '9 sin (n - A ir ). The terms which involve sin (5il 2w 34'542) in Damoiseau's formulae for Table III. of each Satellite are sufficiently accurate as they stand. Damoiseau states that the values of J, , l} SE and Sr which he employs in the formation of the several Tables III., are taken from Bouvard's Tables of Jupiter. Mr Godward, however, has found that the numbers in these Tables do not accurately represent the results given by Damoiseau's formulae. It may be remarked also that the value of Argument 1, or the mean anomaly of Jupiter, employed by Damoiseau slightly differs from Bou- vard's value, except at the Epoch 1750, when the two coincide. In order to be strictly accurate in forming the complete Arguments, the values of J and of J+ corresponding to the actual time should be employed ; whereas Table I. only includes the values of those quantities corresponding to the beginning of the year. 19] CONTINUATION OF DAMOISEAU'S TABLES OF JUPITER'S SATELLITES. 119 The following Table contains the yearly differences of the corrections thus applied to the several mean Arguments, and the correction of any Argument formed from Tables I. and II. will be found with sufficient accuracy by multiplying the corresponding value of A taken from this Table by the Fraction of the year. All the Satellites. Arg' 1. Arg' 3. Arg' 4. ' Arg' 5.' A A A A 1880 o o O o -0013 -049 024 036 1881 -0014 -'073 037 055 1882 -0013 -060 030 045 1883 -0014 -026 013 020 1884 " -0014 007 -003 -005 1885 -0014 030 -015 -023 1886 -0014 042 -021 -031 1887 -'0014 042 -021 -'032 1888 -0014 032 -016 -024 1889 -0015 010 -005 -008 1890 Sat*. I., III., IV. Arg" 6, 7, Satellite in. Satellite IV. Sat. II, Arg" 5, 6 and all the Satellites An-' 4 Arc' !> Arg 1 4. Arc' o. Arg" I., II., Ill TV "'ft * A - i ^ Vt A A *""o * A A 1880 049 028 065 180 049 1881 074 042 098 272 073 1882 061 034 080 222 060 1883 027 015 035 098 026 1884 -007 -'004 -009 -026 -007 1885 -031 -017 -041 -113 -030 1886 -042 -024 -056 -155 -042 1887 -043 -024 -'056 -156 -042 1888 -032 -018 -'042 -117 -032 1889 -on -006 -014 -039 -010 1890 120 CONTINUATION OF DAMOISEAU'S TABLES OF JUPITER'S SATELLITES. [19 M M o in f) fc O ON ON I^vO m rj- rO HH H- M hH N inOQ N . ro O ONOO ON ON in ON r^ o^ fl 00 HH Tt ON M & 1C M - VNO M N ON - ^ \0 ON- M cOvO 1 M HH ro Tj-vO t^ 00 - H- . O N CO -tf- >nvO t-*OO ON O -" 3 ON b Tf ^o O fOCO N L/"l IN H- vn O to N I^-. N t-*. ON N ro N NO ON < oo ao M b b HH HH M N d N fO ro ro *d 00 ONOO r-^r^oONONO min^j-Tj-ri N HH 80 - N ro ^- LTlO t^-00 ON O s 1 I \O M ui V. K t-. GO Ttt^ t->. O^ ON Tj-t^O HH 1* r> vp ON fo ro 1 in r>,oo O fOM3 oo vp ,^vp op vp _* 5 NbV^^lHHO^^>.in*T}^^HH o^oo NO in V O^ C % m in\O l^GO ON O HH p) ro ^- invO t^.OO ON a r-- oo a oo c fo N vO CN - m r^. ON r^ in 4 oo oo CO OO f te' NO t-- I-OO OO GO ONOO ON ON ONOO ON CN ON .9 \O f^ O t^-*O *-.. ro ON \O M A 1 | t^ N M N _ - -- ON ON ON ja OO N inOQ O r)- vri r>.OO ON O N fO Tf PH ^3 W ! "> N rOTj- m'O t^ CO ON O - O B s row OGO r^.infOHH ONt^-m^w Oco S B _ M I w OCO U-JTj- M ON m M QvO g fcl 3 6 oo in r^ i- O\ ^ * N O J^ O " M fO ^f inNO t-xOO ON O HH M ro < ^- | " - M M *t xrjO r^oo ON O O 5 vo r-x oo oo oo oo Eq co QQ - M N M W t- - O ON ON ON !o rOTfnf-.jQoo O O N ^-^-Tt- >nvo TJ -^ IO o ogr . N OO rO M W ^^oo J* S 1 1 w C_| OJ " * N -a- m ^- iri Tt m r^. o*^ f O r^ fOOO HH rou-)nnroHH r^coON^- 3 a -* ro 'IVD ""ICO N <>* 5 - ONOO OO t*- l>.vO m *fr ro N ONOO vO in ^ B ni * * Vi- ij- W (S N rf in in N N vD m m M M ^^ i * MNNNMMNMNN S S" 01 Tf f^OO *J- O' N M vO mvO r^.oo ON O HH N ro TT mvO r>.GO ON H 51 rf m <4 S 02 3 N f> Q fl G\ ON ON -" t>. oo imn VT) Tf CO >N ro Tj- HH t^. . HH ro ro N M cy,b b ON O O HH b ON O " t-*-00 O N ro rh m-O t^-00 GO ON ON ON ON M w M - O O O HH O O - O 1 " NNNMOMNPIMNNNiNWN -3 O HH N ro Th xnvO t^-OO ON O HH N ro ^}- vO O Os r^. M ^* 00 - - r>. M vo T}- r^ ro ON N un \O m ON M a ro * 00 00 p M OO ON ON o\ o o "M HH HH N . _ HH HH o HH ON ONGO 00 NO m TT fo HH o M* Q * rororOrOPONMNNMNMNNN la 1 ll 4 co Tt- m\o NO r-^vo inroHH I^-NVO ONO e H o ' o c! w g- 888 8S' ^88 o o a v O ON N mco HH *T4- fN. o ro moo O N m IN N rororo^^-Tj-minmin S iS-S OH _ - _ -, _ _ r, ; <-i g 03 P ui N Cs r- Tf - CO >O t<-) OOO C S3 HH M i .2. o ; N m t^- N rooo N OO fOOO N U^ HH ON fO HH -fr . mvo tj- -^ON CO t*- S - O'-' l - ll -' 1 - | vNvNMNNrorOro + + s a 8 t>* |-"" H H M N- (N HH J& M ^ B unrfNO ONinfOfO^l^HH t->.inrororO * fO ^J- mvO OO O 0) ^-vo ON HH -^- 1->. o fO \nminmin ** ** w w P-< O HH M to ^f mvO t->.OO ON O HH (S CO Tf J OO CC OO oo oo oo OO OO OO GO CO OO 00 OO OO 00 CO GO OO ON 0000 OO CO 2 'I 31HVJ[ HI aisv.1 BI ap 19] CONTINUATION OF DAMOISEAU'S TABLES OF JUPITER'S SATELLITES. 121 > -" rf- ^\o r O ON -. N 00 O Tf^O <- N 83 (3 VOSOI'I^O'^-P^MHHOON r-.'O xr> ro i ; mmmmmmmmmNNM 1 1 \o m N O \f)\O -rf M p So 1 ^' vo ON moo m ONXO ^-mm^-fN.i-tvo ^ M b b ~ 03 M r*-> i- N Tf U-1SO N m t-.oo ON fo kc 9 E s- 5 M3ONi^jNoo^-) ONb m b oo "i to ^ fimPOP'lfONNMNCINNWNN Jj "i^O r-OO ON O - N fO ^- u^O r*.CO ON a vb 00 ON OO ON i-i N >- M *m 1 00 00 m SB 03 r^^ O r*. TJ- - O r*. Th~ -. Tffs. Tj-r*-'-, *0 T}- t^O id s - m m m m m m m m m m m m m m m 1 | SH t^ Os M io N \O ON m -s^s f^ r^* M w g vc ^J- O ^O nOO w>00 -< ^ t-* ON & 60 I^iON -oo m M -.00 w^ 13 1C v ot**.^ - bt**- r ^ON'O dcow^-'r-'.mb ir>nTj-^}-Tj-mmf^NNN PM | o^ m ON SO ro ^ PH " ^^^-^mmmmmmmmfomro t> a N M N - - b b ON Oi Cv O m ^- u->^o -*oo ON o N m TJ- Hj 60 W ^ >o 01 N ro-^ urO I**- 00 ON O ~ a <1 vO I**. 00 00 oo oo OGG I T3 H *= ir> N- o O oo vb ri n ^- N m ^ ON O "^ N t**. ^- N oco b l? : 5 a vo ON m\o r*-o -" minr>.ONpNO ^ 5 bb^'-'- i ^- | MNNNNmm I I EH W w M fO ^-"I^O r*.oo ON o o J m r*.oo r*. rj-oo O r*. ^^o r- r** 3 s m N N vO M OO \0 "J- O ON U% H u-> ro r*- 'O r-. N OO 00 fO a E - -r b ONOO r* ws ^ H o r* *H M ONVO m Tj-Tj-mfimmf^fifOM N w -* ** W w * '-(- i^ir> M N N T}- invb W N N \O J-O^O N M N ^fr ^*TfTj-Ti-TtTfTl-Tj-^-^--*'trJ--^- H &> 00 M O OH ^ m . u-,\O t>.00 ON O *^ N m ^f wi^O t*.OO ON <^ M M Th ^n 03 ^ CO ^J- r*. ON vO O> ^H r*-.\o Tf r*- -. 00 O N 1000 N ^? <1 00 00 w s w f) f^ fO ^J- ro ONOO t^. - - ^-i vo 10 *a- N ON O m M - O - O j s r-. ui i oo ON ON^ON O*O ^"^ N Tf 5 mmmNWNi-"-^i-"6bb6b + - + i t h- 1 O ^^s; T}- m m m m ji 3-\O r*-ioO N -o oo oo ^too ON t** X a" i| QOQ M M f| i| o o o o o o o o - o M N O O o 1 " r^iMTj-r-Of^ 1 ^ r-.oo ON O >-> ** M N PI N mromfomm^-Tj-^-Tj-rf o> O M M m ^ iO"O t**.00 ON O - 1 N m ^~ 1 p 00 ON CX) N r^- OOQ ON ON N Tj- ir^ GO N *O ON O O t^oo O sSt *S ft P * N N 3 m -"t oo oo 00 00 . r^ -O ON m^O OO O O i~- mO ON W) CO " ON >-" m N t* C4 M HH m > *r~ ut ro so 00 s o N >O O iOOO NOONN ^- I**. O * 0*m * NNmmmrj-^-Tj-iovrj in\b ^b vb o + + i "- > ji o N oo oo m N- moo *** ON m o O m 1 - O ro m *j- -s HH ---i ri TJ- N N fO 1 * ints. Gh H . fe N ^-t^ - N CNM ^t t^ON "- N M ^vC HH S 1 1 + + 81 O " PO tfr'O t^. 00 O " J3 ON.NOOOO -" ONN O M ONi-iOO M 10NO OO IONO r-*. O O O 00 O - N rO Tl- M N lOvD 3 C OU t^.'i-Or^-^ONMO ONOO ON ON mu->u->O*Orf'^-'3-Tj- M * WJVO "* OO ON O M O M r^ O - P* pn Th iovO r^oo ON O "' J ON \o T|- PO Tf *t <* N t^. ON rj- ro ^2-83; Tj- 10 oo r^ 4 00 OO M N ioQO 11 rj- t- O fO "^ 5 N N - 00 t^-^O 10 N ONVO 10 N OO 10 - M xtuvo OO ON O M o ~ N ro o "* t-^ t^. i-^vb vbvb-bvb ioioioioTj-Tj-^- 1 ': hH vO 1^ - vo ON Tf b b ~ 10 M N fO 333 ~ N N Tf "1VO * t^. ON ON N 10 N fO fO t-^oo o ro (> ON OOO N t-* iO\O O 10 PO tO O *" t^. O PTO ONNNO ONP^i^-NNb nnib N mrOWtH iO 1 ^-'^t' cr JPOMN^H-' "* \o *O \O NO NO NO *o 10 10 >o 10 10 10 *o 10 X ui u^ ^ t * t f^ ro ro PO ro IONO r^.00 ON O "- N P^ ^ tovO tN.00 ON -p f>.cib ^ O - N CO ^J- tT) O t>. a 1^. 00 13 &c ^ t^Tj-- O t^Tj- N- o r-- *- - c^r^rOTf-^-iOTj-TfioiO't oo ONOO r^ BQI M in N VO "1 Tf 00 ^ rl- rOH- O i^>. O f) HH N N OS 00 tf'S | 1 | g 03 r^u-iw 00 "1 N >- 00 *n g U^O N PON ONNO POONiOQ^O N t-*.N H OO 10 N O r^ ro O vO M Os o 1 NNHH wiiO-^-rOWN"- 10^- S a t^ s P M N ro^- in\O f- OO O O M i-i O a O " M PO ^ lOxO t-*-OO ON O -" N P*1 ^~ i-j to ^ ^r P OOO^O OO ^C (^1 PO ON O oo in K. V N N O bob a < 00 00 OO OO P g 00 -, _-. POCO M r-^MNO ONp^t->.'-' o r^.00 ^ N -o t^OO ON " t-HB-NNPOPOP'J'd-'tiOiO'O iovb ^b W - \O i-^O HH in o TfOO ro ^~ 1 1 a b? b r-.ro O vo d ON 10 M " N ^ vO P*1 10 fOvO ^J-OO ON\O OvOO PO^O 00 t^- H " OV-N ^^o oo O -" PO 1O t^ 3 B " OOt>.*Or^Of*.POONiOO>oO 'tOO N NNNNH- iriiOTj-fN,PO W N a\^o r^ O ^O N OO ^J- O i^ H &> u b 6 w M . r^\o >O Tf N \O Q^ m O 00 t^. -t- -t H- vO 10 ^* f^ *H lOONP^t^-O -^-r^O rfONN r^- rJ-O\ ' CO ^-^-^- Tj-fOr^ fo f^ f^ rn ro * f-Nb ^bioio'tPOPON i- "H 6 b b b -w OQ p + +11 1 < C N jA J >^J O <- ^"O Ov rou-ioo HH r^ 10 O M ** OO O W r^ ON ^vO a * N-ooio*Hr*-NNbO P'lNO t^- ON ON ON ON rJ-^-iO i-.HHN. r^ ^- O G\ r+) t^. f^OO r*} >. f^ O O 'too GO ^- ON \D fO O\ tOOO PO ON vC N N vO a c . 10 P^ ONvO r^OO N OO N vO O ~rvi M - - N PO ^ lO'O t->. 00 ON S " NNNMNN-'^OOON ONOO oo 1 si CT*00 O\ O N O - rOTf ro-rj- | 10 ^- ro M ON "t r^-vO N 10 r^l vo ro vo "H fc 19 888 000 O O O O *O i-ifOTj-iO NPO^J-xO "-NPO S ,8 O O O O O o o o O O S w>\O l^OO ON O M N PO ^t lONO I^OO ON 01 3 N OO r^ O~> m ^O N OO m ON c a M z .2, z - c^oo M m v> "^ ^t N \n N N ON 10 . *vo Tt" ^O PO 10 p + ^-.-^HH-.H 1 ! > xi oo - N e ^- rhO ov T- ONVO ft?? -nw, N -t m t^* 1 N N N A 01 TfuvO 00 ON O M o - N fO O * N f^ ^- uivo t>.OO ON O -" ^* " Q ON O i 00 OO in o\ N O OO \O xo Tj- ? ^2 H-l '--c N *in ^^ t>.vO NO NO M"> ui T^ rj- f^ f*"l N N "^ O ON 00 ON O M " N ON O O in t^> r>. in ^> ON xn N PC tON 3 E (S ON ON O in ^-OO m>O i N ON N ON xn " o PC r- b ^.hob PCOO '.j- b *b ^t* c* b b S M O N PC *UVO t-'.CO ON O ~ in\O l^.OO ON O ** PC rf xn\O t-.OO ON 3s O N N N N ON ONOO oo 10 B oj w & b cb 'b Tj- CS O \O ^- N O !-- 5 M M M PC ~ r^-oo ON O to S 1 ??^^?b^bb^E- M ^O ?^-5 ?%S> rC i ON CO r^xn ^8 S H | ^O N N N N M W 000 o o> ?N"-.inPc8 N ^M'2i^) rc^2 Pn 03 N PC ^- UVO t- 00 ON O - B gj " O O ON ON ONOO CO 00 CO t^. r^. r-s.'O *O NO & 1 M ** O * W PC ^- mvO t-.OO ON O i N PC ^f >-3 60 N P4 Tf ir,inro co xn PC M a CO OO Ed "^ m ONPC^ HH If) ON O ON PC n M *- C^ N 00 00 CO <" ^-0 ON xn N o O NO PC ** NM TJ- TJ- in m\O xnTl-rc- ONt^-mO -^-ON w - H- l-l " PC V in^b i^.ob ONO*- *-wpcVt^-V g *. ONPcr^ M m ON , m a W g * 2 V 2 ON PCOO ON N t^ O N Perturb. 5 re b -b b ^"b obobob f^.in ONinb H o" vo m Tf N xn in^o t^oo ON o ^ N PC ^- xn^o r^.oo ON 02 CO to o *^\O ON N rj- 2 ^?Cr 1 oo oo CO CO 1 - to PC ON in i- 1 m o PCOO -^ i *s. TJ- n o N -H ONNO r-^ t^-* t-. + " +11 H * < t> a> OONN u-i ON M in O O O ^->O ON incb O O O NCO Perturb. O PC W r^"OO PC PC^O ^"CO ON^O O ^^ * vb PC ON PCO cb cb ^b rcob N inK.K.\b & - O N <*> t^ f*"> ON g.'&s in *- oo O PC in m r>. Oi OVD N ovS 1 d O *< N PC -^ in^o r>.oo ON o M PC ^~ fc * V * 00 00 i- 1 00 ^ CT> O n ~ N N N 0" o M - O i- PC-O^OW^OO-OON ONCO rf O t-x p- " b b b ON ONCO CO r^b rh PC N " O CO *p^ W "g o' ^- I*.OO OO ^'O M N PCPCN "- ONPCPC 15 H a fc o a 880 ?0 858 o"8 o * j^^jy^^s;^^ ^^^^^r ft 3 2 "" o o o o o o o b b o o f " ^- ^- in xnvO >O r- f^OO OO ON ON ON O O 1 g O Tt-00 ^> - ><^-a- 00 N a a o '5" 5 M t^i ul 00 O r^ 00 O rn to ro CO CO CO ^ oo "- m iO ON - 00 - * VO TC N IN N OOOO \O OPCmmMt^- O i?'* 3 ^3000 *^r. coro^ l*^ fO O "" ^< PC xn r^oo O N PC xn^b f**oo ON ON O - M M -.NNNMMNNNMPC S > 1 .; ^Q' TJ-NO coo Oco rj-xn-,^ TJ-ON TJ-VO PC 8 ta tot>1 O ^^O PC t-*. O m o 9 Tj-mnN^- N^J- re p-)inNxn X CM O i- . O i-i M re Tf xn>O r^.00 ON O i W PC ^> 1 00 00 OO OO OO 00 OO OO CO oo oo oo CO OO 00 oo oo oo 00 OX 00 00 I CO CO M - 'I aiavx ' gp 162 CONTINUATION OF TABLES I. AND III. OF DAMOISEAU'S TABLES OF JUPITER'S SATELLITES FOR THE PERIOD 18901900. [Appendix to the previous Paper*.] ON revising the above tables for 1880 1890, and continuing them for the period 1890 1900, it was found that some additional corrections should be applied to the terms which involve sin (5u 2u 34'542) in Damoiseau's formulae for Table III., and hence that the statement in the Introduction to the Tables 1880 1890 (see p. 118) as to the sufficient accuracy of these terms as they stand should be somewhat modified. It appears that Damoiseau's values of these terms are sensibly erroneous both in the Argument and in the Coefficients, and in these tables for 1890 1900, revised expressions have been used for the inequalities in Table III. for Satellites II., III. and IV. depending on the terms referred to. In the case of Satellite I., this inequality is insensible. The approxi- mate values of the adopted expressions appear to be 8. For Satellite II. + 0'84 sin (5u-2u - 16'6), Satellite III + 2'3 sin (5u-2u - 16 0> 6), Satellite IV + 12'6 sin (5H-2U,- 16'6), where w is the mean longitude of Jupiter and u that of Saturn. The above expressions give corrections to times of Conjunctions in seconds of time. The corresponding corrections to the longitudes of the Satellites in seconds of arc would have for their coefficients for Satellite II - &5 Satellite III - 4'8 Satellite IV -11'3 These agree closely with the expressions given by Souillart in his " Thdorie des Satellites de Jupiter." * [For this Appendix and the Tables, which were communicated to the Nautical Almanac Office in Jan. 1890, I am indebted to the kindness of Dr Downing, Superintendent of the Nautical Almanac. ED.] 19] APPENDIX. 125 Tt N fO N M ON ON t-*. "-HO 6= r^MN>-OOONON t^OO 1 H H ~ V^ N N M O fOvO 00 - Tf r^ O n o 1 1 "> fl^io r>.oo ON 2 -i JO *} O OO O 10 10 10*0 r O N r*} r-% ro r** fi\o O u^OO M r a "i * o ^- N -< O ONOO rs.'O vO o ^^-^^^-rofOrorocofO 1 H vbcb ~ M fj U-J - M 8 10 M f>iO vO oo ON N- O ro Tt . O n N fO ^- "^'O t^.oo ON O " ON Q N ro N ON N r- *-^ vn r^. O "-100 tOt^. N - u-)OO OO O O "^ 00 ON | H ^^ ioiriio \o o o t>. r*. ta iO*O *O *O\O *O lOiOiovrt^O^-^-^-rO CO _ Q ^ N r^ Tf u^\o f**. 00 ON q 1 | ON fO ON N 00 >- iO Tf t" fO jj vO >- 10 O "> O 1 . r^oO ''OOO f>OO ^ O 'O r*i TJ- Co \O r^-oo =go O N a * t>.\O ^- rn *- ONOO \O to ro N O ONOO sO N N B g iovO r^*oo ON O N fO ^J- o\O t>.oo ON *0 ^ o ^- o r^ oo 1 c ior>. HI N N * rot-^. O^ vO m O lot^. ON\O *1 ON m -i N O ON ON ON OO OO a N CO vO N O -O M ON 10 OOO <5 1 1 ttf -s J ^ e VO ONO "- ^ O ONl^iOOOv>^ 1 O ~ H B M ro Tj- w-i vO t>.OO ONO 9 - in V -^- ^1 N -" ONOO t>-vb rf r^ O O\ C a "" E S vO Tf N ooo ^o N O i^* ^N PH ** NNNNNNNNNNNNM PH i>. \s-> ^o ^^ * ON t-* 10 O <-" N fO Tt io\O t--OO ON O "- M ro ^* 3 bj> ^ -51 "2S M fO ^- io\O t>*00 | \O t**> ON ON 00 00 W 0! ONOO Cs ON ON ON oo oo r-. \O \D ONOO t-s.u*)tO r *'> O "- "* rOTj-iovr>t^ Q 4> _,, -^ -O ^ ^j- b w \S) H \O M N H-. I-^ N "-" t^ro N M 3 " bbbbbbbbbbbbbbb + +1 i --' H a; * r^co t^ 00 I-.OO OOO O n 2 | f^M Ot^-NtN-O NMt-tOI 1 ^ flOO fO 5 | 3 S <3- o ^O t^- N N N ONO t^ ONOO K. r^.f^ N N N N ON m 00 -OO OO OO OO OO f- t-N.vO NO NtMNMMMMNNNNMN - MMMNNMNMNNM w >? 00 \O -. O O * Tf ONOO "->\O fN-OO ON O N fO Tf iO\O 1^-00 ON , g ^ ^? o\:oo r-*-QO r>. vO r-*vO ioO id r^rOroN O CNON r>.\O \O ^J- N -> ** W 2 oo M O C> OO r>.vC >n^tfO M "H Q + + s > tr)\o r>. . HH o N t-*. O ON ro N M 1^0 xo ro r** a -jj fc ^ ^ 3 N N f) ro Tj- Tj- ^ VO 10 to ^ V H* x <^0 O "-" N ro ^t "1 O t^> . w S 5 MNNNNNNNNNMNMWM + + g ., o> e vu o a ^" OOOOOOON ^ts.0 Tj-00 r^ t^ rx M. a fc H H G 1| 888 888 888 .88 -c * N TfNOOO "- foxOOO O M io f^. O N 10 N N M M fOromr^^-^Tj-Tj-ioiOu^ ^S i 3 iovo rN.oo ON O " M rO ^- io\O t--OO ON B oo ^o m O I-*. < ^- N ONVO co O a a M fl fc o _s_ t r^cX3 ON <^ ^- Tf ON m ro vn Tf -fON N N ^t *>r> j OO ONN i^OO ^fOfO"1t^*- b * N M -*^b f~ b\ O i-a * fi _._--- 1 , O " N rO Tj- uivo r^OO ON O *-" N PO * 1 ON O\ ON 000000 ON O^ ON OO 00 OO ON ON ON OO OO OO 58 oo SS 1 I" & HI ap 126 CONTINUATION OF DAMOISEATJ'S TABLES OF JUPITER'S SATELLITES. [19 t> poo p Ovb \b ob ^ w M Tj- N cJ"8 3 1 1 mtm ^ I ~- CT\ O i- 4 r^- l^roo ON ON- *ON^ 10 N M PO fO O OOSOO 00 Tj- H ~ Jfrjj. O fn^O 00 - * r>. O PH v co ro ro ro fO f) ro ro ro m ro M 05 to <) 10 o o ^ N O *-" W f^ ^" io^O 1^00 ON O Oj Q N 10 M K^ft? ss>; ^g, 00 ON M OVOOO - ro^b ob io 00 S 04MCNicOMtNl>---OO ONOO r^ 00 N mm ^O OO ON z 3 1 1 N :?!! in in in " 00 ON Perturb. . ;^ T , TJ .^;^* CO a in m ! - into t% vn CT* 1^-. a i 00 oin TJ- in E-'S =28?! N CM oo oo 60 *** N- o i- '*" 2 t^* O t^ S3 t^ ON O N rj- ^t>o r>.00 00 ON O ** ** N B O r"* O ro t^ ro r^ O to VO M p I 1 OH ^ W M rf 1^00 M inoo M M ,0 (-^ Q i i HH ON in ** >noo O N fo fi N *^ H g - vO fl O ONNOro O ON B - g, ^^- ? 5- S^> 3 P>^ N ?! ^ PH c rooo "1 m- vn N ONVO flt^ oj t> 3 O Q\CO OO 0000 1^ t-^-vO NO VO to j Q I-.. M PO TJ- inO t^-CO ON O W PO 1 ^- hj a -4 M M N ~^ \o t^-oo ON O a ON ON 00 CO Q sB 00 CO N HH ON n m o oo VO " 83 ^0 ^-- ONt-rf-. HH P0>0 00 ON M Tj- in VO -I CO ^O rj- N ON ** 3 - ,- 1 .-,.- l OOOOOOOOO'-''-Hw + +1 1 H * ^ 2 M ro ^-invO r-oo 4 POONtOTj-mo TtinThH m r^oo vO M ^ d i-3 d W * 00 OOO ro ^- p S""^ p rOOO fO ON Tj- O M M C* 00 r^ ONOO N N 1 in n in >n in in H o* in^O t^oo ON O i- M m -rj- invo r^oo ON """' "~" ri- m CQ >* O * O N m r^- ^H t^ oo N m t-^-OO O t>. "H VO o 1 ? ^bt-dN^ljC^SS^S w 3^ , OOO OOO OOO o o r A - N N rororocorororOMrOf-Nrororo mvO I>OO ON O "- 1 N -i 00 NOO M HH en W ^J- to t^o! 3 PO inoo hH ^"NO ON N* ^~NO 00 O f^o t~* 1 a ft 3 o3 " **.. N N CO ** e 5 i N in O OO ON rn ONOO ON fl ON t**> I s * O NO * Z- -- ri ri PO in'b ob O m moo >-< inoo OQ OH N W N N N N O "-t N CO Tj- m^O t^.OO ON O - CS CO Tf 2 i 00 OO OO ON ^ C^ 00 00 OO ON ON CT> CO OO OO ON O OO ON ON ON *q j a^avJ J ui ep ep 19] APPENDIX. 127 ON -f ON ^J- ON Tf OO ro 1-^. f^t>. O OOOiOflO t^ ^ O ON t-H *%~ *vo .W ONONNOOOO ON N m *f v~i\O t^. tv.00 OO M M a " N r^ N f>. r^OO if M t-, rj- HH lOTj-^-ror^N N N ** M M M h-t NO r^-OO ON O M M N ro if N N N lO^O N N & - VOXO^OVOXOOVONO^VOVO ffl co-f 10 vO r^cc ON O n o A O w fO Tf io\O I^OO ON O 10 O *>. I-* r\o 10 N vO VO \O r^. r 10 ^t ro N O u^-^- a . iosO t^ r>. t>. 00 ON a 1 *" voO^ir^O^i^-b ^ob "i 10 ON fooo NB-I iOTj-^-f)f4NM m >s -4-3 13 v> o |-oO ON - o r^ Si-i N N Tj-,-. ro ^ 10 N N N r^. ^ IM fes t^ g 9 iO\O t>.oO ON O -" N r'j ^t io^O r>.00 ON !"> OO ON ON 00 00 ND OC I-'O U") $ f> N O ON 10 rf vO O COvO CO N Tj-Ot->.ONO " fDTj-iri 3 00 OOO >-" *f N f^. O f*l ^s^s ON N - N o " fo^frJ-rJ-Tj-ioiOiOiO lO^O O O ^O \O 1 1 S -S ^ NH \oioN,vOCO\O N r-O -" O t--rO H 2 \O C4 ON 1^. * t- - 5 O w M ro Tt irivo t>.oo ON O w N f*Tf 1-5 60 O GO ^O t O r^. ^- M Tf NH ^2 ^ H VO OQQ \f) (^ N N N HH OM- N -. HH * N O r*". 10 -O ^ 1 *>.roO r* -.OO CO 0;0 O - N g {> PH H o> *o\O ^.OO ON O ** N en - ^000 O N iO t-N. r^vO jd OOOO'OO NH OO*HO IONO M fO t^. t-* PH M H CO 0) Ov r*JOQ N ON "1 \O Q^ f^ N r*. ro N 5 a B . t>*OO r>, r LO S p * f^ob oboo ONbNbNONbNbNbNONONONON - a a HH N 7 * ^$ N - - * oo NO *n o <^i GO v> M iO -3 00 00 " r^. t^.vo \O 10 10 ^^ ro N r-* N r- moo ro o rj-oo r>* moo N 10 5m fc o J> O !-5'^ ^ W Tj-N ro IH ro O t>. ON S * b -'NNmifififio iO\b vb t** t^. + + s I t> A f -" CO O to -i ^- rf OO \O OO 1000 \O CO s ~ i- 1 * obd^ONifK.b if ob mob if b r-*- if ^^fiOiOU-iiO -(-iMCOfO^- 00 g ^ rf Tj- Th *^" ^" iO >-O iO IO LO 1O 1O 1O *O -w O " M O *- N m 'f lO'O r^-OO ON O "-< N rO ^> 2 ^ O" O^ OO CX5 OO ^ ON O^ OO CX) CO GO GO OO o>5 00 Ov 1 O "- ON ON 'I aisvj, B| ap 'HI aiavx I ap 128 CONTINUATION OF DAMOISEAU'S TABLES OF JUPITER'S SATELLITES. [19 fO <-" O CO I-.U1 Tj- N ON t>. \n id COI^HH\O M M-O ONNVO > inob HH ri - fO^O ON ^'c? 00 H- P 7 ~ " " " i 00 N com O OO ON O " co-^ JD ^- HH ^- fO rs.'O ^"OO ON r* M B rf^O ON - Tf\O "ITt-vO 00 O 1 NHH ^.fTJNHHHH IT-^- M N N N b fo^b * - 2" H- N * fO"t u- O >-" o HH .OO ON O a S 8 Tt-oo xo ro O ** r^i r^ i-r O Hj M gv H M Tl-vb r^ oo b HH N N ~ N it N N N ir\O N M ;d u-j u-> m u-j-O Tj-N HH ONvn^-Hr>. ^-OO 5 u^ioinu^^invoinTj-^-^-ijrcocoN w co Tj-m \O ts.00 OsO 1-1 ~ 1 1 "* u-lO "">O >^O\"^fON COOO ^ COO vO CO ONVO \O ror*. ON'O o ooo vn i-t vo ^S8 T * xob^-ONfof>.Nr^MK.McbTfbt^ >0 ro HH ^- to -i ^J- co HH into H N co ^t t ^O u-i\o ^O ^ t^ .\O \C M <*>,-. HH N ro-rh i^i^o r^ 00 CT> d w^\O t^OO ON O " N fO T}- xo^O ^.00 ON S mu->\O \O I-. 1 s * * Tf CO fO 3 < 0^ % 00 CO 1 t- i/1 CO n ON t^. * o 00 VO HH HH BO r^ N M "-"- VO *i-(S O^O COJ>.HH iriONN uiONO CO a M w N CO *u^O t^oo " OobNOHHw^iNfOrOfOTj-Tj-Vj-imo * * , ^^^,-^^^^HHH. HHHH P5 8 r^ t^, Tf- NOO N i^*vo m *t-5 r>. HH r^. "1 w> ONO ONOO ro Tf N t^GO CO MM f O O Ox ON ONOO CO O H 00 S | vO OONOOCO = o CO CO CO CD Tf \f) r^ t^ t^ ^O 1^00 t^^o o\ o I HH inTfCONHBU-l^J-COHH lOrOCN " cocONNMNfc-H-h-^OOO NNNNfSMNNNNMNN 1 O HH N ro Tj- wi'O r*-OO ON O HH c) ro 1 < 4 - '"s So N HH M fO CO CO - 0>VO ^- a \o *^ ON ON W . g -g + 4-1 I id O ONOO Tj-ONM TTU-JCOHH r>.fOO ^^ V O 3 8 1-3 g w *i- 50 t^O ci c-i in O ro m o to M N 00 11 B " ONinHHsbONNrorONbOsMKbco N fO^-Tj-^u-iinimr^u^Tf^-cOrON * COcococOfOcOfOcocorOcorOcocoro ^ I 1 U^QO ON OO CO I" mrO -. * a io\O t^-oo ON O HH N ro ^ "^O i^CO ON 5 a 02 fO o t- o fO u-)CO 1-1 CO HH ro vc ON a 4 ON ON CO OO HH 2 ^ .2 10 Tf CO *K Tj-oo HH rf r-.DO ON - O ONOO r^. PO O O w t-. Tf l-t OO m N OO "^ ar~ P NN ^H- H-^HH + H o O\ c* "^ ix. O fO o ro^o H ui ff^ i -j-i ,,-j ro rj rri rri ri n D (~i OH & EH D M 000 M m N M O OO m o o o moo ooo M 10 O O 2 1 1 ** t^ob ON OND t-*J">NCQ roJ-.b HH HM fOiON-fOu-i'-iCOiri NfOm HHN " ONONOOOHHHHH-WNNNrOCOtO N-MNNMMMMNNNNNN 0" I O N ro ^f m*O *^OO ON O HH ro < ^- t - vOCC T!- r>. ro HH ro iy-i O VO N CO? CO Tj- H- t>* *n l^ ro ON ro Tj- ON ON v 1-1 o N r^ tt r}- u- T}- U->10 vO^O TJ-M _ t^. HH t^o N Tj-tot^r^r^mO O x CObNbb ^MNMriMNNWN HHHHMMNNNNWMNMMNM >H + + 1 11 5 H fl 25 u _c8 N r^ t-s. 885 bob Th TJ- Tf N ro ^- O O O bob CO m m p p p ooo * t CO 1 ^- $ b o | $ N \O OJO ON-O t>. 7J- ^"O p y N ON*O ^ ^ N HH HH N fou^t-^ONCS Ti-f>ONNij-j M roin com- roiON 3- IT>HH " ^f -^t- rt- \r> \f) mvC *O O t>- 1>,00 CO OO ON o f^-S 1 PH a . II 0! ro r-*. -i u^o < ^- ON COt>. 1 o a 1 HH N JZ- o - r t *- rro fO OO n CO COHH ^- CO CO VD 00 ? 00 00 H g- v -*vO ON - Th^o rd-\O ON S,S CO ON N ro-O ON M Tf N rf rj- f-OO ON t>* g oo " HH w rt- u-j\O I-- ON O N cO TJ- 100 t^ O > -3 82 SO to HH N + MW "" MW + X ij > jf COO xnt^-O'O u^ t^ N* rOtN.HHOOvO irt 8 O cot^. ^ ON fO t>. POVO ON rot^. (H i * fOu^ob N OO Tf M "- 1 N COU->ON co ON\O Tj-^-^-mio -Mco-^-m NCOn 03 CM 4 O "-> N ^O *f *O O HH M PO ^- m\O t*-CO ON O N fO ^~ ^ ON ON ON 00 CO OO ON ON ON CO OO CO ON O^ ON 00 CO OO OsS OO ON a ON O* ^2 *S - I ap "HI aiavx [ ap 20. ON PROFESSOR CHALLIS'S NEW THEOREMS RELATING TO THE MOON'S ORBIT. [From the Philosophical Magazine, Vol. VIII. (1854).] IN the June Number of your valuable Journal, Professor Challis calls attention to some circumstances connected with his withdrawal of a paper, relating to the Moon's motion, which he had communicated to the Cam- bridge Philosophical Society, and of the principal results of which he had given an account in your Number for April (p. 278). Professor Challis mentions that one of the reporters, whose unfavourable judgement led to this withdrawal, had of his own accord communicated to him some of the reasons on which this judgement was based. Professor Challis, however, thinks these reasons to be very unsatisfactory, and con- sequently invites the reporter to discuss with him the questions on which they are at issue, in the pages of the Philosophical Magazine. As I am the reporter thus referred to, I beg that you will allow me to state some reasons which appear to me sufficient to prove, beyond a doubt, that the principal conclusions of Professor Challis's paper are erroneous, in order that he may have the opportunity, which he desires, of replying publicly to my objections*. At the same time, I must decline to enter * It may be proper to mention that the opinion of the other reporter on the paper perfectly agreed with my own. A. 17 130 ON PROFESSOR CHALLIS'S NEW THEOREMS [20 into any prolonged controversy on the subject, submitting with confidence what I have now to say to those who are competent to form a judgement respecting it. The principal results of Professor Challis's paper are embodied in two theorems, which, as already stated, form the subject of an article in the Philosophical Magazine for April last. As my main objections to the paper relate to these theorems, I shall confine my observations almost entirely to' the article in question. It will be convenient, however, to make a few preliminary remarks on the nature of the process usually followed in the lunar theory. Professor Challis objects to the logic of this process, on the ground that the intro- duction of the quantities usually denoted by c and g into the first ap- proximation to the Moon's motion is only suggested by observation. He therefore considers the results of the ordinary process to be hypothetical, until they are confirmed by observation. But surely the sufficient and the only test of the correctness of any solution is, that it should satisfy the differential equations of motion at the same time that it contains the proper number of arbitrary constants to fulfil any given initial conditions. Any process which does this, no matter how it may be suggested to us, must be logical ; and if the results obtained by it should not agree with observation, the conclusion would be that the law of gravitation, which was assumed in forming the original differential equations, is not really the law of nature. If we begin with the supposition that the Moon's orbit is an im- moveable ellipse, the differential equations cannot be satisfied, without adding, to the first approximate expressions for the Moon's coordinates, quantities which are capable of indefinite increase ; and this proves, as is stated by Professor Challis, that an immoveable ellipse is not, or rather does not continue to be, an approximation to the real orbit. But if we introduce the quantities usually denoted by c and g, having assigned values slightly differing from unity, which amounts to supposing the apse and node to have certain mean motions, we find that the differ- ential equations are satisfied by adding to the first approximate expressions for the Moon's coordinates, terms, which always remain small ; and we thus 20] RELATING TO THE MOON'S ORBIT. 131 know that our first approximation was a good one, and that the true and the only true solution of the differential equations has been obtained. On the other hand, no solution can be a true one, which does not contain the proper number of arbitrary constants ; and any person who asserts that one of the constants usually considered arbitrary is not so, is bound to show by what other really arbitrary constant the former is replaced. I will now proceed to consider Professor Challis's two theorems, which are thus enunciated by him. Theorem I. All small quantities of the second order being taken into- account, the relation between the radius-vector and the time in the Moon's orbit is the same as that in an orbit described by a body acted upon by a force tending to a fixed centre. Theorem II. The eccentricity of the Moon's orbit is a function of the ratio of her periodic time to the Earth's periodic time, and the first ap- proximation to its value is that ratio divided by the square root of 2. I will endeavour, in the first place, to show that these theorems cannot possibly be true ; and secondly, to point out the fallacies in the argument by which Professor Challis attempts to establish them. The problem will be simplified by supposing the Moon to move in the plane of the ecliptic, and the Earth's orbit to be a circle. On these sup- positions, Professor Challis's fundamental equations become u,x m'x 3m'r . ,, Sm'r Multiply these equations by y and x respectively, and subtract the results \ and again multiply by x and y, and add the results together ; thus we obtain, after expressing x and y by means of polar coordinates, 172 132 ON PROFESSOR CHALLIS'S NEW THEOREMS [20 Now these equations, which are equivalent to the former, are satisfied to terms of the second order inclusive by putting r = a-!l + ;re 2 ecos (cnt + e CT) - e 1 cos 2 (cnt + e rar) (_ 6 2 / - m 3 cos (2 nt + c - Zrit + e 1 ) _ _ me cos(2nt + e Zn't + e' cnt + e 5 = nt + e + 2e sin (cnt + e - w) + - e 2 sin 2 (cw + e - OT) o m 2 sin ig + me sin (1nt-\- e 2n't + ^ cnt + e nr), 4 u .. fn n f 3 where n=-7, n = r t , m= , c=l -m, a 3 a" TO 4 and a, e, e, and w are the four arbitrary constants required by the complete solution. The fact that the differential equations are satisfied by these expressions for r and 6, whatever be the value of e, is quite sufficient to shew that Professor Challis is mistaken in restricting e to one particular value. The terms of the second order in the value of r, which depend on the arguments and and which constitute the well-known inequalities called the "variation" and " evection," prove the incorrectness of Professor Challis's Theorem I. ; since in an orbit described by a body acted on by a force tending to a fixed centre, and varying, as Professor Challis supposes, as some function of the distance, the expression for the radius-vector in terms of the time cannot possibly contain any terms dependent on the Sun's longitude. I now come to consider the reasoning by which Professor Challis arrives at his theorems. All this reasoning is based on his equation fdr^ (dt. 20] RELATING TO THE MOON'S ORBIT. 133 the truth of which, he says, cannot be contested. In speaking of the truth of this equation, Professor Challis cannot mean that it is anything more than an approximation to the truth, since in forming it he avowedly neglects all quantities of orders superior to the second. Now what I assert is, first, that the degree of approocimation attained by the equation (C) is not sufficient to justify Professor Challis in inferring Theorem I. from it; and secondly, that Theorem II. does not follow from that equation at all. To prove the first of these assertions, I remark that the equation (C) gives an approximate value of (-77) m terms of r, but that it does not pro- dr fess to include terms of the third order. Now -7- is itself a quantity of /drV the first order, and consequently an error of the third order in (-j-\ leads \at I dr to one of the second order in -j~, and therefore to one of the same order in the value of r expressed in terms of t. Hence Professor Challis is not entitled to infer that the relation between the radius-vector and the time in the Moon's orbit is the same, to quantities of the second order, as that which would be given by the equation (C). We may test the degree of accuracy to be attained by the use of this equation in the following manner. By differentiation, the constant C disappears, and the resulting equation dr ~dt (J/V* becomes divisible by -T~; dividing out, we obtain d\ h* p m ' r _o This is a strict deduction from Professor Challis's equation; we will now obtain directly from the equations of motion given above, an expression to be compared with it. Integrating equation (l), and putting, with Professor Challis, nt + t for 6, and a for r in the term of the second order, we find r* -h - dt ~ 4 a' 3 n 134 ON PROFESSOR CHALLIS'S NEW THEOREMS [20 The value of the constant h, expressed in terms of the system of constants before used, is Hence and 'de\* A 2 3 mf a cos (Xnt + e. vn't + <), dt = + a C S putting, as before, a for r in the small term. Substituting this value of PITT) in equation (2), we find d*r A 2 u, m'r m' The equation above deduced from Professor Challis's differs from this by the omission of the last term, which gives rise to the variation inequality. In order to find the evection, which is also an inequality of the second order, it would be necessary to carry the approximation one step still further than we have here done. This shews how unfitted equation (C) is for giving any accurate infor- mation respecting the Moon's orbit. As a matter of fact, it may be observed that this equation would make the Moon's apsidal distances to be constant. A simple inspection of the calculated values of the Moon's horizontal parallax, given in the Nautical Almanac, is sufficient to shew how far this is from the truth. I now proceed to make good my second assertion, viz. that Professor Challis's Theorem II. cannot be inferred from his equation (C). The process by which he attempts so to infer it is of the following nature. He first finds that a method, apparently legitimate, of treating the equation (C) leads to a difficulty. To get rid of this difficulty, he makes the strange suppo- sition that the equation (C) contains the disturbing force as a factor, and then tries to shew that, in order that this condition may be satisfied, the arbitrary constants h and C must have a certain relation to each other, from which it would immediately follow that the eccentricity must have the value assigned to it in Theorem II. 20] RELATING TO THE MOON'S ORBIT. 135 Now it is remarkable that every one of the steps of this process is unwarranted. The difficulty to which Professor Challis is led is purely imaginary ; the supposition that the equation (C) contains the disturbing force as a factor is wholly unsupported by any proof; and even if that supposition were well founded, it would not follow that the constants h and C must have the relation assigned to them by Professor Challis. The supposed difficulty is founded on the inference at the bottom of p. 280 of Professor Challis's paper, " Hence we must conclude that the mean distance and mean periodic time in this approximation to the Moon's orbit are the same as those in an elliptic orbit described by the action of the central force ." But this is not a correct conclusion: if h and C be r supposed to have the same values in equation (C) and in that obtained from it by putting a for r in the small term, the values of the mean distances in the two cases would not be the same, but would differ by a quantity of the second order. This may be readily shewn in the following manner. ci 'Y* At the apsides -j- = 0, and therefore th ing equation for finding the apsidal distances, ci 'Y* At the apsides -j- = 0, and therefore the equation (C) gives the follow- Now if a be the mean distance, and e the eccentricity, the apsidal distances are a (1 + e) and a (1 e). Substituting these values for r in the above equation, and developing the small term to quantities of the fourth order, we obtain h* - 2pa (1 + e) + Co? (1 + 2e + e 2 ) - ^ a 4 (1 + 4e + 6e 2 ) = 0, and fl whence it follows that / h*- 2pa + Ca?(l + e 2 ) - j^- t a 4 (1 + 6e 2 ) = and 136 ON PROFESSOR CHALLIS'S NEW THEOREMS [20 These equations give the relations between the arbitrary constants h and C, and the new constants a and e by which the former may be re- placed. From the second of them, we find _ p. m' a* ~ + ; or, putting for a in the small term its first approximate value ^, p m' p, 3 = C + rf s C" which agrees with Professor Challis's expression in p. 281. Now apply a similar process to the equation _ dt r*~ r '~ 2a' 3 ~ > which differs from the equation (C) in having a put for r in the small term. In this case, we find Co? (1 +e 2 ) -- ^ 4 (1 +e 2 ) = 0, /a and / from the latter of which equations it follows that _ p m' a? _ ~ C 2a' 3 C< ' to the same degree of approximation as before. Hence we see that the values of a, in the two cases supposed, differ by a quantity of the second order. Consequently the difficulty into which Professor Challis is led by the conclusion that these values are the same, disappears, and the solution of the difficulty with it. 20] RELATING TO THE MOON'S ORBIT. 137 But even if we were to suppose, with Professor Challis, that the equation (C) contains the disturbing force as a factor (of which, as already remarked, no proof whatever is given), it would not follow, as is inferred by him, that A 2 C must be equal to fj.". On the contrary, it is evident that the required condition would be satisfied if h?C differed from p." by any quantity involving the disturbing force as a factor ; whence it would follow that e must be some function, indeed, of the disturbing force, but it could not be decided what function. Professor Challis attempts to find the relation between r and t by direct integration of the equation dr dt = /_r_^4-^ V ^ r 2a /3 fdr\ (drV -j-} is a small quantity of the second order at I which vanishes twice in each revolution, and that the difference between the fdrV complete value of I -T- ) and the approximate value n ^ ~T ' --- r 2 ~T ' --- r 7; 7J~ r 2 r 2a' 3 which is used instead of it in the above equation, is a periodic quantity of the third order. Hence it follows that the quantity n h a 2/i mV ^ "~ ~a "1 --- ' 7i rt~ r 2 r 2a /drV may vanish for values of r different from those which make i-^-j vanish, \ctt/ fdr\* and that it may even become negative for actual values of r, which l-j-} \at I itself can never do. Therefore the coefficient of dr in the above differential equation may become infinite, or even imaginary, within the limits of integration, so that it is not surprising that Professor Challis should have met with such difficulties in performing the integration. The relations between r, 6, and t, given in page 281 (which profess to include all small quantities of the second order), are said to be derived from the equations (B) and (C). It is easy to see, however, that they do not A. 18 138 ON PROFESSOR CHALLIS'S NEW THEOREMS [20 satisfy the first of those equations, since the term of the second order cos 2 0^1 in the right-hand member of that equation involves the longitude of the Sun, which does not occur at all in the relations in question. The contradiction to Professor Challis's theory, which is presented by the eccentricity of the orbit of Titan, is supposed by him to be occasioned by the large inclination of that orbit to the plane of the orbit of Saturn. But in page 280 it is remarked that the inclination of the orbit is taken into account ; and even if this were not the case, no proof is offered that the taking it into account would tend to reconcile the discrepancy. At the bottom of page 282, Professor Challis attempts to shew, a priori, that the eccentricity of the Moon's orbit must be a function of the disturbing force in the following manner. If there were no disturbing force, the value of the radius-vector drawn from the Earth's centre in a given direction, would be constantly the same in different revolutions. But if a disturbing force act in such a manner as to cause the apsidal line to make complete revolutions, the value of the above-mentioned radius-vector would fluctuate in different revolutions, between the two apsidal distances. Hence it is argued that, since if there were no disturbing force there would be no such fluctuation of distance, therefore the total amount of such fluctuation, and consequently the eccentricity, must be a function of the disturbing force. But, on consideration, it will appear that this argument is fallacious. No doubt it may be inferred that some of the circumstances of this fluc- tuation of distance will depend on the disturbing force which causes it, but it cannot be asserted, without investigation, that the total amount of such fluctuation must necessarily depend on the disturbing force. As a simple example, we will suppose the principal force to vary in- versely as the square of the distance, and a central disturbing force to be introduced which varies inversely as the cube of that distance. In this case we know, by Newton's 9th section, that the motion would be accurately represented by supposing it to take place in a revolving ellipse, the angular velocity of the orbit being always proportional to that of the body at the same instant ; and the eccentricity of the orbit might be any whatever, and would not at all depend on the disturbing force. 20] RELATING TO THE MOON'S ORBIT. 139 Now, since the orbit would be fixed, were it not for the disturbing force, it might be argued in exactly the same manner as is done by Pro- fessor Challis in the passage above referred to, that the eccentricity of the orbit must be a function of the force which causes the orbit to revolve, but this we know to be a false conclusion. What would depend on the disturbing force in this case, would be, not the total amount of the fluctuation of distance in different revolutions, but the number of revolutions of the body in which such fluctuation would take place, or the time of revolution of the apse. If the disturbing force were increased, the total fluctuation in the value of the radius-vector in question would be the same as before, but the change from one of the extreme values to the other would occupy a shorter time. The objection mentioned by Professor Challis at the top of page 283, is alone quite fatal to the supposition that the eccentricity of the Moon's orbit must have a particular value. Where is the proof that the eccen- tricity would settle down to such a value, as Professor Challis imagines, if it were initially different ? In fact, it is easy to shew, by the method of variation of elements, that there would be no such settlement, but that the non-periodic part of the eccentricity would remain constant. 182 21. ON THE SECULAR VARIATION OF THE MOON'S MEAN MOTION. [From the Philosophical Transactions of the Royal Society, Vol. CXLIII. (1853). Abstract of same, Proceedings of the Royal Society, June 16, 1853 and Monthly Notices of the Royal Astronomical Society, Vol. XIV. (1853).] 1. IN treating a great problem of approximation, such as that pre- sented to us by the investigation of the Moon's motion, experience shows that nothing is more easy than to neglect, as insignificant, considerations which ultimately prove to be of the greatest importance. One instance of this occurs with reference to the secular acceleration of the Moon's mean motion. Although this acceleration, and the diminution of the eccentricity of the Earth's orbit, on which it depends, had been made known by obser- vation as separate facts, yet many of the first geometers altogether failed to trace any connexion between them, and it was only after making repeated attempts to explain the phenomenon by other means, that Laplace himself succeeded in referring it to its true cause. 2. The accurate determination of the amount of the acceleration is a matter of very great importance. The effect of an error in any of the periodic inequalities upon the Moon's place, is always confined within certain limits, and takes place alternately in opposite directions within very moderate intervals of time, whereas the effect of an error in the acceleration goes on increasing for an almost indefinite period, so that the calculation of the Moon's place for a very distant epoch, such as that of the eclipse of Thales, may be seriously vitiated by it. 21] ON THE SECULAR VARIATION OP THE MOON'S MEAN MOTION. 141 In the Mecanique Celeste, the approximation to the value of the ac- celeration is confined to the principal term, but in the theories of Damoiseau and Plana the developments are carried to an immense extent, particularly in the latter, where the multiplier of the change in the square of the eccentricity of the Earth's orbit, which occurs in the expression of the secular acceleration, is developed to terms of the seventh order. As these theories agree in principle, and only differ slightly in the numerical value which they assign to the acceleration, and as they passed under the examination of Laplace, with especial reference to this subject, it might be supposed that at most only some small numerical corrections would be required in order to obtain a very exact determination of the amount of this acceleration. It has therefore not been without some surprise, that I have lately found that Laplace's explanation of the phenomenon in question is essentially incomplete, and that the numerical results of Damoiseau's and Plana's theories, with reference to it, consequently require to be very sensibly altered. 3. Laplace's explanation may be briefly stated as follows. He shews that the mean central disturbing force of the Sun, by which the Moon's gravity towards the Earth is diminished, depends not only on the Sun's mean distance, but also on the eccentricity of the Earth's orbit. Now this eccentricity is at present, and for many ages has been, diminishing, while the mean distance remains unaltered. In consequence of this the mean disturbing force is also diminishing, and therefore the Moon's gravity towards the Earth at a given distance is, on the whole, increasing. Also, the area described in a given time by the Moon about the Earth is not affected by this alteration of the central force ; whence it readily follows that the Moon's mean distance from the Earth will be diminished in the same ratio as the force at a given distance is increased, and that the mean angular motion will be increased in double the same ratio. 4. This is the main principle of Laplace's analytical method, in which he is followed by Damoiseau and Plana ; but it will be observed, that this reasoning supposes that the area described by the Moon in a given time is not permanently altered, or in other words, that the tangential disturbing force produces no permanent effect. On examination, however, it will be found that this is not strictly true, and I will endeavour briefly to point out the manner in which the inequalities of the Moon's motion are modified by a gradual change of the central disturbing force, so as to give rise to such an alteration of the areal velocity. 142 ON THE SECULAR VARIATION OF THE MOON'S MEAN MOTION. [21 As an example, I will take the Variation, the most direct effect of the disturbing force. In the ordinary theory, the orbit of the Moon as affected by this inequality only, would be symmetrical with respect to the line of conjunction with the Sun, and the areal velocity generated while the Moon was moving from quadrature to syzygy, would be exactly destroyed while it was moving from syzygy to quadrature, so that no permanent alteration of area! velocity would be produced. In reality, however, the magnitude of the disturbing force by which this inequality is caused, depends in some degree on the eccentricity of the Earth's orbit, and as this is continually diminishing, the central dis- turbing forces at equal angular distances on opposite sides of conjunction will not be exactly equal. Hence the orbit will no longer be symmetrically situated with respect to the line of conjunction. Now the change of areal velocity produced by the tangential force at any point, depends partly on the value of the radius vector at that point, and consequently the effects of the tangential force before and after conjunction will no longer exactly balance each other. The other inequalities of the Moon's motion will be similarly modified, especially those which depend, more directly, on the eccentricity of the Earth's orbit, so that each of them gives rise to an uncompensated change of the areal velocity. Since the distortion in the form of the orbit just pointed out is due to the alteration of the disturbing force consequent upon a change in the eccentricity of the Earth's orbit, and it is by virtue of this distortion that the tangential force produces a permanent change in the rate of description of areas, it follows that this alteration of the areal velocity will be of the order of the square of the disturbing force multiplied by the rate of change of the Earth's eccentricity. It is evident that the amount of the acceleration of the Moon's mean motion will be directly affected by this alteration of areal velocity. 5. Having thus briefly indicated the way in which the effect now treated of originates, I will proceed with the analytical investigation of its amount. 21] ON THE SECULAR VARIATION OF THE MOON'S MEAN MOTION. 143 In the present communication, however, I shall confine my attention to the principal term of the change thus produced in the acceleration of the Moon's motion, deferring to another, though I hope not a distant, opportunity, the fuller development of this subject, as well as the consideration of the secular variations of the other elements of the Moon's orbit arising from the same cause. In what follows, the notation, except when otherwise explained, is the same as that of Damoiseau's Theorie de la Lune. 6. If we suppose the Moon to move in the plane of the ecliptic, and also neglect the terms depending on the Sun's parallax, the differential equations of the Moon's motion become d*u 1 m'u' 3 3 m'u" . 3mw du . /n , A 3m / c 3/ w\ fu 3 dv . , .. -S-TT- -ffln(2i' 2*0 IT- (W+TTJ^ - sm(2v-2i' / ) 2 AV w ft 2 \ dv V J tt 4 eft 1.3m' [ufdv , . 27 m' 2 T fu"ck . L cZi/ /i-w 2 In the solution usually given of these equations, u is expressed by means of a constant part and a series involving cosines of angles composed of multiples of 2v 2mv, cv CT, and c'mv us' ; also t is expressed by means of a part proportional to v and a series involving sines of the same angles; the coefficients of the periodic terms being functions of m, e and e'. Now if e' be a constant quantity, this is the true form of the solution, but if e' be variable, it is impossible to satisfy the differential equations without adding to the expression for u a series of small supplementary terms de- pending on the sines of the angles whose cosines are already involved in it, and to that for t, similar terms depending on the cosines of the same angles, the coefficients of these new terms involving -j as a factor. The quantity I sin (2v 2z/), which occurs in the above equations, J u is proportional to the variable . part of the square of the areal velocity, and consists, in the ordinary theory, of a series of periodic terms involving cosines of the angles above mentioned. In consequence, however, of the existence of the new terms just described, there will be added to it a 144 ON THE SECULAR VARIATION OF THE MOON'S MEAN MOTION. [21 series of small terms involving sines of the same angles, together with a non-periodic part of the form IHe'de' or ^He'~. The introduction of this term will evidently change the relation between the non-periodic part of -j- and e' 2 , upon which the secular acceleration depends. 7. We must commence by finding the new terms to be added to the ordinary expression for u. For the sake of simplification we will neglect the eccentricity of the Moon's orbit. Let ' denote the non-periodic part of u, and [-Su the complete value. a a Then by substitution in the equation for u, making use of Damoiseau's developments of the undisturbed values of the several functions of u, u', and v v' which occur in it, putting h* = a f , and writing, for convenience, mv instead of I mdv + A., and c'mv instead of c' I mdv + X ra-' (as in Plana, vol. I. p. 322), we obtain d'H ' (a 1 1 = TT- + --- + dv a a. s + S 1 m 2 / 3 ,A 3m 2 .,,,, 3m 2 , , 3 m 2 r , . , - + - (l + o e ) + o erStr + - e cos c'mv - {1 + 3e' cos c'mv} aou d(-\ 3 - a W gin ^ 2y _ 2mv ^ + 1 ^ / L _ 5 , 2 \ cos ^ 2i/ _ 2mj/ ) 2 tt \ 2 / 2 a dv 21 tri* 3 m* + - e' cos (2v 2mv c'mv) - e' cos (2v 2mv + c'mv) 4 a ' 4 a '- I dv \ ( 1 - e'-} sin (2v 2mv) +&' sin (2v 2mv c'mv) a , J L\ ^ / * - e' sin (2v 2mv + c'mv) j- 21] ON THE SECULAR VARIATION OP THE MOON'S MEAN MOTION. 145 - \ ( 1 - e'" } cos (2v - 2mv) +^ e ' c s (2v 2mv c'mv) - e' cos (2v 2mv + c'mv) [ - { ( 1 - 5 e' 2 ) sin (2v - 2mv) + - e' sin (2v - 2mv - c'mv) 2 a, i\ 2 / z -e' sin (2v - 2mv + c'mv) 12 I dv \(l - 1 e' 3 ) sin (2v - 2mv) + I e' sin (2v - 2mv - c'mv) a , j \\ ^ / - e' sin (2v 2mv + c'mv) ^ 3m 2 ( . 8 J Also, integrating by parts, and putting 2 instead of 2 2m, 2 3m, and 2 m in the divisors introduced by integration, since we only want to de' find the terms of the lowest order which are multiplied by -, - , we obtain J dv -I ( 1 - e' 2 ) sin (2v 2mv) + -e' sin (2v 2mv c'mv) , J IV 2 / - e' sin (2v 2mv + c'mv) [ = - ( 1 - e' 2 ) cos (2v 2mv) + e'cos (2v 2mv c'mv) 2 a, \ 2 / 4 a, - - e'cos (2v 2mv + c'mv) 4 a t , 15m 2 (, e'de'ndt /0 . 21m 2 f, de' ndt + -JT- \dv -,- -j- cos (2v 2mv) \dv r - r - cos (2v 2mv c'mv) 2 a f j ndt dv ' 4 a,J ndt dv 3 m 2 f 7 de' ndt ,^ + - \dv 5- -j cos (2v 2mv + c'mv). 4 a, J ndt dv And a'Su' = 3mV sin c'mv [ &' sin c'mv} retaining only the term which will be required. 10. When the proper substitutions are made, the terms involving cosines destroy each other, as in the usual theory, and by equating to zero the terms involving the sines, we obtain 21] ON THE SECULAR VARIATION OF THE MOON'S MEAN MOTION. 147 95 95 or 3a 30 = m 2 .-. 30 = m 2 = .'. a 16 = 3m 3 38 -, = , o 133 133 or 3a a3 = -- m" .-. 0^=- Q -m 2 = 0, o 19 19 or 3^ = m- .'. a M =~ wi 2 . 11. In order to obtain the relation between a and a t , we must sub- stitute the value just found for a8u, in the same equation, and equate to zero the non-periodic part, observing that the terms 12 -- I dv \ ( 1 - - e' 2 ) sin (2v - 2mv) +-e' sin (2v 2mv - c'mv) a, } [\ 2 ] 2 - e' sin (2v 2mv + c'mv) Y a8u give 12m 2 f 7 f95 ,e'de' 931 a e'de' 19 . - m 2 2m 2 f , f! oTJ " [24"" ndt 96 "' ndt ~96"' ndt f 285 m 4 ( , e'de' = I ndt y- nearly, 4 a, J ndt 285m 4 ,, . . ,. = e ' as then- non-periodic part. 8 a. Also the terms 15m 2 f, e'defndt ln , 21m 2 f, de' ndt /n , x dv j- ^ cos (2v 2mi') - \dv y- -y cos (2i - 2mi/ C'TOV) 20,,} ndt dv ' 4 a, J ndt dv 3 m 2 /" , cfe' cZ< , , v H rtv r ^ cos (2z/ 2mi> + c'mv) 4 a J ndt dv 192 148 ON THE SECULAR VARIATION OF THE MOON'S MEAN MOTION. of Art. 9, similarly give 15m 2 f, / 11 ,e'de'\ 21 m 2 f , / 77 a e'de'\ 3 m 2 f , ill -- cfo> - m 2 - -- v ~ m + [21 m 2 f, / 11 ,e'de'\ 21 m 2 f , / 77 a e'de'\ 3 m 2 f , i cfo> - m 2 -IT -- v ~7^ m ^77 +7 ndt dt { 32 J in which the coefficient of m 4 is totally different from that in Plana's result. I have since carried the approximation to the seventh order in m, and find that dn e'de' ( 3771 34047 306865 17053741 ,} r = rp i - 3m 2 + m 4 H - m "H -~r in' + - -mH. ndt dt { 32 32 48 576 J This reduces the coefficient of ( j , in the expression for the acceleration to 5"'7, only about one-half of the value hitherto received*. M. Delaunay has recently verified my coefficient of m 4 ; and he informs me that he shall very soon have carried the approximation to the eighth order in m, and included the terms depending on e~ and y. In my memoir above referred to I mentioned that other elements of the Moon's orbit suffer secular changes which had been overlooked. * The first part of this Paper was communicated to the French Institute in January, 1859, and was published in the Comptes Rendus. 22] ECCENTRICITY AND VARIATION OF THE MOON'S ORBIT. 159 I find the following expressions for the secular variation of the eccen- tricity and inclination of the Moon's orbit, adopting Plana's definitions of e and y : de_ , de' \ 235 di~ ' dt ["64 221 779 199631 *W1* I yw a I IIV f~ f /-V 4~ ni 1TO Il1l f\T dt dt~ f dt^ 64 ' "256" 4096 de I am engaged in carrying on the approximation to the value of -r- to the same extent as I have done in the case of T?, and in finding the part of the secular variation of the mean motion which depends on e* and y". These terms, however, can only very slightly affect the numerical value of the secular acceleration. Supplement to the foregoing. Since I sent my result respecting the secular variations of the eccen- tricity and inclination of the Moon's orbit to the Society the other day, I have found the leading terms of the secular acceleration of the mean motion which depend on the eccentricity and inclination of the orbit. The result is one of remarkable simplicity, considering the nature of the calculations which have led to it ; and I should be glad if you would let it appear in the Monthly Notices as soon as you conveniently can, as a supplement or a note to my former communication. The result is, dn e'de' f 3771 27 27 . , [I have not written down the coefficients of higher powers of m, as given in my former note.] It is curious that the coefficients of e" and /, in this expression, are equal and of contrary signs, although they are found by totally distinct processes. The effect of the terms in e 1 and y 2 on the magnitude of the secular acceleration is, as I anticipated, very insignificant. The term in e 2 increases the coefficient of the square of the number of centuries by 0"'036, and that in y 2 diminishes the same coefficient by 0"'097 ; so that, on the whole, the coefficient 5"'70, which I previously found, must be diminished by 0"-06, or reduced to 5"'64. This value I believe to be within one-tenth of a second of the true theoretical value of the coefficient of the secular acceleration. Whether ancient observations admit of such a small value of the acceleration is a different question. 23. REPLY TO VARIOUS OBJECTIONS AGAINST THE THEORY OF THE SECULAR ACCELERATION OF THE MOON'S MEAN MOTION (WITH POSTSCRIPT.) [From the Monthly Notices of the Royal Astronomical Society (1860). Vol. XX.] IF I have hitherto published no reply to the " Observations " of M. de Ponte"coulant, contained in the Monthly Notices of July last, it is riot be- cause the task presented any difficulty, for the fallacies which pervade M. de Pontdcoulant's communication were perfectly evident to me from the very first. I thought that any competent person who chose to look into my Memoir " On the Secular Acceleration," and into these observations upon it, might be safely left to form his own judgment on the matter. Again, I had some hopes that M. de Pontecoulant might be led to see and ac- knowledge the errors into which he had fallen, and with that object in view I sent to him, on more than one occasion, through a friend, com- munications which appeared to me amply sufficient to expose the fallacies contained not only in his printed " Observations," but also in several private letters which he subsequently wrote upon the subject. I find, however, that M. de Pontecoulant, in a letter which he has lately caused to be circulated among the members of the French Institute, has ventured to ignore these communications of mine altogether, and to speak as if his observations had been admitted without dispute. Under these circumstances, as my further silence might be misconstrued, I beg leave to offer to the Society the following remarks. 23] SECULAR ACCELERATION OF THE MOON'S MEAN MOTION. 161 In order to give a more complete view of the subject, however, and to obviate the necessity of my returning to it in a controversial manner, I shall not confine myself to the observations of M. de Ponte"coulant, but shall likewise say a few words in reply to the objections of M. Plana and those of M. Hansen. I shall also take the opportunity of making some preliminary remarks which may tend to remove certain misapprehensions, which I have reason to believe exist in some minds with respect to the real nature of the matter in dispute. First, then, I would call attention to the fact that the question is a purely mathematical one, with the decision of which observation has nothing whatever to do. It may be simply stated thus : if the eccentricity of the Earth's orbit be supposed to change at a given uniform rate and very slowly, what will be the corresponding rate of change, according to the theory of gravitation, in the mean motion of the Moon ? Now the solution of this question is effected by means of a purely algebraical process, the validity of each step of which admits of being placed beyond all possible doubt. What conclusion must be drawn, then, supposing that ancient obser- vations should shew that the secular variation of the Moon's mean motion is different from that which, according to theory, is due to the known change of the eccentricity of the Earth's orbit ? Why, simply this ; that the mean motion of the Moon is affected by some other cause or causes, besides the variation of eccentricity which has been taken into account. This fact, if established, would be a most inte- resting one, and might put us on the traces of an important physical discovery. It is not difficult to imagine the existence of causes which may affect the mean motion of the Moon, but whether it were so or not, any question respecting the validity of a mathematical process must be decided on mathematical grounds alone, quite independently of the agree- ment or disagreement of theory and observation. In the case before us the mathematical question as stated above may be greatly simplified, without its ceasing to involve the point which is in dispute. The values of the secular acceleration given by M. Plana's theory and mine, differ in terms which are independent of the eccentricity and inclination of the Moon's orbit ; consequently in deciding which of the theories is right, we may suppose the eccentricity and inclination to vanish. A. 21 162 SECULAR ACCELERATION OF THE MOON'S MEAN MOTION. [23 In the next place I would remark that the error which I attribute to M. Plana's theory on this point is not one of calculation which might require long and complicated numerical processes to be gone through for its correction, but that it is an error of principle, about which a mathe- matician ought not to have much difficulty in making up his mind. I am therefore inclined entirely to agree with M. de Pontecoulant's opinion, that the prolonged discussion of this subject would not be creditable to science, and indeed, considering the importance of the question, and the length of time which has passed since the publication of my Memoir, I cannot but think it strange that any controversy respecting it should still exist at all. Some persons appear to be under the impression that the contest lies between two values of the secular acceleration, that M. Delaunay and I agree in one value, and that MM. Plana, de Pontecoulant, and Hansen, agree in a larger value ; but this is by no means the true state of the case. Between M. Delaunay's result and my own, indeed, there is a perfect agreement. He has carried the approximation much further than I have done, but all of the terms which I have calculated have been confirmed by him. Again, before publishing my Memoir in 1853, I had obtained my result by two different methods, and I have since confirmed and extended it by means of a third. M. Delaunay arrived at his result by an inde- pendent method of his own, and he has lately found exactly the same result by following the method given by Poisson. On the other hand, among our opponents there is far from being the same satisfactory agreement. In his theory of the Moon, M. Plana obtained one value of the secular acceleration. In 1856 he printed a paper in which he admitted that his theory was wrong on this point, and actually deduced my result from his own equations. Soon afterwards, however, M. Plana retracted his admission of the correctness of my result, and obtained a third result, differing both from his former one and from my own. Again, M. de Pontdcoulant, in the last communication which I received from- him, gives two different values of the secular acceleration, one of which he has obtained by using the time, and the other by using the Moon's longitude as the independent variable. Strange to say, however, he does not appear at all startled at obtaining two contradictory values, but seems fully inclined to defend both. Indeed, judging from the last paragraph of 23] SECULAR ACCELERATION OF THE MOON'S MEAN MOTION. 163 his letter in the Monthly Notices, he appears to have expected that the results of the two methods would differ from each other. One of the values which M. de Pontecoulant thus obtains agrees with that given in M. Plana's theory, as of course it must do, being found by means of the same principles. But he seems to be quite unaware that this value has been abandoned by M. Plana himself in his last paper above referred to, which is contained in the eighteenth volume of the Turin Memoirs. M. Hansen's value of the secular acceleration is not given in an analytical form, like those of MM. Plana and de Pontecoulant, and therefore we can only compare the final numerical results. This comparison, which I shall presently give, shews that M. Hansen's value of the acceleration considerably exceeds either of those found by M. Plana. Here then we find nothing to inspire confidence ; certainly nothing like the cumulative testimony which there is in support of M. Delaunay's result and mine. I may now be permitted to make some remarks on another point. In the introduction to my Memoir of 1853, I gave some general reasoning to shew that a change in the eccentricity of the Earth's orbit had a tendency to produce a change in the mean areal velocity of the Moon, and that M. Plana was therefore wrong in assuming this velocity to be constant, as in his theory he does. Now this seems to have led some persons to imagine that my analysis in the following part of the memoir depended in some way or other on the validity of the general reasoning which had gone before, and therefore that my conclusions could not be regarded as established with mathematical strictness. But this is quite a mistaken view of the case. I make no assumption respecting the variability of the mean areal velocity. I prove mathematically that this velocity does vary by finding the amount of its variation, and the general reasoning given in the introduction is simply the translation, so to speak, of my analysis into ordinary language, in order to make the nature of my correction to M. Plana's theory more generally intelligible. It may be remarked too that even if I had started with the assumption that the mean areal velocity was variable, no error could have been caused thereby, for if this velocity had been really constant I should have found its variation equal to zero. In mathematics the terms " constant " and " variable " are not looked upon as opposed to each other, but a constant is regarded as a particular case of a variable quantity. 212 164 SECULAR ACCELERATION OF THE MOON'S MEAN MOTION. [23 It may be as well to guard against the idea that the extreme minute- ness of the quantities which we have to deal with in this investigation, gives rise to any uncertainty in the result. The present rate of approach of the Moon to the Earth which accompanies the acceleration of its motion, is less than one inch per annum, but the theory can determine this minute quantity to within, say, a thousandth part of its true amount, just as easily and certainly as if the quantity to be found had been any number of times greater. I will now proceed briefly to explain the principles which I employ in determining the secular acceleration, and to point out the errors which vitiate the several results of MM. Plana and de Pontecoulant which have been already referred to. The principle of my method is simply this, viz., that the differential equations must be satisfied, and that quantities which really vary must be treated as variable in all the differentiations and integrations which occur throughout the investigation. Now if e', the eccentricity of the Earth's orbit, be variable, the differ- entiation or integration of any term which involves e' in its coefficient will produce, in addition to the term which would result if e' were constant, de' another term involving -7- in its coefficient, supposing t to be the independent variable. In consequence of the existence of these supplementary terms, the ordinary expressions for the Moon's coordinates when substituted in the differential equations will not satisfy them, but will leave terms multiplied de' by -5- outstanding. In order to destroy these terms, it is necessary to a,dd terms of the same form to the usual expressions for the Moon's co- ordinates. The values of these new terms may, if we please, be easily found by the method of indeterminate coefficients, each of the coefficients being obtained by means of a simple equation. If n, the Moon's mean motion, be variable, the double differentiation of the Moon's coordinates will produce in the differential equations, terms involving -j- of the same form as those already mentioned which involve -5- . 23] SECULAR ACCELERATION OF THE MOON'S MEAN MOTION. 165 Thus the same system of simultaneous simple equations that gives the values of the indeterminate coefficients, determines likewise the value of fiYt r , which is what we want to find. at If the Moon's longitude v be taken as the independent variable, we must proceed according to the same principles, but there is one additional circumstance to be attended to. In the former case, since e' is supposed to vary uniformly with the de' d'e' time, -j- is considered constant, or -j^- 0. In the latter case the terms dl dt" which are introduced by the consideration of the variability of e' will involve , - instead of ,- as before ; and since the Moon's motion in longi- tude is not uniform, the value of -p- cannot be considered constant, or -5 dv di? de' cannot be neglected. To take this into account we must substitute for -j- its value -? r- , in which -j- is a known function of v, and then the dt dv dv remainder of the process will be exactly similar to that before described. Let us now consider the method followed in M. Plana's theory, and also by M. de Pontecoulant. de! In this method the terms above described involving -5 are ignored, and consequently the differential equations as developed by these astronomers dfL furnish no materials whatever for determining the value of -j- . Hence they are forced to supply the lack of data by means of an assumption, which is that one of the so-called constants introduced by integration is absolutely constant. The value of any one of the constants so employed can be expressed in terms of n, e' and known quantities. If then this so-called constant were dLfb really so, we should be able by differentiating this relation to obtain ^- in terms of - But if on the other hand this supposed constant be dt 166 SECULAR ACCELERATION OF THE MOON'S MEAN MOTION. [23 really variable, we must take its variation into account, in order to obtain the true value of -j- in terms of --*-. dt at In M. Plana's theory, in which v is taken as the independent variable, the constant so employed is Ir, which is added to complete the integral , f j dR j 2 r -j dv, in the equation 4 idvY ,...( .dR that is, by supposing the change in e' to be proportional to the Moon's true motion in longitude, would evidently cause the eccentricity of the Earth's orbit to be affected by all the inequalities of the lunar motion. All attempts to express e' in terms of v, without introducing periodic terms, lead to this absurdity. I have already alluded to the strange notion expressed at the end of M. de Ponte"coulant's paper, that there may be two values of the secular acceleration, one applicable to the true longitude and the other to the mean longitude. The difference between the true and the mean longitudes consists wholly of periodic quantities, and cannot contain any term increasing continually with the time. How M. de Pontecoulant could have so far deceived himself as to imagine that this paper settled the question of the secular acceleration, " sans con- testation possible ddsormais," is, I confess, beyond my comprehension. 23] SECULAK ACCELERATION OF THE MOON'S MEAN MOTION. 179 P.S. In the Compte Rendu of April 9, 1860, which has appeared since the foregoing paper was read, M. de Pontecoulant gives the value of the secular acceleration of the Moon's mean motion, which he has obtained by taking the time as the independent variable, and which he considers to be " desormais a 1'abri de toute objection." This result, however, of M. de Pontecoulant 's is the same as that which he formerly communicated to me, the error of which I have already pointed out. M. de Pontecoulant thus describes his method, " En deVeloppant la formule qui donne 1'expression de la longitude vraie en fonction de la lon- gitude moyenne, et en n'ayant e"gard qu'au premier terme de ce deVeloppe- ment, c'est-a-dire a sa partie non-pe"riodique j'en ai conclu le rapport du moyen mouvement de la lune dans son orbite troublee au moyen mouvement relatif a son orbite elliptique, c'est-a-dire a 1'orbite que cet astre ddcrirait autour de la terre sans 1'action du soleil... En differentiant ensuite cette valeur par rapport & 1'excentricite e' de 1'orbite terrestre qu'elle renferme,... j'ai obtenu une expression de cette forme : n The value of H thus obtained is - 3 5337 which, as I have shewn in p. 9 (see p. 167 above), is the result that would be found by differentiating the relation between n and a, and then neglecting the variation of a. The fallacy of M. de Pontdcoulant's reasoning consists in his treating the Moon's " orbite elliptique, c'est-a-dire, 1'orbite que cet astre de"crirait autour de la terre sans 1'action du soleil," as if it were a real elliptic orbit with an unalterable semi-axis major, whereas the semi-axis major of the elliptic orbit spoken of by M. Pontecoulant, which is the same quantity as that above denoted by the symbol a, is really variable, and its variation must be found by means of the differential equations in the way which I have before described. The numerical value of the coefficient of the secular equation which M. de Pontdcoulant obtains in this paper, when reduced so as to correspond with the value -1270"? of the integral ( (e^-E"*} ndt is 7"'96 which, as 232 180 SECULAR ACCELERATION OF THE MOON'S MEAN MOTION. [23 we see, differs widely from the similarly reduced values of the coefficient according to the theories of M. Plana and M. Hansen, given in p. 14, (see p. 173 above) as well as from the values obtained by M. Delaunay and myself. After giving his formula for the secular equation, M. de Pontecoulant remarks, " En comparant ce resultat tl celui que M. Plana a deduit de ses formules, on voit qu'il en differe d'une maniere notable, et que 1'espece de compensation qui devait s'e'tablir, selon ce geometre, entre les quantites du quatrieme ordre et celles des ordres superieurs, et qui semblait permettre de s'en tenir, comme 1'avait fait Laplace, aux termes re"sultans de la premiere approximation, n'existe pas re"ellement. La consideration des puissances supeVieures de la force perturbatrice altere sensiblement, au contraire, la valeur du coefficient qu'on obtient en faisant abstraction des quantites qui en dependent, et comme tous les termes de la formule, jusqu'aux termes du septieme ordre, sont affectes d'un signe ndgatif, la grandeur du coefficient qu'on s'e"tait habitue" a supposer a liquation seculaire d'apres les indications de Laplace, doit 6tre conside"rablement diminue'e." It is needless for me to point out how totally inconsistent these remarks of M. de Pontecoulant are with the conclusion at which he arrives in his paper in the Monthly Notices, " II rdsulte, je pense, sans contestation possible desormais, de la discussion pre"ce'dente, que les formules employees jusqu'ici pour determiner Fequation sdculaire de la lune, ont toute la correction necessaire a cet important objet." 24. ON THE MOTION OF THE MOON'S NODE IN THE CASE WHEN THE ORBITS OF THE SUN AND MOON ARE SUPPOSED TO HAVE NO ECCENTRICITIES, AND WHEN THEIR MUTUAL INCLINATION IS SUP- POSED TO BE INDEFINITELY SMALL. [From the Monthly Notices of the Royal Astronomical Society. Vol. XXXVIH. (1877).] A VERY able paper has recently been published by Mr G. W. Hill, assistant in the office of the American Nautical Almanac, on the part of the motion of the lunar perigee which is a function of the mean motions of the Sun and Moon. Assuming that the values of the Moon's coordinates in the case of no eccentricities are already known, the author finds the differential equations which determine the inequalities which involve the first power of the eccen- tricity of the Moon's orbit, and, by a most ingenious and skilful process, he makes the solution of those differential equations depend on the solution of a single linear differential equation of the second order, which is of a very simple form. This equation is equivalent to an infinite number of algebraical linear equations, and the author, by a most elegant method, shews how to develop the infinite determinant corresponding to these equations in a series of powers and products of the small quantities forming their coefficients. The value of the multiplier of each of such powers and products as are required is obtained in a finite form. By equating this determinant to zero, an equation is obtained which gives directly, and without the need of successive approximations, the motion of the Moon from the perigee during half of a synodic month. The small quantities 182 ON THE MOTION OF THE MOON'S NODE [24 which enter into the value of the above determinant are of the fourth, eighth, twelfth, &c. orders, considering, as usual, the ratio of the mean motion of the Sun to that of the Moon as a small quantity of the first order ; and the author has taken into account all the terms of lower orders than the sixteenth. The ratio of the motion of the perigee to that of the Moon thus obtained is true to twelve or thirteen significant figures. The author compares his numerical result with that deduced from Delaunay's analytical formula, which gives the ratio just mentioned developed in a series of powers of w, the 1'atio of the mean motions of the Sun and Moon. The numerical coefficients of the successive terms of this series increase so rapidly that the convergence of the series is slow, so that the terms calculated do not suffice to give the first four significant figures of the result correctly, although by induction, a rough approximation may be made to the sum of the remaining terms of the series. I have been led to dwell thus particularly on Mr Hill's investigation because my own researches in the Lunar Theory have followed, in some respects, a parallel course, sed longo intervallo. I have long been convinced that the most advantageous way of treating the Lunar Theory is, first, to determine with all desirable accuracy the inequalities which are independent of the eccentricities e and e', and the inclination 2sin~ 1 y, and then, in succession, to find the inequalities which are of one dimension, two dimensions, and so on, with respect to those quantities. Thus the coefficient of any inequality in the Moon's coordinates would be represented by a series arranged in powers and products of e, e', and y, and each term in this series would involve a numerical coefficient which is a function of m alone and which may be calculated for any given value of m without the necessity of developing it in powers of m. The variations of these coefficients which would result from a very small change in m might be found either independently or by making the calculation for two values of m differing by a small quantity. This method is particularly advantageous when we wish to compare our results with those of an analytical theory such as Delaunay's, in which the eccentricities and the inclination are left indeterminate, since each numerical coefficient so obtained could be compared separately with its analytical development in powers of m. 24] IN A PARTICULAR CASE. 183 It is to be remarked that it is only the series proceeding by powers of in in Delaunay's Theory which have a slow rate of convergence, so that it is probable that all the sensible corrections required by Delaunay's co- efficients would be found among the terms of low order in e, e', and y. The differential equations which would require solution in these suc- cessive operations after the determination of the inequalities independent of eccentricities and inclination would be all linear and of the same form. It is many years since I obtained the values of these last-named inequalities to a great degree of approximation, the coefficients of the longitude expressed in circular measure, and those of the reciprocal of the radius vector, or of the logarithm of the radius vector, being found to ten or eleven places of decimals. In the next place I proceeded to consider the inequalities of latitude, or rather the disturbed value of the Moon's coordinate perpendicular to the Ecliptic, omitting the eccentricities as before, and taking account only of the first power of y. In this case the differential equation for finding z presents itself natur- ally in the form to which Mr Hill reduces, with so much skill, the equations depending on the first power of the eccentricity of the Moon's orbit. In solving this- equation I fell upon the same infinite determinant as that considered by Mr Hill, and I developed it in a similar manner in a series of powers and products of small quantities, the coefficient of each such term being given in a finite form. The terms of the fourth order in the determinant were thus obtained by me on the 26th December 1868. I then laid aside the further in- vestigation of this subject for a considerable time, but resumed it in 1874 and 1875, and on the 2nd of December in the latter year I carried the approximation to the value of the determinant as far as terms of the twelfth order, or to the same extent as that which has been attained by Mr Hill. I have also succeeded in reducing the determination of the inequalities of longitude and radius vector which involve the first power of the lunar eccentricity to the solution of a differential equation of the second order, but my method is much less elegant than that of Mr Hill. Immediately after Mr Hill's paper reached me, I wrote to him expressing my opinion of its merits, and telling him what I had done in the same direction, and I received from him a very cordial and friendly letter in reply. 184 ON THE MOTION OF THE MOON'S NODE [24 The equation which I had obtained by equating the above-mentioned determinant to zero differed in form from Mr Hill's, and on making the reductions required to make the two results immediately comparable, I found that there was an agreement between them except in one term of the twelfth order. On examining my work I found that this arose from a simple error of transcription in a portion of my work, and that when this had been rectified my result was in entire accordance with Mr Hill's. The calculations by which I have found the value of the determinant are very different in detail from those required by Mr Hill's method, and appear to be considerably more laborious. I have not yet had time to copy out and arrange the details of the calculations from my old papers, but I hope soon to do so, thinking that they may not be without interest for the Society. Meantime I now make known the result which I have obtained for the motion of the Moon's node on the suppositions stated in the title of this paper. If nt and n't represent the mean longitudes of the Moon and the Sun at time t, omitting, for the sake of brevity in writing, the constants which always accompany nt and n't, and if 6 and r represent the Moon's longitude and radius vector, I find that, in the case of no eccentricities and inclination, if m = '0748013, which is the value used by Plana, it -01021, 13629, 5 sin 2(n-n')t + 0-00004,23732,7 sin 4(w-w')< + 0-00000,02375,7 sin 6(n-n')t + 0-00000,00015,1 sin %(n-n')t + 0-00000,00000,1 sin \Q(n-n')t; i= 1-00090,73880,5 T + 0-00718,64751,6 cos 2(n-n')t + 0-00004,58428,9 cos 4(w-n')< + 0-00000,03268,6 cos 6(n-ri)t + 0-00000,00024,3 cos 8 (n - n') t - 0-00000,00000,3 cos W(n-n')t; supposing that 6 is expressed in the circular measure, and that the unit of distance is the mean distance in an undisturbed orbit which would be described by the Moon about the Earth in the same periodic time. In 24] IN A PARTICULAR CASE. 185 this case, if p. denote the sum of the masses of the Earth and Moon, we shall have yx = /i 2 . The differential equation which determines z, the Moon's coordinate per- pendicular to the Ecliptic, is Now, the Sun's orbit being circular, we have *-y = n' 2 , and the only *i function of the Moon's coordinates which we require in order to form this equation is . I find that, with the above unit of distance, \= 1-00280,21783,115 r + 0-02159,98364,4 cos 2(n-n')t + 0-00021, 53273,9 cos 4(w-w') + 0-00000,20644,8 cos 6(n-n')t + 0-00000,00192,9 cos 8(n-n')t + 0-00000,00000,3 cos 10 (n-n f ) t. Let + or " (n - nj \r 3 T r, / ' (n - nj \r> ) ' ~ (1 - m) = ^ + 2^! cos 2 (n n')t + 2q^ cos 4 (n n') t + 2q a cos 6 (n n') t + &c. ; then we find, from the above value of -= , that r q*= 1-17804,44973,149, and 2=1-08537,75828,323, 7, = 0-01261, 68354,6, & = 0-00012,57764,3, q, = 0-00000,12059,0. These are all the quantities necessary for finding the motion of the Moon's node, to the order which we require. If g?r denote the angular motion of the Moon from its node in half a synodic period of the Moon, the equation so often referred to above gives A. 24 186 ON THE MOTION OF THE MOON'S NODE [24 COS gTT = COS qiT -j 1 _ 256', that of the Sun. 77, the Moon's mean distance from the ascending node. c= and <7 = T;, so that (1 c)n denotes the mean motion of the ndt ndt Moon's perigee, and (g l)n denotes the mean retrograde motion of the Moon's node, in a unit of time. Also let e denote the mean eccentricity of the Moon's orbit. e', the eccentricity of the Sun's orbit. y, the sine of half the mean inclination of the Moon's orbit to the ecliptic. n' m = , the ratio of the mean motion of the Sun to that of the Moon. n ft, the sum of the masses of the Earth and Moon. 190 ON THE CONSTANT TERM IN THE [25 a = (-^} , the mean distance in the purely elliptic orbit which the Moon \n I if undisturbed would describe about the Earth in its actual periodic time. To fix the ideas, we will suppose the quantities e and y to be defined as in Delaunay's Theory of the Moon. If r denote the Moon's radius vector, and if we omit terms depending on the Sun's parallax, then, as is well known, the value of may be expanded in an infinite series involving cosines of angles of the form 2ijf' + 2% where i, j, f, k denote any positive integers, including zero, and the co- efficient of the term with this argument contains eV^'y 2 * as a factor, the remaining factor being a function of m, e", e' 2 , and y 2 . In particular, there is a constant term in - , corresponding to the case in which i, j, f, and k are all zero, and this term has the form A + Be* + Cy + Ee* + 2Fey + <7y 4 + &c., where A = &c. &c. &c. and A w A l &c., , B^ &c., (7 , C l &c. are all functions of m. Plana and, after him, Lubbock, Ponte"coulant, and Delaunay have developed the functions of m which occur in the coefficients of the several terms of r and of the other coordinates of the Moon, in series of ascending powers of m, and have severally determined, by different methods, the numerical co- efficients of the leading terms in these developments. With respect to the constant term in - , Plana shewed that the quan- tities denoted above by B and (7 C , viz. the coefficients of e 2 and y 2 in the above constant, both vanish when account is taken of the terms involving m 2 and m 3 . Pontecoulant carried the development of the quantities -B and (7 two orders higher, viz. to terms involving m 6 , and found that these terms likewise vanish. 25] RECIPROCAL OF THE MOON'S RADIUS VECTOR. 191 These investigations of Plana and Ponte"coulant, however, while they shew that the coefficients of the above mentioned powers of m vanish by the mutual destruction of the parts of which each of the coefficients is composed, supply no reason why this mutual destruction should take place, and throw no light whatever on the values of the succeeding coefficients in the series. Thinking it probable that these cases in which the coefficients had been found to vanish were merely particular cases of some more general property, I was led to consider the subject from a new point of view, and on February 22, 1859, I succeeded in proving, not only that the coefficients B and C vanish identically, but that the same thing holds good of the more general coefficients B and C, so that the coefficients of e; fW, e*e", &c. y, yV 2 , yV 4 , Ac. in the constant term of -- are all identically equal to zero. r Further reflection on the subject led me, several years later, to a simpler and more elegant proof of the property above mentioned. This new proof was found on February 27, 1868, and I now venture to lay it before the Society. The resulting theorem is remarkable for a degree of simplicity and generality of which the lunar theory affords very few examples. There are also two remarkable relations between the coefficients of e 4 , try 3 , and y 4 in the constant term of - , which we before denoted by E, F, and G. These relations may be thus stated : If the terms of the quantity c or which involve e 2 and y 2 be denoted by dn and similarly if the terms of g or + which involve e 2 and y 2 be denoted by where H, K, M, and N are functions of m and e' 2 , then we shall have E H F _M G ~ N 192 ON THE CONSTANT TERM IN THE [25 These relations are established by means of the same principle which was employed to prove the theorem above mentioned, viz. that JB = and (7-0. They were, however, arrived at much later, namely on August 14, 1877. ANALYSIS. Let x, y, z denote the rectangular coordinates of an imaginary Moon at any time t, the plane of xy being that of the ecliptic, and the axis of x the origin of longitudes. Also let xf, y' be the rectangular coordinates of the Sun, r 1 its radius vector, and p! its mass. Then if we neglect the terms which involve the Sun's parallax, the equations of motion are Now let x u y v z 1 be the rectangular coordinates, and r, the radius vector, of another imaginary Moon at the same time t as before, so that the same equations of motion hold good, and p., p.', x', y', and ?' are unaltered. Hence Multiply the first set of equations by x lf y v z l respectively, and subtract their sum from the sum of the similar equations in x u y v z 1 multiplied by x, y, z respectively. 25] RECIPROCAL OF THE MOON'S RADIUS VECTOR. Thus we find d*z 193 X or d x /I 1\ + p-(xx 1 + yy 1 + zz 1 )\^-^J dx\ d I dy, dy\ ,dfdz,_dz = ; + p. (xx, + yy, + zz>) (--,) = 0. VI ' / Hence the quantity x /I 1\ C, + 2/2/1 + 22.) ^TT~?j is a complete differential coefficient with respect to t, and therefore when developed in cosines of angles which increase proportionally to the time it cannot contain any constant term 4 '". Now 1 3 and rrr 1 1 Hence, if x-x lt y-y z-z v and therefore also r-i\, and - - - be quantities of the first order with respect to any symbol, then 1 will differ from 3 f - - ) by a quantity of the third order only. V*i T l * We may remark here that neither of the quantities can contain any constant term, but no use is made of this in what follows. A. 25 194 ON THE CONSTANT TERM IN THE [25 Hence, in the case supposed, the quantity --- cannot contain any TI T constant term of lower order than the third. More generally, the constant part of - - cannot be of a lower order r, r than the constant part of the product of the quantity ; multiplied by T l V one or other of the quantities or x - x +y-y+ r - r i- Now, as the two systems x, y, z and x v y u z 1 satisfy the same differ- ential equations, the solutions can only differ from each other by involving different values of the arbitrary constants. By applying the principle just stated to four different cases of variation of the arbitrary constants, we shall be able to prove the properties already enunciated, viz, = 0, 0=0, |=f , and^j. Let x = u cos (nt + e) v sin (nt + e), y = u sin (nt + e) + v cos (nt + c) ; and similarly X] = u t cos (nt + e) 1\ sin (nt + e), y l = u, sin (nt + e) + v^ cos (nt + e), where nt + z is supposed to retain the same value as before. Then (x - x,Y + (y- y,)* = (u - utf + (v- rj 2 . Hence, in the statement of our principle, we may replace (x - x,? + (y- y,}" + (z- z,Y- -(r- r,) 2 by (u - u,Y + (v- v,Y + (z- ztf ~(r- r,}\ For the sake of simplicity, we will take the quantity which was before denoted by a as our unit of length, so that, instead of the quantity formerly designated by - , we shall write simply - . 25] RECIPROCAL OF THE MOON'S RADIUS VECTOR. 195 Now it is known, a pmori, that the values of r and u, as well as that of - , may be developed in an infinite series involving cosines of angles in the form *itjj>j'$ 1kn, where i, j, f, and k denote any positive integers whatever, including zero, and that the value of v may be developed in a similar series involving sines of the same angles. Also we know that the coefficient of the term with the above argument occurring in any of these series contains eV-'y 24 as a factor, the remaining factor being a function of m, e 1 , e'" and y 2 . Similarly we know that the value of z may be developed in an infinite series involving sines of angles of the form and that the coefficient of the term with this argument contains e'e^ as a factor, the remaining factor being a function of m, e', e' a and y 3 as in the former case. It is essential to observe that -, r, u, and v involve only even powers of y, while z involves only odd powers of the same quantity. Having made these preliminary observations, we are now in a position to apply our principle to the four cases already alluded to. CASE I. First, suppose that the values of x, y, z are those belonging to the solution in which e and y vanish, therefore all the arguments in the values of -, r, u, and v will be of the form 2i f' and z will vanish. Also let the values of x v y u Zj belong to the solution in which e has a finite value, but y is still =0, while nt + e, and therefore also n, retains the same value as before. Hence z, also vanishes, and therefore z z l = 0. 252 196 ON THE CONSTANT TERM IN THE [25 Then all the arguments which occur in the values of - , r, u, and v will also occur in those of , -,, u lt and i\, but the coefficients of the r t corresponding terms will differ by a quantity which contains e* as a factor. Let the terms with these arguments be called terms of the first class. Also there will be additional terms in the values of - , r lt u,, and v, , *V with arguments of the form where j does not vanish, and the coefficients of these terms will contain e as a factor. Let the terms with these arguments be called terms of the second class. Now, in the formation of the quantities terms with the argument zero can only arise by multiplying together three terms of the first class, one term of the first and two of the second class, or three terms of the second class, one of which at least involves e* as a factor. Such a term formed in the first of these ways would be of the order of e* at least, while one formed in the second or third of these ways would be of the order of e' at least. Hence, by the principle before proved, the value of ---- can contain no constant term of the order of e\ r, r Hence B = generally, and as this holds good for every value of e', we must have = 0, A = 0, .B 2 = 0, &c. CASE II. In the next place, let the values x, y, z, as before, belong to the solution in which e and y vanish, and let the values x l , y lt z, belong to the solution in which e is still equal to 0, but y has a finite value, while nt + e, and therefore also n, retains the same value as before. 25] RECIPROCAL OF THE MOON'S RADIUS VECTOR, 197 Then all the arguments which occur in the values of - , r, u, and v likewise occur in those of -, r 1} tt,, and v lt but the coefficients of the i corresponding terms will differ by a quantity which contains y 2 as a factor. Also there will be additional terms in the value of - , r l} u l} and v 1} r i with arguments of the form where k does not vanish, and these will also contain y 2 as a factor in every term. Hence , r r 1} u u lt and v v, will contain y 2 as a factor in T I r every term. I Also z = 0, and therefore (z z^ 2 = z*, which will also contain y 2 as a factor in every term. /I 1\ 3 Hence ( ) will be of the order of >' at least, while Vi r) ( - tO* + ( - ttf + (z - Zl ) 2 - (r - r,)'} will be of the order of y 4 at least. Therefore, by the same principle as before, the value of can contain no constant term of the order of y 2 . That is, (7=0 generally; and as this holds good for every value of e' we must have C = 0, Ci = 0, C 2 = 0, &c. CASE III. Next, let the values x, y, z belong to the solution in which y vanishes and e is finite, while a;,, y^, z, belong to the general case in which e l and y are both finite, the value of e being now changed to e l while nt + e, and therefore also n, retains the same value as before. 198 ON THE CONSTANT TERM IN THE [25 Then all the arguments which occur in the values of - , r, u, and v, and which are of the form will occur unchanged in the values of , r lt 1; and v lt provided that , ''i and therefore also --TJ or c, remains unchanged, but the coefficients of the corresponding terms will differ by quantities which involve either e e^ or y" as a factor. Let the terms with these arguments be called terms of the first class. Also there will be additional terms in the values of , i\, u lt and v lt ''i the arguments of which are of the form where k does not vanish. The coefficients of these terms will all contain y 2 as a factor. Call the terms with these arguments terms of the second class. And (z z 1 ) 2 = z 1 s , which contains y 2 as a factor in every term. Now the condition that c remains unchanged gives us the following relation between e 2 , e?, and y 2 : He- = He,- + Ky, taking into account only the terms of lowest order in e", e^, and y 2 . Hence, ultimately, If this value of y be substituted for it, we see that every term in the values of -, r-r l} u-u 1} v-v lt and (z-z,)" will be divisible by e-e,. Hence the constant part of will be divisible by (e - e,) 2 , and ?\ T therefore also by (e' e^f, since this constant part involves only even powers of e" and e. 25] RECIPROCAL OF THE MOON'S RADIUS VECTOR. 199 That is, is divisible by (e~ e?}-; or is divisible by (e" e^f. Divide by e 2 e, 2 and then put e, 2 = e 2 , TT therefore - 2Ef + 2F -^ e 2 = 0, E H or = - CASE IV. Lastly, let the values of x, y, z belong to the solution in which e vanishes and y is finite, while a;,, y iy z 1 belong to the general case in which e and y l are both finite, the value of y being changed to y l while nt + e, and therefore also n, retains the same value as before. Then all the arguments which occur in the values of -, r, u, and v, and which are of the form will occur unchanged in the values of -, r lt w,, and v lt provided that 17, *"i and therefore also ?- or g, remains unchanged, but the coeflScients of the corresponding terms will differ by quantities which involve either e 2 or y* y? as a factor. Let the terms with these arguments be called terms of the first class. Also there will be additional terms in the values of , * w,, and ,, 'i the arguments of which are of the form where j does not vanish. The coeflScients of these terms will all involve e as a factor. 200 ON THE CONSTANT TERM IN THE [25 Call the terms with these arguments terms of the second class. Moreover, all the arguments which occur in the value of z, and which are of the form will occur unchanged in the value of z 1; but the coefficients of the cor- responding terms will differ by quantities which involve either (? or y y t as a factor. Let the terms with these arguments be called terms of the first class. Also there will be additional terms in the value of z i; the arguments of which are of the form where j does not vanish. The coefficients of these terms will all involve ey, as a factor. Call the terms with these arguments terms of the second class. Now the condition that g remains imchanged gives us the following relation between e a , y 2 , and y^: taking into account only the terms of lowest order in e 2 , /, and y-. N Hence, ultimately, ) d(y>) in which y 2 is to be put = after the differentiations. The relation thus deduced holds good for all values of e 2 . By equating the coefficients of e 2 on the two sides of the equation dP cfc dP dc E H find -p, = -F? , as before. r K. Also, by equating the coefficients of higher powers of e 2 , we obtain other relations between the coefficients of terms of higher orders in the value of P. Similarly, taking e 2 , y 2 , and y{ to be related as in Case IV., we have, by the same reasoning as before, dP dP = -*-,--* . & + -r7-;\ (y, 2 y") + terms of higher dimensions in e" and y 2 y 2 . ct(e 2 ) (yv Also = -TS . e 2 + , -Pj* (y? y") + terms of higher dimensions in e 2 and y, 2 y 2 . Hence, we have ultimately, when e 2 = and y,* = y 2 , jlP dg Limit of - -^ = g in which e~ is to be put = after the differentiations. The result thus deduced holds good for all values of y 2 . By equating the coefficients of y 2 25] RECIPROCAL OF THE MOON'S RADIUS VECTOR. 203 on the two sides of the equation d(P) dg dP - dg , , F M , , we nnd -^ = -^ , as before. (r I\ Similarly, by equating the coefficients of higher powers of /, we obtain other relations between the coefficients of terms of higher orders in the value of P. It may not be without interest to give here the result which I have obtained for the development of the constant term in the reciprocal of the Moon's radius vector. The expression includes, besides the terms spoken of in the foregoing paper, an additional term depending on the square of the Sun's parallax. Reintroducing the symbol a to denote the length before denned, which in the paper has been taken as the unit of length, I find (Hi The constant term in - r I 179 97 757 6 m 2T8 -48 " i . e 799 192 m 5401 873 T2 m 18527 4039 ' 432 287849 -230T 34751189 1990656 268607 576 31013527 , ,1 J 75 225 |~ S* O I y 2m 2 + m 3 + where e and y have the same significations as in Delaunay's Theory. The method which I employed in obtaining this expression is closely related to my first method, above alluded to, of proving the evanescence of the coefficients B and C. 262 204 ON THE CONSTANT TERM, ETC. [25 The coefficients of e* and y 4 were found independently, and from each of these, by means of the relations proved above, was derived a value of the coefficient of e'-y'. The perfect coincidence of these values supplied a test of the correctness of the calculations. The terms of c and g which are required for this verification are the following : I hope to lay the details of these calculations before the Society on some future occasion. 26. NOTE ON SIR GEORGE AIRY'S INVESTIGATION OF THE THEORETICAL VALUE OF THE ACCELERATION OF THE MOON'S MEAN MOTION. [From the Monthly Notices of the Royal Astronomical Society (1880), Vol. XL.] I LOSE no time in pointing out briefly the reason why the Astronomer Royal, in the investigation which he communicated to the Society at the last Meeting, has failed to find my value of the coefficient of the Lunar Acceleration. It may be useful, in the first place, to recall to mind that, according to my theory, the secular changes of n, the Moon's mean motion, and e', the eccentricity of the Earth's orbit, are connected by the following relation : dn e'de' f 3771 , 34047 1 -T- = j1 -3m? + --m i + - m 5 +...k ndt dt { 32 42 J where m denotes, as usual, the ratio of the Sun's mean motion to that of the Moon. If we stop at the first term of the series within the brackets the result is identical with that found by Laplace. We do not know why Laplace did not carry his investigations further than this first term; but he probably thought that the succeeding terms would prove to be inconsiderable. 206 ON THE THEORETICAL VALUE OF THE [26 It is seen, however, that these terms have very large numerical co- efficients and that their sign is contrary to that of the first term, and on calculation it is found that the sum of the series is less than its first term nearly in the ratio of 3 to 5. Hence the secular acceleration will be diminished in the same ratio, and its amount in a century, instead of being about 10", will be reduced to nearly 6". No investigation of the Moon's secular acceleration can be satisfactory which does not take into account terms of the nature of those which give rise to the terms involving m 4 , m 5 , &c., above referred to. There is nothing to object to in the general principles of the method adopted by the Astronomer Royal, but in the practical application of the method I notice very grave defects. In the first place, the only periodic terms which are included in the T T Astronomer Royal's expressions for T - and P - and for the factors multiplying fk \Al \AJ I f\ Cv \ f\ \M / f\ \ n on the right-hand side of the equations, are those which involve the angle 2D or F; whereas it will be seen by a reference to my paper in the Philosophical Transactions for 1853, that a great part of the co- ft tt efficient of w 4 in the value of j- there obtained arises from the combination nat of terms involving the angles S, FS and F+S in the expressions for the Moon's coordinates with similar terms in * /\ 5 .. O - ) , 0V, &C. In the present investigation terms of the forms last mentioned are simply ignored. In the next place, it is to be noted that, although periodic terms a depending on the angle F are introduced into the assumed values of S - and v, yet in Art. 12, the value of h which is the coefficient of t* in the value of Sv, is found equal to Bb, quite independently of the values of the coefficients e, f, g, k, and I, which occur hi the terms thus introduced. 26] ACCELERATION OF THE MOON'S MEAN MOTION. 207 The result of this is to reduce the secular acceleration practically to its first term only; which accounts for the coincidence of the Astronomer Royal's value with that of Laplace. It may also be remarked in reference to Art. 11, that although terms involving the argument 2F or 4Z> may be properly omitted, we must put and coB 3 F= - + - cos 2F, 2t and the constant terms in these latter quantities should be taken into account. After these general remarks, we will enter a little more closely on the consideration of one or two points in the investigation which are important. Adopting the Astronomer Royal's notation, let cr denote the Sun's mass, A the semiaxis major of the Sun's (or Earth's) orbit, E the eccentricity of the orbit, R the radius vector at any time. Then it may be shewn, as in the paper before us, that the mean value of a- . cr 1 a- /, 3 ^A ls = +^ y> Hence if E receive the variation 8E in the time t, this quantity will be increased in the ratio of 1+3E8E to 1 nearly, or in the ratio of l+bt to 1, calling Having arrived at this point, the Astronomer Royal assumes that the variation of the disturbing forces due to the variation 8E in the eccentricity of the Sun's orbit will be represented by supposing T to be replaced by T(l + bt), and similarly P to be replaced by P(l+bt), 208 ON THE THEORETICAL VALUE OF THE [26 and therefore that the new forces, the effects of which are to be found by the present method, are Tbt and Pbt respectively. On consideration, however, it will appear that this is only true for the non-periodic term in P, and that the periodic terms, whether in P or T, will be changed by any given variation of E in very different ratios. For instance, the periodic terms in both T and P which depend on 5 the angle 2D or F will vary nearly in the same ratio as 1 - E* does, g instead of in the ratio in which 1 + -E" varies as in the above case. & Hence these terms will be changed by the above-mentioned variation of E in the ratio of 1 + b't to 1, where E8E V = 5 - nearly. (/ Again, the periodic terms in T and P which depend on the angles S, FS and F+S will vary nearly in the same ratio as E does, so that these terms will be changed in the ratio of l + b"t to 1, where 8E U' = - Et nearly. Hence we see that the values of b' and b" are quite different from that of b which belongs to the non-periodic term, and that b" is much larger than the other two quantities. The correct way of finding 8T and 8P, the changes of the disturbing forces T and P due to change in the eccentricity of the Sun's orbit, is to express T and P in terms of the Moon's coordinates v and r, the Sun's mean longitude L and its mean anomaly S, and the eccentricity E. Hence 8T and SP may be at once expressed in terms of Sv, 8?', and SE. Thus calling V the Sun's longitude, and employing the other symbols in the sense before explained, we have 1 err 3 err , ,, Tm - 3 Bin (2,- 2 F). 26] ACCELERATION OF THE MOON'S MEAN MOTION. 209 Or, P r= l-% Tr= - Now, by the formulae of elliptic motion, we may find A neglecting terms involving 2S, and powers of E above the second. Substituting, and then taking the variation, we have S (Pr) = ~ rSr + 3 ~ rSr cos (2v- 2V) - 3 ~ r*%v sin (2v- 2 V) 1 ~ T - 5E8E cos (2v - 2L) + 7 - 8E cos (2v - 2L - S) 8 (2V) = l_ in which r'Sl-J may be written for rSr, and the expressions given by A. 27 210 ON THE THEORETICAL VALUE OF THE ACCELERATION, ETC. [26 the ordinary lunar theory in the case of unvaried eccentricity are to be substituted for v and r. Hence, the expressions for &\T-} and S(P-), which are employed \ Ct/ \ Oi] in the paper, are wholly incorrect, except in the case of the non-periodic term, which gives rise to the principal term of the secular acceleration or that found by Laplace. The remark made near the close of the paper, viz. that the magnitudes of the quantities A, B, C, and therefore also that of the secular accelera- tion are proportional to the inverse cube of the Sun's distance, or to the cube of the Sun's parallax, can only be the result of inadvertence, as the Astronomer Royal himself will be the first to acknowledge. In fact, the quantities A, B, C involve the factor -j,- and this is equal to n*, where n' is the Sun's mean motion and is known. The Sun's mass cr is determined by means of the parallax from this equation ; or conversely, if the Sun's mass be known the parallax is thereby determined. The values of A, B, C are approximately as follows where m denotes, as before, the ratio of the Sun's mean motion to that of the Moon. 27. INVESTIGATION OF THE SECULAR ACCELERATION OF THE MOON'S MEAN MOTION, CAUSED BY THE SECULAR CHANGE IN THE ECCEN- TRICITY OF THE EARTH'S ORBIT. [From the Monthly Notices of the Royal Astronomical Society, Vol. XL. (1880).] As the question of the Moon's secular acceleration has lately been again brought before the Society, I have thought that it might not be useless or without interest to communicate an investigation of the two leading terms of that acceleration which I gave many years ago in my lectures on the lunar theory. 1. Let r, 6 be the polar coordinates of the Moon at time t, u = -, H = r 2 -j- , fj. the sum of the masses of the Earth and Moon ; also let m' be the mass of the Sun, /, & its polar coordinates, a' the Sun's mean distance, n' its mean motion, and e' the eccentricity of its orbit, X' = its mean longitude, and <' = n't + e' CD' its mean anomaly. Then the equations to be satisfied are d*u [L 1 m' 3 m' , fi ff . * -Tr*~2Trw*~2~irw* C( (0 3 m' du . ^ ,, ,.,, d(H*) 3m' and \, a '= r ^- dd uY 3 272 -0')= cos2(0-0') = n' 2 {(l - 1 A cos 2(0-\') + l e' cos (20 - 2X' - and = n * i _ ef gin 2 ((9 - X') + e' sin (26 - 2X' - f ) 212 INVESTIGATION OF THE SECULAR ACCELERATION [27 Also, by the formulae of elliptic motion The angles involved in these expressions are formed by combining the angle 20 2X' with multiples of <'. For our present purpose we may omit the terms which involve 2'. Also, for the sake of brevity we may write n't instead of n't + e' a/ or t ~ T m ' ( l ~ \ e ' 2 ) cos ^ ~ 2n '^ ~ TF mV cos _ | e 'A cos (20 - 2') - y mV cos (20 - 3w') + ~ mV cos (20 - n't), or = 1 + - mV 2 + 3mV cos ' - ^ m 2 f 1 - ^ e'^ cos (20 - 2?i') a0 4 \ 2 / - 7 J mV cos (20 - 3n'0 + ^ ?/iV cos (20 - n't), o o since the other terms only give rise to terms of higher orders than we have here taken into account. Hence H'-l-r-j-] = u Xifdt/ {4*1 245 1 + 5m 4 ( 1 - 5e' 2 ) + = m 4 e /2 + - - 44 cos n - 4m 2 (l - - *) cos (20 - 2n't) - 14mV cos (20 - 3n't) + 2mV cos (20 - n't)\ \ . * / J - 6mV cos n'< + ^ m 2 f 1 - ~ e' 3 } cos (20 - 2n't) + ^ m 2 e' cos (20 - 3n't) 2 \ 2 I 4 or, by actual multiplication, m 4 (1 - 5e' 2 ) + - mV 2 - 1 8mV 2 - 1 1m 4 (1 - 5e") o4 - mV 2 - H mV 2 + - m 2 fl - 1 e' 2 ^ cos (20 - 2n'<) 4 4 2 2 21 3 1 ~ m*e' cos (20 - 3n't) - ^ m?e' cos (20 - n't) Y 27] OF THE MOON'S MEAN MOTION, ETC. 215 4 ( 1 - 5e") + mV 2 + m 2 l - e cos (20 - 2n't) 54 Z \ Z ] 21 S 1 + mV cos (20 - Sn't) - - wV cos (26 - n't) \ . Hence the constant part of H* is 4 f 171 , 2421 4 --^ n being the actual mean motion. Hence K\ 01 q 1 / cos ^ 2 ^ ~ 2n '^ ~ T m ' e/ cos ^ 2 ^ " 3n '^ + f m2e/ cos ^ ~ w/ ') 135 4 , . o7> m v 1 "^ I C~A ^'^ " W I 1 ~ * I COS ( i5t/ ! oZ o4 21 31 irfe' cos (20 3n't) + - m?e' cos (20 n't) 4 4 'J and therefore the constant part of -^ is i*_ fj _ 135 4 _ 1881 4 J 3. Also j = - I o ^'e' sin n't 2m? ( 1 - e" ) sin (20 2n'i) at? a (2 \ 2 / - 7mV sin (20 - 3rit) + m?e' sin (20 - n't)\ ^ = 1 J _ 4m 2 ( 1 - jj e' 2 ) cos (20 - 2n'<) - 1 4mV cos (20 - 3n') (Xi/ C6 I \ 2t I \. \ / and 216 INVESTIGATION OF THE SECULAR ACCELERATION also n ^ _ m^ j 1 + m ^ cog n/f _ , f / 1 _ e/s \ cQg ^ 2 _ 2n ,Q [27 CQ q "1 - ^ mV cos (26 - Sn't) + - m'ef cos (20 - n't) I , and ** ^ = ^ j _ 2m 2 (l - J e' 2 ") sin (20 - 2n'<) - 7mV sin (20 - 3n'<) .tt V a [ \ 2 / -n'm. Hence, substituting in the first differential equation and transposing, we find the quantity which is to be equated to /t 2 to be a 2 4 27 4 lt 27 4 ft . "8 -^e 1 - m^l-Sc 1 )- ^ -^ 32 Q 1 A7 m 4 1 - 5e' 2 + - - - - m 2 f 1 - ^ *] cos (20 - 2n'rt - mV cos (20 - 3n') + - mV cos (20 - rit)\ 2 \ 2 / 4 4 o / K \ 01 - 1 m 2 ( 1 - - e' 2 j cos (20 - 2n't) - y mV cos (20 - 3') + - 7u 2 e' cos (20 n't 4 Comparing this with the former expression and observing that % t is nearly =1, we see that the periodic terms agree, and by equating the non- periodic parts, we have 135 3231 27] OF THE MOON'S MEAN MOTION, ETC. 217 which gives the relation between n and a. 4. In the above, e' is considered constant throughout ; if now we consider e' to be variable, we may choose n and a so that the constant (or rather the non-periodic) parts of u and of H" may have the same forms as before, and in this case we shall find the same relation between n and a as that which has just been found, and n will continue to signify the actual mean motion at the time to which 6 belongs, but n and a will now become variable quantities, and, in order to satisfy our equations, it will be necessary to add certain periodic terms to u arid H" which would not exist if e' were constant. Suppose then that u = -\l+8v - 1 wiV cos n't + m* ( 1 - 1 e'*\ cos (20 - 2n't) + -m?e' cos (20 - 3n't) a { 2 \ 2 / 2 -imV cos (20-n't)\, and H* = n'a< {1 + 287; + 1JI m< + 2 -||- mV 2 + ^ m 2 f 1 - e' 2 ) cos (20- 2n't) ^ mV 2 cos (20 - 3n'<) - | mV cos (2^ - ')| . We will suppose e' to vary uniformly with the time, and very slowly, or, in other words, we will suppose -7- to be constant, so that j-^ = 0, and we will neglect (~77/ A. 28 218 INVESTIGATION OF THE SECULAR ACCELERATION [27 de' We must therefore recollect that -^ is not constant, but is equal to de' dt 1 de' lh'de~Hu*'~dt ndt y cos n't - ~ m* cos (26- 2rit) - ~ ni>e' cos (26- 3n't) . J du i 5. In consequence of the variability of e', -^ will contain the additional terms adt 2 dt 3 ,de . , de , Q .. - m --i- cos n 7 5mV -,- cos (26' 2nt) -,- dt ffl? cos (2tf - 3n') - m 2 cos (2^ - 1 d.8v a dd or an\ adt 2 dt 5? cos (26> - 2n') + ^ ' dt 2 dt cos - 3w') I d.Bv to the order of approximation required. u Therefore also -r^ will contain the additional terms atf { lOmV ^ sin (26> - 2n') - 7m 2 ~ sm(2d- 3rit) + m 2 ^ sin (2tf - n't) an \ dt dt dt + 10mV sin (20 - 2n't) - 7m* d ~ sin (26? - 3n't) + m 2 ^- sin (26 - n't)\ I d\Sv 27] OF THE MOON'S MEAN MOTION, ETC. 21 9 d*a /daY , de 1 neglecting -r^ , ( -j- ] and also m 2 -j- Ctv \Ctt / Cvt in the coefficients of the periodic terms. Hence T + u contains the additional terms 1 if 2 , de' . f , 2 oV / /j / \ , tfe' / \\ v y Also -^ contains the additional term -~- t [ 2877]. The other terms which enter into the first differential equation receive no additional terms of the order to which we restrict ourselves. 6. Also differentiating the expression for H 2 , and including terms of de' the order mV-i- in the non-periodic part, but only those of the orders , de' , de' m 2 r and m'e -=- at at in the periodic part, we have the following additional terms in '-,-- , viz. . . 1 (2dn 4da 2421 . ,de' 15 .e'de' na u * \ ~Jt + -JT + ~^T me ^ ~ lT m -7TT cos ( 2 ^ - 2n Hu [ndt aat 32 at 2 at frg O fja^ + - m 3 ^ cos(2^-3w'e)--m 2 -^ cos(20-n' 1 A /^// x ' A fit \ Also the right-hand side of the second differential equation contains the following additional quantity : wrna' ,Va 4 [48w] J3 sin (20 - 2n't) + ^ef sin (20 - 3n') - 1 e' sin (2tf - n'f) j , which, as we shall immediately find, contains non-periodic terms of the order t ,de' me j~ f at 282 220 INVESTIGATION OF THE SECULAR ACCELERATION Hence, taking the periodic parts of this equation, we have [27 4 at \ fp rp 2 (817) - -I iV sin (20 - 2n') - m 2 sin (20 - 3n') o ;7 8 at sm -n't)}. 7. Substitute this in the first equation, putting -- = 1 in the co- WCC -efficients of the periodic terms, as these are only required to the order of m 3 , and we obtain d * 8v + Sv = - - J20mV -j sin (20 - Zn't) - 14m 2 -^- sin (20 - 3n't) n [ at at + 2m 2 -^ sin (20 - n't) + - - mV ~ sin (20 - 2n't) r 8 <7<>' ^ rip' ~\ *^T s^ (20 - Zn't) + J m 2 ^- sin (20 - ') I a^ ; 8 ai n 4 sin (2^ - 2'<) - -- m 2 sin (20 - Zn' 1 [95 2 ,de' . =mes >\ 133 "' 24 ' Substitute this value of Su, and also the value of -^r- , viz. jHn 8 - 1 1 + 3mV cos n't - ^ m* cos (20 - Zn't) - ~ m*e' cos (20 - Zn't) 27] OF THE MOON'S MEAN MOTION, ETC. 221 for that quantity in the second differential equation, and equate the non- periodic parts which result from this substitution, 2dn 4da 2421 . ,de' 165 . ,de' 1617 4 ,de' 33 . ,de' .'. 5- H -- 1-+- me -=- + - mV-s --- -me -^ rT***"5r ndt adt 32 dt 16 dt 64 cfa 64 dt 95 ,cfe' 931 . .tie' 19 . ,cfe' = mV -j --- m e -JT FT m e -JT > 2 dt 8 dt 8 ofa ' 2c?n , da 963 . ,de' 285 . ,de' or -T. + 4 TT + , ,, m*e -j- = --- mV -j- , ndt adt 16 dt 4 rf c?w , da 2103 ,de' ___ I ^ _ -- _ fy\j r _ ndt adt 32 dt ' 8. The substitution of the values of Sv and Srj in the first differential de' equation introduces no non-periodic terms depending on -^-; consequently the value of -( ; remains of the same form as before. wV Hence 3 , , c?e' 1173 4 , c^' _ , i~ ^2~ 3 ^" l \rndt 3 1173 A ,c?e' , J dn\ o m +-^r m e 2 32 n' , since m = , and . . ^ = -- -j- , n mat ndt n' being constant. Hence I A ft A <* , 8 ^ tt Vo 1173 A '^ e (4 - 2m 2 ) - + 6 - y, = - 3m 2 + - m 4 ) e' -j- , 'ndt adt \ 16 / dt also from above , dn , da 6309 /^ o i\ dn U 3963 A .-. (1 -2m 2 )--T. = - 3m 2 -- ^-m } 7 rf< \ 32 / e -j- , dt 222 and INVESTIGATION OF THE SECULAR ACCELERATION dn /' 3771 A ,def = - 3m 2 - -- m 4 e' - ndt \ 32 t \ ,de' 2103 4 ,< 6 dt ' ^T me [27 !^- = 3m 2 - aa \ 32 2937 A ,dc' - or da _ adt 2937 32 ,M_ dt 9. These equations give the rate of variation of the quantities n and a. We will now shew that n denotes the actual mean motion, as it did when e' was constant. From the values of u and H* we find = -=. = - l - 2Sv - d6 HU? n 3mV n' - m a l - e' cos (2tf - 2n't) 4 \ 2 / - ~ mV cos (26 1 - 3n't) + - - mV cos (20 - n't)\ , o o J or ndt ~dd -1+5 mV 2 + 3mV cos n't - -. ^ m 2 1 - - e' 2 ) cos (20 - Zn't) - mV cos (20 - 3n'0 + - wiV cos (2/9 - n't) 8 o - TT V sin 2S - 2 "' 8n 2 - Divide by 1 + - mV 2 + 3mV cos n' Zi and take into account mV" in the non-periodic term, '<} = 1 - -^ m 3 ( 1 - 5 e} cos (20 - 2n') 4 \ ^ / ... nd (i _ Civ 27] OF THE MOON'S MEAN MOTION, ETC. 223 - 7 ~ nre' cos (26 - 3n't) + ^ nftf cos (20 - n't) o 8 24 ndt 595 de' . , ,, , . 85 , cfe' . , a .. -77T m ji 8m (2^ - 8'n - m 2 r- sin (20 - n't), 48 new ' 48 new and therefore {ndt = 6+ 3me' sin n'< - ~ m 2 f 1 - 1 e' 2 ) sin (2^ - 2n') J \ * / - mV sin (20 - 3n') + mV sin (20 - n't) cos n't + wV - cos (20 - 2n't) - m> cos (20 - 3n') 24 wa< ' 48 Hence differs from Jwcfa by periodic terms only, which proves the proposition. The value of 5- above found agrees with that found in my paper published in the Philosophical Transactions for 1853. 28. NOTE ON THE CONSTANT OF LUNAR PARALLAX. [From the Monthly Notices of the Royal Astronomical Society, Vol. XL. (1880).] FROM the report of a discussion which took place at a late meeting of the Society, I have reason to believe that an explanation of the ap- parent discrepancy between the value of the constant of parallax given by me in the Appendix to the Nautical Almanac for 1856, and in the Monthly Notices, vol. xiii. p. 263, and the value of the constant found by Hansen in the Introduction to his Lunar Tables, may not be unacceptable to some of our members. It will be proper to begin this explanation by recalling to mind that my formula, in the article of the Monthly Notices above referred to, does not represent the parallax itself, but rather the sine of that quantity converted into seconds of arc by dividing by sin 1" or, which is the same thing, by multiplying by the number of seconds in the arc equal to the radius. The employment of the sine of the parallax instead of the parallax itself appears to be desirable both on theoretical as well as practical grounds. In the first place, the sine of the parallax, being proportional to the reciprocal of the radius vector, is the quantity given directly by the lunar theory, and, in the next place, it is the same quantity which is wanted in the reduction of lunar observations. What I have called the constant of parallax in the papers above referred to is, then, the constant term in the expression for the converted sine of the parallax, supposing the periodic terms to be expressed in cosines 28] NOTE ON THE CONSTANT OF LUNAR PARALLAX. 225 of angles which increase in proportion to the time. The value found for this constant was 3422"'325. This quantity may also be called very appropriately the mean sine of the parallax, although I do not use the term in the papers referred to. The value of the corresponding constant in the expression of the parallax itself is 0"'157 greater than this, or 3422"'48, which may appropriately be called the mean parallax. The formula in the Introduction to Hansen's Lunar Tables does not give the sine of the parallax, but the logarithm of the sine of the parallax, and the constant which Hansen calls C is a quantity such that the constant term in his expression for the logarithm of the sine of the parallax is log sin C. Now, it is plain that the constant term in the development of log sin parallax is a different quantity from the logarithm of the constant term of the sine of the parallax, and hence my constant of parallax differs from Hansen's quantity ----.. . sin 1" We may readily express the relation between these two constants in the case in which the orbit is supposed to be an undisturbed ellipse. In this case, if the reciprocal of the radius vector, which is proportional to the sine of the parallax, be developed in terms of cosines of multiples of the mean anomaly, then, a being the semi-axis major, and e the eccentricity of the orbit, the constant term in the development will be - . In the same case, the constant term in the development of the logarithm of the reciprocal of the radius vector, expressed in terms of the same form as before, will be very nearly, instead of log - ; so that if c denote the constant term in the Cv former development, and logc' the constant term in the latter, we shall have c' 1 . - = 1 - e very nearly. A. 29 226 NOTE ON THE CONSTANT OF LUNAR PARALLAX. [28 This relation will still be approximately though not exactly satisfied when the Moon's perturbations are taken into account. Hansen himself, in a paper in the 17th volume of the Astronomische Nachrichten, p. 299, in which he gives the results which he had obtained in a preliminary investigation of the lunar perturbations, finds that the number corresponding to the constant term in the logarithm of the sine of the parallax requires to be augmented by 2"' 71 in order to reduce it to the constant term in the sine of the parallax itself. Calling the parallax p, Hansen finds that the value of the constant . , /sin\ . term in log I -. ) is log(3419"-35), and hence he concludes that the constant term in ( ^ -; , ) is 3422"'06. \sm I" I By repeating Hansen's calculation and taking into account some small terms omitted by him, I find the amount of the reduction to be slightly sin T) less than the above, viz. 2"'67, so that the constant term in -.- , according sin 1 to Hansen's preliminary theory would be 3422"'02. This value, however, is not immediately comparable with my own, being founded on different elements. Both values are purely theoretical, depending on the ratio of the Moon's mass to that of the Earth, the ratio of the Earth's equatorial and polar axes, and the ratio of the Earth's radius to the length of the seconds' pendulum in a given latitude. If M denote the mass of the Earth, m that of the Moon, A the Earth's equatorial radius, R the Earth's radius at a point of which the sine of the latitude is 73' P the length of the seconds' pendulum at the same point ; 28] NOTE ON THE CONSTANT OF LUNAR PARALLAX. 227 then the constant term of the sine of the horizontal parallax corresponding to the latitude just specified may be represented by M /2\* ' \M+m ' P and therefore the constant term of the sine of the equatorial horizontal parallax may be represented by AIM R\^ _ / M A 3 \* R (W+^m P] ' = \M+ in ' WF) ' where F is a factor which may be found by theory from elements which may be considered as known with all desirable accuracy. M The values of - , A, R and P employed in finding my constant are the following : m ~ which corresponds very nearly to Dr Peters' constant of Nutation ; ^4 = 20923505 English feet, ^ = 20900320 P = 3'256989 R and P belong to a point the sine of the geographical latitude of which is -fa . A and R are the quantities found from Bessel's latest determination of the figure and dimensions of the Earth as given in Astron. Nachr., Vol. xix., p. 216, supposing that 1 Toise = 6-394564 English feet. P is found thus : according to the formula given in p. 94 of Baily's Report on Foster's Pendulum experiments, (Mem. of the Roy. Astr. Soc., Vol. vii.), the square of the number of vibrations made in a mean solar day, at a point the sine of whose geographical latitude is -j-, by a pendulum which vibrates seconds in London is 7441625711+4 (38286335) = 7454387823. o 292 228 NOTE ON THE CONSTANT OF LUNAR PARALLAX. [28 Also Captain Kater's determination of the length of the seconds' pendulum in London is 39-13929 inches = 3-2616075 feet. Hence as the square of the number of vibrations made at a given place in a given time varies inversely as the length of the pendulum, we derive the value above given for P. The values of the fundamental elements employed by Hansen are the following : *-80, m A =6377157 metres, ^ = 6370063 ^ = 0-992666 and R l and P, belong to a point the sine of the geocentric latitude of which is -T-. V-} The corresponding values of R and P for a point the sine of whose geographical latitude is -^ are the following : v** .ft = 6370126 metres, P = 0'992651 And the constant term of the sine of the equatorial horizontal parallax may be represented either by / M A 3 \ J / M A 3 \* ' ' by \M+mR*Pj '' In my calculation of the factor F, I took into account terms of the order of the square of the Earth's compression. It would otherwise have been useless to distinguish between R*P and R?P^ or between F and F^. At the time when Hansen's paper appeared in the Astron. Nachr. Bessel's latest determination of the figure and dimensions of the Earth was not available. Hansen employed an earlier determination given by Bessel in Astron. Nachr., Vol. xiv., p. 344, in which the results were affected by an error in the calculation of the French arc of the meridian which was discovered later. 28] NOTE ON THE CONSTANT OF LUNAR PARALLAX. 229 Hence the corrections to be applied to the logarithms employed by Hansen in order to make them agree with those employed by me are the following, expressed in units of the 7th decimal : Correction. / M 'Htf+W +987 s) +25 / -150 The correction to be applied to Hansen's value of the logarithm of the constant term in the sine of the parallax is therefore 25 + -(987 -150) = 304 of the same units. o And the corresponding correction of the constant term of the sine of the parallax will be 0"'24, and therefore according to Hansen's preliminary theory, employing rny system of fundamental data, the value of this constant term will be 3422"'26. In my independent transformation of Hansen's expression I found the rather more precise value 3422"'264. This is less than my own value of the same constant by 0"'06 nearly, as stated in my paper in the Appendix to the Nautical Almanac for 1856. I there intimated my belief that Hansen's definitive theory would pro- bably be found to introduce a correction to his former value of the constant term in question, and this turns out to be the case. In Astron. Nachr., Vol. XVIL, p. 298, the constant term in w which denotes the perturbations of the natural logarithm of the reciprocal of the radius vector, divided by sin 1", is given as 1345"'281, but in the Intro- duction to Hansen's Lunar Tables this same quantity is given as 1348"'840. Hence, the correction to the former value is 3"'559, and multiplying this by sin 1" and by 3422" we find the corresponding correction of the constant of parallax to be 0"'059, so that this constant becomes 3422"'323, a result which agrees perfectly with my own. In this connection it may be worth mentioning that the only periodic term in which I found any difference much exceeding 0"'01 between my 230 NOTE ON THE CONSTANT OF LUNAR PARALLAX. [28 coefficients of parallax and those obtained by a transformation of the results of Hansen's preliminary theory was that which has the argument denoted by t + z in Damoiseau's notation. The corresponding term in w is in Hansen's preliminary theory 10"-92cos( + z), whereas in the Introduction to the Lunar Tables this term is 8*78 cos (*+*); the correction to the coefficient is 2"' 19, and multiplying this as before by sin 1" and by 3422" we find the correction to the corresponding term of the sine of the parallax to be -0"-086 cos (+*), and if this be applied to the value of this term in the preliminary theory, viz. 0"'181 cos (t + z), the result is 0"'145 cos (t + z), which agrees perfectly with my own. It should be remarked that, in the Introduction to his Lunar Tables, Hansen still continues to use the same fundamental data as he had done in his earlier paper, so that the value of the constant term in the sine of the parallax according to the data adopted in the Tables is 3422" - 08. Note added June 17, 1880. In Professor Newcomb' s valuable transformation of Hansen's Lunar Theory, which I have just received, it is wrongly assumed that I employed the same data as Hansen for the figure and dimensions of the Earth, and that my value of P, viz. 3 "2 5 698 9 feet, relates, like Hansen's, to a point the sine of whose geocentric latitude is -r- , whereas it should be the geo- V<3 graphical latitude, as that is the latitude which enters into Baily's formula from which my value of P is deduced. In consequence of this, Professor Newcomb finds a discrepancy of 0"'03 between Hansen's value of the constant of parallax and mine when both are derived from the same system of fundamental data ; but it has been shewn above that no such discrepancy exists. By a typographical error, the value of P which Professor Newcomb quotes from me is printed as 3'256 89 feet, instead of 3'256989 feet. 29. NOTE ON THE INEQUALITY IN THE MOON'S LATITUDE WHICH IS DUE TO THE SECULAR CHANGE OF THE PLANE OF THE ECLIPTIC. [From the Monthly Notices of the Royal Astronomical Society, Vol. XLi. (1881).] THE first theoretical explanation of this inequality was given by Hansen in the year 1849, in No. 685 of the Astronomische Nachrichten, just a year after the Astronomer Royal had pointed out, in a letter published in the same journal Beilage zu No. 648 that such an inequality was clearly indi- cated by the observations. In the same paper Hansen shews that there is a small term in the Moon's longitude depending on the same cause, the coefficient of which amounts to about 0"'5, the inequality being proportional to the cosine of the longitude of the Moon's node. The existence of this inequality also had been indicated by the Astronomer Royal from the observations, though he assigns to it a somewhat larger coefficient. The calculation of both these inequalities is given by Hansen somewhat more fully in p. 491, Art. 176 of his Darlegung. In 1853 I communicated to Mr Godfray a simple theoretical explanation of the inequality in latitude, which he inserted in his Elementary Treatise on the Lunar Theory. This explanation is there given in rather too compendious a form, and I propose in the course of this paper to present to the Society the same investigation, with some slight modification, together with some additional remarks, which will, I hope, render it clearer than before. 232 NOTE ON THE INEQUALITY IN THE MOON'S LATITUDE, ETC. [29 At the Meeting of the Society in March last, the Astronomer Royal gave an investigation of the inequality in latitude based upon the equations supplied by the "Factorial Tables" of his "Numerical Lunar Theory." About one portion of this investigation I wish to make a remark which seems to be important. The Astronomer Royal forms his equations with reference to the fixed ecliptic, and, by integrating them, derives the value of the disturbed latitude above the fixed ecliptic, whence the latitude above the variable ecliptic is immediately deduced. The latitude so found contains not only the inequality in latitude required, but also the small residual terms Bt {'003 sin nt-C\ + '005 sin \nt-2Nt+ C\ }, which the Astronomer Royal rejects, attributing them to accidental errors in the last places of the decimals employed. I shall presently attempt to shew that these terms must indeed be rejected, though not for the reason here supposed, but because they are destroyed by other terms which would be found by a more complete in- vestigation. It should .be remarked that if terms of the above form really existed, they would, notwithstanding the smallness of their numerical coefficients, ultimately become much more important than the other terms in which t does not occur in the coefficients. I propose to prove that in the complete solution of the differential equations no terms of the above-mentioned form can occur, supposing the displacements of the plane of the ecliptic to be proportional to the first power of t. The method which I employ for this purpose is the following. Instead of solving the differential equations of motion with reference to the fixed ecliptic and then transforming the results so as to make them apply to the variable ecliptic, I first transform the differential equations of motion, so as to make them refer to the variable ecliptic, and when this is done, it is found that the terms which contain t in their coefficients disappear completely from the differential equations, so that the solution may be effected by the ordinary methods without any difficulty. Employing the same data and notation as the Astronomer Royal, and taking into account only the terms which are independent of the Moon's 29] DUE TO SECULAR CHANGE OF THE PLANE OF THE ECLIPTIC. 233 eccentricity and inclination, I find 8s = - l"-424 cos (nt -C) + 0"'048 cos (-nt + 2Nt - C) - 0"'007 cos (3nt - 2Nt - C). The reason why, in the result found by the Astronomer Royal, the terms which are multiplied by t do not completely destroy each other, as they ought to do, appears to be the following. It is at once seen, from the form of the periodic terms to which the Astronomer Royal confines his attention, that his investigation is only com- plete with respect to the terms which are independent of the eccentricity and inclination of the Moon's orbit. In order to take the eccentricity and inclination into account, other periodic terms must be included, the argu- ments of which involve the Moon's mean anomaly and its mean distance from the node. From the combination of these terms with each other will arise terms with the same arguments as those which are independent of the eccentricity and inclination, while each of their coefficients contains the square of one of these elements as a factor. Hence it is clear that terms of this order are omitted in the investigation. On the other hand, a slight examination shews that the coefficients in the Astronomer Royal's expressions for T - cos I and v, a as well as in the quantities taken from his Factorial Table, include very sensible portions depending on the squares of the eccentricity and inclination. In fact, it is plain that this must necessarily be the case since the quantities in question are functions of the Moon's actual coordinates, in which the numerical values of those elements are essentially involved. Now, if terms depending on the squares of the eccentricity and incli- nation were either wholly neglected, or completely taken into account, the terms which are multiplied by t would be found identically to destroy each other ; but if, as in the present case, such terms are taken into account in one part of the investigation, and omitted in another part, it will follow that some of the terms multiplied by t will remain outstanding. A curious circumstance relating to this inequality of latitude remains to be noticed. In the Mecanique Celeste, tome in. p. 185, Laplace proves that the plane of the Earth's orbit in its secular motion carries the plane of the A. 30 234 NOTE ON THE INEQUALITY IN THE MOON'S LATITUDE [29 Moon's orbit with it, so that the inclination of the Moon's orbit to the variable ecliptic is not liable to any secular variation. In the same place he finds an analytical expression for the perturbation of latitude in reference to the variable ecliptic which is caused by the secular change in that plane. Now the point to be noticed is that this analytical expression given by Laplace requires only the very slightest possible development to furnish for the inequality in question a result which is identical with the value given by the formula of Hansen, in which displacements of the ecliptic varying not only as the first but also as the second power of the time are taken into account. It is true that Laplace imagined that this in- equality would turn out to be insensible, but this was only because he had not attempted to turn his formula into numbers. Analysis. I. Investigation of the inequality in the Moon's latitude which is due to the secular motion of the plane of the ecliptic, making the same sup- positions and employing the same data as the Astronomer Royal. At the time t let x, y, z be the rectangular coordinates of the Moon, and x', y' those of the Sun, referred to the Earth's centre as origin, the variable plane of the ecliptic at the same time being taken as the plane of xy. Also at the time t let f, 77, be the rectangular coordinates of the Moon, and ', r/, ' those of the Sun, taking the fixed plane of the ecliptic corresponding to t = as the plane of gr). For greater simplicity we will suppose, with the Astronomer Royal, that the variable ecliptic intersects the fixed ecliptic in a fixed line, and that the angle between these two planes is proportional to the time. Let this fixed line be taken as the axis of x and also as the axis of f, and let [c_ 3 cos ( 3nt + =0. Whence again, we find c_ 1= 8-69441&), c, = -230-8866 w, from which by substitution we obtain c_ 3 = 0-01097&), c 3 = - 0-34915&), c 6 = - 0-00136w. Hence the solution of the differential equation for Sz is 8z = w {0-01097 cos (-3nt + in't) + 8'69441 cos (-nt + 2n't) - 230'8866 cos nt - 0-34915 cos (3w - 2n't) - 0'00136 cos (5nt - 4n'*)}. Here cu is expressed in terms of the circular measure, and Sz in terms of the unit of length defined before. If s denote the sine of the Moon's latitude, z ~-i r and if 8s be the change in s due to the secular change in the plane of the ecliptic, we have . Sx bs = , r since Sr 0, according to the suppositions made above. Also - = 1-00090,74 + 0-00718,65 cos 2 (nt-rit) + 0'00004,58 cos 4 (nt-n't). Hence by substitution Sz = o> {0-0369 cos(-3n + 4 / ) + 7'8727 cos (-nt + 2n't)- 231 "0661 cosnt -1-1789 cos (3nt - 2rit) - 0'0079 cos (5nt - 4rit)}. 29] DUE TO SECULAR CHANGE OF THE PLANE OF THE ECLIPTIC. 241 Also s being supposed very small, 8s is equal to the circular measure of the change of the Moon's latitude due to the secular change in the plane of the ecliptic, and if we divide 8s by sin 1" we shall find the change of the latitude in seconds = -; , 7 JO'0369 cos (-3nt + 4n't) + 7'8727 cos (-nt + 2n't) - 231 '0661 cos nt Sill 1. - 1-1789 cos (3nt - 2n't) - 0"0079 cos (5nt - 4n't)}. Now, according to the data adopted by the Astronomer Royal, the circular measure of the angular motion of the plane of the ecliptic in 1 year is 0'479 sin 1". o_ Also 1 year is represented in our notation by the time . n o_ Hence to = '47 9 sin 1", (a n' and -^-, = 0-479^ = 0-00616,354. sin 1 2?r Therefore the inequality of latitude expressed in seconds is 0"'0002 cos ( - 3nt + 4w') + 0"'0485 cos (-nt + 2rit) - l"'4242 cos nt - 0"-0073 cos (3nt - 2n't). In this expression the mean longitudes nt and n't are reckoned from the node of the variable ecliptic upon the fixed ecliptic. If the mean longitudes are reckoned from the equinox in the ordinary way, and if C be the longitude of the above-mentioned node, we must replace nt and n't in the above by nt C and n't C respectively, and the expression for the inequality in latitude becomes 0"'0002 cos ( - 3nt + 4n'< - C) + 0"'0485 cos (-nt + Zn't - C) - l"-4242 cos (nt -C)- 0"'0073 cos (3nt - 2n't - C). In the above investigation the quantities at and C are supposed to be constant. If these be subject to small secular variations, the differential equations become a little less simple, but are easily formed, and the above solution will require the following modifications, viz. (1) Instead of the constant value of w we must employ the variable value which is of the form A. 31 242 NOTE ON THE INEQUALITY IN THE MOON'S LATITUDE [29 (2) The coefficients of the above expression will be very slightly changed by quantities which are propoi'tional to dC "dt' (3) The expression for the inequality of latitude will contain extremely small additional terms of the form {g_ 3 sin ( - 3nt + kn't - C)+g_ 1 sin ( - nt + 2n't - C) +g l sin (nt - C) sin (3nt - 2n't - C)}; that is to say, these terms will involve the nines instead of the cosines of the same arguments as before, and the coefficients of these new terms are proportional to dot) Hi' II. Theoretical explanation of- the same inequality, which was originally given, in substance, in Godfray's Elementary Treatise on the Lunar Theory. The general principle of this explanation may be very simply stated. If, for a moment, we suppose the plane of the Moon's orbit to remain fixed, and imagine the plane of the ecliptic to turn through a very small given angle about a line in its own plane, this will give rise to cor- responding small changes in the longitude of the Moon's node and in the inclination of the orbit to the ecliptic, and the magnitude of these changes will depend on the angular distance of the Moon's node from the line about which the ecliptic is supposed to be turning. If now the planes of both orbits be supposed to vary continuously, the total changes in the longitude of the node and inclination of the orbit produced in an indefinitely small time will be found by adding together the changes respectively due to the motion of the plane of the ecliptic, and to the motion of the plane of the Moon's orbit with respect to the ecliptic when the latter is supposed to remain fixed during that small time. The motion last mentioned is given by the formulae of the ordinary Lunar Theory, in terms of the disturbing force of the Sun. In consequence of the action of this force, the Moon's node gradually makes complete revolutions with respect to the line about which the ecliptic is turning, and the summation of all the momentary changes of node and inclination due to the motion of the ecliptic will produce periodic changes in those 29] DUE TO SECULAR CHANGE OF THE PLANE OF THE ECLIPTIC. 243 elements, the magnitudes of which, at any given time, like the momentary changes themselves, will depend on the angular distance, at that time, between the Moon's node and the line about which the ecliptic is turning. The combined effect of these periodic changes in the position of the node and in the inclination is to produce the inequality in latitude which is now under consideration. The motion of the Moon's node is not uniform, but the principal in- equalities by which that motion is affected have periods which are short compared with the time of revolution of the node. Hence the periodic changes of node and inclination above described, will be accompanied by others which are due to the same cause, but which in consequence of the shortness of their periods will be comparatively un- important, and the combined effect of these changes in the elements will be to add other terms which are equally unimportant to the expression of the inequality in latitude. We proceed to find the analytical expressions for the changes in the longitude of the Moon's node and in the inclination of the orbit, due to the motion of the plane of the ecliptic, supposing the Moon's orbit itself to remain fixed. Take C the longitude of the instantaneous axis about which the ecliptic is rotating at the time t, a) the angular velocity of the ecliptic, N the longitude of the Moon's node, and i the inclination of the orbit, at the same instant. Then, in the indefinitely small time 8t, a point of the ecliptic situated in any arbitrary longitude L will move through an angular space o)8t sin (L C) in a direction perpendicular to the ecliptic. Hence the point of the ecliptic originally coincident with the node N will move through the space perpendicular to the ecliptic. 312 244 NOTE ON THE INEQUALITY IN THE MOON'S LATITUDE [29 And if 8N be the consequent increase of the longitude of the node we have evidently from the figure, SN = o>S sin (N- (7) cot i Cl'J.V or dt * / -\~r y"Y\ i = ft> sin (N G ) cot ^. N' Again, the point of the ecliptic 90 in advance of N will move through the space or a>Stcos(N-C), perpendicular to the ecliptic, and this quantity will measure the diminution in the inclination of the Moon's orbit. Hence we have or Si= - (t) Stcos(N-C), -T- = cos (N C\ c c cos i But if denote the Moon's longitude, we have cos i sin i/ = cos B sin (6 N), and cos $ = cos ft cos (6 N}. Hence 8s = '*-. cos ft [sin (6 - N) sin (N-C)- cos (0 - N) cos (N- C)], C COS % ft) or cos ft8ft -- . cos ft cos (6 C), c cos* 248 NOTE ON THE INEQUALITY IN THE MOON'S LATITUDE [29 and therefore 8B= .cos (0C), c cost which is the inequality in latitude due to the motion of the ecliptic, expressed in the circular measure. This value of 8ft agrees exactly with that found in my article inserted in Godfray's Lunar Theory, since c cos i in this formula has the same signi- fication as '-'- in Godfray, viz. the mean angular velocity of the Moon's node, o The steps, however, by which this result is arrived at, are slightly different in the two investigations. In the earlier one, the variation of SN cos i was neglected, and St/ was taken = , whereas in the present investigation the variation of cos i is taken into account, and 8 0-479 x 18'6 , . Hence r = - - , expressed in seconds, c cos i sm 1" 27r which agrees with the value of the coefficient of the principal term found in the former investigation. The form above found for 8/8 suggests a very simple geometrical inter- pretation of this inequality in latitude. If we suppose a fictitious ecliptic to be inclined to the true ecliptic at the angle l"'42. the circular measure of which is - = , and if we also c cost suppose that the longitude of its ascending node on the true ecliptic is 90 + C, then the elevation of the fictitious above the true ecliptic cor- responding to the longitude will be c cost O) .sin (0-90 + C), . cos(0-(7), C COS I = 8/8. Hence the latitude above the fictitious ecliptic will be equal to j8, that is, the expression for the Moon's latitude with respect to the fictitious ecliptic is the same as the expression found for the latitude in the case when the ecliptic is taken to be a fixed plane. This geometrical interpretation of the inequality was first given by Hansen. III. Note on the Mecanique Celeste, tome in. p. 185 (edition of 1802). At any arbitrary point whose longitude is \, Laplace takes the elevation of the variable ecliptic above the fixed plane of reference to be represented by 2 k sin(\ + ^ + e), A. 32 250 NOTE ON THE INEQUALITY IN THE MOON'S LATITUDE [29 and he shews that if s 1 denotes the perturbation of the Moon's latitude with respect to the variable ecliptic which is due to the motion of that plane. ^ Then Sl = S where v denotes the Moon's longitude; - r2* or s i = 2 sw! (fm? very nearly, neglecting i~ compared with i except when it is divided by an additional power of -m\ 9 Or, replacing iv by it ^ T 2ii 4&r , . > s, = sin i/2 .- + ,-, - 2 \ cos ( + e) L|m 2 (fm 2 )-J ^ "_/<'/ tA/t / . \ + COS VZ ^ - + 7^ jrr, Sin (l< + e). |_fm 2 (fm 2 )-J Now, Hansen's expression for the elevation of the variable above the fixed ecliptic at any point whose longitude is X is of the form p cos X + q sin X, where p and q are functions of t, expressed in series of powers of t. Comparing this with Laplace's expression for the same quantity, we have p = 2k sin (it + e), dp _, . ,. x hence ~~Ji = ^"^ cos ( l ^ + )> and -jQ = SH 2 sin (it + e) ; similarly q = 2k cos (it + e), dq ~di f-** Ski sin (it + e), 29] DUE TO SECULAR CHANGE OF THE PLANE OF THE ECLIPTIC. 251 Hence, by substituting for k sin (it + e), k cos (it + e), &c. in Laplace's expression for s l} their values in terms of p, q and their differential co- efficients, we find f 1 dp 1 d z q~\ S 1 = Sin V \ o j -fr 7^ jr 3 - L f m- dt (f m 2 ) 2 c 3 J f 1 da I I nno ,. 2+2 which exactly agrees with Hansen's expression in his Darlegung, p. 490*, except that Hansen's argument f+ ;), which denotes the mean motion of the Moon's node, is equivalent to |w a in Laplace, as the latter takes n, the Moon's mean motion, to be equal to unity. 322 252 NOTE ON THE INEQUALITY IN THE MOON'S LATITUDE, ETC. [29 and ~- = ^ dC . -T = -T7 cos C- o> -j- sin C. of dt at Hence, putting c for -m", and denoting the Moon's longitude by 6 as before, instead of Laplace's v, we have 1 fd n . J s, = smv\ w sin C -. - -57 cos G to ,.- sin C L c c 1 \c a .r 1 \ (do . dC J\~l + cos - - w cos (7+ - -7- sin C + (U-z) Delaunay. -0"0971 0-0158 -0-0199 0-0025 0-0076 -0-0127 0-0185 0-0159 o-ono Adams. -O"l06 0-005 -0-036 0-002 0-014 -0-015 0-032 0-030 0-034 Hansen transformed by Newcomb. -0-106 0-003 -0-037 + 0-011 -0-019 0-043 0-032 0-035 In the above many very small coefficients have been omitted. As stated in my paper in the appendix to the Nautical Almanac for 1856, or in the Monthly Notices, Vol. xni. p. 177, my coefficients of parallax were obtained by comparing the results of the theories of Damoiseau, Plana, and Ponte"coulant, and tracing out the origin of the discordances in the cases where those results did not agree with each other. These coefficients were also compared with those which I obtained by a transformation of Hansen's preliminary results as given in a paper in Vol. xvn. of the Astro- nomische Nachrichten. In Pontecoulant's method the expression for the reciprocal of the radius vector is first found, and then the expression for the longitude is derived from. it. Hence the analytical values of the coefficients of parallax, given by Ponte"coulant, Vol. iv. pp. 149152, 281, 282, 336, 337, are at least as accurate as the values of his coefficients of longitude. In his final expression, however, in pp. 568 572, in which the several terms of the reciprocal of the radius vector are collected together, he neglects all terms of orders higher than the 5th, and the same omission takes place in the conversion of his coefficients of parallax into numbers. Accordingly these numerical values, which are calculated in pp. 599 601, and collected together in p. 635, nearly coincide with the values of Delaunay, but are on the whole still less accurate. It is greatly to be desired that some intrepid and competent calculator would undertake to make the numerous substitutions which would be required in order to find, by Delaunay 's method, the expression for the reciprocal 30] FOR THE MOON'S PARALLAX. 257 of the radius vector to the same order of accuracy as that which Delaunay has already attained in the case of the corresponding expressions for the longitude and latitude. The work would be one of simple substitution, not requiring the solution of any new equations, and consequently its only difficulty would consist in its great length. The fact that Delaunay's determination of the value of the reciprocal of the radius vector is a comparatively rough one, affords a ready explanation of a difficulty which Sir George Airy has recently met with in his Numerical Lunar Theory. The first operation required in this method is the substitution in the differential equations of motion of the numerical values of the Moon's coordinates as obtained in Delaunay's theory. If the theory were exact, the result of the substitution in each equation would be identically zero, so that the coefficient of each separate term in the result of the substi- tution would vanish. In consequence of errors in the coefficients obtained by Delaunay, however, this mutual destruction of terms will not take place, and the result of the substitution will consist of a number of terms the coefficients of which will depend on the errors of the assumed coefficients. If, as is actually the case, these latter errors be so small that their squares and products may be neglected, each of the residual coefficients may be represented by a linear function of the errors of the assumed coefficients, and the formation of the corresponding linear equations constitutes the second operation in Sir George Airy's method. The solution of these linear equations by successive approximations will finally give the corrections which must be applied to Delaunay's coefficients in order to satisfy the differential equations. Now, since the proportionate errors of Delaunay's coefficients of parallax are considerable, and much greater than the errors affecting his coefficients of longitude and latitude, it will be readily understood that the result of the substitutions will be to leave considerable residual coefficients in the two equations which relate to motion parallel to the ecliptic, and much smaller residual coefficients in the third equation which relates to motion normal to the ecliptic, since in this last equation every error in the co- efficients of the radius vector or of its reciprocal will be multiplied by the sine of the inclination of the Moon's orbit. This result, which might thus have been anticipated, is exactly what Sir George Airy has found to take place, according to a memorandum which he has recently addressed to the Board of Visitors of the Royal Observatory. A. 33 258 DELAUNAY'S EXPRESSION FOR THE MOON'S PARALLAX. [30 Since the errors affecting Delaunay's coefficients of parallax are com- paratively large, it will be necessary to determine the factors by which these errors are multiplied in the equations of condition with a much greater degree of accuracy than is required in the case of the factors by which the errors of the coefficients of longitude and latitude are multiplied in the same equations. Otherwise, it will not be possible to deduce these last-mentioned errors from the equations with the requisite degree of pre- cision. It will be necessary to take special precautions in order to determine with accuracy the corrections of the assumed coefficients in the inequalities of longitude which have long periods. 31. KEMARKS ON MR STONE'S EXPLANATION OF THE LARGE AND IN- CREASING ERRORS OF HANSEN'S LUNAR TABLES BY MEANS OF A SUPPOSED CHANGE IN THE UNIT OF MEAN SOLAR TIME. [From the Monthly Notices of the Royal Astronomical Society, Vol. XLIV. (1883).] IN some recent communications to the Royal Astronomical Society Mr Stone contends that the mean solar day in use before 1864 when Le Verrier's Solar Tables were substituted for Bessel's in calculating the sidereal time at mean noon given in the Nautical Almanac differs from the mean solar day adopted since that time. In the Monthly Notices, Vol. XLIII. p. 403, Mr Stone states that the consequent error in our present reckoning in time is increasing at about the rate of l"'46 per annum, and in the same volume, p. 335, he adduces this supposed error in explanation of the increasing errors of Hansen's Lunar Tables. That this view of Mr Stone's is erroneous may, I think, be shewn by very simple considerations. The only mean Sun known to astronomers is an imaginary body which moves uniformly in the equator at such a rate that the difference between its Right Ascension and that of the true Sun consists wholly of periodic quantities. These periodic terms are due to the obliquity of the ecliptic, the eccentricity of the Earth's orbit, and also to the small perturbations of the Earth's motion about the Sun. 332 260 ON ME STONE'S EXPLANATION OF THE LARGE AND [31 The difference between the Right Ascensions of the two bodies at any moment is called the Equation of Time. The instant of Mean Noon is determined by the transit of this imaginary Mean Sun over the meridian of a given place just as the instant of Apparent Noon is determined by the transit of the true Sun over the same meridian. Hence, the mean time, according to the definition of it above given, may be determined by observation of the transit of the true Sun over the meridian, subject only to the small error to which all transit observations are liable, and also to the extremely small error which is possible in the theoretical expression for the equation of time. When this mode of deter- mining the mean time is employed,' no accumulation of error in proportion to the interval of time from a given epoch is possible. If, as it is frequently convenient to do, we wish to determine the mean solar time by means of the sidereal time supposed to be known, without having to make a transit observation of the Sun, we must employ the sidereal time at mean noon calculated from the proper formula or from the Solar Tables. This sidereal time at mean noon is equal to the Sun's mean longitude at mean noon corrected by the equation of the equinoxes in Right Ascension. In order to find the mean time correctly in this way it is necessary to employ the correct value of the Sun's mean longitude, and any error in the assumed value of this quantity will produce an equivalent error in the mean time deduced. Any such error can be at once checked and corrected by observation of the Sun's transit over the meridian. If we wilfully refuse to check our results by solar observations, the error in the determination of the mean time by means of the sidereal time would, no doubt, increase in proportion to the interval of time from a certain epoch. Practically, however, it would be intolerable to use Solar Tables which were grossly erroneous, and long before the error of time became important the tables would be replaced by more accurate ones. For many years previously to 1864 Bessel's formula had been employed in the Nautical Almanac for the calculation of the sidereal time at Green- wich mean noon. 31] INCREASING ERRORS OF HANSEN'S LUNAR TABLES. 261 In 1864 the error of Bessel's formula amounted to rather more than half a second of time, and accordingly in that and subsequent years the sidereal time at mean noon was deduced from Le Terrier's Solar Tables, which gave much more accurate results. Now it is contended by Mr Stone that by the change thus introduced into the Nautical Almanac the unit of mean solar time was practically altered to such a degree that at the end of 1881 the difference in the count of mean solar time amounted to nearly 27 seconds, and that the difference is increasing at the rate of about 1"46 seconds per annum. It is clear, therefore, that if no such change had been made in the Nautical Almanac that is, if Bessel's formula had continued to be employed no such change of the unit of time would have taken place. Let us see then, what difference this would have made in the count of mean solar time as derived from sidereal time when compared with the count found by means of our present Nautical Almanac. Bessel's formula for the sidereal time at Greenwich mean noon of Jan. 1 in any year is given in the prefaces to the Nautical Almanacs from 1834 to 1863 inclusive. In 1864 and subsequent years the sidereal time at Greenwich mean noon is derived from Le Verrier's tables. The following little table shews the sidereal time at Greenwich mean noon of Jan. 1 as calculated for every fifth year from 1860 to 1885 by Bessel's formula, and as taken from the several Nautical Almanacs : By Bessel's From Diff. Formula. Nautical Almanac. h. m. s. h. m. s. s. i860 18 41 28-87 18 41 28-87 Bessel's formulae employed o-oo i86 5 18 44 35-36 18 44 35-92 Le Verrier's Tables employed 0-56 1870 18 43 43-87 18 43 44-44 0-57 1875 18 42 54-47 18 42 55-06 0-59 1880 18 42 5-95 18 42 6-56 ?? 0-61 1885 18 45 1173 18 45 12-37 0-64 Hence we see that the difference of sidereal times at mean noon in consequence of the change from Bessel's formula to Le Verrier's Tables, which amounted to 8 '56 in 1865, had increased to 8 '64 in 1885. That is, the difference increases at the rate of 8 '08 in twenty years, or of 8 '02 in five years. 262 ON MR STONE'S EXPLANATION OF THE LARGE AND [31 But according to Mr Stone's theory as shewn in his tabular comparisons of mean solar times computed from sidereal times by means of the Nautical Almanac and of those sidereal times "corrected to agree with Bessel's sidereal times," the differences would be as follows: 1865 2-0 1875 16'6 1870 9-3 1880 23-9 and at the end of 1881 the difference would have increased to 26 8 '8 ; so that the increase in five years would be 7 8 '3 instead of 9 '02 as above. In fact the difference according to Mr Stone's theory is just 365 times as great as it should be. The origin of this enormous discrepancy between Mr Stone's theory and the fact is readily seen by considering that mean solar time is measured, not by the Sun's mean motion in longitude, as Mr Stone's theory supposes, but by the motion of the mean Sun in hour angle, which is about 365 times greater in amount. Hence any small error in the determination of the Sun's mean motion in longitude causes a proportionate error of only about a 365th part of the amount in the interval of mean solar time as inferred from the interval of sidereal time. In fact, if n denote the Sun's mean motion in longitude in a mean solar day, then the length of the mean solar day will be to the sidereal day in the ratio of 360 + n : 360. If now n + dn denote another slightly different determination of the Sun's mean motion in longitude in a mean solar day, the ratio of the length of a mean solar to that of a sidereal day will become 360 + n + dn : 360. Hence the measure of a mean solar day when expressed in sidereal time will be increased in the ratio of 360 + n + dn : 360 + n, dn Since 360 is nearly 365 times n, this ratio will be + 366 I dn , 6 in error, and therefore the mean solar time inferred by means of it from the sidereal time would be in error to the same amount. The mean longitude found from Le Verrier's Tables is much nearer to the truth, and therefore the mean solar time found from the sidereal time by using this value would be much more nearly correct. It must not be forgotten however that, as we have already stated, the mean solar time may be derived from observations of the transit of the Sun over the meridian, without employing the sidereal time at all. Apparent solar time, which is found directly from observation of the Sun is converted into mean solar time by applying the equation of tune, which is known from the solar theory, without reference to the sidereal time. 32. REMARKS ON SIR GEORGE AIRY'S NUMERICAL LUNAR THEORY. [From the Monthly Notices of the Royal Astronomical Society, Vol. XLVIII. (1888).] IN the Report of the Council on the subject of Sir George Airy's Numerical Lunar Theory, it has been explained that the large discordances which have been found by the author to result from the substitution of the values of the Moon's coordinates, as found by Delaunay, in the differ- ential equations of motion, are caused by the large errors of Delaunay 's coefficients of parallax, which Sir George has employed. It may be useful and not uninteresting to give on this subject some additional details. In the first place it will be well to prevent a possible misapprehension. In speaking of the errors of Delaunay's coefficients it is not intended to imply that there is any mistake in Delaunay's theory. The terms of the analytical expression for the Moon's parallax which Delaunay gives are all correct, but they only extend to the fifth order of small quantities, and are therefore not nearly precise enough to be used for the purpose to which the ex- pression for the parallax is applied by Sir George Airy. Delaunay intended this value of the parallax to be employed merely in reducing the apparent place of the Moon to its place as seen from the Earth's centre, and for this purpose the value is perhaps sufficiently accurate. If the several transformations of the elements given by Delaunay in his great work had been applied to the analytical expression for the reciprocal of the radius vector, and if Delaunay had carried the developments to the same extent as he had done in the case of the Moon's longitude and 32] KEMAEKS ON SIR GEORGE AIRY'S NUMERICAL LUNAR THEORY. 265 latitude, the theory would have been quite competent to give the third coordinate with the same degree of precision as had been attained in the case of the two other coordinates. The following table, which is reduced from the table given in pp. 398, 399, of Vol. XLIII. of the Monthly Notices, K A. S., shews the proportional values of the coefficients of parallax as found by me, mainly after Ponte"- coulant, when compared with those employed by Sir George Airy after Delaunay. My Coefficient. 10000000, 544989, 3 100236,0 82493, 6 29716,6 9029,0 5595, 6 4234, 3380, 7 - 2773, - 2770, - 2074, 6 1835, 1753,2 - 1168,8 675,0 894, 1 1087, 897, 05 Argt. I 2D-1 2D 21 2D + 1 2D-S 2D-1-S l-S D l + S 2f-l 31 W-l S 2D- W-21 2D-21 2D + 21 2D + 1-S 2D-2f 21-S 2D-31 D + l 21 + S 2D-2f-l 2D-2S 821,0 648,7 759,7 423,7 309,7 359,4 338,95 309,7 292,2 251,3 260, 05 Delaunay's. 10000000, 545145, 6 99822, 4 82329,2 29796, 4 8950, 8 5482, 2 4243, 1 3074, 5 - 2739, 9 - 2664, - 2068,25 1844, 7 1458, 2 - 1248,4 1107,0 957, 1 906,9 809,3 790,9 574,7 572,6 440, 3 319, 300, 9 295,7 283,7 267,9 238,4 222,0 A. 34 266 REMARKS ON SIR GEORGE AIRY'S NUMERICAL LUNAR THEORY. [32 Argt. My Coefficient. Delaunay's. 2D-2/+1 140,25 146,4 2D-1-2S 146,1 132,6 41 119,8 121,0 2D + 1 + S 137,3 117,2 4D-1-S 184,1 86,8 3D-1 105,2 58,1 93,5 54,05 51,2 51, W-21-S 87,7 46,5 3D 14,6 46,2 2D + 2f-2l 42,6 43, D + l + S 38,9 38,9 1-2S 38,3 38,3 2D-1 + 2S 37,4 37,4 2D-21-S 37,1 37,1 This table shews at a glance how great the errors of Delaunay's co- efficients of parallax, when reduced to the form in which they are employed by Sir George Airy, in many cases really are. Hence the discordances which he met with in the results of the substitutions should occasion no surprise. In the Introduction to the Numerical Lunar Theory, p. 4, line 20, it is stated through inadvertence that the factor which Sir George Airy calls M is a quantity "depending on the proportion of the masses of the Earth and Moon." This is not the case however, since M is simply the ratio of the sum of the actual masses of the Earth and Moon to the sum of the masses which would be required to make the Moon describe an undisturbed orbit about the Earth in which the periodic time and the mean parallax were the same as in the actual orbit. The theoretical value of M is simply expressed as the cube of the constant term in Delaunay's value of - . This value is given analytically in p. 802 or p. 914 of the second volume of Delaunay's Theory, but only to the fifth order of small quantities, which is not accurate enough. The development of the constant term of has been carried by me to a much greater extent at p. 472 of Vol. xxxvui. of the Monthly Notices (see p. 203 above). Turning this expression into numbers, and cubing it, we find the 32] EEMARKS ON SIR GEORGE AIRY'S NUMERICAL LUNAR THEORY. 267 value of M to be 1 '0027259, which agrees very closely with the value found by Sir George Airy by comparing the constant terms on the two sides of his equation (10). The other two ways of finding M proposed by Sir George in p. 76 of his Theory, viz. by comparing the quantities on the two sides of the equations (10) and (12), corresponding to the arguments 2 and 301 respec- tively, are not satisfactory, as the results will be affected by errors in the theoretical determinations of the mean motions of the Moon's perigee and node respectively. The multiplier M, representing the sum of the masses of the Earth and Moon, must be employed wherever the mutual attraction of these two bodies comes in question. In Sir George Airy's note at p. 254 of the March number of the Monthly Notices, he calls M the coefficient of the solar term, but this is plainly a mistake. I should mention that I have already communicated the substance of this paper to Sir George Airy himself. 342 33. ON THE METEORIC SHOWER OF NOVEMBER, 1866. [From the Proceedings of the Cambridge Philosophical Society. Vol. II.] THE author described the instrument used in the observation of the Meteors, and mentioned the various hypotheses which have been advanced concerning the orbit of these bodies ; he explained the calculations which he had made to determine this, and shewed that the attractions of the Earth, Jupiter, Saturn and Uranus were nearly sufficient to account for a hitherto unexplained change of about 29 minutes in the position of the nodes of the orbit in each period of 33 years. He called attention to the fact that the orbit calculated appeared to coincide very nearly with those of certain comets ; and held that the latter were elongated ellipses with a periodic time of 33 years. [The instrument consists of an axle which is mounted in all respects as the axle of a theodolite. To one end of the axle is fixed a graduated circle, as in the theodolite, which marks when the line of sight of the instrument is horizontal. To the other end of the axle and at right angles to it is a bar to which are attached a V-shaped piece of metal, a, and an eyepiece. On the eyepiece, about 3 in. from the eye towards the V is a thin bar, b, with a notch at its middle point, which can turn about the line in which the instrument is pointing. Attached to the thin bar is a circle divided to degrees, which marks when the bar is exactly parallel to the upper edge of the V with the notch downwards. The circle is provided with a vernier of 12 divisions, so that angles can be read to 5'. The point of the V is on the axis or line of sight about which the thin bar turns. The altitude and azimuth of any point in the line of sight can be read off on the vertical and horizontal circles of the instrument. When the instrument is directed to a meteor, the thin bar can be readily turned with its circle so as to coincide in direction with the apparent path of the meteor across the field of view.] 34. ON THE ORBIT OF THE NOVEMBER METEORS. [From the Monthly Notices of the Royal Astronomical Society. Vol. xxvu. (1867.)] IT is known to the President and to several members of the Society that I have been for some time past engaged in researches respecting the November meteors, and allusion is made to some of my earlier results in the last Annual Report. As my investigations are now in some measure complete, and the results which I have obtained appear to me important, I have thought that they may not be without interest for the Society. In a memoir on the November Star Showers, by Professor H. A. Newton, contained in Nos. Ill and 112 of The American Journal of Science and Arts, the author has collected and discussed the original accounts of 13 displays of the above phenomenon in years ranging from A.D. 902 to 1833. The following table exhibits the dates of these displays, and the Earth's longitude at each date, together with the same particulars for the shower of November last, which have been added for the sake of completeness. No. A.D. Day and hour. Earth's longitude. a. h. O / 1 902 Oct. 12 17 24 17 2 931 14 10 25 57 3 934 13 17 25 32 4 1 002 14 10 26 45 5 IIOI 16 17 30 2 6 1 202 18 14 32 25 7 1366 22 17 37 48 8 1533 24 14 41 12 9 1602 27 10 O.S. 44 19 10 1698 Nov. 8 17 N.S. 47 21 11 1799 11 21 50 2 12 1832 12 16 50 49 13 1833 12 22 50 49 14 1866 13 13 51 28 270 ON THE ORBIT OF THE NOVEMBER METEORS. [34 From these data Professor Newton infers that these displays recur in cycles of 3 3 '2 5 years, and that during a period of two or three years at the end of each cycle a meteoric shower may be expected. He concludes that the most natural explanation of these phenomena is, that the November Meteors belong to a system of small bodies describing an elliptic orbit about the Sun, and extending in the form of a stream along an arc of that orbit which is of such a length that the whole stream occupies about one-tenth or one-fifteenth of the periodic time in passing any particular point. He shews that in one year the group must describe either 1 1 1 2 00.0,;' r l aoZ>*> Or 33-25' "- 33-25' 33'25 revolutions, or, in other words, that the periodic time must be either ISO'O days, 185'4 days, 354'6 days, 376'6 days, or 33'25 years. It is seen that the time of the year at which the meteoric shower takes place becomes gradually later and later, and that accordingly the Earth's longitude at that time, or the longitude of the node of the orbit of the meteors, is gradually increasing. Professor Newton finds that the node has a mean motion of 102"'6 annually with respect to the Equinox, or of 5 2" '4 with respect to the fixed stars ; and he remarks that since the periodic time is limited to five possible values, each capable of an accurate determination, and since therefore from the position of the radiant point the other elements of the orbit can be found, it seems possible to compute the secular motion of the node for each periodic time with con- siderable accuracy, and the actual motion of the node being known, we have thus an apparently simple method of deciding which of the five periods is the correct one. Soon after the remarkable display of these meteors in November last, I undertook the examination of this question. From the position of the radiant point as observed by myself, I calculated the elements of the orbit of the meteors, starting with the supposition that the periodic time was 354 '6 days, the value which Professor Newton considered to be the most probable one. The orbit which corresponds to this period is very nearly circular, and it readily follows from the ordinary theory that the action of Venus would produce an annual increase of about 5" in the longitude of the node, and that of Jupiter an annual increase of about 6". The calculation of the motion of the node due to the Earth's action, presented greater difficulty in consequence of the two orbits nearly intersecting each other. I succeeded, however, in obtaining an approximate solution, applicable 34] ON THE ORBIT OF THE NOVEMBER METEORS. 271 to this case, from which it followed that the Earth's action would produce an annual increase of nearly 10" in the longitude of the node. Thus the three planets above mentioned which alone, in- the case supposed, sensibly affect the motion of the node, would cause a motion of about 21" annually, or nearly 12' in 33 '25 years. It has been already mentioned that the observed motion of the node is 52"'4 annually, or about 29' in 33 '25 years. Hence the observed motion of the node is totally irreconcilable with the supposition that the periodic time of the meteors about the Sun is 354'6 days. If the periodic time were supposed to be about 377 days, the calcu- lated motion of the node would differ very little from that in the case already considered, while, if the periodic time were a little greater or a little less than half a year, the calculated motion of the node would be still smaller. Hence, of the five possible periods indicated by Professor Newton, four are entirely incompatible with the observed motion of the node, and it only remains to examine whether the fifth period, viz. one of 33 "25 years, will give a motion of the node in accordance with observation. The calculations which have been above described were entirely founded on my own determination of the radiant point. In order to have as secure a basis as possible for the subsequent calculations, I adopted for the position of the radiant point the mean of my own and five other determinations, partly taken from published documents and partly privately communicated to me. These determinations are as follows, the several authorities being placed in alphabetical order : B. A. Deol. Adams 148 50 22 10 N. Baxendell 149 33 22 57 Briinnow 150 22 Challis 149 39 23 12 Herschel 148 9 23 48 Herschel, A. 149 24 Mean 149 12 23 1 N. Or with reference to the ecliptic, Long. 143 22' Lat. 9 51'N. Starting from this position of the radiant point, and the assumed period, and taking into account the action of the Earth on the meteors as they were approaching it, I obtained the following elliptic elements of their orbit: 272 ON THE ORBIT OF THE NOVEMBER METEORS. [34 Period 33'25 years (assumed) Mean distance 1CC3402 Eccentricity 0'9047 Perihelion distance 0'9855 Inclination 16 46' Longitude of Node 51 28 Distance of Perihelion from Node 6 51 Motion Retrograde In order to determine the secular motion of the node in this orbit, I employed the method given by Gauss in his beautiful investigation " Deter - minatio attractions, &c." It may be proved that if two planets revolve about the Sun in periodic times which are incommensurable with each other, the secular variations which either of these bodies produces in the elements of the orbit of the other would be the same as if the whole mass of the disturbing body had been distributed over its orbit in such a manner that the portion of the mass distributed over any given arc should be always proportional to the time which the body takes to describe that arc. In the memoir just referred to, Gauss shews how to determine the attraction of such an elliptic ring on a point in any given position. When this attraction has been calculated for any point in the orbit of the meteors, we can at once deduce the changes which it would produce in the elements of the orbit, while the meteors are describing any given small arc contiguous to the given point. Hence, by dividing the orbit of the meteors into a number of small portions, and summing up the changes corresponding to these portions, we may find the total secular changes of the elements produced in a complete period of the meteors. In this manner I have found that during a period of 3 3 '2 5 years, the longitude of the node is increased 20' by the action of Jupiter, nearly 7' by the action of Saturn, and about 1' by that of Uranus. The other planets produce scarcely any sensible effects, so that the entire calculated increase of the longitude of the node in the above-mentioned period is about 28'. As already stated, the observed increase of longitude in the same time is 29'. This remarkable accordance between the results of theory and obser- vation appears to me to leave no doubt as to the correctness of the period of 33-25 years. 34] ON THE ORBIT OF THE NOVEMBER METEORS. 273" In order to attain a sufficient degree of approximation it is requisite to break up the orbit of the meteors into a considerable number of portions, for each of which the attractions of the elliptic rings corresponding to the several disturbing planets have to be determined ; hence the calculations are necessarily very long, although I have devised a modification of Gauss's formulae which greatly facilitates their application to the present problem. In these numerical calculations I have been greatly aided by my assistants, more especially by Mr Graham. I am now engaged in obtaining a closer approximation by subdividing certain parts of the orbit of the meteors into still smaller portions, but the results which have been given above cannot be materially changed. Since I entered upon the foregoing investigation other astronomers have been led, on totally independent grounds, to conclusions which strongly confirm, and are confirmed by, those at which I have myself arrived. In the Bullettino Meteorologico dell' Osservatorio del Collegia Romano, Vol. v. Nos. 8, 10, 11, 12, are published four letters from Sig. Schiaparelli, Director of the Observatory of Milan, "Intorno al corso ed all' origine pro- babile delle Stelle Meteoriche." In these letters the author arrives at the conclusion that the orbits which the Meteors describe about the Sun are very elongated, like those of comets, and that probably both these classes of bodies originally come into our system from very distant regions of space. In his last letter, dated 31st Dec. 1866, Sig. Schiaparelli shews that if the August Meteors be supposed to describe a parabola, or a very elongated ellipse, the elements of their orbit calculated from the observed position of their radiant point, agree very closely with those of the orbit of Comet II. 1862, calculated by Dr Oppolzer. The following table exhibits this agreement : August Meteors. Comet II. 1862. Perihelion distance 0'9643 0'9626 Inclination 64 3' 66 25' Longitude of Perihelion 343 28 344 41 Longitude of Node 138 16 137 27 Direction of Motion ... Retrograde Retrograde Hence it appears probable that the great Comet of 1862 is a part of the same current of matter as that to which the August Meteors belong. In the letter which has just been referred to, Sig. Schiaparelli likewise gives approximate elements of the orbit of the November Meteors, calculated on the supposition that the period is 33 '25 years; but as the calculations A. 35 274 ON THE ORBIT OF THE NOVEMBER METEORS. [34 were founded on an imperfect determination of the radiant point, these elements were not sufficiently accurate, and Sig. Schiaparelli failed to find any cometary orbit which could be identified with that of the meteors. Soon after this, on the 21st January, 1867, M. Le Verrier communicated to the Academy of Sciences a theory of the origin and nature of shooting stars, very similar in its main features to that of Sig. Schiaparelli, and at the same time gave more accurate elements of the orbit of the November Meteors, his calculations being based on a better determination of the radiant point than that employed by the astronomer of Milan. In the Astronomische Nachrichten, of the 29th January, Mr C. F. W. Peters of Altona pointed out that the elements given by M. Le Verrier closely agreed with those of Tempel's Comet (I. 1866), calculated by Dr Oppolzer, and on the 2nd February, Sig. Schiaparelli, having recalculated the elements of the orbit of the meteors on better data than before, himself noticed the same agreement. Dr Oppolzer's elements of Tempel's comet are as follows : Period 33*18 years Mean distance 1-0'3248 Eccentricity 0'9054 Perihelion distance 0'9765 Inclination.... 17 18' Longitude of Node 51 26 Distance of Perihelion from Node 9 2 Direction of Motion Retrograde If these elements be compared with those of the November Meteors which I have given in a former part of this communication, it will be seen that their agreement is remarkably close. The curious and unexpected resemblance which is thus shewn to exist between the orbits of known comets and those of the meteors, both of August and November, opens a wide field for speculation. It is difficult to believe that the coincidences which have been noticed are merely acci- dental ; but whether or not we are disposed to adopt the ideas of Sig. Schiaparelli as to the intimate relations between meteors and comets, I cannot help thinking that my researches respecting the motion of the node of the November Meteors have settled the question as to the periodic time of these bodies beyond a doubt. 35. NOTE ON THE ELLIPTICITY OF MARS, AND ITS EFFECT ON THE MOTION OF THE SATELLITES. [From the Monthly Notices of the Royal Astronomical Society, Vol. XL. (1879).] ONE of the results of Professor Asaph Hall's able discussion of his observations of the satellites of Mars is to shew that the orbits of both the satellites are at present inclined at small angles to the plane of the planet's equator. It becomes an interesting question to inquire whether this state of things is a permanent one. The plane of Mars' orbit is inclined to its equator at an angle of 27 or 28. If then the planes of the orbits of the satellites retain constant inclinations to the orbit of the planet, as they would do if the Sun's disturbing force were the only force tending to alter those planes, their inclinations to the plane of Mars' equator, and still more their inclinations to each other, would in time become considerable. In No. 2280 of the Astronomische Nachrichten, Mr Marth has found the motions of the nodes of the orbits of the satellites on the orbit of the planet due to the Sun's action, and he concludes that, if there is no force depending on the internal structure of Mars which counteracts or greatly modifies the Sun's action, the nodes of the orbits will be in opposition to each other a thousand years hence, when the mutual inclination of the satellites' orbits will amount to about 49. In this case the near approach to coincidence between the planet's equator and the planes of the orbits of the satellites, which is observed 352 276 NOTE ON THE ELLIPTICITY OF MARS, AND ITS [35 to exist at the present time, would be merely fortuitous; but this appears a priori to be very improbable. It is well known that, if there were no external disturbing force, the ellipticity of a planet would cause the nodes of a satellite's orbit to retro- grade on the plane of the planet's equator, while the orbit would preserve a constant inclination to that plane. Laplace has shewn that, when both the action of the Sun and the ellipticity of the planet are taken into account, the orbit of the satellite will move so as to preserve a nearly constant inclination to a fixed plane passing through the intersection of the planet's equator with the plane of the planet's orbit, and lying between those planes, and that the nodes of the satellite's orbit will have a nearly uniform retrograde motion on the fixed plane. The angles which this fixed plane makes with the planes of the planet's equator and its orbit respec- tively will depend on the ratio between the rates of the above-mentioned retrogradations of the nodes produced by the Sun's action and by the ellipticity of the planet. If the latter of these causes would produce a much slower motion of the nodes than the former, as in the case of our Moon, the fixed plane will nearly coincide with the planet's orbit ; but if, as in the case of the inner satellites of Jupiter, the ellipticity of the planet would produce a much more rapid motion of the nodes than the Sun's action, then the fixed plane will nearly coincide with the planet's equator. The ratio of the motion of a satellite's node to that of the satellite itself, when the Sun's action is the disturbing force, varies, ceteris paribus, as the square of the satellite's periodic time, that is as the cube of its mean distance from the planet. On the other hand, the ratio of the same two motions, when the ellipticity of the planet is the disturbing cause, varies inversely as the square of the mean distance. Hence, for different satellites of the same planet, the motion of the nodes caused by the ellip- ticity will bear to the motion caused by the Sun's action the ratio of the inverse fifth powers of the mean distances. Now, the distance of the inner satellite of Mars from the planet's centre is only about 2% radii of the planet, a greater comparative proximity than is known to exist elsewhere in the Solar System, and the distance of the outer satellite from the same centre is only about 7 radii of the planet, while the periodic times of both are very small compared with the periodic time of Mars. Hence the effect of a given small ellipticity of Mars on the motion of the nodes of the satellites will be greatly magnified. It is true that the ellipticity of Mars is still unknown, and is pro- bably too small to be ever directly measureable; but we are not without 35] EFFECT ON THE MOTION OF THE SATELLITES. 277 means of determining, within not very wide limits, its probable amount, and we shall presently see that, in all probability, in the case of both the satellites the motion of the nodes produced by the ellipticity greatly exceeds the motion caused by the Sun's action, so that the fixed planes for both satellites are only slightly inclined to the planet's equator. From measures of the planet's diameter and of the greatest elongations of the satellites, combined with the known time of rotation of Mars and the periodic times of the satellites, it is found that the ratio of the centri- fugal force to gravity at Mars equator is about -%^-Q. Hence it follows that if the planet were homogeneous its ellipticity would be about XTT- If, instead of the planet being homogeneous, its internal density varied according to the same law as that of the Earth, so that the ellipticity would bear the same ratio to the above-mentioned ratio of centrifugal force to gravity at the equator as in the case of the Earth, then the ellipticity would be about ^g. In all probability the actual ellipticity of Mars lies between these limits. The following Table shews the annual motions of the nodes of the two satellites, caused by the Sun's action and by the planet's ellipticity respectively, for the above values of that ellipticity, and also for the ellip- ticity T ^g-, which has been deduced from Professor Kaiser's observations, although I have no doubt that this value is too great. The Table like- wise contains the corresponding inclinations of the fixed planes, so often mentioned above, to the planet's equator. Satellite I. Annual motion of the node due to the Sun's action, 0'06. Supposing ellipticity = 1 1 1 118 176 228 the annual motion of the node due to that ellipticity will be 333 182 113 Corresponding inclinations of fixed plane to planet's equator : 17" 31" 50" Satellite II. Annual motion of the node due to the Sun's action, 0'24. Supposing ellipticity = 1 1 1 118 176 228 the annual motion of the node due to that ellipticity will be 13-4 7'3 4"5 Corresponding inclinations of fixed plane to planet's equator : 27' 50' 1 19' 278 NOTE ON THE ELLIPTICITY OF MARS, ETC. [35 From this it may be inferred that the orbit of the 1st satellite pre- serves a constant inclination to a plane which is inclined less than 1' to the plane of Mars' equator, and that the orbit of the 2nd satellite preserves a constant inclination to a plane which is inclined about 1 to the plane of the same equator. The ellipticity will also cause rapid motions in the apses of the orbits of the satellites, particularly in that of the first ; and as this orbit appears from Professor Hall's determination to have a sensible eccentricity, it will be possible, by future observations, to determine the motion of the apse, and therefore the ellipticity of the planet. If further observations shew that the orbits of the satellites are sensibly inclined to then: fixed planes, the motion of their nodes will supply another means of determining the ellipticity of the planet. 36. NOTE ON WILLIAM BALL'S OBSERVATIONS OF SATURN. [From the Monthly Notices of the Royal Astronomical Society, Vol. XLIII. (1883).] IN No. 9 of Vol. i. of the Philosophical Transactions, a brief account is given of an observation of Saturn made on Oct. 13, 1665, at 6 o'clock, by William Ball, at Mamhead, near Exeter, and it is suggested that the appeai'ance presented by the planet may perhaps be caused by its being surrounded by two rings instead of one. This account has recently given rise to considerable discussion ; and there are some difficulties connected with it which do not appear to have been satisfactorily cleared up. In a few copies of the volume this account is illustrated by a figure, in which the external boundary of the ring, instead of being of a regular elliptical form, has two blunt notches or indentations at the extremities of the minor axis. The plate containing this figure, however, is wanting in by far the larger number of the copies. Now, I think, it may be safely asserted that no telescope, capable of shewing Saturn's ring at all, ever exhibited it in this extraordinary form, and therefore if the above figure faithfully represents William Ball's drawing, he was either a very inaccurate and careless observer, or he must have been provided with very inadequate instrumental means. On the other hand, we have ample proof that he was a careful and assiduous observer, that in particular he made a long series of observa- tions of Saturn, and that these were made with instruments not much inferior to those employed by Huyghens himself in similar observations. 280 NOTE ON WILLIAM BALL'S OBSERVATIONS OF SATURN. [36 It is well known that Huyghens's discovery of the true nature of the appendage to Saturn, which had so puzzled Galileo and others, was con- tested by Father Fabri at Rome, who wrote under the name of " Eustacius de Divinis." Huyghens replied to Fabri's objections in a tract which appeared in 1660, entitled Brevis Assertio Systematis Saturnii sui, and which is contained in the third volume of his collected works. In this tract he repeatedly appeals to Ball's observations in England in confirmation of his own. It is clear that Huyghens was in possession of drawings by Ball which represented the various appearances presented by the planet during the four years from 1656 to 1659 inclusive, and that he had carefully compared them with those which he had himself taken during the same interval. After mentioning the dark band which he had observed on the disk of Saturn at times when the remainder of the ring was invisible, he quotes a letter from Dr Wallis, dated Dec. 22, 1658, in which reference is made to an earlier letter dated May 29, 1656, wherein Dr Wallis had mentioned this band as having been observed by Ball, and had inquired whether his correspondent had likewise perceived it. Huyghens goes on to say that from Feb. 5, 1656, to July 2, when the planet appeared round and without ansse, this band or dark shading was observed by Ball to cross the centre of the disk, as shewn in his drawing, exactly as in Huyghens's own figure. Afterwards, when the ansse had re-appeared, the band was seen with more difficulty, and its position was less accurately laid down in Ball's drawing. From Nov. 5, 1656, to July 9, 1657, when the oblong arms of Saturn were seen apparently united to the disk, Ball gives a figure quite similar to that of Huyghens, except that he makes the arms a little thicker. Again, from Nov. 9, 1657, to June 7, 1658, when the arms were more open, Ball's figure is exactly similar to Huyghens's, except a slight difference in the position of the obscure zone or belt. Also, finally, the same remark applies to the figure of the planet from Jan. 3, 1659, to June 17 of the same year, when the ansse were a little more widely opened. Having made these comparisons between Ball's drawings of the planet and his own, Huyghens remarks that Ball was unacquainted with his hypo- thesis* (respecting the ring), and therefore could not be supposed to be * Huyghens's Systema Saturnium only appeared in 1659. 36] NOTE ON WILLIAM BALL'S OBSERVATIONS OP SATURN. 281 biased by it, while he himself would not dare to represent the phenomena otherwise than they really were, since, if he did, he might at once be contradicted by the English observer. This judgment of so competent an authority as Huyghens, made while he had before him all the materials for forming it, left no doubt on my mind as to the merit of Ball's observations. In order to see whether any further light could be thrown on the subject, I have recently taken an opportunity of consulting the MSS. pre- served in the archives of the Royal Society. Among them I find there is a letter in William Ball's own hand, dated April 14, 1666, in which he makes reference to his observations of Saturn, although the greater part of the letter relates to other subjects. He mentions that the observations were made partly with a telescope thirty- eight feet in length, having a double eye-glass, and partly with another telescope twelve feet in length. In the postscript to this letter he gives a small sketch of Saturn as it appeared at that time (1666), and he men- tions that the same appeai-ance was presented by the planet in 1664. In this figure the external boundary of the ring has the form of a regular oval, without any notches or other irregularities. No allusion is made to the very different appearance which, if the figure in the Philosophical Transactions is authentic, the planet must have presented in 1665. It should be understood that the paper in the Philosophical Transactions which is now in question was not written by Ball himself. It contains, however, a quotation from a letter of Ball to a friend (probably Sir R. Moray), and in what appears to be the last clause of this quotation, the figure is said to be "a little hollow above and below." I cannot help thinking that this clause has been added or altered in some way to correspond with the given figure. The letter of Ball on which this paper was founded is not in the archives; but there is preserved, not a drawing, but a paper- cutting, representing the planet and its ring, which is no doubt the original of the figure engraved in the Transactions. The defect in the paper-cutting probably originated in the following way. In order to make the cutting, the paper was first folded twice in directions at right angles to each other, so that only a quadrant of the ellipse had to be cut. The cut started rightly in a direction perpendicular to the major axis, but through want of care, when the cut reached the minor axis, its direction A. 36 -282 NOTE ON WILLIAM BALL'S OBSERVATIONS OF SATURN. [36 formed a slightly obtuse angle with that axis instead of being perpendicular to it. Consequently, when the paper was unfolded, shallow notches or depres- sions appeared at the extremities of the minor axis. I imagine that the account in the Philosophical Transactions was written by some one inexperienced in astronomical observations, who took for granted that the figure was correct. The mistake being soon discovered, the plate which contained the erroneous figure of Saturn, together with two other figures relating to different subjects, was cancelled, and thus its appearance in only a few of the copies is accounted for. The other figures on the cancelled plate were repeated in a new plate which accompanied No. 24 in the same volume of the Transactions. In Lowthorp's abridged edition of the Transactions the figure of Saturn has been corrected. I find no evidence that Ball, any more than Huyghens, had noticed any indication of a division in the ring. It may be interesting to give the original text of the passages of Huyghens's Brevis Assertio Systematic Saturnii sui, in which reference is made to Ball's observations. The citations are taken from the third volume of Huyghens's Opera Varia, edited by 'S Gravesande, and published at Leyden in 1724. " Credo et fasciam nigricantem in Saturni disco, liquido sibi conspici dixisset Eustacius, ni Fabric visum fuisset earn nimium hypothesi meae annulari favere. Cum autem ne option's quidem suis perspicillis earn cerni affirmet, hinc quoque quanto ilia meis deteriora sint perspicuum sit. Nam ne mihi phenomenon illud confictum credatur, idem et in Anglia pridem observari coepisse sciendum est ; et liquet ex literis viri clar. Job. Wallisii, Oxonia ad me datis 22 Dec. 1658, quibus inter alia hsec scribit. Monebam etiam iisdem literis (nempe datis 29 Maji 1656) de Saturni fascia quam jam ante observaverat D. Ball, et sciscitabar num tu eandem conspexeras, &c. Earn porro fasciam a 5 Feb. 1656 ad 2 Jul., quo tempore rotundus Saturnus absque ansis apparuit, medium planetse discum secare D. Ball adnotavit, ut in schemate ad me misso expressa est. Atque ita mihi quoque fuerat eo tempore observata, ut cernitur pag. 544 Systematis Saturnii, quam figuram hie repeto. Postmodum tamen renatis Saturni ansis cum difficillime conspici eadem fascia ccepisset, minus recte quoque a D. Ball, quantum ad situm attinet, depicta est. At in mearum observationum adversariis, die 26 Nov. 36] NOTE ON WILLIAM BALL'S OBSERVATIONS OF SATURN. 283 1656, et alias adscriptum invenio, lineam obscuram fuisse evidentissimam, eo nempe positu, qui pag. 545 System. Satumii memoratur." Pp. 624, 625. "Non segre nunc fidem habitum iri spero, turn mihi turn Anglis simul observatoribus, qui anno 1657 oblonga Saturni brachia disco utrinque con- juncta spectavimus, qualia exhibet figura Systematis mei pag. 545, quam hie repono; non autem binorum orbiculorum forma a medio disco disjunctorum, ut Eustacius se ilia eodem tempore vidisse dejerat. Adderem hie schema quod mihi a D. Ball, supra memorato, advenit, nisi plane simile esset huic nostro, hoc uno tantillum duntaxat abludens, quod brachia ilia ubique paulo crassiora ille referat. "Earn vero formam a 5 Nov. 1656 ad 9 Jul. 1657 sibi apparuisse scribit. Apertis autem brachiis, qualis pag. 547 Systematis mei et hie repre- sentatur, talem a 9 Nov. 1657 ad 7 Jun. 1658, idem observator depingit,. simillima prorsus figura, nisi quod ad positum zonse obscurse attinet, de quo dixi supra. Ac denique h, 3 Jan. 1659 ad 17 Jun. ejusdem anni, ansis paulo latius adhuc apertis. Et hsec quidem ille, ignarus adhuc mese hypo- theseos, ne ob prseconceptam opinionem aliquid indulsisse sibi existimetur. Neque ego aliter quam se revera habent referre auderem, cum redarguere me, si fallam, autori observationum in promptu sit." P. 626. The following extract comprises all that is material in the Paper in the Philosophical Transactions: " This observation was made by Mr William Ball, accompanied by his brother, Dr Ball, October 13, 1665 at six of the Clock, at Mainhead [Mamhead] near Exeter in Devonshire, with a very good Telescope near 38 foot long, and a double Eye-jglass as the observer himself takes notice, adding, that he never saw that planet more distinct. The observation is represented by Fig. 3 concerning which, the Author saith in his letter to a friend, as follows, This appear'd to me the present figure of Saturn, somewhat other- wise, than I expected, thinking it would have been decreasing, but I found it full as ever, and a little hollow above and below. Whereupon the Person, to whom notice was sent hereof, examining this shape, hath by letters desired the worthy Author of the System of this Planet, that he would now attentively consider the present Figure of his Anses, or Ring, to see whether the appearance be to him, as in this Figure, and conse- quently whether he there meets with nothing that may make him think, that it is not one body of a circular Figure, that embraces his Disk, but two." 362 284 NOTE ON WILLIAM BALL'S OBSERVATIONS OF SATURN. [36 From this it is clear that the suggestion of two rings was made, not by Ball himself, but by his anonymous correspondent. By the kind permission of the President and Council of the Royal Society, I am enabled to make the following extracts from two letters in William Ball's own hand, and likewise to give exact representations of the form of the paper- cutting, and of Ball's small sketch of Saturn, referred to in the foregoing Paper, both of which have been kindly copied for me by our Assistant-Secretary, Mr Wesley. The annexed figure shews the form of the paper-cutting. The writing on the cutting appears to be in Oldenburg's hand. The first letter is dated Mamhead, April 14, 1666, and is probably addressed to Oldenburg. "I have seen J? two mornings this year (with a 12 foot glasse the longest I can use at this time with convenience) and find the figure the same as it was in -64. What his figure was last autumne (by mee observed with 38 foot glasse much better than that at Gresham Colledge) I suppose S r . R. Moray hath communicated. I could not have a second sight, straining very much for that one, for the shadow of the body on his ring I doe not well understand the meaning but I suppose I saw the same thing; for I never had a clearer sight of him in any glasse I ever looked in, one thing I can boast of, sc. I am not prejudiced with any conceit of hypothesis which doth commonly send all observations to favour one side and soe there must bee a little added or diminished as the designe requires," &c. &c. 36] NOTE ON WILLIAM BALL'S OBSERVATIONS OF SATURN. 285 In a postscript is the following, with the little sketch : "I saw t? this morn, at 4 a clock with 12 foot glasse and judge him the same figure as in -64 that is just ovall with two black spotts and I thinke a faint shadow of a belt which I have alwaies scene, but will not be peremptory in itt." The second letter is dated "Mamhead i? September 15, -66," and is addressed "For Sir Robert Moray K' at Whitehall, These." " I designe to send you all the figures of J? . I promised them my L d Brounker and hee was pleased most kindly to accept itt but I (like any thing you please to call mee bad enough) have hitherto shamfully failed, as alsoe of an account of husbandry to Mr Oldenburg. I am still gazing at the starrs though to very little purpose more then to keep my eyes in use," &c. &c. It will be noticed that the passage in Ball's first letter in which he claims to be unbiased by any hypothesis, agrees with the statement of Huyghens respecting him. The passage in the same letter, " for the shadow of the body on his ring I doe not well understand the meaning but I suppose I saw the same thing," I conjecture to refer to an attempted explanation by Huyghens, or some other astronomer, of the phenomenon observed by Ball, by attri- buting it to the shadow of the body of the planet cast on his ring. It is plain that such an explanation would not be applicable, if similar depressions had been observed at the two extremities of the minor axis of the ring. 37. ON THE CHANGE IN THE ADOPTED UNIT OF TIME. [From the Monthly Notices of the Royal Astronomical Society, Vol. XLiv. (1884).] THE December number of the Monthly Notices contains a paper by Major-General Tennant in which the author arrives at conclusions which appear to him to confirm Mr Stone's views respecting a change in the unit of mean solar time. In reality, however, those conclusions are quite consistent with my own as given in the same number of the Monthly Notices, (see p. 259 above) and not at all with Mr Stone's. According to Major-General Tennant (Monthly Notices, p. 43), the factor by which the tabular mean motions should be multiplied in consequence of the change from Bessel's to Le Verrier's determination of the ratio of the mean solar to the sidereal day is what he calls Sidereal Seconds in Le Verrian Mean Day Sidereal Seconds in Besselian Mean Day Now, if n be the Sun's mean motion in a mean solar day as deter- mined by Bessel, the sidereal seconds in a mean solar day will be But if n + Sn be the Sun's mean motion in a mean solar day as determined by Le Verrier, the sidereal seconds in a mean solar day will be 86400 x 360 37] ON THE CHANGE IN THE ADOPTED UNIT OP TIME. 287 and therefore the factor above referred to by Major-General Tennant will be Sn 360+n' whereas, according to Mr Stone's views, this factor should be n + Sn _ 8n 1 H -- > n n where the difference from 1 is nearly 366 times greater than it should be. The same thing may be otherwise shewn thus : If N denote the number of mean solar days in a mean tropical year, according to Bessel's determination, then N + 1 will be the corresponding number of sidereal days in the same interval. Consequently, the ratio of the length of a mean solar to that of a sidereal day will be N N- But if N + &N denote the number of mean solar days in a mean tropical year, according to Le Verrier's determination, then N+8N+1 will be the corresponding number of sidereal days in the same interval. And consequently the above-mentioned ratio will become 1 N+8N Hence the ratio of the length of a mean solar to that of a sidereal day will be changed in the ratio of whereas, according to Mr Stone, the ratio which measures this change would be N where, as before, the difference from 1 is nearly 366 times too great. 288 ON THE CHANGE IN THE ADOPTED UNIT OF TIME. [37 Mr Stone's error appears to arise from his equating two things which are really different, and which are inconsistent with each other, viz. Bessel's and Le Verrier's determinations of the Sun's mean motion in longitude in the same interval of time. Major-General Tennant is wrong in supposing that solar observations are no longer employed in Observatories for the determination of mean solar time. If this were the case, it would only shew that the Observatories had taken a very retrograde step, since the final test whether the mean solar times have been correctly found can only be supplied by solar obser- vations. Whenever the mean solar times are deduced from the observed sidereal times, it is tacitly assumed that the tabular mean longitudes of the Sun which have been employed are correct; and if this is not the case, the mean solar times deduced will require a corresponding correction, which can only be found by solar observations. Thus mean solar time may be determined with reference to a natural phenomenon, viz. the transit of the true Sun over the meridian of a given place ; and the mean solar day is the average of all the apparent solar days defined as the intervals between two successive transits, and therefore has nothing arbitrary about it. To speak of Besselian mean time and Le Verrian mean time, or of the Besselian mean solar day and the Le Verrian mean solar day, can produce nothing but confusion in our ideas of the measure of time. 38. ON NEWTON'S SOLUTION OF KEPLER'S PROBLEM. [From the Monthly Notices of the Royal Astronomical Society, Vol. XLIII. (1882).] OF all the methods which have been proposed for the solution of this problem, that which leads most rapidly to a result having any required degree of precision may be briefly explained as follows : The equation to be solved by successive approximations is x e sin x = z, where z is the known mean anomaly, e the eccentricity, and x the eccentric anomaly to be determined. Suppose x a to be an approximate value of x, found whether by esti- mation, by graphical construction, or by a previous rough calculation, and let x a e sin cc = z . Then if 8^=, Z ~ 2 1 e cos x a and x r = x + 8x , x f will be a much more approximate value of x than x,,. Similarly, if we put xf e sin xf = z', and if - ' z ~ z ' 1 e cos x 1 A. 37 290 ON NEWTON'S SOLUTION OF KEPLER'S PROBLEM. [38 and x" = x! + Sx 1 , x" will be a much more approximate value of x than a/; and so on, to any required degree of approximation. If the error of the assumed value a? be supposed to be of the order i, when e is taken as a small quantity of the first order, then the error of the value x 1 will be of the order 2i + 1 = i' suppose, similarly the error of the value x" will be of the order 2i' + I = 4i 4- 3, and so on, so that the order of the error is more than doubled at each successive approximation. The above explains the immense advantage of this process over the use of series proceeding according to powers of e, when great precision is required in the result; since, in this latter method, the addition of a new term only increases the order of the error by unity. The degree of rapidity of the approximation may be still further increased by the following slight modification of the above process. Starting, as before, with the value x , and calling z z = Sz ot we should obtain a much more accurate value than before of the correction Sx^ to be applied to x,, by putting 1 e cos x a + x 1 e cos Now, e being supposed to be small, 8z is an approximate value of 8a; and may be written for it in the small term in the denominator. Hence, if we put X == XQ ~T~ OX$ , x' will be a nearer approximation to the true value of x than was obtained before by the corresponding operation. Similarly, if x' e sin x 1 = z', and z z' = 8z', Sz 7 and if Sa/ = . , -, /\, 1 e cos (x + % Sz ) then x" = x' + Sx' will be the next approximate value of x, and the process may be continued as far as we please. 38] ON NEWTON'S SOLUTION OF KEPLER'S PROBLEM. 291 If the error of x be of the order i, that of x' will now be of the order 2i + 2, that of x" will be of the order 2 (2i + 2) + 2 = 4i + 6, and so on, so that the degree of rapidity of the approximation is still greater than before. If we chose to take the mean anomaly itself as the first approximate value of the eccentric anomaly that is, if we put we should have z = z e sin z, and the value of Sx given by the first method would be rj e sin z 1 e cos z ' while that given by the second and more accurate method would be j, e sin z 1 ecos(z + e sinz)' and the error of x' = x + 8x would be of the 3rd order in the former case, and of the 4th order in the latter. In practice, however, a much nearer first approximate value of x may be always found by inspection, and of course the smaller the error of this value is, the more rapid will be the rate of the subsequent approximations. The methods above explained have been long known. The first method is given at p. 41 of Thomas Simpson's Essays on Several Subjects in Speculative and Mixed Mathematics, published in 1740; and Gauss' method given at pp. 10 12 of the Theoria Motus, published in 1809, is essentially the same. The second method, or rather the modification of the first, is given by Cagnoli in his Trigonometrie, at pp. 377, 378 of the first edition, published in 1786, and at pp. 418 420 of the second edition, published in 1808. Now, my object in the present note is to point out that the first method explained above is exactly equivalent to that given by Newton in the Principia, at pp. 101, 102 of the second edition, and at pp. 109, 110 of the third edition, when Newton's expressions are put into the modern analytical form. 372 292 ON NEWTON'S SOLUTION OF KEPLER'S PROBLEM. [38 None of the subsequent authors, however, mentions this method as being Newton's, the unusual form in which Newton's solution is given having, no doubt, caused them to overlook it. In the first edition of the Principia a modification of the method is given which was, I have no doubt, intended by Newton to be equivalent to the second method given above ; but by some inadvertence, instead of the denominator of See' being 1 e cos ( x' + ^ 8z' ' when expressed in the above notation, he takes it to be what is equiva- lent to 1 e cos \x' + ^ e s i n \ which is only true for the first approximation when x a is taken = z. In the second and third editions this error is corrected, but Newton contents himself with the more simple expression given by the first method. We need not be surprised that Newton should have employed this method of solving the transcendental equation x e sin x = z, since the method is identical in principle with his well-known method of approximation to the roots of algebraic equations. For convenience of calculation, the approximate values x a , x', x", &c., should be so chosen that their sines may be taken directly from the tables without interpolation ; and, since each approximation is independent of the preceding ones, this may always be done if x' be taken equal, not to x + 8x,, itself, but to the angle nearest to x + 8x which is contained in the tables, and if similarly x" be taken equal to the tabular angle which is nearest to x' + 8x r , and so on. In the first approximation it will be amply sufficient to use 5 -figure logarithms, but in the subsequent ones tables with a larger number of decimal places should be employed. A first approximate value of the eccentric anomaly corresponding to any given mean anomaly may be found by a very simple graphical con- struction, provided we have traced, once for all, a curve in which the ordinates are proportional to the sines of the angles represented on any given scale by the abscissae. 38] ON NEWTON'S SOLUTION OF KEPLER'S PROBLEM. 293 This curve is commonly called "the curve of sines." It will be suf- ficient to trace the portion of the curve for which the ordinates are positive. Let A OB be the line of abscissa? , and let AO be taken equal to OB, and let each of them be divided into 90 equal parts representing degrees of angle. Let AN be any abscissa representing the angle x, and let the corresponding ordinate NP = c sin x ; then the greatest ordinate will be OC=c, corresponding to the abscissa AO. Suppose the curve line APCB to be divided into 180 parts which correspond to equal divisions on the line of abscissae ANOB. Then if E be taken in AO so that ^0 = 6x57-296 divisions, or if AE 90 e x 57 '296 divisions, and if CE be joined and PM be drawn parallel to it through P meeting the line of abscissae in M, then AM will represent the mean anomaly corresponding to the eccentric anomaly repre- sented by AN. For, since the triangles PMN, CEO are similar, MN PN EO ~ CO = X ' and therefore MN = EO sin x = 57 '296 (e sin x). Hence MN represents the number of degrees in x z, and therefore AM represents the mean anomaly z. Conversely, if AM represents any given mean anomaly, then if MP be drawn parallel to EC, it will cut the curve in the point P corresponding to the eccentric anomaly. By the employment of a parallel ruler we may find the eccentric anomaly corresponding to any given mean anomaly, or conversely, without actually 294 ON NEWTON'S SOLUTION OF KEPLER'S PROBLEM. [38 drawing a line. For if we lay an edge of the ruler across the points EC and then make a parallel edge to pass through the point M it will cut the curve in the point P required. Thus we may always find a first approximate value of the eccentric anomaly, without making repeated trials, whether the eccentricity be large or small. I described this graphical method of solving Kepler's problem at the Birmingham meeting of the British Association in 1849. It is referred to in a paper by Mr Proctor in Vol. xxxiu. of the Monthly Notices, p. 390. The construction is so simple that it has probably been proposed before, though I have nowhere met with it. Note on Professor Zenger's solution of the same problem given in Number 9 of Vol. XLII. of the " Monthly Notices." The only peculiarity in this solution is in the mode of obtaining the first approximate value employed. The subsequent approximations are carried on by means of the first method given above. Professor Zenger's process may be represented in a slightly different form as follows : We have x z = e sin x, and therefore sin (x z) = sin (e sin x) = e sin x \ 1 - e 2 sin 2 x + r e* sin 4 x etc. > , I o 1.ZU or sin (x z) =f sin x ; where /= e \ 1 - e 2 sin 2 x + e 4 sin 4 x etc. > . ^ D 1 \J j , . fsinz Hence tan (x - z) = - J f. . 1-/COSZ Now, an approximate value of f is e, and the error in the determi- nation of tan (x z) if we were to put , x e sin z tan(cc z)= . 1 e cos z would be of the 3rd order in e. 38] ON ZENGER'S SOLUTION OF KEPLER'S PROBLEM. 295 If we determine / so that the error in the determination of x shall vanish when _ "" we shall have and the approximate equation for finding x z becomes sin e sin z i \ ai tan (x z) = - sin e cos z The error still remains in general of the 3rd order in e, but the maximum error will be smaller than when f is taken = e. The value of x given by this equation is readily seen to be equivalent to that given by Professor Zenger's equation, e cosec z cot x = cot z 1 3 1 + -7. sin 2 e + sin 4 e + etc. 6 40 where we may remark that the quantity 1 1 3 1 + - sin 2 e + sin 4 e + etc. 6 40 sine 1 1 , orto l-^+ e ~ etc " is equivalent to a series which converges much more rapidly than the series for its reciprocal, employed by Professor Zenger. A still more advantageous result may, however, be obtained by deter- mining f so that the error may vanish both when 7T /' "3' and when x = , 3 that is when sinx= = 296 ON ZENGER'S SOLUTION OF KEPLER'S PROBLEM. [38 so that f =el ~ e * + et ~' 6tC The order of accuracy of the approximation will not be altered by confining ourselves to the first two terms of this value of f, so that we may take e ( 1 - e~ } sin z tan (a; -2) = *- r , nearly. 1 e[ 1 ^e 2 ) cosz The error is still of the 3rd order, but its maximum amount is less than before. If /be taken =e -jl --e 2 sin a zl , , x / sin z and the error in the determination of tan(cc z), and therefore in the determi- nation of x, will be only of the 4th order. There are several misprints and some errors of calculation in Professor Zenger's paper, on which I need not dwell. True anomaly in line 8 of the paper should be eccentric anomaly, and the same error occurs on p. 448. 39. NOTE ON DR MORRISON'S PAPER (ON KEPLER'S PROBLEM). [From the Monthly Notices of the Royal Astronomical Society, Vol. XLIII. (1883).] THE reference to Hansen's paper should be made to Abhandlungen der Sdchsischen Gesellschaft der Wissenschaften, Band iv. p. 249, instead of to Band n. as stated by Dr Morrison. In this paper Hansen's object is not merely to express the coefficients of the series which gives the eccentric anomaly in powers of e, otherwise this might have been done much more simply in the following manner. Calling g the mean, and x the eccentric anomaly, we have g = x e sin x, or x=g + e sinx, which is in the proper form for the application of Lagrange's theorem for developing x or any function of x in terms of g and ascending powers of e. Hence we have . 5 . 7 2 10 . 5 . 7 And in his Astronomy, 1814, vol. n. p. 50, In the coefficient of cos 7g, 355081 , 355081 instead of rf read 40. ON NEWTON'S THEORY OF ASTRONOMICAL REFRACTION, AND ON HIS EXPLANATION OF THE MOTION OF THE MOON'S APOGEE. [British Association Report (1884), p. 645.] 41. ON THE GENERAL VALUES OF THE OBLIQUITY OF THE ECLIPTIC, AND OF THE PRECESSION AND INCLINATION OF THE EQUATOR TO THE INVARIABLE PLANE, TAKING INTO ACCOUNT TERMS OF THE SECOND ORDER*. [From The Observatory, No. 109 (1886).] IF we adopt the values of the precession and nutation employed by Peters in his classical work Numerus Constans Nutationis, I find that the ratio of the sum of the masses of the Earth and Moon to the mass of the Moon is that of 82' 834 to 1, a result which differs slightly from that found by Peters from the same data. The amount of precession caused by the Sun's action depends in a slight degree on the eccentricity of the Earth's orbit. In order to find the precession for an indefinite period, it will be proper to employ the mean value of the square of this eccentricity instead of the value of this quantity at the present time. Taking this circumstance into account, and also introducing the small correction of the coefficient of precession which depends on the square of the coefficient of nutation, I find that if , ON' = <]>', being a fixed point, 6 and & the inclination of the equator and ecliptic respectively to the fixed plane, and cos X = sin 6 cos & cos 6 sin & cos ( '), sin (a sin X = sin & sin (tf> <'), which give CD and X when 6 and are known. From the instantaneous motion of the equator with reference to the ecliptic at time t, supposed for an instant to be fixed, it is easily seen that we have I dt sin -n cos (a sin o> cos X, (7 d0 dt c cos cu sin 01 sin X, 41] ON THE GENERAL VALUES OF THE OBLIQUITY OF THE ECLIPTIC. 305 or, substituting from above for cos w, sin <')} {sin 9 cos 6 tan 0' cos (< - <')}, etc sin c/ ^ = c cos 2 0' {cos e + sin tan & cos (< - 9')} tan ff sin (< - 9'), which are the differential equations for determining 6 and , 0' and (' being supposed to be already known in terms of t. From the above we may deduce the following : d /cos k-g t ' *-1t a ti = - fa (a/ - 1 ) tan h - fat? cot h ; * {(a/ + a/ - 2) tan h + (a, + a,) cot h} ; = - -- ** ~~ 9i ~ 9j a' - \ <-2 [a? - 2a { -a/ + 2a,} tan h + $ - . (a, - a,) cot h. 9i~9j "< W Also & f = a< (a< 1 ) tan h a t cot A ; 1.. = i ai 3 (a, - I) 2 tan 2 h + %a { (a/ + a, - 1) + Ja/ cot 2 h ; r - K+ a - ! +/ + , 1 -,,- ~ ~ k k - 'ij tan h + % - - {a< (a { - 1 ) + a,- (a^ - 1 )} tan 2 h ~ 9i~9i k or the value of this last coefficient may be otherwise expressed thus Also the value of w, the obliquity of the ecliptic, is thus expressed in terms of the same quantities : w = h + 2 (a f - 1 ) y ( cos {(A - gr ( ) + a - /8 Z -} [ - %a t (a t - 1 ) 2 tan h-% (n { - I ) 2 cot h] y/ cos 2 {(A -<&)* + a - A} cot A + a / y) tan h + % (a f + a,- 1 ) cot h x y{fi cos {(2k - g t - g,) t + 41] ON THE GENERAL VALUES OF THE OBLIQUITY OP THE ECLIPTIC. 307 k k ;; r7 ( a < ~ a ;) ( a * + a > - 2) ton A + - - (a t - a } ) cot h ~ ~ - ^ (a/ + a? ttj ctj) tan h -| (a { + a } I ) cot h X yc/j cos {(& - (Jj) Also the value of k in terms of the constant c which, as stated before, is known from the theory of precession is *= -c cos h {1 -2 J (,--!) (3a,- 5) y/} ; A and a are the arbitrary constants which enter into the complete integrals of our equations, and they are determined so as to make the initial values of 6 and , or those of 01 and tf>, equal to the ob- served values. It is to be remarked that one of the values of g is 0, and if the invariable plane of the system be taken as the fixed plane of reference, the corresponding value of y will be also zero, so that the expressions for 6, * ^ % & O t- ~^ / ^...-17"'6266 -18"'9365 -0"'66166 ft... 300 1' 254 43' 20 31' logy,.... 7-59939 8'41184 7'12320 392 308 ON THE GENERAL VALUES OF THE OBLIQUITY OF THE ECLIPTIC. [41 where the quantities g t are expressed in seconds and have reference to a Julian year as the unit of time, and the quantities y t are expressed in circular measure. Now in the figure before given the point N' is the descending node of the invariable plane on the ecliptic of 1850, so that the longitude of N' is 286 14' 18". Also the longitude of the point E, which is the autumnal equinox, is 180. Hence N'E = 253 45' 42". Whence we may find for 1850: 0= 23 3' 43" -(f>' = 257 20 31 or < = 183 34 49 Also, according to Stockwell, the obliquity of the ecliptic in 1850 was w = 23 27' 31"'0. Hence by repeated approximation we may find : h= 23 18' 54" nearly a=177 25 52 also &=-50"'4607 whence by substitution all the terms in 0, <, and w may be found numerically. ADDITION. If we wish to take into account the variability of the eccentricity of the Earth's orbit, the value of k should be taken = 50"'4548 + 24"'034 (e 2 -e 2 ), and the quantity Jet in the above formulae should be replaced by 50"'4548 t+ J24"-034 (e 2 -e 2 )c&. Where e is the eccentricity of the Earth's orbit at time t, and e 2 the mean value of the square of the eccentricity, which, according to Stockwell's determination, is = 0009864. 42. ADDRESS ON PRESENTING THE GOLD MEDAL OF THE ROYAL ASTRO- NOMICAL SOCIETY TO M. PETERS. [From the Memoirs of the Royal Astronomical Society. Vol. XXI. (1852).] IT has already been announced to you that the medal of the Society has been awarded to M. Peters, for his two papers, entitled, "Numerus Constans Nutationis ex Ascensionibus Rectis Stellse Polaris in Specula Dorpatensi Annis 1822 ad 1838 observatis deductus," and "Recherches sur la Parallaxe des Etoiles Fixes," which are published respectively in the third and fifth volumes of the sixth series of the Mathematical and Physical Transactions of the Imperial Academy of Sciences of St Petersburg; and it is now my duty to explain to you the grounds of this award, which (unless their effect be marred by my very imperfect statement of them) will, I doubt not, secure your approval. These papers form part of a series emanating from the astronomers of the Pulkowa Observatory, and having for their object the advancement of sidereal astronomy ; first, by a new and more accurate determination of the elements which affect the apparent places of all the stars, such as precession, nutation, and aberration ; and, secondly, by an examination of the peculiarities affecting individual stars, such as annual parallax and proper motion, by which alone we can gain a knowledge of the scale on which the visible universe is constructed, and of the arrangement in space and of the relative motions of the bodies of which it is composed. 310 ADDRESS ON PRESENTING THE GOLD MEDAL OF THE [42 These important objects have been steadily pursued at the Pulkowa Observatory, under the guiding mind of its illustrious director, with an energy and success which have placed that establishment in a position with respect to sidereal astronomy, similar to that which our own obser- vatory of Greenwich occupies with respect to the observation of the Moon. The order of date, as well as the nature of the subjects treated of, leads me first to speak of M. Peters' paper on the constant of nutation. But before proceeding to give an account of the paper itself, it may not be out of place to advert rapidly to former researches respecting nutation. When Newton traced the precession of the equinoxes to its cause in the attraction of the Sun and Moon on the protuberant equatoreal zone of the terrestrial spheroid, he perceived that the Sun's action would likewise cause a nutation of the Earth's axis, the period of which is half a year. He contents himself with remarking that this nutation can be scarcely sensible. In the same way, of course, the Moon's action produces a small nutation, of which the period is half a month. Abstracting these nutations, the tendency of the Sun's action is to make the pole of the equator move in a circular arc about the pole of the ecliptic; and in a similar manner the Moon's action tends to make the pole of the equator describe a circular arc about the pole of the Moon's orbit for the time being. Now, as this latter pole moves in a circle about the pole of the ecliptic in a period of about nineteen years, it is easy to see that this will give rise to an inequality in the rate of precession, and to a change of the obliquity of the ecliptic, having the same period. It is curious, however, that Newton does not allude at all to this, which constitutes by far the most important part of nutation ; and this is the more remarkable, since the principles which he lays down in treating of precession are quite sufficient to obtain, by means of very simple geometrical reasoning, not only the law, but very approximately, the co- efficients of the inequalities in the precession and obliquity due to this cause. The state of practical astronomy, however, in Newton's time, was not sufficiently advanced to induce him to enter more fully into this subject; and it was, consequently, reserved for the immortal discoverer of aberration to detect these motions of the Earth's axis by means of his observations, and then to trace them to their true cause. While discussing the obser- vations which led him to the discovery of aberration, Bradley noticed that the annual changes of declination of the stars did not exactly correspond 42] ROYAL ASTRONOMICAL SOCIETY TO M. PETERS. 311 with those which would be occasioned by precession, and he made allowance for this by employing in the reduction of his observations the changes deduced from the observations themselves. No sooner, therefore, had Bradley determined the law and the cause of aberration, than a new subject of investigation presented itself, requiring a much longer course of observations for its complete examination. Com- paring his observations of different stars, he found that their changes of declination were such as might be attributed to a real motion of the Earth's axis, and he was not slow in perceiving that the varying action of the Moon upon the equatoreal parts of the Earth, according to the different positions of the nodes of the lunar orbit, was the probable cause of this motion. During the course of the observations, Bradley communi- cated what he had observed to Machin, who was then " employed in considering the theory of gravity and its consequences with regard to the celestial motions," mentioning at the same time what he suspected to be the cause of these phenomena. Machin confirmed this supposition, and shewed that the observed motions might be very nearly accounted for, by supposing that the pole of the equator described a small circle about its mean position as centre, during a period of the Moon's nodes. Bradley remarked that his observations would be more completely represented by supposing the true pole to move about the mean pole in an ellipse instead of in a circle, the major axis being in the solstitial colure ; and this conclusion is perfectly true, the minor axis being, however, a little smaller than he made it. Bradley continued the observations during an entire revolution of the Moon's nodes, and then published an account of his discovery in the Philosophical Transactions for 1748, in a paper which is a perfect model of lucid statement and strict inductive reasoning. In the following year, D'Alembert succeeded in determining the true motion of the Earth's axis by means of analysis, in his "Recherches sur la Precession des Equinoxes et sur la Nutation de 1'Axe de la Terre," and since that tune the subject has been repeatedly treated of by physical astronomers. The most complete and elegant theoretical investigation, how- ever, of the motion of the Earth about its centre of gravity is that given by Poisson in the seventh volume of the Memoires de Vlnstitut. The theoretical investigations with respect to nutation leave nothing to be determined by observation, except the value of one constant. This is 312 ADDRESS ON PRESENTING THE GOLD MEDAL OF THE [42 generally chosen to be the coefficient of the principal inequality in the obliquity of the ecliptic. The accurate determination of this constant is important, not only from its being required for the reduction of star observations, but also from its affording one of the best means we have of determining the mass of the Moon. In precession we see the effect of the joint action of the Sun and Moon, but by means of the observed quantity of nutation, we can ascertain what part of this is due to the Moon's action, and having thus obtained the ratio between the actions of the Sun and Moon, the Moon's mass easily follows. The most trustworthy determinations of the constant of nutation, previous to this of M. Peters, are those of MM. Von Lindenau, Brinkley, Robinson, and Busch ; and M. Peters begins his memoir with a critical examination of their labours. The results of the three latter astronomers present an admirable agreement, while that of Von Lindenau differs from them by about a quarter of a second. Von Lindenau employed about 800 observations of right ascension of Polaris, made at different observatories, and therefore his result is liable to be vitiated by the different personal equations of the several observers. We shall find in the sequel that this remark is important. Brinkley deduced his value of the constant from 1618 observations of ten stars, made about the times of two opposite maxima of nutation in declination with the Dublin meridian circle, the proper motions of the stars being determined by the comparison of his own declinations with those in the Fundamenta. As these observations embrace only half a period of the Moon's nodes, the result is liable to be affected by errors in the supposed proper motions. Dr Robinson's investigation is contained in the eleventh volume of the Memoirs of the Royal Astronomical Society. He employs the declinations of the polar star, and of fourteen others observed at Greenwich between the years 1812 and 1835 with Troughton's mural circle. There can be no doubt of the high value of this investigation, but M. Peters thinks that, in consequence of the way in which the error of collimation is determined, errors of observation may exist with a yearly period, and that these may slightly affect the resulting value of nutation. Baily's coefficient of aber- ration is employed, the annual parallaxes of the stars are neglected, and the equations of condition are not treated by the method of least squares. 42] EOYAL ASTRONOMICAL SOCIETY TO M. PETERS. M. Busch has deduced the constant of nutation from Bradley 's obser- vations at Kew and Wansted. The reductions are made in the most strict manner, except that the annual parallaxes are neglected, and M. Peters regards the result as worthy of the highest confidence. M. Peters then enters upon his own investigations, which are based on 603 right ascensions of Polaris, observed at Dorpat between 1822 and 1838, with Reichenbach and Ertel's meridian circle. Of these observations, the first 249 were made by M. Struve, and the remaining 354 by M. Preuss. These are compared with the right ascensions deduced from the Tabulae Regiomontance, and the equations of condition thence arising are treated by the method of least squares, taking as the unknown quantities the correction of the constant of nutation, the correction of the constant of aberration, the annual parallax, the corrections for the position of the axis of the transit-circle (illuminated pivot east or west), the correction of the star's right ascension, and the personal equation of the two observers. The equations are first solved, giving equal weight to all the obser- vations. The observations are then divided into two groups (one for each observer), and the equations of each group are solved separately. There is a surprising agreement between the results found from the four years' observations of M. Struve, and the twelve years' observations of M. Preuss, the coefficients of nutation deduced differing by less than three-hundi'edths of a second. This investigation supplies a measure of the precision of the separate observations, and it is found that M. Struve's observations are entitled to greater weight than those of M. Preuss. The whole of the observations are then combined, giving the proper relative weights just obtained, and the equations are re-solved. The values found for the unknown quantities differ extremely little from the results given by the supposition of equal weights. One of the most striking results is the constant difference between the right ascension given by the two observers, or the personal equation, which amounts, for Polaris, to more than 0'8 of a second of time. The magnitude of this shews that the personal equation changes with the declination of the stars. Hence, also, we may easily understand that M. Lindenau's results may be vitiated by the omission of the consideration of personal equation, especially as the observations which he employed were made with different instruments, as well as by different observers. A. 40 314 ADDRESS ON PRESENTING THE GOLD MEDAL OF THE [42 While M. Peters was employed in these investigations, M. Lundahl was likewise engaged in discussing the observations of declination of the same star, made also at Dorpat within the same space of time. The value of the constant of nutation which he deduces agrees admirably with those found by MM. Peters and Busch. Finally, M. Peters takes the mean of the three results, giving the proper relative weights to the several determinations, and he finds the most probable value of the constant to be 9"'2231, with the probable error 0"'0154. This value differs very little from Brinkley's, which has generally been employed by English astronomers, but M. Peters' determination un- doubtedly possesses much greater weight. M. Peters next enters upon a theoretical investigation of nutation, far more complete than any that had before appeared. Starting from the equations of Poisson's theoiy, he develops them, taking into account the ellipticities of the orbits of the Earth and Moon, and also the principal lunar inequalities. He thus obtains a great number of small terms which had previously been neglected. Most of these may be safely omitted ; but there are two terms which should be taken into account in delicate investigations, as they have an annual period, and are therefore mixed up with the effect of aberration and parallax. M. Peters takes care to apply the requisite corrections to the coefficients of aberration, and to the parallax of Polaris given by his investigations. Although most of the new terms found by M. Peters are very small, yet these researches are not the less valuable, since it is always satisfactory to know what we really neglect. M. Peters takes into account the effect of a possible difference between the ellipticities of the two hemispheres, which he determines by means of the pendulum experiments collected by Mr Baily in his "Report on the Experiments made by Foster," in the seventh volume of the Memoirs of the Royal Astronomical Society. It fortunately happens that this effect is insensible, as this difference of the two hemispheres is extremely doubtful. The last part of M. Peters' paper contains researches on the obliquity of the ecliptic and the precession of the equinoxes, so that he treats of all the elements which relate to the apparent changes in the places of the stars, due to the motion of the pole of the Earth. He deduces the secular diminution of the obliquity of the ecliptic by comparing the obliquity for 1757, given by Bradley's observations, with that for 1825 given by the observations at Dorpat, both being reduced to the mean by 42] ROYAL ASTRONOMICAL SOCIETY TO M. PETERS. 315 the new value of nutation. The rate of the diminution so found agrees very well with that found by M. Le Verrier from theory, the difference not amounting to one second in a century. The true value of the obliquity of the ecliptic at a given epoch cannot, however, be considered as definitively settled, in consequence of the puzzling constant differences between the declinations determined at different observatories. For instance, the obliquity given by the mean of several years' observations at Greenwich exceeds by rather more than one second the obliquity for the same epoch given by M. Peters' investigations. M. Peters' researches respecting precession are based on the results of M. Otto Struve's paper, which obtained our medal on a former occasion, combined with M. Le Verrier's determination of the secular change in the position of the ecliptic. M. Otto Struve determines, independently, by observation, the values of two constants on which the precessions in right ascension and declination depend. Now, theory establishes a relation between these constants, and M. Peters is thereby enabled to find the most probable values which result from the combination of the observed values, and thence to derive complete formulae for precession applicable to any given epoch. I have no hesitation in regarding M. Peters' results, with respect both to precession and nutation, as definitive for the present state of astronomy. I now come to M. Peters' second paper, which relates to the delicate subject of the parallax of the fixed stars. The first part of this important paper contains an historical and critical review of the researches of astronomers respecting parallax from the time of Tycho to the year 1842. The second treats of the parallaxes of several stars as determined by M. Peters' own observations, made at Pulkowa by means of the great vertical circle of Ertel. In the third part, the results of the two former are applied to determine the mean parallax of stars of the second magnitude. The historical part is drawn up with great care, and contains many curious and interesting discussions on particular points. For instance, M. Peters shews that the coefficient of aberration may be obtained with great accuracy from Flamsteed's observations of the zenith distance of the pole-star. The probable error of a single observation is found to be only 6", which gives a far higher idea of the accuracy of Flamsteed's observations 402 316 ADDRESS ON PRESENTING THE GOLD MEDAL OF THE [42 than has been generally entertained. Bradley himself remarked, that Flam- steed's observations of the pole-star agreed with his theory of aberration. The celebrated controversy between Brinkley and Pond is discussed at considerable length, and the labours of the latter astronomer are criticised with great severity. M. Peters considers that Brinkley was far superior to his opponent in his knowledge of the theory of his instruments, and in the use of precautions to avoid error, though it is certain that Pond was the more correct in his conclusions respecting parallax. The parallaxes determined by M. Struve at Dorpat, from 1818 to 1821, by means of observed differences of right ascension of circumpolar stars having nearly opposite right ascension, deservedly occupy a good deal of attention. The parallaxes thus found, though small, were almost all positive, and M. Peters confirms their reality by the following ingenious consideration. He shews that any diurnal variation of the instrument due to temperature will affect the coefiicients of aberration and parallax in the same direction, and the former probably more than the latter. Now, the coefficient of aberration found from these observations is about 0"'08 less than the definitive value given by the Pulkowa observations, and it is therefore probable that M. Struve's parallaxes should be increased by a few hundredths of a second. It is unnecessary for me to follow M. Peters in his account of Struve's micrometrical measurements of the parallax of a Lyrce, of Bessel's well- known observations of 61 Cygni with the heliometer, and of the parallaxes of a Centauri and Sirius, as determined by MM. Henderson and Maclear at the Cape, as these have been fully discussed by Mr Main in an able paper in the twelfth volume of our Memoirs. The Council is also indebted to Mr Main for a careful report on M. Peters' paper, from which I have derived considerable assistance in drawing up my account of it. The second and most important part of M. Peters' paper consists of an investigation of the parallaxes of eight stars, by means of observations of zenith distance made by M. Peters at Pulkowa, in 1842 and 1843, with Ertel's great vertical circle. The stars selected are Polaris, Capella, i Ursce Majoris, Groombridge 1830, Arcturus, a Lyrce, a. Cygni, and 61 Cygni. The utmost care is taken in the instrumental adjustments, in the equalisation of the interior and exterior temperatures, and in eliminating every imaginable source of error. 42] ROYAL ASTRONOMICAL SOCIETY TO M. PETERS. 317 It would be impossible for me to convey an adequate idea to any one, unacquainted with M. Peters' paper, of the numerous precautions used by him for this purpose. For instance, the observations are made by placing the wire very near the star, and then waiting for the time when the star is exactly bisected by it. The large motions of the instrument are always made without touching either the telescope or the divided circle, or the pieces carrying the microscopes. In making the double observation (face East and face West) the micrometer-screw is always turned finally in the same direction, the reading of the levels is always commenced at the same end of the scale (though they are protected from heat by glasses). The effect of flexure of the telescope-tube is eliminated by an important arrangement, by which the eye-piece and object-glass are capable of being fixed at pleasure at either end of the tube. This trans- position was made after every eight complete observations of the Sun. At every observation the readings of the microscopes are taken for coincidence with both the preceding and succeeding divisions on the limb, and the utmost pains are employed to correct for any inequality in the micrometer-screw and for errors of division. Again, in the reduction of the observations and the elimination of the unknown quantities, the same attention to minute accuracy is observable. Thus, small terms are introduced into the expressions for aberration and nutation which had hitherto been neglected, and an elaborate investigation is entered into respecting the proper motions of the stars observed. The unknown quantities to be determined are the correction to the assumed latitude, the flexure of the telescope-tube, the correction of the thermo- metrical coefficient of refraction, the correction of the assumed mean decli- nation, the annual parallax, and the correction of the coefficient of aberration. Of these, the first three are found by means of the observations of the pole-star. All the equations are solved by the method of least squares, and the greatest care is used in estimating the probable errors of all the results, whether arising from probable errors of observation or uncertainty in the elements employed in the calculation. There are also discussions on some curious points, such as the effect of clouds on refraction, the possible variability of latitude, &c. The resulting values for parallax are all positive, with the exception of that of a Cygni, which comes out a minute negative quantity ; this, of course, only indicates that the real parallax of that star is probably extremely small. 318 ADDRESS ON PRESENTING THE GOLD MEDAL OF THE [42 The constant of aberration obtained by taking the mean of the several results for the different stars is 20"'481, which differs only 0"'036 from the definitive value found by M. Struve. The smallness of this difference gives great confidence as to the accuracy of the results for parallax, as there is no reason why the aberration should be found more accurately than the parallax. Another strong confirmation is afforded by the fact, that the parallax of 61 Cygni determined by M. Peters is absolutely identical with that found by Bessel by means of the heliometer. The last part of M. Peters' paper treats of the mean value of the parallax of stars of the second magnitude. M. Peters finds that there are thirty-five stars whose parallaxes are determined with sufficient accuracy to serve as a basis in this research. Of these, however, he excludes two stars which have very large proper motions, 61 Cygni and 1830 Groombridge, as exceptional, and therefore not properly to be included when an average is the quantity sought. Struve's scale of relative distances of stars of different magnitudes is employed in combining the observed parallaxes for different stars, although the final result is nearly independent of the assumed scale, inasmuch as the second magnitude is nearly the mean of all the magnitudes of the stars employed. M. Peters shews his usual skill in estimating the probable errors which may arise from the defects of the hypotheses employed, such as that of the same absolute brightness of the stars, as well as from the errors of the observed parallaxes ; and he finally arrives at the result, that the most probable value of the mean parallax of stars of the second magnitude is 0"'116, and that the probable error of this determination is only 0"'014. M. Peters closes his paper with a most interesting result, deduced by combining his own researches with those of M. Otto Struve respecting the solar motion. M. Otto Struve finds that the annual apparent motion of the Sun, as seen at right angles from a point at the mean distance of stars of the first magnitude, is 0"'339. Now, according to M. Peters, the mean parallax of a star of the first magnitude is 0"'209 ; so that we are able to turn the former result into absolute measure. Thus the annual motion of the Sun with respect to the great body of the surrounding- stars is equal to T623 times the radius of the Earth's orbit. 42] ROYAL ASTRONOMICAL SOCIETY TO M. PETERS. 319 I cannot but regard this work of M. Peters as a perfect model of excellence, evincing consummate skill in the observer, as well as admirable power of turning the observations to the best account. It shews that it is possible by meridional observations to obtain absolute parallaxes almost as small as the relative parallaxes that can be measured by the heliometer, or by similar means; though to do so requires a most rare union of instru- mental advantages, care and judgment in the observer, and analytical skill in combining in the best manner the results of observation. No one can read the papers of M. Peters, or those of the Russian and German astronomers generally, without being struck with the constant employment of the method of least squares. It is to be wished that this method were more in use among English astronomers, as I believe not a little of the precision of modern determinations is due to it. We seem to entertain a distrust respecting the results of the calculus of probabilities, more particularly with regard to the estimation which it affords of the probable amount of error in any determination. It should be borne in mind, that when we speak of the probable error being of a certain amount, it is not meant that it is improbable that the error should exceed that amount, but only that it is as probable a priori that the error falls short of, as that it exceeds it. If we know by inde- pendent means that the error of any determination is much greater than the probable error given by the observations, we may infer, with great probability, that some constant cause of error has occurred in the obser- vations employed. In the estimation of probable error, only fortuitous causes of error are taken into account. The employment of the method of least squares does not render it less necessary to avoid all sources of constant error: it is not a substitute for, but an auxiliary to good observations, and enables us to obtain from them all that they are capable of yielding. I cannot conclude without congratulating the Society on the improved prospects of that very delicate branch of astronomy which relates to the research of stellar parallax, especially as there is every reason to believe that this country will contribute its full share to the advancement of it. We may hope that the beautiful reflex zenith telescope of the Astronomer Royal, the magnificent heliometer which is in the able hands of Mr Johnson, and the improved method of recording star transits by means of galvanism, will enable us ere long to take many firm, though long-reaching, steps into regions of space hitherto untrodden. 320 ADDRESS ON PRESENTING GOLD MEDAL TO M. PETERS. [42 (The President then, delivering the Medal to Mr Hind, Foreign Secretary, addressed him in the following terms): In transmitting this medal to M. Peters, you will assure him of our high appreciation of the importance of the results at which he has arrived, and of the admirable science and skill which he has shewn in obtaining them ; and you will express our confident hope, that in his new sphere at Konigsberg he will confirm and add to the reputation which he has so deservedly acquired at the Observatory of Pulkowa. 43. ADDRESS ON PRESENTING THE GOLD MEDAL OF THE ROYAL ASTRO- NOMICAL SOCIETY TO MR HIND. [From the Memoirs of the Royal Astronomical Society. Vol. xxu. (1853).] GENTLEMEN, You have heard from the Report which has just been read how much reason we have to congratulate ourselves on the present state and future prospects of our science. Never was there a time when greater vigour and activity were exhibited in the promotion of it. Nor is this activity confined to one country, or devoted merely to one department of astronomy. Whether we regard the introduction of improved instruments and methods of observation, or the more rigorous discussion to which the observations are submitted, the formation of extensive catalogues of stars, the discovery of new members of our planetary system, or the closer and more systematic scrutiny and examination of those which are already known, in every direction we find the most satisfactory evidences of progress. One of the most prominent features of astronomical discovery for several years past, has been the continual addition of new members to the remark- able group of small planets between the orbits of Mars and Jupiter, and the year just ended has been distinguished beyond all precedent in this respect. Since our last anniversary meeting no fewer than eight of these bodies have been brought to light, and the supply seems to be inexhaustible. New discoverers have made their appearance on the field, while those who A. 41 322 ADDRESS ON PRESENTING THE GOLD MEDAL OF THE [43 have already distinguished themselves seem to have acquired a new aptitude in the search. It is gratifying to find that one of our own body has been the very foremost in this noble career of discovery ; and to him, in testimony of our appreciation of his well-directed and successful labours, the Council has awarded the medal, which it is my pleasing duty this day to present. Skilfully using the excellent instrumental means placed at his disposal by the enlightened liberality and scientific zeal of Mr Bishop, and in spite of the interruptions occasioned by a climate, the disadvantages of which are peculiarly felt in researches of this nature, Mr Hind has added no fewer than eight planets to our system, four of which have been found in the course of the past year. After this, I feel that it is unnecessary to add another word in justification of the award of your medal. Mr Hind's discoveries are of a nature to be understood and appreciated by all ; and I shall, therefore, confine myself to a very brief notice of some circumstances connected with them, and to a few remarks on the conclusions to which they seem to point, respecting the constitution of our planetary system. The first five of Mr Hind's planets were found by comparing the heavens with the excellent and well-known star-maps of the Berlin Academy. These, however, are limited to 15 on each side of the equator, and there- fore do not include the whole of the region about the ecliptic, which it is so desirable to examine ; neither do they contain stars smaller than between the ninth and tenth magnitudes. Mr Bishop, therefore, very soon determined to intrust to Mr Hind the formation of a series of ecliptic charts, which should contain all stars down to the eleventh magnitude, which were situate within 3 on each side of the ecliptic. Mr Hind has already begun to reap the fruits of these labours, the planet Fortuna having been detected in the course of preparing one of the charts, while Calliope and Thalia were found by the comparison of two of the completed charts with the heavens. Eight of these valuable charts have now been published, and I under- stand that most of the remaining ones are considerably advanced. Other astronomers, particularly Mr Cooper of Markree, are engaged in the prepar- ation of charts on a similar plan, and the path of future discoverers cannot fail to be singularly facilitated by their means. 43] ROYAL ASTRONOMICAL SOCIETY TO MR HIND. 323 The existence of such a numerous group of small planets in the same part of our system has naturally given rise to much speculation respecting their origin and mutual relations. When, instead of the single planet which was expected to fill up the gap between the orbits of Mars and Jupiter, Ceres and Pallas were found at very nearly the same mean distance from the sun, Olbers threw out the conjecture that they were fragments of a larger planet which had been rent asunder by some internal convulsion, and that many more such fragments probably existed. If this wei'e the case, he reasoned, they would all, after longer or shorter periods, again pass through the point where the explosion took place, and though the pertur- bations which they would suffer, would, in the course of time, prevent them from continuing to pass exactly through the same point, yet it might be expected that they would not stray far from it, and that, therefore, the remaining fragments might be found by carefully watching the parts of the heavens corresponding to the two points in which the orbits of Ceres and Pallas approached towards intersecting. Although the finding of Juno and Vesta appeared to give some counte- nance to this hypothesis, later discoveries have deprived it of much of its plausibility. Several of the orbits are everywhere far distant from each other, and where the contraiy is the case, the points of nearest approach occur in various parts of the heavens. Probably one reason why Olbers did not discover more of these bodies, though he continued his examination for many years after detecting Vesta, was, that he was induced by his theory to confine the search within too narrow limits. Several astronomers have endeavoured to find some general relations between the orbits of this group, similar to that imagined by Olbers ; but it appears to me that they have only succeeded in shewing a kind of general resemblance, indicating rather that similar causes have operated in determining the orbits of these bodies than that they were originally identical. If we allow ourselves to speculate on the formation of our planetary system, and adopt the nebular theory, it seems at least as easy to imagine that the nebulous matter, circulating in any particular region about the Sun, would, in cooling, collect into many small masses, as that it would all coalesce into one. Although, as has been stated, there is no single point through which all the orbits nearly pass, yet many of them, taken two and two, approach very closely to each other. In the case of Astrcea and Hygeia, in particular, 412 ,/* 324 ADDRESS ON PRESENTING THE GOLD MEDAL OF THE [43 the shortest distance between the two orbits is less than xihrth P ar ^ f the Earth's mean distance from the Sun ; so that, as M. D'Arrest remarks, the time of their actual intersection cannot be very distant from the present. One of the most curious circumstances connected with this group is, that there are several cases in which the mean distances are nearly identical with each other. Thus the mean distances of Ceres and Pallas are so nearly equal that their order of magnitude is sometimes changed by per- turbation. The same remark applies to Iris and Metis, and also to the three planets, Astrcea, Egeria, and Irene. It should be noticed that this identity of mean distance would not be at all explained by supposing the planets in which it occurs to have been originally one. There are also some remarkable cases in which the mean motions are nearly commensurable. Thus the mean motions of Juno and Vesta are very nearly in the ratio of 5 to 6, while those of Juno and Flora are as 3 to 4, and consequently those of Vesta and Flora as 9 to 10*. The extreme smallness of the apparent diameters of these bodies makes it very difficult to determine their real diameters by direct measurement. According to Sir W. Herschel's observations, the diameters of Ceres and Pallas would not be far from 140 English miles, while Schroter's obser- vations would make them much larger. Stampfer has attempted to determine their diameter by means of their apparent brightness, supposing the reflective power of their surfaces to be the same as that which obtains in the case of Jupiter, Saturn, Uranus, and Neptune. This supposition is obviously rather precarious, especially as the reflective power of Mars is found to be much less than that of the other planets; but Stampfer's result agrees very closely with the above-mentioned determination of Sir W. Herschel. Several of the more recently-discovered planets appear to be much smaller than these ; and it is not improbable that there are many more which, by their excessive minuteness, elude our telescopes altogether. In this point of view, these asteroids would seem to form a connecting link between the larger planets and the aerolites, the cosmical nature of which appears to be pretty well established. * The mean daily sidereal motion of Juno is 814"-24; that of Vesta, 977"-20; and that of Flora, 1086"-08. Also | x 814-24 = 977-08, and * x 814-24 = 1085-65. 43] ROYAL ASTRONOMICAL SOCIETY TO MB HIND. 325 To the physical astronomer these bodies offer problems of great interest and difficulty. On account of the large eccentricities and inclinations of some of the orbits, methods of approximation which succeed in determining the perturbations of the older planets become quite inadequate to deal with these, and, consequently, astronomers have hitherto been compelled to have recourse to the method of mechanical quadratures in order to calculate their motions. But although this method may be employed in all cases, and the use of it becomes much simplified by applying it directly to the differential equations of motion, in the elegant manner which has been recently devised by Mr Bond and Professor Encke, yet it only enables us to follow the disturbed planet, as it were, step by step, and it is, therefore, very desirable to have a method by which the course of the planet might be traced through an indefinite number of revolutions, and the results of which might be embodied in tables. Professor Han sen has attacked this very difficult problem with his -characteristic originality and skill, and Sir J. Lubbock has also treated the same subject very ably in his tracts on the perturbations of the planets. Much, however, remains to be done before the application of the method of quadratures to these cases can be superseded. It will be quite indis- pensable to take into account the square and higher powers of the disturbing force. It may be remarked, however, that the eccentricities and inclinations of the orbits of several of these new planets are so moderate, that there will be little difficulty in calculating their perturbations by the ordinary methods. The disturbances which these bodies suffer from the action of Jupiter are so large as to afford an excellent means of determining the mass of that planet. It was thus that Nicolai found that the value of this mass which had been employed by Laplace and Bouvard was considerably too small, a result which Mr Airy afterwards confirmed by direct measures of the elongations of the satellites. Considering the great degree of proximity to each other, to which these bodies sometimes attain, it does not seem improbable, notwithstanding their minuteness, that they may occasionally produce a sensible effect on each other's motions ; in which case the astro- nomer would be able to weigh these minute atoms in the same balance which he has already applied to the larger bodies of our system. In examining the heavens in search of small planets, Mr Hind has naturally been led to pay great attention to the variable stars, and he 326 ADDRESS ON PRESENTING THE GOLD MEDAL OF THE [43 has consequently detected a considerable number of these objects among the smaller stars. Two of these I will mention, which are at opposite extremities of the scale, and which seem to imply the operation of totally different causes. The first is that remarkable new star in Ophiuclius which Mr Hind noticed on the 27th of April, 1848, as being of the 6th magnitude, and occurring in a spot where he was certain no star even of the 9 10th magnitude had been visible three weeks before. After attaining to the 4 5th magnitude, so as to be conspicuous to the naked eye, it gradually faded away, and at present it is only of the llth magnitude. The other star to which I will refer appears to vary in a similar way to Algol. Its period, according to Argelander, is about 9 d ll h , but for 9 days of this time it shines as a star of the 8th magnitude, then suddenly descends to the 10 llth, and as quickly returns again to the 8th. Variations of this latter kind appear to be most naturally accounted for by the periodical interposition of an opaque body in its revolution about the star, but those of the kind first mentioned seem to mock all our attempts at explanation. In recording these discoveries, it is doubly gratifying to recollect that they emanate from an observatory founded and maintained by a private individual out of pure love of the science and zeal for its advancement. Of the judgment which Mr Bishop has shewn in the selection of his observers, and the choice of objects of observation, there can be no better proof than is afforded by the admirable double-star observations of Mr Dawes and the planetary discoveries of which we have just been speaking. Mr Bishop may well feel proud in the consciousness that his observatory has been the means of contributing so largely to science, and has thus become known wherever astronomy is cultivated. Another subject of congratulation is the manner in which Mr Hind's services to science have been recognised by the Government of the country. It is sometimes asked, whether the progress of science is best promoted by private or by public means; but the truth is, that there is no such opposition between these modes of advancing it as is implied in the form of the question. In a country where the dignity of science, and the benefits which it confers, are properly estimated, both Government and people will harmoniously co-operate in its support, and each will easily find its appro- 43] ROYAL ASTRONOMICAL SOCIETY TO MB HIND. 327 priate sphere of action. Surely few objects can be mentioned more truly national in their character than the encouragement and reward of scientific discoveries, which at the same time reflect honour on the country, and give so powerful an impulse to the intellectual advancement of the people. (The President then, delivering the Medal to Mr Hind, addressed him in the following terms): Mr Hind, It is with peculiar pleasure that I present you with this Medal, in testimony of our appreciation of your eminent services to astronomy. The whole world will acknowledge how nobly it has been earned, and will join with us in the wish that your health may long be spared, and that thus you may be able to make many more additions to our knowledge in that field of science to which you have devoted yourself with so much energy and success. 44. ADDRESS ON PRESENTING THE GOLD MEDAL OF THE ROYAL ASTRO- NOMICAL SOCIETY TO M. CHARLES DELAUNAY. [From the Monthly Notices of the Royal Astronomical Society. Vol. xxx. (1870).] GENTLEMEN, It has been announced to you that the Society's Medal has been awarded to M. Ch. Delaunay for his great work on the Theory of the Moon. The illness of our excellent President having made it impossible for him to be present on this occasion, the Council have done me the honour to request that I would occupy the chair, and in his stead lay before you the grounds of their award. I have acceded to their wishes with the more readiness because I have given some attention to special branches of the Lunar Theory, and my study of M. Delaunay's work has led me to form the highest opinion of its merits. Of all the problems presented to us by physical astronomy none has so much engaged the attention of mathematicians as that of the deter- mination of the motion of our satellite. The theoretical interest as well as the great practical importance of the results, has proved an irresistible attraction, and the mathematical difficulties have merely acted as a stimulus to the invention of various methods of surmounting them. It is fortunate that this has been the case, as the excessive labour involved in any theory of the Moon approaching to completeness, might otherwise have proved too great for human perseverance. The foundations of the theory were laid by 44] ADDRESS ON PRESENTING GOLD MEDAL TO M. DELAUNAY. 329 Newton in his Principia ; and although his investigations are only fragmentary, being simply intended to shew how some of the leading lunar inequalities may be deduced from theory, yet they form one of the most admirable portions of that immortal work. Towards the middle of the eighteenth century the theory was more systematically entered upon by Clairaut, D'Alembert, and Euler, who severally shewed that the theory was competent to give very approximate values of all the inequalities which were then recognised by observation. Still the theory was far from being sufficiently perfect to serve as a foundation for lunar tables accurate enough for the uses of navigation. This degree of accuracy was first attained by the tables of Mayer, who not only carried the approximations to the values of the coefficients of the various lunar inequalities further than his predecessors had done, but also corrected the theoretical coefficients thus obtained by comparison with his own observations. The theory was greatly advanced by Laplace, not only by his more accurate theoretical determination of the coefficients, but also by several important discoveries, especially that of the cause of the Moon's secular acceleration. The improvements in the lunar tables, however, which were made successively by Biirg and Burckhardt, were founded, not on theory, but on comparison of the former tables with observations ; and the empirical tables thus produced were far more accurate than any that could have been formed at that time by theory alone. Dissatisfied with this state of things, and wishing to see astronomy founded exclusively on the law of attraction, only borrowing from observation the necessary data, Laplace induced the Academy of Sciences to propose for the subject of the mathematical prize which it was to award in 1820 the formation, by theory alone, of lunar tables as exact as those which had been constructed by theory and obser- vation combined. The prize was divided between two memoirs one by M. Damoiseau, the other being the joint production of MM. Plana and Carlini. Damoiseau's memoir is printed in the third volume of the Recueil des Savants Strangers. Plana's great work on the lunar theory, which appeared in 1832, is the development of the joint memoir by himself and Carlini. By these important works an immense advance was made in the theory, the approximations being carried to such an extent that the resulting coefficients were comparable in accuracy with those given by observation. In 1824 Damoiseau published tables founded entirely on his theory, which were found to be quite as exact as those of Burckhardt. A. 42 330 ADDRESS ON PRESENTING THE GOLD MEDAL OF THE [44 Both Damoiseau and Plana, following the example of Laplace, start from differential equations in which the Moon's longitude is taken as the independent variable ; and after the equations have been integrated, they obtain the values of the Moon's coordinates in terms of the time by reversion of series. An important innovation, however, was introduced by Plana in the mode of conducting the investigation and exhibiting the results. The values of the Moon's coordinates being developed in series of sines and cosines of angles which vary uniformly with the time, the coefficients of the several terms of these series will depend on the eccentricities of the orbits of the Sun and Moon, the inclination of the Moon's orbit to the plane of the ecliptic, the ratio of the mean motions of the Sun and Moon, and the ratio of their mean distances from the Earth. Now Damoiseau, in common with all previous writers, having assumed certain values of the quantities just mentioned as given by observation, contented himself with determining the numerical values of the coefficients. Although this is all that is required for the construction of tables, yet, from a theoretical point of view, it leaves the mind unsatisfied, inasmuch as any coefficient in its numerical form shews no trace of its composition, that is of the manner in which its value depends on the value of the assumed elements. The several coefficients are far too complicated functions of the elements to be represented analytically, except in the form of infinite series, and Plana, accordingly, developes these coefficients in such series, proceeding by powers and products of the eccentricities, the tangent of the inclination, the ratio of the Sun's mean motion to that of the Moon, and the ratio of the Moon's mean distance to that of the Sun, all these quantities being assumed to be small, and the last mentioned ratio, which is much smaller than the others, being considered as a quantity of the second order. In this mode of development, the numerical factor which enters into any term of the coefficient of any of the lunar inequalities is an ordinary fraction which admits of being determined not merely approximately, but with absolute accuracy. It is easy to see what great facilities are afforded by this circumstance for the verification of the work by a comparison of the results obtained by different methods. The greater or less degree of approximation will thus depend on the greater or less number of terms taken into account in the several series. The numerical values of the several elements are not substituted in the formulae until the work is completed, and this is attended with the important advantage that when a comparison of the theory with observation 44] ROYAL ASTRONOMICAL SOCIETY TO M. CHARLES DELAUNAY. 331 has supplied more accurate values of the elements, their corrected values can be at once substituted in the same formulae, without requiring any additional work. On the other hand, if the numerical values of the elements be intro- duced into the calculations from the first, then if it is desired to introduce corrected values of the elements, much additional investigation will be required for the purpose. No doubt the labour required in order to obtain a given amount of numerical accuracy by this method is very much greater than is required when each coefficient, instead of consisting of a series of terms, is reduced to a simple numerical quantity, but the great theoretical advantage of knowing the composition of every coefficient in terms of the elements well repays the additional labour. The degree of convergence of the series obtained for the several co- efficients is in general sufficiently rapid, but in some few of the coefficients, on the contrary, the convergence is so slow, at least in the leading terms, that it is necessary to take into account terms which are analytically of a higher order than those to which the approximation is in general limited. Thus Plana, who proposed to himself to determine the lunar inequalities completely to the fifth order, found it necessary in special cases to carry the approximation to the seventh and even to the eighth order, and in several cases he also added an estimated value of the remainder of the series founded on the observed law of diminution of the calculated terms. Soon after the publication of Plana's great work, Sir John Lubbock formed the plan, which he partly carried out in his various tracts on the theory of the Moon, of verifying Plana's results by a totally different method, starting from differential equations in which the time is taken as the independent variable, and thus avoiding the necessity of reversion of series. Later, M. de Ponte"coulant undertook the same work on a similar plan, and carried it out more completely in the fourth volume of his Theorie Analytique de Systeme du Monde. These works, while they corrected some errors which had crept into Plana's computations, confirmed their wonderful general accuracy, and with some few exceptions they do not extend the approximation beyond the order to which Plana restricts himself. 422 332 ADDRESS ON PRESENTING THE GOLD MEDAL OF THE [44 Meantime, M. Hansen had undertaken a completely new investigation of the lunar theory, by a remarkable method peculiar to himself and explained in his Fundamental nova investigationis orbitce verce quam Luna perlustrat, which appeared in 1838. In applying the method described in this work to the case of the Moon, M. Hansen throughout employs numerical values of the elements of the Moon's orbit, and consequently the coefficients of the lunar inequalities as obtained by him are also purely numerical. The process is one of successive approximations, which are repeated again and again until the values of the inequalities which are found from the last approximation sensibly coincide with those which were assumed in entering upon that approximation. The numerical values of the coefficients thus finally obtained are un- doubtedly very exact. The slight corrections which these coefficients still require are probably chiefly due to the small corrections required by the numerical elements on which the calculations are based, and in the method employed no provision is made for taking into account the effect of these corrections. From his formulae, M. Hansen constructed tables of the Moon, which were published in 1857, at the expense of the British Government; and these tables, having been found far superior in accuracy to all others, are now exclusively employed in the calculation of ephemerides. A detailed account of the calculations leading to M. Hansen's last approximation, was given by him in the two parts of his Darlegung der Theoretischen Berechnung der in den Mondtafeln angewandten Storungen, which severally appeared in 1862 and 1864. After the great works, to which we have thus briefly referred, had been either completed or were in progress, it might have been supposed that the matter was exhausted. Our Associate M. Delaunay, however, was not of this opinion. Having devised, so long ago as 1846, a perfectly original and singularly beautiful method of integrating the differential equations of the Moon's motion, he determined to apply this method to the complete re-investigation of the theory, and to carry on the approximation to a much greater extent than had been done by his predecessors. The principal fruits of his labours, to which he has devoted himself with almost unexampled perseverance for so many years, are contained in the magnificent volumes which the Imperial 44] ROYAL ASTRONOMICAL SOCIETY TO M. CHARLES DELAUNAY. 333 Academy of Sciences have done both M. Delaunay and themselves the honour of publishing among the volumes of their Memoirs. It is for this great work that your Council have awarded to M. Delaunay the Society's medal. Strongly impressed with the advantages of determining the coefficients of the lunar inequalities in the analytical form, both as affording a solution more complete in itself and more satisfactory to the mind, as well as one offering facilities for the comparison of the results of different investigations, M. Delaunay did not hesitate to follow the example set in this respect by M. Plana, notwithstanding the immense length of the necessary calcu- lations. M. Delaunay 's results are thus obtained in a form which makes them directly comparable with those of M. Plana, while the methods employed in obtaining them are wholly different. M. Delaunay chooses the time as the independent variable, and takes as his starting-point the differential equations furnished by the theory of the variation of the arbitrary constants. In an able Memoir which appeared in 1833, Poisson had advocated the employment of these equations in the theory of the Moon's motion, and he applied them to the discussion of some special points of that theory. These equations had been long used, almost exclusively, for the determination of the perturbations of the planets, and they offer peculiar advantages in the treatment of the secular in- equalities and those of long period. In the case of the Moon, however, in consequence of the large perturbations caused by the disturbing force of the Sun, the ordinary mode of integrating these equations by successive approximations soon leads to calculations of inextricable complexity. In fact, these equations give the differential coefficients of the several elliptic elements taken with respect to the time, in terms of the elements them- selves. In the case of the planets, where the disturbing forces are so small compared with the predominant central force of the Sun, very approximate values of the disturbed elements may be found by substituting in the values of the differential coefficients, the undisturbed instead of the disturbed values of the elements, and then integrating. The perturbations of the elements thus found are said to be due to the first power of the disturbing force. If now the approximate values of the disturbed elements be substituted in the differential equations, and these be again integrated, we shall obtain a second approximation to the values of the disturbed elements, and the additional terms thus found are said to depend on the square of the disturbing force. In the theories of the 334 ADDRESS ON PRESENTING THE GOLD MEDAL OF THE [44 planets it is only in special cases that terms depending on the square of the disturbing force need be taken into account, and it is scarcely ever necessary to consider terms of the next order of approximation. In the case of the Moon, however, it would be necessary to repeat the process of approximation at least four or five times, in order to obtain results of the accuracy required in the present state of the theory. If we consider that the disturbing function consists of a great number of terms, and that each term gives rise to a corresponding term in the value of each of the disturbed elements, while powers and products of the corrections of all the elements in every possible combination, up to a certain order, have to be taken into account, it may be readily imagined how impracticable it would be by such a process to carry on the approximation to a greater extent than has been already done by Plana. Every process in which the approximations require to be repeated several times, is subject to the inconveniences that have been described, and these inconveniences are much greater when, as in the present case, we have to make successive approxi- mations to the values of the six elements of the orbit, instead of to the values of the three coordinates of the Moon. It was with the view of avoiding this excessive complication of the method of successive approximations that M. Delaunay devised his method of integrating the differential equations of the Moon's motion. The funda- mental idea of this method consists in attacking the difficulty by small portions at a time, and in replacing these extremely complicated successive approximations by a much greater number of distinct operations, each of which is comparatively simple, so that it may be carried out to any degree of exactness that may be desirable, while the mind is relieved by being able readily to embrace the whole of each operation in one view. It is difficult, without the use of algebraical symbols to give an idea of M. Delaunay's beautiful method, but I must endeavour, in some measure, to fulfil this task, and I must crave your indulgence should I fail in the attempt. The theory of the variation of the arbitrary constants gives, as is well known, the differential coefficients of the elliptic elements with respect to the time, in terms of the elements themselves and the partial differential coefficients of a certain function, called the Disturbing Function, taken with respect to those elements. By a proper choice of elements, the differential equations may be reduced to their simplest, or to what is called their canonical form. In this form the six elements are divided into three pairs, 44] ROYAL ASTRONOMICAL SOCIETY TO M. CHARLES DELAUNAY. 335 the elements of each pair being conjugate to each other. Then the differ- ential coefficient of any element with respect to the time is simply equal to the partial differential coefficient of the disturbing function taken with respect to the element which is conjugate to the former, the partial differential coefficients which occur in the two equations corresponding to a pair of conjugate elements being affected with opposite signs. The disturbing function may be readily developed in a series of periodic terms involving cosines of angles, each of which is formed by the combination of multiples of the Moon's mean longitude, the distance of the Moon's perigee from its node, and the longitude of the node, together with angles which depend on the position of the disturbing bodies. The disturbing function likewise contains a non-periodic term, which, as well as the co- efficients of the periodic terms, are all functions of the major semi-axis, the eccentricity and the inclination of the Moon's orbit. Since the mean longitude of the Moon involves the time multiplied by the mean motion which is a function of one of the elements, it is obvious that the differentiation with respect to this element will give rise to terms in which the time occurs without its being included under a sine or a cosine. Such terms would render the equations very inconvenient for the determination of the lunar inequalities ; and M. Delaunay accordingly avoids the introduction of them by taking the mean longitude itself instead of the epoch of mean longitude, as one of his elements, while by the simple yet novel expedient of adding to the disturbing function a non-periodic term which is a function of the major semi-axis alone and is independent of the disturbing forces, he preserves to the differential equations the same very simple form which they had at first. After this modification of the disturbing function, the time no longer enters into it explicitly except in so far as it is introduced by the values of the coordinates of the disturbing bodies, and consequently the difficulty which was before met with completely disappears. The six elements employed by M. Delaunay are thus, the Moon's mean longitude, the distance of the perigee of its orbit from the node, and the longitude of the node, which for distinction may be called the three angular elements, and three other elements which are respectively conjugate to the former, and which are determinate functions of the major semi-axis, the eccentricity and the inclination of the orbit. The three coordinates of the Moon at any time are given in tenns of the three angular elements and of the quantities last mentioned. 336 ADDRESS ON PRESENTING THE GOLD MEDAL OF THE [44 Now let us imagine, for a moment, that the disturbing function con- tained no periodic terms, but was reduced simply to its non-periodic part. Consequently the partial differential coefficients taken with respect to the angular elements would all vanish, and therefore the three conjugate elements would be all constant, as well as the major semi-axis, the eccentricity and inclination, of which those elements are functions. Hence, again, the partial differential coefficients taken with respect to the conjugate elements would be functions of those elements, and would therefore be constant. Hence each of the angular elements would consist of an arbitrary constant and a term proportional to the time, the multiplier of the time in each case being a known function of the three constant elements. The object of M. Delaunay's method is, by means of a series of changes of the variables, to cause all the more important periodic terms to disappear from the disturbing function, one by one, while the differential equations continue to retain their canonical form, so that after each transformation we approach more nearly to the conditions of the ideal case which has just been considered. In order to effect any one of these transformations, M. Delaunay supposes, for the moment, that the disturbing function is reduced to its non- periodic part, together with one of the periodic terms selected from among those which have the greatest influence in producing the lunar inequalities. With this simplified form of the disturbing function, the equations admit of being easily integrated. The elements with which we start may thus be expressed in terms of three new angular elements which vary uniformly with the time, and three new constant elements. M. Delaunay shews how the constant elements may be so chosen that they may be considered as respectively conjugate to the three new angular elements, so that, in fact, the quantities which are multiplied by the time in the expressions of these angular elements are respectively equal to the partial differential coefficients of a function of the new constant elements taken with respect to these elements. Having thus found the relations between the old set of elements and the new ones by means of the simplified form of the disturbing function. M. Delaunay now restores the complete value of that function, and chooses new elements which are connected with the old ones by exactly the same relations as in the case just considered. Of course the three new angular elements will no longer vary uniformly with the time, and the three elements respectively conjugate to these will no longer be constant. 44] ROYAL ASTRONOMICAL SOCIETY TO M. CHARLES DELAUNAY. 337 When, by means of the proper formulae of transformation, the new variables have been substituted for the old ones in the disturbing function and in the expressions of the Moon's coordinates, M. Delaunay shews that 1st. One of the important terms of the disturbing function disappears, viz., the periodic term which was selected in the preliminary investigation. 2nd. Various inequalities corresponding to this term are introduced into the values of the three coordinates of the Moon. 3rd. The values of the six new variables in terms of the tune are determined by differential equations of exactly the same form as those which determined the values of the six variables for which they have been sub- stituted. One of the periodic terms having been in this manner caused to dis- appear from the disturbing function, a new operation of exactly the same kind causes another term of this function to disappear; similarly a third term may be taken away by means of a third operation, and so on to any number of terms. In this way, after a suitable number of operations of this kind have been effected, the disturbing function will have been simplified by the removal from it of its most important periodic terms, after which the further process of integration becomes simple enough to be treated in the same manner as if we were concerned with the perturbations of a planet or of the Sun. The whole difficulty in the determination of the lunar inequalities is caused by the great magnitude of the disturbing force of the Sun. M. De- launay has therefore at first confined his attention to the investigation of the irregularities which are produced by this disturbing force, and the two magnificent volumes before us are entirely occupied with this investi- gation. Thus he has provisionally left out of consideration the very small inequalities due to some secondary causes, such as the attraction of the planets and the figure of the Earth; and, besides, he has omitted to consider the perturbations of the Sun's apparent motion about the Earth, intending in a supplementary volume to take into account the effects due to these several causes. By means of repeated applications of the beautiful method of trans- formation which I have above attempted to describe, M. Delaunay proceeds to get rid of all the periodic terms of the disturbing function due to the Sun's disturbing force, which are capable of producing inequalities in the A. 43 338 ADDRESS ON PRESENTING THE GOLD MEDAL OF THE [44 coordinates of the Moon of an order inferior to the fourth. For this pur- pose fifty-seven such operations are required to be performed. When these have been effected, the periodic terms which remain in the disturbing function are so small that their powers and products may be neglected, and consequently the differential equations which determine the six elements last introduced in terms of the time, may be integrated at once. Since the values of the Moon's coordinates are known in terms of the elements just mentioned and the time, we have only to substitute the values of the elements that have been found, in order to determine the Moon's coordinates in terms of the time. The values of the elements, however, that would be found in this way are very complicated, and therefore the substitutions which would be required in order to find the Moon's coordinates would be excessively long. M. De- launay, accordingly, prefers to get rid of the remaining periodic terms in the disturbing function, one by one, by means of transformations exactly similar to those which have been already effected. In order to carry on the approximation to the extent which he desires, M. Delaunay finds it necessary to perform no less than 448 of these secondary operations, but each such operation becomes very simple, since the squares of the coefficients of the periodic terms under consideration may be neglected. Thus, at length, by means of 505 transformations, all the periodic terms of the disturbing function are removed, and the problem is reduced to the ideal case which was considered at the outset of our account of M. Delaunay's method. After each transformation, by making the proper substitutions in the expressions for the Moon's coordinates, those coordinates are obtained in terms of the system of elements last introduced, so that finally the three coordinates are known in terms of the three final constants and angles which vary uniformly with the time. It has been already mentioned that Plana, in his great work on the Lunar Theory, determined the analytical values of the coefficients of the lunar inequalities as far as terms of the fifth order inclusive, and that he only carried on the development to a greater extent in cases where the slowness of the convergence of the series appeared to him to render it necessary to take into account terms of higher orders than the fifth. M. Delaunay has proposed to himself to carry on the approximation so as to include all terms of the seventh order, and in cases where the series 44] ROYAL ASTRONOMICAL SOCIETY TO M. CHARLES DELAUNAY. 339 converge slowly to take into account terms of the eighth, and even of the ninth order. Those who have had any experience in. calculations of this nature will readily understand how enormously the labour required has been increased by thus adding two orders more to those which Plana has considered. It is not merely that the terms of higher orders are far more numerous than those of the lower, but also that each of the terms of the former kind is much more difficult to calculate, since it arises from a much greater number of combinations of terms of the inferior orders. This enormous labour, which has occupied M. Delaunay for nearly twenty years, has been performed by him without assistance from any one. Indeed, from the nature of the calculations which are required, it would not have been easy to obtain any effective assistance. In order to insure accuracy, M. Delaunay has omitted no means of verification, and he has performed all the calculations, without exception, at two separate times, with a suf- ficient interval between them to prevent any special risk of committing the same error twice in succession. The volumes before us are perfect models of orderly arrangement. Not- withstanding the great length and complication of the calculations, the whole work is so disposed that any part of it may be specially examined with the utmost readiness by any one who may wish to test its accuracy. Finally, the analytical expressions which have been obtained for the Moon's coordinates are converted into numbers, by substituting for the elements the most accurate numerical values which the comparison of theory with observation has made known. Such is an imperfect sketch of M. Delaunay's labours on the Theory of the Moon contained in these two magnificent volumes, the former of which appeared in 1860, and the latter in 1867. As I have already stated, they do not include a complete theory of the Moon, but only that which is by far the most difficult and complicated part of that theory, viz., the investigation of the perturbations due to the direct action of the Sun supposing its apparent motion about the Earth to be purely elliptic. Of the investigations which are required to take into account the remaining very small causes of disturbance, and which are intended by M. Delaunay to be included in a supplementary volume, some of the most important have been already completed by him, particularly the calculation of the 432 340 ADDRESS ON PRESENTING GOLD MEDAL TO M. DELAUNAY. [44 Secular Variation of the Moon's Mean Motion, and the investigation of the long inequalities due to the action of Venus. I understand also that M. Delaunay is engaged in the construction of new Lunar Tables founded upon his theory. Your Council, however, has decided that we ought not to await the appearance of M. Delaunay 's supplementary researches before we mark em- phatically our sense of the value of his labours. The present work is complete in itself; in it the very difficult and complicated problem of determining the Moon's motion is attacked by a perfectly original method, and that one as powerful and beautiful as it is new. The work has been planned with admirable skill and has been carried out with matchless perseverance. The result is an enduring scientific monu- ment of which our age may well be proud, and which we are happy to distinguish, on this occasion of our fiftieth anniversary, with the highest marks of our approval which it is in our power to bestow. (The Chairman, then delivering the Medal to M. Delaunay, addressed him in the following terms) : M. Delaunay, il ne me reste plus maintenant qu'a vous presenter cette me'daille au nom de la Societe' Royale Astronomique, qui de'sire par ce tribut vous exprimer la haute appreciation qu'elle a de vos travaux. Notre President regrette vivement que 1'etat de sa sant^ 1'empe'che de remplir cette tache agrdable. II m'a prie" de le remplacer dans cette circonstance, et je le fais avec d'autant plus de plaisir que depuis bien long-temps j'ai la plus grande estime pour vos hauts talents, et que j'ai e'tudie' vos belles recherches avec la plus grande admiration, aussi je suis heureux de vous exprimer que notre Socie'te' vous a suivi dans votre immense travail avec le plus vif inte"r6t; et quoique ce travail ne soit pas entierement termine", elle sent qu'elle ne peut tarder plus long-temps a reconnaltre la haute valeur de vos recherches. Nous sommes heureux de vous voir au milieu de nous a cette occasion, et nous faisons des vceux pour que votre sante et vos forces puissent durer de longues anne"es afin d'enrichir la science de plus en plus du fruit de vos grands talents. 45. ADDRESS ON PRESENTING THE GOLD MEDAL OF THE ROYAL ASTRO- NOMICAL SOCIETY TO PROFESSOR H. D' ARREST. [From the Monthly Notices of the Royal Astronomical Society, Vol. xxxv. (1875).] IT has been already announced to you that the Council have awarded the Society's Medal to Professor H. L. D' Arrest, Director of the Observatory of Copenhagen, for his Observations of Nebulae contained in his. Resultate aus Beobachtungen der Nebelflecken und Sternhaufen and in his later and much more extensive work, Siderum Nebulosorum Observationes Havnienses, as well as for his other recent astronomical labours. It now becomes my duty to lay before you the grounds of this award ; and I feel confident that a plain statement of the nature and extent of the work accomplished by Professor D' Arrest will be sufficient to convince you that he richly deserves our medal. Professor D' Arrest has been long well known for his contributions to our science. No reader of the Astronomische Nachrichten can fail to have been struck by the untiring activity shewn by his numerous communications to that periodical, so indispensable to the astronomers of all countries. Among his discoveries I may refer to that of the interesting periodical comet which bears his name, and likewise to that of the minor planet Freia, the 76th member of the group of small planets between Mars and Jupiter, the known number of which now amounts to 142, and is yearly increasing at a rate which shews no signs of slackening. But of all the labours of Professor D' Arrest, unquestionably the most important are his observations of nebulae contained in the two works men- tioned at the commencement of this address. 342 ADDRESS ON PRESENTING THE GOLD MEDAL OF THE [45 These works would, in the opinion of your Council, even if they stood alone, amply justify the award of your medal. Nearly forty years have elapsed since the Society's medal was awarded to Sir John Herschel for his Catalogue of Nebulae and Clusters of Stars, printed in the Philosophical Transactions for 1833. In his address on that occasion, the Astronomer Royal gave an able sketch of the history of our knowledge of the nebulae up to that time, which makes it quite unnecessary for me to go over the same ground, necessarily much more feebly. I may merely recall that the three catalogues of Sir William Herschel, published in the Philosophical Transactions for 1786, 1789, and 1802, contain the places and descriptions of 2500 nebulae and star-clusters. Sir John Herschel's catalogue contains the results of his observations made at Slough, with his 20-foot reflector, between the years 1825 and 1833. These observations were undertaken for the purpose of reviewing the nebulae and star-clusters discovered by his father. The catalogue comprises 2307 of these objects, about 500 of which are new. Not content with having made this, survey of the heavens visible in this latitude, Sir John Herschel resolved to undertake a similar survey of the southern heavens ; and for this purpose he transported to the Cape of Good Hope the same instrument which he had employed in the northern hemisphere, "so as to give a unity to the results of both portions of the survey, and to render them comparable with each other." The observations required in order to carry out this grand plan were made in the years 1834, 1835, 1836, 1837, and 1838, and the fruits of these prolonged labours appeared in 1847, in the magnificent work, Results of Astronomical Observations made- at the Cape of Good Hope. The survey included the double-stars of the southern hemisphere, as well as the nebulae and star-clusters. The work contains a catalogue of 1708 of these latter objects, entirely similar in its arrangement and construction to the Catalogue of Northern Nebulae in the Philosophical Transactions for 1833, and reduced to the same epoch (1830), in order to facilitate the union of the two catalogues into one general one. Of these objects 89 are common to the two catalogues, so that the number of distinct nebulae and clusters which they contain is 3926. Both of these works of Sir John Herschel contain engraved representations of some of the most remarkable nebulae, whether of typical or of exceptional form, by means of which future observers may be able to ascertain whether any secular changes are perceptible in them. 45] ROYAL ASTRONOMICAL SOCIETY TO PBOF. H. D'ARREST. 343 The latter work also comprises valuable chapters on the apparent distri- bution of the nebulae over the heavens, and on their classification, together with many general remarks on the phenomena presented by them, which have been suggested by the author's long experience. By these labours of Sir William and Sir John Herschel, and by them almost exclusively, astronomers had now obtained a considerable amount of knowledge respecting the apparent distribution of the nebulae over the heavens, and respecting their forms and physical structure as seen through powerful telescopes. Their distances from us, however, and therefore their real distribution in space and their actual magnitudes remained matter of speculation only. Sir William Herschel, having found that many nebulse, which in inferior instruments shewed no traces of stellar composition, were, when viewed by his powerful telescopes, resolved entirely into stars, was at first inclined to believe that all nebulae were so resolvable. Hence he was inclined to regard them as so many galaxies, similar in their nature to our Milky Way, and owing their nebulous appearance to the enormously greater distances from us at which they were situated. Longer experience, however, induced him completely to change his views. Already in 1791, in a paper on Nebulous Stars, he had arrived at the conclusion that there exists a diffused self-luminous matter " in a state of modification very different from the construction of a sun or star," and that a nebulous star is one " which is involved in a shining fluid of a nature totally unknown to us," and "which seems more fit to produce a star by its condensation than to depend on the star for its existence." Again, in his paper on the Construction of the Heavens, in the Philo- sophical Transactions for 1811, he shews that although the appearances presented by diffused nebulous matter and by a star are so totally dissimilar, yet that these extremes may be connected by a series of such nearly allied intermediate steps as to make it highly probable that every succeeding state of the nebulous matter is the result of the action of gravitation upon it while in a foregoing one, and that by such steps the successive conden- sation of it has been brought up to the condition of planetary nebulae, and from this again to a stellar form. From the appearances presented by the planetary nebulae he infers that the nebulous matter is partially opaque, since the superficial lustre which 344 ADDRESS ON PRESENTING THE GOLD MEDAL OF THE [45 these objects exhibit could not result "if the nebulous matter had no other quality than that of shining, or had so little solidity as to be perfectly transparent." He also suggests that comets may be composed of nebulous matter in a highly condensed state, and that the faint nebulous branches which are often seen appended to a nucleus may be similar to the Zodiacal Light in relation to our Sun. In the same paper he finds reason to conclude that the distance of the faintest part of the great nebula in Orion probably does not exceed that of stars of the 7th or 8th magnitude, but may be much less, perhaps even not exceeding the distance of stars of the 2nd or 3rd order, and consequently that "the most luminous appearance of this nebula must be supposed to be still nearer to us." These views of Sir William Herschel respecting the gradual formation and growth of stars by the condensation of nebulous matter were still further confirmed and developed in his paper in the Philosophical Trans- actions for 1814. Sir John Herschel's graphic description of the two Nubeculse, or Magel- lanic clouds, likewise clearly shews that irresolvable nebulae, resolvable nebulae, and clusters of stars represent luminous matter in different conditions, but not necessarily at very different distances from us. The direct measurement of the distance of a nebula by determining its annual parallax must be regarded as nearly hopeless. The nearest known fixed star has a parallax of scarcely one second. Now the error to which we are liable in the determination of the place of a nebula, although, as we shall see, it may under favourable circumstances be made much smaller than has been commonly supposed, still considerably exceeds one second. Hence, unless a nebula were much nearer to us than the nearest fixed star, there would be no chance of our being able to determine its parallax. There is one method, however, by which we may expect ultimately to throw great light on the mutual relations of the nebular and sidereal systems, and on their relative distances from us : I mean by the study of their proper motions. Of course, no definite conclusion respecting the distance of an individual nebula could be drawn from the observation of its proper motion. For a nebula comparatively near to us might still have a very small proper motion, simply because its motion in space was nearly equal 45] ROYAL ASTRONOMICAL SOCIETY TO PROF. H. D' ARREST. 345 and parallel to our own. If a large number of instances, however, were taken, it might be asserted with a high degree of probability that those bodies which had a large proper motion were on an average nearer to us than those whose proper motion was small. Now we know, at least approximately, the proper motions of many of the fixed stars, and materials are gradually accumulating which will give us a much more accurate and extensive knowledge respecting them ; but of the proper motions of the nebulae we know little or nothing. Unfortunately for this object, the instruments of Sir William Herschel were not well adapted for the very accurate determination of the places of nebulae. He himself estimates that after 1785 the uncertainty of his places might amount to 1-J- minute of space in R. A., and from 1 to 2 minutes in Declination, and that his earlier observations were liable to much greater errors. Hence these observations can scarcely be employed in such a delicate research as that of the determination of proper motions. The degree of accuracy attained in Sir John Herschel's two catalogues is much greater. The author considers the probable error of a single obser- vation in his northern catalogue not to exceed 1 seconds of time in R. A., and 30" in Declination. In his Cape Observations he estimates that the error of a single observation will seldom exceed 30" of space in the direction of the parallel, or 45" in that of the meridian. Both of these catalogues give the results of the separate determinations of the place of a nebula, and therefore afford the means of calculating the probable errors of the observed places. Professor D'Arrest has thus found that the probable error of a single position is nearly 15" in R. A. and 19"'5 in Declination. Considering the comparatively recent date of these observations, however, it is plain that a considerable time must elapse before the comparison of Sir John Herschel's observations with later ones of a similar degree of accuracy can be expected to yield trustworthy results respecting the proper motions of the nebulas. M. Laugier was the first who attempted to determine the places of certain selected nebulae with much greater precision than is attained in Sir John Herschel's catalogues, in order that they might furnish a secure foundation to future investigations respecting proper motion. In the Comptes A. 44 346 ADDRESS ON PRESENTING THE GOLD MEDAL OF THE [45 Rendus of December 12, 1853 (tome xxxvii. p. 874), he gives a catalogue of the places of 53 nebulae for the beginning of 1850, selecting such as had well-defined centres or points of greatest brilliancy. It is to be regretted that no details are given respecting either the number of observations on which the places in the catalogue are founded, the mode of observation, or the telescope employed, so that the catalogue itself affords us no means of judging of the degree of accuracy of the places contained in it. Professor D'Arrest's first series of observations on the nebulae began in May 1855, and, like M. Laugier's, had for their object the accurate determi- nation of positions for the express purpose of affording means in due time -of studying the proper motions of the nebulas, and thence arriving at more certain conclusions respecting the relations between the nebular and sidereal systems than could be attained by the mere contemplation and examination of the objects themselves, even with the aid of the most powerful telescopes. The results of these observations were published in the Transactions of the Royal Saxon Society of Sciences for 1856. The number of nebulas observed amounts to 230. The observations were made at the Leipzig Observatory, of which Professor D'Arrest was then the Director, with the Fraunhofer refractor of 4J French inches in aperture and 6 feet focal length, by means of a Fraunhofer's double ring-micrometer. The magnifying power usually employed was 42 times. The nebulas were thus directly compared with neigh- bouring stars out of Bessel's and Argelander's Zones. In one night usually three and sometimes four transits of a nebula and its comparison-star were observed, the transits being taken alternately in the northern and southern halves of the ring-micrometer. In order to guard against the uncertainty which may still remain in the places of the stars of comparison, Professor D'Arrest often gives, in his description, the observed differences of right ascension and declination. He also often gives the position of the nebula with respect to the nearest stars, frequently those of the 10th and llth magnitude, which must ultimately prove most useful for the determination of the nebula's proper motion. In this last point he followed the excellent practice of Sir John Herschel ; but he was able to make more repeated measures of this kind, since, on account of the comparatively small power of the instrument, the description of the objects was of secondary importance. It should be remarked that all these measures were taken with the ring- micrometer, no mere estimations being admitted except when they are expressly mentioned. The results derived from each night's observations are given separately. The places given in the catalogues of Sir William and 45] ROYAL ASTRONOMICAL SOCIETY TO PROF. H. D' ARREST. 347 Sir John Herschel and in the small catalogue of Laugier are likewise reduced to the same epoch (1850) for the sake of comparison. We are so much accustomed to think of the observations of nebulae in connection with the most powerful instruments, that it will be no doubt a matter of surprise that a refractor of scarcely 4^- inches aperture should have been found suitable for such work. Professor D'Arrest, however, from his experience with such an instrument, estimates that it is capable of shewing nearly a thousand nebulae, that is about a third part of all that have been observed in our latitudes with the most powerful telescopes. He remarks also that the small nebulae of Herschel, mostly round or elliptical in form, can have their places determined more accurately than the majority of tele- scopic comets. Besides, in observing nebulae, there is the immense advantage of being able to repeat the observation of one and the same place on different nights. The prevailing central condensation in nebulae, which some- times attains a degree of concentration almost stellar, and which very frequently offers a well-defined nucleus, gives a great degree of definiteness to the observation. Those nebulae which, for various reasons, cannot be observed accurately are, according to Professor D'Arrest, comparatively less numerous. Of the 53 nebulae observed by Laugier, 31 have been re-observed by Professor D'Arrest. Excluding one of Laugier's right ascensions, which is evidently affected with a large error, and three of the declinations, which appear to be about 1' in error, perhaps through mistakes in copying, and assuming the probable error of one of Laugier's positions to be equal to that of the mean of three of his own single positions, Professor D'Arrest finds each of these probable errors to be about 6" both in right ascension and declination. By a provisional calculation of the probable error of his obser- vations, founded on a comparison of the several determinations with their mean, Professor D'Arrest finds that the probable error of a definitive position, that is of the mean of the observations of three nights, generally depending on 9 transits, does not exceed 4 or 5 seconds of space in each coordinate. Professor D'Arrest makes an interesting use of his comparisons of his own places with those of Sir John Herschel. The mean epoch of Sir John Herschel's observations is nearly 25 years earlier than that of his own. Hence the difference between the places of a nebula as given by the two authorities, and reduced to the same epoch, will include not merely the errors of the observations, but also the proper motion for 25 years and the difference of the star-places used in the reductions. Now, from the probable errors of Sir John Herschel's and Professor D' Arrest's places which have been 442 348 ADDRESS ON PRESENTING THE GOLD MEDAL OF THE [45 already ascertained, we can at once obtain the value of the mean of the squares of the differences between those places, supposing the differences to be entirely due to casual errors of observation. The actual mean of the squares of the differences is found to be greater than the above-mentioned mean, and the excess is due partly to the proper motions of the nebulae in the interval, partly to the differences in the star-places employed, and, very probably also partly to constant differences in the mode of observing the same nebula by the two observers. Hence Professor D'Arrest concludes that the probable amount of the annual relative motion of the nebulae with respect to the sidereal system is less than 0"'4 measured in arc of a great circle. I may appropriately conclude my remarks on Professor D' Arrest's Resultate mis Beobachtungen der Nebelflecken und Sternhaufen by a quotation from one who has himself done much in the same line of research. Speaking of Laugier's and D' Arrest's observations, Dr Schultz says : " These works have the high merit of having originated a new and important branch in the study of the nebulae ; and D'Arrest has done especial service to this study by shewing that, when what is required is simply good determinations of positions, a much greater number of nebulae than has been usually supposed may be advantageously observed with instruments of but very moderate dimensions. But his series of observations is chiefly and especially important as proving beyond the possibility of a doubt that the positions of nebulae in general are determinable with far greater accuracy than it had been previously usual to suppose; and D' Arrest's work thus made an epoch in the study of nebulae, by freeing it from the deterring prestige which had before that period been attached to it." Many other observers have since followed up the work thus begun by Professor D'Arrest. Very accurate positions of nebulae have been observed by Auwers, Schmidt, Schonfeld, Vogel, Riimker, Stephan, Schultz, and others. I may particularly mention Schonfeld's Mannheim Observations of 235 Nebulae, which appear to be extremely accurate and are published in a form that leaves nothing to be desired. This work also enjoys the immense advantage that the places of all the stars of comparison have been newly determined by the meridian observations of Professor Argelander. But a still more extensive work in the same field, and which promises to attain even a greater degree of accuracy, is that by Dr Schultz, from whom I have quoted above. This work consists of micrometrical observations of 500 nebulae made 45] ROYAL ASTRONOMICAL SOCIETY TO PROF. H. D'ARREST. 349 at the University Observatory of Upsala, with the Steinheil 13-foot refractor, employing a parallel wire-micrometer with bright spider-lines on a dark field. By means of the various series of observations to which I have referred, future astronomers will be provided with a rich store of materials for the study of the proper motions of the nebulae, and we may hope that even in our own time some valuable results may be arrived at respecting them. Professor D'Arrest's observations of nebulae were interrupted for a time by his appointment as Director of the Observatory of Copenhagen. In no long time, however, his new position gave him the opportunity of resuming his observations with the aid of greatly increased optical power. In the year 1861, the Observatory acquired a magnificent refractor, by Merz, of 15 feet focal length and 10-J French inches in aperture, of which Professor D' Arrest has given an elaborate description in a separate publication, De Instrumento magno ceqiiatorio. He considers this instrument to be intermediate, as regards optical power, between Sir John Herschel's 20-foot reflector in its best con- dition, and the excellent telescope with which Mr Lassell made his observations at Valletta. Finding that with this instrument he could not only perceive the very faintest of the nebulae discovered by the two Herschels, but could make sufficiently precise observations of them, he resolved no longer to continue the work begun in Leipzig, where he confined his attention to selected nebulas, but to enlarge his plan of operations and make a survey of the nebulas of the whole of the northern heavens. At first, indeed, it was his intention to observe all the nebulae he should meet with, whether previously known or not, with the utmost attainable precision, and that not once or twice only but repeatedly. He soon found, however, that to carry out such a plan, especially in such a climate, was beyond human powers, the number of the nebulas far exceeding all expectation. After labouring assiduously and perseveringly at these observations for more than six years, Professor D' Arrest was at length compelled by failing health to bring his work to a close. He estimates that in those six years he had not been able to make more than about one-eighth of the total number of observations which would be required in order to form a catalogue of the approximate positions of those nebulas which could be accurately observed with the Copen- hagen refractor. The results of these prolonged labours have been published in the great work, Siderum Nebulosorum Observations Havnienses, 1867. This volume contains about 4800 single positions of 1942 different nebulas. Of these 350 ADDRESS ON PRESENTING THE GOLD MEDAL OF THE [45 about 390 have either not been previously observed, or have not had their places determined. Sir John Herschel's Northern Catalogue of Nebulae and Clusters of Stars contains a larger number of objects, viz., about 2300. The difference between these numbers partly arises from the fact that D' Arrest has designedly omitted those objects in Herschel's catalogue which, in his judgment, should not be classed with the nebulae, viz., clusters and collections of stars belonging to Sir William Herschel's sixth, seventh, and eighth classes. These clusters appear to have no necessary connection with true nebulae, and they are distributed over the sphere in a totally different manner. The number of such clusters, especially near the Milky Way, might be easily greatly increased ; and in making his sweeps, Professor D'Arrest has often been surprised to find certain clusters inserted in Herschel's catalogue, while several others in the same neighbourhood were omitted. The selection appears to him arbitrary and by no means natural. He thinks too that the intro- duction of these objects would tend to vitiate any inquiries into the law of distribution of the nebulae. By far the greater number of the nebulae cannot be observed at all with bright wires, or at any rate can only be so observed by great expenditure of time and trouble. Hence Professor D'Arrest did not attempt to define therr places with all the precision of which his instrument was capable, but brought each nebula into the centre of the ring-micrometer, the smallest radius of which was 3' 40". The power employed in determining all these approximate positions was 123. The hour circle was read off to integral seconds of time, and the declination circle to tenths of a minute of arc. In fact, nearly the same method was followed which astronomers are accustomed to employ in finding the places of very faint cornets. Thus everything was scrupulously avoided which would interfere with the keen- ness of vision, and the more precise definition of place was generally left to micrometrical observations and comparisons with minute stars situated in the immediate neighbourhood of the nebula. The nebulae were generally observed in zones of about 4 or 5 in breadth, and in each zone 4 or 5, or even sometimes 7 fixed stars of the 7th or 8th magnitude were included, whose places were taken from Bessel's or Argelander's zones, or sometimes from those of Lalande. The work contains about 4000 micrometrical measures, chiefly made with the ring-micrometer. More rarely nebulae were compared with the stars and with each other by means of the wire-micrometer. Bright and small nebulae, 45] ROYAL ASTRONOMICAL SOCIETY TO PROF. H. D'ARREST. 351 having stellar nuclei, or at least an entirely regular form, were observed with all possible precision, and the differential determinations of their positions referred to neighbouring stars will, without doubt, be found of the greatest importance in the future study of their proper motions. Excluding a few nebulae, whose places do not admit of any accurate determination, Professor D' Arrest finds, from 1627 observations of declination of 525 nebulae, that the probable error of a single observation of declination is 17"'58, while from 1552 right ascension observations of 497 nebulae, he h'nds the probable error of a single observation of right ascension to be 9 '809 sec S. These probable errors are slightly less than the corresponding probable errors of Sir John Herschel's catalogues. Following the excellent example set by Sir John Herschel, Professor D'Arrest gives the results of each night's observations of a nebula separately, both as regards its place and its description. The use of an equatorially- mounted telescope has no doubt rendered this catalogue comparatively free from incidental errors and mistakes in the identification of nebulae, which will occasionally happen, in spite of the greatest care, when the observations are made with an instrument not so mounted. Lord Rosse's valuable selection from the observations of nebulae made with his gigantic reflector of 6-feet aperture appeared in the Philosophical Transactions for 1861, but, curiously enough, did not reach Professor D' Arrest's hands till 1864, when his own work was considerably advanced. This work contains sometimes brief and sometimes full descriptions of about 800 nebulae, many of them being illustrated by figures. Professor D'Arrest found that not a few of the nebulae which he had detected in the interval between 1861 and 1864 had been already observed by Lord Rosse and his assistants, and that his descriptions were generally confirmed by theirs. Very many "new" nebulae, however, still remained which had not been observed by Lord Rosse ; while, on the other hand, many which occur in Lord Rosse's work had escaped the notice of Professor D'Arrest. After this period he derived the greatest assistance from Lord Rosse's work. It is not surprising to find occasional differences and discrepancies in the descriptions of nebulae given in these two works. Professor D'Arrest mentions that he has found and observed by far the greater part of those nebulas which had been 352 ADDRESS ON PRESENTING THE GOLD MEDAL OF THE [45 observed by Herschel, but had been inserted by Lord Rosse in a list of "nebulae not found." He also succeeded in verifying the existence and determining the places of many very faint nebulae, which had been first discovered by means of Lord Rosse's telescope. In the Philosophical Transactions for 1864, Sir John Herschel published his General Catalogue of Nebulae and Clusters of Stars, and thereby laid astronomers under another very heavy obligation. This excellent catalogue contains all the nebulse and clusters of stars, both northern and southern, actually known at that date, 5063 in number, arranged in order of light ascension, and reduced to the common epoch 1860. A short description of each nebula or cluster is given in abbreviated words, made out from an assemblage and comparison of all the descriptions of each object given in his father's and in his own observations. It is not easy to over-estimate the boon which such a catalogue offers to an observer of nebulse, by enabling him "at once to turn his instrument on any one of them, as well as to put it in his power immediately to ascertain whether any object of this nature which he may encounter in his observations is new, or should be set down as one previously observed." As Sir John Herschel remarks, " For want of such a general catalogue, a great many nebulas have been from time to time, in the Astronomische Nachrichten and elsewhere, introduced to the world as new discoveries, which have since been identified with nebulae already described and well known. Many a sup- posed comet, too, would have been recognised at once as a nebula, had such a general catalogue been at hand, and much valuable time been thus saved to their observers in looking out for them again." While Sir John Herschel was engaged in the preparation of this catalogue, an important work by Dr Auwers appeared, entitled, William Herschel's Verzeichnisse von Nebelflecken und Sternhaufen, bearbeitet von Arthur Auwers, Konigsberg, 1862. This contains a complete and most elaborate reduction to 1830, from the observed differences in right ascension and polar distance with known stars, recorded in the Philosophical Trans- actions, of all the nebulse and clusters in Sir William Herschel's three catalogues; together with a separate catalogue of all those collected by Messier from his own observations or those of Mechain and others (101 in number), similarly reduced ; another of Lacaille's southern nebulas ; and one of fifty "new nebulae, comprising nearly all those observed by other 45] KOYAL ASTRONOMICAL SOCIETY TO PROF. H. D'ARREST. 353 astronomers (Lord Rosse excepted) in this hemisphere, all brought up to the same epoch." Sir John Herschel states that a comparison with Dr Auwers' results led him to the detection of several grave errors in his own work which would otherwise have escaped notice, and whose rectification has added materially to its value. Sir John Herschel's general catalogue contains the places and descriptions of 125 of the new nebulae discovered by Professor D' Arrest, and reduced by him to the epoch of that catalogue. At the end of his own work Professor D' Arrest gives a catalogue of the mean places of his 1942 nebulae, reduced to the epoch 1860 for com- parison with Herschel's general catalogue. He also gives a comparison of his own positions with the places of 223 nebulae contained in the very accurate special catalogue by Schonfeld, which has been already mentioned. In the above rapid sketch I have omitted to mention the many excellent descriptions and delineations of particular nebulae which we owe to Mr Lassell, Professors W. C. Bond and G. P. Bond, Mr Mason, Otto von Struve, Padre Secchi, and others. I must not terminate this very imperfect account of the principal additions to our knowledge of the Nebulae which have been made in recent years, without referring to the entirely new mode of investigation to which they have been subjected by means of the spectroscope. By observations of this kind, Mr Huggins and others have thrown much additional light on the nature and constitution of these mysterious bodies. Already the spectra of about 140 nebulae have been examined, and the light from many of them has been proved to emanate from glowing gas. This entirely confirms the mature view of Sir William Herschel, viz., that the condition of the luminous matter in many of the nebulae is widely different from its condition in the fixed stars. Professor D'Arrest has himself contributed to the spectroscopic obser- vations of the nebulae, and he has made the suggestive remark, that almost all the gaseous nebulae are found either within or near the borders of the Milky Way, and that there is an entire absence of them in the regions near the poles of the galaxy, in which the other nebulae so abound. I believe that a similar remark was made about the same time by Mr Proctor. A. 45 354 ADDRESS ON PRESENTING GOLD MEDAL TO PROF. H. D' ARREST. [45 It is worth mentioning that one of the most remarkable of these gaseous nebulae, viz. the planetary nebula numbered 4373 in Sir John Herschel's General Catalogue was observed as a fixed star by Lalande in 1790, and that by comparing its place so determined with the very accurate modern determinations of Schonfeld, D'Arrest, and others, it has been shewn that the proper motion of this nebula is quite insensible. I trust that the statement, however bald and imperfect, which I have just laid before you respecting the labours of Professor D'Arrest, will have convinced you that your Council have been fully justified in awarding to him the Society's medal. (The President then, delivering the Medal to the Foreign Secretary, addressed him in the following terms): Mr Huggins In transmitting this medal to Professor D'Arrest, you will express to him the admiration we feel for the skill and perseverance which he has shewn in his observations of the nebulse, and our high appreciation of the value of his labours. You may assure him of our ardent wishes that health and strength may long be spared to him, so that he may be able to make many further contributions to the progress of Astronomy. 46. ADDRESS ON PRESENTING THE GOLD MEDAL OF THE ROYAL ASTRO- NOMICAL SOCIETY TO M. LE VERRIER. [From the Monthly Notices of the Royal Astronomical Society, Vol. xxxvi. (1876).] IT has been already announced to you that the Council have awarded the Society's medal to M. Le Verrier for his theories of the four great planets, Jupiter, Saturn, Uranus, and Neptune, and for his tables of Jupiter and Saturn founded thereupon. It now becomes my pleasing duty to explain to you the grounds of this award. I need not, on the present occasion, enter into any detail respecting the previous achievements of our distinguished Associate, and the numerous and valuable researches with which he has enriched our science. These will be fresh in your recollection, and they have already been eloquently described to you from this chair. It is not many years since our medal was awarded to M. Le Verrier for his theories and tables of the four planets nearest the Sun, viz. Mercury, Venus, the Earth, and Mars. Long before this he had been occupied with the larger planets, but before proceeding further with their theories he found it necessary to establish on solid foundations the theory of the motion of the Earth, on which all the rest depend, and this again naturally led him to investigate the theories of the three nearer planets which, with the Earth, constitute the inferior portion of the planetary system. 452 356 ADDRESS ON PRESENTING THE GOLD MEDAL OF THE [46 By the comparison of these theories with observation, M. Le Verrier was led to two interesting results. He found that in order to bring the theories of Mercury and Mars into accordance with observation, it was necessary and sufficient to increase the secular motion of the perihelion of Mercury, and also the secular motion of the perihelion of Mars. Hence M. Le Verrier inferred that there existed, on the one hand, in the neighbourhood of Mercury, and on the other, in the neighbourhood of Mars, sensible quantities of matter, the action of which had not been taken into account. This conclusion has been verified with respect to Mars. The matter which had not been considered turns out to belong to the Earth itself, the mass of which had been taken too small, having been derived from too small a value of the solar parallax. A similar increase of the mass of the Earth is indicated by the theory of Venus, and a corresponding increase of the solar parallax is likewise derived from the lunar equation in the motion of the Sun. With respect to Mercury, a similar verification has not yet taken place, but the theory of the planet has been established with so much care, and the transits of the planet across the Sun furnish such accurate observations, as to leave no doubt of the reality of the phenomenon in question ; and the only way of accounting for it appears to be to suppose, with M. Le Verrier, the existence of several minute planets, or of a certain quantity of diffused matter circulating about the Sun within the orbit of Mercury. The results which M. Le Verrier had thus obtained from his researches on the motions of the interior planets added to the interest with which he now entered upon similar researches on the system of the four great planets which are the most distant from the Sun. Such researches might furnish information respecting matter, hitherto unknown, existing in the neighbourhood of these planets. Possibly they might afford indications of the existence of a planet beyond Neptune, and at any rate they would provide materials which would facilitate future discoveries. As I shall have occasion to explain later on, the theories of the mutual disturbances of the larger planets are far longer and more complicated than those of the smaller, so that all that M. Le Verrier had yet done might be almost regarded as merely a prelude to what still remained to be done. Increased difficulties, however, far from deterring, seemed rather to stimulate him to greater exertions. 46] ROYAL ASTRONOMICAL SOCIETY TO M. LE VERRIER. 357 On the 20th of May, 1872, M. Le Verrier presented to the Academy an elaborate memoir, containing the first part of his researches on the theories of the four superior planets, Jupiter, Saturn, Uranus, and Neptune. This memoir contains an investigation of the disturbances which each of these planets suffers from the action of the remaining three. Throughout this investigation the development of the disturbing function, as well as that of the inequalities of the elements is given in an algebraical form, in which everything which varies with the time is represented by a general symbol, so that the expressions obtained hold good for any time whatever. Thus the eccentricities and inclinations, the longitudes of the perihelia and of the nodes are all left in the condition of variables. The mean parts of the major axes, which suffer no secular variations, are alone treated as given numbers. At the end of the resume of the contents of this memoir, given in the Comptes Rendus, M. Le Verrier lays down the following almost appalling programme of the work still remaining to be done. It would be necessary, he says, 1. To calculate the formulae, and to reduce them into provisional tables. 2. To collect all the exact observations of the four planets, and to discuss them afresh, in order to refer their positions to one and the same system of coordinates. 3. By means of the provisional tables, to calculate the apparent positions of the planets for the epochs of the observations. 4. To compare the observed with the calculated positions, to deduce the corrections of the elliptic elements of the four planets, and to examine whether the agreement is then perfect. 5. In the contrary case, to find the causes of the discrepancy between theory and observation. Extensive as is this programme, it has already been completely carried out as regards the planets Jupiter and Saturn, and partly so as regards Uranus and Neptune. Having received from the Academy the most effectual encouragement to pursue his researches, M. Le Verrier lost no time in bringing them gradually to completion, so that they might become available for practical use. 358 ADDRESS ON PRESENTING THE GOLD MEDAL OF THE [46 Accordingly, on the 26th of August, 1872, he presented to the Academy a memoir containing a complete determination of the mutual disturbances of Jupiter and Saturn, and thus serving as a base for the theories of both these planets, which are closely connected with each other. Again, on the llth of November, 1872, he presented his determination of the secular variations of the elements of the orbits of the four planets, Jupiter, Saturn, Uranus, and Neptune. These variations are mutually depen- dent on each other, and must be treated simultaneously. Their determination consequently involves the solution of sixteen differential equations, which are very complicated in form, and can only be integrated by repeated approxi- mations. This part of the work forms a necessary preliminary to the treatment of the theory of any one of these planets in particular. On March 17, 1873, M. Le Verrier presented to the Academy the com- plete theory of Jupiter; and on July 14 in the same year he followed it up by the complete theory of Saturn, On January 12, 1874, he presented his tables of Jupiter, founded on the theory which has just been mentioned, as compared with observations made at Greenwich from 1750 to 1830 and from 1836 to 1869, and with obser- vations made at Paris from 1837 to 1867. Again, on November 9, 1874, he presented to the Academy a complete theory of Uranus. Already in 1846, in his researches which led to the dis- covery of Neptune, M. Le Verrier had given a very full investigation of the perturbations of Uranus by the action of Jupiter and Saturn. In the memoir just mentioned he gives a fresh investigation, including a full treatment of the perturbations of Uranus by the action of Neptune. On December 14, 1874, he presented a new theory of the planet Neptune, thus completing the theoretical part of the immense labours which he had undertaken with respect to the planetary system. Finally, on August 23, 1875, he presented to the Academy the com- parison of the theory of Saturn with observations. Such is a bare enumeration of the various labours for which our science is already indebted to our illustrious Associate. That any one man should have had the power and perseverance required thus to traverse the entire solar system with a firm step, and to determine 46] ROYAL ASTRONOMICAL SOCIETY TO M. LE VERRIER. 359 with the utmost accuracy the mutual disturbances of all the primary planets which appear to have any sensible influence on each other's motions, might well have appeared incredible if we had not seen it actually accomplished. I will now proceed to give a brief outline of the investigations relating to the motions of the four larger planets, with which we are now more particularly concerned. The most important parts of these investigations are printed in full detail in the volumes of Memoirs which form part of the Annals of the Observatory of Paris. As in his former researches, M. Le Verrier here also exclusively employs the method of variation of elements, and the investigations are based on the development of the disturbing function given by him, in the first volume of the Annals of the Paris Observatory, with greater accuracy and to a far greater extent than had ever been done before. The 18th Chapter of M. Le Verrier's researches, which forms nearly the whole of the 10th Volume of the Memoirs, is devoted to the determination of the mutual action of Jupiter and Saturn, which forms the foundation of the theories of these two planets. These theories are extremely complicated, and I shall endeavour briefly to point out, and to explain as far as I can without the introduction of algebraical symbols, the nature of the peculiar difficulties which M. Le Verrier has had to encounter in their treatment, and which he has so successfully overcome. These difficulties either do not present themselves at all, or do so in a very minor degree in the theories of the smaller planets. First, then, the masses of Jupiter and Saturn are far larger than those of the interior planets, the mass of Jupiter being more than 300 times and that of Saturn being nearly 100 times greater than the mass of the Earth. For this reason it is necessary to develop the infinite series in which the perturbations are expressed to a much greater extent when we are dealing with Jupiter and Saturn, than when we are concerned with the mutual disturbances of the interior planets. Also Jupiter and Saturn are so far removed from these latter planets that the disturbances which they produce in the motion of these planets are extremely small, in spite of the large masses of the disturbing bodies. But the great magnitude of the disturbing masses is far from being the only reason why the theory of the mutual disturbances of Jupiter and Saturn is so complicated. 360 ADDRESS ON PRESENTING THE GOLD MEDAL OF THE [46 Another cause which aggravates the effect of the former is the near approach to commensurability in the mean motions. Twice the mean motion of Jupiter differs very little from five times that of Saturn. In other words, five periods of Jupiter occupy nearly the same time as two of Saturn, so that if at a given time the planets were in conjunction at certain points in their orbits, then after three synodic periods they would be again in conjunction at points not far removed from their positions at starting. Hence, whatever uncompensated perturbations may have been produced in the motions of the two planets during these three synodic periods will be very nearly repeated in the next three synodic periods, and again in the next three, and so on. Hence the disturbances will go on accumulating in the same direction during many revolutions of the two planets, and will become very important. The inequalities of long period thus arising will affect all the elements of the orbits of the two planets ; but the most important are those which affect the mean longitudes of the bodies, since these are proportional to the square of the period of the inequalities, whereas the inequalities affecting the other elements are proportional to the period itself. The principal terms of the inequalities of mean longitude are of the third order, if we consider the eccentricities of the orbits and their mutual inclination to be small quantities of the first order. Terms of the same period, however, and those far more numerous and more complicated in expression, occur among those of the fifth and of the seventh order of small quantities, and M. Le Verrier has included these terms also in his approximations. But the circumstance which contributes in the highest degree to cause the superior complexity of the theories of the larger planets is the necessity, in their case, of taking into account the terms which depend on the squares and higher powers of the disturbing forces. I will endeavour to point out the nature of these terms and the manner in which they arise. By the theory of the variation of elements we are able to express at any given time the rate of variation of any one of the elements in terms of the mean longitudes and the elements of the orbits of the disturbed and the several disturbing bodies. If this rate of variation were given in terms of the time and known quantities, we should at once find the value of the 46] ROYAL ASTRONOMICAL SOCIETY TO M. LE VERRIER. 361 element for any given time by a simple integration. But this is not the case. The method of variation of elements gives us, not a solution, but merely a transformation of our original differential equations of motion. The rates of variation are given in terms of the unknown elements themselves ; and in order to find the elements from the equations so formed, we must employ repeated approximations. Let us consider this matter a little more particularly. The terms which express the rate of variation of any element may be divided into two classes: 1. Those which involve the mean longitudes of one or both of the planets concerned, as well as the elements of their orbits. 2. Those which involve the elements only. The first are called periodic terms, since they pass from positive to negative, and vice versd, in periods comparable with those of the planets themselves. The second are called secular terms, and vary very slowly, since the elements on which they depend do so. Each of the terms in the expression of the rate of variation of any element will involve the mass of one of the disturbing bodies as a factor. Hence, if all these masses be very small, all the periodic inequalities of the elements will be likewise very small, and we shall obtain a value of the rate of variation which is very near the truth if we substitute for the com- plete value of any element its value when cleared of periodic inequalities. Then the periodic inequalities in the element under consideration may be found by direct integration, supposing the elements to be constant in the terms to be integrated, and the mean longitudes only to vary. Also the secular variation of the element considered, that is the rate of variation of the element when cleared of periodic inequalities, will be given by the secular terms taken alone. If the disturbing masses, however, are not very small, this process is not sufficiently accurate, and the periodic inequalities thus found can only be regarded as a first approximation to the true values. A. 46 362 ADDRESS ON PRESENTING THE GOLD MEDAL OF THE [46 In order to find more correct values, we must substitute for the elements in the second member of the equation their secular parts augmented by the approximate periodic inequalities before found. Now, if in any periodic term we increase any element by a periodic inequality depending on a different argument, that is involving different multiples of the mean longitudes, the result will evidently be to introduce new periodic terms which will involve the square of one of the masses or the product of two of them as a factor. Similarly, if in any periodic term any element be increased by a periodic inequality depending on the same argument, the result will also introduce new terms of the second order which do not involve the mean longitudes, and which therefore constitute new secular terms. These will be particularly important if the inequality in question be one of long period. Also in the secular terms the result of increasing any element by a periodic inequality will be to introduce a new periodic term depending on the same argument. Lastly, it should be remarked that in finding the periodic inequalities of any element by integration of the corresponding differential equation, we must take into account the secular variations of the elements which were neglected in the first approximation. The new terms thus introduced, like the others which we have just described, will evidently be of the second order with respect to the masses. If the disturbing masses be large, as in the case of the mutual disturb- ances of Jupiter and Saturn, it may be necessary to proceed to a further approximation, and thus to obtain new terms, both periodic and secular, which involve the cubes and products of three dimensions of the masses. The number of combinations of terms which give rise to these terms of the second and third orders is practically unlimited, and the art of the calculator consists in selecting those combinations only which lead to sensible results. This is the chief cause of the great complexity of the theories of the larger planets, and more especially of those of Jupiter and Saturn. M. Le Verrier lays it down as the indispensable condition of all progress that we should be able to compare the whole of the observations of a planet 46] ROYAL ASTEONOMICAL SOCIETY TO M. LE VERRIER. 363 with one and the same theory, however great may be the length of time over which the observations extend. In order to satisfy this condition, he develops the whole of his formulae algebraically, leaving in a general symbolical form all the elements which vary with the time, such as the eccentricities, the inclinations, and the longitudes of the perihelia and nodes. He treats in the same way the masses which are not yet sufficiently known. All the work is given in full detail, and is divided as far as possible into parts independent of each other, so that any part may be readily verified. All the terms which are taken into account are clearly defined, so that if it should ever be necessary to carry on the approximations still further, it will be easy to do so without having to begin the investigation afresh. The whole work is presented with such clearness and method as to make it an admirable model for all similar researches. After the development of the disturbing functions, and the formation of the differential equations on which the variations of the elements depend, the first step to be taken is to determine by integration of these equations the periodic inequalities of the elements of the orbits of Jupiter and Saturn which are of the first order with respect to the masses. As we have already said, the expressions of these periodic variations of the elements are given with such generality that, in order to obtain their numerical values at any epoch whatever, it is sufficient to substitute the secular values of the elements at that epoch. The calculation of the various terms under this general form is very laborious, and it requires great and sustained attention in order to avoid any error or omission of importance. On the other hand, by substi- tuting from the beginning the numerical values of the elements at a given epoch, the calculation is rendered much shorter and admits much more readily of verification ; but the result thus obtained only holds good for the given epoch, and is thus entirely wanting in generality. In the determination of the long inequalities of Jupiter and Saturn, the approximation is carried to terms which are of the seventh degree with respect to the eccentricities and the mutual inclination of the orbits. In the next place the terms of the first order in the secular variations of the elements of the orbits are determined. After this the periodic inequalities of the second order with respect to the masses are considered. These are determined in the same form as the 462 364 ADDRESS ON PRESENTING THE GOLD MEDAL OF THE [46 terms of the first order, in order that their expressions may hold good for any epoch whatever. The formulae relating to these terms are necessarily very complicated. The coefficient belonging to a given argument depends, in general, on a great number of terms which are classed methodically. Next are determined the terms of the second order in the secular variations of the elements of the orbits. Afterwards, M. Le Verrier takes into account the influence of the secular inequalities on the values of the integrals on which the periodic inequalities depend. The last part of this chapter is devoted to the completion of the differential expressions of the secular inequalities by the determination of certain secular terms in the rates of variation of the eccentricities and the longitudes of the perihelia, which are of the third and fourth orders with respect to the masses. The 19th Chapter of M. Le Verrier's researches, which forms the first part of the llth Volume of the Annals of the Paris Observatory, contains the determination of the secular variations of the elements of the orbits of the four planets, Jupiter, Saturn, Uranus, and Neptune. In the first place are collected the differential formulae which are esta- blished in the previous chapter, and which give the rates of secular change of the various elements at any epoch in terms of the elements themselves, which by the previous operations have been cleared of all periodic in- equalities. The terms of different orders which enter into these formulae are carefully distinguished. If we were to confine our attention to the terms of the first degree with respect to the eccentricities and inclinations of the orbits, and of the first order with respect to the masses, the differential equations which determine the secular variations would become linear, and their general integrals might be found, so as to give the values of the several elements for an indefinite period. In the present case, however, the terms of higher orders are far too important to be neglected, and when these are taken into account the equations become so complicated as to render it hopeless to attempt to determine their general integrals. 46] ROYAL ASTRONOMICAL SOCIETY TO M. LE VERRIER. 365 Fortunately, however, these are not needed for the actual requirements of Astronomy, and for any definite period the simultaneous integrals may be determined with any degree of accuracy that may be desired by the method of quadratures. In this way M. Le Verrier has determined the values of the elements for a period of 2000 years, starting from 1850, at successive intervals of 500 years. The first steps in this integration were attended with some difficulties, because the determination of the numerical values of the rates of change of the several elements at the various epochs depends on the elements them- selves which are to be determined. Hence several approximations were necessary in order to obtain the requisite precision. After this work of M. Le Verrier, however, the extension of the investi- gation to other epochs, past or future, is no longer attended with the same difficulties. In fact, from his results we may at once find, by the method of differences, very approximate values of the elements at an epoch 500 years earlier or later than those which he has considered. His general formulae will then give the rates of change of the several elements at the epoch in question, and having these we can determine by a direct calculation the small corrections which should be applied to the approximate values of the elements first found. This process may evidently be repeated as often as we choose. It is important to remark that in the formulae which give the rates of change of each of the elements at the five principal epochs considered, as well as in those which give the total variations of the elements at the same epochs, the masses of the several planets appear in an indeterminate form, so that it may be at once seen what part of the variation of any element is due to the action of each of the planets, and what changes would be produced in the value of any element at any epoch by any changes in the assumed values of the masses. Consequently, when the astronomer of the future, say of 2000 years hence, has determined the values of the elements of the planetary orbits corresponding to that epoch, it will be easy for him, by comparing those values with the general expressions given by M. Le Verrier, to determine with the greatest precision the actual values of the masses, provided that all the disturbing bodies are known ; and should there be any unknown disturbing causes, their existence would be indicated by the inconsistency of 366 ADDRESS ON PRESENTING THE GOLD MEDAL OP THE [46 the values of the masses which would be found from the different equations of condition. By means of the work which has just been described everything has been prepared which is required for the treatment of the theories of the several planets. The remainder of the llth Volume of the Annals is accordingly occupied by the complete theories of Jupiter and Saturn, the former theory being given in Chapter 20 and the latter in Chapter 21 of M. Le Verrier's researches. The coefficients of the periodic inequalities of the mean longitudes and of the elements of the orbits are not only exhibited in a general form, but are also calculated numerically for the five principal epochs considered in Chapter 19 of these researches, viz. for 1850, 2350, 2850, 3350, and 3850. The long inequalities of the second order with respect to the masses, depending on twice the mean motion of Jupiter plus three times the mean motion of Uranus minus six times the mean motion of Saturn, are also determined in a similar form. Chapter 22 of M. Le Verrier's researches, forming the first part of the 12th Volume of the Annals, contains the comparison of the theory of Jupiter with the observations, the deduction of the definitive corrections of the elements therefrom, and finally the resulting tables of the motion of Jupiter. The observations employed are the Greenwich observations from 1750 to 1830 and from 1836 to 1869, together with the Paris observations from 1837 to 1867. To the results given in the Astronomer Royal's "Reduction of the Greenwich Observations of Planets from 1750 to 1830" M. Le Verrier has applied the corrections which he has found to be required by his own reduction of Bradley's observations of stars and his redetermination of the Right Ascensions of the fundamental stars, published in the 2nd Volume of the Annals (Chapter 10). The equations of condition in longitude, for finding the corrections of the elements and of the assumed mass of Saturn, are divided into two series corresponding to the observations made from 1750 to 1830, and into two other series corresponding to the observations made from 1836 to 1869. 46] ROYAL ASTRONOMICAL SOCIETY TO M. LE VERRIER. 367 Moreover, in each of these series the equations are subdivided into eight groups, corresponding to the distances of the planet from its perihelion, to 45, 45 to 90, and so on. From these are formed four final equations, the solution of which gives the corrections of the epoch, of the mean motion, of the eccentricity, and of the longitude of the perihelion, in terms of the correction required by the mass of Saturn, which is left in an indeterminate form. The substitution of these expressions in the thirty-two normal equations corresponding to the several groups above mentioned gives the residual differ- ences between theory and observation in terms of the correction of the mass of Saturn. No conclusion can be drawn from the ancient observations; but from the modern observations M. Le Verrier finds that the mass of Saturn assumed which is that of Bouvard should be diminished by about its ^yth part. This correction is very small, but M. Le Verrier regards it as well established. On the other hand, Bessel's value of the mass of Saturn, founded on his observations of the Huyghenian satellite, exceeds Bouvard's by about its The equations of condition in latitude are treated in a similar manner, being grouped according to the distances of the planet from its ascending node. From these equations the corrections of the inclination of the orbit and longitude of the node are found separately from the ancient and from the modern observations. The results differ very little, but the second solution is employed in the construction of the tables. After the application of these corrections to the elements, the agreement between theory and observation may be considered perfect; so that the action of the minor planets on Jupiter appears to be insensible, and there is no indication of any unknown disturbing causes. There are some peculiarities in the mode of tabulating the perturbations caused by the action of Saturn. The perturbations of longitude and of radius vector are not, as usual, exhibited directly, but instead of them M. Le Verrier gives the perturbations, both secular and periodic, of the mean longitude, of the longitude of the perihelion, of the eccentricity, and of the semi-axis major of the orbit, and then from the elements corrected by these 368 ADDRESS ON PRESENTING THE GOLD MEDAL OP THE [46 perturbations he derives the disturbed longitude and radius vector by the ordinary formulse of elliptic motion. Where the perturbations are large, M. Le Verrier considers this prefer- able to the ordinary method of proceeding. The perturbations of latitude being small, he applies to the inclination and longitude of the node their secular variations alone, and then determines directly the periodic inequalities of latitude. All these perturbations, whether of the elements or of the latitude, are developed in a series of sines and cosines of multiples of the mean longitude of Saturn, including a constant term, the coefficients multiplying these several terms being functions of the mean elongation of Saturn from Jupiter, which for a given elongation are developed in powers of the time reckoned from the epoch 1850. These coefficients only are tabulated with the mean elongation as the argument, and the perturbations are thence calculated by means of the ordinary trigonometrical tables. The intervals of the argument are so small, that the requisite interpo- lations are very simple, and the coefficients which relate to the four elements, and depend on the same argument, are given at the same opening of the tables. The tables have been calculated specially for the 500 years included between the years 1850 and 2350. Nevertheless they may be applied to epochs anterior to 1850, by simply changing the sign of the time reckoned from 1850. For one or two centuries before 1850 this extension will have all the rigour of modern observations, while for still earlier times the accuracy of the tables will greatly surpass that of the observations which we have to compare with them. M. Le Terrier's Tables of Jupiter are now employed in the computations of the Nautical Almanac, beginning with the year 1878. The 13th Volume of the Annals is devoted to the theories of Uranus and Neptune. These theories are not unattended with difficulties. In the first place, these planets are disturbed by the actions of the two great masses, Jupiter and Saturn, interior to their orbits, and these actions are modified by the great inequalities of Jupiter and Saturn depending on 46] ROYAL ASTRONOMICAL SOCIETY TO M. LE VERRIER. 369 five times the mean motion of Saturn minus twice the mean motion of Jupiter. In the next place, twice the mean motion of Neptune differs very little from the mean motion of Uranus, and thus arise inequalities of long period in the elements of their orbits which are large enough to produce very sensible terms of the second order. Lastly, the mean elliptic elements of the two planets are not yet suf- ficiently well known. In a preliminary chapter, the 24th, M. Le Verrier investigates formula? which are specially applicable to the case of a planet disturbed by another which is considerably nearer to the Sun. In this case it is easily seen that, by the direct action of the disturbing planet on the Sun, perturbations of large amount may be produced in the elements of the orbit of the disturbed planet, while the corresponding pertur- bations of the coordinates of the planet are comparatively small. Hence arises the advantage of considering this case apart. We have seen how closely the theories of Jupiter and Saturn are related to each other. In a similar manner the theories of Uranus and Neptune are also closely related in consequence of the great perturbations introduced into the elements of their orbits by the near approach to commensurability in their mean motions. Hence, before entering upon the separate theories, M. Le Verrier devotes Chapter 25 of his researches to the determination of the mutual actions of Uranus and Neptune, and this forms the base of the theories of both planets. The method employed is similar to that adopted in the case of Jupiter and Saturn, and the results are exhibited in the same general form. It is important to remark that the elements of Uranus and Neptune as determined from observations severally differ from their mean elliptic values by the amount of their perturbations of long period corresponding to the mean epoch of the observations. The apparent elements of Uranus and Neptune for the epoch 1850 have been carefully determined by Professor Newcomb in his excellent work on the theory of those planets which obtained the Society's Medal in 1874. A. 47 370 ADDRESS ON PRESENTING THE GOLD MEDAL OF THE [46 By the application of his own general formulae, M. Le Verrier deduces from these elements the values of the mean elliptic elements corresponding to the same epoch. It may be remarked that the mean elements thus determined will depend on the assumed masses of the two planets, and will therefore require small corrections when more accurate values of the masses have been obtained. When the secular variations of Uranus and Neptune given in Chapter 1'J were found, the elements were less accurately known, and M. Le Verrier has therefore recalculated the values of the eccentricities and longitudes of the perihelia of the two planets for the same five epochs as before, starting from the mean elliptic values of the elements above referred to. Chapter 26 contains the completion of the theory of Uranus. The last chapter, which contains the completion of the theory of Neptune, is not yet printed. The 23rd Chapter also, which contains the comparison of the theory of Saturn with observations, together with the tables of the planet, and which will form the latter part of the 12th Volume of the Annals, is not yet printed. The results of this comparison of the theory with observations have, however, been fully published in the Comptes Rendus, and I under- stand that the tables will be used for computing the place of Saturn in the forthcoming volume of the Nautical Almanac. Although the comparison of the theory of Saturn with observations shews in general a satisfactory accordance, there occur some discrepancies in indi- vidual years which are larger than might be desired. During the thirty-two years over which the modern observations extend, viz. from 1837 to 1869, the discrepancy between theory and observation, however, remains constantly less than 2" - 5 of arc, excepting in two instances, viz. in the years 1839 and 1844, when the differences amount to 4"'5 of arc. In the ancient observations only, made in the time of Maskelyne, rather larger differences occur, amounting in two instances to nearly 9" of arc. In order to test whether these discrepancies could be due to any imper- fections in the theory, M. Le Verrier has not shrunk from the immense labour of forming a second theory of the planet independent of the former, employing methods of interpolation instead of the analytical developments. 46] ROYAL ASTRONOMICAL SOCIETY TO M. LE VERRIER. 371 I learn directly from M. Le Verrier that this second investigation entirely confirms the accuracy of the first as regards the periodic inequalities, but that the secular variations of the eccentricity and longitude of the perihelion are slightly changed. The effect of these changes is to bring the theory into very satisfactory accordance with the observations of Bradley, but the discrepancies above mentioned in the time of Maskelyne and in the modern observations still remain unaffected. The character of the discrepancies shewn by the modern observations makes it very improbable that they can be due to any errors in the theory. In fact, the error appears to change almost suddenly from a positive one of 4"'4 in 1839 to a negative one of 5"'0 in 1844, a variation of nearly 9"'5 in five years. Now no terms or group of terms due to the action of the planets could thus suddenly disturb the motion in five years, at a given epoch, and then leave the motion unaffected during the following twenty-five years. M. Le Verrier is therefore inclined to think that the discrepancies arise from errors in the observations, notwithstanding that the Greenwich and Paris observations are mutually confirmatory of each other. He suggests that it is possible that the varying aspects presented at different times by the ring may affect the accuracy of the observations of the planet, and may cause changes in the personal equations of the observers, which, from being rather large in the case of the ancient observations, have gone on diminishing as the system of observation has become more perfect. One unlooked-for result follows from M. Le Verrier's comparison of his theory of Saturn with the observations. Considering that the influence of Jupiter on the longitude of Saturn may amount to 3800", it might have been expected that from observations of the planet extending over 120 years the mass of Jupiter could have been determined with great precision. M. Le Verrier has found, however, that this is not the case. The equations of condition furnished by the comparison of the heliocentric longitudes of Saturn as deduced from theory and observation contain five unknown quantities, viz. the corrections of the assumed values of four elements and the correction of the assumed mass of Jupiter. 472 372 ADDRESS ON PRESENTING THE GOLD MEDAL OF THE [40 On solving the equations with respect to the first four unknown quan- tities, the corrections to be applied to the elements are found to be greatly influenced by the indeterminate correction of the mass of Jupiter, and after they have been substituted in the equations of condition, the coefficients of the correction of the mass of Jupiter in great part destroy each other, nowhere amounting in the resulting equations to one-tenth part of their values in the primitive equations. Hence these equations are insufficient to determine the mass of Jupiter with any precision. Consequently, in the formation of the Tables of Saturn, M. Le Verrier has employed the value of the mass of Jupiter determined by the Astronomer Royal from his observations of the 4th satellite. The result which has just been noticed will appear to be less paradoxical if we consider that by far the larger part of the disturbances which Jupiter produces in the motion of Saturn is represented by the inequalities of long period which affect the mean longitude and the elements of the orbit. Now in the course of 120 years these inequalities have run through only a small part of their whole period, and therefore, during this interval, the greater part of their effects may be represented by applying changes to the several mean elements equal to the mean value of the corresponding long inequalities during the interval. It is only from the residual disturbances, which are comparatively small in amount, that any data can be obtained for the cor- rection of the mass of Jupiter. In the course of a few centuries, when these long inequalities, as well as the secular variations of the elements of Saturn, shall have had time to develop themselves, it will be possible to determine the mass of Jupiter from them with all desirable precision. I trust that the review which I have just given, however hasty and imperfect, of the work of our distinguished Associate has been sufficient to convince you that your Council have done well in according him your Medal. In conclusion, I may be allowed to express the great satisfaction I have felt in becoming the mouthpiece of the Council on this occasion, and in thus joining in doing honour to the eminent Astronomer whose untiring labours have added so greatly to our knowledge of the motions of the principal members of our Solar System. 46] ROYAL ASTRONOMICAL SOCIETY TO M. LE VERRIER. 373 (The President then, delivering the Medal to the Foreign Secretary, addressed him in the following terms) : Dr Huggins In transmitting this Medal to M. Le Verrier, you will express to him the interest with which we have followed his unwearied researches, and the admiration which we feel for the skill and perseverance by which he has succeeded in binding all the principal planets of our system, from Mercury to Neptune, in the chains of his Analysis. You can tell him how sorry we are not to see him among us on the present occasion, and how glad we shall be to welcome him if he is able to visit us later in the session. We hope that he will then have finished the printing of his "Tables of Saturn" and his "Theory of Neptune," and thus be able to rest awhile and re-establish his health shaken, we fear, by his too arduous labours until he goes forth again, with fresh vigour, to win new triumphs in the fields of Physical Astronomy. 47. ASTRONOMICAL OBSERVATIONS MADE AT THE OBSERVATORY OF CAMBRIDGE, UNDER THE SUPERINTENDENCE OF PROFESSOR ADAMS. [Extracts from the Introduction to Vol. xxi. (1861 18C5).] Corrections for Collimation, Level, and Azimuth. UP to the end of 1863 the corrections for Collimation, Level, and Azimuth were applied in the usual way, by the aid of Professor Challis's calculating machine : thence forward, they were thrown into the form cotan N.P.D. +ccosecN.P.D. where c denotes the Collimation error, considered positive when the angle between the line of sight and the eastern half of the axis is less than a right angle ; n, the elevation of the west end of the axis above the plane of the equator ; and m, the deviation of the west end of the axis southward in the plane of the equator. m, n, and c are expressed in seconds of time. It is easy to see that, if a and l> denote the deviations of the axis horizontally and vertically, or the azimuthal and level errors, expressed in seconds of time, and ^> the latitude, m = asui(f> + b cos = l> sec n tan , n= a cos + b sin , consequently a = m sin < n cos = l> tan n sec . 47] ASTRONOMICAL OBSERVATIONS MADE AT CAMBRIDGE. 375 The collimation and level errors were found by observing the reflec- tion of the wires in a trough of mercury, with a Bohnenberger's eye- piece, before and after reversing the Instrument. The deviation of the line of sight from the vertical, in one position of the Instrument, which was assumed to be illumination West, being b + c, in the other position, illumination East, it will be b c. The value of c thus obtained at any reversal of the Instrument was, up to the end of 1863, in most cases supposed constant till the next reversal and used for finding b by means of intermediate observations of the reflection of the wires. Subsequently mean values of c were generally taken. This method assumes that the position of the Y's is unaltered during the process of reversal, a supposition which was by no means borne out by the examination of the pivots in May, 1864, and it was thought better to adopt some mode of determining the errors independently for each position of the Instrument. In default of Collimating Telescopes, a star near the pole, usually Polaris, was observed both directly and by reflection at the same cul- mination ; from the times of transit reduced to the centre wire and corrected for irregularity of Pivots, the level error was easily found thus, if a be the star's Right Ascension, S its Declination, T the time of the direct observation, reduced to the centre wire and corrected for irregularity of Pivots, T' the time of the reflected observation, E the Clock correction, a, b, c the Azimuth and Level errors, and the Collimation error of the centre wire, cos o cos o cos c T' i F \ ~ 7 - -L T -" T I* S U 5, c. j COS COS O COS whence T - T' + 2b- = 0, cos o 7 1 I-TI V\ COS 8 and b ~ - ( T - f) ---- ,-.- 5,-, . 2 v cos (< 8) 376 ASTRONOMICAL OBSERVATIONS MADE AT [47 The observation of the reflection of the wires gave b + c or b c ; thence c was obtained. This mode was adopted almost exclusively from September 24, 1864, till the Instrument was finally dismounted. . The coefficient for diurnal aberration, 0",19= O s ,013, is, in every case, incorporated with the Collimation error. Correction for Curvature of Star's path. When the object is not bisected precisely on the meridian a small correction is necessary for curvature of path. For stars near the pole the correction (C) may be calculated from the formula where A is the North Polar Distance, and 6 the hour angle. Differentiating, and expressing c?A in seconds of arc, we have a dC = 2 cos 2 A sin 2 - c?A. So that, for the Polar Distance \ 9 A + n", C= -. -j-, sin 2 A sin 2 - + 2 cos 2 A sin 2 - . n". sin \. Zi Zi For Polaris, A = l 25' + n", (7- [4 -00842] sin 2 1+ [0-30050] sin 2 .n". * 2* For 51 Cephei, A -2 46' W, C= [4-29861] sin 2 1 + [0'29900] sin 2 f . n". 2i 2* For 8 Urs. Min., A = 3 25' + n", C= [4-38991] sin 2 6 - + [0'29793] sin 2 . n". 2i For \ Urs. Min., = l6' +n", C= [3-89862] sin 2 + [0-30071] sin 2 . w". 2 2i 47] THE OBSERVATORY OF CAMBRIDGE. 377 For convenience of calculation these quantities are given in Tables I, II, III, at the end of this Introduction, for values of the hour angle taken at intervals of 10" and extending to a sufficient distance from the meridian. When the star is not very near the pole, since 6 is very small, we may write which gives sin 2 6 for sin 2 - correction = - : -7. sin A cos A sin 2 6. 2 sin I" But if E be the equatorial interval corresponding to the apparent distance from the meridian of the point at which the bisection was made, then sin A sin 6 = sin E ; f . ,, sin 2 ^ theretore sin- 6 = -.. - . sin' A and correction = - -j, cot A sin 2 E ; sin i or, if E be expressed in seconds of time, sin 2 15" , correction = - ; r. cot A 2 sin 1" = sin l.& cot A. Zi In the Mural Circle, one equatoiial interval of the wires =16*' 6. Hence, if / be the number of intervals in the distance of the point of bisection from the meridian, 225 correction-- - sin l"(16'6) 2 / 2 cot A 2t = [9"-l7694]/ 2 cotA = 0"'15037 2 cotA. In practice, the middle wire is always so nearly in the meridian that / may be taken to be the number of intervals in the distance of the point of bisection from the middle wire. A. 48 378 ASTRONOMICAL OBSERVATIONS MADE AT [47 The values of the correction for different values of / and A are given in Table IV. at the end of this Introduction. Correction for Change of Declination. In the case of the Sun and Planets a small correction is required for the motion in Declination in the interval between the time of crossing the meridian and the time of observation. This interval is 16*'6/cosec A, where / has the same signification as before, and therefore the correction will be i fi^'fi - - - /cosec A x Var. of Decl". in 1 hour of longitude. The last factor is obtained from an Ephemeris. The multiplier of / in this expression, or the value of the correction for one interval, is given by means of Table V. at the end of this Introduction, so that the correction may be deduced by multiplying the number taken from the Table by /, the number of intervals stated in the eleventh column. The sign to be given to the correction is stated in the precept at the foot of the Table. The Micrometer-wire was always so nearly adjusted equatorially that no correction for error of its position has been thought necessary. The Pointer, which is used for setting the Telescope to observe an object either directly or by reflection, the setting angle to the nearest minute having been previously computed, is placed below Microscope A at an interval of 10 45' nearly from the zero of its reading. The graduation proceeding in the direction from the microscope downwards, the Pointer reading is the number of degrees and minutes of that division which in the order of graduation comes next before the position of the Pointer. It is unnecessary to place the Pointer reading in a separate column, as it may be at once inferred from the concluded Circle reading, the minutes being always an integral number of 5'. The concluded Circle reading in the twelfth column is the Pointer reading added to the mean of the Microscope readings with all the above- mentioned corrections applied. It is therefore the reading which would have been given by the Circle, if the microscopes had been in accurate adjustment for runs, and the object had been bisected by the fixed wire 47] THE OBSERVATORY OF CAMBRIDGE. 379 at the middle vertical wire. For the Polar stars the concluded reading applies to the time of meridian passage. The Circle reading corresponding to the position of the Telescope when directed exactly to the zenith is called the Zenith Point. The adopted Zenith point is obtained by means of the collimating eye-piece, and is therefore more strictly the Circle reading corresponding to the Nadir point increased by 180. The Collimating eye-piece employed is of the same form as that used by Professor Challis, and consists of a common inverting microscope of three lenses, to which is attached, beyond the third lens, a piece of plate- glass, inclined at an angle of 45 to the axis of the microscope. The eye-piece of the Telescope being removed, this apparatus is put in 'its place, so that the plate-glass is between the wires and the microscope ; and when the Telescope is directed vertically to a trough of mercury, the wires and their images by reflection become visible as dark lines on a bright ground, by throwing the light of a lamp on the plate-glass. The Micrometer reading for coincidence of the micrometer-wire with its image is deduced from at least six readings for coincidence, or for alternate contact. The Microscope readings for the determination of the Zenith point are inserted among those for the observations of the celestial objects named in the second column. The concluded Circle reading obtained by reducing an observation of Nadir point in the same manner as the other observations are reduced, and then increasing the result by 180, is in general the adopted Zenith point. The limits within which any value is used are indicated by bars across the column of " concluded circle readings." If two observations of Zenith point occur within the same limits, the value used is the mean between the two results. The temperature of the Circle room at the times of taking the Zenith point is given in the Table of observations of Runs. The apparent Zenith distance in the direct observation of any object is the algebraic excess of the concluded Circle reading above the adopted Zenith point, and for a reflection observation it is the algebraic excess of the Nadir point above the concluded Circle reading. The object is South or North of the zenith according as the excess is in either case 482 380 ASTRONOMICAL OBSERVATIONS MADE AT [47 positive or negative. The apparent Zenith distance thus obtained is used with the data in the three next columns for the calculation of refraction. The thirteenth column contains the height of the barometer, as shewn by a cistern-barometer constructed by Dollond and attached to the Circle pier. The lower surface of the mercury is raised by a screw pressing the bag till the light seen below a brass edge is excluded ; and a brass slider is brought to the upper surface to shut out the light in the same way. Before calculating the refraction, a correction of +0'01in. was applied to these Barometer-readings [see Introduction to Vol. xx., p. cxvi.] for Index-error ; but a comparison with a very fine Standard Barometer by Adie, which was mounted in the Transit Room in July, 1872, seems to she"w that this correction is too small. A large number of comparisons made between August, 1872, and the end of the year, shew that the reading of Adie's Barometer exceeds that of Dollond's by 0'055 in., and the correction of Adie's Barometer, by comparisons with the Standard Barometer at Kew, is only O'OOl in. Probably the error of the old Barometer had been gradually increasing. The fourteenth column contains the reading of the thermometer whose bulb is plunged in the cistern of the barometer. The fifteenth column contains the reading of an external thermometer, which is fixed to a stage near the north shutter-opening at a distance of four feet from the wall of the building and nine feet from the ground. It is protected from radiation and from the weather, and contiguous parts of the building prevent the direct rays of the Sun from falling upon it. The refraction is calculated by Bessel's Tables, using the convenient form in which they are given in the Appendix to the Greenwich Obser- vations for 1836. In this mode of calculation the reading of the attached is supposed to be the same as that of the external thermometer. The former reading, though not made use of, is inserted in the printed columns, to allow of correcting for the error of this supposition, if it is thought necessary. By adding the refraction to the apparent Zenith distance North or South, the true Zenith distance is found, and by adding algebraically the true Zenith distance, considered negative when north of the Zenith, to the assumed co-latitude of the Observatory, viz. 37 47' 8"'00, the 47] THE OBSERVATORY OF CAMBRIDGE. 381 Apparent N.P.D. from the observation, given in the seventeenth column, is obtained. Accordingly, when a circumpolar star is observed below the pole, in which case S.P. is appended to the name of the star in the second column, this apparent N.P.D. is affected with the negative sign. Occultations of Fixed Stars by the Moon. The following are the formulae employed in obtaining the Equations of Condition given in this volume. Let T= mean local time of observation. 1 = assumed longitude of place of observation, + when West. y+ = = approximate time on first meridian. a, 8, the Moon's Right Ascension and Declination. TT, a-, the horizontal equatorial parallax and semi-diameter, all calculated from the Ephemeris for the time t. Up to the end of 1861 the quantities given in the Nautical Almanac are sin TT T sin ' = geocentric latitude. 6 = sidereal time corresponding to time T. a', 8', the Right Ascension and Declination of the star occulted. -f-,. , sin (a a') 5 r md x = ; , .. cos b, sm 1 sin (8 S') . 5 , , A u = ; , ' + x sm 8 tan ^ (a a ), sin 1" f s i n ^ a to A t = T, . p cos sm (v a), sm 1" r 382 ASTRONOMICAL OBSERVATIONS MADE AT [47 3111 TT r . * ri,j it ' C 1 / //i /\"i 77 = -r - p \sm (b cos o cos A sin o cos Iff a Ik sin 1 cos x sn x Also let A be the correction of the assumed longitude in seconds of time ; A T the correction of T in seconds of time ; Aa, AS, &c. the corrections of a, 8, &c. in seconds of arc ; 7 7SJ - and -r- , the changes of a and 8 in a second of time, at at estimated in seconds of arc ; ,,(l + p) and -77(1+5), the sines of the true horizontal sm 1 sin 1 equatorial parallax and semi-diameter, each divided by sin 1". Calculate the following quantities : (a) = cos 8 [cos x + sin x sm ' sin ( a ')]> (8) = sin x cos x sin 8 sin (a a'), tn p sin TT cos $' [cos x cos (6 a!} + sin x sin 8' sin (6 a')], (8') = p sin TT sin x [sin ' sin 8' + cos <$>' cos 8' cos (6 a')] sin x> = (l)- (1-00274) 15m, ') = p sin TT [sin x cos <' cos 8' + sin x sin (' sin 8' cos (0 a') cos x sin ' sin (6 a')], sn , . _ "shTT 7 '' Then the final equation of condition will be Hill. 47] THE OBSERVATORY OF CAMBRIDGE. 383 Correction for Refraction. The seventh and eighth columns contain the excess of the Comet's refraction above that of the Star, in Right Ascension and North Polar Distance respectively. If the Transits of the two objects be observed across a wire placed accurately in the apparent circle of declination, which is usually the case in these observations, we shall have Excess of Comet's refraction in R.A. in seconds of time = AS x k sec 2 (8' - PQ) iau ZQ CQS ( 2 g, _ P Q^ C0gec2 g^ J. Excess of Comet's refraction in N.P.D. = AS x k sec" (8' PQ). Where the symbols have the following significations : AS is the excess of the Comet's N.P.D. in seconds of arc, PZM being the spherical triangle formed by the pole, the zenith and the middle point between the true places of the Comet and the Star, ZQ is the perpendicular from Z upon PM. 8' is the N.P.D. of the point M, or the mean of the N.P.D. of the two bodies. k is a quantity depending on the zenith distance of M, and on the state of the barometer and thermometer. PQ and ZQ are found from the hour angle (h) by means of the equations tan PQ = cot ^ cos h cos cos h sin d> cosZQ = . nr - = - j , sin PQ cos PQ where < is the latitude of the Observatory. Also , the zenith distance of M, is given by the equation cos = cos ZQ cos (8' - PQ). These formulae are equivalent to those of Bessel in his Untersuchungen, Band i. p. 168, PQ being the quantity there denoted by N, and ZQ being the complement of n. 384 ASTRONOMICAL OBSERVATIONS MADE AT [47 Professor Challis has constructed Tables similar to Bessel's, and specially adapted to facilitate the calculation of refraction for this Observatory. These tables, together with the precepts for their use, are printed at the end of this Introduction. By their means the total refractions in R.A. and N.P.D. may be found if required, as well as the differential refractions spoken of above. When the Comet is compared with a Star in N.P.D. only, with the Clock going, it is usual to bisect the two objects alternately, beginning and ending with the Star. The micrometer readings for the Star will vary in consequence of the variation of the refraction in N.P.D. From two consecutive readings, the reading corresponding to the intermediate time of bisection of the Comet may be deduced on the supposition that the readings vary proportionally to the time, and the result may be treated as if the bisections of the Comet and the Star had been simultaneous. In this case, if Aa and AS denote the approximate excesses of the Comet's R.A. and N.P.D. respectively, we have Excess of the Comet's refraction in N.P.D. 15k k = ' sin <4 cos sin h x Aa + "-- [I cos 2 6 sin 2 Ji\ x AS, cos 2 cos- 4 L where the other symbols have the same signification as before. For the observations of Mars made in 1862, for the purpose of de- termining the Sun's Parallax, the micrometer-wire was adjusted so as to be at right angles to the apparent diurnal path of a star across the field of view. In this case, we have True excess of the planet's R.A. above that of the star -D 2k tunZQ B in(8'-P<2) = apparent excess of planets R.A. ;-,-> nTvC ~ T^ " s/~ x Ad, cos" (8' PQ) 1 5 sm 8' employing the same notation as before. The ninth and tenth columns respectively contain the excesses of the Comet's R.A. and N.P.D. above the R.A. and N.P.D. of the Star, as given by the observations when cleared from the effects of refraction. In the same columns are placed the coefficients for finding the Comet's Parallax in R.A. and N.P.D. respectively. From the nature of the case, no confusion can arise from placing two such different quantities in the same column, half of the space in which would otherwise be wasted. 47] THE OBSERVATORY OF CAMBRIDGE. 385 In cases in which each comparison with a Star is complete in itself, the differences of R.A. and N.P.D. are placed opposite to the name of the Star, and the coefficients of Parallax opposite to that of the Comet ; but in the cases in which the observations are made with the clock going, and each bisection of the Comet is compared with the result obtained from combining the two bisections of the Star which immediately precede and follow it, the differences of K.A. and N.P.D. are placed opposite to the Comet and the coefficients of Parallax opposite to the Star, and usually in the line above the former quantities. These coefficients represent respectively Comet's Parallax in R.A. x A and Comet's Parallax in N.P.D. x A, where A is the distance of the Comet from the Earth, considering the Earth's mean distance from the Sun to be unity. Hence, to find the Parallax in R.A. and in N.P.D. respectively, these coefficients must be divided by A. If PZ'C be the spherical triangle formed by the pole, the geocentric zenith and the apparent place of the Comet, and if Z'Q 1 be a perpen- dicular from Z' upon PC, then the values of these coefficients will be as follows : v r( en pir cos ' cos h, sin Z'Q = cos <$>' sin h, or cos Z'Q' = - ' COS A. 49 48. ON THE MEAN PLACES OF 84 FUNDAMENTAL STARS, AS DERIVED FROM THE PLACES GIVEN IN THE GREENWICH CATALOGUES FOR 1840 AND 1845, WHEN COMPARED WITH THOSE RESULTING FROM BRADLEY'S OBSERVATIONS. [From Appendix II. to Astronomical Observations made at the Cambridge Observatory. Vol. xxii. (18661869.)] INTRODUCTION. THE present Appendix contains the formulae and instructions which I drew up, many years ago, for the formation of a proposed New Fundamental Catalogue, to be used in the computation of the Star places given in the Nautical Almanac. The proposed plan was eagerly accepted by my friend, the late Lieutenant Stratford, who was then the superintendent, and my instructions were ably carried out by Mr R. Farley, then the principal assistant in the Nautical Almanac Office. The mean places were thus calculated for the beginning of each of Bessel's so called fictitious years from 1830 to 1870. The results for the years from 1857 to 1870 inclusive have already appeared in the several volumes of the Nautical Almanac. It has been thought desirable to collect together these results as well as those for the previous years, so as to exhibit at one view a set of mean places of each star, for the beginning of each year from 1830 to 1870, founded on consistent elements. It should be remarked that in all these calculations the actual proper motion of each star is supposed to be uniform and to take place in a fixed great circle. Hence no attempt is made to take into account the variability in the observed proper motions of 48] ON THE MEAN PLACES OF 84 FUNDAMENTAL STARS. 387 Sirius and Procyon. Indeed one of the principal objects which I had in view in the formation of this Catalogue was to test how far the observed proper motions of those stars which had been long and carefully observed, could be reconciled with the hypothesis that the proper motion, when referred to the equator or ecliptic of a given date, was really uniform. The rule laid down in my instructions to Mr Farley embodies a very simple mode of representing the apparent variability of proper motion arising from the change of position of the great circles to which the star is referred, whenever the star is not very near to the pole. When the star is very near the pole, the Right Ascension and Decli- nation for the time 1800 + 1 when referred to the Equator and Equinox of 1800 is first found by adding the proper motions in R. A. and Decl. for t years to the Right Ascension and Declination for 1800, and then this Right Ascension and Declination is converted into the corresponding Right Ascension and Declination referred to the Equator and Equinox of 1800 + t by the proper Trigonometrical formulae given below. These formulae are founded upon the elements of precession given by Dr Peters in his classical work Numerus Constans Nutationis. It should be noticed that the corresponding formulae given by Mr Carrington at p. xxx of the Intro- duction to his valuable Catalogue of Circumpolar Stars are not sufficiently accurate. The quantities which he denotes by z + v, z' v' and 0, and which he employs in reducing the place of a star from one epoch 1800-M to another 1800 + t', ought to vanish identically when t = t', whereas, according to Mr Carrington's Table of Precession Constants, when t = t' 55, the value of z + v is -0"73 and that of z'-v' is +0"'73. In the rule which I gave to Mr Farley for forming the value of the secular variation of the Precession to be employed in reducing the observed Right Ascension and Declination from 1840 to 1845, it is not taken into account that different Elements of Precession are employed by Argelander and Bessel from those which are employed in the Nautical Almanac. The slight inaccuracy thence arising will, however, scarcely be appreciable. It should be remarked that the Polar Star 51 Cephei was not observed by Bradley, and consequently that this star, although included among the 84 Stars to which Mr Farley's calculations refer, does not, properly speaking, fall within the scope of my plan. The coordinates of this star for 1800, which I gave to Mr Farley as part of his fundamental data, were the means of two discordant determinations of those elements by Piazzi. Hence it is not surprising that the predicted places of this star when tested by 492 388 ON THE MEAN PLACES OF 84 FUNDAMENTAL STARS. [48 comparison with more recent observations, should prove to be sensibly in error. The following Table gives the places and the proper motions for 1800 of the remaining 83 stars embraced in the calculations. MEAN PLACES AND ANNUAL PROPER MOTIONS FOR 1800, DEDUCED FROM PLACES FOR 1755 AND 1845 AND PRECESSIONS FOR 1755, 1800 AND 1845. Name of Star Mean E.A. 1800.0 Annual Proper Motion Mean Decl. 1800.0 Annual Proper Motion h. m. i. s. ' /' // y Pegasi o. 2.57,112 - 0,00087 14 . 4 . 16,02 -0,0193 a Cassiop. o . 29 . 14,688 +0,00610 55 . 26 . 18,02 -0,0393 |3 Ceti o . 33 . 32,660 +0,01291 -19. 5. 11,77 + 0,0207 Polaris 0.52.25,375 +0,08822 88 . 14 . 24,49 + 0,0055 1 Ceti i . 14 . 1,762 - 0,00665 - 9 . 13 . 10,81 - 0,2204 a Arietis i -55-55.763 + 0,01290 22 . 30 . 34,96 0,1487 7 Ceti 2 . 32 . 57,049 - 0,01047 2-23. 6,73 -0,1823 a Ceti 2.51 .50,367 -0,00277 3.17. 47,92 -0,1114 a Persei 3.10. 7,011 +0,00288 49 . 8 . 1 1 ,48 - 0,0487 7; Tauri 3-35-37.3!9 -0,00031 23 . 28 . 30,58 -0,0600 7 1 Eridani 3 . 48 . 42,288 + 0,00259 -14- 5-J3-39 -0,1162 o Tauri 4.24.27,571 + 0,00423 16. 5 .39,11 -o,i747 a Aurigse 5 . i . 56,233 + 0,00863 45 . 46 . 38,07 -0,4294 j3 Orionis 5. 4-55.9i8 -0,00090 - 8.26.37,83 - 0,0202 (3 Tauri 5 13 39.578 + 0,00157 28 . 25 . 25,58 -0,1980 S Orionis 5.21 .47,582 + 0,00113 - 0.27.32,14 - 0,0380 a Leporis 5 . 23 . 54,707 + 0,00167 -17.58.33,48 + 0,0042 e Orionis 5 . 26 . 4,201 -0,00091 - i . 20 . 29,95 -0,0148 o Orionis 5 . 44 . 20,863 + 0,00108 7.21 .24,58 - 0,0026 fi. Geminorum 6 . 10 . 51,481 + 0,00540 22 . 36 . 7,IO -0,1269 a Can. Maj. 6 . 36 . 20,106 -0,03520 -16.27. 7,75 - 1,2273 c Can. Maj. 6 . 50 . 46,005 + 0,00075 - 28 . 42 . 32,65 -0,0109 8 Geminorum 7 . 8 . 9,908 + 0,00007 22 . 2O. 13,49 -0,0160 a a Geminorum 7 . 21 .48,902 0,01238 32. 18.43,73 -0,0758 a Can. Min. 7 . 28 . 49,438 - 0,04674 5 43 35.76 -1.0351 /3 Geminorum 7 33 3.462 -0,04772 28 . 29 . 46,37 -0,0619 15 Argus 7.59. 1,667 -0,00615 -23.44. 11,91 + 0,0668 e Hydras 8 . 36 . 10,303 -0,01223 7- 8.34,54 - 0,0384 i Ursfe Maj. 8.45.26,714 - 0,04659 48 . 48 . 57.75 -0,2769 a Hydras 9.17.45.456 -0,00214 - 7-47-56,3i + 0,0322 6 Ursae Maj. 9 . 19 . 23,808 -0,10677 52 . 34 . 46,70 -0,5656 e Leonis 9 . 34 . 28,320 0,00402 24.41 . 15,97 -0,0182 o Leonis 9.57 .42,369 -0,01770 12 . 56. 19,30 +0,0086 a Ursa; Maj. 10.51 . 15,542 -0,01647 62 . 49 . 38,58 -0,0888 5 Leonis ii . 3 . 27,011 + 0,01167 21 . 37 . 1,82 -0,1441 8 Hyd. & Crateris II . 9 . 21,011 - 0,00876 -13.41 .52,38 + 0,1777 /3 Leonis 1 1 . 38 . 50,858 -0,03532 15 .41 . 21,56 -0,1022 y Ursffi Maj. ii .43. 14,559 + 0,01142 54.48.24,17 - 0,0042 /3 Corvi 12 . 23 . 54,679 -0,00737 -22 . 17 . 19,90 -0,0673 12 Can. Yen. 12 . 46 . 38,984 -0,02185 39.24. 4,58 + 0,0573 a Virginia 13 . 14 . 40,472 - 0,00445 - 10 . 6.46,22 - 0,0386 ?) Ursaa Maj. 13 39 38,578 -0,01176 50. 18.57,68 - 0,0231 7) Bootis 13-45- 9.590 - 0,00362 19 . 24 . 20,17 - o,3543 a Bootis 14. 6.32,585 0,08003 2O. 13 .46.04 -1.9747 e Bootis 14.36. 15,145 - 0,00467 27 . 55 . 28,28 +0,0046 a 2 Libra 14-39-50.3" -0,00927 -15.12. 6,88 - 0,0592 (3 Ursse Min. 14 . 51 . 26,890 - 0,00565 74 . 58 . 23,66 -0,0361 48] ON THE MEAN PLACES OF 84 FUNDAMENTAL STARS. 389 MEAN PLACES AND ANNUAL PROPER MOTIONS FOR 1800, DEDUCED FROM PLACES FOR 1755 AND 1845 AND PRECESSIONS FOR 1755, 1800 AND 1845. Name of Star Mean E.A. 1800.0 Annual Proper Motion Mean Decl. 1800.0 Annual Proper Motion h. m. s. 8. O / // // /3 Libras 15 . 6 . 15,726 - 0,00768 - 8.38. 7,19 - 0,0146 a Cor. Bor. 15 .26. 13,406 + 0,00813 27.23.44,55 - 0,0730 a Serpentis lS-34-25,570 + 0,00744 7- 3-5I-9I + 0,0553 jS 1 Scorpii 15 -53-49,729 -0,00131 -19. 14-44,95 - 0,0202 S Ophiuchi 16. 3.52,659 - 0,00524 - 3 . 10 . 6,94 -0,1222 a Scorpii 16 . 17 . 10,043 -0,00195 -25.58.28,37 - 0,0287 e Ursffi Min. 17. 6.57,962 + 0,01472 82 . 20 . 33,63 -0,0012 a Herculis 17- 5-3I.976 -0,00193 14 . 37 . 44,02 + 0,0441 /3 Draconis i7-25-5S,273 - 0,00284 52.27. 17,71 + 0,0027 a Ophiuchi 17-25 -39,375 + 0,00604 12 . 42 . 58,90 -0,2101 7 Draconis 17-5" -57,881 +0,00077 51.31. 5,12 - 0,0396 fi. 1 Sagittarii 18 . i . 48,341 -0,00313 -21. 5.48,16 - O,OO63 a Lyree 18. 30 . 10,051 + 0,01747 38 . 36 . 19,75 + 0,2854 S Ursffi Min. 18.36.38,748 + 0,03237 86 . 33 . 43,42 + O,023I /S Lyras 18 . 42 .41,906 -0,00181 33. 8.21,26 O,O22 Aquite 18 . 56 . 13,274 -0,00571 "3 -34 -35-13 -0,0732 S Aquilse 19. 15.24,798 + 0,01465 2 -43 -37," + 0,0983 7 Aquilffi 19 . 36 . 45,029 - 0,00054 10. 8. 9,47 + O,OO28 o Aquilce 19.41. 1,390 + 0,03526 8 . 21 . 1,96 + 0,3785 j3 Aquilffi 19.45 -29,267 + 0,00076 5 55 2,55 - 0,4769 o 2 Capricorn! 20. 6.56,817 + 0,00170 -13- 9- 13-73 -O,OOO3 o Cygni 20 . 34 . 37,004 - 0,00043 44. 34. 18,28 + O,OOO5 X Ursse Min. 20.51 .33,984 -0,05293 88 . 41 . 16,41 + 0,0123 61 1 Cygni 20 . 57 . 56,873 + o,33999 37 . 46 . 24,22 + 3-2233 f Cygni 21 . 4.25,881 - 0,00264 29 . 24 . 47,98 - 0,0695 a Cephei 21 . 13.47,721 + 0,02174 61 .44.31,83 + O,0052 ft Aquarii 21 . 21 . 1,193 + 0,00014 - 6 . 26 . 36,99 + O,OO53 /3 Cephei 21 . 26 . 1,574 + 0,00084 69 . 41 . 6,41 -O,O4I2 f Pegasi 21 .34.21,675 + 0,00282 8 - 57 52-37 + O.OO2O a. Aquarii 21 .55.30,413 - 0,00098 - I . 17. 8,49 -O,OI3O f Pegasi 22.31 .29,549 + 0,00177 9 . 47 . 28,06 + O,OO25 o Pise. Aust. 22 . 46 . 34,099 +0,02319 - 30 . 40 . 42,46 -0,1745 o Pegasi 22 . 54 . 48,447 + 0,00307 14 7 54,oo -0,02 1 8 i Piscium 23 . 29 . 4O,O32 + 0,02554 4 . 32 . 36,90 -0,4512 y Cephei 23.31 . 15,471 -0,01994 76 . 31 . 0,21 +0,1516 o Andromedse 23.58. 4,639 + 0,00886 27-59- 8,39 -0,1542 Mr Farley has remarked that one of these stars, viz. e Ursae Mlnoris, is too near the pole to allow the treatment of it as an ordinary Non-polar Star to be quite satisfactory. In this case it would be preferable to use the formulae for the reduction of star places which are specially appropriate to the Polar Stars. In two other cases, viz. ft Ursse Minoris and y Cephei, the polar distances, though larger, are sufficiently small to make it expedient to use the same formulas when the greatest degree of accuracy is required. 390 ON THE MEAN PLACES OF 84 FUNDAMENTAL STARS. [48 ON A PROPOSED NEW FUNDAMENTAL CATALOGUE. I have frequently felt great inconvenience from the changes which have been made from time to time, in the Fundamental places of the Standard Stars in the Nautical Almanac. At present, also, different astronomers use different Fundamental places, so that it is impossible accurately to compare the observations made at different observatories, or at the same observatory in different years, without a troublesome preliminary investigation of the mean differences of the several catalogues employed to determine the Clock error. The appearance of the Greenwich Twelve-year Catalogue seems to me to afford an excellent opportunity for the formation of such a catalogue as astronomers in general would be likely to employ in the reduction of their observations. By comparing the places in the Greenwich Catalogue with those of Bradley given in Bessel's Fundamenta, places would be obtained, which for many years to come, might be more depended on, than those given by a year or two's observations, however near these might be to the time for which the places were wanted. In order, however, to ensure this general assent of astronomers and to do justice to the excellence of the materials, the most scrupulous accuracy should be attended to in the reduction of the places to the proposed epoch, and in the calculation of the coefficients of the 1st and 2nd powers of the time which are required and wanted in order to find the places for any other epoch. A short Appendix should be added to the Nautical Almanac in which the proposed Catalogue is given, fully explaining the method employed in its formation, in order that astronomers might use it with confidence. I proceed to point out the method which it appears to me most desirable to adopt for this purpose. The R.A. for 1840 and 1845 given in the Greenwich Catalogue are not referred to the same Fundamental position of the Equinox. The mean corrections of the R. A. of the Fundamental Catalogue in the Nautical Almanac for 1834, given by the observations of the first 6 years and of the last 6 years, differ by O s> 067. Part of this difference, however, arises from the proper motions having been omitted, except in a few cases, in the Nautical Almanac Catalogue, so that the mean corrections would vary with the time. By the comparison of the R.A. for 1840 and 1845, of the 30 stars common to the Greenwich Clock List and the Tabula Regiomontance, using as a basis Bradley's places for 1755, I find that in 48] ON THE MEAN PLACES OF 84 FUNDAMENTAL STARS. 391 order to refer the R.A. to the most probable position of the Equinox as determined from the observations of the whole 12 years, the R.A. for 1840 must be increased by O s- 028 and those for 1845 diminished by the same quantity. The mean epoch of the observations on which the Catalogue for 1840 depends is the beginning of 1839, and the observations may be looked upon as giving the places for that time, independently of any assumed proper motion. The proper motions for 1 year should therefore be added to the places for 1840 of those stars whose proper motions have not been taken into account, and to the places of the other stars should be added, for the sake of uniformity, Adopted proper motion for 1 year - Proper motion employed in the reductions. The proper motions employed may be those given in the Fundamental Catalogue in the Nautical Almanac for 1848, which are those of Argelander as far as he gives them, the rest being taken from the B. A. Catalogue. The proper motions used by the Astronomer Royal in his reductions are those given in the Nautical Almanac for 1834. For two stars, proper motions are mentioned in the notes to the Catalogue of 1439 stars, which are not given in the Nautical Almanac, viz. for a Aquilce, a proper motion of - 0"'32 in N.P. D., and for i Piscium, a proper motion of + 8 '025 in R.A., both being taken from Baily. These however are not included in the Annual Precessions of that Catalogue, and I am not quite certain that they have been used in obtaining the places for 1840. The Astronomer Royal should be consulted on this point. The R.A. for 1755 given in the Fundamenta should be diminished by 8> 020 in consequence of Bessel having employed too large a value of the coefficient of nutation in his reductions. The next step is to reduce the places for 1840 to the epoch 1845. If a denote the R.A. for 1755, a, that for 1840, and half the secular variation of the precession in R.A. be denoted by p, as in the Nautical Almanac Catalogue, then the R.A. for 1845 will be a, a 9 a ' + ~% + 2 P ' and similarly for the Declination. The value of p may be taken at once from the Nautical Almanac for 1848. The value there given, however, does not include the small terms 392 ON THE MEAN PLACES OF 84 FUNDAMENTAL STARS. [48 due to proper motion, and they are only partially included in the secular variations of precession given by Argelander and Bessel. To be rigorously exact, we should take for the value of p Secular Variation of Precession from Argelander or Bessel Value of p given in Nautical Almanac. Argelander gives the secular variation in his Catalogue ; and for stars not in that Catalogue, it may be deduced from the change of precession for 45 years, given in the Fundamenta, bearing in mind that Bessel' s precessions in R.A. are expressed in arc. From the places thus reduced to 1845 and those given for the same epoch in the Greenwich Catalogue, the final places are to be deduced, giving to each determination a weight proportionate to the number of observations on which it depends. The precessions should be calculated for 3 epochs, viz., 1755, 1800 and 1845. M. Peters' elements of precession should be employed; these are given by M. Struve in the Astron. Nachr. No. 486, and are founded on Otto Struve's investigations respecting precession combined with Le Verrier's determination of the changes of the plane of the Ecliptic. The constants to be employed are : For 1755. m = 46"'0495 log n =1 '302430, ^ = 3'06997 log ^ = 0-126339. 15 15 For 1800. m = 46"'0623 log n =1 '302346 || = 3-07082 log ^7 = 0-126255 For 1845. m = 46 //> 0751 log ft =1-302262, ^ = 3-07167 log ^ = 0-126171. 15 3 15 48] ON THE MEAN PLACES OF 84 FUNDAMENTAL STARS. 393 If a denote the R.A. in 1755 and a! the R.A. finally adopted for 1845, the R.A. for 1800 will be p having the same signification as before. Similarly, the Declination for 1800 may be found. Hence the precession in R.A. for 1800 may be calculated. Let this = c. Then the proper motion in R.A. for the same epoch will be a! - 90 " C> and similar formulae hold for the Declination. In consequence of the change of the plane to which the stars are referred, the proper motions in R.A. and Declination will not be strictly uniform, even if the actual proper motions be so. This variability of the proper motion may be very conveniently taken into account in the following manner. To the R.A. and Declination for 1845 add the proper motions for 45 years just found, and with the places thus obtained calculate the precessions. These combined with the proper motions found for 1800 will give very approximately the annual variations for 1845. Similarly, from the R.A. and Declination for 1755 subtract the proper motions for 45 years, and with the places thus obtained calculate the precessions. These combined with the proper motions for 1800 will give very approximately the annual variations for 1755. Now let c, be the annual precession calculated in this way for 1755, c that for 1800, and c' that for 1845, and let the differences of these quantities be taken according to the following scheme, Ac, c A'c Ac c'. A. 50 394 ON THE MEAN PLACES OF 84 FUNDAMENTAL STARS. [48 Then one-half the secular variation of precession for 1850, or Annual rate of variation for 1850, 127 a' and a being as before the R.A. for 1845 and 1755 respectively. Also, R.A. for 1850, Similar formulae, of course, hold for the Declination. If the difference between the determinations for 1845 exceed 0*'05 for R.A. or I" for Declination, it should be ascertained whether the places have been rightly derived from those given in the several volumes of the Greenwich Observations. I found, for instance, a discrepancy in the R.A. of a Ceti, and on examination it appeared that the R.A. for 1840 should be 2 h 53 m 55"'23 instead of 2 h 53 m 55 8 '32 ; the correction -0 8 '09 mentioned in the Introduction to the Catalogue having apparently been omitted. The calculation of the Fundamental places should be carried to 3 places of decimals in R.A., and 2 in Declination, and the calculation of the Precessions and Secular Variations should be carried to 5 places in R.A. and 4 in Declination. I may mention here that the Secular Variations of Precession given in the British Association Catalogue do not include the terms which depend on the variation of m and n. Also that for Bradley's Stars the proper motions are calculated by using Bessel's old values of the precession given in the Funaamenta, and therefore ought not to be combined with the annual precessions given in the same Catalogue, which are founded on his later elements. Consequently, with the Precessions, Secular Variations, and proper motions of the Catalogue, we cannot reproduce the places for 1755, which were taken as the basis of calculation. 48] ON THE MEAN PLACES OF 84 FUNDAMENTAL STARS. 395 EXAMPLE OF THE APPLICATION OF THE METHOD JUST EXPLAINED TO FIND THE PLACE &c. OF a CANIS MAJOBIS FOR 1850. Prop, motion (Arg.) Do. employed by Airy Difference const. Gr. Catalogue 1840 a t Adopted place 1840 Do. 1755 Sec. Variation from] ArgelanderJ (p) Naut. Almanac Difference =p adopted E.A. -0035 -0-034 -o-ooi + 0-028 638 5-89 Decl. ii -1-23 -1-14 -0-09 -1630 6-98 638 5-917 -1630 7'07 6 34 20-953=BeBsei'sE.A. -0-020 - 16 23 53-80 17 J~ -6 13-27 ~-21-96 -0-379 -0-1919 17) 344-964 13-233 + 0-0004 + 0-00061 Place in Cat. for 1845 R A. diminished by O s> 028 Adopted place for 1845 Do. 1755 Mean -0-00021 li. m. s. 638 5-917 13-233 -o-ooi 638 19-149 m obs. 6 38 19-172 127 obs. Place 1800 638 19-160 6 34 20-953 6 36 20-057 + 004 6 36 20-061 or 995'0"-91 ff -0-1871 a 7-07 -1630 -21-96 0-84 -16 30 29-87 234 obs. -163027-02 58 obs. -163029-30 -1623 53-80 -1627 11-55 + 3-79 -1627 776 502 396 ON THE MEAN PLACES OF 84 FUNDAMENTAL STARS. [48 CALCULATION OF PRECESSION FOR 1800. n 15 sin a tan S Precession in R.A. Proper motion 1800 Do. in 45 years 0-126255 9-994519 -9-470270 -9-591044 -0-38998 3-07082 2-68084 m. s. 358-207 2-64674 -0-03410 -1-534 n T302346 cos a -9-198314 PrecessionJ in Decl. J 8' -8 -0-500660 -3"'1671 -635-50 -4-3944 -1-2273 -55-23 CALCULATION OF PRECESSION FOR 1755. Correction Place to be used in 1 calculating Precession J h. m. s. 6 34 20-953 + 1-534 63422-487 n 15 sin a tan S or 98 35' 37"'30 0-126339 9-995097 -9-468336 -9-589772 -0-38884 3-06997 -16 23' 53-80 + 55-23 -162258-57 n 1-302430 cos a -9-174427 Precession! in Decl. J -0-476857 -2-9982 Precession in R.A. 2-68113 48] ON THE MEAN PLACES OF 84 FUNDAMENTAL STARS. 397 CALCULATION OF PRECESSION FOR 1845. Correction Place to be used in "I calculating Precession/ h. m. s. 6 38 19-160 -1-534 6 38 17-626 n 15 sin a tanS or 9934'24"-39 0-126171 9-993909 -9-472258 -9-592338 -0-39115 3-07167 -16 SO* 29*80 -55-23 -1631 24-53 n T302262 cos a -9-220923 Precession ) in Decl. J -0-523185 -3-3357 Precession in R.A. 2-68052 COLLECTING AND DIFFERENCING THE RESULTS. B.A. 1755 2-68113 1800 2-68084 1845 2-68052 Decl. -2-9982 -3-1671 -3"3357 9 398 ON THE MEAN PLACES OF 84 FUNDAMENTAL STARS. CALCULATION OF PLACE FOR 1850 AND ANNUAL VARIATIONS &c. FOR SAME TIME. - 0-1684 [48 Ac + A 1 . p = - 0-00034 a' a - 0-00038 + 2-64674 + 0-00002 p> S'-S - 0-1871 - 4-3944 0002 Half Sec. variatioi Annual variation. Place for 1850. 90 127 A 90 k' 5k' k 5k 1 -4* a' + 2-64638 4-5817 + 13-232 ooo -22-91 + 0-05 + 13-232 638 19-160 -22-86 -163029-30 6 38 32-392 -16 30 52-16 [Here follows Table of Elements for calculating the Mean Places of the Standard Stars, extracted from Mr Farley's Calculations of Fundamental Stars for 1850.] The Right Ascension for the time 1850 + t is and the Declination for the time IS50 + 1 is (DecL where A"c and AV are the 2nd differences of the respective precessions given in the Table. By these formulae the places were calculated for every 5th year from 1830 to 1870, the results differenced, and then interpolated for every year. 48] ON THE MEAN PLACES OF 84 FUNDAMENTAL STARS. 399 BESSEL'S FICTITIOUS YEAR. The value of the precession given by Dr Peters refers to the tropical year as the unit of time, and the places of the Stars given by him and all the other German Astronomers correspond to the beginning of Bessel's fictitious year, viz. to the instant when the Mean Longitude of the Sun = 280. It seems desirable for the sake of uniformity to adopt the same usage, and therefore the places of the Stars found from Airy will require a small correction. Greenwich Times at the commencement of the Fictitious Years d. d. d. d. 1830 Jan. o+,36i 1840 Jan. o + ,783 1850 Jan. O + .2O5 1860 Jan. o + ,628 i o + ,603 I O + ,026 I 0+448 I o-,i3o 2 + ,846 2 Q+,268 2 o + ,69O 2 O+.II2 3 o + ,o88 3 o + ,5io 3 o - ,068 3 o+,354 4 o + ,33o 4 o+,752 4 + .I74 4 o + ,597 5 + .572 5 o-,oo6 5 + 417 5 o-,i6i 6 o + ,8i4 6 o + ,237 6 o+,659 6 o + ,o8i 7 o+,057 7 0+479 7 o-,099 7 o + ,323 o+,299 8 o+,72i o + ,i43 8 o+,565 9 o + ,54i 9 o - ,037 9 o+,38s 9 o-,i92 1870 Q+,050 The Epochs to which the Greenwich Catalogues of 1840 and 1845 most nearly correspond follow the beginnings of the several fictitious years by O d> 580 and O d '627, that is by O y '001588 and O y '001716, respectively. Hence we have y. Correction to the Greenwich Place for 1840= -0'001588 x (Ann. Var. for 1840) 1845= -0-001716 x (Ann. Var. for 1845). LE VERRIER'S CORRECTIONS OF THE RIGHT ASCENSIONS OF MASKELYNE'S 35 FUNDAMENTAL STARS FOR 1755. Mr Farley's preliminary calculations were completed when Le Verrier published in the Comptes Rendus the corrections which a new and more complete reduction of Bradley's observations of these Stars shewed to be required to be applied to the Right Ascensions for 1755 as given in the TabulcB Regiomontance. 400 ON THE MEAN PLACES OF 84 FUNDAMENTAL STARS. [48 The same corrections were subsequently published in the Monthly Notices for January, 1853, and Mr Farley made the modifications which were required in order that the results might coincide with those which would have been found if the above mentioned small corrections to the places for 1755, 1840, and 1845 had been first applied, and the calculations before described had been made with the places so corrected. These modifications are as follows : As explained before, in the preliminary calculations Mr Farley applied the constant correction 8 '02 to the Right Ascensions for 1755 given in the TabulcB Regiomontance. Hence the correction to be further applied to the Right Ascension for 1755 will be =Le Verrier's correction + 5> 02. The corrections of Declination for 1755 will be 0, as well as the corrections of Right Ascension for the same date of Stars not included in Le Verrier's list. Again the correction of the place for 1845 as deduced from that for 1840 correction for 1840 correction for 1755 = correct ion lor 1840 -\ - -, and the mean of this value and of the correction for 1845 derived independently, as before mentioned, is to be taken according to the number of observations on which they respectively depend, and we shall have the adopted correction for 1845. Also, Adopted correction for 1845 + $$ (adopted correction for 1845 correction for 1755) = correction for 1857 to be applied to former results. The correction of the Proper Motion before found will be = -$ (adopted correction for 1845 correction for 1755). [Here follows a table shewing the results of calculations made in conformity with the above.] 48] ON THE MEAN PLACES OF 84 FUNDAMENTAL STARS. 401 POLAR STARS. Adopted places and proper motions of the 4 Polar Stars for the beginning of 1800, to be employed in obtaining the places for every 5th year from 1830 to 1870. B.A. 1800 Polaris 51 Cephei 8 Ursse Min. \ Ursae Min. 13 6 20-631 90 41 51-950 279 9 41-220 312 53 29762 Annual Proper Motion in B. A. Deol. 1800 Annual Proper Motion in Decl. + f -32332 88 14'24'-493 + o'-00549 -1-90675 87 16 34-340 -0-09101 + 0-48557 86 33 43-415 + 0-02306 -0-79394 88 41 16-413 + 0-01234 Constants and formula- to be employed in reducing the above places to other epochs. If denote the inclination of the Equator of 1800 + t to the fixed Equator of 1800, and if 90 z denote the Eight Ascension of the inter- section of the Equator of 1800 + with that of 1800, reckoned upon the latter, and 90 + z' denote the Eight Ascension of the same intersection reckoned on the Equator of 1800 + , then ' 2 -0"-04184025 and the as follows : = 33'26"'077 ( ,10 = - ^ 5 tano Then and tan ((f> d)' As a check the following formula may be employed, sin (a + z) cos 8 = sin (a' z') cos 8'. But as a more severe check, and in order to find still more accurately the places for 1800 + Z, we may employ the following. Let a + z = A, a' z' = A'. Then sin %(A'-A) = sm% (A 1 + A) tan (8' + 8) tan $ 6, /s , 2X cos%(A' + A) ,, tan -1- (8' 8) = rrij it tan 1 9. ' cos^(A A) The differences A' A and 8' 8 may be more accurately found from the logarithmic tables by these formulse than A' and 8' themselves can be by the formulse given before. The above was the process followed by Mr Farley, except that he calcu- lated the values of 6, z and z' for each 4th year, differenced the results and interpolated the places for every year. [Here follow the star places thus found for every year from 1830 to 1870.] PUEE MATHEMATICS. 49. ACCOUNT OF SOME TRIGONOMETRICAL OPERATIONS TO ASCERTAIN THE DIFFERENCE OF GEOGRAPHICAL POSITION BETWEEN THE OBSERVATORY OF ST JOHN'S COLLEGE AND THE CAMBRIDGE OB- SERVATORY. [From the Cambridge Philosophical Society's Proceedings. Vol. I. (1852).] THE observations, especially those of eclipses and occupations, which were made during many years by the late Mr Catton at the Observatory of St John's College, and which have recently been reduced under the superintendence of the Astronomer Royal, render it a matter of some importance to determine the exact geographical position of that Observatory. The simplest and most accurate means of doing this appeared to be, to connect it trigonometrically with the Cambridge Observatory. For this purpose, a base was measured along the ridge of the roof of King's College Chapel, by means of two deal rods terminated by brass studs, the exact lengths of which were determined by comparison with a standard belonging to Professor Miller. The extremities of the base were then connected by a triangle, with a station on the roof of the Observatory at St John's, from which, as well as from the two former points, a signal post on the roof of the Cambridge Observatory could be seen. The angles at the extremities 512 404 ACCOUNT OF SOME TRIGONOMETRICAL OPERATIONS, ETC. [49 of the base, combined with the corresponding ones at the station at St John's, furnished two determinations of the distance of the Cambridge Observatory, which served to check one another. The meridian line of the transit instru- ment at St John's passes through King's College Chapel, so that by observing the point at which it intersected the base, the azimuths of the sides of the triangles could be immediately found. The result thus obtained is, that the transit instrument of the Cam- bridge Observatory is 2313 feet to the north, and 4770 feet to the west of that at St John's College. Hence it follows that the difference of latitude is 22"'8, and the difference of longitude 5"'10; and the latitude of the Cambridge Observatory being 52 12' 5l"'8, and its longitude 23"'54 east of Greenwich, we have finally for the geographical coordinates of the Observatory of St John's College, Latitude 52 12' 29"'0 Longitude 0' 28"'64 E. of Greenwich. These operations, of course, furnish incidentally a very exact determi- nation of the orientation of King's College Chapel. The line of the ridge of the roof points 6 20''3 to the north of east. 50. PROOF OF THE PRINCIPLE OF AMSLER'S PLANIMETER. [From the Cambridge Philosophical Society's Proceedings. Vol. I. (1857).] Let be the fixed point, P the tracer, Q the hinge, W the centre of wheel, M the middle point of PQ, OQ = a, PQ = b, MW=c. The area of any closed figure whose boundary is traced out by P, is the algebraical sum of the elementary areas swept out by the broken line OQP in its successive positions. Let and i| be the angles which OQ, QP at any time make respec- tively with their initial positions. s the arc which the wheel has turned through at the same time. If now OQP take up a consecutive position, and (f>, \jj, s receive the small increments 8, 8\p, 8s, we see that 8s = motion of W in direction perpendicular to PQ. 406 PROOF OF THE PRINCIPLE OF AMSLER'S PLANIMETER. [50 Hence motion of M in the same direction =8s + cty, and therefore the elementary area traced out by QP = b (Ss + cSi/>). Also elementary area traced out by Hence the whole area swept out by OQP in moving from its initial to any other position is a 2 < + bc\jt + bs. If OQP returns to its initial position without performing a complete revolution about 0, the limits of (f> and /; are 0, and the area of the figure traced out by P is bs. If OQP has performed a complete revolution, the limits of and \fj are Zir, and the area traced out is 77 (a 2 + 2bc) + bs. 51. NOTE ON THE RESOLUTION OF x n + -2 cos net INTO FACTORS. x n [From the Cambridge Philosophical Society's Transactions. Vol. XL, Part 2 (1868).] THE relation between successive values of x m + corresponding to tV successive integral values of m is XI \ X" when m = 1 this becomes x] \ x, An exactly similar relation holds good between the successive values of 2 cos md, thus 2 cos (m + 1) 6 = (2 cos 0) (2 cos m0) - 2 cos (m - 1) 0, when m = 1 this becomes 2 cos 20 = (2 cos 0) (2 cos 6) - 2. Now let v , v 1} v 2 &c. v n be a series of quantities, the successive terms of which are connected by the same relation as that which we have seen to exist between the successive values of x m H and of 2 cos md. viz. x 408 NOTE ON THE RESOLUTION OF x" + \-2cosna INTO FACTORS. [51 3C Also as in those cases let v^2, but let v l be any quantity whatever, thus we have &c. &c. Then it is evident (1) that v n is a definite integral function of v, of w dimensions, and that the coefficient of v" in it is unity. (2) that if v l = x + -, then v n = 05 n + . (3) that if v 1 = 2 cos 9, then v n = 2 cos nd. Hence v n 2 cos na will vanish when v, is equal to any one of the n quantities, 2cosa, 2cos(a + ), 2cos(a+2 ), 2cos(a+ 1 ), \ nj \ n] \ n I and therefore v n - 2 cos na = [v, - 2 cos al v 1 -2cos(a-] -] \\V 1 - 2 cos (a + 2 ) J L \ /JL \ /J x \v 1 2 cos (a + n 1 J , for all values whatever of v r Now, put v, = 03 + - ; 03 .*. 03" + r 2 cos na x = 03-1 2 cos a 03 H 2cos(a + ) 03 H 2 cos (a + 2) L 03 JL x \ nj_\l x \ n/J r i / 27r\~i x cc H 2 cos a + n 1 - , L 03 \ n/J which is the required resolution. Similarly, if we put v l = 2 cos 6, we have 2 cos nd 2 cos na = [2 cos 0- 2 cos a] [~2 cos 0-2 cos ( a + ) 1 2 cos 6-2 cos (a + 2 ) J L V /JL \ TC /J x 2 cos 2 cos ( a + n 1 ) . V n/J 51] NOTE ON THE RESOLUTION OF x n + ~-2co S na INTO FACTORS. 409 Hence we see that the two equations just found are particular cases of the general equation from which they have been derived, v, being in one case numerically not less than 2, and in the other not greater than 2. If either x = I or 6 = 0, v 1 becomes = 2, and either of the equations gives 2-2cosa = [2-2cosa] [2 -2 cos fa + 2 -)l [2 -2 cos (a + 2\] . L \ /JL V /J xr2-2cos( / a+r^T 277N )]. L \ /J Similarly, if either x= -1 or 6 = IT, v,= -2, and either of the equations gives n f- 2 - 2 cos w/J \ n A. 52 52. ON A SIMPLE PROOF OF LAMBERT'S THEOREM. [From the British Association Report (1877).] THE following proof of Lambert's Theorem, which I find among my old papers, appears to be as simple and direct as can be desired. Let a denote the semiaxis major and e the eccentricity of an elliptic orbit, n the mean motion, and p. the absolute force. Also let r, / denote the radii vectores, and u, u' the eccentric anomalies at the extremities of any arc, k the chord, and t the time of describing the arc. Then r = a (1 e cosw), r' = a(l -ecosu'), J i ^.' _ jf j = 1 cos (/3 a) = - (y8 a) 2 ultimately ; ''' 1 /a 3 \ 4 f /r + / + *\ /r + r' - ultimately a a = -i= {(r + r' + k)* - (r + r' - k)*}, 6v> which is Lambert's theorem in the case of the parabola. 53. ON THE ATTRACTION OF AN INDEFINITELY THIN SHELL BOUNDED BY TWO SIMILAR AND SIMILARLY SITUATED CONCENTRIC ELLIPSOIDS ON AN EXTERNAL POINT. [Abstract."] [From the Cambridge Philosophical Society's Proceedings. Vol. n. (1871).] No problem has more engaged the attention of mathematicians, or has received a greater variety of elegant solutions, than that of the determi- nation of the attraction of a homogeneous ellipsoid on an external point. Poisson's solution, which was presented to the Academy of Sciences in 1833, is founded on the decomposition of the ellipsoid into infinitely thin shells bounded by similar surfaces. By a theorem of Newton's, it is known that such a shell exerts no attraction on an internal point, and Poisson proves that its attraction on an external point is in the direction of the axis of the cone which envelopes the shell and has the attracted point for vertex, and that the intensity of the force can be expressed in a finite form, as a function of the coordinates of the attracted point. In 1834, Steiner gave, in the 12th volume of Crelle's Journal, a very elegant geometrical proof of Poisson's theorem respecting the direction of the attraction of a shell on an external point. He shews that if the shell be supposed to be divided into pairs of opposite elements with respect to the point in which the axis of the enveloping cone meets the plane of contact, then the resultant of the attraction of each pair of such elements acts in the direction of the axis of the cone, and consequently the attraction of the whole shell acts in the same direction. 53] ON THE ATTRACTION OF AN INDEFINITELY THIN SHELL. 415 About three years later, M. Chasles shewed that Poisson's solution might be greatly simplified by the consideration that the axis of the enveloping cone is identical with the normal to the ellipsoid which passes through the attracted point and is confocal with the exterior surface of the shell. This mode of enunciating the direction of the attraction has the advantage of making known the level surfaces with respect to the attrac- tion of the shell on external points. In 1838, M. Chasles presented to the Academy of Sciences a very simple and elegant investigation, in which he arrives at Poisson's results respecting the attraction of a shell on an external point, by a purely synthetical method. M. Chasles' method is founded on Ivory's well-known property of cor- responding points on two confocal ellipsoids, and on some elementary propositions in the theory of the Potential. Struck by the simplicity and beauty of Steiner's method of finding the direction of the attraction of a shell on an external point, the author of the present paper was induced to think that by means of the same method of decomposing the shell into pairs of elements employed by Steiner, a correspondingly simple mode of determining the intensity of the attraction might probably be found. The author has been fortunate enough to succeed in realizing this idea, and the result is the method contained in the first part of the present paper. This method is throughout quite elementary. It requires the knowledge of only the most simple properties of ellipsoids, including Ivory's well-known property respecting corresponding points on two confocal ellipsoids. The proof of the theorem respecting the direction of the attraction differs from that given by Steiner, and harmonizes better with the method employed for determining the intensity of the force. No use is made in this method of the properties of the Potential. The second part of the present paper is devoted to what the author considers to be an improvement on M. Chasles' method of determining the attraction of a shell on an external point. Its novelty consists in the mode in which the intensity of the attraction of the shell is found. M. Chasles first compares the attractions of two confocal shells on the same external point. He then takes the outer surface of one of these shells to pass 416 ON THE ATTRACTION OF AN INDEFINITELY THIN SHELL. [53 through the attracted point, and having found the attraction of this shell by a method applicable to this particular case, he deduces from it the attraction of the general confocal shell. Now it may be remarked on this that the method of finding the attraction of the shell contiguous to the attracted point does not seem free from objection, and also that it may be doubted whether it is legitimate to include this limiting case under the general one without a special examination. If, in order to remove these objections, special considerations are introduced, the proof is thereby deprived of its simple and elementary character. Whether these criticisms on M. Chasles' method are well founded or not, the author thinks that mathema- ticians will not be displeased to see a direct determination of the attraction of a shell on an external point without the intervention of another shell whose outer surface passes through that point. In order to make the paper more complete, the author briefly shews how from the expression for the attraction of a shell, we may pass to the expression the integral of which gives the attraction of a homogeneous ellipsoid on an external point. ON THE ATTRACTION OF AN INDEFINITELY THIN SHELL BOUNDED BY TWO SIMILAR AND SIMILARLY SITUATED CONCENTRIC ELLIPSOIDS. WE shall find it convenient to consider the relations between two systems of points. A system of points is said to be related to another system of points when if x, y, z and x', y', z' be corresponding points, then x_ y _i . z ~ ~~ \Ai * \J . *~ O * x, y, z, where a, b, and c are constants. If a = b = c, the systems are similar. Volumes bounded by corresponding surfaces are in the ratio of abc : I ; for the ultimate corresponding elements are in this ratio, and therefore, by Newton's fourth Lemma, the whole volumes are in the same ratio. The shells will be supposed to be contained between two similar and similarly situated concentric surfaces ; the ratio of similitude between the inner and outer surfaces being 1 : 1 + 1, where t is indefinitely small. We may without ambiguity designate any shell by the same symbols which denote its inner bounding surface. If the principal sections of two ellipsoids be confocal the ellipsoids themselves will be said to be confocal. A. 53 418 ON THE ATTRACTION OF AN INDEFINITELY THIN SHELL. [53 Let E be an ellipsoid whose principal semi-axes are a, b, c ; and let ! 1 be a confocal ellipsoid whose principal semi-axes are a lt b v Cj. Then a t -b* = a 1 t -b l *; &c. or a? a? = b* lf = c, 2 c 2 . First Solution. Let a, b, c be the semi-axes of E the interior surface of the attracting shell, and let l+t be the ratio of similitude between the inner and outer surfaces. Let MI (whose coordinates are x v y v z,) be the attracted point, a 1} 6,, Cj the semi-axes of a confocal ellipsoid through M iy then a b c a, Xl> 6~ y " c/ 1 will be the coordinates of a point (M' suppose) on the ellipsoid E. The equations to the normal to the ellipsoid E 1 at M l are a?X /2 b'Y , c?Z or a," -- = &/ - - = Cj 2 - . x, y, z, Take -X", Y, Z the coordinates of a point M on this normal such that a* we see that the relation of M to M' is such that M is a corresponding point to M' in the system of points whose relation is a b c M is the point in which the normal to the external ellipsoid at M 1 meets the plane of contact of the cone of which M 1 is the vertex and which envelopes the attracting shell E. Let the attracting shell be divided into pairs of elements by means of double cones of indefinitely small solid angle having their vertices at the point M. Let one of these cones of solid angle So intercept a pair of elements of the shell E at P and Q. 53] ON THE ATTRACTION OF AN INDEFINITELY THIN SHELL. 419 Let P' be the point on the ellipsoid E l which corresponds to P on E. Join P'M' and produce it to Q', so that M'Q' : P'M' :: MQ : PM. Then since M and M' correspond in the above system of points so also do P and P', and the lines joining them both are divided in the saine ratio, therefore Q and Q' will be corresponding points in the same system and therefore Q' is also on the ellipsoid E l . Now by the property of corresponding points on confocal ellipsoids we have ' and QM l = Q'M'. Since the portions of the line PQ intercepted by the shell at P and Q are equal, the volumes of elements at P and Q are in the ratio of MP* to MQ', i.e. are as M'P H to M'Q' 2 or as M.P 2 to Mff; therefore the masses of these elements have attractions so that the attraction of the element P on M' = the attraction of the element Q on M', and therefore the resultant attraction of these elements will bisect the angle between Mf and M^Q, i.e. will be in the direction M : M, for since MP : MQ :: M,P : M,Q, the angle PM$ is bisected by Hence the attraction of every such pair of elements will be in the direction M^M, and therefore the resultant attraction of the shell E on M l is in this direction. We have now to find the magnitude of this attraction. Let p be the perpendicular on the tangent plane at P, then the thickness of the shell at P is pt. Hence if PN be the normal to the surface at P drawn inwards, the elementary surface intercepted by a cone whose solid angle is Seo will be Su . MP 2 sec MPN, therefore the volume of the element is C- l,rr 1^71TT Pt . 8(1) . MP 3 pt So) MP sec MPN= ^p cos 532 420 ON THE ATTRACTION OF AN INDEFINITELY THIN SHELL. [53 Hence if p=l, the attraction of the element on M l resolved in the direction .. cos PMM _pt.8a>. MP 3 M,P cos PM,N 1 ~ MP cos M PN ' " M.P 3 M.P 3 ' MP caalfiPN ' Let x, y, z be the coordinates of P, then the direction cosines of PN are y , 77 , ^ and the projection of MP upon the normal PN will be px I a 2 \ py / b~ ~~ I XX, 1111, ZZ,\1 or JlfP cos MPN=p 1 - (-^ + TT + t) Similarly Jl/jP cos PM,M is the projection of M,P upon The direction cosines of M,M are -V , -^ 1 , - 1 / , where p l is the per- Ctj Oj Cj pendicular from origin on the tangent plane at M v The projection of Mf upon M t M is Hence attraction of element at P on M l resolved in the direction M^M is MP 3 MP 3 t.&a.p, -- = t . So> . p, t (since M,P = M'P'). Let So/ be the solid angle of a cone whose vertex is M' and base the element of E' which corresponds to the element E at P. Then the volume of this cone is ultimately - So/ . M'P' 3 . But the volume of the corresponding cone will be - S 53] ON THE ATTRACTION OF AN INDEFINITELY THIN SHELL. Sr' 423 theretore ST7 M f, fl 1\ M ( , Sr' ov=- \cfcr (. --- = - latrs 4ir) \r rj 4?rJ r'- _M f , s ,do-_Ma'Sa' (do- ~ Q '' ~~ " 4?r Now the volume of the cone whose base is So- and vertex and radius unity is - Scr ; hence the volume of the corresponding cone enveloping o the element at P, or P' is -a'b'c'oo-; therefore if oa> be the solid angle o of the cone or and we have Scr _ 8d) ^f ab'c" 2 T , If ,,. , 8V = a' da' -rrr, 47T lao'c' MSa' Hence it follows that the attraction of shell E at P 1 in the direction p p , . 8V . M 8a' M a'8a' i ' L e> p D/ ' 1S T777 7>~~JP' ~ a'l)' c' ' P P' ' Now if x = a 1 l, y = b 1 m, z = c^i be the coordinates of P 1} those of P' will be a'Z, 6'w, c'n and the projections of P,P' on the axes will be I8a', m8b', * Putting for I the value =-^. a' and so for m and n, we get ct X U "Z - but the direction cosines of the normal are as - ,, : frr : ,~ a' 1 b'- c'~ Hence P,P' is ultimately in the direction of the normal at P,. Hence attraction of shell E at P l which has been shewn to act in the direction of this normal = ,,-f 1 , , where p 1 is the perpendicular from ct/ u c on the tangent plane at P,. If we call p the density of shell E, the volume of the shell is lirtabc, and we have M = 4tirptabc, therefore the attraction of the shell = y- . t . ,. 424 ON THE ATTRACTION OF AN INDEFINITELY THIN SHELL. [53 We may regard a homogeneous ellipsoid as made up of indefinitely thin shells. Let X, Y, Z be the components in the direction of the axes of the attraction of an ellipsoid whose semi-axes are a, b, c on the point P l , and let X + BX, Y+8Y, Z+8Z be the attractions of a similar ellipsoid whose semi-axes are a + 8a, b + 8b, c + 8c, where 8a = at, Sb = bt, Sc = ct, k-rrpabc p,x lirpbc ?v s then 8X = ^ . tp l . L V = r- - \ x a - aA^i of o l c l a," Let u = , then Su = . Se^ ultimately, a, ar and a 1 8(t 1 = p l 1 t, hence 8t* = j . J*, 2 . 8a ; 9j hence SA r = 477/30; . , . Sw, c a SZ = 47T02 . , - . -. ; . 6U. &iC, C,- We have now to substitute for the quantities -, , j\ , &c. OjCj (>, Since a l '-a t = b 1 > -l? = c l *-+l on the right-hand side of the resulting equation, we shall find in which C r n denotes the coefficient of of in the expansion of ( 1 + x) m+ \ This equation gives B n when B v B^.-.B^ are known. Now let B n /. + (- !)"(/-!), where ( l) n f n is the fractional part of B n given by Staudt's Theorem, so that / is an integer. 542 428 ON THE CALCULATION OF THE [54 Substituting in the above equation, and writing for simplicity C r instead of (7,", as we may do without ambiguity, we have Now by Staudt's Theorem the fraction ^ occurs in each of the fractions / ; hence the quantity arising from this fraction in CJ\ + C t f 3 + &c. + C n f n will be Also, by the same Theorem, if 2r+l=p be an odd prime number, the fraction will occur in each of the fractions f r , f^, f 3r , &c. P Hence the part of and a test of the correctness of the work is supplied by the divisions by n r and 2n 2r+l being performed without leaving any remainder. I have proved that if n be a prime number, other than 2 or 3, then the numerator of the nth number of Bernoulli will be divisible by n. This forms another excellent test of the correctness of the work. I have also observed that if q be a prime factor of n, which is not likewise a factor of the denominator of B n , then the numerator of B n will be divisible by q. I have not succeeded, however, in obtaining a general proof of this proposition, though I have no doubt of its truth. TABLE I. Formation of the quantities f n . n I 2 + 3 + 7 42 3 2 + 3 + 5 = 3 4 2 + 3 + i7 = 66 5 2 + 3 + 5 + 7 + TS ~ 2730 I + 1 = S 6 7 2 + 3 + s + l7~sio 8 2 + 3 + 7 + i9 = 798 9 / 1 + I + I + = 37i 2 3 7 ii 33 1 + I + = II 2 3 23 138 2 3 5 7 13 2730 - - = a 3 6 . 2 3 5 29 870 1 i i i i ! 574S - + -H --- h H -- = 3 2 3 7 ii 3i 14322 2 3 5 17 51 I + I-5 23 6 n 10 , , . = 2 3 5 7 13 19 37 1919190 18 54] BERNOULLIAN NUMBERS FROM M TO TABLE I. (continued). /. n 2 + 3~6 i i , i i , i I554I '9 20 2 3 S ii 4i 13530 lit i _ 1805 2 3 7 43 ~ 1806 21 1 + i i + i _ 743 2 3 5 23 690 22 2 + 3 + 47 = 282 23 I 2 1,1,1,1,1 60887 24 3 ' 5 ' 7 ' 13 ' 17 46410 i i i 61 25 ill i i673 26 2 ' 3 ' 5 ' 53 1590 i i i i 821 27 i i i i 929 - + -+- + = s- z 2 3 5 29 870 28 2 + 3 + 59~354 2 9 I + I 5 + 7 i i i i _ 79085411 ii i3 T 3i T 6i 56786730 30 5 + H 31 2 3 5 17 "510 32 ill i i 66961 33 2 ' 3 ' 7 ' 23 ' 67 64722 1 + 1 + 1 = 31 2 3 5 30 34 i i i i 4397 2 3 ii 71 4686 35 i i , I I I I 188641729 36 5'7 13 19 37 73 140100870 i i _5 37 1 + i + i _3 ! 2 3 5 ~ 3 38 i i - + - l + l + i + J_-38i i 3 7 79 33i8 i i i i 277727 J 1 1 L ''' ' 39 ^O 2 3 T 8l '+1 + 1 + 1 + ^ + ^ + i . 2 + 3 + 5 + 7 + i3 + 2 9 + 43' 2 + 3 + 5 + 2l + 8T 1 i i i i i 2 + 3 + 7 + IT + r 9 + 3l : 1 + 1 + 1 + -L: 2 3 5 47 -H 1 1 1 H 1 : 2 3 5 7 13 i? 97 2 + H + n + ior i+H+4 1 I I I - + - + -H : 2 3 5 53 2 3 107 2 + 3 + 5 + 7 + i3 + i9 + 37 + io9 : i i i i 2 + 3: + -n + 23 2 + 3 + 5 + '7 + 29 + M3 421 "498 4462547 ' 3404310 _ 66817 "61410 . 313477 "272118 = I487 ,5 . 5952449 " 4501770 .378oi " 33330 .4265 "4326 .1673 "1590 541 "642 280724077 '209191710 1469 "1518 1897709 1671270 2 3 7 42 2 3 5 59 i . i 1770 ; S l + l + i + l + -L + -L + J_ + J. i ^3299288581 23 5 7 ii 13 3* 41 01 2328255930 2 3 i ' 6 31 431 n 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 432 ON THE CALCULATION OF THE [54 TABLE II. Values of I n , or the Integral parts of Bernoulli's numbers. The values of I I to 7 6 are zero. 7 55 529 6192 86580 H 25517 272 98231 6015 80874 i 51163 15767 42 96146 43061 1371 16552 05088 48833 23189 73593 19 29657 93419 40068 841 69304 75736 82615 40338 07185 40594 55413 21 15074 86380 81991 60560 1208 66265 22296 52593 46027 75008 66746 07696 43668 55720 50 38778 10148 10689 14137 89303 3652 87764 84818 12333 51104 30843 2 84987 69302 45088 22262 69146 43291 238 65427 49968 36276 44645 98191 92192 21399 94925 72253 33665 81074 47651 91097 20 50097 57234 78097 56992 17330 95672 31025 2093 80059 11346 37840 90951 85290 02797 01847 2 27526 96488 46351 55596 49260 35276 92645 81470 262 577IO 28623 9576O 47303 04973 61582 O2O8l 44900 32125 082IO 27180 32518 20479 2 342 64985 24352 I041I 41 59827 81667 94710 9I39I 70744 95262 35893 66896 030II 5692 06954 82035 28002 38834 56219 I2IO5 86444 80512 97l8l 8 21836 29419 78457 56922 90653 46861 73330 14550 89276 28860 1250 29043 27166 99301 67323 39829 70289 55241 77196 36444 84775 2 OOI55 83233 24837 02749 25329 19881 32987 68724 22OI3 28259 I59I5 336 74982 91536 43742 33396 67690 33387 53016 21959 89471 93843 67232 59470 97050 31354 47718 66049 68440 51540 84057 90715 65106 90499 04704 no 11910 32362 79775 59564 13079 04376 91604 63051 14442 23148 86269 99497 2I 355 2 5954 5 2 535 01188 65838 50190 41065 67897 32987 39163 46921 18045 934 43 32889 69866 41192 41961 66130 59379 20621 84513 68511 80910 91449 86557 88033 9188 55282 41669 32822 62005 55215 50189 71389 60388 91627 19959 59100 44871 13437 20 34689 67763 29074 49345 50279 90220 02006 59751 40253 37827 70239 36918 42141 08241 4700 38339 58035 73107 85752 55535 00606 06545 96737 36975 90579 15139 76356 41204 83354 II 31804 34454 84249 27067 51862 57733 93426 78903 65954 75074 79181 78993 54166 54911 76373 2838 22495 70693 70695 92641 56336 48176 47382 84680 92801 28821 28228 53171 44648 65111 0702! 7 40642 48979 67885 06297 50827 14092 09841 76879 73178 80887 06673 11610 03487 48532 84412 10855 2009 64548 02756 60448 34656 19672 71536 31868 67270 82253 28766 24346 13019 89213 56500 97796 98883 5 66571 70050 80594 14457 19346 03051 93569 61419 46828 75104 20621 38756 44521 52460 86197 22777 9 S 4 CC 1658 45111 54136 21691 58237 13374 31991 23014 94962 61472 54647 27402 46681 55898 78137 71265 07431 49939 5 03688 59950 49237 74192 89421 91518 01548 12442 37426 49032 14141 52565 13225 28310 97674 29893 27917 85387 1586 14682 37658 18636 93634 01572 96643 87827 40978 41277 89638 80472 86451 42973 11365 09885 00683 12009 45 IZI 5 17567 43617 54562 69840 73240 68250 71225 61240 84923 59305 50859 06216 69403 18108 29579 66515 49771 87766 32444 1748 89218 40217 11733 96900 25877 61815 9 I 45 I 41476 16182 65448 72627 34721 58762 12289 52384 00153 32666 64382 79521 6 11605 !9994 95218 52558 24525 26426 41677 80767 72684 67832 00716 84324 01127 35747 50763 44103 14895 29605 90861 82635! 2212 27769 12707 83494 22883 23456 71293 24455 73185 05498 77801 50566 55269 30277 36635 00257 26591 02528 03139 11549 56836! 8 27227 76798 77096 98542 21062 45998 45957 31204 65051 84335 66283 84885 29885 84472 02350 07188 81721 85613 01633 96614 2740: 3195 89251 11415 70958 35916 34369 18081 48735 26276 67109 91122 73184 50424 31195 31118 14531 48045 43981 20342 28242 29698 2030: 1 54] BERNOULLI AN NUMBERS FROM TO J3 433 TABLE III. Formation of the quantities 2 (C*+ C 2 " + <7 3 +&c.) for the several values of n. n = 3i, 271+1 = 63 1537 22867 461 16860 219 60409 88 02459 49 71096 53 92944 25 89740 2 55909 21 70028 27 76703 13 22870 2 29252 64231 2155 i 28091 29301 20574 87155 61155 89900 79873 52292 10967 95355 69063 74991 7403S 40869 46673 02320 58743 95452 24385 25380 27886 76823 23061 24995 24512 17745 26589 88805 16186 84091 i I9I33 651 2503 77529 91803 05258 98382 29514 14054 5046 4273 2795 2789 206 578 102O 1431 578 26O 34, 63505 79051 66215 32955 25378 37334 19219 03495 54913 23484 80684 54780 12226 =69 2 5 5944O II254 3 97842 62172 13977 07736 16639 71972 50266 01161 82703 16095 9OH4 54858 9I33I 374II 18937 09120 9290 75285 95667 53612 99020 40010 94479 58946 12302 51040 78584 62742 20632 59664 31680 29452 40848 50408 782 60957 23943 31633 55153 62 96488 i8 88946 8 99498 3 43444 2 67607 i 02597 I 72198 49571 10859 25221 88366 71771 47291 10849 224 64382 593 '3 37768 74927 5346 07084 71642 71709 28203 99066 93'39 19061 09459 39080 48794 50294 3737 61936 41141 94562 37166 51546 29344 27205 72401 57538 57622 35033 96015 43110 42452 39750 72571 J8745 18261 90131 31180 00353 66685 47375 86775 36225 24900 48160 52525 01910 67150 77200 47800 10900 59280 61525 43090 925 71=32, 271+1=65 6148 91469 12365 17205 1844 67440 78004 51891 878 28402 58774 41612 335 39464 88292 66520 223 60238 81321 77796 212 48412 70622 02255 132 24055 48295 17680 7 92100 73035 54800 67 77266 56354 24360 97 06794 53315 77584 67 77266 45331 22640 15 89482 13224 66632 5 28067 15357 03200 26216 09982 26400 30 98315 75760 117 99840 i 35408 10037 56709 02711 61583 n=35, 3 93530 i 18059 56222 20307 17631 9476 11647 1592 3091 5979 3091 1592 212 I 54023 I62O7 22228 54192 27285 97I3I 24748 56135 15922 80932 89919 80924 12768 00697 63547 98 4 9I37I 73838 77668 87532 44797 56938 49522 43305 26676 29302 00693 17633 76557 72275 48259 19761 i 01141 04179 00142 84501 09166 08047 93318 29223 24496 88224 26746 85528 04840 07416 88820 16760 86616 94327 i 6 43767 96768 67906 53491 n = 38, 271+1=77 251 85954 75 55786 35 97897 14 07667 10 03408 3 49129 6 i533i 2 73908 27132 68271 3 I53I9 3 '53'9 2 32560 68269 2189 57530 37256 O2I34 77399 02996 42731 21952 '3542 08236 07470 07470 91224 49998 50580 6o593 80416 9 47744 39445 94197 60778 5I53I 82888 58309 34485 19627 61443 41356 4I35J 02591 54229 45903 02390 06944 97266 338 73045 51219 IIOOO 65828 07890 65295 56115 77900 04320 48460 61005 27010 30220 70080 64876 75400 70980 73130 70540 70655 n=33, 271+1=67 24595 7378 3513 1286 993 790 628 34 202 322 322 loo 38 2 65876 69762 26846 9:831 86927 52070 02997 86773 22050 24808 24808 12378 91854 76018 652 49460 86248 20946 91837 65468 97251 66186 84299 48830 87800 90141 92181 07956 41677 7247 142 68821 27187 84437 73500 89636 51119 40508 47509 20584 26624 73456 70152 32584 12240 52432 06840 56528 _ 40210 35660 25758 34158 71 = 36, 271+1 = 73 15 74122 4 72236 2 24885 83272 69855 31238 45962 8380 4226 8998 23596 I5 3 88 1587 16095 64829 31546 34533 72448 21785 07685 47061 65484 38364 77424 10558 08896 09213 35897 77780 08934 33429 46682 2464 141 79339 12155 44061 27639 75084 85271 98364 42427 93052 87858 42745 243 04565 69043 14875 40526 72825 25359 39887 96540 64436 13424 62297 68878 84128 32448 54376 02336 95216 18636 876 2 5 72769 09012 71820 60672 1=39, 271+1 = 79 1007 43818 302 23145 143 91684 56 93504 37 45oi7 12 88839 21 08252 14 16901 66909 I 78887 10 75856 13 11063 10 75856 3 98368 19218 126 4 30121 49042 5748o 74720 22130 38881 54612 66512 30148 63430 11203 52II3 11203 04821 99539 26129 48900 393 90978 07049 60840 88405 58920 13420 02295 98796 "'43 03495 04663 82460 03879 03942 58487 07932 05828 92035 28987 397 92181 49043 57721 43478 48100 56975 24587 73665 06968 63160 02965 544io 58270 62910 58356 91940 36780 88635 53715 08955 102 74937 93013 77043 72350 A. 410 18153 91632 42827 74968 1637 37464 41384 23961 32815 55 434 ON THE CALCULATION OF THE [54 TABLE III. continued. 71 = 40, 271 + I = 8 1 4029 75273 I2O8 92581 575 67896 226 98241 141 76943 52 94048 69 78059 68 90498 i 71499 4 54952 35 20983 51 80299 47 04148 21 69264 i 53373 1616 223 20487 96157 17212 27456 58308 29982 5 '623 33065 19492 4747 63937 76644 18215 26375 943i 54458 12155 17 63915 28686 51987 80596 47224 14478 76241 47086 05921 69994 92297 86795 71619 51721 01230 25306 04209 25678 07629 35738 68725 33395 95532 495 6 7 55020 06223 31890 33520 89432 82928 95070 32061 22800 20720 96240 95200 10320 77880 46120 05950 1080 6538 29904 13950 88360 49673 64476 19342 9210 2228 1117 733 1377 23 27 336 703 771 534 75 i 04371 81311 94154 87500 32141 20270 36454 41466 31422 68771 48147 98290 01108 89146 86550 35781 63 27802 38296 39608 60249 19003 21381 62045 64740 25211 06277 32426 29570 82585 27241 72146 53889 00500 49788 25330 IOI2 = 85 22650 68748 40030 60409 23160 48307 71240 74261 55900 76233 91174 64982 76217 38093 36606 61232 13875 85837 27486 05436 624 99605 78771 39887 35310 II92I I003I 08370 26650 40425 60744 51900 41697 79895 45200 40770 86640 81208 49550 15080 09050 79080 II9O I 04478 37801 39109 42509 92974 ! = 44, 271+ I =! 10 31616 3 09485 i 47373 55040 38738 24298 9106 22571 958 217 2805 8112 10565 10565 2805 144 36 69940 44835 00982 13626 11079 28532 08418 38412 58806 16065 88226 22690 03076 89093 86365 74402 80431 79852 13966 91956 64150 28911 58924 75574 23250 84466 23250 84466 64129 80453 75977 94637 78322 26506 17508 53706 159 56413 10 38259 46 62415 60910 349o8 01106 67132 02645 28574 44302 81358 71022 83105 37239 93118 93108 37554 80681 32796 50690 23245 76512 22743 830 93685 82547 53892 33185 80456 66543 66705 47263 87972 97744 85971 62197 50014 84720 04356 98496 24208 09460 72580 56435 82376 15284 16119 4835 2302 895 552 237 226 3i6 5 ii no 195 195 no II 71 = 41, 271 + 1=83 01092 81950 55662 70327 84563 17675 74189 90252 77359 26860 35488 24264 56703 26238 03053 79729 89500 47040 02262 92052 24854 39806 70248 92075 27201 22882 99685 28345 04317 73975 84095 58302 97205 22214 95595 22219 22214 95595 22219 84095 58259 34545 22428 42267 02556 18341 52357 21398 3006 80714 29737 3 04149 55035 745 00808 22 II2II 74901 56915 52586 03267 36700 52495 53175 74305 82328 79328 39990 81861 79620 4476o 91408 46320 91365 41620 20870 67524 26125 62920 85503 12590 23891 2 57904 77371 36843 13857 9232 5267 2476 5706 141 70 987 2436 2916 2436 477 17 3 17485 25245 45355 40525 15088 91263 02233 32862 50675 75337 27623 26275 50059 26275 15962 18929 81358 1131 6 11208 53450 01601 25853 46612 99989 35946 79462 07117 53558 50780 66719 79649 66719 83192 96554 23279 16952 96768 36058 90603 98421 63274 21747 27229 14060 97563 75705 49587 75815 70454 09039 23361 14089 29356 38680 85610 51623 92805 23984 15869 38688 53385 70137 83465 46485 51960 49260 53016 08606 28502 84040 25667 84328 94847 95555 81071 25105 05108 71010 64926 17730 445179 4 18145 43696 3H77 87672 59286 ?i=45, 2+i=gi 41 26466 12 37940 5 89493 2 20630 1 62213 1 08277 37462 85510 6422 9 6l 7736 26055 36789 43701 15504 1 122 323 2 79761 03928 14657 12287 29752 69370 25916 48319 80070 33966 77333 72781 64355 85980 71078 92967 38991 3111 248 79342 5345 92002 04558 88085 36146 39891 92768 93125 71106 34621 86366 3341 76658 49873 95912 04394 53093 50058 63589 2426 49663 90527 13642 50794 98248 90754 87356 37134 63558 06982 36509 96212 65386 66950 91746 53584 43657 81919 03291 11221 94050 94429 74741 03539 I79I7 81101 78116 98767 06005 47486 81296 74309 50687 20581 3448o 23160 03020 71290 22864 79129 65310 93575 74740 88555 1365 16 74442 26979 22910 86006 56090 67 06615 69773 00887 27623 92033 54] BERNOULLIAN NUMBERS FROM TO 435 TABLE III. continued. 1 = 46, 271 + 1=93 165 05867 49 5176 23 5798i 8 90369 6 70564 4 63925 I 69682 3 "125 40919 5874 20738 80889 i 23440 i 72946 19047 8072 2624 29 02721 77988 32843 85063 61924 "3732 58249 02986 12959 98810 09258 92982 98810 19623 56653 12836 52612 5065 17369 98654 14507 30852 02481 42664 61544 18866 34174 44613 06490 32488 61491 41683 78697 65961 34213 63953 26061 66881 07875 86518 51892 49346 92343 39345 39542 81973 4556i 73035 17166 09807 43559 46904 94972 53142 64617 64749 06829 62892 98880 49039 73 44928 5 98965 3'923 99852 67976 08710 70159 50961 33429 09380 745 8 4 4743" 42181 65723 25699 99658 39796 594 07162 77060 57685 02264 18878 83947 268 57701 29884 00347 68309 72537 71 = 48, 2n+i=97 2640 93875 792 28162 377 27753 145 35370 108 37153 75 70844 38 98331 36 89710 13 95226 2 74144 I 37716 7 09528 12 54050 23 97892 18 81075 3 40042 i 37689 3207 103 '3 04754 51426 4"39 87283 23049 36360 48805 59446 33260 94233 87279 53534 43294 91849 64941 89785 50126 53662 87145 87324 883 77919 46190 47394 90942 46940 72332 90928 81610 05743 I2I22 44213 9IOO6 26386 23159 23320 53571 27429 06535 35487 20835 9M03 86439 78479 68520 77289 28670 95462 4H53 36984 89003 38174 46552 57261 40803 36622 43800 28471 18151 64235 06333 73538 69068 07628 56766 60578 83445 66099 87875 39798 69225 59503 12732 60540 84136 60144 99972 82586 12744 I052O 03856 32640 04528 24II2 18608 45636 27760 01648 4301 II894 89589 OI9I5 63549 89088 42255 12676 6036 2332 1650 1077 847 387 382 109 8 55 112 287 362 112 55 2 5O6OO 42630 40675 55774 80364 23245 69809 18490 36921 35049 51258 50057 06270 48989 50057 51258 50117 13478 2384 4 76076 22822 18154 80026 27959 70990 74107 84777 75383 78929 02617 72040 92849 08615 01185 92663 09166 10192 44762 88513 07561 1963 46716 82755 66017 76219 741 12 16504 40633 64144 05882 24186 48321 54203 50087 98253 99829 54749 76687 66943 37287 89017 23027 92871 9169 55677 96796 70390 30376 30704 89535 42330 31298 81257 20447 32155 56059 25307 49201 34186 79000 72529 84312 82612 12951 97855 62196 8 35J25 36275 50000 28267 43270 42415 51665 43990 01390 07920 53435 3949 5935 52180 96300 89200 41720 574io 67760 40300 47600 47900 73100 16585 660 23468 198 07040 94 31943 36 01280 27 21489 19 10138 8 09432 10 90183 2 45931 40278 54II9 2 43214 3 99970 6 56639 3 99970 54H7 19695 320 7 76188 62856 10085 10129 38551 9357 09753 22208 38824 41554 67806 98215 49121 98215 65191 27659 34031 83051 89389 32 !71 + I = 95 69479 94619 62251 35874 12489 68876 91093 31574 23348 79466 08342 38240 24179 80247 51631 01527 25376 36023 55964 39178 34542 83174 51991 41425 59908 19665 08468 27389 59726 60692 38358 30015 81500 88956 97173 28274 55059 32025 44641 47491 46333 74700 4204 50055 1241 95861 34291 62822 44388 93346 73487 1 022 1 39355 35540 70344 01615 57861 45535 98895 93990 06290 30158 26055 75243 52425 26535 93465 58255 1075 09243 47821 06065 90738 25977 n = 49, 2n+ 1 = 99 10563 75500 3169 12650 1509 10785 583 82981 424 86750 289 86148 184 54457 121 08374 74 93UI 17 80340 3 42381 20 11644 38 11653 84 41348 84 41348 20 11644 9 01392 29469 1241 191 19019 05705 74H5 17671 38753 7493' 78596 64846 34372 09554 24335 04748 28964 72830 72830 02133 40300 24270 08478 73532 20418 66 11679 67874 58811 80971 08304 92631 26980 89911 18060 71787 38721 25270 87513 64039 64039 69968 34630 22540 11948 00780 41411 5 OI 34 141 I39I9 24222 30559 16015 66471 73108 02288 76890 80202 07391 18019 68663 29567 53318 50150 05063 49263 89436 28654 44305 06213 87293 62980 33781 38003 90540 20138 82105 59023 77522 37535 11496 82844 20820 98986 72032 24728 79376 51752 43408 65279 53368 07636 21256 72018 46436 1617 17198 83414 90506 38151 84000 91699 71=51, 271+1 = 103 69020 50706 24145 9280 6406 3923 375i 1225 1857 635 20 149 322 944 1493 596 322 19 i 08003 02400 71033 96457 10079 60246 26391 38382 11456 71623 25925 31277 93'99 82479 45834 93489 93199 72770 34095 26941 71 04305 91291 51046 79446 41121 19656 25299 04060 43367 53966 34400 13811 12194 17455 72886 18042 06481 47510 23743 50889 36397 49126 4 86866 98577 33298 49859 18390 4273 20662 70387 11227 52292 09747 78700 11427 99171 03533 75682 64948 83471 17227 14205 27411 58743 22709 40501 86626 50675 33411 12161 14416 04892 88586 55300 74075 78895 00059 62105 72538 64195 50681 68146 03250 10920 61103 75445 65674 81490 69940 87170 86172 69300 50712 08756 62912 87240 12723 35720 03149 92905 66656 24135 18414 95260 03780 34476 92070 62470 08756 45660 2042 62905 175' 68751 79439 41361 97624 09049 22648 2 74823 95356 96505 59860 50467 12474 552 436 ON THE CALCULATION OF THE [54 TABLE III. continued. n=52, 2^+1 = 105 = 53, 2/t + i = 107 6 76080 2 02824 96582 36877 25037 14148 15995 3908 8622 355 51 391 902 3015 5917 3015 I78l MS 12 2 32OI2 09603 90287 10835 84627 00851 40243 04814 19227 II558 4IO28 94805 82063 04579 47647 04579 01885 36203 35 21 78599 17223 65167 45317 4344 05650 62482 03127 66848 76628 62111 06172 80960 38685 94921 03889 94920 7S'4 50078 00237 69422 10622 60195 184 47464 49275 63923 91277 79797 34271 46512 96974 76968 89576 6i397 60322 34517 2559 16461 91853 81034 46985 54210 12235 20967 48099 17185 3 90837 46878 55833 34623 10589 68151 74215 01424 63879 63660 83649 78441 70537 32578 82066 48292 76231 49525 58727 65427 25472 15832 72134 09798 9 62005 65651 94572 62825 9436o 67247 01615 03350 63560 09904 06225 30429 88100 38600 76205 58400 79300 86180 35075 02960 45400 2135 73925 56046 1820 27 04321 8 11296 3 86331 i 46700 99091 65721 13046 3 8 349 18386 '57 1005 2461 9343 22597 14539 9343 1005 104 26 2 384H 74994 93377 51201 37629 28746 86125 93950 32280 16857 34094 34684 98590 39'77 34684 30256 77904 55359 15505 200 68893 60667 98818 65344 98154 94150 11152 03658 88648 12099 21727 82822 48540 64057 13976 62922 64041 16213 81151 43853 94183 41228 6107 89859 26885 24837 64131 97282 95429 84484 67371 71806 25600 70559 29033 59022 86311 73572 21854 81913 4'774 80372 5393 62328 57567 82808 319 63350 89750 23262 19517 10154 53653 09096 53"4 52040 44582 IOOI2 40776 86704 59743 38047 67829 17031 07757 73341 07627 32886 75619 3376o 43066 2581 48021 40307 06626 91810 89600 65519 86395 10605 80934 84745 47809 02755 50120 58457 28900 28080 039IS 89830 74135 40360 48878 9635 36885 77946 32122 10 98816 27096 33035 42876 78820 09984 43 95161 50952 42417 49056 15321 06993 108 17285 32 45185 15 45326 85306 97684 1 90771 2 60640 47721 I 63807 92O22 697 2522 6552 28159 83340 67I2O 46764 6552 832 234 I I2I94 53658 86134 54848 55721 17282 80664 74052 85912 24231 68647 81178 84249 21123 69236 67450 40436 83596 29349 67636 96275 336o 75575 42670 77725 03542 46477 31079 90582 16645 94599 94899 92119 81227 81581 30110 56808 60831 89923 64487 15060 11894 21214 75986 7"93 59438 87687 30205 89787 68242 72954 29871 40096 42478 90221 65569 67512 25696 73232 93889 51244 59473 46048 07657 79327 63794 88443 69570 24104 4 53401 575" 93121 2OI2I O3IO2 44149 74052 68666 04060 11892 64477 38778 70770 20531 11768 42073 57182 97074 09703 97142 71762 9"57 834H 72931 22 1 2O 2892 92085 09427 98167 51950 62561 4539 I2O25 89825 63196 42904 61607 39679 86148 52250 82399 09416 07380 12230 66180 47085 08944 58108 43410 37245 94I7I 89052 1962 432 69140 129 80742 61 81305 23 43031 16 14023 7 37429 10 00236 1 96775 6 73431 4 40618 4330 6203 17052 82649 2 97366 2 97366 2 23918 40409 6196 "933 22 48779 H633 78397 8l8dO 64612 25606 96391 OI827 78568 44879 96649 83300 57037 99258 99409 99409 97150 I55I2 52654 46718 69432 50535 41 02302 70694 00328 13614 93655 25888 16728 94590 19304 54858 73527 79206 399'4 76274 26252 26252 79987 64339 34685 63181 14044 56403 30978 37754 31614 91107 61240 98961 90494 82415 29161 83954 88694 22949 64609 05781 44754 81819 81819 09479 33968 08227 25126 76724 30948 01517 468 4 13607 2IIOI 73372 53224 2 9 865 36307 35982 24819 85615 95723 89855 69871 24102 68852 50425 44017 33801 65290 53342 34756 42249 33539 38913 83295 55424 90586 175 88532 57245 26240 26553 83862 64446 68341 26899 52620 97101 39475 45359 64441 18297 72068 38504 27907 50403 59665 98085 82595 59060 22396 42085 50645 79425 '5915 48890 51850 5"245 52981 01735 97801 2035 704 14188 02606 57029 63487 47957 17823 54] BERNOULLIAN NUMBERS FROM B 31 TO 437 TABLE III. continued. n=56, 2n+ i = 113 1730 76561 519 22968 247 25219 94 02296 65 88250 29 82642 37 27607 9 02333 26 70098 20 23381 30554 15008 43425 2 36548 10 28272 12 67163 10 28272 2 36548 43423 I492O 241 6 95116 58534 39768 00824 79051 34192 22031 46348 80697 43671 71855 27461 49159 68984 31618 86438 31618 68976 72089 70775 36078 87718 871 09209 82770 27527 42074 91915 84858 41051 71041 09858 16553 21946 94451 72479 35027 47720 93150 47720 88010 37638 00261 29364 38544 36102 652 51016 05880 97569 54503 76250 27229 89638 47755 43818 12549 50609 74918 20287 19128 12772 05625 12761 49170 09594 49187 74285 17014 47471 11800 38012 564 54430 90367 95480 71544 78660 59659 98004 48453 91752 33027 92436 85874 51794 07988 69454 19692 43029 80071 4978o 15049 71685 18145 35482 345H 96492 44I5 1 3609 73365 14803 45372 45401 19116 456i5 53826 26400 43757 50704 90702 79703 37840 95840 50226 46688 13296 95480 04520 99270 17496 99088 48356 72635 62356 23256 37368 87748 6923 06247 2076 91874 989 00881 377 62158 268 82951 125 45289 135 59805 44 03434 i 02 29421 89 3H74 2 16071 362O9 1 08336 6 60944 34 51267 S 2 04423 45 38939 13 18517 2 87517 i 08311 2375 85 80464 34139 32892 96919 56468 96465 29162 20285 74735 09963 11886 83855 36838 87040 29779 01445 41588 56924 66712 44994 55545 37867 16272 83899 80460 33201 61707 82285 79570 87144 89906 10502 54918 33823 71023 37432 61152 63168 70160 99209 62031 36765 45562 85332 24997 23 04066 00067 57140 14312 44962 74050 56224 22979 2I2OI 69240 62595 57461 47381 483H 39259 01674 64411 52138 81709 09547 59974 56681 54059 85679 73095 47434 6 17722 97241 79931 76549 82169 50221 52331 61971 61941 10236 97502 69645 78434 22970 12278 91594 56603 26421 02331 91974 28523 74141 06232 89731 09610 79632 57206 4019 2819 65747 03883 19929 99419 40647 80228 93461 02451 46597 04698 68996 64751 42361 83291 77525 03804 85510 18231 31590 47500 63690 06040 34448 48275 51140 19950 00580 98905 70580 17675 92796 42860 79090 61340 2185 11291 46539 06528 22156 05672 89171 60320 71 = 58, 271+1 = 1 71=59, 2n+ 1 = 119 27692 8307 3956 1515 1091 540 485 218 379 379 14 2 18 112 206 192 70 a 7 24991 67497 03570 49252 i6739 69111 15774 95202 33043 75523 69761 65186 04898 59759 41270 99024 17615 04898 42425 21755 973 2 21857 36557 17408 86452 64229 85293 28013 78106 51955 24136 42173 34395 21340 18359 55905 93869 35852 86267 13981 75690 08678 74737 71989 8 47352 24176 21050 11467 52861 5933 03401 05354 65173 66131 29659 47784 80116 82892 20874 03552 4I5I7 47066 83568 39740 63011 06533 11986 07787 1184 30 16264 82575 30894 23346 61660 88023 85137 40258 29082 42396 52645 68210 13206 12601 30499 06642 80637 11615 75002 41810 28185 745 2 4 74987 70891 10465 65643 810 95841 87475 74961 92323 61203 45696 11780 56709 21410 31527 56033 48864 99914 88695 H730 89095 24996 28623 55947 86937 18460 68170 60439 12244 87368 57697 14 73845 80979 67692 82993 61410 80687 49077 27180 60486 82304 39037 91311 52003 80732 34977 08703 76168 04466 H930 52680 03814 93980 355 39071 38552 78268 12590 82741 I 10768 33230 15824 6071 4393 2348 1726 1078 1363 1558 95 2 4 8 357 791 791 357 107 48 i 99964 69989 14381 25206 30863 32141 49475 72509 30552 23921 79406 39335 21990 552H 92548 92548 87429 46228 62765 40499 08378 33172 09181 9438i 50719 88619 3215 94693 77979 92125 I55i8 22980 22980 75671 21990 86271 10265 t 63236 04553 29795 28571 06743 224 45210 69849 27257 80209 35823 13172 90671 89408 65058 83566 96880 24125 17373 76468 28826 29627 81747 88410 47263 40276 48576 91485 63673 10119 10778 95 '53 57885 26004 86723 02780 58227 56412 91686 59494 49099 97920 90071 40527 83982 31394 89394 28641 73123 76304 07197 86255 78826 08430 35042 07880 13131 60500 60021 34338 63895 60021 34338 63895 07802 03305 63887 79282 47781 28999 45067 04486 60465 92929 52427 82082 97557 69578 24001 24858 00182 84546 16912 06937 35271 48617 53678 41198 1582 63826 19616 72989 02772 967 23482 4957 9538i 50963 28702 40323 63926 18127 92901 37368 48932 21304 28815 63231 16407 80662 88819 24089 19448 01637 28015 43130 42154 41859 79212 76835 60921 40247 09226 19898 29741 i 80973 11279 28451 63429 01611 99280 92273 438 ON THE CALCULATION OF THE [54 TABLE III. continued. 4 43075 I 32922 63296 24273 17537 10137 6204 99859 79957 57425 81412 60713 05071 23779 343 34096 13082 53139 41976 38862 4760 6182 587 12 no6 49118 2943 87045 3141 73715 1748 03254 613 57942 297 68407 14 97596 i 00574 564 n=6o, 497'9 84915 57930 00370 24716 52329 06828 01421 29195 39396 92560 91981 24844 21104 92012 64689 92808 53643 00077 934 '6 48193 86413 67719 5424 16 57634 87405 13606 26578 69781 47854 94196 41639 28268 39322 62591 71104 64897 75154 50819 28281 93698 73796 71077 18569 68783 66797 89272 80777 67192 So 60235 67285 10058 31642 I54I5 56603 96000 04478 48740 38734 55473 62393 87591 04676 86169 18432 39507 82827 69719 16473 59718 90199 91670 24112 99721 7 24182 73411 81733 68035 40727 96988 07883 70025 71217 46669 90027 9 34267 64887 72248 67473 48645 27665 75526 54545 95473 86709 89583 62737 35153 88336 56844 42163 38731 56780 34537 42669 69234 76046 12518 93576 10009 81525 19155 56875 53817 77625 37487 57365 85975 39656 07479 90507 44663 41070 83215 29526 755'7 96892 07756 24425 08680 28740 39165 79407 06126 43820 37912 36660 64435 28627 17 72303 5 3'69i 2 53186 96908 69521 43087 22914 24160 16158 23750 3456 77 38 323 3340 10619 12069 8217 3340 1751 "3 9 99437 98878 19831 39663 28491 14125 22257 13794 24129 43529 87536 3258 64302 15945 92864 33451 27469 28051 41878 23710 63969 65595 71763 91169 85881 95584 49230 88336 84643 56448 16347 83109 86887 15228 72442 84893 84643 56446 78325 27847 49966 06458 20260 00685 7120 62690 I 15962 498 24 30538 48930 47219 41643 60542 60993 60828 62038 77738 79199 28115 06828 53414 48107 32271 36800 13412 77144 49344 5'444 05280 24372 19969 87773 45356 00699 2784 64 40941 37071 93852 31907 77277 33867 91001 02179 32025 95149 25317 26629 59575 36644 48569 69036 72920 43387 97963 29403 53771 86303 61040 82569 30520 41284 71201 80821 46499 64905 73364 76316 85572 9273" 30360 42686 42601 05102 68000 09448 49261 96831 70327 80823 46409 21219 24083 10560 06712 75090 61640 64838 20326 36050 33089 43000 1363 80209 26101 68435 32300 40292 36725 31215 72261 83900 46656 90604 75827 55063 73841 19420 48791 34069 41617 47231 82506 89381 66996 52971 31521 35871 85626 75626 40101 82476 04651 28 97042 79681 94647 52859 15592 96072 72074 70 89215 21 26764 10 12745 3 86684 2 745 '4 i 79389 88326 i 08964 53416 88493 19438 556 118 813 9852 37222 44971 37222 17435 9852 813 78 9775' 79325 11239 50786 07044 57698 29369 57287 87275 42679 93661 33273 47201 74880 19173 30527 86718 30527 39385 I9I73 71172 98134 82860 955'3 58653 41912 22868 94746 25603 48517 53990 74718 27412 66785 68635 08489 81117 3738i 22341 95681 22341 60579 373>3 06336 05659 14788 13577 12876 735 i 22153 96184 97668 24914 36352 72166 19801 4"33 42927 45701 70819 58511 43336 96046 17857 74674 74975 74674 34289 66978 64129 62224 35981 09957 71698 39217 26184 3665 i 63765 48285 92269 46058 67454 86433 40706 84138 I93I3 79031 99699 04739 79218 73622 75824 81250 19308 21871 65763 91256 10300 64844 83442 97916 29635 40502 51541 36028 24498 00057 66591 26240 78456 82090 66565 34669 09197 40435 42525 93054 34116 79412 19756 93668 00107 19894 18335 I439I 40079 39847 49861 75890 65083 59009 91493 25414 35505 97693 04405 1253' 65000 33440 90650 16175 89925 50100 ooooo 30000 85875 61575 25125 20375 40750 46625 41450 87750 61900 37375 99000 16750 43375 08375 12005 75500 40250 64625 72375 115 86892 47169 98580 24228 40420 37749 33281 54] BERNOULLIAN NUMBERS FROM B^ TO B M . 439 TABLE IV. Values of odd powers of 2. Power Index 2 i 8 3 32 5 128 7 512 9 2048 i i 8192 13 32768 15 i 31072 17 5 24288 19 20 97152 21 83 88608 23 335 54432 25 1342 17728 27 5368 70912 29 21474 83648 31 85899 34592 33 3 43597 38368 35 '3 74389 53472 37 54 97558 13888 39 219 90232 55552 41 879 60930 22208 43 3518 43720 88832 45 H073 74883 55328 47 56294 99534 21312 49 2 25179 98136 85248 51 9 00719 92547 40992 53 36 02879 70189 63968 55 144 11518 80758 55872 57 576 46075 23034 23488 59 2305 84300 92136 93952 61 9223 37203 68547 75808 63 36893 48814 74191 03232 65 i 47573 95 2 58 96764 12928 67 5 90295 81035 87056 51712 69 23 61183 24143 48226 06848 71 94 44732 96573 92904 27392 73 377 78931 86295 71617 09568 75 1511 15727 45182 86468 38272 77 6044 62909 80731 45873 53088 79 24178 51639 22925 83494 12352 81 96714 06556 91703 33976 49408 83 3 86856 26227 66813 35905 97632 85 15 47425 04910 67253 43623 90528 87 61 89700 19642 69013 74495 62112 89 247 58800 78570 76054 97982 48448 91 990 35203 14283 04219 91929 93792 93 3961 40812 57132 16879 67719 75168 95 15845 63250 28528 67518 70879 00672 97 63382 53001 14114 70074 83516 02688 99 2 5353 12004 56458 80299 34064 10752 101 10 14120 48018 25835 21197 36256 43008 103 40 56481 92073 03340 84789 45025 72032 105 162 25927 68292 13363 39157 80102 88128 107 649 03710 73168 53453 56631 20411 52512 109 2596 14842 92674 13814 26524 81646 10048 in 10384 59371 70696 55257 06099 26584 40192 113 41538 37486 82786 21028 24397 06337 60768 115 i 66153 49947 3"44 84112 97588 25350 43072 117 6 64613 99789 24579 36451 90353 01401 72288 119 26 58455 99156 98317 45807 61412 05606 89152 121 106 33823 96627 93269 83230 45648 22427 56608 123 425 35295 86511 73079 32921 82592 89710 26432 125 440 ON THE CALCULATION OF THE [54 TABLE V. Values of F n = 2 - ((7; + C + <7 3 " r + &c.) - 2 2 "- 1 + n. 197 93228 99666 11337 31 814 19505 34163 85807 32 3316 86845 5'567 39S9 33 13383 28684 34869 42259 34 53472 15732 80850 01814 35 2 11585 84869 23594 53860 36 8 30204 96439 84139 44995 37 32 39222 05336 71210 65438 38 126 21736 96201 37492 94582 30 493 66994 33219 42486 96625 46 1947 11281 62577 29096 11580 41 7764 31244 47406 08533 43608 42 31289 17468 64664 51766 61697 43 I 27017 22068 55657 42382 65606 44 5 16915 50130 31873 53128 29966 45 20 98900 51313 24292 70327 24135 46 84 74040 33538 01845 98808 32232 47 339 71082 32456 85035 95830 13968 48 1353 20164 61977 70633 13121 91076 49 5369 26438 27247 27549 25533 20010 50 21293 83352 40046 79561 16403 01773 5' 84695 79078 07200 21679 42563 67028 52 3 38679 58879 39076 64266 70295 35014 53 13 62604 88953 12876 87396 03759 76372 54 55 10477 29438 03576 06856 27545 65366 55 223 50904 11209 06115 72894 59001 70236 56 906 87167 35831 66898 99573 62587 20185 57 3672 32362 44471 59181 11426 06835 29961 58 14819 61331 97306 79316 04023 73930 49260 59 59568 73622 57154 31583 50374 95586 56399 60 2 38586 80524 96330 07051 54180 90465 82983 61 9 53068 50542 05310 40997 94772 15321 76735 62 " 54] BERNOULLIAN NUMBERS FROM , TO . 441 CALCULATION OF BERNOULLI'S NUMBER FOR n = 32. Table of the values of the alternate binomial coefficients for the index 2n+ 1 = 65, or of the values of C r n for n = 32. = 32, 271+1=65 C r T 2OSO I 6 77040 2 825 98880 3 50473 81560 4 17 90137 99328 5 402 78104 84880 6 6099 25587 71040 7 64804 59369 42300 8 4 98105 89663 01600 9 28 33960 39082 73840 10 121 45544 53211 73600 ii 397 3753 30616 65800 12 IOO2 59642 18786 64480 13 1965 40727 14605 56560 14 3009 10630 52706 45216 15 3609 71421 70081 32870 16 3397 37808 65958 89760 17 2507 58858 77255 37680 18 1448 19483 16025 15360 19 651 68767 42211 31912 20 227 06887 60352 37600 21 60 72772 26605 86800 22 12 32I6 69166 40800 23 1 86789 7II23 63100 24 20737 46998 21536 25 1642 10735 15280 26 89 50689 96640 27 3 19667 49880 28 6961 90560 29 82 59888 30 43680 31 65 32 18446 74407 37095 51615 Sum = 2 64 -i A - 56 442 ON THE CALCULATION OF THE [54 Formation of the several values of - (C?+ C+ 1 + &,c.), ivhen n = n=32, r=i, p 3 ) 18446 74407 37095 51615 6148 91469 12365 17205 77040 81560 84880 42300 73 8 40 65800 56560 32870 3768o 31912 86800 63100 15280 49880 402 64804 28 33960 397 37053 1965 40727 3609 71421 2507 58858 651 68767 60 72772 i 86789 1642 3 50473 78104 S9369 39082 30616 14605 70081 77255 42211 26605 71123 "735 19667 82 5 ) 9223 37203 90022 59455 1844 67440 78004 51891 r=3, p=2r+l = 825 402 78104 4 98105 89663 01600 397 37053 36i6 65800 3009 10630 52706 45216 2507 58858 77255 37680 227 06887 60352 37600 i 86789 71123 63100 89 50689 96640 _ 82 59888 7 ) 6147 98818 11420 91284 878 28402 58774 41612 17 90137 99328 28 33960 39082 73840 3009 10630 52706 45216 651 68767 42211 31912 20737 46998 21536 82 59888 II ) 3689 34113 71219 31720 335 39464 88292 66520 r=6, p = 402 78104 84880 397 37053 30616 65800 2507 58858 77255 37680 i 86789 71123 63100 82 59888 13) 2906 83104 57183 11348 223 60238 81321 77796 r=8, 2> = 64804 59369 42300 3609 71421 70081 32870 i 86789 71123 63100 65 17 ) 3612 23016 00574 38335 212 48412 70622 02255 r=9, p = 2r+i = i() 4 98105 89663 01600 2507 58858 77255 37680 89 50689 96640 19) 2512 57054 17608 35920 132 24055 48295 17680 r=n, p = 121 45544 53211 73600 60 72772 26605 86800 23 ) 182 18316 79817 60400 7 92100 73035 54800 r=i4, p = 2r+ 1 = 29 1965 40727 14605 56560 3 19667 49880 29 ) 1965 40730 34273 06440 67 77266 56354 24360 3009 10630 52706 45216 82 59888 31 ) 3009 10630 52789 05104 97 06794 53315 77584 r=i8, y = 37 ) 2507 58858 77255 3768o 67 77266 45331 22640 r=2o, p=2r+i=4l 41 ) 651 68767 42211 31912 15 89482 13224 66632 r=2i, p = 2r + i = 43 43 ) 227 06887 60352 37600 5 28067 15357 03200 r=23, p = 47 ) 12 32156 69166 40800 26216 09982 26400 r=26, p = 53)1642 10735 15280 30 98315 7576o 59 ) 6961 90560 117 99840 61 ) 82 59888 i 35408 54] BERNOULLIAN NUMBERS FROM B~, TO 443 The following extract from the calculations for B 3l supplies the further data which are required in making the similar calculations for B.&. Table of the products P for n = 3I, and calculation of the quantities I 31 and B 3l . 3 5091 51 78275 32629 66196 119 34005 74495 23247 08675 60794 21 42517 90194 62IO8 785 22331 21935 "8l6 8645 5'75' 16414 13997 16774 64384 84868 76711 p r 142 26491 58999 84925 71881 92280 03855 56131 58235 85806 91869 3738 37926 85674 25656 36404 IO2I2 86701 76884 24755 "5793 97596 76074 72655 00199 90972 79046 11409 66003 49365 80981 35639 23718 31329 95811 92825 33475 87920 97257 58306 86655 9993 ' 36525 47600 18520 64045 62624 r 7 9 ii 13 15 17 19 21 23 25 27 29 135 I 71790 1385 12625 6 71133 42236 1813 75138 15056 2 47319 34636 79761 147 61606 58640 58443 3102 83046 11851 35783 15767 53321 46412 01553 8498 13384 95367 52252 2 7135 63249 03065 98357 IOI39 43708 76272 98042 30953 86990 O3OIO 56476 44902 71972 28992 65576 93686 74OIO 79085 I92I9 48845 44735 48537 64196 05779 73079 64048 54645 00391 68787 95545 20139 79801 33372 05036 67795 29205 18500 33947 74789 90688 86060 35175 57165 64369 18261 52967 8 10 12 H 16 18 20 22 24 26 28 30 26227 04351 88775 49492 30619 77419 37134 85683 Sum 27518 60498 94566 69639 20928 76040 64824 28921 Sum 197 93228 99666 1 1 337= ^31 27518 60498 94566 69639 21126 69269 64490 40258 26227 04351 88775 49492 30619 77419 37134 85683 63 ) 1291 56147 05791 20146 90506 91850 27355 54575 p si 20 50097 57234 78097 56992 17330 95672 31025 =7 31 Also = /+ 1- Hence the numerator of B ai is 123 00585 43408 68585 41953 03985 74033 86151 and the denominator is 6 As a test, this numerator should be divisible by 31. By actual division we find the quotient to be 3 96793 07851 89309 20708 16257 60452 70521 without any remainder. Hence the test is satisfied. Table of the factors by which the quantities P r n for n 31 must be multiplied in order to find the corresponding quantities for n = 32. r= 7, Factor = 65 . 64 _ 13 . 32 51. 50" 51.5 65 .64 13 . 16 r=io, Faetor = " = 45-44 9- II r= 13, Factor^ 6 ! = S -^- 3 - 2 39 38 3-19 r=i6, Factor = 65 . 64 _ 65 ^ 2 33 32 ~ 33 r=i9, Factor = - 3 = 5_l_2_ 27 . 26 27 r = 22, Factor = 65 . 64 _ 13 . 16 21 . 20" 21 ** StS-^ , = 2 8) Factor=|^ = 6JL_8 65 .64 65 . 32 - = 3i, Factor=-^ ? = -^ i- 3-2 3 r= 8, r=ii, 65 . 64 _ 65 . 4 "49. 48~49- 3 65 . 64 _ 65 . 32 "43. 42" 43- 21 65 . 64 _ 65 . 16 "37 36~ 37 9 r=,7,Factor = ^^ = i3^? 31-30 31 3 65 . 64 13 . 8 r = 20, Factor = " = 25 24 5-3 19 . i8~ 19 . 9 6c . 64 $ . 16 r=a6 , ]**=_= r = 2 9 ,Factor = 6 -i^ = 6 -^!? r=2 3 , 32 17 .46 47 . 23 65 .64 13 , r =12, Factor = = r=i S , r=i8, 41 . 40 41 65 . 64 _ 13. 32 "35-34 7-17 r=24, Factor = 29 . 28 29 . 7 65 . 64 _ 65 . 32 "23 . 22 "23 . II . 64 _ 65 . 4 r=27, 17 . 16 17 = 65 . 64 _ 13 . 32 II . 10 II r=3O, Factor=-^- ! - = 13 . 16 5 4 562 444 ON THE CALCULATION OF THE [54 The general equation for finding I n is Hence putting TO = 32, the equation for finding 7 32 is P. = 65 / = (C? 2 /, + C -/, + &c. + C^/ 31 q/" Z/i,e products P r n = C r n l r for n = 32, awe? calculation of the quantities I 3 , and S. 3 , P r r 6099 25587 71040 7 273 95824 31465 88000 9 7 52052 11742 87069 31200 n 14292 18243 52720 81535 36160 13 181 02208 01083 62 555 55 8 5 r 9 8 7 8 4 '5 i 4595 6 33740 16156 32779 21049 5536o 17 707 20034 04420 28331 06379 22168 88480 19 i 91122 29427 92298 81501 97313 85143 24000 21 260 61036 46811 76525 19023 36205 50468 48000 23 15554 89989 86905 19795 15515 50951 74607 85920 25 3 26957 75316 75298 38464 45063 81824 21193 67520 27 16 61488 53356 44144 55275 75671 54091 08961 07520 29 8 95482 61960 15233 01854 18130 16189 66511 72000 31 28 99746 33490 63992 38881 71047 04677 31614 73984 Sum 4 53632 '5585 96100 8 14991 65046 74768 61360 10 344 04340 75247 90249 64000 12 5 36521 41705 40998 03634 45360 14 5456 55799 32924 09847 00119 61290 16 34 38319 oiiii 06135 63711 54717 15840 18 12575 34291 17724 63047 26474 10262 50016 20 24 49639 24021 61449 84035 26184 7 OI 63 48400 22 2257 65747 79208 93849 11233 52764 25664 03700 24 82742 14563 16036 20888 25426 35887 94710 49840 26 9 11013 03017 92694 23112 48336 95819 28170 55080 28 17 67611 84070 36444 68422 26180 95706 47598 17136 30 27 63649 29648 26477 79222 58362 41320 15644 68122 Sum 814 19505 34163 85807 F 32 27 63649 29648 26477 79222 59176 60825 49808 53929 28 99746 33490 63992 38881 71047 04677 31614 73984 65 ) i 36097 03842 375H 59659 11870 43851 81806 20055 ^32 2093 80059 11346 37840 90951 85290 02797 01847 I Also = /- l Hence the numerator of B S2 is 10 67838 30147 86652 98863 85444 97914 26479 42017 and the denominator is 510 54] BERNOULLIAN NUMBERS FROM B^ TO JB m . 445 CALCULATION OF BEENOULLI'S NUMBER FOR n = 62. Table of the values of the alternate binomial coefficients for the Index 2n+l = 125, or of the several values of the quantities C r n when n = 62. 11 = 62, 7750 I 96 91375 2 46906 25500 3 117 61743 44125 4 17736 70910 94050 5 17 61577 70018 40875 6 1224 97403 15126 27000 7 6232 55385 32048 98625 8 23 9758 36588 21178 64750 9 7iS S93'5 48903 94232 15775 10 16914 02002 46820 45487 36500 II 3 21917 92460 01984 96178 00125 12 50 02109 28994 15458 63688 94250 13 641 93735 88758 3!7i 9 17341 42875 14 6870 94331 70709 71228 66992 39600 15 61852 34256 19392 87370 99034 37125 16 4 71665 45718 35584 15994 82476 00750 17 30 65825 47169 31297 03966 36094 04875 18 170 77912 58485 10610 53673 21400 13500 19 819 08296 12811 25889 76655 76099 87825 20 3396 19764 43363 75640 49548 27731 20250 21 I22I6 97736 11804 29497 46947 97853 36375 22 38244 45086 97822 14079 03489 32410 53000 23 1 04460 24213 63466 32604 17243 44642 59125 24 2 495' 74978 85308 13877 39472 91774 87510 25 5 22166 16188 77624 49856 53874 31881 80875 26 9 58946 66208 31863 85900 05857 23959 04500 27 15 4739 1 20472 51416 68156 91269 63661 18625 28 21 96116 01106 18163 05805 27355 455" 77250 29 27 43283 89856 36586 73522 85866 05169 97175 30 30 17467 21788 07033 53213 93231 82841 64000 31 29 23171 36732 19313 73425 99693 33377 83875 32 24 93894 45323 96897 03202 59878 22881 79250 33 18 73'57 774H 09609 66716 26273 77063 54125 34 12 37912 96378 01133 34525 53015 70928 94900 35 7 19209 99656 23897 89425 04392 92969 28375 36 3 66927 57321 84276 67466 75695 46727 75750 37 i 64151 80907 14018 51235 12811 13009 78625 38 64283 22593 00594 66217 95226 73626 21000 39 21990 55925 01247 73095 44506 36136 05475 40 6555 45 I2 6 69748 64608 39825 74457 90250 41 1698 09882 21681 87820 24774 13865 60125 42 380 96881 92005 23669 65886 40046 45500 43 73 74553 16164 02309 09540 70604 60375 44 12 26330 18867 72518 81586 54437 61950 45 I 74311 14722 00107 18954 60915 04625 46 21056 11661 68303 95700 76267 02000 47 2147 16978 65846 78508 95935 12375 48 183 41067 39645 23348 33526 12250 49 13 00548 41538 48019 24559 12505 50 75745 39402 35761 16747 76500 51 3577 96577 445'9 7n6o 78875 52 '35 oi757 63944 14006 06750 53 3 99584 72764 70196 44125 54 9064 80783 31934 39800 55 '53 '2175 39390 78375 56 i 85429 23159 83250 57 1529 02664 73625 58 7 97406 33500 59 2345 31275 60 3 17750 61 125 62 212 67647 93255 86539 66460 91296 44855 13215 Sum=2 124 -i 446 ON THE CALCULATION OF THE [54 Formation of the several values of (C? + C + C 3 " + &c.) when n 62. 7i = 62, 2+I = I25 r=i, p = 3 ) 212 67647 93255 86539 ta4j5o_9_ 70 89215 97751 95513 22153 63765 48285 04405 30 819 12216 i 04460 5 22166 15 47391 27 43283 29 23171 18 73157 7 19209 i 64151 21990 1698 73 i 3 641 61852 65825 08296 97736 24213 16188 20472 89856 36732 774H 99656 80907 55925 09882 74553 74311 2147 13 715 21917 93735 34256 47169 12811 11804 63466 77624 51416 36586 19313 09609 23897 14018 01247 21681 l6l6 4 14722 16978 00548 3577 3 17 62320 59315 92460 88758 19392 31297 25889 29497 32604 49856 73522 73425 66716 89425 51235 73095 87820 02309 00107 65846 41538 96577 99584 153 117 61577 55385 48903 01984 31719 87370 03966 76655 46947 17243 53874 91269 85866 99693 26273 04392 12811 44506 24774 09540 i 8 954 78508 48019 44519 72764 I2I75 1529 96 91375 61743 44125 70018 40875 32048 98625 94232 I577S 96178 00125 17341 42875 99034 37125 36094 04875 76099 87825 97853 36375 44642 59125 31881 80875 63661 18625 05169 97175 33377 83875 77063 54125 92969 28375 13009 78625 36136 05475 13865 60125 70604 60375 60915 04625 95935 I 2 375 24559 12505 7"6o 78875 70196 44125 39390 78375 02664 73625 2345 __ __ _ 5 ) 106 33823 96627 93269 80924 6i347_30290 626JI5 21 26764 79325 58653 96184 92269 46058 12531 r =3. p = 30 3396 i 04460 9 58946 27 43283 24 93894 7 19209 64283 1698 6870 65825 19764 24213 66208 89856 45323 99656 22593 09882 26330 2H7 23 21917 94331 47169 43363 63466 3 '863 36586 96897 23897 00594 21681 18867 16978 75745 3 17 9758 92460 70709 3 I2 97 75640 32604 85900 73522 03202 89425 66217 87820 72518 65846 39402 99584 i 61577 36588 01984 71228 03966 49548 17243 05857 85866 59878 04392 95226 24774 81586 78508 3576i 72764 85429 46906 70018 21178 96178 66992 36094 27731 44642 23959 05169 22881 92969 73626 13865 54437 95935 16747 70196 23159 2345 7 j_7o_892ij_7867S 9339Q 10 12745 " 2 39 4 I 9 12 83680 72184 05035 97668 67454 86433 25500 40875 64750 OOI25 39600 04875 20250 59125 04500 97175 79250 28375 2IOOO 60125 61950 12375 76500 44125 83250 Ji275 55000 65000 17736 715 59315 48903 6870 94331 70709 71228 8l9 08296 I28II 25889 76655 2 49510 74978 85308 13877 39472 27 43283 89856 36586 73522 85866 12 37912 96378 OII33 34525 53015 21990 55925 01247 7395 4456 12 26330 18867 72518 81586 13 00548 41538 48019 9064 80783 )_4_2_53529 __ 3 86684 50786 22868 7458_ 24914 70910 94232 66992 76099 91774 05169 70928 36136 54437 24559 3 '934 2345 25521 94050 15775 39600 87825 87510 97175 94900 05475 61950 12505 39800 31275 67840 47775 40706 84138 33440 17 61577 70018 3 21917 92460 01984 96178 30 65825 47169 31297 03966 36094 1 04460 24213 63466 32604 17243 44642 27 43283 89856 36586 73522 85866 05169 7 19209 99656 23897 89425 04392 92969 1698 09882 21681 87820 24774 13865 2147 16978 65846 78508 95935 3 99584 72764 70196 2345 13^35 68682 91584 31702 72578 51079 2 74514 07044 94746 36352 19313 79031 p=:2r+i = i7 55385 87370 17243 99693 44506 78508 I2i75_ 17 ) 30 49622 80870 35263 26838 94883 62320 61852 34256 19392 i 04460 24213 63466 32604 29 23171 36732 19313 73425 21990 55925 01247 73095 2147 16978 65846 153 32048 99034 44642 33377 36136 95935 J593_9o 80565 40875 00125 04875 59125 97175 28375 60125 12375 44125 31275 78450 90650 98625 37125 59125 83875 05475 12375 .78375 74975 I 79389 57698 25603 72166 99699 04739 16175 r=9, j> = 2r+i = i9 23 97508 36588 30 65825 47169 31297 03966 9 58946 66208 31863 85900 05857 7 19209 99656 23897 89425 04392 12 26330 18867 72518 81586 3 99584 72764 19 ) 16 78199 58020 21826 76234 05155 88326 29369 48517 19801 79218 21178 36094 23959 92969 54437 70196 98835 73622 64750 04875 04500 28375 61950 44125 08575 89925 r=u, p = 16914 02002 46820 45487 36500 12216 97736 11804 29497 46947 97853 36375 24 93894 45323 96897 03202 59878 22881 79250 73 74553 16164 02309 09540 70604 60375 9064 80783 3193139800 23 ) 25 06185 17613 41779 46076 4397 68761 52300 i 08964 57287 53990 433 75824 81250 50100 54] BERNOULLI AN NUMBERS FROM B m TO B K n=62, 271+1 = 125 447 641 93735 88758 31719 17341 42875 15 47391 20472 51416 68156 91269 63661 18625 1698 09882 21681 87820 24774 13865 60125 '53 I2I75 3939Q 78375 29 ) 15 49089 30996 66834 44888 59938 34259 ooooo 53416 87275 74718 42927 19308 21871 ooooo 6870 94331 70709 71228 66992 39600 27 43283 89856 36586 73522 85866 05169 97175 12 26330 18867 72518 81586 54437 61950 _ 2345 3I27S 31 ) 27 43296 23057 49786 16751 3868i^28945jogoo 88493 42679 27412 45701 65763 91256 30000 37 r=i8, j) = 2 30 65825 47169 31297 03966 36094 04875 7 19209 99656 23897 89425 04392 92969 28375 3 99584 72764 70196 44125 7 '9240 65481 71071 20306 81123 99259 7737S "9438 93661 66785 70819 10300 64844 85875 27 43283 89856 36586 73522 85866 05169 97175 _ 2345 31275 6 1 ) 27 43283 89856 36586 73522 85866 07515 28450 44971 86718 95681 74975 78456 82090 41450 '=33. ? = 03202 _ __ 37222 30527 22341 74674 66565 34669 87756 67 ' = 35. 2> = 71 ) 12 37912 96378 01133 34525 53015 70928 94900 '7435 39385 60579 34289 09197 40435 61900 73 ) 7 19209 99656 23897 89425 04392 92969 28375 9852 19173 37313 66978 42525 93054 37375 r =39i p= 79 ) 64283 22593 00594 66217 95226 73626 2IOOQ 813 71172 06336 64129 34116 79412 99000 r = 20, p = 819 08296 12811 25889 76655 76099 87825 21990 55925 01247 73095 4456 36136 05475 2345 31275 41 ) 22809 64221 14058 98985 21162 14581 24575 ~~S5~6 33273 68635 585H 83442 97916 61575 r=2i, j> 3396 19764 43363 75640 49548 27731 20250 1698 09882 21681 87820 24774 '3865 60125 43 ) 5094 29646 65045 63460 74322 41596 80375 118 47201 08489 43336 29635 40502 25125 7=23, jp = 38244 45086 97822 14079 03489 32410 53000 i 74311 14722 00107 18954 60915 04625 47 ) 38246 19398 12544 14'86 22443 93325 57625 813 74880 81117 96046 51541 36028 20375 r=26, p = 2 5 22166 16188 77624 49856 53874 31881 80875 3577 96577 445'9 76o 78875 53 ) 5 22166 16188 81202 46433 98394 03042 59750 9852 19173 37381 17857 24498 00057 40750 21 96116 01106 18163 05805 27355 455 22 7725O 1529 02664 73625 59) 21 96116 01106 18163 05805 28884 48187 50875 37222 30527 22341 74674 66591 26240 46625 r=4i, #=2) - +i=83 83 ) 6 55S 4SI26 69748 64608 39825 74457 90250 78 98134 05659 62224 19756 93668 16750 r=44, p=2r+i=8g 89 ) 73 74553 16164 02309 Q9S4Q 70604 60375 82860 14788 35981 00107 19894 43375 97 ) 2147 16978 65846 78508 95935 12375 22 13577 09957 18335 I439I 08375 r=5o, 2> = 2 101 ) 13 00548 41538 48019 24559 12505 12876 71698 40079 39847 12005 103 ) 75745 394Q2 35761 16747 76500 735 39217 4986i 75890 75500 107 ) 135 QI757 63944 14006 06750 i 26184 65083 59009 40250 '=54. P = zr+ 1 = 109 109 ) 3 99584 72764 70196 44125 3665 91493 25414 64625 '=56, p = 2 113) 153 12175 3939Q 78375 1 35505 97693 72375 448 ON THE CALCULATION OF THE [54 Table of the Factors by which the quantities P r n for n = 6 1 must be multiplied in order to find the corresponding quantities for n = Q2. 124 3IOO 125 , 124 31000 124 31000 f 7, III . no , 124 22 . Ill 3100 52 . 42 31000 109, 125 108 . 124 216. 109 31000 107. J2C I06 212 . 107 124 310 r-io, 105 , 104 124 I0 3 . IO2 124 204 . 103 31000 101 . 125 . IOO 124 202 31000 99, rv/vf/x*. * 2 5 98 " 124 196 . 99 31000 97 .96 . 124 192 97 3100 r 15, factor 94 124 1 88 . 95 31000 r-i6, 93 125 , .92 , 124 184 . 93 31000 9i T? I2 5 .90 . 124 18.91 3100 r ,8, Factor -^ 88 124 176.89 31000 r-ig, r = 22, Factor 87 . 86 T7.M I25 I24 172.87 1550 .84 * . 124 42-34 31000 125 82 124 164-83 31000 r actor Ol i ,80 .124 648 310 ~ 23 ' ~ 79 -78 . 124 156.79 31000 r 24, Factor ;; 76 . 124 I52-77 3100 _ 75 125 74 . 124 in 31000 73 125 .72 . 124 144-73 31000 I32T67 3100 .70 124 64 - 124 994 3100 1040 69 .68 . 124 136 . 69 IOOO 67 , ., !?,,*- I2 5 .66 . 124 65, 832 31000 t 3't Factor ^ .62 .124 252 31000 61 .60 . 124 732 3oo 594 31000 59. 58 . 124 _ 116.59 31000 57 56 .124 112.57 310 51 3100 r 35, F^tor ^ 125 54 . 124 30, rwnur 125 .52 . 124 104 53 31000 92.47 3100 '-371 Si .50 .124 -3 ' ~ 49 .48 . 124 96.49 31000 47 .46 . 124 r-ep, 45 44 . 124 396 31000 43 .42 . 124 84-43 31000 41 .40 . 124 328 3100 ' 43> . 39 .124 76.39 31000 64 33 31000 ? -44, Factor- 36 . 124 72-37 100 T 45, -racior - r 48 Factor* 2 34 .124 238 31000 r 40, '=49, 33 32 .124 31 -, *wt . I2S 3 . 124 6 31 12 3IOOO 29 .28 . 124 56-29 31000 .26 .124 52.27 3100 ' 84 3100 25 125 .24 .124 S ' 23 .22 .124 44-23 31000 544 3100 r-52, 21 WftM . 125 . 20 . 124 19 .18 " . 124 684 3IOOO 3" 31000 54, * j_ . 16 . 124 r -SS> .124 42 31000 r-56, Factor- . 12 .124 57 ruu.ur ^ . IO . 124 22 3100 r-58, i actor .8 .124 144 31000 r 59, Factor - .6 ' 8 4 4 4 3 . 2 12 54] BERNOULLIAN NUMBERS FROM 32 TO B 6a . 449 The following extract from the calculations for S 6l supplies the further data which are required in making the similar calculations for B,. Table of the products P? for n 61, and calculation of the quantities / 61 and .B 61 . rt-61 Pr 964 96341 45012 37140 7 964 89657 65941 44480 78045 9 709 87762 29656 03133 39385 12416 ii 446 32904 46564 67643 64169 13414 48289 13 238 13879 23030 96548 04209 62125 83207 39024 IS 107 06939 54705 19032 5 8 969 747oi 13695 04939 02335 '7 40 25652 28249 IOOO2 12635 2 3598 54178 27001 58963 45042 19 12 55182 69448 74874 88060 27995 11438 03444 11435 21755 26595 21 3 21576 40020 43518 49711 48609 93756 17416 30520 44444 26034 94720 23 67013 45301 04531 62370 59301 99455 73390 66066 99036 42526 75425 75320 25 11231 92690 01350 11344 11384 88426 IOHI 02042 63460 59712 86463 90646 44690 27 1495 24216 89836 15461 98946 23048 42766 90639 91405 60304 98194 27552 08337 44768 29 155 88977 5894O II502 64044 87592 68543 79442 43154 07109 74300 17982 62640 392IO OI2OO 31 12 52735 11130 65823 80947 92554 38160 79807 93537 05108 42554 57225 79654 88308 53904 41190 33 76200 72588 67366 51664 72332 20669 '6734 05384 05765 11897 06029 79475 60838 23923 30496 01586 35 3436 04646 97275 63168 89528 79432 85080 48564 67069 85420 40166 55083 75038 77057 44549 30465 42575 37 112 10695 64594 13504 42139 70913 71749 31071 09518 52483 41211 65312 62173 47670 44666 56845 56543 86100 39 2 57214 84207 01260 94423 04144 67701 47558 15044 83117 16659 84555 02098 71789 07406 75788 13556 63504 40576 41 4011 14723 17325 72699 40249 05126 47523 41384 39220 48875 47894 38625 57212 13956 40395 37641 30013 25696 79394 43 40 79437 80361 27635 97401 30647 51948 92803 78672 37883 93735 67400 74226 47306 81895 8 3 2 79 "4464 97593 68619 21663 45 25705 59787 87797 26191 94794 17115 04630 03419 37495 12225 69774 989'5 60970 10378 11301 69297 37355 62468 49910 70920 47 94 01591 50554 71453 90027 46633 07601 64638 39147 39457 95157 51563 74099 01857 13757 96469 23049 28627 81245 98033 10917 49 18314 01945 38397 81142 51756 35540 10858 11425 38938 64976 87217 55946 03250 40950 35753 00762 30167 14245 14457 60953 83690 51 i 87872 33205 42521 24918 73620 52220 43451 02967 59560 37380 91077 17414 07192 86078 91797 33570 62616 16951 13720 73193 08800 53 8 59772 65027 85766 57268 29020 42958 36182 03055 01314 74215 62227 37323 466 33589 83982 33494 20456 03407 40119 24748 31132 55 92549 60879 33598 99438 14748 05752 18187 4959' 64300 62439 20770 82079 77657 98879 69766 67926 76502 85769 43176 51024 62060 57 5 03881 30741 91631 57225 17932 37777 11664 03691 29995 67059 46701 18170 63553 01176 06592 67910 74055 83529 45784 76509 00402 59 1 62484 19137 85710 31655 46598 32491 46638 75498 17026 16147 84648 68636 76909 76794 00194 28931 54031 89899 63234 14415 30589 Sum A. 57 450 ON THE CALCULATION OF THE [54 3 31320 18836 89997 72019 2 66693 72038 65962 93743 29234 i 81615 2 93 2 3 22841 13916 65975 29500 1 05278 19363 60383 00590 50922 84521 85704 51610 87066 84400 26912 81306 11708 43921 18731 21241 16304 99659 31737 15459 55819 22481 50881 09376 7280 72679 73513 70244 59338 72132 63345 64463 36534 23148 2060 26085 99438 35927 83133 80731 79420 81422 03431 02817 20660 476 68199 69471 46106 12987 81829 82540 47419 26264 52866 50466 79083 89 21905 44597 87914 19855 84927 97629 57046 17662 79931 87388 03141 65854 '3 349" 68626 82125 54426 67263 91301 27021 31250 59921 40779 48817 63530 01597 i 57559 69464 90906 02362 51676 66589 32518 95763 16754 35575 45346 98938 05686 49552 14452 36699 93578 17658 78109 10421 03333 08814 63116 74540 96649 31618 72384 39026 45205 1012 89026 20578 31483 39325 62887 20505 88928 48887 08759 34066 15643 40587 25845 19619 98600 53 '9595 '9372 08958 57180 51919 28203 98358 57851 14947 21065 36321 97658 24055 41400 85654 11943 2 04708 84567 33692 18114 19357 71372 08907 73807 56096 74038 64311 32708 35093 50374 72353 40000 91120 5622 61071 38888 61163 17544 00695 22634 93248 12401 70831 89557 90085 04790 54505 86023 95045 20787 48165 106 85138 02205 80828 46327 85065 45356 25477 14586 95071 39929 55833 42462 87931 96338 41268 67903 68058 56640 i 35335 92369 91826 49507 34982 43435 24030 17278 74067 66018 60929 96796 89030 65303 15111 96691 82485 08229 42176 1091 20035 25033 87814 43607 29215 15664 30791 00522 59675 08269 86975 14649 40960 64551 39142 55673 78451 76458 52536 5 28717 88547 23215 68582 24341 28297 28291 56152 30058 71218 77129 87692 21 121 55475 27071 50118 75363 73234 51919 59398 1428 87056 91956 77753 33416 02807 48407 75074 495'9 5 2 39Q 64650 96383 18658 97527 67379 12344 56280 09008 07936 35403 32328 i 94837 80347 27439 3797 92842 05773 52990 57895 78308 19504 95599 03524 06683 41684 70820 08271 58896 19768 45591 52241 61715 116 29171 13142 05062 45676 86666 84610 98972 68332 81333 06936 16842 53516 90349 91744 44337 00067 76452 46815 30223 10462 17076 2444 40510 00611 34277 34348 19912 07637 33051 80859 05140 55988 60047 79577 24735 84269 57353 60541 04881 16412 80092 85863 28123 12421 63859 33413 08160 25365 40853 53199 76473 81555 06485 50749 40725 16944 15022 61000 66767 25271 99148 83192 35078 83636 21782 6694 81687 21005 57718 07023 47327 93924 32961 18028 34525 45701 69446 00742 51674 66882 21113 54067 02661 35387 62270 41926 67156 21679 "499 65386 68639 52347 37280 90301 99388 13670 18402 89434 37562 09528 52868 66850 50666 32631 09490 13249 10031 07505 18421 2 38586 80524 96330 07051 54180 90465 82983 21679 "499 65386 68639 52347 37280 90301 99388 13670 18402 89434 37562 09528 52868 66852 89253 13156 05820 20300 64211 97971 01404 20661 62484 19137 85710 31655 46598 32491 46638 75498 17026 16147 84648 68636 76909 76794 00194 28931 54031 89899 63234 14415 30589 123)1017 49015 46248 82929 20691 90682 57810 52749 38172 01376 73286 52913 40891 75958 90058 89058 84224 51788 30401 00977 83555 70815 8 27227 76798 77096 98542 21062 45998 45957 31204 65051 84335 66283 84885 29885 84472 02350 07188 81721 85613 01633 96614 27405 Also = / + !-$ = / + *. Hence the numerator of B S1 is 49 63366 60792 62581 91253 26374 75990 75743 87227 90311 06013 97703 09311 79315 06832 14100 43132 90331 13678 09803 79685 64431 and the denominator is 6. The numerator should be divisible by 61. By actual division we find the quotient to be 81366 66570 37091 50676 28301 22557 22553 17823 40824 77147 77011 52611 66874 01751 34657 38412 01480 83830 78849 24257 14171 without any remainder. Hence the test is satisfied. 54] BERNOULLIAN NUMBERS FROM B M TO -B M . 451 Putting n = 62 in the general formula for I n , the equation for finding J 62 is p, = 1254 = (C 2 /! + c?i 3 + &c. + cs/ 6I ) - (a 62 /, + cr/ 4 + &c. + c;/ M ) - ^ 82 . o/" iAe products P r n or C r n l r for n = 62, ancZ calculation of the quantities I and B w . P r r 1224 97403 15126 27000 7 1318 62960 12351 64825 61250 9 1047 31611 99283 12256 57764 08000 it 713 05918 28669 60263 49682 70245 77250 13 413 34280 85888 01399 17721 06713 37034 10400 15 202 63438 70321 17827 24545 91925 22866 08858 95750 17 83 39696 65578 86264 76322 66209 22l8l 66068 51635 O55OO 19 28 58555 94542 40458 51444 95140 21053 41372 87282 67294 53750 21 8 08898 76714 82398 03720 87545 28273 40141 63107 25233 04696 80000 23 i 87154 68858 77520 75089 04356 92173 67127 07033 93705 33182 82720 57200 25 35029 14828 00991 29946 43152 05353 03*63 14217 47211 11780 56376 36824 93500 27 5241 12474 42890 18466 94632 87483 18156 27525 70508 10657 44462 06932 90192 32000 29 618 61022 18016 32946 98590 77748 75173 78739 80770 12340 25000 71359 62858 69881 ooooo 31 56 74282 35688 25327 01546 71125 92487 54244 00883 779i 98596 10461 67343 85967 95884 97500 33 3 97680 55597 45515 49091 99040 13593 29756 84664 27393 71853 34498 92886 17169 26199 06629 03900 35 20885 77265 91283 25144 26547 57336 93626 48138 19444 21182 83365 30901 22784 68388 39417 34201 60750 37 803 72702 36451 93949 36709 29307 41033 44866 77861 76453 69463 74812 96803 37137 79986 96163 85952 75000 39 22 07547 09417 88230 69522 22725 63329 38621 99997 16675 57158 14287 27868 28754 51165 41924 75153 84450 88000 41 41951 94473 13460 70742 73859 84791 07026 25815 16813 47887 93767 19768 12947 47857 12637 20270 00813 41633 350o 43 53' 35534 4'68o 49039 99764 91627 35469 23074 53295 69076 49498 27488 65975 06937 55786 05736 75804 30842 13107 44350 45 4 28426 63131 29954 36532 46569 51917 43833 90322 91585 37094 96249 81926 82835 06301 88361 54956 22593 74474 98511 82000 47 2075 84997 62960 22130 27671 97738 85791 33753 65790 05125 71141 72703 68283 17358 45083 26599 81857 46055 68821 50303 69250 49 5 61002 57220 26019 91519 80678 87098 18776 22714 49701 72216 44016 14947 63599 50060 35912 28884 73499 42291 97812 14989 07500 51 '64 97137 85625 99647 25849 15549 98294 54651 92975 82414 60245 95602 92158 23068 25214 11867 55393 87574 93399 49331 41792 ooooo 53 158 40362 28246 63723 22183 32460 27879 08673 68346 20850 01629 26306 11944 17753 36392 93934 24572 24135 84831 99277 79042 02600 55 '2i 31990 32997 34403 75375 32680 83261 99147 15186 05997 07342 90433 83968 51807 51230 12577 533i6 89039 03874 47599 17105 63000 57 )78 60958 69040 54508 80720 94091 79650 18870 76551 17449 86231 87340 86782 15992 43548 13964 90868 53939 21585 63425 21180 05500 59 516 23278 09475 67117 87425 96660 10529 35902 77702 23226 56866 92973 03712 27159 85467 35342 46671 19785 35941 92741 85579 38750 61 H5 16809 96562 49020 79663 81837 01215 24471 83370 44978 75762 49844 39975 60548 91277 44873 82476 22450 02809 87250 27463 53200 Sum 572 452 ON THE CALCULATION OF THE [54 Pr 4 36243 87697 24342 90 3 78548 77893 70185 48811 4497 2 78716 53911 88518 57990 91348 2250 1 75 2 37 543 10 12235 45871 22992 27887 5412 93497 954io 94928 02378 28453 11066 01306 4987 42037 54178 68324 74709 63945 75484 93164 37520 0400 15805 49934 99924 70418 93522 43425 18467 43582 93596 7210 . 4928 09310 63471 38407 62125 6i935 61886 20685 42157 55195 4787 1262 57193 31306 84320 74728 50027 73304 40020 26162 02910 25672 4637 263 10794 21854 47616 07261 35156 70330 73480 92422 63878 24298 71041 8012 44 09874 49641 03355 936 70628 86864 80995 38445 07413 00528 99973 00663 9487 5 87061 36227 42558 50148 80045 26955 41837 46232 95599 16206 61749 59985 54841 5097 61205 37936 88650 74784 45544 02056 28869 63559 23035 39722 67230 71062 90152 47243 1712 49i8 48341 538io 73932 51737 85949 77393 88593 85259 98048 17677 13807 26535 87280 87127 1250 299 17897 49734 17582 67887 53537 32642 14280 82979 977O7 46194 89474 10632 33983 64192 04513 3712 13 49059 14453 11321 77198 12944 11678 30812 04939 28358 62074 39126 51776 97257 35887 84641 87931 1750 44015 38690 16552 26277 38349 54937 37798 71386 82942 66613 32397 70867 79925 98404 46147 08687 23336 3462 1009 87584 96457 33439 74440 05191 78671 92009 61035 20491 88358 63059 80594 28625 26369 14429 58845 76163 2800 15 74854 96797 09692 69792 73444 24359 02753 51216 57694 24390 72383 26089 93975 3' 68 o 81257 87329 79368 45012 0400 16016 67184 07220 74927 80220 67078 53027 24678 58049 47882 36915 70184 44191 14479 '7184 24914 42181 49623 43851 4612 loo 92521 21283 37245 2 3429 52327 44590 99161 58079 62943 39767 18612 18262 65251 35303 81290 97094 43519 52136 75805 057$ 36912 48970 42216 75294 46580 72526 67200 22757 79254 36758 36816 56565 65356 86131 57294 02234 53902 32708 71689 14585 8514 71 90442 74720 84069 69923 54885 46404 07985 65201 51850 05540 04250 IIOO7 22840 38364 23122 10022 92597 77169 20639 51773 9662 6626 91737 26844 79662 05850 85794 53935 07634 52788 99494 76141 95070 93794 13322 50142 91262 90626 28725 20724 94331 32954 5837 2 42873 58365 99203 92941 17930 04084 51145 01942 52021 13324 86046 84236 11841 88497 19092 24236 43501 64474 64092 39995 56928 5837 26 74102 75273 31982 84499 05053 22635 36060 46446 40326 46185 64108 50557 31032 34034 09865 95728 01609 27873 53908 85027 27241 3362 51 88483 07587 79322 31504 43191 79152 91355 44914 71967 57229 18813 20655 7545 47868 33713 62994 01945 62549 25407 59574 93170 4590 81 12158 60420 69598 79225 90120 85689 41029 32563 98786 56239 85421 01781 36936 61135 OI 499 7537 26254 19489 19026 62190 99866 3896 9 53068 50542 05310 40997 94772 15321 7673' 81 12158 60420 69598 79225 90120 85689 41029 32563 98786 56239 85421 01781 36936 61135 01509 28438 76796 24799 60024 56963 15188 15701 85 11645 16809 96562 49020 79663 81837 01215 24471 83370 44978 75762 49844 39975 60548 91277 44873 82476 22450 02809 87250 27463 5320 125)3 99486 56389 26963 69794 89542 96147 60185 9'907 84583 88738 90341 48063 03038 99413 89768 16435 05679 97650 42785 30287 12275 375 3195 89251 11415 70958 35916 34369 18081 48735 26276 67109 91122 73184 50424 31195 31118 14531 48045 43981 20342 28242 29698 2030 AlSO B n = 4 - 1 + f i = I 6 , + glfr . Hence the numerator of B^ is 95876 77533 42471 2 8 75 7749 3 I0 75 42444 62057 88300 13297 33681 95535 12729 35859 33544 35944 41363 19436 10268 47268 90946 0900 and the denominator is 30. This numerator should be divisible by 31. By actual division we find the quotient to be 3092 79920 43305 52540 34757 75195 98143 37485 73816 13332 17215 54694 68152 55995 46243 36643 36818 16756 00331 24105 44869 2287 without any remainder. Hence the test is satisfied. 54] BERNOULLIAN NUMBERS FROM B M TO 453 Table of Bernoulli's Numbers expressed as Vulgar Fractions. Numerator Denominator n i 6 i 4 t 20 24576 660 41 60378 67701 '9593 71461 i 1505 58279 54961 34152 246 55088 48463 65575 42994 79457 41491 85145 52903 60231 94176 78653 121 52331 123 10 67838 47260 OO22I 7877 31308 38134 73333 66994 41104 41728 92211 82593 53727 40082 82951 64693 55749 83682 31545 13116 01117 57384 78474 29 2479 84483 40483 00585 30147 26335 58718 67003 38277 68014 07687 79035 69019 09786 31009 26261 2 15 2 7 8 5964 560 94033 49 50572 80116 57181 14996 36348 39292 93132 61334 88800 75557 20403 43408 68585 86652 98863 65405 16194 72814 19091 80307 65673 24464 10673 33007 373H 19604 05851 54954 20734 04684 97942 26990 02077 98991 739" 49627 78306 2 86 1 770 257 26315 27155 2 92999 61082 71849 2OO97 64391 33269 57930 5IIII 59391 68997 81768 05241 07964 35489 95734 84862 42141 26753 68541 41862 04677 04994 07982 41953 03985 85444 97914 28551 93234 49208 47460 77857 20851 65282 48830 72635 18668 99904 36526 92199 37537 57872 75128 36027 57025 98090 87728 86653 38893 i i I 691 7 3617 43867 i 74611 8 54513 2363 64091 85 53103 37494 61029 58412 76005 93210 41217 76878 58367 30534 77373 39138 41559 64491 22051 80708 02691 10242 35023 21632 77961 62491 27547 82124 77525 79249 91853 81238 12691 57396 63229 59940 36021 02460 41491 74033 86151 26479 42017 22418 99101 62443 47001 14381 60235 18442 60429 83077 83087 78288 65801 24004 83487 89196 56867 34148 81613 10839 32477 17619 96983 3 2 42 3 3 4 66 5 2730 6 6 7 510 8 798 9 330 10 138 " 2730 12 6 13 870 14 14322 15 510 16 6 17 19 19190 18 6 19 13530 20 1806 21 690 22 282 23 46410 24 66 25 1590 26 798 27 870 28 354 29 567 86730 30 6 31 510 32 64722 33 30 34 4686 35 1401 00870 36 637 30 38 33"8 39 2 3OOIO 40 498 41 34 04310 42 6 43 13114 26488 67401 7579 955" 42401 93118 43345 75027 55720 28644 29691 98905 74047 61410 44 "7 90572 79021 08279 98841 23351 24921 50837 75254 94966 96471 16231 54521 57279 22535 2 72118 45 129 55859 48207 53752 79894 27828 53857 67496 59341 48371 94351 43023 31632 68299 46247 1410 46 122 08138 06579 74446 96073 01679 41320 12039 58508 41520 26966 21436 21510 52846 49447 6 47 2 11600 44959 72665 13097 59772 81098 24233 67304 39543 89060 23415 06387 33420 05066 83499 87259 45 01770 48 67 90826 06729 05495 62405 11175 46403 60560 73421 95728 50448 75090 73961 24999 29470 58239 6 49 945 98037 81912 21252 95227 43306 94937 21872 70284 15330 66936 13338 56962 043" 39541 51972 47711 33330 50 32040 19410 86090 70782 43020 78211 62417 75491 81719 71527 17450 67900 25010 86861 53083 66781 58791 4326 51 3' 95336 31363 83001 12871 03352 79617 42746 71189 60607 82727 38327 10347 01628 49568 36554 97212 24053 1590 52 3637 39031 72617 41440 81518 20151 59342 71692 31298 64058 16900 38930 81637 82818 79873 38620 23465 72901 642 53 34 69342 24784 78287 89552 08865 93238 52541 39976 67857 60491 14687 00058 9I37I 50126 63197 24897 59230 65973 38057 2091 91710 54 7645 99294 04847 42892 24813 42467 24347 50052 87524 13412 30790 66835 93870 75979 76062 69585 77997 79302 17515 1518 55 26508 79602 1 5509 97133 52597 21468 51620 14443 15149 91925 09896 45178 84276 80966 75651 48755 15366 78120 35526 00109 16 71270 56 2173783231936 91633 33310 76108 66529 91475 72115 66790 90831 36080 61101 14933 60548 42345 93650 90418 86185 62649 42 57 > 95 539 1 65 7 1 84297 69125 '345 8 03384 14168 69004 12806 43298 44245 50404 57210 08957 52457 19682 71388 19959 57547 52259 '77 58 > 69631 19969 71311 '5349 47151 58558 50066 84606 36108 06992 04301 05944 06764 14485 04580 64618 89371 77635 45170 95799 6 59 ' 910906184399685 78499 83274 09517 03532 62675 21309 28691 67199 29747 49229 85358 81132 93670 77682 67780 32820 70131 23282 55930 60 560792 62581 91253 26374 75990 75743 87227 90311 06013 97703 093" 79315 06832 14100 43132 90331 13678 09803 79685 64431 6 61 542471 2875077490 31075 42444 62057 88300 13297 3368i 95535 12729 35859 33544 35944 41363 19436 10268 47268 90946 09001 30 62 55. ON SOME PROPERTIES OF BERNOULLI'S NUMBERS. [!N 1872 a paper on this subject was communicated to the Cambridge Philosophical Society. The paper contained a comparatively simple proof of the theorem given above as Staudt's theorem, which was there attributed to Clausen : another property of Bernoulli's numbers was also established, viz.: "That if n be a prime number other than 2 or 3, then the numerator of the nth number of Bernoulli will be divisible by ."] ON THE CALCULATION OF BERNOULLI'S NUMBERS. [A table of the values of the first sixty-two numbers of Bernoulli, as given above, was printed in Vol. 85 of Crelle's Journal. A paper on this subject was also published in the Report of the British Association in 1877, of which the greater part is contained in the above paper, and the remainder is given below.] Thirty-one of the numbers of Bernoulli are at present known to Mathe- maticians, and are to be found in a communication by Ohm in Crelle's Journal, Vol. xx. p. 11. Of these numbers the first fifteen are given in Euler's Institutiones Calculi Differentialis, Part 2, Chap. 5, and Ohm states that the sixteen following numbers were calculated and communicated to him by Professor Rothe of Erlangen. I find, however, that the first two of these had been already given by Euler in a memoir contained in the Acta Petropolitana for 1781. It may be sometimes useful to have the values of Bernoulli's numbers expressed in integers and repeating decimals. 55] ON THE CALCULATION OF BERNOULLI'S NUMBERS. 455 It readily follows from Staudt's theorem that if the fractional part of the nth number of Bernoulli be converted into a repeating decimal, then the number of figures in the repeating part will be either 2 or a divisor of 2n, and the first figure of the repeating part will occupy the second place of decimals. Table of Bernoulli's Numbers expressed in Integers and Repeating Decimals. No. No. 1 -16 I 2 '03 2 3 '02380 95 3 4 "03 4 5 '075 5 6 -25311 35 6 7 1 -16 7 8 7 -09215 68627 45098 03 8 9 54 -97117 79448 62155 3884 9 10 529 -124 10 11 6192 -12318 84057 97101 44927 536 n 12 86580 -25311 35 12 13 14 25517 -16 13 14 272 98231 '06781 60919 54022 98850 57471 2643 14 15 6015 80873 '90064 23683 84303 86817 48359 16771 4 15 16 1 51163 15767 '09215 68627 45098 03 16 17 42 96146 43061 '16 17 18 1371 16552 05088 '33277 21590 87948 5616 18 19 48833 23189 73593 '16 19 20 19 29657 93419 40068 '14863 26681 4 20 21 841 69304 75736 82615 '00055 37098 56035 43743 07862 67995 21 57032 11517 165 22 40338 07185 40594 55413 '07681 15942 02898 55072 463 22 23 21 15074 86380 81991 60560 '14539 00709 21985 81560 28368 23 79432 62411 34751 77304 96 24 1208 66265 22296 52593 46027 '3ll93 70825 25317 81943 54664 24 94290 02370 17884 07670 7606 25 75008 66746 07696 43668 55720 '075 25 26 50 38778 10148 10689 14137 89303 '05220 12578 6163 26 27 3652 87764 84818 12333 51104 30842 '97117 79448 62155 3884 27 28 2 84987 69302 45088 22262 69146 43291 '06781 60919 54022 28 98850 57471 2643 456 ON THE CALCULATION OF BERNOULLI'S NUMBERS. [55 No. No. 29 238 65427 49968 36276 44645 98191 92192 '14971 75141 24293 29 78531 07344 63276 83615 81920 90395 48022 59887 0056 30 21399 94925 72253 33665 81074 47651 91097 '35267 41511 61723 30 87457 42183 07692 65988 72659 15822 23522 99560 12610 6 31 20 50097 57234 78097 56992 17330 95672 31025 '16 31 32 2093 80059 11346 37840 90951 85290 02797 01847 '09215 68627 32 45098 03 33 2 27526 96488 46351 55596 49260 35276 92645 81469 '96540 33 58898 05630 23392 35499 52102 83983 80766 97259 04638 29918 72933 46929 94" 34 262 57710 28623 95760 47303 04973 61582 02081 44900 '03 34 35 32125 08210 27180 32518 20479 23042 64985 24352 19411 '06167 35 30687 15322 23644 89970 12377 29406 74349 12505 33504 05463 08151 94195 47588 5 36 41 59827 81667 94710 91391 70744 95262 35893 66896 03011 36 34647 07892 24934 86300 26351 72786 57869 86190 73528 95096 22602 62909 14538 93184 246 37 5692 06954 82035 28002 38834 56219 12105 86444 80512 97181 37 16 38 8 21836 29419 78457 56922 90653 46861 73330 14550 89276 38 28860 '03 39 1250 29043 27166 99301 67323 39829 70289 55241 77196 36444 39 84775 -01115 12959 61422 54370 10247 13682 94153 10427 96865 58167 57082 57986 73899 93972 27245 3285 40 2 00155 83233 24837 02749 25329 19881 32987 68724 22013 40 28259 15915 '20745 61975 56627 97269 68392 67857 91922 09034 38980 91387 33098 56093 21333 85504 97804 44328 5 41 336 74982 91536 43742 33396 67690 33387 53016 21959 89471 41 93843 67232 '15461 84738 95582 32931 72690 76305 22088 35341 36 42 59470 97050 31354 47718 66049 68440 51540 84057 90715 65106 42 90499 04704 '31085 21256 87731 14081 85506 02030 95487 77872 75541 88660 84463 51830 47372 30158 24058 32606 31376 43 110 11910 32362 79775 59564 13079 04376 91604 63051 14442 43 23148 86269 99497 '16 44 21355 25954 52535 01188 65838 50190 41065 67897 32987 39163 44 46921 18045 90304 '08804 75492 59078 32600 55365 57563 91467 18775 44373 55] ON THE CALCULATION OF BERNOULLI'S NUMBERS. 457 No. No. 45 43 32889 69866 41192 41961 66130 59379 20621 84513 68511 45 80910 91449 86557 88032 "84801 07894 36935 44712 22043 37824 03222 13157 52724 92080 64148 64139 82169 49999 63251 23659 58885 48350 3 46 9188 55282 41669 32822 62005 55215 50189 71389 60388 91627 46 19959 59100 44871 13437 '05460 99290 78014 18439 71631 20567 37588 65248 22695 03 47 20 34689 67763 29074 49345 50279 90220 02006 59751 40253 47 37827 70239 36918 42141 08241 "16 48 4700 38339 58035 73107 85752 55535 00606 06545 96737 36975 48 90579 15139 76356 41204 83354 '32224 63608 75833 28335 29922 67485 89999 04482 01485 19360 16278 04174 80235 55179 40721 09414 74131 28613 85632 76 49 11 31804 34454 84249 27067 51862 57733 93426 78903 65954 49 75074 79181 78993 54166 54911 76373 '16 50 2838 22495 70693 70695 92641 56336 48176 47382 84680 92801 50 28821 28228 53171 44648 65111 07028 '13414" 51 7 40642 48979 67885 06297 50827 14092 09841 76879 73178 51 80887 06673 11610 03487 48532 84412 10855 '01410 07859 45446 13962 08969 02450 30050 85529 35737 40175 68192 32547 38788 71937 12436 43088 30328 24780 39759 59315 765 52 2009 64548 02756 60448 34656 19672 71536 31868 67270 82253 52 28766 24346 13019 89213 56500 97796 98883 '05220 12578 6163 53 5 66571 70050 80594 14457 19346 03051 93569 61419 46828 53 75104 20621 38756 44521 52460 86197 22777 98400 '15732 08722 74143 30218 06853 58255 45171 33956 38629 28348 9096 54 1658 45111 54136 21691 58237 13374 31991 23014 94962 61472 54 54647 27402 46681 55898 78137 71265 07431 49939 '34194 64710 14554 06621 99281 22390 70085 52107 53810 46409 53506 23597 84716 13430 57045 61619 57851 96268 05479 05077 11801 7726 55 5 03688 59950 49237 74192 89421 91518 01548 12442 37426 55 49032 14141 52565 13225 28310 97674 29893 27917 85387 03227 93148 88010 54018 445 A. 58 458 ON THE CALCULATION OF BERNOULLI'S NUMBERS. [55 No. 56 57 59 61 62 1586 14682 37658 18636 93634 01572 96643 87827 40978 41277 89638 80472 86451 42973 11365 09885 00683 12009 45121 13548 91788 87911 58819 34098 02126 52653 37138 82257 20559 81379 43001 43005 02013 43888 18084 45074 70366 84676 92234 04955 51287 344 5 17567 43617 54562 69840 73240 68250 71225 61240 84923 59305 50859 06216 69403 18108 29579 66515 49771 87766 32444 -02380 95 1748 89218 40217 11733 96900 25877 61815 91451 41476 16182 65448 72627 34721 58762 12289 52384 00153 32666 64382 79521 -05028 24858 75706 21468 92655 36723 16384 18079 09604 51977 40112 9943 6 11605 19994 95218 52558 24525 26426 41677 80767 72684 67832 00716 84324 01127 35747 50763 44103 14895 29605 90861 82633 '16 2212 27769 12707 83494 22883 23456 71293 24455 73185 05498 77801 50566 55269 30277 36635 00257 26591 02528 03139 11549 56836 "41706 43950 64162 89896 44622 10131 68427 75098 18261 25962 01999 15049 7 8 27227 76798 77096 98542 21062 45998 45957 31204 65051 84335 66283 84885 29885 84472 02350 07188 81721 85613 01633 96614 27405 '16 3195 89251 11415 70958 35916 34369 18081 48735 26276 67109 91122 73184 50424 31195 31118 14531 48045 43981 20342 28242 29698 20300 "03 No. 56 57 59 60 61 62 56. NOTE ON THE VALUE OF EULER'S CONSTANT; LIKEWISE ON THE VALUES OF THE NAPIERIAN LOGARITHMS OF 2, 3, 5, 7 AND 10, AND OF THE MODULUS OF COMMON LOGARITHMS, ALL CARRIED TO 260 PLACES OF DECIMALS. [From the Proceedings of the Royal Society, Vol. xxvu. (1878).] IN the Proceedings of the Royal Society, Vol. xix., pp. 521, 522, Mr Glaisher has given the values of the logarithms of 2, 3, 5, and 10, and of Euler's constant to 100 places of decimals, in correction of some previous results given by Mr Shanks. In Vol. xx., pp. 28 and 31, Mr Shanks gives the results of his re- calculation of the above-mentioned logarithms and of the modulus of common logarithms to 205 places, and of Euler's constant to 110 places of decimals. Having calculated the value of 31 Bernoulli's numbers, in addition to the 31 previously known, I was induced to carry the approximation to Euler's constant to a much greater extent than had been before practicable. For this purpose I likewise re-calculated the values of the above-mentioned logarithms, and found the sum of the reciprocals of the first 500 and of the first 1000 integers, all to upwards of 260 places of decimals. I also found two independent relations between the logarithms just mentioned and the logarithm of 7, which furnished a test of the accuracy of the work. On comparing my results with those of Mr Shanks, I found that the latter were all affected by an error in the 103rd and 104th places of decimals, in consequence of an error in the 104th place in the determination 81 of log . With this exception, the logarithms given by Mr Shanks were 80 found to be correct to 202 places of decimals. 582 460 NOTE ON THE VALUE OF EULER'S CONSTANT, ETC. [56 The error in the determination of Iog e 10, of course entirely vitiated Mr Shanks' value of the modulus from the 103rd place onwards. As he gives the complete remainder, however, after the division by his value of log, 10, I was enabled readily to find the correction to be applied to the erroneous value of the modulus. Afterwards I tested the accuracy of the entire work by multiplying the corrected modulus by my value of log, 10. Mr Shanks' values of the sum of the reciprocals of the first 500 and of the first 1000 integers, as well as his value of Euler's constant, were found to be incorrect from the 102nd place onwards. Let S n , or S simply, when we are concerned with a given value of n, denote the sum of the harmonic series, n Also let R n , or R simply, denote the value of the semi-convergent series, B l B, , B 3 _ 2n 2 4n 4 6n' where B lt B t , B 3 , &c., are the successive Bernoulli's numbers. Then if Euler's constant be denoted by E, we shall have and the error committed by stopping at any term in the convergent part of R n will be less than the value of the next term of the series. I have calculated accurately the values of the Bernoulli's numbers as far as B^, and approximately as far as B lw , retaining a number of significant figures varying from 35 to 20. When ^ = 1000, the employment of the numbers up to B 61 suffices to give the value of R Jm to 265 places of decimals. When n=500, it is necessary to employ the approximate values up to B 7t , in order to determine RIM with an equal degree of exactness. In order to reduce as much as possible the number of quantities which must be added together to find S m and S lax , I have resolved the reciprocal of every integer up to 1000 into fractions whose denominators are primes or powers of primes. Thus SgQ and S ltm> may be expressed by means of such fractions, and by adding or subtracting one or more integers, each of these fractions may be reduced to a positive proper fraction, the value of which in decimals 56] NOTE ON THE VALUE OF EULER'S CONSTANT, ETC. 461 may be taken from Gauss' Table, in the second volume of his collected works, or calculated independently. Thus I have found that : _249 _2_ 3 120 _3_ .86, 205 _58_ J_ _3_ 21 30 11 '"256 81 5 343 121 169 289 861 28 29 81 87 41 15 26 32 24 33 27 67 28 38 73 72 33 61 43 47 53 59 61 67 71 73 79 83 89 97 101 ^5_ JL1 102 _68____23_ HI , H 6 . , .25 126 _27_ 28 103 107 109 113 127 131 137 139 149 151 157 29 85 88 91 92 97 98 100 101 107 113 h 163 167 173 179 181 191 193 197 199 211 223 115 116 118 121 122 f 227 + 229 233 239 241 + (the sum of the reciprocals of the primes from 251 to 499) 19. Similarly I have found that : c _ 249 3K) 181 _75_ 62_ 35 220 11 300 726 32 34 IOOO ~5T2 + 729 625 343 121 169 289 361 529 841 961 37 21 1.0 40 48 28 ,56 7_ 31 40 45 25 49 44 *" 41 43 47 58 59 61 67 71 73 79 88 89 97 J59_ _82_ _9?_ ,104 _12_ _67_ 84 121 85 144 10 h TOT 103 107 109 113 127 131 137 139 149 151 _ .. _ - L + + j... _ h 157 163 167 173 179 181 191 193 197 199 211 j)5_ _21_ 212 138 22_ 223 211 2^6 221 226 ^7_ f 223 + 227 + 229 233 239 241 251 257 263 + 269 + 271 j48_ 236 ^9_ 246 ^ 261 ^4_ 266 _57_ 170 175 f 277 + 281 283 293 307 311 313 317 331 337 347 176 178 181 185 188 191 193 196 200 202 206 f 349 + 353 + 359 367 373 379 383 389 397 + 401 + 409 211 212 217 2J.8 221 223 226 230 232 233 235 f 419 + 421 + 431 + 433 439 443 449 457 461 + 463 + 467 241 245 247 251 f + + + 499 + (the sum of the reciprocals of the primes from 503 to 997) 43. 462 NOTE ON THE VALUE OF EULER'S CONSTANT, ETC. [56 This mode of finding $ 500 and S ltm is attended with the advantage that if an error were made in the calculation of the former of these quantities, it would not affect the latter. The logarithms required have been found in the following manner : 10 . 25 7 , 81 i 50 , , , 126 Let log = a, log -6, \og = c, log -d, and 1 og^=e- Then we have = 7a 2& + 3c, los: 3 = lla 36 + 5c, Ioof5 = ' O O Also Iog7=4(39a-106+17c-d); Zi or again, log 7 = 1 9a 4& + 8c + e, and we have the equation of condition a - 26 + c = d + 2e, which supplies a sufficient test of the accuracy of the calculations by which a, b, c, d, and e have been found. 10 / 1 ' Since log = -log 1- 9 = 10 25 50 126 If we have settled beforehand on the number of decimal places which we wish to retain, and have already formed the decimal values of the reciprocals of the successive integers to the extent required, then the formation of the values of a, b, c, d, and e, will only involve operations which, though numerous, are of extreme simplicity. In this way have been found the following results : 56] NOTE ON THE VALUE OP EULER'S CONSTANT, ETC. 463 Log 10 -s- 9 = 10536 05156 57826 30122 75009 80839 31279 83061 20372 98327 40725 63939 23369 25840 23240 13454 64887 65695 46213 41207 66027 72591 03705 17148 67351 70132 21767 11456 06836 27564 22686 82765 81669 95879 19464 85052 49713 75112 78720 90836 46753 73554 69033 76623 27864 87959 35883 39553 19538 32230 68063 73738 05700 33668 65 Log 25 * 24 = 04082 19945 20255 12955 45770 65155 31987 01772 11747 63352 02297 28561 42083 06828 16287 62241 55690 62020 38337 10701 85958 13391 57612 02856 02344 55254 44440 90711 64191 09254 90615 87090 13793 32587 08185 56690 89768 86470 69797 42768 97243 12354 16791 64980 33118 36535 36811 73829 09383 64151 16223 48133 67972 69296 Log 81 + 80 = 01242 25199 98557 15331 12931 28631 20890 67623 60339 58145 90685 43409 40510 22236 97287 99924 04408 75833 17607 39941 83907 88915 98331 57135 00593 07313 64880 85644 69078 59065 10006 71375 61155 92285 64823 02773 78467 95356 20673 20672 56121 24774 48623 61600 82118 41837 57253 45313 78157 48027 60627 91715 42041 36587 2 Log 50 H- 49 = 02020 27073 17519 44840 80453 01024 19238 78525 33383 73356 83210 27195 49256 65918 71880 87170 92908 14086 00703 48551 55810 69865 22995 29709 68602 61790 51909 27000 19877 96234 68586 52194 37909 61418 83597 32774 05301 16399 74760 65371 30928 59153 97434 74168 79079 46094 49807 56880 62620 29129 95963 65850 08854 45 Log 126 + 125 = 00796 81696 49176 87351 07973 39067 84478 84307 61916 78206 21803 11515 15228 34251 08036 00862 32503 51700 93221 55597 11104 32429 31908 69430 97326 52573 22928 44338 63827 35942 41437 63883 38664 80785 92159 70835 21671 40563 92519 30299 88730 07233 43319 67047 32333 55315 84852 90164 08154 11413 00140 51668 01463 4832 All these are Napierian logarithms. The above-mentioned equation of condition is satisfied to 263 places of decimals. 464 NOTE ON THE VALUE OF EULER'S CONSTANT, ETC. [56 Whence have been deduced the following : Log, 2= -69314 71805 59945 30941 72321 21458 17656 80755 00134 36025 52541 20680 00949 33936 21969 69471 56058 63326 99641 86875 42001 48102 05706 85733 68552 02357 58130 55703 26707 51635 07596 19307 27570 82837 14351 90307 03862 38916 73471 12335 01153 64497 95523 91204 75172 68157 49320 65155 52473 41395 25882 95045 30081 06850 15 Log, 3 = 1-09861 22886 68109 69139 52452 36922 52570 46474 90557 82274 94517 34694 33363 74942 93218 60896 68736 15754 81373 20887 87970 02906 59578 65742 36800 42259 30519 82105 28018 70767 27741 06031 62769 18338 13671 79373 69884 43609 59903 74257 03167 95911 52114 55919 17750 67134 70549 40166 77558 02222 03170 25294 68992 45403 15 Log, 5 = 1-60943 79124 34100 37460 07593 33226 18763 95256 01354 26851 77219 12647 89147 41789 87707 65776 46301 33878 09317 96107 99966 30302 17155 62899 72400 52293 24676 19963 36166 17463 70572 75521 79637 49718 32456 53492 85620 23415 25057 27015 51936 00879 77738 97256 88193 54071 27661 54731 22180 95279 48521 29282 13604 17624 80 Log,7 = 1-94591 01490 55313 30510 53527 43443 17972 96370 84729 58186 11884 59390 14993 75798 62752 06926 77876 58498 58787 15269 93061 69420 58511 40911 72375 22576 77786 84314 89580 95163 90077 59078 24468 10427 47833 82259 34900 84673 74412 50497 37048 53551 76783 55774 86240 15102 77418 08868 67107 51412 13480 93879 74210 03537 95 Log,10=2'30258 50929 94045 68401 79914 54684 36420 76011 01488 62877 29760 33327 90096 75726 09677 35248 02359 97205 08959 82983 41967 78404 22862 48633 40952 54650 82806 75666 62873 69098 78168 94829 07208 32555 46808 43799 89482 62331 98528 39350 53089 65377 73262 88461 63366 22228 76982 19886 74654 36674 74404 24327 43685 24474 95 M= -43429 44819 03251 82765 11289 18916 60508 22943 97005 80366 65661 14453 78316 58646 49208 87077 47292 24949 33843 17483 18706 10674 47663 03733 64167 92871 58963 90656 92210 64662 81226 58521 27086 56867 03295 93370 86965 88266 88331 16360 77384 90514 28443 48666 76864 65860 85135 56148 21234 87653 43543 43573 17247 48049 05993 55353 05 where M denotes the modulus of common logarithms. 56] NOTE ON THE VALUE OF EULER'S CONSTANT, ETC. 465 50 In these calculations the value of log - - has been determined with less 4y 126 accuracy than that of log - , and therefore the value of log 7 found by * Aw means of the latter quantity has been preferred. If now in the formula which gives Euler's constant we take n = 500, we find the following results : -=o-ooi 2n R iw = -00000 03333 33200 00025 39671 87309 34479 09501 49853 06920 81561 41982 03143 98353 10049 47690 35814 25947 82825 73530 80967 33251 23444 83365 27221 32891 79715 39888 78668 70158 11997 43277 84264 18919 84678 56672 58294 26067 37401 94207 08483 64907 04495 03811 66583 11699 18899 16275 81704 82573 08004 99446 91635 500 = 6-79282 34299 90524 60298 92871 45367 97369 48198 13814 39677 91166 43088 89685 43566 23790 55049 24576 49403 73586 56039 17565 98584 37506 59282 23134 68847 97117 15030 24984 83148 07266 84437 10123 70203 14772 22094 00570 47964 42959 21001 09719 01932 14586 27077 01576 02007 28842 06850 09735 01135 74118 52998 6631 Log, 500 = 6-21460 80984 22191 74263 67422 42594 91605 47278 04331 52606 36739 79303 69340 93242 07062 36272 51021 28288 27237 62074 83901 87110 62880 60166 54305 61594 90289 71296 61913 55661 26910 65179 94054 14829 26073 41092 64585 48079 22114 05716 58115 31635 24264 74180 14925 98528 81625 94504 71489 68628 97329 77937 00975 E= -57721 56649 01532 86060 65120 90082 40243 10421 59335 93992 35988 05767 23488 48677 26777 66467 09369 47063 29174 67495 14631 44724 98070 82480 96050 40144 86542 83622 41739 97644 92353 62535 00333 74293 73377 37673 94279 25952 58247 09491 60087 35203 94816 56708 53233 15177 66115 28621 19950 15079 84793 74508 5697 A. 59 466 NOTE ON THE VALUE OF EULER'S CONSTANT, ETC. [56 Again, if in the same formula we take w = 1000, we find the following: - = 0-0005 2n -^000= '00000 00833 33325 00000 39682 49801 59487 73237 84632 11743 88611 32124 18782 98862 06644 51967 06850 04241 14869 65631 43736 78499 44114 24665 37423 82138 50259 70190 89962 61572 33894 07843 88131 36054 55889 69002 08034 44545 27898 47738 31546 74821 27649 54293 18527 10448 88349 55931 43201 82238 86978 52223 81562 S 1000 = 7-48547 08605 50344 91265 65182 04333 90017 65216 79169 70880 36657 73626 74995 76993 49165 20244 09599 34437 41184 50813 96798 01438 22544 03715 81484 21958 84703 40431 40398 43368 92966 39178 33827 35905 57913 00071 54692 68403 25933 79804 87809 56515 86955 67800 24804 71415 08712 32350 00711 42865 21027 95267 06455 Log, 1000 = 6-90775 52789 82137 05205 39743 64053 09262 28033 04465 88631 89280 99983 70290 27178 29032 05744 07079 91615 26879 48950 25903 35212 68587 45900 22857 63952 48420 26999 88621 07296 34506 84487 21624 97666 40425 31399 68447 86995 95585 18051 59268 96133 19788 65384 90098 66686 30946 59660 23963 10024 23212 72982 31056 E= -57721 56649 01532 86060 65120 90082 40243 10421 59335 93992 35988 05767 23488 48677 26777 66467 09369 47063 29174 67495 14631 44724 98070 82480 96050 40144 86542 83622 41739 97644 . 92353 62535 00333 74293 73377 37673 94279 25952 58247 09491 60087 35203 94816 56708 53233 15177 66115 28621 19950 15079 84793 74508 56961 It will be seen that the two values found for E agree to 263 places of decimals, which supplies another independent verification of the value obtained for log, 2. 57. SUPPLEMENTARY NOTE ON THE VALUES OF THE NAPIERIAN LOGARITHMS- OF 2, 3, 5, 7, AND 10, AND OF THE MODULUS OF COMMON LOGARITHMS. [From the Proceedings of the Royal Society. Vol. XLII. (1886).] IN Vol. xxvu. of the Proceedings of the Royal Society, pp. 88 94, I have given the values of the logarithms referred to, and of the Modulus, all carried to 260 places of decimals. These logarithms were derived from the five quantities a, b, c, d, e, which were calculated independently, where ! 10 , . 25 . 81 , . 50 ,126 a = log, b = \og , c = log , d = \og , and e^ , , , and a complete test of the accuracy of these latter calculations is afforded by the equation of condition a - 2b + c = d + 2e. In the actual case the values found for a, b, c, d, e satisfied this equation to 263 places of decimals. Although this proved that the values of the logarithms found in the above paper had been determined with a greater degree of accuracy than was there claimed for them, yet I was not entirely satisfied with the result, since the calculation of the fundamental quantities had been carried to 269 places of decimals, and therefore the above-cited equation of con- dition shewed that some errors, which I had not succeeded in tracing, had crept into the calculations so as to vitiate the results beyond the 263rd place of decimals. 592 468 ON THE VALUES OP THE NAPIERIAN LOGARITHMS, ETC. [57 Of course in working with such a large number of interminable decimals, the necessary neglect of decimals of higher orders causes an uncertainty in a few of the last decimal places, but when due care is taken, this uncer- tainty ought not to affect more than two or three of the last figures. The Napierian logarithm of 10 is equal to 23a 66+ lOc, and the Modulus of common logarithms is the reciprocal of this quantity. Since the value found for the logarithm of 10 cannot be depended upon beyond 262 places of decimals, a corresponding uncertainty will affect the value of the Modulus found from it. In the operation of dividing unity by the assumed value of log 10, however, the quotient was carried to 282 places of decimals. This was done for the purpose of supplying the means of correcting the value found for the Modulus, without the necessity of repeating the division, when I should have succeeded in tracing the errors of calculation alluded to above, and thus finding a value of log 10 which might be depended upon to a larger number of decimal places. Through inadvertence, the values of the logarithms concerned, and the resulting value of the Modulus, were printed in my paper in the Pro- ceedings above referred to exactly as they resulted from the calculations, without the suppression of the decimals of higher orders, which in the case of the logarithms were uncertain, and in the case of the Modulus were known to be incorrect. Although it was unlikely that this oversight would lead to any mis- apprehension as to the degree of accuracy claimed for my results in the mind of a reader of the paper itself, there might be a danger of such misapprehension if my printed results were quoted in full unaccompanied by the statement that the later decimal places were not to be depended on. My attention has been recalled to this subject by the circumstance that in the excellent article on Logarithms which Mr Glaisher has con- tributed to the new edition of the Encyclopaedia Britannica, he has quoted my value of the Modulus, and has given the whole of the 282 decimals as printed in the Proceedings of the Royal Society, without expressly stating that this value does not claim to be accurate beyond 262 or 263 places of decimals. 57] ON THE VALUES OF THE NAPIERIAN LOGARITHMS, ETC. 469 I have now succeeded in tracing and correcting the errors which vitiated the later decimals in my former calculations, and have extended the com- putations to a few more decimal places. The computations of the fundamental logarithms a, b, c, d, e have now been carried to 276 decimal places, of which only the last two or three are uncertain. The equation of condition, a 26 + c = d + 2e, by which the accuracy of all this work is tested, is now satisfied to 274 places of decimals. The parts of the several logarithms concerned which immediately follow the first 260 decimal places as already given in my paper in the Proceedings, are as follows : a 05700 33668 72127 8 b 67972 72775 92889 4 c 42038 01732 39184 3 d 08865 93150 99834 1 e 01463 48349 12851 7 Whence a-2b+ c = 11792 89849 25533 3 and d + 2e = 11792 89849 25537 5 Difference = 42 Also the corresponding parts of the logarithms which are derived from the above are log 2 30070 95326 36668 7 log 3 68975 60690 10659 1 Whence log 5 log 7 log 10 13580 59722 56777 3 74183 10810 25196 7 43651 55048 93446 And the correction to the value of log 1 which was formerly employed in finding the Modulus is -(263) 33 69426 01554 where the number within brackets denotes the number of cyphers which precede the first significant figure. The corresponding correction of M, the Modulus of common logarithms, will be found by changing the sign of this and multiplying by M 2 , the approximate value of which is 0-18861 16970 1161 470 ON THE VALUES OF THE NAPIERIAN LOGARITHMS, ETC. [57 Hence this correction is (264) 6 35513 15874 7 And finally the corrected value of the Modulus is M= '43429 44819 03251 82765 11289 18916 60508 22943 97005 80366 65661 14453 78316 58646 49208 87077 47292 24949 33843 17483 18706 10674 47663 03733 64167 92871 58963 90656 92210 64662 81226 58521 27086 56867 03295 93370 86965 88266 88331 16360 77384 90514 28443 48666 76864 65860 85135 56148 21234 87653 43543 43573 17253 83562 21868 25 which is true, certainly to 272 and probably to 273 places of decimals. 58. NOTE ON SIR WILLIAM THOMSON'S CORRECTION OF THE ORDINARY EQUILIBRIUM THEORY OF THE TIDES. [From the Report of the British Association, 1886, p. 541.] IN Art. 806 of Thomson and Tait's Treatise on Natural Philosophy it is pointed out that if the Earth's surface is supposed to be only partially covered by the Ocean, the rise and fall of the water at any place, according to the equilibrium theory, would be falsely estimated, if, as is usually done, it were taken to be the same as the rise and fall of the spheroidal surface that would bound the water were there no dry land. In the articles which immediately follow the above, it is shewn that in order to satisfy the condition that the volume of the water remains unchanged, the expression for the radius vector of the spheroid bounding the water must contain, in addition to the terms which would be sufficient if there were no land, a quantity a which depends on the positions of the Sun and Moon at the time considered, and which is the same for all points of the sea at the same time. This quantity a contains five constant coefficients which depend merely on the configuration of land and water. The values of these coefficients in the case of the actual oceans of our globe have been carefully deter- mined very recently by Mr H. H. Turner of Trinity College, in a joint paper by Professor G. H. Darwin and himself, which is published in Vol. XL. of the Proceedings of the Royal Society. 472 NOTE ON SIR WILLIAM THOMSON'S CORRECTION. [58 It should be remarked that every inland sea or detached sheet of water on the globe has in the same way a set of five constants, peculiar to itself, which enter into the expression of the height of the tide at any time in that sheet of water. By taking such constants into account the formulae which apply to the Oceanic tides are rendered equally applicable to the tides of such a sea as the Caspian, which are thus theoretically shewn to be very small, as they are known to be practically. In the work above cited reference is made to a passage in a memoir by Sir William Thomson on the Rigidity of the Earth, published in the Philosophical Transactions for 1862, as being the only one known to the writers in which any consciousness is shewn that such a correction of the ordinary equilibrium theory as that above mentioned is required. However just this remark may be in reference to modern writers on the equilibrium theory, it is only fair to Bernoulli, the originator of the equilibrium theory, to point out that in his prize essay on the Tides he distinctly recognises the fact that when the sea is supposed to have only a limited extent the rise and fall of its surface cannot be the same as if the Earth were entirely covered by it. In particular, he shews that the Tides are so much the smaller as the sea has less extent in longitude, and thus explains why they are altogether insensible in the Caspian and in the Black Sea and very small in the Mediterranean, of which the com- munication with the Ocean is almost entirely cut off at the Straits of Gibraltar (see Bernoulli, Traite sur le Flux et Reflux de la Mer, Chap. xi. sect. ii.). It may be as well to mention that this treatise of Bernoulli, as well as the dissertations of Maclaurin and Euler on the same subject, is published in the 3rd volume of the Jesuit's edition of Newton's Principia and also appears in the Glasgow reprint of that edition. 59. ON CERTAIN APPROXIMATE FORMULAE FOR CALCULATING THE TRAJECTORIES OF SHOT. [From the Proceedings of the Royal Society, Vol. xxvi. (1877) and Nature, Vol. XLI. (1890).] IN the postscript to a paper by Mr W. D. Niven, "On the Calculation of the Trajectories of Shot," which is published in the Proceedings of the Royal Society, Vol. xxvi. pp. 268 287, I have given, without demonstration, some convenient and not inelegant formulae applicable to a limited arc of a trajectory when the resistance is supposed to vary as the nth power of the velocity. In these formulae, the angle between the chord of the arc and the tangent at any point is supposed to be always small. The index n is not restricted to integral values, but may take any value whatever. As the proof of these formulae is not altogether obvious, and a similar method of treatment may be found useful in other problems, I think it may not be unacceptable to your readers if I shew here how the formulae may be demonstrated. Analysis. Investigation of formulae applicable to a small arc of a trajectory, when the resistance varies as the nth power of the velocity. Let x and y denote the horizontal and vertical coordinates at time t, u the horizontal velocity, and the angle which the direction of motion makes with the horizon at the same time. A. 60 474 ON CERTAIN APPROXIMATE FORMULAE FOR [59 Hence the velocity at time t is u sec <, and we may denote the resistance by ku n (sec the independent variable, the fundamental formulae are (3) --(*) tan*; . , dt u. ... From the first of these equations 1 du _ k . T^+i 77T ~" 7. \ 8ec ' u <> g and therefore, by integration between the limits = a and ) 2 d ; 9 J P and 1 f a =- 9 J p =i f S' J/3 /3 and we wish to compare the two former of these definite integrals with the following known one, viz. : du , , kn-2 59] CALCULATING THE TRAJECTORIES OF SHOT. 475 and the last with 1 1 , x ( a 1 du . k(n-l) (" -==i--==i,^(-l) -3-rr<& * -- - u (s q n p n 'j ft u n d(f> g ) p v sec This may be done by means of the following lemma, which follows immediately from Taylor's theorem : Lemma. If F( only, or of and u, where u is a function of given by the above differential equation (l), and if a and ft be the limiting values of in the integral and y=-(a + ft), then, m putting for a moment (f> = y + w, f JP ^ where 1 and F(y), F'(y), F"(y), &c., are what F(), F'(), F"(4>), &c., become when y is substituted for , and the corresponding value of u (u suppose) is put for u. In what follows, the last of the terms above written, which is of the 5th order in (a ft), is neglected, together with all terms of the same order of small quantities. All the definite integrals with which we are here concerned are included in the two forms fa fa u 1 (sec (j)) m d(f), and u 1 (sec ) m tan < d) is a function of only, viz. when F (<) = (sec <)" +1 . 602 476 ON CERTAIN APPROXIMATE FORMULAE FOR [59 Hence F($) = (n + !)[( + 1) (sec <)" +1 (tan is known. In the next place, let F (<) = u l (sec <) m . Hence tan I~H ~l or F'($) = F($) - u n (sec ) n+1 + m tan < ; \-9 J and - u n (sec <}>) n+1 + m tan i tlflm flit Tfl 1 w"- 1 ^ (sec <^) n+1 + - (n + 1) w" (sec <^) n+1 tan ^ + m (sec ? , *S T *J ' or tlsp /./ ~] 5- w 2 " (sec <) 2n+2 + 2 ^^ u n (sec <^.) n+1 tan ^ + m 3 (sec <)' - m j SF V V (sec c^,) 2 "- 1 - 2 4. _(+ 1) (sec <^) n+1 tan < + m (sec ^) 2 m (sec ' ,y M 59 1 CALCULATING THE TRAJECTORIES OF SHOT. 477 Since ^T = -w" +1 (sec<6) +1 , = (a- ft) u a < (sec y )- {l + -L (a - 0)' [j (Z + n) ~ \ud/ wnere ^gr denotes what ^- becomes when w = 0, or when y is sub- stituted for (f), and for u, that is The factor u l may be eliminated from this expression, and the expression itself simplified, by means of the formula for, putting m = n+l in the above expression, we have > (sec A)- ' cfy = ( a - /8) < (sec y )-+ l + -L ( a - ^ z (Z + w ) z (sec <) m d+u l (sec < J /3 1 L. 478 ON CERTAIN APPROXIMATE FORMULAE FOR [59 It will be noticed that the term involving \j has disappeared by this division. Now make m = 2, and this formula becomes (seotf d* - - (cosy)- 1 Divide throughout by gr, and put 1 = 2, then, from before, Similarly, divide throughout by g, and put Z=l, then Lastly, let ^(<^>) = w* (sec <^) m tan ^ =/(^>) tan <^> suppose, so that then ' F'($) =f ($} tan <^, +f() (sec )* tan = (a- ft) \F (y) + ^ (a - ft)" F" (y) I approximately, J/3 = (-) l/(y) tan y + -L (a - 0)*[f" (y) tany + 2/(y) (secy) 2 + 2/(y) (secy)'* ^ also f " /(^) ^ = (a - )8) j/(y) + ^r (a - ^) 2 /" (y)} approximately ; J/3 - 1 59] CALCULATING THE TRAJECTORIES OF SHOT. 479 and therefore JV() d = tan y + (a - /3) [^ (sec y) + (sec y) 2 tan y ] ; in which the term involving f"(y) has disappeared. Now, since /() = u l (sec <) m , we have, as before and therefore fM_,(*> W\M-JJL +tany. /(y) Hence F ()d(j> -=- f() d = tan y -\ (a /8) 2 (sec y) J I ( ,, ) + m + 1 tan y ; and in the particular case where 1 = 2, and m = 2, we have Hence the angle which the chord of the arc makes with the axis of x is y + 12 ( a - & [ 2 (udt), + 3 tan ?] = y suppose - Multiplying by the value of X found above, we have {(s|).[ 4(K - 1)(tanr)! - 4(fleer) '] _ _ ^ M, - 1 n + 4 (sec y) 2 - 6 (secy) 2 - n- 1 n + or [4 (w- 2) (sec y) 2 - 4 (n - 1 )1 + tan y f^^2 rT+5 (sec y) 2 -^ , L 480 ON CERTAIN APPROXIMATE FORMULAE FOR [59 Considering -^ ^r a , s=i s=i > an( ^ a ~ to be small quantities of the first order, the above expressions for , X, Y, and T are true to the fourth order. The quantity ( rrl which occurs as a factor in some of the terms \ud 2 + (ti) 7T j, WA 2 In this make ca = -(a /3) and -(a/8) successively; therefore and Hence we have to the first order of small quantities p-q (du\ a-/3 and and therefore \~TI) l \ i o\ t the first order. Making this substitution for ( f-.} the expressions for X, Y, and T \uda)/ a become X = 59] CALCULATING THE TRAJECTORIES OF SHOT. 481 {tan y - i- |-~| (a - /3) [n=2 (sec y)> - =T] (a /8) 3 tan y [ 2 TO + 5 (sec y) 2 n 1 n + 3] >- ; and these values are still true to the fourth order, considering - - -- and ' P + V a /3 to be small quantities of the first order as before. The angle which the chord of the arc makes with the axis of x becomes, in like manner, which is true to the third order. The above expressions for X and Y may be transformed by introducing this angle y into them instead of y, thus (cos y)- ' = (cos y )-> - (n - 1 ) (cos y )- ' sin y [1 ^| (a - ft) + \ (a - ft tan y] = (cos y)- jl - ^i Ti (o _ ^ tan y - ^ (a - /J)> (tan y) j . Hence we find and (a - ^) 2 [n=2 (sec y) 2 - ri^ A. 61 482 ON CERTAIN APPROXIMATE FORMULAE FOR [59 or putting Q for 1 (a /3) 2 [n 2 (sec y) 2 n 3], we have 1 /I 1 x ^y {-?=-, ~ Tjs=iJ (cos y)'" 2 sin y Similarly, if we have and therefore where ^ has the same value as before. Hence the values of X, Y, and T are as stated in my postscript to Mr Niven's paper. Although the method of finding the expressions for X and T given above, is perhaps the plainest and most straightforward that can be taken, the following leads to simpler operations. Let Then /(^^^tt^sec^'^^^^-"- 1 ^ by equation (l) = Tn \ u " + const. K (L n) Hence 59] Now let then and CALCULATING THE TRAJECTORIES OF SHOT. )'" = u l (sec <)" l+ " +1 , 483 sec <' tan F" (<) =f" (4>) (sec <) + 2/' (<) (sec <)'" tan < + m/(<) [ m (sec ^)" (tan <^) 2 + (sec = /' (0) (sec <)'" + 2wi/ (<) (sec <^,) m tan + mf(tjt) [m + 1 (sec ^,) m+2 - m (sec Hence, by the lemma, = ( - ^) {/(r) (sec y) 1 " + ^ (a - /S) 2 (sec y] m \J" (y) + 2mf( 7 ) tan y + f(y) [m+1 (sec y) 2 - m]l ] ~ But from above + m f(y) [m+l (sec y) 2 - m]l ] . k(l->, Hence, by division, = (sec 7 ) m l + ( a - / S) 2 tan y + m [7^+! (sec y) 2 - It will be noticed that in this division the quantity /" (y) has disappeared. 612 484 ON CERTAIN APPROXIMATE FORMULA FOR [59 Now, from above, /(# = ' (sec # + ', and therefore 7 du and = l +(w+l)tany. /(y) Hence ~j"j + m [m + l (sec y) 2 m\ = (sec y) m J 1 + (a - /8) 2 2Zm ( ^- ) tan y + m (?n, + 2i + 3) (sec y) 2 [24 L \ud,

p +1 3 1 Again, we have P* = g f*^ ~ 2 ' 3 n+l p ,3 re 1 p l / " K+1 + 22n+l" / * "- 2^ Substitute for ^P n+1 and /aP,,., their equivalents obtained by writing n+l and n-l successively for n in the above formula, _3 (re+l)(n + 2) ^~2f2 [3 h + 3) 2 2 (2-/1- 3 (re-l)re By a slight reduction the coefficient of P n becomes n TJ p p _ V"" r x ; y + 2) p TO (n + 1 ) ^V ~ 2 (2TO+l)(2TO + 3) ^" +2 * (2w - 1) (2re + 3) , 3 (n-l)n _ f Again, putting TO = 2 in our original formula, we have P.=- uP - P ; O O K O PP --U.PP --PP * * ~~ o r** * a 5 (n-1) n p _ 2 re + 1 p _ 2 n f 2 2re"^l2^+ 1 r *"' 3 2n + 1 " +1 3 2n + 1 60] EXPRESSED BY A SERIES OF LEGENDRE'S COEFFICIENTS. 485* Substitute for juP n+2 , ftP,, and /uP n _ 2 their equivalents as before, p _5 n 2 (5 (M+l)(n + 2) n + 2 5 2n+ 1 3 {5 n(n + l) n 5 (w l)n nl 2 3 (2-l)(2wT3) 2n+l + 2 (2w-l)(2w+l) 2^3 ~ 3 5 -- 2 2n- "~ a By reduction the coefficient of P n+l in this expression becomes 3 n(n+l)(n + 2) 2 2n- and similarly the coefficient of P n _j becomes (n-l)n(n+l) 2 (2-3)(2n+l)(2n Hence we have P * Pn ~ 2 (2n+l)(2w + 3)(2ft + 5) in+J 3 n(n + l)(n+2) p 2 (2n-l)(2n+l)(2 + 5) n+1 3 (n-l)w(n+l) p f 2 (2w-3)(2n+l)(2n + 3) "-' 5 ( n -2)(n-l)n f 2 (2n - 3) (2 - 1) (2n + 1) n - 3 ' 7 3 Again, since P 4 = - /iP 8 - P,, we have P 4 P = ^ (P,P B ) - \ (P,P). A. 62 490 THE PRODUCT OF ANY TWO LEGENDRE'S COEFFICIENTS [60 Whence by substituting the values found above for P 3 P n and P z P n and again for pP n+3 , pP n+1 , &c., we obtain p _5.7 (n+l)(n + 2)(n + 3) Jj^+j4_p . n + S p \ 3. 7 n(n + l)(n + 2) (n + 2 p + n+ 1 p \ 3. 7 (n-l)n(n + l) [ n p + n-l p \ .5.7 (n-2)(n-l)n jn-2 n-3 \ ~+T) \2n-5 2 5 J 3.3 (n+l)(n + 2) p _3 p 2.4 (27i+l)(2n + 3) ?t+2 ' 4 (2-l)(2 + 3) 3. 3 (n-l)n p ~2T4(2-l)(2ri+l) "- 2 By reduction, the coefficient of P M+a in this expression becomes 2 (2n-l)(2w+l)(2n + 3)(2w + 7)' Similarly, the coefficient of P n _ 3 becomes 5 (n-2)(n-l)n(n+l) 2 (2n and finally, the coefficient of P n becomes \2J (2n - 3) (2 - 1) (2n + 3) (2n + 5) ' Hence, collecting the terms, we have 1.3.5.7 4 " ~ 1 . 2 . 3 . 4 (2w + 1) (2 + 3) (2n + 5) (2n + 7) 1.3.5 1 u p h !.2.3' 1 (27i-l)(2n+l)(2n + 3)(2n + 7) 1. 3 1. 3 (n-l)n(n+l)(n + 2) p h 172 'l.2 (2-3)(2w-l)(2 + 3)(2n+5) 1 1.3.5 (n-2)(n-l)n(n+l) p h l ' 1 . 2. 3 (2-5)(2n-l)(2n+l)(2n "" 1.3.5.7 (ro-3)(n-2)(n-l)n p h 1.2.3.4 (2n - 5) (2n - 3) (2n - 1) (2n + 1) "- 4> 60] EXPRESSED BY A SERIES OF LEGENDRE'S COEFFICIENTS. 491 where the law of the terms is obvious, except perhaps as regards the succession of the factors in the several denominators. With respect to this it may be observed that the factors in the denominator of any term P p are obtained by omitting the factor 2p + 1 from the regular succession of five factors (n +p - 3) (n +p - 1) (n +p + 1) (n +p + 3) (n +p + 5). For instance, where p = n + 4, 2p + I = 2n + 9, so that the factor 2n + 9 is to be omitted, and we have 2n + l, 2n + 3, 2n + 5 and 2n + 7, as the remaining factors, and so of the rest. Hence by induction we may write, supposing to fix the ideas that m is not greater than n, P P = -----~ 1.2.3...m ' (2n+l)(2w + 3)...(2n 1.3. 5...(2w-3) !_ n(n+l)...(n + m- !.2.3... m-1 ' T ' 2n + &c., &c. 1.3.5...(2m-2r-l) 1 . 3 . 5...(2r- 1) 1.2.3...(m-r) 1.2.3...r (n r + 1) (n r + 2)...(n r + m) xx V_ x [(2 + 2m - 4r + 1) P n+m .J + &c., &c. 1 1.3. 5. ..(2m- 3) (n-m + 2)(n-m + 3)...(n + l) ' 1 .3. 5...(2m-l) (n-m+l)(n-m + 2)...n 1.2.3...m ' (2w-2m+l)(2n-2m + 3)... 622 492 THE PRODUCT OF ANY TWO LEGENDRE'S COEFFICIENTS [60 And it remains to verify this observed law by proving that if it holds good for two consecutive values of m, it likewise holds good for the next higher value. If the function - - be denoted by A (m), the general 1.2.3.. .fti term of the above expression for P m P n may be very conveniently represented by A(m-r)A(r)A(n-r) -r) \2n + 2m-2r+l r being an integer which varies from to m. The fundamental property of the function A is that 2m + 1 . , x A(m) f l m + We may interpret A (m) when m is zero or a negative integer, by supposing this relation to hold good generally, so that putting m = 0, we have Similarly A (- l) = -f-A (,0) = 0; and hence the value of A (m) when m is a negative integer will be always zero. We will now proceed to the general proof of the theorem stated above. Let Q m denote the quantity of which the general term is. A(m r)A (r) A(n r) i2n + 2m 4r + 1\ A (n + m - r) \2n + 2m-2r+l) n + m "*' In this expression r is supposed to vary from to m, but it may be remarked that if r be taken beyond those limits, for instance if r 1, or r = m+l, then in consequence of the property of the function A above 60] EXPRESSED BY A SERIES OF LEGENDRE'S COEFFICIENTS. 493 stated, the coefficient of the corresponding term will vanish. Hence prac- tically we may consider r to be unrestricted in value. Similarly, let Q m _ l denote the quantity of which the general term is A (m r)A (r l)A (n r+1) /2n + 2m 4r + 3\ p A(n + m-r) ' \2n + 2m-2r+l) n + m ~* r + 1 ' writing m I for m and r I for r in the general term given above. Also let Q m+ i denote the quantity of which the general term is A (m - r + 1) A (r) A(n r) /2n + 2m-4r + 3\ A(n + m-r+l) ~ \2n + 2m - 2r + 3/ "+-*' writing m + 1 for m in the general term first given. In consequence of the evanescence of A (m) when m is negative, we may in all these general terms suppose r to vary from to m+l. Let us assume that Q m ^ 1 = P m ^ l P n , and also that Q m =P m P n , then we have to prove that Q m+l = P m+1 P n . As before, (m+ l)P m+1 + mP m _ 1 -(2m+ l)pP m = 0, .-. (m+l)P m+l P n + mP m _ 1 P n -(2m+l) l j,P m P n = 0. Hence our theorem will be established if we prove that Now Q m = ...... A(m-r+l)A(r-l)A(n-r+l) ' A(m-r)A(r)A(n-r) /2n + 2m-4r+l\ A (n + m - r) ' \2n+~2m-2r + l) n+m ~ w Multiplying by p, and substituting for ^P n+m -^+i and )u,P n+m _ 2r , &c., in terms of -P n+m _ 2) . +1 , &c., we find the coefficient of P n+m .y,^ in p,Q m to be A(m-r+l)A(r-l)A(n-r+l) fn + 2m-2r + 2\ A(m r)A(r)A(n r)/ n + m 2r+] A(n + m-r) ' \2n + 2m-2r+l/ ' 494 THE PRODUCT OF ANY TWO LEGENDRE'S COEFFICIENTS [60 Hence the coefficient of P n+m . v+l in (m + 1) Q m+l + mQ m .,-(2m+ I) pQ m will be A (m-r+ l)A(r)A(n-r) ^ + ] A L^(r-lM(n-r+l) / n + m-2r + 2 \ (w + m r+1) \2n + 2m 2r + 3/ _ A(m-r)A(r)A(n-r) ^ + ^ / n + m-2r+l \ /2n + 2m - 4r + 3\ *2n+2a-2r+l/' The sum of the first two lines of this expression is Suppose for a moment that n r + 1 = q, then the quantity within the brackets becomes Now this quantity evidently vanishes when q = r, and therefore it is divisible by q r. It also vanishes when m + 1 = r, and therefore it is likewise divisible by m r+1. Hence it is readily found that this quantity or " =- - r(n r+1) v So that the sum of the first two lines of the expression for the coefficient OI " +m-2r+i IS A(m-r+l)A(r-l)A(n-r) f(m-r+ 1) (n-2r+ I)} A(n + m r+1) \ r(n r+l) }' 60] EXPRESSED BY A SERIES OF LEGENDRE'S COEFFICIENTS. 495 Again, the sum of the other two lines of the expression for the co- efficient of P w+m _ 2r+1 is A (m r)A(r\}A (n r) A(n + m-r) (2n + 2m-2r+l) Or _ 1 2?? _ 2r-l-1 r n r+l As before suppose n r+l=q, and the quantity within the brackets becomes O _ 1 O0 _ 1 (2m + 1 ) (m + q - r) + - -- m (2m + 1 + 2q - 2r). Now this quantity evidently vanishes when q = r, so that it is divisible by q r. It also vanishes when m= q, and therefore it is likewise divisible by m + q. Hence it is readily found that this quantity or = n-2r+l ( n + m -r+l)(2m-2r+l), r - v and therefore the sum of the last two lines of the expression for the coefficient of P n+m -^+i IS A(m-r)A(r-l)A(n-r) f(rc-2r + l) _ (n + m-r+ 1) (2m-2r+ 1)1 l /' A(n + m r) ' \r (n r+l) 2n + 2m 2r+ Hence the whole coefficient of -P n+ ,_2r + i is ^4 (m-r)^4(r-l)^4 (re-r) (n-2r + l) A(n + m r+1) r(n r+1) x{(2m-2r+l)-(2m-2r+l)} = 0. And the same holds good for the coefficient of every term. Hence we finally obtain which establishes the theorem above enunciated. The principle of the process employed in the above proof may be thus stated : Every term in the value of Q m gives rise to two terms in the value of p.Q m or in that of (2m +1) p.Q m ; one of these terms is to be subtracted from the corresponding term in (m+l)Q m+1 , and the other from the cor- responding term in mQ m _ 1 , and it will be found that the two series of terms thus formed identically destroy each other. 496 THE PRODUCT OF ANY TWO LEGENDRE'S COEFFICIENTS, ETC. [60 Hence we can find at once the value of the definite integral j l P m P n P p dn, for if p = n + m 2r we have A im+p n\ . /n + mp\ . /n+p m\ PP { 2 m n n+m+p+l + &c. Hence P P m P n P p dp fm+p-n\ . fn + m-p\ / n +p-m\ A (~^) A (2) A (*-) 2p+l ( . (n + m+p\ n + m+p+l J _/ p ' ^ r~2~ -) /m+p-n\ . in + m-p\ / n +p-m\ 9 \ -- 2 -- /I 2 -- / I -- 2 -- / _ _ * _ \^/\^/\^/ . ~ n + m + p+l . /n + m +p\ A \~*~) or if L- 2 A(s-m)A(s-n)A(s-p} : A a where as above 1 3 5 / _1\ \ o O 9**\ ty I / ctm \ I It is clear that, in order that this integral may be finite, no one of the quantities m, n, and p must be greater than the sum of the other two, and that m + n+p must be an even integer. I learn from Mr Ferrers that, in the course of the year 1874, he likewise obtained the expression for the product of two Legendre's co- efficients, by a method very similar to mine. In his work on " Spherical Harmonics," recently published, he gives, without proof, the above result for the value of the definite integral I P m P n P p d^. J -i 61. SUR LES ETOILES FILANTES DE NOVEMBER (LETTRE A M. DELAUNAY.) [Paris Academy of Sciences, Compt. Rend. XLiv., 1867.] Observatoire de Cambridge, 23 Mars, 1867. JE me suis occupe des mete"ores de Novembre et j'ai obtenu quelques resultats qui me paraissent importants. Si vous pensez qu'ils puissent interesser 1' Academic, je vous serai obligd de les lui communiquer a sa prochaine seance. Je les ai fait connaitre verbalement a la seance de la Societe philosophique de Cambridge de lundi dernier, mais Us n'ont pas encore ete imprimis. Adoptant la position suivante du point radiant : Decl. = 231'N. qui est la moyenne de ma propre determination et de cinq autres, et tenant compte de 1'action de la Terre sur les mdte"ores lorsqu'ils se sont approch^s de nous, je trouve les elements suivants de 1'orbite : Pe"riode .................................... 33'25 annees (admise) Moyenne distance ........................ 10'3402 Excentricite .............................. 0'9047 Distance perihdlie ........................ 0'9855 Inclinaison ................................. 1 6 46' Longitude du noeud ..................... 51 28' Distance du pe"rihe"lie au noaud ...... 6 51' Mouvement retrograde A. 63 498 SUR LES ETOILES FIL ANTES DE NOVEMBRE. [61 L'accord de ces elements avec ceux de la comete de Tempel (i., 1866) est encore plus grand que celui que presentent les elements calcules il y a quelque temps par M. Le Verrier. Avec les elements, j'ai calculi la variation seculaire du nceud de 1'orbite des me"teores due a 1'action des planetes Jupiter, Saturne et Uranus. J'ai em ploy 6 la mdthode de Gauss donnde dans sa Determinate Attrac- tionis etc., et j'ai trouve que, dans une pdriode totale des me"te"ores, c'est- a-dire en 33 '2 5 annees, le mouvement du nosud est Par 1'action de Jupiter, de 20' Saturne, de 7'f Uranus, de l'^ De sorte que le mouvement totale du nosud en 33'25 anne"es serait de 29 minutes, ce qui s'accorde presque exactement avec la determination du moyen mouvement du noeud d'apres 1'observation faite par le pro- fesseur Newton dans son Memoire sur les pluies d'etoiles de Novembre, inse're' dans les nos. Ill et 112 du Journal Americain de Science et Arts. Cela me parait mettre hors de doute 1'exactitude de la pe"riode de 3 3 '25 anndes. 62. THE LUNAR INEQUALITIES DUE TO THE ELLIPTICITY OF THE EARTH. [From the Observatory, No. 108 (1886).] IT is well known that M. Delaunay was unfortunately prevented by a premature death from completely carrying out his purpose of determining all the sensible inequalities of the Moon's motion by means of his very original and beautiful method of treating that subject. Happily the two magnificent volumes in which he determines the inequalities which are caused by the disturbing force of the Sun, on the supposition that the motion of the Earth about the Sun is purely elliptic, are complete in themselves. The small effects due to the action of the planets and the spheroidal figure of the Earth, as well as those which arise from the disturbances of the Earth's motion, remained to be determined. Mr G. W. Hill, who is already well known for his skilful treatment of special portions of the lunar theory, has, in the paper now to be noticed, produced a valuable supplement to Delaunay's work by applying the same method to the determination of the lunar inequalities which are due to the ellipticity of the Earth. This paper forms part 2 of vol. ill. of the valuable series of astronomical papers prepared for the use of the American Ephemeris and Nautical Almanac. The author begins by developing the terms of the disturbing function which are introduced by the ellipticity of the Earth, by substituting for the Moon's coordinates their disturbed values as already given by Delaunay's work. Some idea of the length and complexity of this substitution may 632 500 THE LUNAR INEQUALITIES DUE TO THE [62 be formed when it is stated that the development so obtained contains one constant term accompanied by 121 periodic terms. The next process is by a series of transformations of the variables involved gradually to remove these periodic terms from the disturbing function, so that it is at length reduced to the form of a constant term. The number of such operations required to effect this reduction amounts to 103, although each operation is individually sufficiently simple. By the essential principle of Delaunay's method the differential equa- tions throughout these transformations always preserve their canonical form, and therefore when the disturbing function has been reduced to the above- mentioned simple form, the integrals are at once obtained. In the next place the transformations indicated in the 103 operations above mentioned are also made in Delaunay's expressions for the three coordinates of the Moon, so that finally the values of these coordinates are found in terms of three arbitrary constants and three angles, each of which consists of a term proportional to the time joined to an arbitrary constant. The coordinates thus expressed are the longitude, the latitude, and the reciprocal of the radius vector. As this last quantity is only intended to be employed in finding the Moon's parallax, it is given by Delaunay with much less precision than the other two coordinates, a circumstance which is to be regretted as an imperfection from a theoretical point of view. The expressions thus found are purely analytical, that is the coefficients are expressed in series of powers and products of Delaunay's constants m, e, e', y, each term also involving as a factor a constant quantity which depends on the figure of the Earth. In order to make his work more complete, Mr Hill determines the numerical value of this last-mentioned factor by a very elaborate discussion of the results of numerous pendulum experiments. Finally, by the substitution of the known values of the constants employed, the numerical expressions for the perturbations of the Moon's coordinates produced by the figure of the Earth are obtained. It will be remarked that comparatively few of the coefficients so found amount to an appreciable quantity, by far the larger number being utterly insensible. 62] ELLIPTICITY OF THE EAKTH. 501 The quantity m denoting, as in Delaunay, the ratio of the mean motion of the Moon to that of the Sun, it is found that the analytical expressions of most of the coefficients involve negative powers of in. This circumstance, which never happens in the case of the perturbations due to the Sun's action, has given rise to a difficulty in some minds as to the admissibility of Mr Hill's results. Mr Stockwell, in particular, in an article in the twenty-ninth volume of the American Journal of Science, asserts that the value given to the coefficient of the principal equation of latitude leads to a manifest absurdity, and "justifies the suspicion that the entire solution is erroneous." The difficulty thus noticed by Mr Stockwell, however, admits of an easy explanation. He applies Mr Hill's formulae to a case in which they are not applicable, and for which they were not intended. The form of development in series adopted by Mr Hill is founded on the supposition that the perturbations due to the Earth's figure which he wishes to deter- mine are very small compared with those due to the action of the Sun, and therefore he expressly neglects quantities which are proportional to the square of the first-named perturbations. Now, in the case of our Moon, which is that treated by Mr Hill, the above-mentioned supposition certainly holds good, and consequently his formulae are sufficiently accurate. If, however, the Sun's distance from the Earth were very much greater than it is, or if the Moon's distance were very much less than it actually is, then the perturbations arising from the Earth's figure might be much greater than those which arise from the Sun's action, and a different form of development would have to be adopted. In this latter case it would be better to refer the motion of the Moon, not to the ecliptic, but to a fundamental plane passing through the line of intersection of the equator and ecliptic, and occupying a definite intermediate position between those two planes. If the perturbations due to the action of the Sun are much greater than those due to the Earth's figure, this fundamental plane nearly coincides with the ecliptic, whereas if the latter perturbations are much greater than the former, the funda- mental plane nearly coincides with the equator. In Mr Hill's formula, the principal term in the expression for the latitude nearly represents the dis- tance of the fundamental plane from the ecliptic corresponding to the actual longitude of the Moon at the time. A simple analytical illustration of the change of form of the coefficient of this term of the latitude in different circumstances may be given. 502 LUNAR INEQUALITIES DUE TO ELLIPTICITY OF THE EARTH. [62 If m have its usual meaning as before stated, and if c be a small positive constant depending on the elliptic! ty of the Earth, then the value of the coefficient in question is approximately proportional to- rn + c Now, if, as in the case of our Moon, c is very much smaller than m 2 , so that we may neglect the square of c compared with that of m 2 , f> the quantity just mentioned becomes approximately = ; whereas if w 2 is small compared with c, the same quantity becomes nearly = 1, and the coefficient becomes nearly independent of the ellipticity of the Earth, as it should do, since in this case the coefficient of this term is approximately equal to the sine of the obliquity of the ecliptic. Mr Stockwell's second objection, that Mr Hill has omitted to take into account the modification of the Sun's disturbing force which is caused by the alterations of the Moon's coordinates due to the ellipticity of the Earth, seems to arise from a misapprehension on his part of the spirit of Delaunay's method. These alterations of the Moon's coordinates are implicitly involved in the variables a, e, y, I, g, h, throughout the series of operations by which Delaunay gradually removes from R the periodic terms arising from the action of the Sun. CAMBRIDGE I PRINTED BY J. AND C. F. CLAY, AT THE UNIVERSITY PRESS. 14 DAY USE RETURN TO DESK FROM WHICH BORROWED LOAN DEPT. This book is due on the last date stamped below, or on the date to which renewed. Renewed books are subject to immediate recall. J 5t. ?7Apr'64IVI F REC'D LD ftPR!5'ti4-aPM ULblo titi ^ e_ ^ ^ . DEC 4 '67 -4PM * MS! DFPT. jfC 1 1 1968 9 4 RECEIVED **- 3'68-llPM General Library University of California Berkeley '