fyxmll IKnweMitg Jitotg THE GIFT OF Q.ai/-yt^iji--^j-T.^ .-j,^]A'^ .Aj^...TT^f .MMA^' The dateshows w^en this volume was taken, j All books not in use] for instruction or re-^ , search are limited to'| all borrowers. -l Volumes of periodi- ' cals and of pamphlets comprise so many sub- jects, that they are held in the library as much as possible. For spe- cial purposes they are given out for a limited time. Graduates and sen- iors are allowed five volumes for two weeks. Other students may have two vols, from the circulating library for two weeks. Books not needed during recess periods should be returned to the library, or.arrange- hients made for their return during borrow- er's absence, if wanted. Books needed by more than one person are held on the reserve list. Books of , special value and gift books, when the giver wishes it, are riot allowed to circulate. Cornell University Library QB 3.H64 V.1 The collected mathematical works of Geor 3 1924 020 314 757 XI Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924020314757 THE COLLECTED MATHEMATICAL WORKS OF GEORGE WILLIAM HILL VOLUME ONE THE COLLECTED MATHEMATICAL WORKS OF aEORGE WILLIAM HILL VOLUME ONE Published by the Carnegie Institution of Washington JuifE, 1905 c \ .'nH^J-^1 CAENEGIE INSTITUTION OF WASHINGTON Publication No. 9 (Volume One) THE FniEDENWALD COMPANY BALTIMORE, MB., u. B. A. EEEATA IN FIEST VOLUME. (Lines counted from the bottom of the page are noted as negative.) Page Line Page Line 10 14 for T read T^, 252 6 for aj_ read ai_y 13 2 for 3^ read gy, 252 18 for -. + m read -, + m' 13 12 for ^q read ^q^ 252 -3 for U_ read U_ 13 15 for 2 F/' read 2 F," 259 ■-7 for m^ read m' 14 9 for-ae,-K) 264 4 for TT^j read, ttSJ , read +(|-e,-|eD 265 -1 for 4^/ read '4 5" 18 2 for m^ F„ read in' V„ 266 11 for S read S 18 -5 V V for — 5%r read ^ 374 -1 for aP read aP^ v nf F„ rre F„ 278 13 for aP^ read aP^, 19 21 for - 1."3679 read + l."3679 293 13 add the exponent i to the right 60 -3 '"•S^-^'w) member 293 17 add the exponent J to the 63 -9 add the term 6 (j!'' to the factor denominator of tan 8 for +^ read+^) 396 17 for {liny read {im)i 65 -9 304 23 for (2C) read (3C)I 69 14 for Ja read A^a 308 3 for (us) read (Ms)i 78 20 for A read X^ 308 -3 for a(a_j__i read aia_i_j_i 79 21 for an read and 309 10 for aja_( i read aja_i_y_, 124 15 for cos e' read cos 6 310 -1 for + [3, 3] read + [3, - 3] 133 11 for cos

r" and add the exponent i to the 193 -3 for 5— read 5—, 334 -6 last factor for ao read aj 201 204 4 16 add V before(a'— 2aZcos^ + ?) for / read / t 334 336 -1 23 for [y read [£,]' for r sin read r sin v 210 19 for established read establishes 336 37' forrcos, rsino readrcosuj^sino 210 20 for / read / 336 326 31 36 u it c( ii ti a It tc 11 a 11 « 219 -3 for 3 E Els read 3 E^ El, for ^ read ^, for - read -r 338 6 224 12 r r^ 1 2 331 -9 for^ffj read 9^ a* and transpose an arm of paren- 340 -9 for — ^n read — f m' thesis to one line lower 346 8 for (Z . e' read t^ . e" 225 10 for^£"''-read-e'"'-' d CI 346 -5 for w' read n' ' ,7"-l ,,/'*-" 349 -9 for J read J « 226 4 for'^f read*^f 3 3 230 2 for = read — 353 -7 add exponent i to last factor 230 3 for = read — in denominator for 9^ read 1^ y 3y for wv + "wv' + -vrv" 359 -9 for A' read A„ 238 19 360 36 for i V)* read J ij* 242 -7 361 15 for -^ 5; e' read -g- >? e' read wi* + w'v' + w"/' 363 33 for r; read f, 251 9 for af read of 363 4 for Z2 read ?2 [Insert in Piest Voujmb Mathematical Works of G. W. Hill.] CONTENTS. MBMOm p^Qj, Introduction by H. Poincarfi vii-xvm No. 1. On the Curve of a Drawbridge 1 2. Discussion of the Equations which Determine the Position of a Comet or other Planetary Body from Three Observations 2-4 3. On the Conformation of the Earth 5-19 4. Ephemeris of the Great Comet of 1858 20-22 5. On the Reduction of the Rectangular Coordinates of the Sun Referred to the True Equator and Equinox of Date to those Referred to the Mean Equa- tor and Equinox of the Beginning of the Year 23-24 6. Discussion of the Observations of the Great Comet of 1858, with the Object of Determining the Most Probable Orbit 25-58 7. On the Derivation and Reduction of Places of the Fixed Stars 59-76 8. Determination of the Elements of a Circular Orbit 77-84 9. New Method for Facilitating the Conversion of Longitudes and Latitudes of Heavenly Bodies, near the Ecliptic, into Right Ascensions and Declina- tions, and Vice Versa 85-88 10. Correction of the Elements of the Orbit of Venus 89-104 11. On the Derivation of the Mass of Jupiter from the Motion of Certain Aster- oids 105-108 12. On the Inequality of Long Period in the Longitude of Saturn, whose Argu- ment is Six Times the Mean Anomaly of Saturn Minus Twice that of Jupiter Minus Three Times that of Uranus 109-112 13. Charts and Tables for Facilitating Predictions of the Several Phases of the Transit of Venus in December, 1874 113-150 14. A Method of Computing Absolute Perturbations 151-166 15. On a Long Period Inequality in the Motion of Hestia Arising from the Action of the Earth 167-168 16. Solution of a Problem In the Theory of Numbers 169 17. A Second Solution of the Problem of No. 8 170-172 18. Remarks on the Stability of Planetary Systems 173-180 19. Useful Formulas in the Calculus of Finite DifEerences 181-185 20. Elementary Treatment of the Problem of Two Bodies 186-191 21. The Differential Equations of Dynamics 192-194 22. On the Solution of Cubic and Biquadratic Equations 195-199 23. On the Equilibrium of a Bar Fixed at one End Half Way between Two Centers of Force 200-202 24. The Deflection Produced in the Direction of Gravity at the Foot of a Conical Mountain of Homogeneous Density 203-205 25. On the Development of the Perturbative Function in Periodic Series 206-226 YI CONTENTS. MEMOIB PAGE No. 26. Demonstration of the Differential Equations Employed by Delaunay in the Lunar Theory 227-232 27. Solution of a Problem in the Motion of Rolling Spheres 233-235 28. Reduction of the Problem of Three Bodies 236-242 29. On the Part of the Motion of the Lunar Perigee which is a Function of the Mean Motions of the Sun and Moon 243-270 30. Empirical Formula for the Volume of Atmospheric Air 271-281 31. On Dr. Weiler's Secular Acceleration of the Moon's Mean Motion 282-283 32. Researches in the Lunar Theory 284-335 33. On the Motion of the Center of Gravity of the Earth and Moon 336-341 34. The Secular Acceleration of the Moon 342-347 35. Note on Hansen's General Formulas for Perturbations 348-350 36. Notes on the Theories of Jupiter and Saturn 351-363 INTRODUCTION Par M. H. Poincaeb M. Hill est une des physionomies les plus originales du monde scienti- fique am^ricain. Tout entier a ses travaux et a ses calculs, il reste 6tranger a la vie fi^vreuse qui s'agite autour de lui, il recherche I'isolement, hier dans son bureau du Nautical Almanac, aujourd'hui dans sa ferme tranquille de la vallee de I'Hudson. Cette reserve, j'allais dire cette sauvagerie, a 6te une circonstance heureuse pour la science, puisqu'elle lui a permis de mener jusqu'au bout ses ingenieuses et patientes recherches, sans en etre distrait par les incessants accidents du monde exterieur. Mais elle a empech6 que sa reputation se repandit rapidement au dehors; des annees se sont ^coulees avant qu'il eut, dans I'opinion du public savant, la place a laquelle il avait droit. Sa modestie ne s'en chagrinait pas trop et il ne demandait qu'une chose, le moyen de travailler en paix. M. Hill est ne a New York le 3 mars 1838. Son p^re, d'origine an- glaise, etait venu en Amerique en 18 20 a I'age de 8 ans; sa mere, d'une vieille famille huguenote, lui apportait les traditions des premiers colons de la terre am6ricaine. Quoique n6 dans une grande ville, M. Hill est un campagnard ; peu de temps apres sa naissance, son pere quitta New York et vint s'6tablir a "West Nyack, N. Y. ; c'etait une ferme, pres de la riviere Hudson, a 25 milles en- viron dela grande Cit6. C'est la que M. Hill passa son enfance; il aimatou- jours cette residence; il y revenait toutes les fois qu'il le pouvait, et quand il eut quittS le Nautical Almanac, c'est encore la qu'il s'etablit d6finitivement; c'est la qu'il poursuit tranquillement ses travaux, 6vitant le plus qu'il peut les voyages a New York. Ses aptitudes exceptionnelles pour les math^matiques ne tarderent pas a se manifester et on d6cida de I'envoyer au college. En octobre 1855, a I'age de 17 ans, il entra au College Rutgers, New Brunswick, N. J. Son pro- fesseur de mathematiques etait le Dr. Strong, ami de M. Bowditch, le traduc- teur de la M6canique Celeste de Laplace. Le Dr. Strong etait un homme de tradition, un laudator temporis acti ; pour lui Buler etait le Dieu des Mathematiques, et apres lui la decadence avait commenc6 ; il est vrai que c'est la un dieu que I'on peut adorer avec yill INTRODUCTION profit. De rares exceptions pr&s, la bibliotheque du Dr. Strong 6tait im- pitoyablement fermSe a tons les livres posterieurs a 1840. Heureusement on a 6crit d'excellentes choses sur la Mecanique Celeste avant 1840; on trouvait la Laplace, Lagrange, Poisson, Pontecoulant. Tels furent les maitres par lesquels Hill fut initie au rudiment. En juillet 1859 il reput ses degres au College Rutgers et se rendit a Cambridge, Mass., dans I'espoir d'accroitre ses connaissances matli6matiques, mais il n'y resta pas longtemps, car au printemps de 1861 il obtint un poste d'assistant aux bureaux du Nautical Almanac a Washington. II resta au service de cette 6phemeride pendant trente annees de sa vie, les plus fructueuses au point de vue de la production scientifique. Les bureaux du Nautical Almanac etaient a cette epoque a Cambridge (Massachusetts), oil ils pouvaient profiter des ressources scientifiques de I'Uni- versite Harvard et ils etaient diriges par M. Runkle. Ce savant avaitfonde un journal de mathematiques elementaires. The Mathematical Monthly, dans le but de favoriser les etudes mathematiques en Amerique en facilitant la publication de courts articles et en proposant des prix pour la solution de problemes mathematiques. L'un des premiers articles publics rev61ait la main d'un maitre, et gagna ais6ment le prix. II s'agissait des fonctions de La- place et de la figure de la Terre. L'auteur etait M. Hill, qui venait de sortir du college. C'est ainsi que 1' attention de M. Runkle fut attiree sur ce jeune homme et qu'il songea a utiliser ses services pour les calculs de I'ephemeride ameri- caine. On I'autorisa n6anmoins a continuer sa residence dans sa maison fami- liale de West Nyack (village qui s'appelait alors Nyack Turnpike). II y resta encore quand en 1886 les bureaux du Nautical Almanac furent trans- fer's a Washington. Mais en 1877 M. Simon Newcomb prit la direction de r6phemeride. II voulut entreprendre une tache colossale, la reconstruction des tables de toutes les planfetes ; la part de M. Hill etait la plus difficile ; c'etait la theorie de Jupiter et de Saturne, dont il avait commence a s'occuper depuis 1872. II ne pouvait la mener a bien qu'aupr&s de son chef et de ses coll&gues. II fallut done se resigner a I'exil; I'importance de I'oeuvre a accomplir lui fit facilement accepter ce sacrifice. Ses services furent hautement apprecies; en 1874 il fut eiu membre de I'Academie Nationale des Sciences. En 1887 la Societe Royale Astrono- mique de Londres lui accorda sa medaille d'or pour ses recherches sur la theorie de la Lune. II fut president de la Societe Mathematique Americaine pendant les annees 1894 et 1895. L'universite de Cambridge (Angleterre) INTRODUCTION jj- lui confera des degres honoraires, et il en fut de meme de plusieurs universites americaines. En 1892 il prit sa retraite et quitta les bureaux du Nautical Almanac; il eut hate de s'installer pour ses dernieres annees dans cette ch&re maison ou il avait pass6 son enfance ; au debut, il la quittait encore plusieurs fois par semaine pour venir professer a I'Universite Columbia a New York; mais il ne tarda pas a se lasser de cet enseignement et depuis il y vit seul avec ses livres et ses souvenirs. Le travail quotidien du Nautical Almanac, qui est fort absorbant, lui lais- sait cependant assez de temps pour ses recherches originales, dont quelques- unes portent sur des objets etrangers a ses etudes habituelles. Dans les premieres ann6es surtout, on trouve frequemment son nom dans ces recueils periodiques, ou les amateurs de mathematiques pures se proposent de petits problemes et se complaisent dans I'^legance des solutions, par exemple, dans " The Analyst." Mais il ne tarda pas a se sp^cialiser. Non seulement ses fonctions I'y contraignaient, mais ses gouts I'y portaient. Le travail courant, n6cessaire pour la preparation de l'ephem6ride, lui foumissait deja des occasions de se distinguer. Nous citerons des tables pour faciliter le calcul des positions des 6toiles fixes et qui sont pr^c^d^es d'une note de M. Hill ou la theorie de cette reduction est exposee d'une fapon simple et claire. A cette epoque le prochain passage de Venus preoccupait tons les astro- nomes. En vue des expeditions projetees, le bureau de I'ephemeride dut se livrer a de longs travaux preliminaires. M. Hill fut ainsi conduit a refaire les tables de Venus. C'etait son premier ouvrage de longue haleine, et on pent y voir deja le germe des qualit^s que I'on admirera plus tard dans tous ses Merits. Dans cette premiere p6riode de sa vie scientifique, il revint a plusieurs reprises sur le calcul des orbites. C'est la un probleme qui se pr^sente constamment au calculateur astronomique et qui devait naturelle- ment retenir I'attention d'un praticien constamment aux prises avec les diflB- cultes qu'il fait naitre. Citons une el6gante discussion de I'equation fonda- mentale de Gauss et diverses notes relatives au meme sujet. Les progres de I'astronomie d'observation avaient d'ailleurs fait entrer la question dans une phase nouvelle; les decouvertes de petites planfetes se multiplient et devien- nent de plus en plus fr6quentes. EUes se succ^dent avec une telle rapidite que les calculateurs sont distances par les observateurs. Ceux-ci fournissent aux premiers plus de besogne qu'ils n'en peuvent faire, et ils veulent etre servis promptement, parce que d^s qu'une nouvelle planete est d^couverte ils craignent de la perdre. La question aujourd'hui est done avant tout de faire vite ; il faut des m^thodes rapides, qui n'exigent pas de trop longe cal- INTRODUCTION culs et permettent d'utiliser les premieres observations. On a et6 ainsi con- duit a n^gliger d'abord l'excentricit6 des ellipses et a calculer des orbites circulaires. Tel est le point de vue ou s'est place M. Hill dans une s^rie de notes qui ont paru dans divers recueils entre 1870 et 1874. Mais j'ai hate d'arriver a son oeuvre capitale, a celle ou s'est d6voil6e toute I'originalite de son esprit, a sa theorie de la Lune. Pour en bien faire comprendre la portee, il faut d'abord rappeler quel 6tait I'etat de cette the- orie au moment oii M. Hill commenpa a s'en occuper. Deux oeuvres de haute sagacit6 et de longue patience venaient d'etre raenees a bonne fin ; je veux parler de celle de Hansen et de celle de Delau- nay. Le premier, par une voie inutilement detourn^e, etait arrive le premier au but, devanpant de beaucoup ceux qui avaient pris la bonne route. Ce ph6- nom^ne, au premier abord inexplicable, n'etonnera pas beaucoup les psycho- logues. Si sa methode, qui nous parait si rebarbative, ne I'efifrayait pas, c'est precis6ment parce qu'il etait infiniment patient, et c'est pour cela aussi qu'il est alle jusqu'au bout. Et c'est aussi parce qu'elle etait etrange qu'elle lui semblait avoir un cachet d'originalite, et c'est dans le sentiment de cette originalit6 qu'il a puise la foi solide qui I'a soutenu dans son entreprise. Une autre raison de son succes, c'est qu'il n'a cherche que des valeurs purement numeriques des coefficients sans se preoccuper d'en trouver I'expression analytique ; ce qui chez les autres representaient de longues formules, se reduisait pour lui a un chiffre, et cela d&s le debut du calcul. Quoi qu'il en soit, c'est encore sur les tables de Hansen que nous vivons et il est probable que les nouvelles theories plus savantes, plus satisfaisantes pour I'esprit, ne donneront pas des chiffres tres difif6rents. Delaunay est a I'extreme oppose ; ses inegalites se presentent sous la forme de formules alg^briques; dans ces formules ne figurent que deslettres et des coefficients numeriques formes par le quotient de deux nombres en- tiers exactement calculus. II n'a done pas fait seulement la theorie de la Lune, mais la theorie de tout satellite qui tournerait ou pourrait tourner autour de n'importe quelle planMe. A ce point de vue il laisse Hansen loin derriere lui. La methode qui I'avait conduit a ce resultat constituait le progres le plus important qu'eut fait la Mecanique Celeste depuis Laplace. Perfectionnee aujourd'hui et alleg^e, elle est devenue un instrument que chacun pent manier et qui a rendu deja bien des services dans toutes les parties de I'Astronomie. Telle que Delaunay I'avait d'abord conpue, elle etait d'un emploi plus penible. Peut-etre aurait-il abrege considerablement son travail s'il en avait fait un usage moins exclusif, mais il faut beaucoup pardonner aux inventeurs. INTRODUCTION XI II mena a bonne fin sa tache d'algebriste, mais les formules demandaient a etre r^duites en chifFres; quand un accident imprevu I'enleva a ses admi- rateurs, il etait sur le point de commencer ces nouveaux calculs. Sa mort arreta ce travail, et ce n'est que dans ces derniers temps qu'il put etre repris et termini. Malheureusement les series de Delaunay ne convergent qu'avec une de- sesp6rante lenteur. Elles precedent suivant les puissances des excentricites de I'inclinaison, de la parallaxe du soleil, et de la quantite que Ton appelle m et qui est le rapport des moyens mouvements. Cette quantite est de tw environ, et si les coefficients numeriques allaient en d^croissant, la convergence serait suffisante. Malheureusement il n'en est pas ainsi, ces coefficients croissent, au contraire, tres rapidement par suite de la presence de petits diviseurs. Aussi desesperant de pousser assez loin le calcul des series, Delaunay fut-il oblig6 d'ajouter aujuge des termes complementaires. M. Hill s'assimila promptement la m6thode de Delaunay, et en a fait I'objet de plusieurs de ses ecrits, mais celle qu'il proposa etait tout a fait dififerente et tres originale. C'est dans un memoire de 1' American Journal of Mathematics, tome 1, que nous en voyons les premiers germes. Les series de Delaunay, nous I'avons dit, dependent de cinq constantes, qui sont les excentricites, I'inclinaison, la parallaxe du soleil et enfin la quan- tite m. Si nous supposons que les quatre premieres sont nulles, nous aurons une solution particuliere de nos equations difFerentielles. Cette solution par- ticuliere sera beaucoup plus simple que la solution generale, puisque la plu- part des inegalites auront disparu, et qu'une seule d'entre elles subsistera, celle qui est connue sous le nom de variation. D'autre part cette solution particuliere ne represente pas exactement la trajectoire de la Lune, mais elle pent servir de premiere approximation, puisque les excentricites, I'inclinaison et la parallaxe sont eflfectivement tres petites. Le choix de cette premiere approximation est beaucoup plus avantageux que celui de I'ellipse Keple- rienne, puisque pour cette ellipse le perigee est fixe, tandis que pour I'orbite reelle il est mobile. Les equations difierentielles sont d'ailleurs elles-memes plus simples, puisque I'excentricite et la parallaxe etant nulles, le Soleil est suppose decrire une circonference de rayon tres grand. M. Hill simplifie encore ces equations par un choix judicieux des variables. II prend non pas les coordonnees polaires, mais les coordonnees rectangulaires, et c'est la un grand progres. Que ces dernieres soient plus simples a tout egard, c'est de toute evidence, et cependant les astronomes repugnent a les adopter. Je comprends a la rigueur cette repugnance pour la Lune, puisque ce que nous observons, ce que nous avons besoin de calculer c'est la longitude, mais j'avoue que je me XII INTRODUCTION I'explique difficilement en ce qui concerne les plan&tes, puisque ce n'est pas la longitude hSliocentrique, mais la longitude geocentrique qu'on observe. En tous cas, pour la Lune, elle-meme, M. Hill a juge que les avantages I'emportent sur les inconvenients, et qu'on peut bien se resigner a faire a la fin du calcul un petit changement de coordonnees, pour ne pas trainer pen- dant toute une th6orie, un encombrant bagage de variables incommodes. Les variables de M. Hill ne sont pas d'ailleurs des coordonnees rectan- gulaires par rapport a des axes fixes, mais par rapport a des axes mobiles animes d'une rotation uniforme, 6gale a la vitesse angulaire moyenne du Soleil. D'ou une simplification nouvelle, car le temps ne figure plus explicite- ment dans les equations. Mais I'avantage le plus important est le suivant. Pour un observateur lie a ces axes mobiles, la Lune paraitrait decrire une courbe ferm^e, si les excentricit6s, I'inclinaison et la parallaxe etaient nulles. Comme les equations differentielles sont d'ailleurs rigoureuses, c'etait Ih le premier exemple d'une solution periodique du prohleme des 3 corps, dont I'existence 6tait rigoureusement d^montree. Depuis ces solutions peri- odiques ont pris une importance tout a fait capitale en M6canique Celeste. Mais I'auteur ne se borna pas a demontrer cette existence, il ^tudia dans le detail cette orbite (ou plutot ces orbites periodiques, car il fit varier le seul parametre qui figurat dans ces Equations, le param&tre m); il determina point par point ces trajectoires fermees et calcula les coordonnees de ces points avec de nombreuses decimales. Les developpements de Delaunay furent remplaces par d'autres plus convergents et pour de grandes valeurs de m, quand les series nouvelles elles-meme ne suffirent plus, M. Hill eut re- cours aux quadratures m6caniques. II arrive finalement au cas, on, pour I'ob- servateur mobile dont nous parlions, I'orbite apparente aurait un point de rebroussement. Une derni&re remarque ; M. Hill, dans le memoire que nous analysons, trausforme ses equations de fapon a les rendre homogenes et il tire de ces equations homogenes un remarquable parti; il serait aise de faire quelque chose d'analogue dans le cas general du probleme des trois corps; il sufiirait d'eliminer les masses entre les equations du mouvement; I'ordre de ces equa- tions se trouverait ainsi augmente, mais on arriverait a n'avoir plus dans les deux membres que des polynomes entiers par rapport aux coordonnees rectangulaires et a leurs derivees. Les equations ainsi obtenues ne pourraient servir a I'integration, mais elles pourraient rendre de pr6cieux services comme formules de verification. Par ce memoire les termes qui ne dependent que de m se trouveraient enti&rement determines avec une precision infiniment plus grande que dans aucune des theories anterieures ; les termes les plus importants ensuite sont INTRODUCTION XIII ceux qui sent proportionnels a I'excentricite de la Lune et ne dependent d'ailleurs que de m.. Ces termes dependent des memes equations differen- tielles; mais comme on connait deja une solution de ces equations et que celle que I'on cherche en differe infiniment pen, tout se ramene a la conside- ration des " equations aux variations." Or ces Equations sont lin^aires, elles sont a coefficients p6riodiques; elles sont du 4eme ordre, mais laconnais- sance de I'integrale de Jacobi permet de les ramener ais6ment au 2eme ordre. La theorie g6n6rale des equations lin6aires a coefficients periodiques nous apprend que ces Equations admettent deux solutions particuli^res suscepti- bles d'etre representees par une fonction periodique multipliee par une expo- nentielle. C'est I'exposant de cette exponentielle qu'il s'agit d'abord de determiner et cet exposant a une signification physique tres simple et tres importante, puisqu'il represente le moyen mouvement du perigee. La solution adoptee par M. Hill est aussi originale que bardie. Notre Equation diflferentielle doit etre resolue par une serie. En y substituant une s6rie S a coefficients indetermines, on obtiendra une autre s6rie 2 qui devra etre identiquement nulle. En egalant a z^ro les difFerents coefficients de cette s6rie X, on obtiendra des equations lineaires ou les inconnues seront les coefficients indetermin6s de la serie S. Seulement ces equations de meme que les inconnues etaient en nombre infini. Avait-on le droit d'egaler a zero le determinant de ces equations ? M. Hill I'a ose et c'^tait la une grande hardiesse; on n'avait jamais j usque-la considere des equations lineaires en nombre infini; on n'avait jamais etudie les determinants d' ordre infini; on ne savait meme pas les definir et on n'etait pas certain qu'il fut possible de donner a cette notion un sens precis. Je dois dire cependant, pour etre complet, que M. Kotteritzsch avait dans les Poggendorf s Annales aborde le sujet. Mais son memoire n'6tait guere connu dans le monde scientifique et en tout cas ne I'^tait pas de M. Hill. Sa methode n'a d'ailleurs rien de commun avec celle du geometre americain. Mais il ne suffit pas d'etre hardi, il faut que la hardiesse soit justifi6e par le succ^s. M. Hill 6vita heureusement tous les pieges dont il etait envi- ronne, et qu'on ne dise pas qu'en operant de la sorte il s'exposait aux erreurs les plus grossieres; non, si la methode n'avait pas ete legitime, il en aurait ete tout de suite averti, car il serait arrive a un rSsultat numerique absolu- ment different de ce que donnent les observations. La meme methode donne les coefficients des diverses inegalites proportionnelles a I'excentricite et dont les plus importantes sont I'equation du centre et I'evection. Comparons ce calcul avec celui de Delaunay; la methode de Hill avec deux ou trois appro- ximations donne un grand nombre de decimales ; Delaunay pour en avoir moitie moins devait prendre huit termes dans sa serie, et ce n'etait pas ^jy INTRODUCTION assez, il fallait evaluer par des precedes approches le reste de la s6rie ; s'il avait fallu attendre qu'on arrive a des termes negligeables, la plus robuste patience se serait lassee. A quoi tient cette difference? Le mouvement g du perigee nous est donn6 par la formule cos gTf=

tV 1 this value of tan 6 is imaginary, and the left member of the equation differentiated is not susceptible of a maximum or minimum value, and the equation in sin 6 has only two real roots, which are among those rejected. Hence we conclude, that when the data are taken from observation, the q uantity — ; will always be contained between the ' ^ •' 1 -\- m cos c 4 COLLECTED MATHEMATICAL WORKS OF G. W. HILL limits ± I . If we substitute for 6 in the equation sin* 6 — A sin {6 — (3) = , the result is A sin ^ , and for 6 = 7t , the result is — A sin ^ ; showing the existence of and odd number of roots between the limits 6 = and Q-=7t, which odd number is three, since c and the root which the problem demands are within these limits. If we make Q=^ c -\- dc, there results the quantity or, smce the quantity [4 siv? c cos c — A cos (c — /S)] dc ; J sin'c , a w! sin c A = -. — J , tan /? = ; sin (c — /?) 1 + m cos c (3 cos c — m) sin' cdc. And — A cos /? , the result, on putting 0=7t, is equal to — m sin* c. Therefore, sin c being always positive, 6 has two real values, or only one (between the limits c and n), and, consequently, the problem two or one answer, according as m and m — 3 cos c have the same or opposite signs. It is evident that A cos /3 must be positive, in order that the equation in sin 6 may have three positive real roots ; so the quantity 1 + m cos c is always positive, and tan (3 has the same sign as m . If |3 be taken between the limits zfc -— , J. is always positive. Since the equation in sin 6 must have no root greater than one, unity substituted for sin 6 in the first derived function of its equation must render it positive ; that is, the expression 4 — 5 A cos (i + A^ ia positive, which gives J. <; 2 and A cos /? <^ f . Accord- ing as m is positive or negative, the equation for finding d presents itself under two shapes, sin* 0=. A sin {d — (3) and sin* 6 = A sin (6 + ^), in which A is always positive and less than 2, and ^ never exceeds 36° 53'- From the expression for p in terms of r, it is clear that r is less or greater than E , according as m is positive or negative. Therefore, in the first case, S is contained between the limits c and n — c ; and, in the second case, if c is in the first quadrant, between n — c and n — (3 ; but if c be in the second quadrant between c and n — ^. These remarks may be of use to shorten the tentative process of finding 6 . With regard to 6, it is clear it is the angle subtended at the comet by its radius vector and the line joining it and the earth prolonged beyond the comet. CONFORMATION OF THE3 EARTH MEMOIE No. 3. On the Conformation of the Earth.* (First Prize Essay, Eunkle's Mathematical Monthly, Vol. Ill, pp. 166-182, 1861.) 1. All the particles which compose the mass of the earth are animated by the attraction of gravitation. The law of this force is, that the attrac- tion of any atom for a spherical surface of material points, described about it as a center, is constant. Hence, if the attraction of an atom for a mate- rial point be represented by A, and r be the radius of the spherical sur- face and N the number of material points in a unit of surface, the attrac- tion of the central atom for the spherical surface is inNr^A = a constant M = — 4nNM. Whence A = ^ ; that is, the attraction varies inversely as the distance squared. The constant M is called the mass of the attract- ing atom. We have given A the negative sign because it represents a force tending to decrease the line r . 2. Making A = -^- , then V= — . V is called the potential function, and has this property: that if the partial derivative of it be taken with respect to any of the three spaceal coordinates of which it is necessarily a function, the result will be the partial force in the direction of that coordi- nate axis. M 3. If the attraction of a single atom give ^^^ -jj > D denoting the distance, the attraction of an indefinite number or assemblage of atoms will give F= S . -jY • If ^ . 4' > 4> represent any three lines at right angles with each other, then -^^ , -=-t . o— ^.re the forces acting in each of these direc- o^ 0-4' o(p tions respectively. In a rectangular system D^\(x'- xy + (y'-yy + (z'-zy }i, *Thi3 memoir, written at the end of 1859 and beginning of 1860, was designed to show how all the formulae connected with the figure of the earth could be derived from Laplace's and Poisson's equa- tions, combined with the hydrostatic equilibrium of the surface, without any appeal to the definite integrals belonging to the subject of attraction of spheroids. Some of the assumptions are quite unwarranted, nevertheless I allow them to stand. g COLLECTED MATHEMATICAL WORKS OP G. W. HILL where the accented letters pertain to the attracting atom, and the unaccented to the attracted. Consequently, dV_„ M,._. dV_g ^(v'-y) ^=S.-^(z'-z). (1) 4. Differentiating again, 15=..^ {^.(.'-.,--1}, (2) In this differentiation M has been regarded as independent of x^y, z; but, in order to render equations (2) altogether general, the attracting mass must be considered as extending into the point x ,y , z . Let p be the den- sity of the atom occupying this point, which becomes zero when the attract- ing mass does not reach the point. This atom may be regarded as spherical, then for it ilf= t7tp2>'; substituting this value in equations (1), the results are Hence, we must add the term — f Ttp to the right members of equations (2), and then we can regard D as having always a finite value. By adding these equations, there results The integration of this gives F; p is a function of x , y , z; in the case of solid bodies, as the earth, a limited function. 5. Transform (3) to terms of polar coordinates ; put X = r )^/ (1 — fi'') COS u) , r=^l {x* + y' + «') , z = r p., to = tan~' -^ . X CONFORMATION OF THE EARTH 7 Then dx' ^ dy' "^ 82;' "■ 1 dx' ^ df dz' J dr' ^ \ dx' ^ df dz' i dr + a r^^Z a fl — a^-l^^ " " dv 9(u Hence (3) becomes i?^ + M + ?^l 9'^, /9V , av , avi dv I dx' "^ dy' dz' i dp' \ dx' dy' dz' ) 3/^ 1 aar" "^ ay' a«' ; d'^' \ dx' "*" ay' a^' / ^rara^ ara/i ara^i ^^ • I ax aa; a?/ dy dz dz) drdfi jj f a/i So) 3^ aw a^ a(u 1 a'F I dx dx dy dy dz dz J diJ^d>" 2 ( a<« ar au; ar ac> ar •) d'v 1 aa; a* ay ay az az J d<«dr d'V ) + 1 - fiO dr + aA a.r^ar a.(i-/.')9r ^^ -~aF-+ a^ + i^^, + 4-/'»" = o. 6. To show the application of (4), take the simple case when the sur- faces of equal density are concentrically spherical. Placing the origin of coordinates at the common centre, p becomes a function of r alone, either continuous or discontinuous as the case demands ; and evidently ^ = , 97 --— = ; therefore, (4) becomes ou ^ ' a ^>9^ dr , - , „ —Q^ + i7:pr' = . By integration lf={G-i-fpr'dry-'. Between the limits and r, in J pr^dr is equal to the mass contained within the sphere whose radius is r ; denoting this by M, it is clear that C= 0, since the expression must agree with that for the attraction of a dV M single atom. Thus ^— ■= ^ . Or the principle may be stated : The force acting on any point, wherever sittiated, equals the mass of all the particles nearer the center than the point attracted, divided hy the square of the distance of the point from that center, taken with the negative sign. g COLLECTED MATHEMATICAL WORKS OF G. W. HILL 7. The earth revolves about a constant axis ; hence, to remove our problem from dynamics to statics, it is necessary to introduce the force of pressure called the centrifugal force. Making the coordinal axis of z coin- cide with the earth's axis, and T denoting the period of the earth's rotation, the potential of the centrifugal force is Since this force animates every particle, include its potential in V and make F the potential of both gravitating and centrifugal force. It then becomes g_,3 necessary to add to (4) the term =^ r^ , whence 8. To apply this equation to the solution of the earth's conformation, we must combine it with some condition of equilibrium. From the manner in which the atmosphere and ocean cover the earth, we may conjecture it was once fluid, and in solidifying, preserved the form it had taken by the laws of hydrostatics. In passing from the solid earth to the ocean, and from the ocean to the atmosphere, there occur two faults in the continuity of the earth's density ; hence p is strictly represented by a discontinuous function. But, as the mass of the ocean and atmosphere is about rinnr of the whole, its influence may be neglected, and p supposed continuous from center to surface. 9. If^ is the pressure, then dp = pd V, and F+ 5 = is the equation to surfaces of level, B having a different value for each surface. Let p be a function of p, and thus of V. In (5) make 4.(,-^,)=/(F). 10. The centrifugal force being small compared with gravity, may be regarded as a perturbing force. Supposing at first this force is zero, the particles would arrange themselves symmetrically about a center, since there is no reason why they should accumulate more in one place than in another. Take the origin of coordinates at this center, then .5— =: 0, >r— = . oix oa Thus (5) becomes CONFORMATION OF THE EARTH 9 Consequently F is a function of r alone, and the general equation of sur- faces of level F+ 5 = 0, when solved gives r = a constant ; these surfaces are then concentrically spherical. 11, The term -=2 r^ (1 — n^) , which the centrifugal force introduces into F, and which causes a departure from the spherical form, does not con- tain 6), and so whatever derangement it may produce, cannot introduce w into F; that is, the earth is a solid of revolution. Consequently (5) becomes 1 2. From the form of this same term, it may be concluded that V= F, + Yy +Y,u.^ + . . .^ S.Y,u?' , (7) where Yi is a function of r alone, and a quantity of the order of the i^^ power of the centrifugal force. Substitute this expression of F in (6) and put the coeflBcient of /tt^' resulting equal to zero, and let the coefficient of y?* in rV(^) be ?7,; then J idYt a . r -T-' _5f!l J. r2i: + n (ii + 2^ f,,. - 2?;c2?: + 1^ f + rz = o. (^) dr + (2t + 1) {%% + 2) F,+, - 2i (2^ + 1) F, + R = . This equation has the inconvenience of introducing F^ + j; let us therefore assume more generally F= 2 . YiM^, F being a function of r of the same order as Y^ and Mi a function oi ^ . Making these substitutions in (6) , which may be written d.r" ^ d.(l—/j.)-j— /Q-v 10 COLLECTED MATHEMATICAL, WORKS OP G. W. HILL •In order that the left member may be arranged in a series of the same form as 2 . ViMi, we must have 3-f = »«'• in which n is independent of ^u. We may determine n from the considera- tion that, Vi being of the same order as F* , Mi cannot contain any higher power of (J. than (i^\ Making J^ = 2 . V^'- ^^^^ relation results : '^•+'- (2s + l)(2s + 3)*'- To make this series end at ki, n must equal — 2i (2i + 1) ; and J. _ (2i - 2s)(2t + 2 s + 1) i. . •+' ~ (2s + l)(2s + 2) "■ ' hence, putting A;o= 1, which is allowable, ^ _ 1 _ 2t (2t + 1) (2i - 2) 2» (2i + l)(2t + 3) , ' 1.2 '^ ^ 1.2.3.4 '^ ' • • '■^"-' ^ 2. . .2t(2i + l). . .(4i + l) « 1.3.3...2i '^ • For Fj we have from (9), by rejecting the sign 2 and dividing by Mi, the equation — 3^- 2z (3i + 1) F; + T = 0. <^^^) (12) From the expression (10) we easily deduce Y - ± (^^ + 1)(^^' + 3) ■ • . (4t - 1) r ^ , (^• + l)(4t + 1) ^ 1 . 3 . . . (2i - 1) I '^' + 1 . (2i + 1) '^'+^ (t + l)(^• + 2)(4t + 1)(U + 3) p. , -1 1.2.(2i + l)(2i+3) ''H2 + ••■]-. The inversion of which is v-± i.3...(2t-i) r J, (i + iX2i + i) y "^'-^ {2i + l){2i + 3) . . . (4t-l) t ^' + 1 . (4* + 3) ^'+' (13) (t + l)(t + 2)(2^^ + l)(2i + 3) ^ , 1 . 2 . (4t + 3)(4i + 5) •+' "^ }■ * The complete integral of this equation when « = — 2i (3t + 1) , i being an integer, is where K and JS"' are the arbitrary constants. CONFORMATION OF THE EARTH 11 The upper sign is to be used when i is even, the lower when odd. From (10) we obtain Ml=\-i^M, + llM,. (14) From this and the equation rV (2 . ViM,) = 2 . TiMi , pursuing the approxi- mation to quantities of the second order, we get these expressions for Ti , ,--y„=/(F.) + |yiV"(F.)+. . ., \ r-^T, = rj' ( F.) - f T^Y" ( F„) + . . . , I (15) r-y, = VJ' ( r„) + f^ V,r' ( F„) + . . . J If quantities of the second order are neglected, the two differential equa- tions to be integrated are -^-^_6F + r'F/'(Fo) = 0. (16) To pursue the analysis farther would require a knowledge of the form /(F). 13. However, when the point r, jw, o is without the surface of the earth, (11) can be integrated. Supposing the point not to partake in the motion of rotation, the centrifugal force must be neglected, and, since p = , generally 7'^ = ; consequently (11) becomes -i^-2U2^+l)F=0. ^''^ The integral of this is F=a,r-« + 5, + ^_|c,(25.c,-65. + If^ =0, - he, + h + U<^ (2*oC. - 65, + ^^ = . From the equation ^ ~ — I = 0, we also have (33) dr F.'+|c.(Frc. + 2FO -h+^, + fcJ- 66„c, +345, + ^' 37" F,"c, + F;+f^ c,( F."'c, + 3 F") = 35oC, - 55, + f^ c, (- 65,c + 345, + ^ . If we make ^^=.^[5,(1 -3c,) + 35,], equations (22) and (23) can be reduced to the following simpler forms : 5.= - (1 + |y + i? -f ?c, + ^«-0[F.' + f c,(F."'c + 3F,")], 1 K = [(1 -4x^)0.- W - i? + iV?^] *o, 5, = (c,-A5'C + |4c,')5o, K"ci + T7 - f c C Fo"'Ci+ 2 F") _ [(1 _ |i 5) c, + J^ c,^ - 1 g - ^5^ g^] F.' = , F."c, + F; + fl c, ( F."'c, + 3 F,") - (3c, - f | g'c, + || c,») F„' = (23) (34) :::} f^') 15. In this article we shall neglect quantities of the order of g'^- Let i w denote the quantity obtained by dividing the second member of the second equation of (23) by the second member of the first equation ; then w= 3 — 35, + ^,+ 35.c, - = 6c, + 5-9^; _ Q ^1 from the second equation of (24), 3ci=3~+ \q:=z e, the approximate value of the earth's compression, as is clear from the equation 7-=l + C, (1— 3/^^)+ .... Hence, by addition, e + w=6c, + fg'-6-^=|^. (26) The relation enunciated by this equation is known as Claieaut's Theorem. 14 COLLECTED MATHEMATICAL WORKS OF G. W. HILL 16. The equation of the earth's surface is = (1 + c, + c,-) {1 - [3ci (1 — cO + 10c,-] ix' + s^c,!J.* + . . .] I (27) = a [1 + ei sin' +6^ sin' 6» + . . . J ; J in which 6 is the geocentric latitude. If e represents the compression, or part of its own length by which the equatorial radius exceeds the polar, e = — ei — 62 = 3ci (1 — Ci) — | Cj . (28) 17. Denoting the normal or astronomical latitude by 6', we have e' = 0- tan-^ = 0-(er + e, — ^e,') sin 20 - (ie, — i O sin i0 , (29) the inversion of which is 0=e' + (e, + e,-^ 61') sin 20' + (i e^ + f e^') sin 4:0' . (30) 18. A line geodetically measured on the earth's surface is clearly the shortest possible ; hence, if ds is the element of the curve, by the principles of the Calculus of Variations 3 C ds + ISV=0, /I being the indeterminate multiplier of ^F. Also, ds' = dr" + 4^2 + J-" (1 — /»') dw' ■ 1. fJ. The coefficients of br , hfi, Sa each equal zero; retaining only that of ^o , as sufficient for our purpose, we have That is, the sine of the angle made by the curve with the meridians varies inversely as the distance from the earth's axis. Hence, CONFORMATION OF THE EARTH 15 Taking account only of terms of the first order with respect to q , (1 — /J.') /^1—fi' - fi'~ {l — fi' - fj.')i' which, integrated, gives 1 L±_^ . 1 — ^ — p/* - i sin- iLj + ^e. l'^-' vi-A'- Vl-l'-/>^' ] (31) This, then, is the equation to the curve ; if O/ , W/^ are the extreme values of 0) , and jtfy , (III those of ^tf , h can be found from the expression for an — Oy . If e is the angle made by the curve with the meridian at the commencement, then h = cos 6j sin e{1 + e^ sin^0,) , and, as affording an approximate value of e , we have , tan d,, cos S, ■ „ , / V cot e= -. — j^ ( — sm e^ cot (««„—<«,). sin (u>i^ — u>i) ' ^ II u For the length of the curve, which, integrated, gives If ^ = , this expression gives the length of any arc of the meridian, but in this case ds = i^W+¥W = r[l + I' sin^2^ \de,^a{l + e^ smH + e^ sin* e + ^ sin»2« \ d0, which, integrated, gives 8= G + a "( 1 + -| + ^^ + I .,) ^ - ^-L + ^ Sin 28 - ^^l^ sin 4/ (33) 19. All areas on the earth's surface, bounded by lines whose equations are (31), can be divided into a finite number of parts, each contained by an arc of a meridian, an arc of a parallel of latitude, and a line whose equation is (31). Let A denote the area of this, then ds being the element of the meri- dian, A = r Cr cos Q da ds , or, neglecting quantities of the second order, A= r Ct^ d(i da. If this is integrated along a meridian, the result is A=a' y [(1 + I eyj /.„-(l + I /x») ^ ] ^o, 16 COLLECTED MATHEMATICAL WORKS OF G. W. HILL The arbitrary constant G in all these formulas is determined by the condi- tion that the length or area must vanish when the beginning and end of the geodetic line coincide.* 20. Let F denote the force of gravity at the surface, then ^=lFVWiJ=IFti.i8*'(i-.-)]. If ^0. ^1. ^2 are the numerical values of the members of equations (23), then L Aa + A-i Ao J = Foll + w,fi' + w,/x'+ . . .]. 21. Thus far the general theory of the subject. We shall now assume some particular law of density. Suppose that the matter of which the earth is composed is compressible inversely as its density. This gives <^P = ^ — , m being a constant. Substituting for dp its value pd V, and • • tyi^ mtegratmg, p = _ F. No constant is added, because it may be supposed contained in F. Then /•(F) = m^F-^, and from (15) generally T, = m'r' Vr, but and thus (11) becomes — ^^ — [2i (3i + 1) — m'r'-] F, = . (36) In integrating, the part of Fj involving negative powers of r may be neglected, since it belongs to F. If a^ is an arbitrary constant, and /^' denotes the operation -j- — performed 2i times, the integral of (36) is F; = a,r"-'/"(sinmr). * Equation (34) in the original memoir is erroneous ; the correct form is given here. GONFORMATION OF THE EARTH 17 It may also be obtained thus : put (36) under this form d'rVt r , 2i(2i + l)~\ ^ „ -^ + [m'- ^^ ^ y F; = . (37) Assume rVi = P sinmr + P' coamr; which, by substitution, gives Make P ± P'= /?o ± A»-' + Aj--" ± /?.»—+ . . . ; then this equation results ± 3m(w + 1) /J„+, + [w(w + 1) - 3i"(2t + 1)] ^„ = , whence (w-at)(n + 2t + i) /J. + i-T 2(w + l)m '^"' the upper sign being taken when n is even, the lower when it is odd. Then making f}Q=±m^'a(, in order to agree with the expression Vi =: ajr^'-y^* (sin mr), we have (2i-n + l)...{2i + n) . , '*»-* 1.2...W.2" "* ''*•' ^ '' the upper sign having place when 2i — n is of the forms 4v + 2 , 4v + 3 , the lower when it is of the forms 4r , Av + 1 . V= J^, + a„r - y (sin mr) + a,rf' (sin mr) (1 - 3/.') (39) + a^^-T^Csin mr) (1 — 10/j?+ ^//i*) + • • ■ > or, expanding/^' (sinmr) by using (38), ,^ 8tc' , am mr 3 )^-^^-^52i^^](l-3.') (40) + a. [(^-i^+ ..)^^-(i-^ -i^)Mr](l - W+ ¥.^) + . . . 22. Since Fcontains a constant, the term —3™ may be neglected except in finding the value of the density. Moreover, for simplicity, let Oq = 1 and %r = — E; then, from (36) and (40), we derive Vo"=(2H-l)m'r,, F„"'= [2 + (m'- 6)H-\ m'V,, Fi =a,(3H—l)m'r,, F/ = a, [3 + (m^- 9)5"] m' F„ , Fi" = Oi [m'— 12 - <5m''- 36)5'] m' F„ , F, = a, [m=— 35 - (lOm'- 105)ir] m' V, , F,' = a, [- lOm^ + 175 - (m*- 65m' + 525)5-] m' F„ . J (41) 18 COLLECTED MATHEMATICAL WORKS OF G. W. HILL By substituting these expressions in (21) and (25), neglecting quantities of the second order, they become, after removing the factor WgFo, {%H-l)(h—Hc^ =0, [3 + {m?- 9)fi] «!+ (3 J5" - 1) Ci = f qH. Whence, _s g' _s n{ZH- 1) ,,„. «i - ¥ ^m^m- (_dB - 1) ' "' - ^^m'H'— (3R- 1) * '- -* 23. Represent the volume of the earth by v, its superficial density by B, its mean density by B'. Then, neglecting quantities of the second order, V = — ; and, if ( V) denote the value of F at the surface, B = — — ( V). Since L is the mass of the earth, B' = —^ =^ — - . Hence — ^^ — ^ = ^^ , or, putting for b^ its value from (24), V' Sf m?iV)— 3i2(l + |9;- If, in the expression for F, we make Z^^=- 1, and, consequently, r-:=\ then H R' 1 + 2qH-3R(l + |g)' or "^~-3R-2q(R'- Ry R' To find m we have, since H= mW,' i(i-cotm) = l?. (44) 24. In order to test the preceding theory by numerical calculation, we adopt the following values for q, B, B', the best we can find: q=^, R = 2.56, 5' = 5.67. OONFOJIMATION OF THE EARTH 19 We shall mark with an accent the numbers of the formulas from which the numerical values of the following quantities are obtained :* (43)' H =0.7432817, [ F„"'= 2.39744m= F, , (44)' \m = 146° 27' 56".2 , t =2.556307, V, = 1.3398451m^F„ai, F' = 1.1675917m'' F„«i , (42)' Jo, =0.0006715666, Ic, =0.0011111845, (41)' F/'=- -2.99279wi'F„ai, F = 1.00801m' F^a , (41)' F„"=0.48656m=F„, [ F' = 3.40342m' Fa^. To obtain the values of aj, a^, Cj, c^ true to quantities of the order of 5^ by substituting the preceding in (21) and (25), we have (21)' 0.7432817ci— 1.3298451ai + 0.0000006695 = , (25)' 1.2398451ci + 1.1675917«,— 0.0021543100 = , (21)' 0.74328cj - 1.00801aj — 0.0000009039 = , (25)' 2.71641c, + 3.40342a2 - 0.0000054038 = . The solution gives (h = 0.0006730400 , c^ = 0.0011127213 , «2 = 0.0000002964 , c, = 0.0000016180 . (34)' F = 5o [r~-'+ 0.0005328715r-' (1 - 3//) + 0.0000010445r-^ (l-iOf/+ ^/,*)] , (27)' r = a [1 - 0.003350630 sin' 8 + 0.000018877 sin* i?] , (28)' e = 0.003331753 = gQQ^ , (39)' d' = d + 688."3811 sin 20 - l."3679 sin id , (30)' = 6'— 688."3811 sin %0' + 3."6836 sin 4:0', (33)' s=a [0.998334571 + 0.000832938 sin %0 — 0.000000112 sin 4e] , (35)' F= F, [1 + 0.005406990 sin' - 0.000041419 sin* e] . The following table contains the values of p and of e, the compression of the surfaces of level, for every tenth of the equatorial radius, calculated from the equations R sin mr f -^' — rsinm - , aui r a f e 0.0 11.800 0.1 11.673 1 — 3773'" 0.2 11.393 ISSO'" 0.3 10.677 1 — 1242'" 0.4 9.848 919'" 0.5 8.839 733"' and e = ■ 3F F 7i=3ai ~3 _r' m «! — — coi mr mr r a P s 0.6 7.688 1 - 587'" 0.7 6.437 1 — 489'" 0.8 5.133 1 - 413'" 0.9 3.832 1 - 351'" 1. 3.550 1 — 300'" * The numbers following this In the original memoir are erroneous ; they are here rectified. 20 COLLECTED MATHEMATICAL WOKKS OF G. W. HILL MEMOIR No. 4. Ephemeris of the Great Comet of 1858. (Astronomische Nachrichten, Vol. 64, pp. 181-190, 1865.) The coordinates given in the following ephemeris are unaffected with aberration ; the constant intended to be used is that of Struve. The columns Aa, A8, contain the excess of the present ephemeris over that used for comparison.' f Wash. Oh True a Aa Trues Ai Logr Log A 1858, June 6 141° 14 9.88 41 —2.66 +24 13 51.41 11 —8.47 0.33754 0.39511 9 141 15 38.45 2.56 24 33 33.59 7.65 0.32900 0.39669 12 141 20 1.24 2.51 24 52 32.08 6.95 0.32024 0.39795 15 141 27 11.39 2.52 25 10 52.97 6.22 0.31124 0.39887 18 141 37 2.40 2.45 25 28 42.01 5.34 0.30198 0.39942 21 141 49 28.00 2.30 25 46 4.62 4.60 0.29247 0.39959 24 142 4 22.65 2.39 26 3 5.98 3.82 0.28267 0.39936 27 142 21 41.87 2.34 26 19 51.10 3.29 0.27259 0.39871 30 142 41 22.12 2.31 26 36 25.07 2.48 0.26219 0.39762 July 3 143 3 20.57 2.29 26 52 52.10 1.93 0.25147 0.39609 6 143 27 35.01 2.21 27 9 17.53 1.32 0.24041 0.39407 9 143 54 3.55 2.15 27 25 46.19 0.72 0.22897 0.39155 12 144 22 44.48 2.22 27 42 23.55 —0.17 0.21716 0.38851 15 144 53 36.34 2.33 27 59 15.31 +0.48 0.20493 0.38492 18 145 26 38.65 2.45 28 16 27.20 1.23 0.19226 0.38074 21 146 1 52.22 2.55 28 34 4.66 1.90 0.17914 0.37595 24 146 39 19.33 2.65 28 52 13.27 2.52 0.16552 0.37052 27 147 19 4.27 2.87 29 10 58.54 3.07 0.15138 0.36440 30 148 1 13.43 3.12 29 30 26.24 3.69 0.13669 0.35756 Aug. 2 148 45 55.13 3.29 29 50 42.07 4.40 0.12142 0.34994 '5 149 33 19.76 3.58 30 11 52.17 5.08 0.10552 0.34149 8 150 23 40.13 4.02 30 34 2.98 5.75 0.08896 0.33214 11 151 17 12.15 4.45 30 57 21.02 6.35 0.07170 0.32181 14 152 14 15.79 5.03 31 21 52.34 6.97 0.05372 0.31041 17 153 15 17.36 5.59 31 47 41.92 7.72 0.03498 0.29785 20 154 20 51.06 6.08 32 14 52.80 8.41 0.01545 0.28402 23 155 31 41.08 6.75 32 43 25.42 9.06 9.99514 0.26878 26 156 48 45.37 7.59 33 13 15.90 9.74 9.97404 0.25196 29 158 13 19.50 8.16 33 44 14.36 10.59 9.95219 0.23337 Sept. 1 159 47 1.10 9.15 34 16 0.63 11.49 9.92968 0.21279 2 160 20 37.81 9.30 34 26 41.02 11.50 9.92205 0.20544 3 160 55 35.02 —9.55 +34 37 21.35 +11.61 9.91437 0.19783 * It sbould be stated that this ephemeria is constructed from the final theory of Memoir No. 6, pp. 35-58. EPHEMERIS OF THE GREAT COMET OF 1858 21 Wash. Oh True a Aa True S AS Logr Log A 1, Sept. 4 161 31 58.87 — 9.83 +34 47 59!31 +11.79 9.90665 0.18994 5 162 9 56.20 10.14 34 58 32.25 11.90 9.89889 0.18177 6 162 49 34.55 10.44 35 8 57.37 12.12 9.89112 0.17330 7 163 31 2.11 10.75 35 19 11.02 12.34 9.88334 0.16451 8 164 14 27.99 11.01 35 29 8.94 12.57 9.87557 0.15540 9 165 2.13 11.25 35 38 46.22 12.76 9.86782 0.14595 10 165 47 55.45 11.53 35 47 57.10 12.95 9.86011 0.13614 11 166 38 19.94 11.79 35 56 34.79 13.14 9.85247 0.12595 12 167 31 28.74 12.04 36 4 31.28 13.38 9.84492 0.11537 13 168 27 36.18 12.33 36 11 37.43 13.60 9.83748 0.10439 14 169 26 57.89 12.55 36 17 42.34 13.64 9.83019 0.09299 15 170 29 50.86 12.82 36 22 33.31 13.72 9.82307 0.08114 16 171 36 33.54 13.08 36 25 55.69 13.86 9.81616 0.06884 17 172 47 25.71 13.29 36 27 32.35 13.98 9.80949 0.05607 18 174 2 48.55 13.40 36 27 3.32 14.15 9.80310 0.04281 19 175 23 4.32 13.56 36 24 • 5.57 14.25 9.79703 0.02906 20 176 48 36.37 13.68 36 18 12.45 14.34 9.79133 0.01480 21 178 19 48.74 13.74 36 8 53.06 14.42 9.78602 0.00003 22 179 57 5.84 13.83 35 55 32.06 14.56 9.78115 9.98476 23 181 40 51.78 13.72 35 37 28.78 14.63 9.77677 9.96898 24 183 31 29.48 13.64 35 13 57.05 14.74 9.77291 9.95272 25 185 29 19.90 13.49 34 44 4.69 14.74 9.76960 9.93601 26 187 34 40.58 13.27 34 6 53.07 14.74 9.76687 9.91889 27 189 47 44.44 12.94 33 21 17.77 14.79 9.76477 9.90143 28 192 8 38.03 12.56 32 26 8.32 14.73 9.76329 9.88372 29 194 37 19.97 12.03 31 20 10.26 14.76 9.76247 9.86587 30 197 13 39.26 11.52 30 2 6.42 14.61 9.76230 9.84804 Oct. 1 199 57 13.94 10.91 28 30 43.24 14.45 9.76280 9.83042 2 202 47 29.98 10.13 26 44 44.87 14.19 9.76395 9.81324 3 205 43 40.69 9.41 24 43 20.34 13.68 9.76574 9.79678 4 208 44 47.46 8.52 22 25 51.80 13.22 9.76816 9.78137 5 211 49 40.83 7.58 19 52 13.03 12.47 9.77118 9.76736 6 214 57 2.54 6.40 17 2 57.06 11.59 9.77477 9.75514 7 218 5 29.17 5.79 13 59 22.54 10.65 9.77890 9.74507 8 221 13 35.77 4.75 10 43 36.87 9.56 9.78352 9.73748 9 224 19 58.83 3.63 7 18 32.39 8.37 9.78862 9.73264 10 227 23 20.16 2.69 3 47 36.33 7.16 9.79413 9.73070 11 230 22 30.47 2.01 +0 14 34.17 5.45 9.80002 9.73171 12 233 16 30.95 1.28 —3 16 49.39 4.80 9.80625 9.73558 13 236 4 34.32 0.74 6 43 9.47 3.80 9.81279 9.74211 14 238 46 5.66 —0.31 10 1 35.21 2.94 9.81958 9.75101 15 241 20 41.21 +0.09 13 9 57.92 2.17 9.82660 9.76194 16 243 48 7.80 0.39 16 6 52.25 1.76 9.83381 9.77453 17 246 8 21.80 0.55 18 51 33.65 1.24 9.84118 9.78843 18 248 21 27.23 0.53 21 23 50.16 0.96 9.84868 9.80331 19 250 27 33.93 0.62 23 43 55.37 0.90 9.85628 9.81886 20 252 26 56.04 +0.46 25 52 21.11 0.85 9.86395 9.83485 21 254 19 51.19 —0.01 27 49 49.80 1.00 9.87168 9.85107 22 256 6 39.54 0.40 29 37 10.18 1.12 9.87944 9.86734 23 257 47 42.34 0.64 31 15 13.14 1.22 9.88722 9.88355 24 259 23 20.88 1.15 32 44 48.82 1.35 9.89500 9.89958 25 260 53 56.93 —1.58 —34 6 45.50 +1.54 9.90277 9.91538 22 COLLECTED MATHEMATICAL WORKS OF G. W. HILOL. Wash. Oh True a Aa Trues AS Logr Log A 1858, Oct 26 262° 19' 51.45 —1.98 —35° 21 48.24 u +1.68 9.91051 9.93087 27 263 41 24.71 2.57 36 30 38.26 1.87 9.91821 9.94602 28 264 58 56.27 3.21 37 33 53.30 2.15 9.92586 9.96081 29 266 12 44.50 3.86 38 32 7.34 2.35 9.93346 9.97521 30 267 23 6.42 4.47 39 25 50.85 2.44 9.94100 9.98923 31 268 30 18.13 5.16 40 15 30.68 2.68 9.94847 0.00285 Nov. 1 269 34 34.73 5.84 41 1 31.33 2.85 9.95587 0.01608 4 272 32 6.07 8.02 43 1 0.91 3.36 9.97760 0.05349 7 275 10 36.95 10.27 44 38 22.12 3.71 9.99857 0.08767 10 277 34 14.07 12.42 45 59 3.78 3.80 0.01876 0.11892 13 279 46 8.19 14.54 47 7 0.96 4.17 0.03815 0.14754 16 281 48 46.92 16.56 48 5 4.46 4.12 0.05676 0.17382 19 283 44 5.42 18.49 48 55 19.62 4.03 0.07462 0.19801 22 285 33 34.41 20.64 49 39 20.44 3.94 0.09176 0.22034 25 287 18 26.11 22.73 50 18 18.84 3.82 0.10821 0.24100 28 288 59 38.31 24.46 50 53 10.72 3.58 0.12400 0.26015 Dec. 1 290 37 56.18 26.31 51 24 39.95 3.41 0.13918 0.27794 4 292 13 55.21 27.98 51 53 22.34 3.24 0.15377 0.29450 7 293 48 3.40 29.38 52 19 46.72 2.98 0.16782 0.30992 10 295 20 42.09 31.00 52 44 16.96 2.51 0.18135 0.32431 13 296 52 8.76 32.41 53 7 12.21 2.35 0.19440 0.33775 16 298 22 37.88 33.86 53 28 48.88 1.92 0.20699 0.35031 19 299 52 22.11 35.15 53 49 20.28 1.67 0.21915 0.36205 22 301 21 32.66 36.28 54 8 67.88 1.09 0.23090 0.37305 25 302 50 19.37 37.35 54 27 51.54 0.68 0.24227 0.38334 28 304 18 50.80 38.34 54 46 10.16 +0.27 0.25328 0.39298 31 305 47 13.68 39.38 55 4 1.96 —0.23 0.26394 0.40200 1859, Jan. 3 307 15 33.18 40.37 55 21 34.49 0.78 0.27429 0.41045 6 308 43 53.24 41.23 55 38 54.59 1.51 0.28432 0.41835 9 310 12 17.04 41.91 55 56 8.15 2.13 0.29407 0.42573 12 311 40 47.05 42.66 56 13 20.59 2.77 0.30354 0.43264 15 313 9 26.42 43.33 56 30 36.49 3.42 0.31275 0.43908 18 314 38 18.98 43.70 56 48 0.05 4.09 0.32171 0.44510 21 316 7 28.53 44.10 57 5 35.13 4.78 0.33044 0.45071 24 317 36 59.54 44.34 57 23 25.61 5.56 0.33894 0.45594 27 319 6 55.45 44.53 57 41 35.30 6.36 0.34722 0.46079 30 320 37 19.46 44.73 58 8.20 7.16 0.35530 0.46530 Feb. 2 322 8 14.64 44.72 58 19 8.24 8.00 0.36319 0.46949 5 323 39 43.18 44.61 58 38 39.00 8.87 0.37088 0.47335 8 325 11 47.56 44.48 58 58 43.52 9.79 0.37840 0.47692 11 326 44 30.89 44.19 59 19 24.71 10.72 0.38575 0.48022 14 328 17 56.99 43.80 69 40 44.85 11.74 0.39294 0.48325 17 329 62 10.91 43.29 60 2 46.12 12.77 0.39996 0.48604 20 331 27 18.49 42.63 60 25 30.70 13.82 0.40684 0.48860 23 333 3 25.63 41.78 60 49 0.77 14.89 0.41357 0.49094 26 334 40 38.55 40.75 61 13 18.80 15.92 0.42017 0.49308 Mar. 1 336 19 3.23 39.57 61 38 27.12 17.08 0.42662 0.49503 4 337 58 45.67 38.18 62 4 27.82 18.24 0.43296 0.49681 7 339 39 52.15 —36.55 —62 31 22.53 —19.44 0.43916 0.49843 RECTANGULAR COORDINATES OF THE SUN 23 MBMOIE No. 5. On the Reduction of the Rectangular Coordinates of the Sun Referred to the True Equator and Equinox of Date to those Referred to the Mean Equator and Equinox of the Beginning of the Tear. (AstronomlBche Naciricliteii, Vol. 67, pp. 141-143, 1866.) In computing an ephemeris of any planetary body, it is quite the easiest plan to get the heliocentric rectangular coordinates referred to fixed planes, as those defined by the mean equator and equinox of the beginning of Bessel's fictitious year, either of the current year or of the nearest tenth year. Then, by the addition of the sun's coordinates referred to the same planes, to obtain the geocentric rectangular coordinates, and from thence to proceed to the corresponding polar coordinates, which may be very readily changed to the true equator and equinox of date by using the three star constants f,ga.To.d G. But the coordinates of the sun hitherto published in the various ephem- erides have not been rigorously reduced to these planes. The following method of reduction is offered as being quite simple, since it involves only the star constants in addition to the coordinates them- selves. Let B denote the sun's radius vector and a, h its true right ascension and declination referred to the mean planes of the beginning of the year, and a', h' the same referred to the true planes of date, and let X, Y, Z, X\ T', Z' be the corresponding rectangular coordinates. Whence result these relations X= R cos 5 cos a , X'—R cos &' cos a', F= jR COS 3 sin a , F'= i? cosS' sin a', Z-R&\-a.S, Z'=R sin d'. Through subtraction, in which we can neglect all but quantities of the first order with respect to the small differences a' — a and B' — S, since the error which results in the values of X, Y and Z is less than half a unit in the seventh decimal place, we get X- X'= R cos S' sin a' («'- a) + R sin d' cos a' (d'— d) , Y— Y'=—R COSd' COSa'(a'—a) + R Sin 3' sina'(5'— 3), Z - Z'=- RcosS' (5'— g) . 24 COLLECTED MATHEMATICAL WORKS OF G. W. HILL But, from the well known formulas for the reduction of the fixed stars, we have a'—a = aA-irbB-i-E and S'- 8 = a'A ^VB , in which a = wi + « sin a' tan S', a'= n cos a', b — cos a' tan 8', 5' = — sin a'. Making these substitutions, we shall obtain X-X'={mY'+nZ')A+ Y'E, Y- Y'= — mX'A - Z'B — X'E, Z—Z'=-nX'A+Y'B. Since mA -|- ^ is usually denoted by /, and we may write A' instead of nA = g cos G and B =:■ g sm G, our equations may be written X- X' = fY'+ A'Z', Y- Y'=:—fX'—BZ', Z-Z'=-A'X'+ BY'. In most of the ephemerides /, log B and log A are given ; then to the last add log of n expressed in seconds of arc ; /, A' and B being thus expressed in seconds of arc, it will be most convenient to add to their logs the constant log 1.68557, whence the reductions above will be expressed in units of the seventh decimal place. If it is required to reduce the coordinates to the equator and equinox of the beginning of a year previous to or following the current one, it is only necessary to increase, in the first case or diminish in the second, the value of A by the requisite number of units. This, however, must not be too large, otherwise the quantities of the second order may become sensible. In computing the ephemeris of a planet, if we have not the mean co- ordinates but only the true coordinates of the sun, it will evidently be a saving of labor, to employ the formulas above to reduce the heliocentric co- ordinates of the planet from the mean to the true equinox and equator of date, and not those of the sun in the opposite direction. ORBIT OP THE GREAT COMET OF 1858 ' 25 MEMOIE No. 6. Discussion of the Observations of the Great Comet of 1858, with the Object of Determining the Most Probable Orbit. (Memoirs of the American Academy of Arts and Sciences, Vol. IX, pp. 67-100, 1867.) Communicated hy T. H. Bafford, April 12, 1864. The interesting physical aspect of this comet attracted to it, in an unusual degree, the attention of astronomers, a large part of whose energies were ex- pended in obtaining observations for position. Consequently, we have a large mass of material for determining its orbit, not a little of which is of very good quality. Added to this, the long period of the apparition of the comet (nine months), would enable us to obtain the elements with considerable pre- cision. Moreover, hints were thrown out that some other force besides gravity might affect its motion. Although these seem to have had no foun- dation other than the fact that the orbits derived from three normals did not well represent the intermediate observations, yet it is a matter of some interest to clear up the suspicion. As the first step in the work, I determined to reduce the observations to uniformity, in respect to the places adopted for the comparison stars ; which last I proposed to derive from all the material accessible to me. The desirableness of this course is evident when we consider that the observers at Bonn, Kremsmiinster, Ann Arbor, and the two observatories in the south- ern hemisphere reobserved their comparison stars, in consequence of which their observations agree much better among themselves ; while the rest contented themselves with places from Lalande, Bessel's Zones, or the British Association Catalogue, and their results exhibit larger probable errors. And as the comet was observed nearly simultaneously in Europe, the same comparison star was frequently used by a dozen observatories for the same night's work ; and thus the stars of the latter class of observatories mentioned above are often found among those reobserved by the former. The result of this labor has convinced me that it has not been wasted ; the good effect is apparent, particularly in the Liverpool and Gottingen obser- vations. 4 26 COLLECTED MATHEMATICAL WOEKS OF G. W. HILL A catalogue of all the stars used for comparison having been formed, the following authorities were consulted for material : Baily's Lalande, Piazzi, Bessel's Zones (Weisse's Reduction), Struve Catalogus Generalis, Taylor, Riimker, Argelander's Southern Zones (Oelt- zen), Robinson's Armagh Catalogue, Johnson's Radcliffe Catalogue, Green- wich Twelve Year and Six Year Catalogues, Madler, Greenwich Observa- tions, 1854-1860, Henderson Edinburgh Observations, Challis Cambridge Observations, Leverrier Paris Observations, 1856-59. Leverrier commenced, in 1856, to reobserve the stars of Lalande ; hence quite a number of the stars the observers had taken from this source, were found in the Paris Observations. The searching them out and reduc- ing them entailed considerable labor. In addition to the material before mentioned, that furnished by the observatories at which the comparison stars were reobserved, was, of course, not omitted. All this material was reduced to 1858.0, and to the standard of Wolfer's Tabulae Reductionum, by applying the systematic corrections given by Au- wers, in Astr. Nachr., No. 1300, with the modifications suggested by Mr. Safford, in No. 1368. The systematic corrections for Robinson are found in Astr. Nachr., No. 1408. Also, the following, kindly furnished by Mr. Saf- ford, were employed : B. A. DEC. Greenwich Six Year Catalogue, . . . -|-0'.017 Greenwich Observations, 1854-60, . . .-(-0.027 -1-0. "70 Paris Observations, 1856-59, .... -1-0.056 +0. 19 In a few cases, mostly Piazzi stars, where the observations indicated proper motion, it was taken into account. With regard to the stars used in the southern observations, those common to the northern being excepted, they were retained without change, or when the same star had been used at both observatories, the observations were combined, allowing a weight of 3 to the Cape and of 2 to the Santiago observation. However, the place of the Santiago star, No. 57, equivalent to Cape No. 95, is wrong, seemingly an error of reduction ; hence the Cape place has been adopted. And San- tiago, No. 49, differing 7". 5, in declination, from its equivalent. Cape No. 87, the Cape declination appearing the better, has been retained. ORBIT OF THE GREAT OOMET OF 1858 27 S 1858.0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 h. m s 9 11 35.277 9 23 19.992 9 25 52.434 9 29 26.949 +25 59.98 25 2 12.96 24 5 6.93 25 1 49.81 9 29 41.987 25 18 21.93 9 30 47.635 26 34 35.94 9 32 23.230 26 38 48.99 9 33 27.857 9 37 42.094 9 37 47.087 9 38 33.273 26 33 26.79 27 41 55.28 24 25 33.49 27 34 38.43 9 38 42.482 9 44 17.169 9 45 49.118 9 45 51.470 9 46 34.633 27 48 43.19 28 26 25.61 28 21 41.66 27 57 29.24 28 1 10.89 9 48 45.001 9 49 3.501 9 50 10.023 9 51 24.453 28 46 15.38 29 14 1.93 29 15 28.20 30 19 26.44 9 53 8.109 9 56 54.727 29 27 50.85 30 26 9.00 9 58 59.212 30 12 16.94 10 3 36.641 10 6 0.703 30 50 50.42 32 7 41.13 10 6 56.647 32 10 17.05 10 8 9.965 10 9 27.359 30 58.15 31 35 38.88 10 9 50 10 10 27.200 31 8 36 31 19 36.16 10 12 33.258 10 12 45.450 32 8 25.35 31 22 26.94 10 14 12.375 10 14 47.246 32 15 26.28 31 2 47.57 10 14 56.648 10 16 57.222 31 33 9.66 31 5 41.78 10 23 37.058 31 46 9.62 10 23 47.154 10 25 56.094 10 26 29.210 33 6 25.56 33 14 35.93 32 24 43.22 10 27 27.290 32 30 36.72 10 29 41.545 33 28 13.95 10 29 46.127 33 25 30.26 10 30 43.132 32 42 45.11 10 34 4.401 10 34 13.347 33 53 25.44 32 26 21.00 10 35 34 10 10 35 11.612 10 36 27.312 10 37 50.279 34 6 20.68 33 21 49.84 33 20 33.31 10 38 50.569 34 18 20.60 10 39 45.817 10 44 6.512 34 20 17.23 + 32 7 11.86 No. 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 a 1868.0 S 1868.0 h m s 10 44 8.277 10 45 21.594 +33° 34 47 57"50 58 45.71 10 47 3.944 34 47 31.18 10 47 51.964 34 15 49.07 10 52 35.910 35 13 36.25 10 56 35.054 35 7 11.70 10 59 36.820 35 36 32.64 11 45.884 35 29 1.63 11 1 29.939 37 4 43.39 11 1 58.610 35 40 37.52 11 2 24.790 36 6 12.73 11 4 16.855 35 46 40.87 11 4 37.860 35 33 27.80 11 10 16.610 36 13 5.54 11 10 48.100 33 52 6.00 11 11 4.961 36 15 52.34 11 13 48.264 36 25 24.60 11 14 24.766 36 6 48.63 11 17 49.011 35 56 46.68 11 19 30.334 36 32 58.19 11 20 16.048 36 9 7.63 11 22 8.300 36 25 12.10 11 27 39.248 36 11 24.50 11 28 14.746 36 42 40.78 11 29 52.400 36 23 30.10 11 30 28.154 36 23 31.78 11 31 6.811 36 23 1.60 11 33 33.925 35 12.15 11 38 8.137 36 40 53.08 11 41 22.161 35 37 17.78 11 42 18.698 35 43 13.14 11 48 39.684 36 7 52.28 11 48 57.507 36 14 16.51 11 54 23.109 36 50 12.34 11 55 23.490 36 31 5.11 11 57 25.064 36 21 29.55 11 59 22.626 36 7 52.04 12 8 41 36 2 12 9 21.473 33 51 20.73 12 14 5.054 35 28 35.44 12 18 0.818 35 33 5.54 12 23 36.015 34 32 7.24 12 24 3.679 34 40 32.25 12 24 38.593 34 42 4.90 12 26 38.907 34 1 58.92 12 30 5.468 33 48 31.59 12 40 14.318 33 20 42.67 12 44 8.808 32 15 8.42 12 48 56.000 32 46 19.64 12 49 22.827 39 5 10.26 12 53 28.505 31 33 8.05 12 53 38.459 32 32 45.83 12 55 34.619 +31 7 17.08 28 COLLECTED MATHEMATICAL WORKS OF G. W. HILL No. a. 1858.0 S 1858.0 1 No. a 1858.0 : 1858.0 h m B / « b m s o / // 107 12 57 5.635 +31 31 16.16 160 14 59 35.593 + 6 19 28.56 108 12 57 16 31 14 161 14 59 57.698 6 54 50.52 109 12 57 26.035 30 58 58.19 162 15 33.375 6 49 9.03 110 12 59 23.612 29 47 28.59 163 15 4 21.645 3 22 6.91 111 13 21.817 28 23 16.41 164 15 5 11.046 7 10 34.03 112 13 2 21.623 31 9.03 165 15 8 54.017 6 59 39.72 113 13 2 45.467 31 11 36.59 166 15 12 35.554 + 3 51 2.73 114 13 7 53.393 30 9 19.48 167 15 17 4.230 — 2 17.19 115 13 9 5.077 30 5 55.53 168 15 20 28.516 — 6 57.69 116 13 10 14.299 29 47 44.00 169 15 20 44.108 + 23 21.61 117 13 12 20.788 29 18 25.90 170 15 23 56.400 — 14 16.79 118 13 18 20.109 24 35 44.78 171 15 30 20.818 3 7 57.60 119 13 20 10.842 26 59 50.88 172 15 33 46.943 3 31 59.42 120 13 21 46.769 28 5 9.80 173 15 37 0.458 — 3 23 9.11 121 13 22 2.800 29 11 20.02 174 15 37 16.575 + 6 52 30.92 122 13 23 8.620 28 24 36.87 175 15 41 30.788 — 3 22 46.67 123 13 23 45.303 28 23 16.90 176 15 43 44.425 + 4 54 29.08 124 13 25 28 20 177 15 44 11.680 — 7 36 47.93 125 13 30 3.869 26 36 19.12 178 15 44 33 6 53 126 13 33 22.650 26 38 49.62 179 15 46 54.770 7 40 54.50 127 13 37 33.182 26 5.60 180 15 52 4.738 6 53 37.22 128 13 40 7.651 26 24 59.35 181 15 52 26.783 6 42 53.62 129 13 44 19.729 24 20 51.41 182 15 53 7.954 8 23.10 130 13 45 56.310 24 15 58.17 183 15 55 1.103 10 13 57.47 131 13 46 12.072 24 2 8.33 184 15 56 33.959 10 58 40.88 132 13 46 46.651 24 51 40.80 185 16 21.617 13 22 56.05 133 13 51 39.354 24 38 30.49 186 16 41.310 9 42 57.87 134 13 51 59.705 22 23 26.35 187 16 2 59.572 14 27.35 135 13 54 25.217 22 39 58.50 188 16 3 6.618 13 36 59.00 136 13 55 20.693 22 14 33.72 189 16 4 24.098 13 22 3.31 137 14 7 56.650 19 9 59.50 190 16 4 41.570 10 6 50.01 138 14 9 11.160 19 55 24.82 191 16 5 42.980 12 40 2.27 139 14 9 23.644 19 34 29.31 192 16 5 59.266 16 22 13.69 140 14 11 14.667 19 6 2.59 193 16 6 12.084 13 37 42.35 141 14 13 2.053 16 57 35.06 194 16 6 29.082 10 3 1.35 142 14 17 27.790 16 55 11.16 195 16 6 59.660 14 16 29.96 143 14 20 0.953 17 3 22.25 196 16 8 32.770 13 17 23.70 144 14 21 31.064 16 45 49.85 197 16 10 3.544 13 5 23.42 145 14 23 11.387 16 50 40.76 198 16 11 34.281 16 8 19.19 146 14 28 12.905 13 43 16.55 199 16 14 44.967 16 40 51.94 147 14 33 46.108 13 52 14.60 200 16 20 9.964 15 53 23.81 148 14 33 55.070 14 8 48.84 201 16 23 0.959 16 17 57.48 149 14 34 22.174 14 20 23.45 202 16 23 43.537 21 9 29.66 150 14 34 54.307 12 16 29.77 203 16 30 18.736 18 32 9.72 151 14 39 4.926 13 42 18.62 204 16 34 32.419 21 29 43.20 152 14 42 33.357 10 38 27.29 205 16 34 36.256 21 4 1.58 153 14 42 47.985 10 47 39.30 206 16 37 12.274 18 52 12.21 154 14 44 10.403 10 18 35.81 207 16 40 7.990 21 41 3.72 155 14 44 35.229 10 35 48.49 208 16 41 6.008 24 23 11.15 156 14 51 54.337 7 10 15.57 209 16 41 7.071 21 35 54.42 157 14 57 3.620 6 3 17.78 210 16 41 50.606 24 15 51.12 158 14 58 1.431 7 15 39.93 211 16 52 37.309 26 25 39.09 159 14 58 12.582 + 6 51 17.26 212 16 53 3.366 —27 43 31.09 ORBIT OF THE GREAT OOMET OF 1858 29 No. 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 a 1863.0 S 1858.0 16 53 10.777 13 20 26.52 16 55 4.596 28 2 57.96 16 55 31.065 28 22 0.37 16 57 44.663 28 3 54.25 16 58 39.016 27 55 53.96 16 59 24.776 27 54 39.07 17 5 10.403 29 52 34.77 17 5 37.478 29 41 14.99 17 6 47.473 30 2 30.62 17 8 16.677 30 8.39 17 9 20.175 29 42 53.91 17 10 7.169 31 12 16.83 17 12 14.840 31 25 56.61 17 13 4.999 31 26 22.48 17 17 17.581 32 50 3.02 17 19 44.388 32 52 53.83 17 23 0.424 34 10 1.01 17 23 54.929 34 16 20.54 17 29 16.449 35 21 48.14 17 31 4.961 35 33 46.60 17 33 12.654 36 52 6.28 17 34 26.669 36 42 1.87 17 40 15.178 37 28 49.04 17 41 32.567 37 45 43.95 17 44 36.555 38 35 8.95 17 45 57.092 38 38 45.01 17 50 27.549 39 13 45.64 17 50 39.138 39 39 2.66 17 54 38.324 40 38 8.40 17 55 11.130 40 26 50.86 18 2 23.867 41 44 28.49 18 5 14.187 41 56 26.36 18 5 36.073 43 12 19.51 18 7 1.414 42 30 48.85 18 7 5.615 42 15 28.83 18 8 31.282 42 20 5.76 18 10 43.566 43 49 49.10 18 10 52.913 43 1 59.55 18 11 7.779 42 37 40.29 18 12 9.145 42 59 37.64 18 12 36.869 42 39 18 13 58.237 44 10 30.94 18 18 12.611 44 14 43.36 18 18 54.896 43 55 46.90 18 21 39.243 44 41 8.96 18 27 50.763 45 34 44.15 18 33 11.685 46 18 24.55 18 35 45.167 46 43 42.28 18 36 12.466 46 31 17.59 18 41 53.432 46 45 22.59 18 43 23.840 —47 26 22.74 No. 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 2S3 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 a 1868.0 18 44 10.419 18 44 28.573 18 46 26.978 18 46 31.226 18 48 13.570 18 49 54.520 18 52 54.066 18 53 55.641 18 54 10.142 18 56 41.428 18 59 22.395 19 3 52.330 19 5 52.858 19 6 44.688 19 12 11.637 19 14 33.567 19 19 10.963 19 19 18.450 19 23 22.183 19 23 34.615 19 26 53.202 19 29 44.020 19 30 15.827 19 30 35.665 19 33 0.830 19 33 15.841 19 34 26.520 19 38 11.319 19 39 33.763 19 40 57.485 19 42 1.927 19 45 4.527 19 45 36.756 19 50 33.215 19 50 43.777 19 56 47.461 19 57 18.613 20 15.512 20 2 33.653 20 5 17.695 20 6 48.770 20 9 15.916 20 11 41.803 20 15 45.780 20 16 25.769 20 17 13.505 20 18 46.916 20 19 7.030 20 21 57.156 20 22 58.968 S 1868.0 -47° 49' 47.56 47 47 17.12 47 45 18.57 47 34 3.31 48 9 23.70 48 28 21.45 48 54 32.28 48 36 18.11 48 51 11.31 49 14 21.17 49 32 0.29 49 46 24.01 50 13 41.25 49 42 18.51 50 30 20.14 50 46 57.51 51 3 5.09 51 16 7.09 50 51 50.34 51 34 45.35 51 45 , 7.15 51 51 59.82 51 50 49.35 52 5 43.17 52 8 8.42 52 16 22.64 52 21 40.38 52 25 21.16 52 35 4.72 52 47 37.75 52 40 19.22 53 10 20.41 53 4 53.37 53 21 50.75 53 12 39.53 53 30 37.25 52 58 52.81 53 45 3.25 54 1 31.30 54 11 0.36 54 14 53.47 54 29 49.18 54 42 28.61 54 16 41.37 54 39 2.12 54 45 46.37 55 33 8.81 55 2 3.04 54 59 26.98 -54 56 2.85 30 COLLECTED MATHEMATICAL WORKS OF G. W. HILL No. 314 a 1859.0 li m 8 20 25 26.390 1 1859.0 o / // —55 3 20.79 No. 339 a 1869.0 h m 8 21 23 0.430 » 1859.0 -68° o' 17.76 315 20 27 7.650 55 18 29.95 340 21 23 22.890 57 42 4.33 316 20 27 13.290 55 24 33.43 341 21 25 2.280 58 4.65 317 20 31 23.834 55 36 23.14 342 21 28 35.340 58 20 30.78 318 20 33 40.930 55 36 2.15 343 21 29 54.810 58 4 24.05 319 20 34 34.990 55 41 47.50 344 21 30 52.172 58 22 23.85 320 20 38 21.560 55 43 23.42 345 21 32 9.160 58 15 1.52 321 20 39 58.730 55 53 23.14 346 21 33 18.394 68 32 14.83 322 20 43 12.100 56 6 49.19 347 21 33 39.750 58 28.20 323 20 44 29.930 55 59 27.04 348 21 33 57.470 57 56 21.35 324 20 45 35.640 55 45 12.47 349 21 35 8.380 58 41 34.64 325 20 47 1.230 56 14 47.53 350 21 37 51.400 58 40 326 20 47 55.460 56 20 9.53 351 21 40 18.106 58 67 17.60 327 21 1 2.480 57 5 12.43 352 22 8 41.970 60 32 20.95 328 21 2 14.520 67 5 6.31 353 22 8 48.310 60 57 36.81 329 21 4 50.880 57 8 13.18 354 22 9 36.790 60 49 14.47 330 21 8 3.260 57 18 3.90 355 22 11 13 61 8 11.00 331 332 21 10 45.750 21 11 6.720 57 12 15.77 57 26 34.46 356 357 22 12 6.080 22 16 36.870 60 39 16.90 61 5 50.15 333 334 21 12 39.970 21 14 20.600 57 23 55.34 57 51 22.09 358 359 22 18 40.420 22 21 12.300 61 17 31.70 61 13 40.79 335 21 18 19.350 57 45 21.06 360 22 23 53.980 61 32 27.09 336 21 20 26.300 57 29 5.95 361 22 25 40.970 61 40 32.46 337 21 20 48.940 57 46 28.59 362 22 27 25.610 61 43 53.15 338 21 21 55.210 —57 55 14.83 r. _ . 363 22 30 54.250 —61 57 58.99 The following are the authorities for the observations and the places of the comparison stars : Altona. Astr. Nachr., L. 187. Ann Arbor. Astr. Nachr., XLIX. 179. Brunnow's Astr. Notices, I. 6, 53. Armagh. Monthly Notices, XIX. 305. Batavia. Astr. Nachr., L. 107. Berlin. Astr. Nachr., XL VIII. 333, LI. 65. Bonn. Astr. Nachr., XLIX. 253, LI. 187. Brbslau. Astr. Nachr., L. 37. Cambridge, Eng. Astr. Nachr., L. 243. Cambridge, U. S. Astr. Nachr., LI. 273. Briinnow's Astr. Notices, I. 71. Cape of Good Hope. Mem. Astr. Soc, XXIX. 59-83. The observations were made with two different instruments ; those made with the larger have been denoted in the list of observations which follows by "Cape 1," and those made with the smaller by "Cape 2." Christiania. Astr. Nachr., LII. 277. Copenhagen. Oversigt kgl. danske Videnskabernes Selskabs, 1858. Dorpat. Beob. Kaiserl. Sternw. Dorpat, Vol. XV. These observations are published in a crude form, and I was unable to reduce and use them, from a want of the instru- mental constants. Durham. Astr. Nachr., L. 11. ORBIT OF THE GREAT COMET OF 1858 rLOEENOE. Astr. Nachr., XLVIII. 347, 355, XLIX. 57, L. 97. The observation of Octo- ber 13 is erroneous as regards the comparison star, which it seems should be Piazzi XV. 227. Geneva. Astr. Nachr., XLIX, 115, L. 21. GoTTiNQBN. Astr. Nachr., XLIX. 235, L. 11. GEBBirwiCH. Greenwich Observations for 1858. Monthly Notices, XIX. 12. KoNiGSBEEG. Astr. Nachr., L. 71, LIII. 289. Kkemsmunster. Astr. Nachr., XLIX. 68, 79, 257, LI. 23. Leyden". Astr. Nachr., L. 157. The observer is mistaken in the comparison star of his last observation ; it should be Weisse, XV. 369. Liverpool. Astr. Nachr., XLIX. 267. Monthly Notices, XIX. 54. Markree. Observations on Donati's Comet, 1858, at Markree. Padua. Astr. Nachr., XLVIIL 357. Paris. Annales de I'Observatoire Imperial, Paris. Tome XIV. Observations. PuLKOVA. Astr. Nachr. L. 307. Beobachbungen der Grossen Cometen 1858. Otto Struve. Santiago. Astr. Nachr., LIII. 131. Astr. Jour., VI. 100. Vienna. Astr. Nachr., XLVIII. 349, XLIX. 43, 53, L. 227, LII. 57. WiLLiAMSTOWN. Astr. Nachr., L. 7. As the latitude and longitude of the place are uncertain, I have not reduced these observations. Washington. Astr. Nachr., XLIX. 55, 113, 363. Astr. Jour., V. 150, 158, 166, 180. The comparison star of October 1 is mistaken. The typographical errors to be met with are so numerous I cannot undertake to mention them. To render the reduction of the comparison stars from mean to apparent place uniform, the elements of reduction in the British Nautical Almanac for 1858 were adopted as the standard ; and the same will be used in reducing our normals from apparent to mean places. Consequently, it becomes necessary to add to the observations in which the elements of the Berlin Jahrbuch were used, quantities easily obtained from this small ephemeris. June 15 + 0.09 + 0.18 Sept. 18 + 0.08 + 0.03 July 15 + 0.02 + 0.22 Oct. 3 + 0.07 -0.04 Aug. 14 + 0.03 + 0.18 Oct. 18 + 0.04 -0.19 Sept. 3 + 0.05 + 0.10 Nov. 2 + 0.14 -0.23 For the reduction of the observations for parallax, and the computation of the perturbations, and for comparison, an ephemeris was computed from these elements published by Searle in the Astronomical Journal, V. 188, Searle's own ephemeris not being sufficiently exact for the purpose of com- parison. T= Sept. 29.75230 1858 Washington Mean Time ^ — gj = 129 6 24.8] g2 = 165 18 46.2 [ Mean Equinox and Ecliptic 1858.0 i = 116 57 46.1 J 9'= 85 21 21.2 log ^ = 9.7622362 32 COLLECTED MATHEMATICAL WORKS OF G. W. HILL In the following list the observations of the comet are given reduced for parallax, and are made to accord with the places of the comparison stars given in the foregoing catalogue. Gould's list of Longitudes (in the Ameri- can Ephemeris) has been used in getting the Paris M. T. of Observation. The comparisons in the last two columns are Obs. — Cal. The declinations of the southern observations have generally been reduced to the time of observing the right ascension ; that observation of right ascension being selected which was nearest in time and which had the same comparison star. Paris M. T. Place of S Number ot a „ Oomp. Star " ° Ad of Observation Observation a June 7.41071 Florence 141° 14 47.79 +24° 2l' 54.73 3 +21.69 + 6.26 8.37659 " 141 15 36.99 24 27 52.30 10 +39.17 —15.66 9.42802 " 141 16 20.54 24 34 48.42 10 +27.71 — 7.33 10.39044 " 141 17 25.48 24 41 10.00 10 +23.44 + 5.67 11.40973 " 141 19 3.43 24 47 35.12 1 +28.89 + 5.05 12.37591 Padua 141 20 31.82 24 53 36.67 4 +11.55 + 5.14 12.41803 Florence 141 20 21.71 24 53 56.68 1 — 3.35 +10.15 13.37729 Padua 141 22 34.98 24 59 27.99 4 + 6.81 —14.10 13.40557 Florence 141 22 16.33 25 14.83 1 —15.72 +22.34 13.43268 Berlin 141 22 43.08 24 59 50.15 2 + 7.30 —12.31 14.41069 " 141 24 58.54 25 5 52.65 2 — 0.89 — 7.69 14.41609 Vienna 141 25 15.40 25 5 55.69 2-5 +15.13 — 6.61 15.39007 Florence 141 28 20.08 25 11 23.30 5 +37.49 —31.86 15.40675 Vienna 141 27 58.18 2-5 +14.65 15.44201 Berlin 141 27 36.29 25 12 2.13 5 — 9.96 — 4.80 16.39944 Kremsmiinster 141 30 54.96 25 17 48.71 5 +10.41 — 8.58 16.41628 Berlin 141 30 39.41 25 17 49.11 5 — 8.36 —14.19 17.39261 Florence 141 34 31.47 25 23 26.85 5 +28.40 —23.47 19.37441 " 141 41 42.14 25 35 32.96 5 +12.05 + 7.07 19.38451 Padua 141 42 8.91 25 35 39.64 5 +23.63 +10.24 28.38292 Florence 142 29 25.62 26 26 8.67 8 +26.28 — 6.86 28.61976 Cambridge, U. S. 142 30 24.43 26 27 36.77 6 — 7.25 + 2.80 29.38224 Florence 142 35 56.97 26 31 43.74 8 +22.16 — 2.41 29.41947 Berlin 142 36 2.15 26 31 52.88 6 +13.31 — 5.57 30.37599 Florence 142 42 46.34 26 37 8.61 8 +22.98 — 5.44 30.38577 Vienna 142 42 24.72 26 37 20.26 8 — 2.72 + 2.98 ruly 2.37816 Florence 142 56 57.17 26 48 14.56 7 + 4.77 + 1.86 8.38159 " 143 46 55.06 27 20 55.54 11 +34.65 — 9.74 9.38324 Vienna 143 55 34.04 27 26 44.86 12 + 6.24 + 6.85 9.60789 Washington 143 57 42.00 27 27 56.51 11 + 9.40 + 6.19 10.37333 Florence 144 5 0.20 27 32 2.33 11 +16.80 — 1.59 10.59343 Washington 144 6 50.62 27 33 16.24 9 + 1.75 — 0.72 10.59343 " 144 6 59.45 27 33 18.41 11 +10.58 + 1.45 11.59576 " 144 16 28.58 27 38 51.84 9 — 0.61 + 1.42 12.37144 Florence 144 24 2.74 27 43 13.41 9 — 5.59 + 3.93 13.37158 " 144 34 23.43 27 48 47.14 12 +10.19 + 2.20 13.59089 Cambridge, U. S. 144 36 26.15 27 50 6.11 12 — 1.67 + 7.56 14.36879 Florence 144 44 34.08 27 54 18.74 12 + 3.24 — 2.50 14.58534 Washington 144 46 51.57 27 55 38.77 9 + 4.72 + 4.34 15.58781 Cambridge, U. S. 144 57 23.93 +28 1 20.09 15 — 1.44 + 4.97 ORBIT OF THE GREAT COMET OF 1858 33 ParlB M. T. ol Observation 1868 Place of ObserTatlon a s Number of . " Oomp. Star " " A To pass to any new system, we shall have the known equations x' ^= ax -^ hy -^ Gz , "j y'=a!x-\-Vy^dz, \ (2) But in the case where we wish to obtain the differentials of a; , ?/ , z for an infinitesimal time dt, a, V and c" are each unity, being the cosines of angles infinitely small ; and all the other constants will contain dt as a factor. Hence we may write 4§ = al'x + V'y . (3) The equation x^ + f + z^ = 1 gives 'isx^ + 2/^ + 2^ = 0- Sub- stituting in this the above values of ~ , etc., there result these three equa- tions of condition between the six remaining constants b + a'=0, c + a"=0, c'+h"^0. (4) 60 Hence, COLLECTED MATHEMATICAL WORKS OF G. W. HILL dx _ by + cz, dy 'di~ — bx + dz. dz dt~ — ex — c'y . (5) It belongs to Celestial Mechanics to deduce the values of the three remaining coeflBcients of these equations. When precession alone is con- sidered, c' = 0, and — b and — c are the quantities usually denoted by m and n . Thus we have, the unit of t being one year, dx ^^=-my-nz, dy dz ~dt = nx. (6) If the values of x, y and z are now substituted in these equations, we find that da ■ J. » ~ = m + w sm a tan d , dS dt i.t) m and n are functions of t which admit of being expressed by power series. Differentiating (7) and always eliminating ^ and -=- by means of the at €vt primitive equations, we obtain d'a _ dm n' dt'-~dt'^ -g- sin 2a + ( dn . dt ") sin a + mn COB a] tan S + n' sin 2a tan'' S , dn TTj- = — mn sin a + -Tj cos a.—vi? sin" a tan & , d?a mn'' , , , n . X <^^ ■ a -j^ = -g- + f mw cos 2a + f « TT sm 2a (2n^ — m'+ Sn' cos 2a) m sin a +I2m-^ + w-yrjcosa tan 5 + 3mn' cos 2a + 3w ^ sin 2a tan'' S + 2m' sin a (1 + 2 cos 2a) tan' S , dt 5 /„ dn dm . , , , • , ^ i= — I 2m-T7 + w-^sina — (m'+ w" svo.^a)n cos a - f OTw' sin 2a + 3wtj sin' a tan S — Sw' sin' a cos a tan' <5 . (8) DERIVATION AND REDUCTION OF PLACES OF THE FIXED STARS gj In writing these equations, it has been assumed that -^ and -^ vanish. The right ascension and declination of a star, as far as regards preces- sion, are then found by the formulas dtt\^^dti\<.^ 1 «=-+(a).' + i(S).'' + i(S).''+ \. (9) 2. Let us next consider the effect of proper motion. If the values of -5- and -5- for any star are obtained from observation for a certain epoch, we may compute the functions m + re sin a tan h and n cos a, and subtract them from these quantities, the remainders ^ and ^i' are the effect of proper motion in right ascension and declination at that epoch. But, to deduce the values of ^ and ^' for any time in general, we may adopt the assumption that the proper motion is uniform on the arc of a great circle, and on this supposition derive the rigorous values of the differential coefficients of a and h with respect to the time. Considering now the effect of proper motion only, let p denote the velocity of the star's motion on the arc of a great circle, j(^ the angle of position of this arc, a' and h' the right ascension and declination of the star at the end of the time t. The consideration of the spherical triangle formed by the pole of the equator and the two positions of the star will give these equations, sin 5'= sin S cos {pt) + cos 8 sin {pt) cos/, -v cos 8' cos (a' — a) = COS 8 COS {pt) — sin 8 sin {pt) cobx> \ (10) cos 8' sin (a'— a) = sm{pt') sin;^ . 3 Eliminating p and x by means of the equations ;o sin;^ = iOi cos5, pcoax = p-', we derive from the first and third of the preceding equations the following values of a' and S' in series arranged according to the powers of t : a'=a + fjLi + fip.' ta.nS.f-^\_/M' sm''8-p.p.'\l + 3tanU)-]f+ . . ., 1 8'=S + ,x't-\fi'sm28.t'-}tiy{l + 2sm'8)f+... J ^^^^ 62 COLLECTED MATHEMATICAL WORKS OF G. W. HILL (12) 3. In order to have the combined effect of precession and proper motion, a' and 8' should be substituted for a and 8 in the series whicn give the effect of precession. Hence, we obtain da ■ , ,. -jj z= m + n sm a tan S + /j. , dd ^ — n COS a + fi' ; and, (I and fi' being considered as variable quantities, dfi di'- dt^' ■ 1 2 ■ OS. -TT = — n/i Sin a — ^ At'' SID 2ff . It may be useful to note the rate of variation of the angle of position ;u through the effects of precession and proper motion ; it is nix cos a tan S + w/ sin a sec" S + 3^/ tan d , (13) -^ =n sm a sec o + fj. am (14) By differentiating the values of -^ and -=- , and eliminating jt > ^ > -Jr du' and -^ by means of their values just given, we obtain €tt (fa dm n' . „ n / • + jT sin a + (m + 2/i) n cos a + %ixn' tan 8 + 2w sin a (n cos a + fi') tan' d , d'S . dn „ . m" . ., J ™ = — mm sin a + -^ cos a — 3w^ sin a — -^ sin is — n' sin" a tan 5 , ^ = '^' + 2/./.'"+ 3 §/ sin a + 3V (m + 2/i) cos a + f (wi + 2fj.) n" cos 2a + f w ^j sin 2a — "Up.' sin" 8 (2 w"— to"— 6m"— 3TO/i + 3>i" cos 2a) n sin a (^ dn dm „sin3asin3G] In these formulas, terms multiplied by x^e have been neglected, as also the terms in 6' — 8 multiplied by x^ which are not also multiplied by tan S . Substituting for x Struve's value 20". 4451, these formulas become a' — az=— 30".44:51 sec S [sin a sin O + cos o cos = — [1.31059] sin © , Aq= — [9.53457 + 0.4?;] sin Q, + [7.6128] sin 2Q, , Bq_^ - [0.96490] cos Q, + [8.9518] cos %Q, , IIq= — [7.4951 — 6.6;;] sin Q. . (43) The term Eq being neglected, we write cO + dD + /IT , ' (44) Jga = aAQ + IBq + cC + dD + /tr , Aq8 = u'Aq + I'Bq + c'G + d'D + ii't. DERIVATION AND REDUCTION OF PLACES OF THE FIXED STARS 69 To Aqk and Aq8 should be added the terms of the second order in aberration, and to A^a and Aq5 the terms of the second order in nutation whenever they are sensible. If we make p Q^— [1.31059] d + [6.5942] a - [7.9609] b , q ©=—[1.27313] c + [7.4644] a — [7.2370] 5, p ,o== — [8.4012 ] a + [5.7922] sec' d cos 2a , q ,0= — [9.7410 ] b — [5.7938] sec" 8 sin 2a , p' Q= - [1.31059] d' + [6.5942] a'— [7.9609] b', ?' ©= - [1.37313] c' + [7.4644] a'— [7.2370] b', yjQ= — [8.4012 ] a'- [6.6673] tan d sin 3a , g',0= - [9.7410 ]b' — [6.6688] tan S cos 2a + [5.6042] tan d , we shall have, terms of the second order included, :^ O sin o + g cos O + PiQ sin 2© + g^Q cos 5 , ■ ^ , , ■ c, t c,^ d8 ,d'^d , ■■ p'q sm O + g-'o cos © + p'iQ sin 3© + ? 2© cos 2© + -^^t + | ^^ t . A a =p Q sin Q + q qcos Q + p^Q sin 2© + q^Q cos 2© + ^^ t + 1 ^, r% zf©5 = Let us make Pq=1cq cos Eq , qQ—lcQ sin Eq , q q— k q am PiQ= he cos E^Q , p'2Q= Jc\q cos E' q^Q= k,Q sin E^Q , q\Q= k\Q sin p' Q— k' cos E' © q' Q— k' © sin X' © , q\Q= k\Q sin E\q . Then equations (46) take the form J©a = f?r + l^rH * © sin (© + ^©) + ^^O sln (2© + Z,©) , (XT " Cut /I©^ = ^r + i^^'^- k'Q sin (© + E'q) + Td^Q sin (2© + ^,©) . (45) (46) (47) (48) 10. To compute the variations of Aga and Aq^ for a certain interval of time as 10 years, we compute the variations ofp©, g©, etc., in that interval; calling them 5^©, ^g-©, etc., and certain very small terms being neglected, we have evidently these equations : 5 . Jga = 5po sin © + 8qQ cos © + 8p^Q sin 2© + Sq<,Q cos 2© + 10^"r+ kQ C0S(©+ ^©)5©, 5 . Aq8 = dp'Q sin © + Sq'Q cos © + 8p\Q sin 2© + Sq'^Q cos 2© + 102r + ^'GCOs(©+^'©)5©. (49) (50) 70 COLLECTED MATHEMATICAL WORKS OF G. W. HILL The value of 50 is [6.0057] sin (0 — 15°) ; substituting this, we have a . Jo« = 10 ^ r - [5.7047] h Q sin {Xq+ 15°) + dpQ sin O + ^fe cos © + [5p,0 + [5.7047] A ©COS (^0-15°)] sin 2© + [.Sq^Qi- [5.7047] ^©sin (_Kq- 15°)] cos 2© , S.AqS = 10^,t -[5.7047] ^q sin (K'q+ 15°) + S^'q sin © + Sq'q cos© + [.¥^Q + [5.7047] k'Q cos (E'q- 15°)] sin 2© + [VsO +[5.7047] ¥q sin (Z'©- 15°)] cos 2© . As in the case of Aga and A©5 , these quantities can be made to take the form d . AQa = a + lz + TiQ sin (© + Hq) + A,© sin (2© + H,q) , -i ,..^-. 8.AQS = a' + Vt + A'© sin (© + H'q) + A',© sin (2© + ^',©) .] ^ ^ Except for stars near either pole, the first and last terms of these equations may be neglected, and regard be had in computing S^q, Iq^, etc., only to the variations o^ c , d, d and dJ in the formulas for 2>© , g'© , etc. Then 5./I©a = 10£r+ A© Sin (©+5-©), d^ 5./J©5 = 10^r+ A'o sin (© + ^'©) , (52) 11. In computing A©a and A©5, we may either suppose Tcq, Kq, Jc'q and S^Q constant throughout the year, and afterwards add to A©a and Aq5 thus obtained, the proper fractional part of Aq sin (0 + Sq) and A'© sin (0 + H'q) for the fraction of the year ; or, we may make them vary from date to date. For a star, whose declination is within the limits ±65°, there is, however, no need to attend to this correction. Having formed a table of for every 10 sidereal days, beginning with the fictitious year, we can readily get for the time of the star's transit „ 18*40'" over the fictitious meridian with the constant interpolation factor . — . ^ 240* ' and thus form the arguments + ^q, 20 + K^q, + Eq, etc. Terms with small coefficients can be most readily formed by means of a Traverse Table. 1 2. We can reduce Aj^a and Aq5 to the forms, terms of the second order included, ^fi« = ^n sin (Q + Eq) + h^ sin (_2Q, + K^^) , \ ■A^d = k'a sin (SJ + Z'n) + Jc',a sin (2£J + K',,^) . / ^^^'' DERIVATION AND REDUCTION OF PLACES OP THE FIXED STARS 71 by making k QCoaK Q = — [9.53457 + 0.4;;] a — O'.OOSl, * n sin ^ n = - [0.96490] b , h ,n cos K^a= [7-6128 ] a + [5.0114] cos 3a tau' S , h 2Q sin ^j n = [8.9518 ] i - [5.0294] sin 2a tan'' $ , k' Q cos ^' n = - [9.53457 + 0.4^;] a', y Q sin ^' j^ = — [0.96490] V, *',Q cos ^',n = [7.6128 ] a'— [5.8865] sin 2a tan 3 , Tc\^ sin Z^'jfj = [8.9518 ] V— [5.9031] cos 2a tan 5-[5.3617] tan < But perhaps it will be as well to adopt the formulas (54) or, ■} (55) (56) ^n«=/n+^nsin(G'n+a)tand, -, ^n^= 5'ncos((?n+ a). j 13. Tor stars near either pole, it will be well to construct tables giving, with the arguments Q + Q, and © — Q,, the values of the small terms in (33) and (34). These will be most readily computed with the aid of a Tra- verse Table, when they have been reduced to the forms ^G+Q« = ^ o+a sin (O + £2 + iT Q+jj) , AQ+^d = k'Q+n sin (Q+Q,+ X's^.^) , ^G-a« = *o-a sin (0—^ + ^0-n) , Aq_^S = k'Q_n sin (0- ^ + E'Q^a) , 14. Tables for A^a and A^h may be computed in the same way. by making /fc d cos ^ + d) = a,, r sin (w + 0-) = Oi , r sin 11} COS. -«xY- r sin {u> + d) = a„, r p.OH til sin ^ , cos 0- sm 1 , there will be only two real roots . We will now show how to arrive at a direct solution of the problem by the employment of trigonometric formulas. If tan c is taken for the unknown quantity, the equation, on which the solution of the problem depends, is [c cos /J tan — c' (1 + sin 2/3) , always negative; — c" sin 2/3, negative or positive, according to the sign of sin 2/3; + c" (1 — sin 2/3) , always positive . Moreover, it is plain that there is one real value of fi, which makes sin 2(1 and sin 2/1? have like signs ; this value we shall adopt. Making, according as c^ is greater or less than unity, c^ =: sec^ y or c? = cos^ y', the above cubic is solved by these formulas (see Chauvenet's Trigonometry, p. 96), it being necessary to make three different cases. Case I. . 2sinVtan)' j. , j. i 9> a 3, ,„, tan «> = j^^ . — ^ , tan = tan » -— , sm 2/* = -7^ tan r cot 24> . V 27 sin 2/3 2 V 3 Case 11. 2 sin / tan" /.,,,*' .„ 2., siny= /-g^g^^g^ . tani& = tani-|-, sin 2/t = --;^ sm / cosec 2^ . Cflwe ZZ7. '"^ ^' = 2W^' ' ""^ ^'^ = 73 ^^"^ '-' «i" (^ ± ««°) • When ^ is impossible in Case TI, the formulas of Case III must be used ; and the upper or lower member of the double sign in the second equation must be taken according as sin 2^ is positive or negative, in order that sin 2^ may have the same sign with sin 2/3. All the auxiliary angles ^, 1^ and fi may be taken between the limits ± 90°. Since sin 2/if sin 2^1 is always positive, tan /? tan (i and tan /? cot (i are so likewise, since they are respectively equivalent to sin 3/3 sin 2/t , sin 2/3 sin 2/* and 4 cos" /3 cos" II 4 cos" ^ sin" ix ' 11 82 COLLECTED MATHEMATICAL WORKS OF G. W. HILL Let US take two auxiliary angles 6 and B', determined by the equations tan* /? tani fi cos /J cos 2fi sm 29 = ■ sin25'=. sin fi cos (1^ + fi) ' tan* j9 cot* /t cos /? cos 2fi cos /^ sin (/3 — /t) ' or by the equations . „^ co8 2fi /sin3/S s^° ^^ = T cos (/? + ^) Visa;;:' . „„, cos2/i /sin2/J sm2g = T gi„(^_^) Vsm2:;r' where the upper or the lower of the signs must be taken according as -. — ^ in the first and 1- in the second are positive or negative ; and 2d and 26' cos jU may also be taken within the limits ± 90°. The four values of x or tan a are then tan ? denoting the corresponding latitude. The formulas used are given in Watson's Theoretical Astronom'jf, pp. 153-159. In the following equations we have put x=AL^ -'Jls.in'^AQ,', y = lQOAn', z=Ae', u = e' (a::' ..^2 ain'-^ AQ'V all expressed in seconds of arc ; and x', y', z' and u' denote the corresponding quantities in reference to the solar elements. In the computation of the coefficients of the last, roughly approximate formulas have been used. A mean of the Transits of 1761 and 1769 gives + 0.992a; - 0.839y + 1.61^ + 1.17w + 1.00a;'- 0.84/ + 0.832'- 1.82m'= + 1".745 . THE ORBIT OF VENUS 95 The indeterminate correction of the Sun's semi-diameter nearly disappears from this mean. The following equations of condition are numbered with the same num- bers as the normals from which they are derived. The last column contains the residuals which remain after the elements have been corrected as shown in the sequel. Equatio;ns of Condition, No. Residuals. X — 0.40ar +0.05J/ — 0.36« —1.44m +1.43i»' —0.191/' —0.21s' — 3.O61*' = +l'.()l +o'.'97 2 —1.37 +0.18 —0.87 —2.97 +2.41 —0.32 —1.45 —4.69 = —0.95 —1.02 3 —2.05 +0.28 -0.87 —4.16 +3.08 —0.41 —2.17 —5.74 = —0.69 —0.74 4 —2.07 +0.28 —0.02 —4.28 +3.11 —0.41 —2.65 —5.57 = +3.37 +3.37 5 —0.80 +0.11 +0.31 —2.15 +1.80 —0.24 —2.22 —3.16 = +1.44 +1.43 6 —0.31 +0.04 —0.42 +1.41 +1.30 —0.16 +1.86 +2.32 = +1.27 +0.68 7 —0.98 +0.12 —0.93 +2.31 +1.98 —0.24 +3.56 +2.59 = —0.23 —1.25 8 —2.27 +0.27 —2.31 +4.06 +3.27 —0.39 +5.84 +3.12 = +1.48 —0.38 9 —2.44 +0.29 —3.04 +3.85 +3.40 —0.40 +6.26 +2.85 = +3.13 +1.22 10 —1.70 +0.20 —2.53 +2.69 +2.70 —0.32 +5.18 +2.00 = +1.13 —0.27 11 —0.90 +0.11 —1.76 +1.56 +1.91 —0.22 +3.95 +1.01 = +0.96 +0.08 12 —2.06 +0.21 +3.53 —2.38 +3.08 —0.32 —6.12 —0.14 = —1.66 —2.07 13 —2.51 +0.26 +4.64 —2.02 +3.51 —0.36 —7.01 +0.39 = +0.29 —0.34 14 —2.00 +0.17 —4.12 —0.54 +3.00 —0.26 +4.57 —3.99 = +0.22 —0.91 15 —2.09 +0.18 —4.05 —1.48 +3.10 —0.27 +4.28 —4.47 = +4.08 +3.09 16 —1.12 +0.10 —2.39 —1.18 +2.12 —0.14 +2.72 —3.49 = —0.59 —1.14 17 —2.69 +0.19 +3.74 +3.87 +3.69 —0.26 —1.82 +7.26 = —2.09 —4.20 18 —1.58 +0.11 +1.80 +2.98 +2.58 —0.18 —0.65 +5.38 = —0.63 —2.10 19 —0.27 +0.01 —0.63 —1.18 +1.27 —0.07 +0.21 —2.76 = —0.81 —0.95 20 —2.40 +0.13 —0.41 —4.82 +3.40 —0.18 —2.47 +6.18 = —0.57 —1.14 21 —0.47 +0.02 —0.47 +1.64 +1.47 —0.06 +2.19 +2.48 = —0.98 —1.79 22 —1.54 +0.06 —1.31 +3.16 +2.54 —0.10 +4.37 +3.00 = —1.19 —2.93 23 —1.95 +0.07 —2.61 +3.15 +2.95 —0.11 +5.79 +2.51 = +1.84 —0.20 24 —0.40 +0.01 +1.03 —1.15 +1.40 —0.03 —2.79 —1.18 = +0.42 +0.24 25 —2.29 +0.05 +3.87 —2.60 +3.28 —0.07 —6.58 —0.29 = —2.95 —3.92 26 —0.55 +0.01 +1.78 —0.38 +1.55 —0.03 —3.33 +1.04 = +3.15 +2.75 27 —2.22 +0.02 —4.51 —0.53 +3.22 —0.02 +4.94 —4.09 = +2.47 +0.72 28 —1.22 +X).01 —2.59 —1.14 +2.22 —0.01 +3.02 —3.46 = +0.04 —0.85 29 —1.55 —0.01 +2.84 +1.97 +2.55 +0.02 —2.07 +4.94 = —2.24 —3.81 30 —2.73 —0.03 +3.75 +3.93 +3.72 +0.04 —2.06 +7.28 = —0.41 —3.18 31 —0.88 —0.01 +1.17 +2.02 +1.88 +0.02 —1.96 +4.10 = +4.28 +3.11 32 —2.18 —0.05 —1.24 —4.31 +3.18 +0.08 —1.77 —6.06 = —0.38 —1.64 33 —1.26 —0.03 +0.04 —2.92 +2.26 +0.06 —2.00 —4.22 = +2.07 +1.48 34 —0.44 —0.01 +0.19 —1.60 +1.44 +0.04 —1.92 —2.50 = +1.59 +1.39 35 —0.68 —0.03 —0.51 +1.96 +1.68 +0.07 +2.58 +2.70 = +0.46 —0.68 36 —1.37 —0.06 —1.02 +2.96 +2.37 +0.10 +3.95 +3.09 = +0.71 —1.20 37 —2.43 —0.10 —2.15 +4.44 +3.42 +0.14 +5.83 +3.70 = +2.52 —0.59 38 —0.54 —0.03 +1.14 —1.33 +1.54 +0.09 —3.10 —1.20 = —0.19 —0.52 39 —2.27 —0.13 +3.71 —2.77 +3.27 +0.19 —6.49 —0.59 = +0.25 —1.20 40 —2.32 —0.13 +4.26 —2.04 +3.27 +0.19 —6.59 —0.07 = +1.34 —0.27 41 —0.46 —0.03 +1.64 —0.39 +1.46 +0.09 —3.12 +0.99 = +2.17 +1.73 42 -1-0.13 +0.01 —0.89 +0.28 +0.87 +0.06 +1.91 +0.52 = +1.01 +0.85 43 —0.25 —0.02 —1.34 +0.16 +1.24 +0.09 +2.71 —0.68 = —0.68 —1.18 44 —1.37 —0.10 —3.10 +0.07 +2.37 +0.17 +4.12 —2.66 = +0.01 —1.57 96 COLLECTED MATHEMATICAL WORl SS OF ( G. W. HILL No. Beslduals. 45 —2.17a! —0.162/ — 4.30Z — 1.15W +3.17a;' +0.233/' +4.76«' —4.18k .' = +i:'79 -d'.3l 46 —0.86 —0.06 —2.06 —0.91 +1.86 +0.14 +2.50 —3.03 = +0.55 —0.31 47 —0.41 —0.03 —1.38 —0.72 +1.41 +0.11 +1.57 —2.63 = +1.49 +1.01 48 +0.28 +0.02 +0.81 —0.23 +0.72 +0.06 —1.45 —0.59 = —2.32 —2.18 49 +0.13 +0.01 +0.92 +0.17 +0.87 +0.08 —1.91 +0.43 = —0.21 —0.26 50 —0.05 0.00 +1.06 +0.48 +1.04 +0.09 —1.96 +1.41 = —0.98 —1.26 51 —0.80 —0.07 +1.95 +1.04 +1.79 +0.16 —2.15 +3.30 = +0.98 —0.08 52 —2.13 —0.19 +3.57 +2.60 +3.10 +0.28 —2.37 +5.96 = +4.52 +1.97 53 —2.58 —0.23 +3.49 +3.83 +3.54 +0.32 —2.09 +6.93 = +3.16 —0.06 54 —1.28 —0.12 +1.70 +2.46 +2.26 +0.20 —0.80 +4.79 = +0.13 —1.67 55 —0.48 —0.04 +0.81 +1.49 +1.47 +0.13 +0.19 +3.29 = +1.94 +1.03 56 +0.09 +0.01 —0.55 —0.78 +0.92 +0.09 +1.02 —1.78 = +0.55 +0.58 57 —0.17 —0.02 —0.62 —1.07 +1.18 +0.12 +0.55 —2.81 = +0.44 +0.27 58 —1.03 —0.11 —1.08 —2.31 +2.05 +0.21 —0.57 —4.25 = +0.95 +0.14 59 —2.27 —0.24 —1.37 —4.41 +3.29 +0.35 —1.62 —6.32 = +1.02 —0.62 60 —0.52 —0.05 +0.10 —1.72 +1.52 +0.16 —1.83 —2.76 = +2.87 +2.49 61 —0.06 —0.01 +0.16 —1.09 +1.07 +0.11 -1.89 —1.47 = +2.61 +2.58 62 +0.07 +0.01 —0.02 +1.02 +0.92 +0.11 +0.26 +2.12 = —0.58 —0.88 63 —0.12 —0.01 —0.16 +1.21 +1.11 +0.13 +1.00 +2.37 = +0.07 —0.48 64 —0.62 —0.07 —0.36 +1.89 +1.61 +0.19 +2.35 +2.74 = +0.72 —0.48 65 —2.12 —0.26 —1.60 +4.07 +3.07 +0.37 +5.13 +3.73 = +1.87 —1.35 66 —2.11 —0.26 —2.45 +3.60 +3.07 +0.37 +5.44 +3.15 = +4.21 +1.04 67 -0.23 —0.03 —0.97 +0.86 +1.22 +0.15 +2.71 +0.18 = +0.87 +0.23 68 +0.03 0.00 —0.80 +0.55 +0.96 +0.12 +2.05 —0.56 = +0.23 —0.06 69 +0.09 +0.01 +0.47 —0.82 +0.92 +0.12 —1.17 —1.65 = —0.45 —0.54 70 —0.17 —0.02 +0.73 —1.00 +1.18 +0.16 —2.14 —1.49 = +0.89 +0.73 71 —0.41 —0.05 +0.98 —1.22 +1.41 +0.19 —2.79 —1.34 = +3.22 +2.87 72 —1.19 —0.16 +1.98 —2.02 +2.20 +0.30 —4.49 —1.05 = —1.61 —2.63 73 —2.29 —0.31 +3.67 —2.89 +3.29 +0.45 —6.57 —0.78 = +0.77 —1.19 74 —1.13 —0.16 +2.56 —0.97 +2.14 +0.29 —4.55 +0.41 = +3.09 +1.95 75 —0.31 —0.04 +1.41 —0.36 +1.31 +0.18 —2.77 +1.06 = +3.64 +3.24 76 +0.13 +0.02 —0.88 +0.33 +0.97 +0.15 +1.93 +0.85 = —1.06 —1.21 77 —0.48 —0.07 —1.66 +0.23 +1.64 +0.25 +3.39 —0.84 = +0.13 —0.74 78 —1.09 —0.17 —2.64 +0.21 +2.25 +0.34 +4.25 —1.90 = +1.03 —0.58 79 —2.05 —0.32 —4.24 —0.17 +3.16 +0.48 +5.41 —3.27 = +1.30 —1.37 80 —2.51 —0.39 —4.99 —0.64 +3.50 +0.54 +5.80 —3.80 = +3.15 +0.05 81 —1.84 —0.28 —3.75 —1.02 +2.67 +0.41 +4.44 —3.24 = +3.72 +1.53 82 —0.78 —0.12 —2.03 —0.71 +1.64 +0.25 +2.66 -2.47 = +2.94 +1.95 83 —0.45 —0.07 —1.47 —0.65 +1.27 +0.20 +1.89 —2.22 = +1.89 +1.30 84 —0.07 —0.01 —0.98 —0.51 +0.94 +0.15 +0.91 —2.01 = +1.51 +1.32 85 +0.04 +0.01 +1.02 +0.20 +1.08 +0.18 —2.25 +0.55 = +0.69 +0.53 86 —0.24 —0.04 +1.28 +0.45 +1.39 +0.23 —2.49 +1.68 = +0.95 +0.41 87 —0.51 —0.09 +1.62 +0.66 +1.67 +0.28 —2.61 +2.47 = —0.10 —1.00 88 —1.36 —0.23 +2.76 +1,48 +2.52 +0.43 —2.92 +4.38 = +2.78 +0.78 89 —2.36 —0.40 +3.97 +2.75 +3.45 +0.58 —3.24 +6.23 = +5.62 +2.27 90 —1.95 —0.33 +2.75 +3.06 +2.73 +0.46 —2.12 +5.43 = +4.99 +2.07 91 —0.20 —0.03 +0.64 +1.14 +1.02 +0.17 +0.16 +2.49 = +2.58 +2.00 92 +0.24 +0.04 —0.01 +0.86 +0.66 +0.11 +1.23 +0.92 = +0.53 +0.47 93 +0.04 +0.01 —0.58 —0.82 +0.97 +0.18 +0.98 —1.93 = —1.45 —1.45 94 —0.14 —0.02 —0.64 —1.01 +1.15 +0.21 +0.65 —2.46 = +0.68 +0.49 95 —0.35 —0.06 —0.76 —1.27 +1.36 +0.25 +0.34 —2.95 = +1.31 +0.90 96 —0.81 —0.15 —1.05 —1.93 +1.83 +0.34 —0.20 —3.87 = +0.68 —0.18 97 —1.75 —0.32 —1.47 —3.45 +2.78 +0.51 —1.00 —5.52 = +2.86 +1.13 THE ORBIT OF VENUS 97 No. BesldualB. 98 — 2.47i» —0.46.1/ — 1.39.« — 4.75W +3.490!' +0.652/' —1.61s' — 6.68it' ■ = +2'.'36 +0"02 99 —2.10 —0.39 —0.73 —4.28 +3.12 +0.58 —1.72 —5.98 = +2.60 +0.64 100 —1.05 —0.20 —0.11 —2.57 +2.06 +0.38 —1.67 —3.99 = +0.79 —0.19 101 —0.60 —0.11 +0.02 —1.85 +1.61 +0.30 —1.70 —3.02 = +1.48 +0.92 102 —0.35 —0.06 +0.06 —1.47 +1.35 +0.25 —1.75 —2.40 = —0.92 —1.26 103 —0.09 —0.02 +0.11 —1.13 +1.09 +0.20 —1.82 —1.62 = +0.20 +0.12 104 +0.12 +0.02 +0.09 +0.96 +0.87 +0.17 —0.09 +2.01 = +0.46 +0.23 105 —0.09 —0.02 —0.10 +1.21 +1.10 +0.22 +0.87 +2.40 = +0.79 +0.26 106 —0.32 —0.06 —0,20 +1.48 +1.31 +0.26 +1.49 +2.60 = +1.84 +0.95 107 —1.24 —0.25 —0.70 +2.84 +2.22 +0.44 +3.45 +3.30 = +5.70 +3.37 108 —2.63 —0.53 —2.15 +4.76 +3.58 +0.72 +5.86 +4.25 = +4.49 —0.01 109 —1.44 —0.29 —1.91 +2.58 +2.42 +0.49 +4.44 +2.42 = +3.11 +0.51 110 —0.63 —0.13 —1.27 +1.40 +1.62 +0.33 +3.34 +1.13 = +2.10 +0.78 111 —0.42 —0.09 —1.09 +1.13 +1.41 +0.29 +3.02 +0.72 = +4.67 +3.68 112 —0.20 —0.04 —0.92 +0.87 +1.20 +0.24 +2.65 +0.23 = +1.74 +1.08 113 +0.14 +0.03 —0.76 +0.46 +0.86 +0.17 +1.70 —0.82 = +0.76 +0.63 114 +0.24 +0.05 —0.80 +0.21 +0.75 +0.15 +1.09 —1.20 = +1.98 +2.03 115 +0.39 +0.08 —0.49 —0.22 +0.68 +0.14 +0.24 —1.39 = +0.96 +1.27 116 +0.37 +0.08 —0.55 —0.61 +0.64 +0.13 —0.42 —1.24 = —0.50 —0.12 117 +0.38 +0.08 —0.29 —0.77 +0.62 +0.13 —0.77 —1.00 = —1.55 —1.11 118 +0.40 +0.08 +0.15 —0.82 +0.60 +0.12 —1.07 —0.59 = —1.13 —0.63 119 +0.41 +0.08 +0.33 —0.77 +0.60 +0.12 —1.16 —0.34 = —0.66 —0.17 120 +0.41 +0.09 +0.60 —0.59 +0.59 +0.12 —1.19 +0.28 = —2.75 —2.27 121 +0.42 +0.09 +0.81 —0.23 +0.58 +0.12 —1.08 +0.48 = —2.23 —1.81 122 +0.42 +0.09 +0.84 +0.12 +0.58 +0.12 —0.85 +0.80 = —0.45 —0.09 123 +0.42 +0.09 +0.43 +0.73 +0.57 +0.12 —0.12 +1.16 = —0.44 —0.21 124 +0.42 +0.09 +0.22 +0.82 +0.57 +0.12 +0.14 +1.16 = —0.25 —0.05 125 —0.62 —0.08 +1.24 —1.44 +1.63 +0.22 —3.30 —1.23 = +0.50 —0.03 126 —1.88 —0.26 +2.98 —3.12 +2.88 +0.39 —5.80 —0.87 = +0.49 —1.02 127 —1.65 —0.24 +3.29 —1.47 +2.65 +0.37 —5.52 +0.04 = +3.61 +2.02 128 —0.43 —0.06 +1.57 —0.43 +1.43 +0.20 —3.07 +0.93 = +0.82 +0.31 129 +0^8 +0.03 —0.85 +0.36 +0.81 +0.12 +1.70 +0.76 = +0.51 +0.43 130 +0.06 +0.01 —1.00 +0.24 +0.96 +0.14 +2.18 +0.27 = —0.35 —0.56 131 —0.15 —0.02 —1.22 +0.19 +1.15 +0.17 +2.58 —0.38 = +0.88 +0.42 132 —0.57 —0.09 —1.81 +0.19 +1.56 +0.24 +3.19 —1.29 = —1.10 —2.05 133 —1.62 —0.25 —3.52 +0.03 +2.67 +0.41 +4.51 —2.92 = +0.68 —1.50 134 —2.41 —0.37 —4.84 —0.46 +3.57 +0.55 +5.46 —4.03 = +3.34 +0.29 135 —2.21 —0.34 —4.40 —1.02 +3.22 +0.49 +4.97 —4.07 = +1.74 —0.93 136 —1.18 —0.18 —2.60 —0.95 +2.18 +0.34 +3.20 —3.21 = +1.44 0.00 137 —0.35 —0.05 —1.32 —0.63 +1.36 +0.21 +1.51 —2.54 = +1.09 +0.56 138 +0.02 0.00 —0.83 +0.52 +0.97 +0.15 +0.35 —2.12 = +1.32 +1.23 139 +0.15 +0.03 +0.92 +0.09 +0.84 +0.14 —1.89 +0.22 = —0.60 —0.59 140 —0.07 —0.01 +1.10 +0.35 +1.07 +0.18 —2.09 +1.26 = —0.20 —0.50 141 —0.30 —0.05 +1.35 +0.52 +1.29 +0.22 —2.13 +1.99 = +1.23 +0.64 142 —1.01 —0.17 +2.28 +1.15 +2.00 +0.34 —2.30 +3.70 = +2.00 +0.49 143 —2.22 —0.38 +3.80 +2.57 +3.21 +0.54 —2.66 +6.05 = +5.49 +2.34 144 —2.54 —0.43 +3.66 +3.57 +3.51 +0.60 —2.36 +6.78 = +7.55 +3.84 145 —0.61 —0.10 +1.02 +1.61 +1.60 +0.27 —1.65 +3.56 = +3.16 +1.98 146 +0.42 +0.03 +0.30 +0.80 +0.58 +0.05 +0.38 +1.13 = 0.00 —0.06 147 +0.41 +0.03 —0.80 —0.24 +0.59 +0.05 +1.02 -0.58 = —0.11 +0.11 148 +0.39 +0.03 —0.04 —0.83 +0.62 +0.05 +0.06 —1.26 = +1.08 +1.40 149 +0.34 +0.03 +0.56 —0.61 +0.66 +0.06 —0.81 —1.12 = —0.51 —0.25 150 —0.02 13 0.00 +0.38 +1.00 +1.04 +0.10 +1.03 +2.11 = +0.79 +0.40 98 COLLECTED MATHEMATICAL WORKS OF G. W. HILL No. Beslduals. 151 +0.16® -fO.Oli/ +0.12a +0.89m +0.83af' +OMy' +1.432:' +1.18w' = +3'.'21 +3'.'00 152 +0.38 +0.04 —0.83 —0.03 +0.62 +0.06 +0.61 —1.11 =+0.34 +0.49 153 +0.40 +0.04 —0.58 —0.59 +0.60 +0.06 —0.13 —1.19 =—0.29 0.00 154 +0.42 +0.04 +0.06 +0.83 +0.59 +0.06 —0.83 —0.83 =—0.08 —0.03 155 +0.42 +0.04 +0.72 +0.43 +0.58 +0.06 —0.81 +0.85 =+1.04 +1.21 156 +0.41 +0.04 —0.25 —0.81 +0.59 +0.06 —0.10 +1.21 =+0.38 +0.75 157 +0.39 +0.04 —0.44 +0.72 +0.60 +0.06 +0.65 +1.08 =—0.61 —0.57 158 +0.35 +0.04 —0.85 +0.04 +0.65 +0.07 +1.38 +0.24 =—1.16 —1.04 159 +0.19 +0.02 —0.61 —0.65 +0.81 +0.08 +1.26 —1.29 =—0.19 —0.07 160 +0.21 +0.02 +0.35 —0.80 +0.77 +0.08 —1.76 —0.15 =+1.50 +1.69 161 +0.34 +0.04 +0.81 —0.24 +0.66 +0.07 —0.94 +1.06 =—0.53 —0.30 162 +0.34 +0.06 —0.63 +0.52 +0.66 +0.11 +1.25 —0.55 =+1.72 +1.80 163 +0.38 +0.07 —0.82 —0.05 +0.62 +0.11 +0.60 —1.10 =+0.98 +1.22 164 +0.42 +0.07 —0.02 —0.85 +0.59 +0.10 —0.75 —0.90 =—0.97 —0.50 165 +0.42 +0.07 +0.83 —0.13 +0.58 +0.10 —1.13 +0.28 =+0.52 +0.89 166 +0.41 +0.07 +0.64 +0.55 +0.58 +0.10 —0.70 +0.95 =+0.01 +0.24 167 +0.17 +0.03 +0.29 —0.85 +0.83 +0.16 —1.81 —0.43 =+1.38 +1.57 168 +0.35 +0.07 +0.82 —0.10 +0.65 +0.12 —1.16 +0.75 =+0.61 +0.90 169 +0.38 +0.07 +0.75 +0.38 +0.61 +0.12 —0.07 +1.29 =+0.85 +1.09 170 +0.42 +0.08 —0.79 +0.24 +0.58 +0.11 +1.10 —0.35 =—0.73 —0.48 171 +0.42 +0.08 —0.35 —0.76 +0.59 +0.11 +0.18 —1.15 =—0.27 +0.20 172 +0.42 +0.08 +0.27 —0.81 +0.58 +0.11 —0.50 —1.05 =—3.06 —2.55 173 +0.38 +0.07 +0.79 —0.27 +0.62 +0.12 —1.27 —0.50 =—0.26 +0.10 174 +0.31 +0.06 +0.81 +0.20 +0.64 +0.13 —1.35 +0.18 =—2.86 —2.65 175 +0.32 +0.06 +0.65 +0.55 +0.68 +0.13 —1.25 +0.80 =—0.73 —0.60 The equations derived from the latitudes y; contain two more unknown quantities, v= M', w = Bmi'.^Q,', but, in them, the variation of the solar elements will be neglected. The mean of the Transits of 1761 and 1769 gives - 0.059a; + 0.050^ — 0.0952 - 0.069m + O.OOOi; + l.OOOw = - 1".165 . From this mean the indeterminate correction of the Sun's semi-diameter is nearly eliminated. No. 1 Uquations of Condition. — O.OliB +0.002/ — 0.01» +0.00tt +0.61« +1.24W = +0"82 2 —0.10 +0.01 —0.21 —0.08 —0.36 +1.95 = +0.41 3 —0.12 +0.02 —0.31 —0.11 —1.09 +2.04 = —0.49 4 +0.17 —0.02 —0.41 +0.25 —2.13 +0.88 = —0.14 5 +0.20 —0.03 —0.37 +0.17 —1.60 —0.40 = -1.51 6 +0.09 —0.01 —0.14 —0.10 +0.12 —1.35 = +0.02 7 +0.20 —0.02 —0.23 —0.35 +1.17 —1.42 = +5.62 8 +0.19 —0.02 —0.30 —0.49 +2.32 —0.77 = +1.54 9 —0.14 +0.02 —0.54 —0.16 +2.42 +0.46 = +0.64 10 —0.23 +0.03 —0.54 —0.07 +1.88 +1.10 = —1.70 11 —0.18 +0.02 —0.36 —0.10 +1.05 +1.38 = — L48 THE ORBIT OF VENUS 99 No. 12 — 0.22i» +0.02J/ —0.012! —0.58m —2.341; —0.09m; = -1:49 13 +0.11 —0.01 —0.33 —0.36 —2.06 —1.55 =: +1.04 14 +0.12 —0.01 +0.21 —0.24 +1.57 +1.68 = +0.34 15 —0.03 0.00 —0.02 —0.09 +0.06 +2.34 = +0.63 16 +0.02 0.00 +0.01 +0.05 —0.75 +1.69 = —0.77 17 +0.01 0.00 —0.07 +0.04 +0.27 —2.68 = +2.21 18 —0.15 +0.01 —0.12 +0.32 +1.60 —1.45 = +0.21 19 +0.01 0.00 +0.02 0.00 +0.78 +0.97 = +1.13 20 +0.10 0.00 —0.38 +0.18 —2.05 +1.36 = —0.77 21 +0.11 0.00 —0.17 —0.11 +0.33 —1.45 = +0.53 22 +0.23 —0.01 —0.28 —0.45 +1.63 —1.35 = +0.73 23 —0.23 +0.01 —0.57 —0.06 +2.13 +0.86 = —4.07 24 —0.13 0.00 —0.12 —0.25 —0.86 +1.09 = +0.95 25 —0.17 0.00 —0.07 —0.56 —2.43 —0.35 = —0.67 26 +0.07 0.00 —0.09 —0.11 +0.09 —1.54 = +0.53 27 +0.10 0.00 +0.18 —0.24 +1.52 +1.83 = —0.65 28 +0.01 0.00 0.00 +0.02 —0.62 +1.79 = +3.82 29 —0.06 0.00 +0.14 —0.05 —1.03 —1.91 = —2.56 30 0.00 0.00 —0.07 +0.07 +0.49 —2.67 = —0.52 31 —0.15 0.00 —0.15 +0.29 +1.60 —0.73 = —0.13 32 —0.10 0.00 —0.30 —0.06 —1.04 +2.13 = +0.55 33 +0.21 +0.01 —0.38 +0.27 —1.92 +0.18 = —0.52 34 +0.16 0.00 —0.30 +0.10 —1.27 —0.63 = —0.79 35 +0.14 +0.01 —0.21 —0.20 +0.59 —1.54 = —0.38 36 +0.22 +0.01 —0.27 —0.39 +1.41 —1.47 = +1.29 37 +0.16 +0.01 —0.37 —0.44 +2.39 —0.81 = —0.72 38 —0.16 —0.01 —0.12 —0.30 —0.98 +1.12 = +0.33 39 —0.18 —0.01 —0.09 —0.56 —2.43 —0.21 = +2.84 40 +0.17 +0.01 —0.42 —0.29 —1.88 —1.59 = —1.00 41 +0.06 0.00 —0.08 —0.10 +0.20 —1.44 = +0.10 42 +0.06 0.00 —0.06 —0.11 +0.31 —0.86 = +0.03 43 +0.13 +0.01 +0.04 —0.25 +1.18 —0.46 = +0.64 44 +0.18 +0.01 +0.22 —0.32 +1.78 +0.86 = —0.92 45 —0.05 0.00 —0.03 —0.14 +8.33 +2.36 = —0.04 46 +0.02 0.00 +0.01 +0.05 —0.80 +1.48 = +0.66 47 +0.05 0.00 —0.01 +0.10 —1.04 +0.90 = +0.64 4S —0.03 0.00 —0.05 —0.05 +0.03 +0.70 = +0.07 49 —0.07 —0.01 —0.02 —0.13 —0.68 +0.61 = —1.75 50 —0.09 —0.01 +0.09 —0.16 —1.15 +0.14 = —0.90 51 —0.11 —0.01 +0.19 —0.12 —1.37 —1.06 = —3.16 52 —0.03 0.00 +0.09 —0.03 —0.71 —2.35 = —4.41 53 —0.03 0.00 —0.07 +0.13 +0.84 —2.49 = —1.78 54 —0.14 —0.01 —0.10 +0.29 +1.54 —1.26 = +3.67 55 —0.12 —0.01 —0.14 +0.19 +1.45 —0.21 = +0.35 56 +0.04 0.00 +0.06 —0.05 +0.93 +0.14 = +1.39 57 +0.03 0.00 +0.05 —0.02 +0.90 +.079 = +1.57 58 —0.05 0.00 —0.10 —0.02 +0.18 +1.80 = +2.67 59 —0.08 —0.01 —0.30 —0.03 —1.11 +2.13 = +2.72 60 +0.16 +0.02 —0.31 +0.13 —1.37 —0.50 = +0.58 61 +0.10 +0.01 —0.19 —0.01 —0.63 —0.90 = —0.01 62 0.00 0.00 —0.01 0.00 —0.70 —0.73 = —1.35 63 +0.04 0.00 —0.07 —0.02 —0.37 +1.15 = —1.50 64 +0.13 +0.01 —0.20 —0.17 +0.44 —1.54 = —0.74 65 +0.22 +0.03 —0.34 —0.47 +2.04 —1.19 = —3.38 100 COLLECTED MATHEMATICAL WORKS OF G. W. HILL No. 66 — 0.21iB —0.031/ —0.592 — 0.02tt +2.26!; +0.54W = +2.46 67 —0.08 —0.01 —0.14 —0.09 +0.16 +1.25 = —0.29 68 —0.03 0.00 —0.03 —0.05 —0.40 +0.89 = +3.00 69 —0.04 0.00 —0.07 —0.03 +0.26 +0.90 = +0.62 70 —0.07 —0.01 —0.09 —0.11 —0.36 +1.15 = —2.16 71 —0.14 —0.02 —0.13 —0.25 —0.80 +1.15 = —3.21 72 —0.23 —0.03 —0.07 —0.49 —1.74 +0.76 = —0.32 73 —0.17 —0.02 —0.12 —0.56 —2.41 —0.21 = +2.40 74 +0.16 +0.02 —0.28 —0.18 —0.74 —1.75 = —2.59 75 +0.04 +0.01 —0.05 —0.07 +0.33 —1.30 = —0.70 76 +0.06 +0.01 —0.07 —0.10 +0.26 —0.86 = —0.80 77 +0.16 +0.02 +0.09 —0.30 +1.43 —0.21 = +0.69 78 +0.19 +0.03 +0.21 —0.34 +1.75 +0.50 = +0.68 79 +0.13 +0.02 +0.20 —0.30 +1.72 +1.54 = —0.50 80 +0.01 0.00 +0.03 —0.17 +1.22 +2.20 = +0.45 81 —0.04 —0.01 —0.03 —0.09 +0.07 +2.22 = +0.26 82 +0.02 0.00 +0.01 +0.04 —0.76 +1.50 = —0.82 83 +0.04 +0.01 0.00 +0.08 —0.99 +1.02 = —0.72 84 +0.05 +0.01 —0.04 +0.10 —1.06 +0.30 = —0.07 85 —0.08 —0.01 +0.01 —0.16 —0.88 +0.49 = +0.83 86 -0.11 —0.02 +0.12 —0.17 —1.29 —0.09 = —1.35 87 —0.12 —0.02 +0.17 —0.16 —1.41 —0.56 = +0.45 88 —0.10 —0.02 +0.20 —0.09 —1.26 —1.63 = —0.07 89 —0.03 0.00 +0.07 —0.04 —0.61 —2.49 = +0.58 90 —0.09 —0.02 —0.05 +0.23 +1.23 —2.00 = +1.14 91 —0.11 —0.02 —0.17 +0.15 +1.24 +0.17 = +0.70 92 —0.04 —0.01 —0.07 —0.03 +0.01 +0.75 = —0.49 93 +0.04 +0.01 +0.06 —0.05 +0.96 +0.26 = +0.83 94 +0.03 +0.01 +0.06 —0.02 +0.95 +0.69 = +0.99 95 +0.02 0.00 +0.04 —0.01 +0.82 +1.06 = +0.89 96 —0.02 0.00 —0.04 0.00 +0.44 +1.61 = +0.92 97 —0.09 —0.02 —0.21 —0.05 —0.44 +2.14 = +0.44 98 —0.03 0.00 —0.31 +0.05 —1.44 +2.02 = +1.56 99 +0.14 +0.03 —0.33 +0.27 —1.96 +1.25 = —1.15 100 +0.20 +0.04 —0.34 +0.26 —1.79 +0.12 = —0.91 101 +0.17 +0.03 —0.31 +0.17 —1.48 —0.35 = +0.19 102 +0.14 +0.03 —0.27 +0.09 —1.18 —0.62 = —0.52 103 +0.10 +0.02 —0.20 0.00 —0.72 —0.86 = —0.60 104 —0.01 0.00 +0.01 0.00 —0.79 —0.52 = +0.02 105 +0.03 +0.01 —0.06 —0.01 —0.43 —1.12 = +0.74 106 +0.07 +0.01 —0.13 —0.06 —0.08 —1.38 = +1.18 107 +0.20 +0.04 —0.28 —0.33 +1.14 —1.61 = +0.54 108 +0.05 +0.01 —0.46 —0.30 +2.47 —0.62 = +0.96 109 —0.24 —0.05 —0.53 0.00 +1.77 +0.94 = +0.22 110 —0.16 —0.03 —0.31 —0.07 +0.88 +1.27 = +1.94 111 —0.12 —0.02 —0.23 —0.09 +0.56 +1.28 = +1.03 112 —0.08 —0.02 —0.14 —0.09 +0.16 +1.20 = +1.81 113 —0.01 0.00 —0.01 —0.03 —0.57 +0.65 = +0.34 125 —0.17 —0.02 —0.12 —0.33 —1.12 +1.08 = +1.69 126 —0.23 —0.03 —0.07 —0.49 —2.24 +0.23 = +1.42 127 +0.19 +0.03 —0.39 —0.22 —1.30 —1.73 = +0.59 128 +0.06 +0.01 —0.08 —0.10 +0.16 —1.43 = +0.76 129 +0.05 +0.01 —0.07 —0.08 +0.11 —0.85 = —0.11 130 +0.08 +0.01 —0.05 —0.16 +0.56 —0.85 = +0.64 THE ORBIT OF VENUS 101 131 +0.12a! +0.02J/ +0.01« —0.24m +1.03t? — O.eito = +0'.'74 132 +0.17 +0.02 +0.12 —0.31 +1.51 —0.08 = +1.60 133 +0.18 +0.03 +0.24 —0.33 +1.81 +1.08 = +1.68 134 +0.07 +0.01 +0.13 —0.29 +1.47 +1.98 = —0.50 135 —0.06 —0.01 —0.05 —0.18 +0.46 +2.34 = +0.90 136 —0.01 0.00 0.00 —0.01 —0.51 +1.80 = —0.11 137 +0.05 +0.01 —0.01 +0.09 —1.03 +0.84 = -0.35 138 +0.05 +0.01 —0.06 +0.09 —0.99 +0.03 = +0.88 139 —0.06 —0.01 —0.02 —0.12 —0.56 +0.66 = —0.91 140 —0.09 —0.02 +0.07 —0.17 —1.13 +0.23 = —0.84 141 —0.11 —0.02 +0.14 —0.17 —1.34 —0.22 = —1.90 142 —0.11 —0.02 +0.21 —0.11 —1.37 —1.26 = —1.37 143 —0.04 —0.01 +0.11 —0.05 —0.77 —2.38 = —0.65 144 —0.03 —0.01 —0.05 +0.12 +0.77 —2.51 = +3.52 145 —0.14 —0.02 —0.14 +0.24 +1.48 —0.50 = +0.28 To apply to these equations the rigorous method of least squares would be very laborious; hence the method of "Equivalent Factors" has been used ; the equations have been multiplied either by whole numbers or by fractions which are ready multipliers. In this way the following Normal Equations were derived from the equations of condition which have cos j? .A0 for their absolute terms : +195.84ic — 44.809J/ +127.71« + 73.19W -251.9037 +43.027^ — 85.48a +119.25U = — 8.77 — 44.78 +47.099 — 83.68 — 62.84 + 41.04 —48.460 + 41.17 — 96.06 = —113.43 +120.94 —83.889 +427.28 +133.17 —136.59 +82.936 —410.76 +400.15 = +162.30 + 70.03 —62.965 +135.64 +365.81 — 73.13 +63.350 +114.76 +508.04 = +197.06 —255.15 +42.172 —138.12 — 80.06 +425.64 —27.182 + 91.22 —132.67 = + 92.63 + 40.68 —48.373 + 82.84 + 61.99 — 26.27 +51.815 — 41.45 + 94.13 = +121.18 — 83.42 +41.537 —422.53 +119.76 +102.83 —40.091 +644.06 —111.82 = — 23.87 +112.81 —95.792 +406.68 +505.65 —126.69 +94.621 —120.34 +902.21 = +264.18 If u is eliminated from these equations, the result is +181.830! — 32.213J/ +100.57^ -237.27® +30.3522/ -108.442 + 17.60it = — 48.20 — 32.75 +36.284 — 60.38 + 28.48 —37.577 + 60.88 — 8.78 = — 79.58 + 95.45 —60.971 +377.90 —109.97 +59.874 —452.54 +215.20 = + 90.56 —239.82 +28.394 —108.43 +409.63 —13.317 +116.34 — 21.48 = +135.76 + 28.81 —37.705 + 59.85 — 13.S8 +41.080 — 60.90 + 8.04 = + 87.79 —106.35 +62.147 —466.94 +126.77 —60.831 +606.49 —278.15 = — 88.38 + 16.01 — 8.770 +219.18 — 25.60 + 7.053 —278.97 +199.94 = — 8.21 and if from these a is eliminated, the result is +156.43a; -15.9872/ — 208.00a>' +14.418J/' +11.99i2:' —39.67m' = — 7^30 — 17.50 +26.542 + 10.91 —28.055 —11.42 +25.60 = — 65.11 —212.43 +10.900 +378.08 + 3.863 —13.51 +40.27 = +161.74 + 13.69 —28.049 + 3.54 +31.598 +10.77 —26.04 = + 73.45 + 11.59 —13.190 — 9.11 +13.151 +47.33 —12.25 = + 23.52 — 39.35 +26.593 + 38.18 —27.674 —16.60 +75.13 = — 61.46 10 2 COLLECTED MATHEMATICAL WORKS OF G. W. HILL It is evident now, that since the principal coefficients of z' and u' have fallen from 644.06 and 902.21 to 47.33 and 75.13, no very reliable values of these quantities can be obtained from these equations. The elimination of y gives +145.89» —201.43a;' — 2.480j/' + S.llis' — 24.25-m' =— 111.'52 —205.24 +373.60 +15.384 — 8.82 +29.76 = +188.48 — 4.80 — 15.07 + 1.950 — 1.30 — 1.01 = + 4.64 + 2.89 — 3.69 — 0.791 +41.65 + 0.47 =— 8.84 — 21.82 + 27.25 + 0.435 — 5.06 +49.48 = + 3.78 The elimination of x from these gives +90.233?' +11.895J/' — 1.632' — 4.35«' = +3l'.'63 + 8.44 + 1.868 — 1.13 + 0.21 = + 0.97 + 0.30 — 0.742 +41.55 + 0.95 =— 6.63 — 2.88 + 0.064 — 4.30 +45.85 =—12.89 The elimination of x' from these gives +0.7552/' — 0.983 + 0.62«' =— l^^g —0.782 +41.56 + 0.96 =— 6.74 +0.444 — 4.35 +45.71 =—11.88 The only condition, relative to the solar elements, which can be obtained with any weight from these equations is a/ +0.132/= +0".335. That is, the mean longitude of the Sun of Hansen and Olufsen's Tables ought to be increased by a third of a second at the epoch 1863. As, how- ever, these Tables will, probably, be used for a long time to come in com- puting the solar coordinates of the American Ephemeris, y\ z' and u' will be put severally equal to zero ; and, as it has been decided to use the Pulkova constant of aberration, x' will be put equal to +0".19. With these assump- tions, the values of aj, y , z and ware a; = — 0".502, ?/ = — 2".863, z = — 0".040, m=+0".195. The equation of condition derived from the Transits of 1 76 1 and 1769 being excluded, the normal equations, determining the corrections of the inclination and the longitude of the ascending node, are +2.51® +0.3902/ +1.843 —0.67m +163.26); — 0.42W = +2^.'02 —4.46 —0.105 —0.29 —1.06 — 5.86 +188.58 = +24.11 From these are obtained the following values of v and w : V- + 0".18, w = + 0".13 or Agi'= + 2".o . THB ORBIT OF VENUS 103 But, from the equation furnished by the Transits in 1761 and 1769, If the first result is supposed to belong to 1855.0, and the second to 1765.4 the proper value of the correction is JQ,'= +0".9 + 0".222t. The origin of the pretty large correction — 0".02863, of the mean motion of Venus, is easily shown. In his investigation, Leverrier {Annales, Vol. VI, p. 72) found the following value of An' : An'= + 0".00035 + 0".0689>/ + 0".0959k'+ 0".1207/'; but the value of this quantity used in forming his Tables is the first term only. If the values of r , v', v" corresponding to the change from Leverrier's values of the masses to those here adopted, be substituted in this expres- sion, the correction of Leverrier's mean motion, from this cause, is found to be z)w'= — 0".01588. Moreover, a comparison of the values of the Sun's mean longitude in the Tables of Hansen and Olufsen and of Leverrier gives Han.-Lev. = — 0".93 — 0".010r4^ . From the way in which An' and An" are involved in the equations of con- dition, it may be concluded, that if An" were left indeterminate in the solu- tion, the value of An', obtained, would be roughly M'= {An') + l.Szlw", (An') denoting the value of An' on the supposition of An" = . Thus, on making An" = — 0".01074, the correction of the mean motion of Venus from this cause is An' = — 0".01289. The sum of these two corrections is An' = — 0". 02877, which is almost identical with that derived from the equations of condition. The increment of the motion of the node, 0".222, requires that the mass of Venus should be reduced from tu-stsz to Tjrmi- This agrees with Lever- rier's result: setting out with the mass 0.0000024885, he found that it should be multiplied by the factor 0.948, which would make the mass ] The corrections to be added to the elements, with which we set out, to obtain the elements, from which the Tables are constructed, are AL'= — 0".502, /l7r'= 4- 28".46, AQ,'= + 0".90 + 0".222t, Ai'= +0".18, Je'= — 0.000000196, An' = — 0". 02863. 104 eOLLBGTED MATHEMATICAL WORKS OF G. W. HILL The Tables have been compared with the occultation of Mercury by Venus, observed at Greenwich, May 28, 1737. The observations made are Qreenwloli M. T. 9" 40°" 3'.9. Mercury distant from Venus not more than a tenth part of the diameter of Venus. 9 48 10.2. Mercury wholly occulted by Venus. The position of Mercury being derived from Prof. Winlock's Tables, the apparent position of the two planets, as seen from Greenwich, and in longi- tude and latitude, are Greenwich M. T. I b V b' I' — li' — b May 28- 8 89° 24' 23"o5 +2°9'l2'.'90 89°3l' 49'.'97 +2°10' 9'.'98 -)-446.'92 +57'.'08 9 89 27 56.68 +2 9 5.67 89 31 14.38 +2 9 42.02 +197.70 +36.35 10 89 31 30.35 +2 8 58.43 89 30 39.63 +2 9 14.28 — 50.72 +15.85 and, interpolating, Greenwloh M. T. I' — l b'—b Dlst. of Centers, h m s ,/ „ /, 9 40 3.9 +31.73 +22.64 38.96 9 48 10.2 — 1.79 +19.87 19.95 With the addition of 0".57 for irradiation, the semi-diameters of Mercury and Venus are respectively 3". 98 and 26".97 ; hence, at the first observation, the distance of the limbs of the planets is 8".01, 2".6 more than a tenth part of the diameter of Venus ; at the second observation, the distance of the cen- ters is less than the difference of the semi-diameters ; hence, the Tables are verified by the statement of the observer. Venus being, at the time, a thin crescent, and about half of Mercury's disc being illuminated, it is plain that it would be diflScult for the observer to estimate the distance in fractional parts of the apparent diameter of Venus. Leverrier's remarks on this occultation are impaired by a mistake made in the last line of his computation. DERIVATION OF THE MASS OF JUPITER 105 MBMOIE No. 11. On the Derivation of the Mass of Jupiter from the Motion of certain Asteroids. (Memoirs of the American Academy of Arts and Sciences, Vol. IX, New Series, pp. 417-420, 1873.) The object of the present note is to show that the discussion of the observations of certain asteroids, provided they extend over a suflBcient period of time, will furnish a far more accurate value of the mass of Jupiter than can be obtained from measurements of the elongation of the satellites, or from the Jupiter perturbations of Saturn. It is to be hoped that observers will hereafter pay particular attention to those asteroids which are best adapted for the end in question. The magnitude of the Jupiter perturbations of an asteroid depends at once on the magnitude of the least distance of the two bodies, and the greater or less degree of approach to commensurability of the ratio of their mean motions, and also on the magnitude of the eccentricity of the asteroid's orbit. Those asteroids which lie on the outer edge of the group, and whose mean motions are nearly double that of Jupiter, will best fulfil the two first conditions named above. For they will have inequalities of long period whose coefficients will be of the order of the first power only of the eccen- tricities, while all other classes of long-period inequalities are necessarily of higher orders, and hence demand longer periods in order to have their coeffi- cients brought up to an equal magnitude. In order to exhibit the relative value of these asteroids for the purpose in view, I have computed the terms of the lowest order in the coefficients of these inequalities of long period for all the asteroids, yet discovered, whose daily mean motion lies between the limits 550" and 650" ; and have appended herewith tables, by which the value of these terms can be readily computed for any which may hereafter be discovered between these limits. The formulas for computing these terms are found in the Mecanique Cileste, Tom. I, pp. 279-281. Here i must be put equal to 2, in the terms which involve the simple power of the eccentricities. We will employ the usual notation for the designation of the elements of orbits, and make some reductions in Laplace's formulas for the sake of ready computation. 14 106 COLLECTED MATHEMATICAL WORKS OF G. W. HILL If we put y = ^ ^ or in Laplace's notation — . and recollect (I ^ n that we need the formulas only for the case of an inferior perturbed by a superior planet ; and moreover make ri?'"" = -^, and rG^'^=J, F^^^ and G^^'> being Laplace's symbols, we shall have H If, in the next place, K and /? are derived from the equations -S'cos(/3 — tt) = jy sin y> — / sinv"' C0S(7r'— ?:) , -ff'sill(^— 7r)=: — Jsinp' Sill(7r' — tt), the inequality in longitude we are computing is m ~KamlL—2L' + ^]. H and J may be regarded as functions of a , and are positive between the limits corresponding to ju = 550" and yi = 650". The common loga- rithms of these quantities are here tabulated for every 0.001 of a between the limits above mentioned ; the values of if' and hf and their differentials were obtained from Kunkle's Tables of the Goefficients of the Perturbative Function. a logH log J u. logfl- log J 0.595 0.3153369 9.871828 0.610 0.3323864 9.889836 .596 .3165277 .873131 .611 .3334562 .890910 .597 .3177113 .874420 .612 .3345169 .891967 .598 .3188875 .875695 .613 .3355683 .893007 .599 .3200561 .876956 .614 .3366103 .894030 .eoo .3212173 .878202 .615 .3376427 .895036 .601 .3223707 .879434 .616 .3386652 .896022 .602 .3235163 .880652 .617 .3396777 .896990 .603 .3246540 .881855 .618 .3406801 .897939 .604 .3257838 .883043 .619 .3416723 .898869 .605 .3269054 .884214 .620 .3426539 .899780 .606 .3280187 .885370 .621 .3436248 .900671 .607 .3291236 .886511 .622 .3445848 .901542 .608 .3302199 .887636 .623 .3455337 .902392 0.609 0.3313075 9.888745 0.624 0.3464714 9.903221 DERIVATION OF THE MASS OF JUPITEE 107 a logff log J' a log H log J 0.625 0.3473975 9.904028 0.643 0.3618323 9.914446 .626 .3483119 .904814 .644 .3624928 .914764 .627 .3492144 .905578 .628 .3501047 .906320 .645 .3631366 .915051 .629 .3509827 .907040 .646 .3637632 .915306 .647 .3643722 .915528 .630 .3518480 .907736 .648 .3649632 .915717 .631 .3527005 .908408 .649 .3655358 .915871 .632 .633 .634 .3535399 .3543659 .3551782 .909056 .909679 .910277 .650 .651 .652 .3660897 .3666246 .3671400 .915991 .916076 .916125 .635 .636 .3559767 .3567612 .910850 .911396 .653 .654 .3676354 .3681103 .916136 .916108 .637 .638 .639 .3575313 .3582866 .3590269 .911916 .912409 .912874 .655 .656 .657 .3685644 .3689972 .3694082 .916040 .915933 .915785 .640 .641 .3597519 . .3604614 .913310 .913718 .658 .659 .3697969 .3701628 .915595 .915362 0.642 0.3611550 9.914097 0.660 0.3705053 9.915085 The values of the elements of Jupiter's orbit for the epoch 1850.0 which we shall use are : 299.1286, loga'=: 0.7162372, ¥.' = 2°45'54".55, n' = 11° 55' 2". The values of the corresponding elements of as many of the asteroids as lie between the limits above mentioned are contained in the following table. The longitudes of the perihelia are referred to the mean equinox of 1850.0. fi log a (^ TT Hygea 634;'3118 0.498 4692 5° 4^ 56'.4 234° 58 40:6 Themis 636.7634 0.497 3523 6 42 52.9 139 56 11.2 Euphrosyne 633.8508 0.498 8680 12 44 10.3 93 27 51.5 Doris 647.1295 0.492 6769 4 23 42.9 74 10 11.3 Pales 655.6209 0.488 9025 13 43 18.3 32 3 13.1 Europa 650.0877 0.491 3564 5 49 14.3 101 45 37.6 Mnemosyne 632.6897 0.499 2106 5 58 17.1 52 58 47.8 Erato 640.8591 0.495 4961 9 46 4.3 33 55 38.0 Cybele 560.8775 0.534 0920 6 54 36.4 258 11 24.3 Freia 569.0505 0.529 9038 10 49 12.0 93 2 36.6 Semele 652.9848 0.490 0690 11 49 36.5 28 25 39.1 Sylvia 543.5800 0.543 1620 4 39 22.6 337 8 6.1 Antiope 632.3591 0.499 3618 11 39 2.7 293 49 3.5 108 COLLECTED MATHEMATICAL WORKS OF G. W. HILL The expression of which result from the formulas, are the inequalities, and the length of their periods substitution of these values of the elements in the Hygea 14676.2 sin [L- -2L' + 228 58 1.4L 97.96 years Themis 14606.2 sin [L- -2L' + 146 4 4.5], 91.72 " Euphrosyne 28996.5 sin IL- -21,'+ 97 58 58.4], 99.23 " Doris 5086.7 sin [L- -2L'+ 85 41 49.4], 72.27 " Pales 11639.2 sin [L- -21,'+ 33 36 12.6], 61.57 " Europa 6584.4 sin [L- -2Zy' + lll 29 19.2], 68.14 " Mnemosyne 12956.0 sin [L- -2L'+ 60 9 1.9], 102.58 " Erato 13654.9 sin [L- -2L'+ 36 21 16.9], 82.91 " Cybele 13145.4 sin [L- -2L' + 251 13 31.6], 94.49 " Freia 32243.5 sin [L- -2L'+ 98 15 25.5], 120.93 " Semele 10860.7 sin [L- -21,'+ 29 55 45.1], 64.54 " Antiope 28567.8 sin [L- -2L' + 288 44 3L6], 103.57 " These expressions can be regarded as rough approximations only to the actual values of these inequalities, since all terms of the third and higher orders with respect to the eccentricities and inclinations, and of the second and higher orders with respect to the disturbing masses, have been neglected. Yet they are suflBciently exact to show the order of magnitude of the Jupi- ter perturbations of the asteroids in question. The effect of these inequalities at the time of opposition will be magni- fied in the proportion roughly of a to a — 1 . Thus in the case of Freia, the determination of the mass of Jupiter will depend on the observation of an arcof 12°.7. INEQUALITY OF LONG PERIOD IN THE LONGITUDE OF SATURN 109 MEMOIE No. 13. On the Inequality of Long Period in the Longitude of Saturn, whose Argument is Six Times the Mean Anomaly of Saturn Minus Twice that of Jupiter Minus Three Times that of Uranus. (AstronomiBche Naohrichten, Vol. 83, pp. 83-88, 1873.) This inequality is proportional to the product of the masses of Jupiter and Uranus. In its coefficient we shall have regard only to the part which is divided by the square of the motion of the argument. Employing the notation in general use, the quantities having no accent, or one, or two, according as they belong to Jupiter, Saturn or Uranus, p designating / ndt, and putting P _ m r 1 r' cos (p~\ m" f 1 r' cos g'—'2g and g'-3g" 4g'-2g " 2g'-3g" 3g'- 2g " 3g'- 3g" belonging to terms of the several factors involved in the expression. The values of the factors proportional to Jupiter's action on Saturn have been derived from Hansen's Untersuchung uber die gegenseitigen Storung- en des Jupiters und Saturns ; the values of those proportional to the action of Uranus have been specially computed. The values of the masses adopted are m = T~stT!, m'-=-^iT5^, m" ^= YrhiF ■ In the following expressions the common logarithms are written in place of the coeflScients, and the values of n'Sz' and Sir' are in seconds of arc. "^'•^^'"^' I^ =+ 8.00004 cos (5ff'- 23 + 66=51/.6) («'(J3')2= + I.47638 3in( s''-3ff'/ + 316°12'.2) *" ^^ + 8.80451 cos (V-23 + 97 39.5) +1.40046 sin (33'-3sf'/+ 60 7.5) + 9.50716 cos (3^'- 2^ +137 52.9) +0.17010 ain (3sr'-3sf'/+ 112 48.0) + 0.00653 COB (2s'-2sr +158 3.7) +8.6777 sin (4s/-3ff'' + 129 ) + 9.79703 cos ( ff'- 2^ +258 50.7), +7.1505 sin (5s'-3s'/ + 135 ), 112 COLLECTED MATHEMATICAL WORKS OF G. W. HILL a'a + mn \ Br' I m 3g ■ a'-B, = + 7.96411 + 8.80930 + 9.59413 + 0.33413 + 9.82355 " " +94.72630 + 95.25666 + 94.36848 + 93.1858 ag" = +93.98809 + 94.85117 + 95.39557 + 94.1683 + 92.9164 sin [Sg'- sin {ig'- sin &g'- sin (2g'- 3in( g'- cos ( g'- cos {2g'- cos (3gr'- cos l,ig'- cos (5^'- sin ( g'- sinCSjr'- sin {.ig'- sin (.ig'— sin (5,0'— -2g +228°48/.3) {Slr'\-- -2g +264 16 .2) ^gr +300 16.3) -2g +335 43.0) •3^ +256 31 .4), -3g"+ 35° 16'. 9) {n'Sz%-- -3s'"+238 47.0) -3/' + 125 24.6) -3ji" + 146 30.5) -3ff" + 148 34 ), -3g''+ 40° 2'.0) ((!«»•')„= ■3s'" + 350 37.4) ■3sr" + 134 50 .1) ■35" + 153 36 ) 3sr"+130 43 ), = +0.1370 cos( sr'-3s'" + S20°58' ) + 1.08430 cos (Sg''-3sr" + 240 30.9) + 0.05350 cos (3fir'-3ff"+ 289 25 ) + 8.7413 cos(4sr'-3s" + 303 36 ) + 7.4637 C03(5sr'— 3sf" + 301 )^ = + 3.46348 sin (5ff'-3^ +346°53'.9) + 3.83450 sin (4sr'-3sr +277 10.8) + 1.41347 sin (3sr'-3sf +135 15.1) + 1.50550 sin (3^'- 3^ +156 17.9) + 0.4363 8in( 5r'-2^ +350 30 ), = + 1.73766 cos (Sjr'-S^ + 63° 43'.0) + 3.52156 cos (49''-2^ +277 2.4) + 1.30103 cos (3ff'—2sr +141 32.0) + 1.48499 cos (25r'-23 +156 1.7) + 0.4146 cos( fl''-3ir + 97 54 )■ In the next place 3 mnn' log 2il + m')(Qn' - 2n - 3n"y = 1.01570, log 3n'n" 3(6w'— 2w — 3n"y 3.18681 . Thus we get, the coefiScients still replaced by their logarithms. + 0.50212 sin (e^r' - -2g- -3^" + 103° 3'.8) + 0.25546sin(6i?'- -2g- -3y + 102°10'.8) + 1.22067 sin ( CC + 337 47.0) + 0.74761 sin( (C + 335 57.8) + 0.69296 sin ( iC + 60 41.0) + 9.8569 sin( i( + 80 40 ) + 9.6999 sin( a + 107 ) + 8.9608 sin( a + 123 ) + 7.9632 6in( ti + 214 ) + 6.8089 sin ( « + 219 ) + 9.1068 sin( a + 269 46 ) + 8.9026 sin( (( + 282 45 ) + 0.90920 sin( it + 324 37.1) + 0.55954 sin ( « + 347 29.8) + 0.66333 sin( Cl + 49 41 .3) + 9.7834 sin ( a + 86 22 ) + 9.9910 sin( C( + 98 19 ) + 8.8401 sin( 11 + 129 28 ) + 8.3019 sin( u + 17 30 ) + 6.5178 sin( (1 + 48 36 ) By the addition of these terms is obtained Sp'= + 34".752 sin (6^'- 2g — Bg"') + 1".312 cos (6/- 2g - Sg") = + 34".776 sin (6/— 2g — 3g" + 2° 9' 43") . The inequality in the mean longitude of Uranus, having the same argu- ment, has been calculated by Leverrier (Additions aux Connaissance des Temps, 1849, p. 85). He found dp"= + 32".74 sin (6/- 2g — 3g" + 181° 1' 58") . Thus, contrary to what might be expected, the inequality in the case of Saturn is larger than in the case of Uranus. CHARTS AND TABLES OF THE TRANSIT OF VENUS 113 MEMOIE No. 13. Charts and Tables for Facilitating Predictions of the Several Phases of the Transit of Venus in December, 1874. (Papers relating to the Transit of Venus in 1874, Part II, 1873.) CONTENTS. PAGE. Constants and elements employed, 114 Hourly ephemerides of the snn and Venus, 115 Axis, diameters, &c., of the enveloping cones, 116 Curves represented on the charts, 116 Problem I. To find the point of the earth's surface at which contact takes place at a given time and altitude, 117 Peoblem II. To find the point on the earth's surface where the contact takes place at a given point of the sun's limb and at a given altitude, 121 Values of quantities required in the computation of the curves, 134 Times, &c., of the beginning and end of each contact, and of its occurrence in the zenith, 127 Approximation of the curves to circles, 128 Tables of positions of points used in drawing the curves, 135 Explanation of the charts, and their use, 139 Tables and formulas for computing more accurate values of the times of contact, Example, 142 Corrections to be applied for determinate changes in the elements, 145 Tables and formulas for finding the position of the planet on the sun's disc, . . . 147 Localities favorable for the determination of parallax, 150 CHAETS. No. I. Ingeess, exteeioe contact. No. III. Egeess, interioe contact. II. Ingeess, inteeioe contact. IV. Egeess, exteeioe contact. All the constants and elements which have been used in the computa- tions on the transit are given below. The quantities having no terms mul- tiplied by t are either constant or may be regarded as such for the duration of the transit ; and the quantities which vary may be regarded as varying uniformly. The unit of t is an hour. 15 114 COLLECTED MATHEMATICAL WORKS OF G. W. HILL Epoch: 1874, Decemher 8*^ 11", Washington Mean Time. VENUS. Orbit longitude, referred to the mean equinox of date, Longitude of the ascending node, Log sine of inclination, . Periodic perturbations of the latitude, Log radius-vector, .... Semi-diameter at mean distance, 76° 58' 13". 84 + 243". 332i( 75° 33' 24". 1 8.7722486 + 0". 11 9.8575310 — 27.6(5 8". 546 THE SUN. True longitude, referred to the mean equinox of date. Latitude, Log radius-vector, .... Semi-diameter at mean distance, True obliquity of the ecliptic, Equation of the equinoxes in longitude, Sidereal time, at Washington, in arc, . Constant of solar parallax, . Constant of aberration. Eccentricity of the earth's meridians, . Horizontal refraction, .... 256° 58' 41". 62 -1- 152". 533i! — 0". 41 9.9932845 — 21.3< 959". 788 23° 27' 27". 67 — 7". 43 63° 44' 9". 6 + (15° 2' 27". 84)^ 8". 848 20". 4451 0.0816967 35' The elements of the heliocentric position of Venus are from the new Tables of Venus,* and may be readily deduced from the first example given in pages 16-19 of the introduction. The apparent position of the sun which results from the above elements coincides with that derived from the tables of Hansen and Olufsen, but the true longitude is 0".19 greater, owing to the adoption of Struve's value of the constant of aberration, 20". 445, instead of the value 20". 255. The value of the sun's semi-diameter is adopted from Bessel. (See Astronomische Nachrichten, No. 228, and Astrunomische Untersuchungen, Vol. II, p. 114.) This value is used in the cotoputation of eclipses for the Ameri- can Bphemeris. Hansen has also used it in his disquisition on the transit of Venus. In the British Nautical Almanac the value 961".82 is used, and is the same as that given for the reduction of meridian observations of the sun. Leverrier states {Annates, Vol. VI, p. 40) that the value, deduced from the previous transits of Venus, is 958". 424. Hence, it is probable that predictions from the elements of the British Nautical Almanac will be found to be considerably in error from this cause. * Tables of Venus, prepared for the use of tlie American Ephemeris and Nautical Almanac, by George W. Hill, Washington, 1872. CHARTS AND TABLES OF THE TRANSIT OF VENUS 115 From the data given above are obtained the following hourly ephemer- ides. For the sake of completeness they are expressed in terms of longi- tude and latitude, as well as in right ascension and declination. Wash. M. T. a = App. R. A. TENUS. App. dec. App. geocentric longitude. App. geocentric latitude. Logr = log distance from the earth. 1874. Dec. 8a 8h 255° 58' 56'.'03 —22° 38' 9'.'96 257° 4 53'.'34 +11 40'.'84 9.4221505 9 57 21.96 37 22.29 3 22.30 12 19.91 482 10 55 47.90 36 34.60 1 51.27 12 58.99 467 11 54 13.86 35 46.90 257 20.24 13 38.06 460 12 52 39.84 34 59.18 256 58 49.21 14 17.13 461 13 51 5.83 34 11.44 57 18.18 14 56.21 470 14 255 49 31.85 —22 33 23.67 256 55 47.16 +15 35.28 9.4221488 Wash. M. T. App. B. A. THE SUN. App, dec. App. longitude. App. latitude. Log r'= log distance Irom the earth. 1874. Dec. 8a 81 255°42'l6"80 — 22°48'24'.'39 256° 50' 35'.'86 -6'ax 9.9932909 9 45 1.47 48 39.36 53 8.39 0.41 888 10 47 46.15 48 54.28 55 40.93 0.41 867 11 50 30.84 49 9.15 256 58 13.44 0.41 845 12 53 15.54 49 23.98 257 45.99 0.41 824 13 56 0.25 49 38.77 3 18.52 0.41 802 14 255 58 44.98 —22 49 53.51 257 5 51.05 —0.41 9.9932781 From these quantities the position of the center of the sun, as seen from the center of Venus, is derived. Wash. M. T. 0=E. A. i = dec. Log G — log distance. 1874. Dec. 8a Sn 255° 36' 9'.'50 —22° 52' 9'.'48 9.8575394 11 255 49 8.82 54 3.58 309 14 256 2 8.52 55 56.63 227 In the next place are obtained the following quantities, which are desig- nated by the eclipse notation* of Chauvenet's Spherical and Practical As- tronomy, which, for the most part, is identical with that of Bessel's Analyse der Finsternisse. It must be remembered that Venus here takes the place of the moon. * The plane of reference passes through the center of the earth perpendicular to the axis of the enyel oping cones ; a and cJ are the right ascension and declination of the vanishing point of the axis; ^1, the hour-angle of that point at the first meridian ; (?, the distance of the sun and planet ; x, y, the coordinates of the axis in the plane of reference, y being taken positive toward the north, x positive dx toward that point whose right ascension is 90° + a; ■— and _i. are the hourly changes of x and y ; / is the angle of the cone ; I, the radius of the cone in the plane of reference ; i = tan /. 116 COLLECTED MATHEMATICAL WORKS OF G. W. HILL Wash. M. T. 1874. Dec. 8d 8h 11 14 +37.6744 + 8.4134 —20.8759 dz dt -9.74895 9.75838 -9.76782 +25.0318 32.7602 +40.4042 dt +2.59020 2.56207 +2.53393 122 36.6 166 55 0.8 211 49 24.6 Wash. M. T. 1874. Dec. 8a 8h 11 14 Exterior contacts. / I log ! log i 22'24'.'272 41.1254 1.614110 7.8141 .299 62 19 41 .324 68 24 41 Interior contacts. I log I .570 .595 38.4845 54 59 1.585286 296 301 lost" 7.8063 63 63 CURVES REPRESENTED ON THE CHARTS. Having novr the necessary data, I proceed to explain the computations which have been made for the purpose of drawing the charts. These charts are designed to give the principal circumstances attending each of the four contacts at any point of the earth's surface where it is visible. These circum- stances may be taken to be the time at which the contact occurs, and the position of the point of contact on the sun's limb. Hence, two classes of curves have been plotted on the charts — first, curves upon which contact occurs at the same instant ; and, secondly, curves upon ivhich contact takes place at the same point on the sun^ s limb. These curves are evidently limited, in both directions, by the curve upon which contact takes place in the hori- zon. The readiest method of drawing them will be to compute a suflBcient number of positions conveniently distributed on these curves, and through these positions, plotted on the chart, draw the curves. As convenient formulas for the purpose are not found in the treatises on practical astronomy, I will develop them here. It will be amply sufficient to determine the position of these curves to within a minute of arc. Hence, as the horizontal parallax of Venus is only 33", the effect of parallax on the right ascension and declination of the point of contact may be neglected. Then the position of this point can be found by the equations, the upper sign being used for the exterior contacts, and the lower for the interior. With sufficient approximation, these equations may be written £^'=5±^('V-5). CHARTS AND TABLES OF THE TRANSIT OF VENUS 117 The exterior contacts last about 21 minutes on the earth's surface, and the interior contacts about 25 minutes. The quantities a' and d' vary so slowly that they may be computed for the middle of the duration of each contact on the earth's surface, and then supposed constant for this duration. In this way the following values have been obtained : Wash. M. T. a' d' b m „ ; „ , For exterior contact at ingress 8 40 255 57 —22 38 For interior contact at ingress 9 10 255 57 22 37 For interior contact at egress 12 48 255 51 22 34 For exterior contact at egress 13 18 255 51 — 22 34 The investigation to be made is conveniently divided into two problems Problem I. — Tv find the point 6f the earth's surface at which contact takes •place at a given time and at a given altitude. Let 0) = the longitude of the required point west from the first meridian ; y = its latitude ; /I = the sidereal time at the first meridian ; h = the given altitude ; 6 =: the parallactic angle at the point of contact ; if— fj. — a'— (o = the hour-angle of the point of contact. The general formulas of spherical trigonometry, applied to the triangle formed by the zenith, the pole, and the point of contact, give these equa- tions : cos ^ sin S' = cos hsind, cos f cos !?'= cos d' sin h — sin d' cos h cosO , sin ?> = sin d' sin h + cos d' cos h cos9. As soon as 6 is known, these three equations, together with the equa- tion, oj = /J. — a'— y= jUi — ^' give the position of the required point. To obtain B, resort must be had to the equation defining the condition of contact, viz. : = x'+y' — 2ixS + yyj) + p' — C^ In place of x and y make the usual substitutions, a; = m sin M, y = m cos M, then S Bin M + rj COS M= ^ 2^^ . 118 COLLECTED MATHEMATICAL WORKS OF G. W. HILL The numerical value of each member of this equation is always less than unity, and it will be determined, to a suflScient degree of precision, with four decimals. The average value of the denominator 2m is about 80 ; hence, in the numerator it will be sufficiently accurate to put p" =: 1 , and 2U^ = 2mi^, and neglect the term — ^*f^; and if terms multiplied by i and e^ are neglected, it is plain that ^ = sin A. Thus simplified, the equa- tion becomes c • j# , Tir '"^' — ^^ + ^ . ■ ■ I 1-21. ^ 8\nM + T] cos M = 5 h t sin h — q— sm^ h . The right hand member of this equation is a known quantity, and it only remains to discover the expressions of £ and y; in terms of 6 to have the equation determining 6. The known expressions for ^ and t^ are $ = P cos {»' sin I? , 7]^p cos d sin ^' — p sin d cos y' cos i? . But if terms of the order of e* are neglected, , cos f p COBp — —— , , (1 — e^) sin

' = sin S , cos

= Ml — ^'. It is worthy of remark that the equation determining 6 remains the same if h, instead of being exactly zero, is a small positive or negative angle ; for sec h will be sensibly unity, and, B and G being small, the terms B sin h and C sin^ h may be neglected. Hence, in taking into account the effect of refraction on the position of points, where contact takes place in the horizon, 6 may still be derived from the equation, cos (0 — y)=:Ap, but it will be necessary to make A =: — (the horizontal refraction) in the equations determining ^ and S-'. The particular case where h = 90°, or contact in the zenith, requires notice. Here the equation determining 6 reduces to A + B + C = 0. This determines the time at which the phenomenon takes place ; and the equations for the position of the point reduce to

? cos iV— OTo cos (ilfo — -A'^) + (^ — i sin h) cos (Q — iV^) . The values of ^ and >7 found in the first problem must be substituted in these equations. The first member of the first of these equations is obtained simply by writing iV-f- 90° for M in the first member of the corresponding equation of the first problem. Hence, making L'= 1— e'coa' d' sin' JSr, X' = — \e^ cos^ d' sin 2N + v sin d', yf ^N+k' + 90°, these quantities are constant for the duration of each contact on the earth's surface, and there is obtained the equation p\^ cosN—Tj siniV] = i^'cosAcos(» — /) + ^' cos(iV+'<') sinA. Consequently, if A' = '^ sin {M, -N)-^, sin {Q -N), B' = ^ sin iM„ -N)—^ cos {N + x') , where Q has been put equal to M^ in the term multiplied by i, the equation determining 6 in this problem becomes cos {e — r')=P sec h [A' + B' sin A] . CHAKTS AND TABLES OF THE TRANSIT OF VENUS 123 The equation giving the value of nt is only needed for the purpose of obtaining ii[, which it is necessary to have in order to get co from S''. In this it will be sufficiently accurate to put for £ and vj their approximate values, f = cos A sin e , rj = cos hsinO, and neglect the term multiplied by i ; then nf = cos h cos (0 — W) — nio cos (J/"„ — W) + I cos (Q — N). If 1^0 denote the value of ^( at the epoch from which t is counted, fi' the motion of /u{ in a unit of time, and the expression for iii[ is M f4 = A" + '=- cos h cos (5 — iV") . 71 After 6 and ^[ have been determined from the equations just given, the position of the point on the earth's surface is found by means of the same equations as in the first problem. Thus it appears that the solutions of the two problems are quite similar, the only differences being that the term cor- responding to O sin^ h, in the factor of the equation which determines 6, is wanting, and that a separate computation must be made for n[ ; and the remarks to be made regarding the solution of the equation determining 6 , and the limits between which Q and h must be assumed, in order that solu- tion may be possible, are quite similar to those made in the first problem. While B' and y' are constant for the duration of each contact on the earth's surface. A' and A" involve the variable Q, and may be tabulated with Q as the argument within its limiting values. The equation determining 6 gives two values for this quantity, corresponding to the two points on the earth's surface, which satisfy the conditions of the problem ; and p must be deter- mined separately for each. The condition of contact taking place at a given point on the sun's limb, and at the maximum altitude, is cos A = ± p \_A' + B' sin A] , and the equations A'p=±\, 124 COLLECTED MATHEMATICAL WORKS OF G. W. HILL give the limiting values of Q. In finding the points on the curves of the second class, which are common to the curve of contact on the horizon, 6 is derived from the equation cos (e — /) = A'p , but h=^ — (the horizontal refraction) in the equations which determine ^ and S''. In computing the value of [I'l for each of the two solutions of the problem, it will be noticed that, with sufficient approximation, the second terra has the same numerical value but opposite signs in the two solutions ; and, in the case of maximum altitude for a given value of Q, the equation becomes simply K = A". In this case also, it will be advantageous to compute four auxiliary quantities from the equations, ^ cos £ = cos d', y sin e' = sin d', ^ sin £ = sin d' cos d , p' cos e' = cos d' cos 8', by means of which the equations determining (p and ^ take the form, cos y sin &' = cos h sind, cos ^ cos &' =p sin (h — e) , sin 9> =p' cos (h — e') . As & is constant in this case, p,p', e, e', are so likewise, provided that after the point of maximum altitude has passed the zenith, h be supposed to increase from 90° to 180°, or, in other words, that 180° — h be used instead of h. VALUES OF THE QUANTITIES EMPLOYED. Denoting the four contacts in their order by the symbols I, II, III, and IV, the values of the various quantities employed in the foregoing discus- sion are : I II III IV Epochfrom which; is counted, 8" 40°" 9" 10°" 13" iS" 13" 18"" V +18' +16' -6' -8' logX', 7.9187 7.9158 7.9359 7.9417 /, 54° 20' 58° 33' 100° 46' 104° 14' logZ, 9.9989 9.9986 9.9977 9.9977 ^ +3' +4' -3' -1' logB, W7.1880 7.2238 m7.3475 M7.3411 N, 284° 50' 30".5 284° 48' 49".5 284° 36' 36".5 284° 34' 65".6 CHARTS AND TABLES OF THE TRANSIT OF VENUS 125 L', m, -ji-siniM.-N), log[-^], . r logB'. .^ ^'cos(if,-iV), log log "V. in minutes of arc — in minutes n logi?, logi>' , e', . of arc , I II III IV 9.9977 9.9977 9.9977 9.9977 -2' -1' + 7' + 8' -34.0289 + 34.0187 + 34.0188 + 34.0290 1.616412 Ml.587591 wl.587600 1.616423 14° 49' 14° 48' 14° 44' 14° 43' 7.3788 7.3517 7.3349 m7.3605 166° 23' 166° 26' 166° 43' 166° 45' 3.5657 3.5369 3.5368 3.5656 1.9516 1.9516 1.9515 1.9515 9.99758 9.9978 . 9.9979 9.9979 31° 7' 21° 56' -21° 54' -21° 54' 9.9876 9.9876 9.9877 9.9877 -156° 40' -156° 41' -23° 15' -23° 15' The quantities which vary with the time and with Q are given in the following tables. I. — For Exterior Contact at Ingress. Wash. M.T. A log c 7 /*'! Wash. M. T. A log C y /^'i h m V / b m 8 29 +1.0339 •n 8.0752 51 29 128 56 8 40 - -0.0313 n 8.0864 49° 25 131 41 30 0.9360 .0762 51 18 129 11 41 0.1268 .0874 49 13 131 56 31 0.8383 .0772 51 7 129 26 42 0.2220 .0884 49 1 132 11 32 0.7408 .0782 50 56 129 41 43 0.3170 .0894 48 50 132 27 33 0.6435 .0793 50 44 129 56 44 0.4117 .0904 48 38 132 42 34 0.5464 .0803 50 33 130 11 45 0.5061 .0914 48 26 132 5T 35 0.4495 .0813 50 22 130 26 46 0.6003 .0924 48 14 133 12 36 0.3529 .0823 50 11 130 41 47 0.6942 .0934 48 2 133 27 37 0.2565 .0833 49 59 130 56 48 0.7878 .0943 47 50 133 42 38 0.1603 .0844 49 48 131 11 49 0.8811 .0953 47 38 133 57 39 +0.0644 .0854 49 36 131 26 50 0.9742 .0963 47 26 134 12 8 40 —0.0313 n 8.0864 49 25 131 41 8 51 - -1.0670 n 8.0973 47 14 134 27 e A' A" 1 A' A" « A' A" 46°50' —1.0360 133° 54' 48° 30' —0.3841 132° 26 50° 10' +0.2969 130° 58 47 0.9721 133 45 40 0.3173 132 17 20 0.3666 130 49 10 0.9079 133 36 50 0.2502 132 8 30 0.4366 130 40 20 0.8435 133 27 49 0.1828 131 59 40 0.5069 130 31 30 0.7788 133 19 10 0.1152 131 51 50 0.5774 130 23 40 0.7138 133 10 20 —0.0472 131 42 51 0.6482 130 14 50 0.6484 133 1 30 +0.0211 131 33 10 0.7193 130 5 48 0.5827 132 52 40 0.0897 131 24 20 0.7907 129 56 10 0.5168 132 43 50 0.1585 131 15 30 0.8624 129 47 20 0.4506 132 35 50 0.2276 131 7 40 0.9343 129 39 48 30 —0.3841 132 26 50 10 +0.5 !96g 130 58 51 50 +1.0065 129 30 126 COLLECTED MATHEMATICAL WORKS OP G. W. HILL II For Interior Contact at Ingress. Wash. M.T. A log c 7 l^\ Wash. M.T. A log C y Z''. h m ^ 1 o i h m • / a / 8 57 +1.0501 n 8.1034 46 1 135 57 9 10 - -0.0238 n 8.1155 43 13 139 13 58 0.9651 .1044 45 48 136 12 11 0.1037 .1164 43 139 28 59 0.8807 .1053 45 36 136 27 12 0.1832 .1173 42 47 139 43 9 0.7967 .1063 45 23 136 42 13 0.2623 .1181 42 33 139 58 1 0.7130 .1072 45 10 136 57 14 0.3410 .1190 42 20 140 13 2 0.6296 .1081 44 57 137 12 15 0.4193 .1199 42 7 140 28 3 0.5466 .1091 44 45 137 27 16 0.4972 .1208 41 53 140 43 4 0.4640 .1100 44 32 137 42 17 0.5746 .1217 41 39 140 58 5 0.3817 .1109 44 19 137 57 18 0.6515 .1225 41 26 141 13 6 0.2998 .1118 44 6 138 12 19 0.7280 .1234 41 12 141 28 7 0.2183 .1127 43 53 138 28 20 0.8040 .1243 40 58 141 43 8 0.1372 .1137 43 39 138 43 21 0.8795 .1251 40 44 141 58 9 +0.0565 .1146 43 26 138 58 22 0.9546 .1260 40 30 142 13 9 10 —0.0238 n 8.1155 43 13 139 13 9 23 - -1.0292 n 8.1268 40 17 142 28 e A' A" Q. A' A" <2 A' A'l 39° 50' —1.0401 142° lO' 42° O' —0.3965 140° 13 44° 10 +0.2964 138°19' 40 0.9924 142 1 10 0.3449 140 4 20 0.3517 138 10 10 0.9444 141 52 20 0.2930 139 55 30 0.4073 138 2 20 0.8961 141 43 30 0.2409 139 47 40 0.4632 137 53 30 0.8474 141 34 40 0.1885 139 38 50 0.5194 137 44 40 0.7985 141 25 50 0.1358 139 29 45 0.5758 137 35 50 0.7493 141 16 43 0.0827 139 20 10 0.6325 137 26 41 0.6997 141 7 10 —0.0294 139 11 20 0.6895 137 18 10 0.6499 140 58 20 +0.0242 139 3 30 0.7468 137 9 20 0.5998 140 49 30 0.0781 138 54 40 0.8044 137 1 30 0.5494 140 40 40 0.1322 138 45 50 0.8623 136 52 40 0.4987 140 31 50 0.1866 138 37 46 0.9204 136 44 50 0.4477 140 22 44 0.2414 138 28 10 0.9788 136 35 42 —0.3965 140 13 44 10 +0.2 964 138 19 46 20 +1.0375 136 27 III. — For Interior Contact at Egress. Wash. M.T. A log c 7 V-'^ Wash. M.T. A log c 7 y-'^ h m • 1 o ; h m / a / 12 35 —1.0188 n 8.1276 —10 53 190 41 12 48 —0.0100 ■n 8.1162 —13 50 193 57 36 0.9440 .1268 11 7 190 56 49 +0.0706 .1153 14 3 194 12 37 0.8687 .1259 11 21 191 11 50 0.1515 .1144 14 16 194 27 38 0.7929 .1251 11 34 191 26 51 0.2329 .1135 14 29 194 42 39 0.7166 .1242 11 48 191 41 52 0.3147 .1126 14 42 194 57 40 0.6399 .1233 12 2 191 56 53 0.3968 .1116 14 55 195 12 41 0.5627 .1224 12 16 192 11 54 0.4793 .1107 15 8 195 27 42 0.4850 .1216 12 29 192 26 55 0.5622 .1098 15 21 195 42 43 0.4069 .1207 12 43 192 41 56 0.6454 .1089 15 34 195 57 44 0.3284 .1198 12 56 192 56 57 0.7290 .1079 15 46 196 12 45 0.2494 .1189 13 9 193 11 58 0.8129 .1070 15 59 196 27 46 0.1700 .1180 13 23 193 27 12 59 0.8972 .1060 16 11 196 42 47 0.0902 .1171 13 36 193 42 13 0.9819 .1051 16 24 196 57 12 48 —0.0100 71 8.1162 —13 50 193 57 13 1 +1.0669 n 8.1041 —16 36 197 12 CHAKTS AND TABLES OF THE TRANSIT OF VENUS 127 10 30 —1.0148 191 4 40 0.9669 191 13 60 0.9187 191 22 11 0.8702 191 31 10 0.8215 191 40 20 0.7724 191 49 30 0.7230 191 58 40 0.6733 192 7 50 0.6233 192 16 12 0.5731 192 25 10 0.5226 192 34 20 0.4717 192 43 30 0.4205 192 51 -12 40 —0.3690 193 -12 40 —0.3690 193 50 0.3173 193 9 13 0.2653 193 18 10 0.2131 193 27 20 0.1605 193 36 30 0.1076 193 44 40 0.0545 193 53 50 —0.0010 194 2 14 +0.0528 194 11 10 0.1069 194 20 20 0.1612 194 29 30 0.2158 194 37 40 0.2706 194 46 -14 50 +0.3258 194 55 -14 50 15 10 20 30 40 50 16 10 20 30 40 -16 50 +0.3258 0.3813 0.4371 0.4931 0.5494 0.6059 0.6628 0.7200 0.7775 0.8353 0.8933 0.9515 +1.0101 194 55 195 .4 195 13 195 21 195 30 195 39 195 47 195 56 196 5 196 13 196 22 196 30 196 39 IV. — For Exterior Contact at Egress. Wash. . M. T. -* log a y H-\ Wash. M.T. A log C 7 /''i h m 13 7 —1.0536 n 8.0983 —17° 48 198* 43 h m 1318 —0.0144 n 8.0874 —19° 59 201° 29 8 0.9605 .0974 18 1 198 58 19 +0.0817 .0864 20 10 201 44 9 0.8671 .0964 18 13 199 13 20 0.1780 .0854 20 22 201 59 10 0.7735 .0954 18 25 199 28 21 0.2745 .0844 20 33 202 14 11 0.6795 .0944 18 37 199 44 22 0.3713 .0834 20 45 202 29 12 0.5853 .0934 18 49 199 59 23 0.4683 .0823 20 56 202 44 13 0.4908 .0924 19 200 14 24 0.5655 .0813 21 8 202 59 14 0.3960 .0914 19 12 200 29 25 0.6629 .0803 21 19 203 14 15 0.3010 .0904 19 24 200 44 26 0.7605 .0793 21 30 203 29 16 0.2057 .0894 19 36 200 59 27 0.8583 .0783 21 41 203 44 17 0.1102 .0884 19 47 201 14 28 0.9563 .0772 21 53 203 59 1318 —0.0144 n 8.0874 —19 59 201 29 13 29 +1.0545 n 8.0762 —22 4 204 14 Q A' A" e A' A" « A' A" —17° 30 —1.0021 199° 18 o ; —19 10 —0.3487 200° 48 — 20°5o' +0.3341 202° 16 40 0.9380 199 27 20 0.2817 200 57 21 0.4039 202 25 50 0.8737 199 36 30 0.2144 201 6 10 0.4739 202 34 18 0.8091 199 45 40 0.1468 201 15 20 0.5443 202 42 10 0.7443 199 54 50 0.0790 201 24 30 0.6150 202 51 20 0.6791 200 3 20 —0.0109 201 33 40 0.6859 203 30 0.6136 200 12 10 +0.0575 201 41 50 0.7572 203 8 40 0.5478 200 21 20 0.1262 201 50 22 0.8288 203 17 50 0.4817 200 30 30 0.1952 201 59 10 0.9006 203 26 19 0.4153 200 39 40 0.2645 202 8 20 0.9727 203 34 —19 10 —0.3487 200 48 —20 50 +0 3341 202 16 —22 30 +1.0451 203 43 BEGINNING, ETC., OF EACH CONTACT. From the foregoing data are readily derived the times, and position of the places, at which the following phenomena occur. 128 COLLECTED MATHEMATICAL WORKS OF G. W. HILL Wash. M. T. Longitude. Latitude. h m p , o / Contact I begins on the earth 8 29.335 55 27 +35 24 occurs in the zenith 8 39.530 131 34 —22 38 ends on the earth 8 50.292 244 25 —38 24 Contact II begins on the earth 8 57.572 65 53 +40 15 occurs in the zenith 9 9.520 139 6 —22 37 ends on the earth 9 22.630 257 24 —44 22 Contact III begins on the earth 12 35.216 36 40 —64 33 occurs in the zenith 12 48.314 194 2 —22 34 ends on the earth 13 0.244 235 18 +62 48 Contact IV begins on the earth 13 7.548 58 15 —61 occurs in the zenith 13 18.300 201 33 —22 34 ends on the earth 13 28.471 251 17 +59 20 APPROXIMATION OF THE CURVES TO CIRCLES. The curves to be drawn on the charts approximate so closely to circles of the sphere that it has been deemed sufficient to compute the positions of three points on each curve, namely, the two at which contact occurs on the horizon, and the one for which the altitude is a maximum, and then regard the curve as a circle of the sphere passing through these points ; and, as the stereographic projection has been chosen for the delineation of the charts, the projected curves will also be circles. But it will be of interest to determine beforehand how great an error can be produced by this assumption. And first, in the case of the time- curves, let a be the radius of the circle of the sphere passing through the three points, and adopt the subscripts (0), (l), (2), (3), for the quantities which refer respectively to the pole of this circle, the points of contact on the horizon, and the point of maximum altitude. Then (T and the position of the pole of this circle are determined by the equations, sin fi sin ^o + cos fi cos f^ cos {■ft[ — '9j) = cos „ = sin y^ , sin y„ = cos >-„ cos {d' — r) , <"a = All — ''o • CHARTS AND TABLES OF THE TRANSIT OP VENUS 13 j If the distance of any point on the time-curve from this pole be denoted by a', then a' — a may be taken as a sufficiently exact measure of the error committed by our method of drawing the curve. But cos a' = xXo + yyo + zz^ , X = cos h sin e , y = cos d' sin h — sin d' cos h cos d , z = sin d' sin h + cos d' cos h cos , whence cos 0-' = cos h sin sin y^ + cos h cos cos y^ cos t + sin A cos ^„ sin t . or, as cos r may be put equal to unity, cos 0-' = (A3 — ;3) sin h + cos h cos (0 — ;-„) . The quantity a' — a is composed of two parts independent of each other ; the first depending on the curvature of the cone enveloping the sun and Venus, and proportional to the quantity we have denoted by G ; the second due to the non-sphericity of the earth and proportional to e'. These parts can then be determined separately. First, from the equations, cos A3 = ±{A + Bsmh, + G sin' h,) , cos ^ = ± A , is obtained, with sufficient exactness, h,-p = :f (5 + C sin /?) . But cos h cos (6 — Ya) = ± (_A + B sin h + C sin7t) , cos a — ± A , thus cos a' — cos (T = ± C sin A (sin h — sin /J) . Secondly, from the equations, cos 7*3 = ± Ap^ = ± ^ Pi — -^ (sin d' sin /3 ± cos d' cos /S cos yf , cos /5 = ± A\l—\^ cos' d' {A"" cos %r + sinV)] , we find that the part of A3 — ^ proportional to ^ is A3 — /? = I e' cos /9 [sin' d' sin /3 ± sin M' cos /3 cos >- — cos' d' sin ji sin' •,-] . 132 COLLECTED MATHEMATICAL WORKS OF G. W. HILL Also COS h cos (0 — j-„) = ± Ap ^: ^e'' cos'' d' sin 2/ cos' /3 cos h sin (0 —)■) , = ± ^ 1 — -^ (sin d' sin h + cos d' cos h cos ^)' , q: ~ cos" (^' sin 2r cos= /? V (sin' /? - sin' k) , cos A cos e = ^ COS >- — V (sin' /9 — sin' h) sin ;- , where the sign of sin (0 — y) must be attributed to the radical is/{s\v? ^ — sin^A). After some reductions it will be found that e' cos a' — cos ff = -„ (sin' d' — cos' d' sin' y) cos /3 sin h (sin ^ — sin h) ') cos /J q= C (sin ^ — sin A) sin h e' + ^ sin M' sin >- cos /J sin A V(sin' /? — sin' h) . Al It will be seen that this expression vanishes when ^ =: and when h = ^ , as it should. Differentiating it with respect to the variable sin h , in order to obtain its maximum value, we arrive at an equation of the fourth degree in sin h. Hence we are obliged to content ourselves with a superior limit to the maximum value, which, however, for practical purposes, may be regarded as identical with it. The first term of the expression has its maximum value when sin h^= i sin j3 , and the second when sin h = sin /3. Substituting these values in their respective terms, we obtain "' — 0- = TTT (cos' d' sin' y ± 2 sin 2d' sin r — sin' d') sin 2/J ± -^ sin /J , Id 4 where the ambiguous signs, in both cases, must be so taken that the largest numerical value of the expression will be obtained. Replacing e" and C by their values, and taking for the factor which involves d' and y the greatest value it can have, it results that a' — a cannot exceed 11' sin ^ + 2' sin 2^ , and the maximum value of this with regard to the variable /3 is less than 1 2.' Having regard to the scale on which the charts have been constructed, this CHARTS AND TABLES OP THE TRANSIT OP VENUS 5^33 quantity may be considered as within the unavoidable errors produced by imperfection of drawing. It is worthy of remark that, in our method of drawing the curves, the error is only a fourth part of that which results from neglecting altogether the curvature of the cones enveloping the sun and the planet, as has gene- rally been done in treatises on practical astronomy. The investigation of the error in the case of the second class of curves differs somewhat from that of the first class, on account of [i[ not being con- stant for all points on the curve. The equations determining a and the position of the pole are sia ,, cos (wj — w,,) = cos (t , sin w. sin «>„ + cos v^ cos «>„ cos (w, — <«„') = cos «■ . where sin ^2 sm ^0 + cos y^ cos „ cos (w^ — <«„) = cos «■ , , = A"+'Lsin(d,-r')-^', If we put 0,3 = A"- K 9 = ^sin(e,-/)^±^V(l-^'^), = A"-K, g is 2u small angle, whose square may be neglected, and the equations, using the notation given in the case of the first class of curves, take the shape (^1 + 9yi) a^o + iVi - c/^i') % + 2, 2o = cos cos = (^3 — i3) sin A + cos A cos (5 — ^o) + '^ V (sin" /J — sin" h) [cos <^' sin y^ sin A + sin d' cos A sin {6 — ;-„)] , — {]h — /5) sin A + cos A cos (ff — ^-j) ± — sin d' (sin" /3 — sin" A) ± — cos d' sin / sin A V (sin" /3 — sin" h) , where the upper or lower sign is taken according as A' is positive or nega- tive, and the sign of sin (0 — y') is assigned to the radical V (sin" /3 — sin' h) . The part of cos a' — cos a which involves the factor ^ will be found to be ^ n ± — sin d' sin h (sin B — sin K) , ± — cos d' sin / sin h V (sin" /J — sin" h) . The part proportional to e^ is obtained from the analogous expression in the case of the first class of curves, simply by changing y into y', and thus is -^ (sin" d' — cos" d' sin" /) cos /? sin A (sin /J — sin h) Z e" + sin SfZ' sin / cos /J sin h V (sin" /5 — sin" A) . CHARTS AND TABLES OF THE TRANSIT OP VENUS 135 Combining these two parts, we have cos 0-' — COS = the latitude of the place, positive when north, 01 = its longitude from Washington, positive when west, loge = 8.9123, log (1 -«'') = 9.99709, sin;f = esin^, h = secxcosp, ^= (1 — e') sec/ siny>, a = A — h sin (fi — w), 1 = B— Eh + Gh cos(fi — (u), c— — C+ Fk — Hh cos(At — w), m=^ i\l be, (usually with the same sign as a). CHARTS AND TABLES OP THE TRANSIT OP VENUS ^45 If m = a, the time Tq is correctly chosen. If m differs from a, a cor- rection of the assumed time may be obtained in seconds, by the formulas, log/ =9.8617, a' = A' — ij.'h cos (/i — (u), V = B' - ii'Ohsin{[i — a>), , _ 10000 (m — a) a' + b' cot Q ' and the actual Washington time of contact will be and the local mean time of the phenomenon will be T, + t-w. Q must be taken of the same sign with a, and is a suflSciently near approximation to the angular distance of the point of contact, reckoned from the north point of the sun's limb toward the east. To find F, the angular distance of the point of contact from the vertex of the sun's limb, positive toward the left, we have the formulas, ^ sin P = sin = + 21° 18' 12" w ==80= ' 51' 45" (1) loge = 8.91220 (3) log sin

= 9.96926 (4) + (5) logA = 9.96945 Prom chart No. 2 the Washington mean time of contact is found to be nearly S^ 58™ 24^ which will be taken as the value of Tq. 19 146 COLLECTED MATHEMATICAL WORKS OF G. W. HILL Computation of t , tfie correction of T^ . fi = 136°32'.8 (9) f^—a>= 55°4r.O (1) log sin (m — )- 9.31684?j (7) + (8) \ogHhco&{ix—u)) - 9.30367m (5) + (7) log t^'h cos {ii-w)- 9.5823 (4) + (5) + (6)log/ffAsin(M - <«) = 9.3444w (30) log 5 =1.8163035 (21) log c = 1.0596051 (33) = i [(30) + (31)] log m = 1.4379043 (32) — (30) log tan i e = 9.6317108 Angle from north point, Q = 45°35'10" (10) (11) (9) + (10) (10) + (11) (12) (13) (14) (15) (16) (17) (18) (19) \ogE= 9.96333 log^= 9.55755 logi?'= 9.96558 log Ek= 9.53078 log Fk = 9.52313 A= + 28.1840 - Asin(/i — «<) = — 0.7698 B= + 66.0334 -Ek = — 0.3317 Gh cos {iJ. — w)=:— 0.2074 - C = + 10.9364 Fk- + 0.3335 - Hh cos {ti—m)= + 0.3012 (13) + (13) a = + 37.4143 (14) + (15) + (16) 5 = + 65.4943 (17) + (18) + (19) c = + 11.4711 OT = + 27.4097 m—a = — 0.0045 (29) log cot Q = 9.9936 (33) A' = — 27.08 (30) log V = 0.8686 (34) — fi-'h cos (/i — m) = — 0.38 (29) + (30) log V cot Q = 0.8623 (35) ^'= + 7.17 (26) — ix'Gh s\n{ix — w) = + (35) + (36) h'= + 0.32 7.39 (31) \og{m—a) + 4 = 1.6533W (27) = = (23) + (24) a' = - 27.46 (32) log (a' + V cot Q) = 1.3050W (28) V cot G = + 7.28 (31) -(33) log if = 0.3483 (27) + (38) a' + 5' cot § = — 20.18 Assumed time, .... , , h m B . To = 8 58 24.0 Correction of the assumed time, . . ;;= +2.3 Washington time of the contact, . , 8 58 36.2 Honolulu time of the contact. . 3 34 59.2 We have F= — 9° 36.0. G = 55° 1'.2, and the angle from the vertex, CHARTS AND TABLES OF THE TRANSIT OP VENUS 147 The corrections which should be applied to the times of the four con- tacts for determinate changes in the elements, exclusive of the effect of a change in the constant of solar parallax, are given by the following formulas. In these Sq =the correction of the sun's longitude, dL=the correction of the orbit longitude of Venus, 5g2=the correction of the longitude of the node of Venus, SB=the correction of the sun's latitude, 5s=the correction of the semi-diameter of Venus at the mean distance, 3s'=the correction of the semi-diameter of the sun at the mean distance. All these quantities being expressed in seconds of arc, the corrections of the times of the four contacts, in their order, are 8 S B S dT,= + 48.4 (5© - dL) + 3.00 {$L -dQ,+ 16.2SB') - 97.4 Ss - 26.1 ds', ST, = + 50.9 (5© - SL)+ 3.94 (5i -dQ,+ U.9SB) + 116.3 ds - 31.3 Ss', ST, = + 30.3 (SQ-SL)- 4.68 {8L -8Q,+ 16.955) - 116.3 Ss + 31.3 Ss', ST^ - + 30.1 (5© - SL)— 3.75 {dZ -d5i+ W.9SB) + 97.4 Ss + 36.1 Ss'. These expressions have been computed for the center of the earth, but they may be taken as approximately exact for any point on the surface. An approximate value of the co-efficient of the correction of the con- stant of solar parallax, for any place, may be found by subtracting from the ascertained Washington mean time of contact at the place, the Washington mean time of the same contact occurring in the zenith, given on page 128. Thus in the example for Honolulu, given above, one finds that dT,= (8''58""26'.3 — 9''9"'.530)^, = — 665=.0— % To where Tto denotes the constant of solar parallax. It must be understood, however, that this method gives quite rude approximations. POSITION OF THE PLANET ON THE SUN S DISC. All that precedes relates to the contacts; but it may be desired to find the position of the planet, when on the sun's disc, relative to the center of this body. For this purpose the following tables of data are appended. 148 COLLECTED MATHEMATICAL WORKS OF G. W. HILL Waeli. M. 1 h m 8 30 D. Change of x in 1 minute. y Change of y in 1 minute. V- d +32.7995 —0.16251 +26.3257 +0 .04309 129°29'.7 — 22°52'.5 40 31.1744 16252 26.7565 4306 131 59.4 52.6 50 29.5492 16252 27.1870 4304 134 29.0 52.7 9 27.9239 16253 27.6173 4301 136 58.7 52.8 10 26.2985 16254 28.0473 4298 139 28.4 52.9 20 24.6730 16255 28.4770 4296 141 58.1 53.0 30 23.0474 16256 28.9065 4293 144 27.8 53.1 40 21.4217 16257 29.3357 4291 146 57.5 53.2 50 19.7960 16258 29.7647 4288 149 27.1 53.3 10 18.1702 16259 30.1934 4286 151 56.8 53.4 10 16.5443 16260 30.6219 4283 154 26.5 53.5 20 14.9183 16260 31.0501 4281 156 56.2 53.7 30 13.2922 16261 31.4780 4278 159 25.9 53.8 40 11.6660 16262 31.9057 4275 161 55.6 53.9 50 10.0397 16263 32.3331 4273 164 25.3 54.0 11 8.4134 16264 32.7602 4270 166 55.0 54.1 10 6.7870 16265 33.1871 4267 169 24.7 54.2 20 5.1605 16266 33.6137 4265 171 54.3 54.3 30 3.5338 16267 34.0401 4262 174 24.0 54.4 40 1.9071 16268 34.4662 4259 176 53.7 54.5 50 + 0.2803 16268 34.8920 4257 179 23.4 54.6 12 — 1.3465 16269 35.3176 4254 181 53.1 54.7 10 2.9734 16270 35.7429 4252 184 22.8 54.8 20 4.6004 16271 36.1680 4249 186 52.5 54.9 30 6.2276 16272 36.5928 4247 189 22.2 55.0 40 7.8549 16273 37.0173 4244 191 51.9 55.1 50 9.4822 16274 37.4416 4242 194 21.6 55.2 13 11.1096 16275 37.8656 4239 196 51.3 55.3 10 12.7371 16276 38.2894 4236 199 21.0 55.4 20 14.3647 16276 38.7129 4234 201 50.6 55.5 13 30 —15.9924 —0.16277 +39.1361 +0. 04231 204 20.3 —22 55.6 The distance B in seconds of arc of the center of Venus from the center of the sun, and the angle of position Q, of this distance, counted from the north point toward the east, are obtained by the formulas, J sin § = « — p cos 55' sin * , id cos Q = y — ;o sin = + 13° 4'.3, w = 203° 43'.6, whence for this place AsinQ =x— [9.9886] sin ■» , AcosQ = y — 0.2070 - [9.5787] cos ^ , . ^_ [8.0589JsinO» — 34°1') ^■^ 1 + [8.3896] cos (^ — 5° 50') " Assume ll'' 4". 6 as an approximate value of the time; for which M= 168°3'.8, 1? = - 34° 38'.8 , a; = + 7.6653, y-+ 33.9566 , JsmQ-+ 8.2190 , Aco8Q= + 33.4378 , Q= 14°13'5".6, ^=-36'0".6, Q — E = 14° 49' 6".2 . 150 COLLECTED MATHEMATICAL WORKS OF G. W. HILL The error is, then, — 6' 22". 9, and the correction to the assumed time, — 383".9 1025" X 1"°=: + 0^3?'34. If the computation be repeated for the time ll*" 4™.9734, the error of the value of Q — E will be found to be only 13". Regarding this result as suflBciently accurate, we compute, for this time, Q and D, and find Q = 14° 6' 33", D = 819".42 = 13' 39".42 . These distances and angles of position are, it must be remembered, actual, not apparent. To obtain the last, the effect of refraction would have to be considered. LOCALITIES FAVORABLE FOR THE DETERMINATION OF PARALLAX. A list of localities favorably situated for observations of the contacts, with a view to the determination of the parallax, may be given in a few words. For the ingress accelerated by parallax, we have, in the first place, the Hawaiian Islands; next, the most southerly and westerly of the Aleutian Islands, the southern part of Kamchatka, and Japan, especially the northern islands; also the Marquesas Islands, and, if more stations are desired, per- haps in the long series of islands stretching west-northwest from the Ha- waiian Islands some might be found available. We may mention the small islands lying between the Hawaiian and Marquesas Islands. For the ingress retarded by parallax, we have the islands of Saint Paul, New Amsterdam, Kerguelen, Bourbon, Mauritius, Diego Rodriguez, Crozet, Prince Edward, and Madagascar, where, however, only the interior contact will be visible, and on the eastern coast at an altitude from 5° to 6°. For the egress accelerated by parallax, we have New Zealand and the small islands to the southward and eastward. With respect to the latter, we may note that on some maps may be found a group of small islands, called the Nimrod Islands, and placed in longitude 80° west from Washing- ton and in latitude 57° south. Here the interior contact occurs at an alti- tude of 9°, and if these islands are of a suflBcient size for the establishment of an observing station on them, it would be a tolerably good one, as far as geographical position is concerned. To these we may add Norfolk Island, New Caledonia, the Fiji Islands, Van Diemen's Land, and the southeastern part of Australia. For the egress retarded by parallax, Southwestern Siberia, the region immediately east of the Caspian Sea, Persia, the Caucasus, Asia Minor, Syria, Arabia, and Egypt contain the best stations. CARNEGIE INSTITUTION OF WASHINGTON CHART No. TRANSIT OF \a^:NUS. DEC. 8.1871 CtlARTXO.l. INGRESS, EXTKJ^rOR CONTACT Scale for filtitudeH is° 2o° 25" :w LEG END The broken lines in blue are for synchronism of runiaci. The broken lines in red ai-e for contact at Uie same point oftlie sular disU. CARNEGIE INSTITUTION OF WASHINGTON CHART NO. 2. TRANSIT OF VKXl^S, DEC. 8. 1874 CHART NO. 2. INGKESS, INTERIOR COXTACT Scale for altitudes 5" 10° 1G° 20° 2S° LEGEN D The broken Lines in blue are for syuchronismof tonlact. The broken lines in red Qi-e for contiujt altlie some poinl of the solijr disk. CARNEGIE INSTITUTION OF WASHINGTON CHART NO. 3. TRANSIT OF VKX US, DEC. 8. 1874 CHART XO. 3. TiGRESS.rNTERlOR COXTACT Scale for altitudes B* iO" 15° LEGE N D The broken lines in blue fire for synchronism of euntat't. TKo broken, lines in red are for contact at tlie same poini of lUe soloi- disU. A METHOD OF COMPUTING ABSOLUTE PERTURBATIONS 151 MEMOIE No. 14. A Method of Computing Absolute Perturbations. (ABtronomisohe Nachrlchten, Vol. 83, pp. 309-324, 1874.) The object of this article is to call the attention of astronomers to the notable abbreviations which are produced in some parts of the formulas for perturbations by the introduction of the true anomaly as the variable according to which the integrations are to be executed. Prof. Hansen, in his later disquisitions, has substituted the eccentric anomaly as the indepen- dent variable in place of the mean anomaly, or what is the same thing, the time; and he regards this step as constituting a remarkable amelioration of the method. The method here explained will, as far as coordinates are con- cerned, be the same as that of Laplace, but the same use will be made of the true anomaly in the elliptic orbit, as independent variable, as that which Hansen has made of the eccentric anomaly. The following notation and equations are so familiar that they seem to need no explanation: E = m'[^-^^)+m"[^-^-^) + cPx II _ dR dt- d''z fi (1) Let us now suppose that each coordinate of the disturbed planet is sep- arated into two parts, such that x = x, + Sx, y — yo + Z =:Zo + Sz, the first of which, Xq, y^, Zq satisfy the differential equations where rl=^XQ + yl +zl, and the second, 8x, 8y, Sz are of the order of the disturbing forces. 152 COLLECTED MATHEMATICAL WORKS OP G. W. HILL It is evident that this separation is, to a certain extent, arbitrary, as certain functions of t might be added to Xq, «/o, Zq without their ceasing to satisfy the differential equations determining them, and then hx, hy , hz would necessarily be diminished by the same functions. This indetermina- tion is eliminated in different ways according to the circumstances attending the computation of the perturbations. If ccq, ^oi 2o are derived from the elements osculating for a certain epoch, it is plain that hx, 8y, 3z ought to vanish at this epoch, as also their first dif- ferentials with respect to the time. This will be accomplished by taking all the integrations, which 8x, Sy, Sz involve, between the limits ^ = and t^t. If the perturbations are computed from so called mean elements, the six arbitrary constants which 8x, 8y, Sz involve, must be determined in accordance with the suppositions upon which the mean elements have been derived. We will now write r = r„ + dr, or ox ^ oy 9z dR is then the differential of jB when the coordinates of the disturbed planet alone vary. The last equation is evidently correct, when, in the first mem- ber, we suppose B to be expressed in terms of r and two other coordinates which make — , — , — independent of r. r r r By multiplying the equations which determine x, y, z, severally by 2dx, 2dy, 2dz, adding the products and integrating, df raj' ^ ' where ^ is the constant added to complete the integral, and we suppose that it is such that the equation dx\ + dy\ + dzl ^+ A = W r„ a is satisfied; if there is any residual constant part, it must be supposed con- tained in the term 2 / dR. By multiplying the differential equations deter- A METHOD OP COMPUTING ABSOLUTE PERTURBATIONS 153 mining x, y,z, severally by these quantities and adding the products to equation (2), we get dr r a J Qr By using the equation r =.ra + Sr, this can be readily transformed into tSrfil + lirJr = 2rdB + r^^-iiP-\'^. dv r, J dr dt' rlr In like manner equations (1) can be transformed into df rl dx \ri r'J d _ dR + dt ■ ri oz \r% r I For the sake of brevity put J dr df rir Then our differential equations take the form (3) d\r,br) df + -^roSr=Qr, dt' ^ rl Q'. (4) The problem of elliptic motion being supposed completely solved, -4- is a known function of t, and df ^ ii^-^' (5) is a linear differential equation. According to the theory of this class of differential equations, the value of q has the form q = K^q^ + K,q, , 20 154 COLLECTED MATHEMATICAL WORKS OF Gk W. HILL K^ and E^ being the arbitrary constants and q^ and q^ any particular solu- tions independent of each other. Then there must necessarily exist the two equations ^ + Aff =0 ^ + ^ J iia{l — e') cos i . Denoting by /I the longitude measured in the plane xy, so that tan ;i = -^ , and putting X ^^ dy ^dx-d^' we shall have Supposing ^=:2,o + 6l^, where tan ;\,„ = ^ , the following equation is obtained for the determination of S^ : Or, if V is made the independent variable, and for brevity we put j9 = a (1 — e") , the expressions for the perturbations are dr= — I Q^rl sin (v — v)dv , dz= ^ CQA sin (v — v)dv. (U) These formulas are absolutely rigorous, since no terms have been neglected, and also perfectly general, as no restriction has been put upon the position of the plane ofajyfrom which the coordinate z is measured. By adopting the plane of the elliptic orbit of the disturbed planet as the plane A METHOD OF COMPUTING ABSOLUTE PERTURBATIONS 159 of 032/, the ^*s^ equation is somewhat simplified. For then i = 0, and Zo= 0, and z = Sz; thus "=/[/i^*- '^"--ir~"' ]f^- w Perturbations of the first order with respect to the disturbing forces. Since, in this case, elliptic values are to be substituted for the coordi- nates in the functions Qr,Qg, Qk> there is no need any further of making a distinction between ro and r; hence the (o) will be omitted from the former. If we put and h^ is the latitude of the disturbed planet measured from the plane of its elliptic orbit, and S/l the perturbation of the longitude measured in this plane, our formulas, in this case, reduce to Sr= r TBin{v -v)dv, di3 = C Z sin (v — v)dv, SX= Tf C Ydv — 2—~\dv. Put now fip dr ' then it will easily be found that ±dB = r-'[tmix+v\. up L p J Thus the shape, in which we shall employ our equations, is dr= r[x+%r^ r r-'(?-^^X+ v) dv~] sin (v — v) dv , dX:^ rr r rdv — % ^~\ dv, [Z sin (v — vy] dv . /' The chief thing now to be done is to expand X, Y and Z in periodic series as functions of w. The elliptic values of the coordinates of the dis- turbed planet are readily expressed in terms of this variable, but the coor- dinates of the disturbing bodies will naturally be expressed in terms of their mean anomalies t,', ^". etc. These last variables must be eliminated by means of the identities n n n n n n 160 COLLECTED MATHEMATICAL WORKS OF G. W. HILL Let us then put so that n n n n M' n' diV n" . -5- = — , -5- = — , etc. av n dv n Then ^', f", etc., will be replaced by the following values : r = *'-i^(i;-C), :"=.*"— ^(w-O, etc. In the development of X, Y and Z in periodic series from particular values of these quantities, it will be better to make the differences of S'', ^", etc., from v, the variables to be employed. Thus we shall put w' = ^' — v, w"^^'i — v, etc. The formulas, to be written now, will be confined to the case of the action of one planet. The expressions for X, Y and Z are ^=^^*[i-^3]/co«^'cos(.-.)-^^, where h^ = iia{l — e'), and A' = r' + r" — 2rr' cos /J' cos (/ — A). If the inclinations of the orbits of the two planets to some fixed plane, as the ecliptic, are denoted by i, i', and the longitudes of their ascending nodes by Q,, Q,', and the longitudes of their perihelia by 7t, 7^, we compute /, 0, 0', n and n' from cos I = cos i cos i' + sin i sin i' cos (^2' — Q,) , sin / cos (^ — Q) = — sin i cos i' + cos * sin i' cos (Q' — Q) , sin / sin (6» - Q,) = sin i' sin (Q; - Q) sin 7 cos (8' — Q') = cos i sin i' — sin i cos i' cos (Q,' — Q) , sin / sin {ff — Q,') = sin i sin {Q' — Q,) , n = 7: -9, n' =n' -8' . The circumference being divided into A; equal parts with reference tow, pute for each of following quantities : compute for each of the h values oi v, 0,-=-7t, ^n .... — L^ 1 ji ^ the A METHOD OP COMPUTING ABSOLUTE PERTURBATIONS Jgl tan u -Vl + .t^" V C =:M — e sinu, V = V- n' . P ■0, 1 + e cos V ' Kcos(n'—J)= cos(v + Z7), ^'cos(/7'-^') = cos/cos(«+ //), ^8iii(/7'-^) = cos/sm(i; + /y), K' sin (^n' - A') = sm(v + n), a = 2Kr, G' = ^r'E, G" = ^r', G"'=~r'K', G"" = "^ sin I . r\ Several of these quantities, as «, ^, F, r, will need to be computed only k . . . . Jc ■ ■ — times, if ^ is a multiple of 2 ; and K, K', A , A! only — times in the same case. The circumference being divided into h' equal parts with reference to the variable w', compute for each of the hh' values of v and w' the following 2 4 2 (Id 1^ quantities, w' taking in succession the values , -^ 7t, -=j Tt .... '■ .^ — 'n: Z' — Vi- w', u'- e' sin u' = Z', Vr' cos-^ = Va'(l — e') cos-s-, V*"' sin-s-= V' (1 + «') sin-s-. If we have tables of the disturbing planet giving the true anomaly or the equation of the center and the radius vector or its logarithm with the argument mean anomaly, we can derive log r' and v' by means of their aid, and thus dispense with computing the last three equations. We now com- pute kh' times A' — r'+ r"— Gr' cos (v'+ A) , X= G' [4r - ^] r' cos ^v'+A)-^, Y^ G'" [^ - ^t] r' sin (v' + A') , Z= G"" [^ - ^] r' sin {v' + 77') . From these M' special values of each of the quantities X, Fand Z, we deduce their developments in periodic series of the form ^u, \K\^ cos {iv - i'w') + K\tl sin {iv - i'lu'y] . 21 162 COLLECTED MATHEMATICAL WORKS OP G. W. HILL This process is so well known that we need not here insert the formulas required for it ; they will be found in Hansen's Auseinandersetzung, Part I, p. 159. A double application of these formulas will be necessary, the first relative to v, the second relative to w'. After these series are obtained, w' can be replaced by S'' — v. The series X is now to be multiplied by — sin v, which, for every periodic term in X, will give two periodic terms, which will be added to T. This result is next to be multiplied by /I + e cos vV l + U' 2e e' = r 1- —2- COS V + -^r-r COS 2v . \ P / p^ p Ip^ There is now an integration to be effected. A table of logarithms of the integrating factors 1 li, i'-\ = ■ n will now be made for all combinations of i and i' which occur in the periodic series. If the last result contains a term sin "^ ^^ the corresponding term of the integrated result will be A multiplication by 2r^ is now to be made. We have -^ =^Eo — Eicoav + E^ cos 2v — E^co&Sv + . . . , where the rigorous value of the coefficients is given by the equation This multiplication accomplished, the product is to be added to X. If this result has a term ^sTn(*"^-^''»')' then 8r has its corresponding term - [t- 1, t'][t + 1, i'] Kl?^ (iv-W) , A METHOD OF COMPUTING ABSOLUTE PERTURBATIONS jgs except in the case where i = 1, i' = 0, when we have, instead of this, 2 cos Having thus obtained Sr, we multiply it by 1 1 e — — 1 — cos V . r P p The result, which is the perturbation of the natural logarithm of r, must be doubled and then subtracted from / Tdv. Another integration being executed on this result, we have h'k the perturbation of the longitude measured in the plane of the fixed elliptic orbit. Finally, ^^ will be obtained by treating Z to the same kind of integra- tion as that last used in obtaining hr ; that is, in general, each coefficient of Z will be multiplied by the proper value of — \i — 1, i'][i + 1, i'] which corresponds to it. Perturbations of the second order with respect to the disturbing forces. Calling the parts of the perturbations of r, ^, ^, which are of two dimensions with respect to the planetary masses, 5V, 8^(3, 6^^, so that we have, with errors of the third order, where hr, 5/3, h7u are the perturbations which have just been determined, we shall have dv , Sr S'r = J -^ SQ, sin (v - v) 8'^ = J" ^SQ, sin (v-v) pr P r' 3V /drV Sr d.SX~\ dv. where, as before, there is no need of any distinction between rg and r. The following are the expressions for 6Qr and 8Q^, 164 COLLECTED MATHEMATICAL WORKS OF G. W. HILL Bearing in mind that X, Y, Z are homogeneous functions of r and r', it will be easy to deduce the following equations : r' ( dR\ I dX ^^\Sr I dX „^\5r' (5A _ SX') + r(r 1^- 3^) 5/? + |^5/S'. ,(|?U^.F (4f-^)^-(^i?-^) ?-!?(---) - w^^ - w^^'' , 5r -a „ »•' r^*K. o,r ,rJ^ »• ^d.dX ^d.S^ e . r' J dR\ r-^-l, n Sr where the differential coefficients — L, — ^ — , ' " are complete with dv dv dv respect to the independent variable v. In computing the values of these functions, — , 5^' and ^(3' must be expressed as functions of «. Hence, if they are at first expressed in terms oft, it must be eliminated by means of the equation nt + c = v — Bi sinv + ^E, sin 2v — ^^3 sin3t> + . . ., where the rigorous value of ^j is We may have given only the perturbation of the orbit longitude and the latitude above the elliptic orbit of the disturbing planet ; in this case, calling the latter Sri', the values of 6X and 5/3' will be given by the equations ^^, cos J , , ain I COS (v' + n') ^ , ' cos/S ' cos /J We see that, in order to obtain the perturbations of the second order it will be necessary to have, expressed in periodic series in terms of v, the following nine quantities : dX dZdZdXdYdZdXdYdZ ^ dr ' '^dr' 3/J' dX ' dX ' QX > dfi" d[i" W A METHOD OF COMPUTING ABSOLUTE PERTURBATIONS 155 For six of these whose expressions are *■ dr dZ m' ^/3r''sin»/3' 1 [ 4J= 2/)" +4^/' „_ 3 m' //' — r' 1\ , . ^, dl3~ h' ' \ A dX m! ^f3r' — r" 1 1 8/9'~ A' ^\2 — r'^ 11 1 \ ji — + 2^+ r^j'" sm/3'cos(A' — ;), 9F m' ,/3r» + /^ 11 15\ , . . . ^„ ,^ 9Z m' r/1 1\ , „, 3rr''sin'/3'cosa'— /I)"! 9^ = IF ^ Ll^~^ J "■ ''°' '^ ^^ ^^ J ' the same method must be used as that which has been given for X, Y, Z. The remaining three, X, Y and Z being considered as functions of the two variables v and S'', can be obtained from the equations 9X_9X w^/ r' \gX e_ . ,aX aA - Sv + w V~a'^'T^^I a*' i? sin?;, r ^^ , 8F_aF «;_/ r- ^ar e_ . ,dY a^_az jj[^/ r' \dz _e_ . a^ 3^ -at; + « V aV"T^^V9*'~i'^''''' S*"* The factor where 1 „ ,_ = E, cos V — E, cos 2v + E, cos 3v - a V 1 — « Moreover, we have the relation ,a^_3^ o„Tr The factor r is given by the equation -— = \E^ — EiCosv + Ei cos 2i; — E^ cos 3« + where 166 COLLECTED MATHEMATICAL WORKS OP G. W. HILL The values of ^' — ^ and /?', necessary for the computation of the first six quantities, can be obtained from the equations cos /3' cos (A' — X) = K cos (i;' + A) , cos/3' sin (>(' — X) = K' sin (v' + A'^ , sin /9' = sin Iam(v'+ 11'). The terms to be integrated in the second approximation have the gen- eral form (iG+G'v)f^^Civ-i'&'-i"n. If these terms are integrated with respect to v, we have T Li, i', i"](C + G'v) ^?f {iv — i'»' — i"»") + [i, i', i"T G' ^|f„ {iv — i'»' — i"»") , SlU COS where [i, l', *"] = - —7 — TT - I l' 1 n n If they are integrated after having been multiplied by the factor sin (w — «) , the result is — \i—\, i', i"Ji + 1, i', i"XG + G'v) ^"^ (iv — i'»' — i"»") COS except in the case where i z= 1 , i' = , i" =: , when we shall have ±(lG'-^Gv-l G'v') 2 v + i(G + G'v) f^^ v . The labor of computing perturbations of the second order is, in some sort, measured by the number of multiplications to be made of two periodic series, each involving double arguments. In this method, in the case of one disturbing planet, there are 22, or 25 if one thinks that the multiplications involving 8X' ought to be considered as distinct from those involving 5/1 . If all the terms involving sin /as a factor be neglected, the number of these multiplications is diminished by 1 2. It is my intention to illustrate this method by applying it to the com- putation of the perturbations of the first order of Ceres by Jupiter. LONG PERIOD INEQUALITY IN THE MOTION OF HBSTIA 157 MEMOIE No. 15. On a Long Period Inequality in the Motion of Hestia Arising from the Action of the Earth. (Astronomische Nachrichten, Vol. LXXXIV, pp. 41-44, 1874.) While the attention of all is directed to the more exact determination of the constant of solar parallax from the approaching transit of Venus, it may be of interest to notice another source from which, at least in the future, can be obtained the value of this constant. Several of the asteroids have periods of revolution approximating quite closely to four years ; hence, in their longitudes are long period equations of the form Ic sin [45* — g' ^-K'\, g and g' being the mean anomalies of the asteroid and the earth. Should It, be quite large, after the inequality has run through a considerable portion of its period, we can, from this source, determine a pretty exact value of the earth's mass, and thence, by the known formula, the corresponding value of the constant of solar parallax. In order to see what may be expected in this direction, I have com- puted this inequality, as far as the first power of the disturbing force is con- cerned, for Hestia. This asteroid has been selected on account of its large eccentricity and the near approach of its period to four years. The ele- ments employed (as many as we have need of), from the Berlmer Jahrbuch for 1875, and from Leverrier's Annales de V Ohservatoire, Tome IV, are as follows : Hestia. The Baeth. Osculating, 1865, July 26. Mean Elements for the same epoch. ■K = 354° 14' 18".7 95 = 9 36 55.8 gi = 181 30 35.3 i— 2 17 30.0 fi = 883".56391 log a = 0.4035124 M. B. 1870.0 7:' = 100° 41' 35".0 p' = 57 38.1 / = 3548".19386 *" — 422800 These elements give fi' — 4(1 = 13". 937 22, whence the period of the in- equality, in this case, is 254.6 years. 168 COLLECTED MATHEMATICAL WORKS OF G. W. HILL By a quite rigorous process, similar to that employed in Hansen's Aus- einandersetzung, the terms of — depending on the argument Ag — g' have been found to be — 0.001 74933 cos (4^ — g') + 0.011041 88 sin (4^ - g') . And, in like manner, the second part of the disturbing function ^ cos i^ contains the terms + 0.00257586 cos (ig—g') — 0.00872291 sin (ig —g') . Thus aB contains the terms + 0.00082663 cos {ig—g') + 0.00331897 sin ij^g—g') . Multiplying these by the factor ■^^''"'"-U X 306364".8 , (4/.-/^')" we have the inequality sought, ^ndt = 75".869 sin {^g—g'+ 109° 37' 10") , /» The eflfect of this inequality on the geocentric position of Hestia at opposition is got, somewhat roughly, by multiplying the preceding expres- sion by , and hence, at a maximum, may amount to about 125". It must be confessed that the determination of the earth's mass from this source is attended with the inconvenience of having to compute very accurately the perturbations of Hestia by Jupiter; and among these is a very large inequality having the argument g — Zg", whose period is nearly the same as that of the inequality just determined. Hence it will be neces- sary to proceed with a very accurate value of Jupiter's mass obtained from other sources. It will be noticed from the expressions given above that the portions of the inequality, contained in the two parts of the disturbing function, have a strong tendency to cancel each other. This is always the case where either one of the mean anomalies is involved in the argument only to the simple multiple. This tendency does not occur in the inequalities having arguments of the form Ig — 2g', and perhaps quite large coeflBcients might be obtained for these in some of the asteroids whose periods approach 3^ years, especially if their eccentricities are large. Melpomene would seem to afford the best chance, and the period of the inequality would have the recommendation of being much shorter than that of the one here computed, namely about 80 years. PROBLEM IN THE THEORY OP NUMBERS 169 MEMOIE N"o. 16. Solution of a Problem in the Theory of Numbers. (The Analyst, Vol. I, pp. 37-38, 1874.) The following problem appeared in the Mathematical MontJdy, Vol, I, p. 29, and no solution was published in that periodical : " Show that the product of six entire consecutive numbers cannot be the square of a commensurable number." Since the square root of every integer, not an exact square, is a surd, it will be sufficient to show that the product cannot be the square of an inte- ger. Let the six numbers be denoted, n being an odd integer, by n — 5 n — 3 n — 1 w + 1 w + 3 w + 5 Then it is required to prove the impossibility of —^ — . — -— . — - — = n. Let us put — - — := X, where x is integral since it is the product of two integers. Then it will suffice to prove the impossibility of a; (a; + 2)(a; — 4)=n. Let us suppose x =■ l_! = r sin (;^o — v) + R-^ sin L_i, ^0 cos X„ = r cos Xo + -^0 cos A , /!„ sin A„ —r sin /„ + R„ sin i, , J, cos X^ = r cos {xa+ v) + Ri cos Li , Ai sin X-i zzzr sin (/« + i?) + R\ sin L^ . These equations contain the six unknowns A_i, Aq, A], r, %, >?. If we eliminate A_i, Aq, Aj from them, we shall have the three equations of the first solution. But by retaining Aq as the unknown, we shall arrive at an elegant solution. Let us first eliminate A_ j and Aj ; this we do by putting P = ;i,_ 1 for the first two equations, and P = ;ii for the last two. The equa- tions for determining the four remaining unknowns, are = r sin (;^o — 5; — A_,) + R_, sin (i_i — A_,) , Jo cos A„ = r cos Xt + Ri cos i„ , J„ sin Ao = r sin/„ +i2„sinZo, 0=:»-sin(/o+ 1? — Ai) + i?i sin (A — ■*!) . If, in the second and third of these equations we put successively P = >7 -f ;i_i and P = — )7 + ;ii, we get A SECOND SOLUTION OP THE PROBLEM OF NO. 8. jyj AsinC^o — v — -i_i) = j-sin(;ifo— 5?— -»_i) + i?„ sin (i, — ij — >l_ i) , ^0 sin (>lo + 1? — -^i) = »• sin (/„ + ly — ;,) + i?„ sin (Z„ + 5; — Ai) . If, from these equations we subtract the first and last of the preceding four, we get /)„ sin(A„ — r, — A_,) = ^0 sin (£, — ■>] — X_^) + E_, sin (Z_ 1 — /i_ , Jo sin (/i„ + t; — Ai) = R, sin (/,„ + ^ — X,) — ij^ sin (A — X,) . Two equations with two unknowns are thus arrived at without complicating the form of the original equations. It is very easy to eliminate Aq from these, and we get [^„ sin (i„ - , — A_ ,) — i?_ , ain (Z_ 1 — A_ 1)] sin (/l„ + ^ — X^) — [i?„ sin (i„ + )j - Xi) — El sin (A — ■*!)] sin (-)» — )j — X_ ,) . But we prefer to keep Ao as our final unknown. Let us put for the sake of brevity ^-1 = ^-1— A_i, 4\ = Li — K- All these are known quantities with the exception of cr, which will take the place of >7 as an unknown. Our two equations can now be written A sin (d — s) = Ba sin (5' — a) + B_i sin (p_ ^ , /(„ sin (3 + (t) = E„ sin (d' + a) + E^ sin ^1. Or, by taking in succession half the sum and half the difierence /)„ sin5 cos -...). 184 COLLECTED MATHEMATICAL WORKS OF G. W. HILL The expressions given for D^" and i)^" + ■* are equally applicable when n is negative ; they then give the formulas to be used in mechanical quadra- tures, thus : If these expressions are expanded in powers of A, we obtain n-i_J^-i ±a,1La^ 191 2497 14797 ,, ■^ -'*r 12 TOO ~ 60480"^ + 3628800^ ~95800320'^ 92427157 ,„ \ 4- , 3615348736000 --J' ~ '* r 13 240 + 60480 3628800 + 22809600 " " j" These are the expressions to be used in computing the values of the inte- grals / ydx and / / ydaf. It must be noticed that A~^ virtually contains an arbitrary constant G, and A~^ an arbitrary expression Cx + C In fact, the quantities in the columns to the left of that of the function y can- not be written until we know one quantity in each column. These constants G and G' are usually determined from the given values of / ydx and / / yd'^^ for x = a. If we denote them by D^'^ and D^^, and if, in gene- ral, the subscript („) denote values which obtain when a; = a, it will be seen that ^0 - h + 12 ^« 720 '^» + • ■ ■ ' D~' 1 1 ~ h' 12 " + 240 " Having thus the sum and difference of the quantities A~^y_^ and A~^3/j, it will be easy to get the quantities themselves. The preceding formulas give the values of the integrals for the series of values of cc, .. a — h, a, a + h^ . . It is generally preferable to compute them for the values, .... a — ^h, a-\-\h, a -\- ^h, .... Formu- las for this purpose can be obtained by the simple consideration, that in the scheme, given at the beginning of this article, it is allowable to treat the odd orders of differences as if they were even, and the even as if they were odd. OAliCULUS OF FINITE DIFFERENCES jgs In this way all the quantities obtained will correspond to the middle of the intervals of the former supposition. Thus, calling D"^ and D~^ in this case Dj ^ and D^ ^, it is evident we must have = - Vi + i^nj Vi + i^V ' or, expanded in powers of A , iJi -n\^^ +24^~5760 +967680 464486400'^ + •• 7' ^4 -A ^^ 24+1920 193536^+66355200'^ • The diflferences of the first formula, although they are of odd orders, are to be taken as equivalent to the simple numbers standing in the original scheme, while the diiFerences of the second, although of even orders, are all the averages of two adjacent numbers of the same scheme. It is plain we have d .Df dA In using the method of mechanical quadratures, it is usual to multiply the values of y by A , if the single integral only is wanted, but by h^ if the double is also to be obtained ; in the last case then it is necessary to divide the results obtained by Ji in order to have the single integral. These formulas appear to have been first obtained by Gauss ( WerTce, Vol. Ill, p. 328). Bncke has given them in the Berlin Jahrhuch for 1838. For use they are much superior to the formula given by Laplace {Mecanique Celeste, Vol. IV, p. 207). 186 COLLECTED MATHEMATICAL WORKS OF G. W. HILL MEMOIR No. 20. Elementary Treatment of the Problem of Two Bodies. (The Analyst, Vol. I, pp. 165-170, 1874.) The deduction of the motion of the planets, in accordance with the laws of Kepler, from the principle of universal gravitation, is important, not only on account of the extensive role this theory plays in Astronomy, but also for its interest, in a historical point of view, as Newton's principal discovery. Hence it is desirable that the demonstration should be made as elementary and as brief as possible, in order that it may be brought within the compre- hension of the largest number of persons. The polar equation of the conic section, referred to a focus as pole r = . =^ — 1 + e cos (A — o) ' is well known ; a denotes half the greater axis, e the eccentricity and o the angle made by the axis with the line from which ^ is measured. It will be advantageous to replace a(l — e^) by ^, p being the semi-parameter, also to put o = e cos 10 , /9 = e sin (u . Thus the equation becomes r + ar cos A + /?r sin X =p . Hence it is plain that the equation, in terms of rectangular coordinates, the origin being at a focus, but the axes of coordinates having any direction we please, is ^x'+f+ax + ^y=p. (1) "We take for granted the following theorems, since they are demon- strated in the most elementary treatises on mechanics : In determining the relative motion of one body about another, it suf- fices to regard the latter as fixed, and to attribute to it a mass equal to the sum of the masses, and then to suppose the moving body without mass. When a body describes a plane curve, and the radius vector, drawn from a fixed point in the plane of the curve, passes over equal areas in PKOBI/EM OF TWO BODIES 187 equal times (which we shall express by saying that the areolar velocity about the fixed point is constant), the force acts always in the direction of the radius ; and the converse. Now let a body describe a conic section about another occupying a focus, the areolar velocity about this focus being constant ; it is required to deter- mine the force acting. In the figure, let PP"T be an arc of the conic section so described, S being the focus. Let P and P" be any two points on the curve at an indeterminate but small distance from each other. Draw 8P, and PP a tangent at P, P"P' parallel to, and' P'A and P"B per- pendicular to SP- Let SP be taken as the axis of cc, and SY perpendicular to it, as the b ^ axis of y. The coordinates of P are then 33 = SP = ro, 2/ = ; substituting these in the equation of the curve, we get (1 + a)r„=^. (2) Since the ordinate y can here be supposed always very small, the term Va;^ + y^ in (1) can be expanded, by the binomial theorem, in a series of ascending powers of y . Neglecting y^ and higher powers, we get or, as X differs from r^ only by a quantity of the order of y, by neglecting if Or, by (2), il! p-^y-i^ ' n 1 + a x = r„ 1 + a^~^ ^ p This is the value of a; from (1) expanded in a series of ascending powers of y, the cube and higher powers being omitted. The equation x==r„ 1 + a- belongs to a right line, which can be nothing else than the tangent PP. Hence it is plain, from the figure, that taking P"B = P'A = y, 188 COLLECTED MATHEMATICAL WORKS OP G. W. HILL tan PF'A ^ ^4- ' (3) 1 + a PA = 1 + a' P'P"^AB = \^, (4) the last equation being only approximate, but more and more nearly true as P"B or y becomes smaller. Let F denote the force acting on the moving body, and t the small inter- val of time in which the latter passes from P to P". Then we have P'P"-^i^=\Ft\ If we denote double the areolar velocity by h, since P"B = y is very small, we have SP.P"B = r,y = M. Eliminating t from these equations, we get Since there is no limit to the supposed smallness of ^ and t, this equa- tion is rigorously exact. The force is then inversely as the square of the radius-vector, and its intensity at the unit of distance is found simply by dividing the square of double the areolar velocity by the semi-parameter. It is evidently attractive except when, the motion being in a hyperbola, the focus, about which the areolar velocity is constant, is the exterior, in which case it is repulsive. Taking up the inverse problem, let a body start from P towards P' with a velocity ■«, which would carry it to the latter point in the time t, and let it be subjected to the action of a force varying inversely as the square of its distance from a second body supposed fixed sX S: it is required to find the curve described. Let the masses of the bodies, measured by the velocities they are able to communicate by their action, in the unit of time and at the unit of dis- tance, be denoted severally by m and M. The force acting at P is then M -\- m _M + m SP' - ~~P~' and, if at the end of the time t , the body is at P" instead of P', we must have p, p„ _ J M+ m PROBLEM OF TWO BODIES jgg But, as before, the constancy of the areolar velocity gives rx-=-}it. Whence This equation coincides with (4) if we suppose K ni V = M^- W Let now a conic section, having this value for its semi-parameter, be described with 8 as focus and touching PP' at P. That this is possible is evident from the general equation (1) ; here are only two unknowns, a and /3, to be determined, and they are given by equations (2) and (3), whence we see the solution is always unique. A body, moving upon this conic sec- tion, would have, at the point P, the same velocity, and the same direction of motion, and be subjected to the action of an equal force having the same law of variation, as the moving body in the problem. Hence, if the path of the latter is thoroughly determinate, and it would be absurd to suppose otherwise, the conic section just described must be the curve sought. We can easily find the elements of this conic section. Thus, let the angle P'P>S' be denoted by i^, then evidently, h^rv mi

I 1/ ai-LA. y M + m whence we derive „ rv^ sin' (h r^if sin' ^ 1 3 M + m '^ (^M+my a r M + m' Consequently the greater axis, and the species of conic section described, are independent of 4^. We have an ellipse, a parabola, or a hyperbola, according as v^ is less, equal to, or greater than 2 . From the last equation v^ = (M+m)[-^-^), (6) 190 COLLECTED MATHEMATICAL WORKS OF G. W. HILL which may evidently be taken as a general expression for the square of the velocity, if r denote the general radius vector. Also from (5), Thus, in different orbits, the areolar velocities are as the square roots of the parameters, and as the square roots of the sums of the masses. In an ellip- tic orbit, if T denote the time of revolution, the double of the area of the whole ellipse Whence i\/ M + m' Thus the theorem that, provided the sum of the masses remains the same, the squares of the periods in different orbits are as the cubes of the greater axes. The mean angular velocity is usually denoted by n ; thus 2;r /, 2-7: I M + m n = It is customary with astronomers to assume the earth's mean distance from the sun as the linear unit. If M and m are the masses severally of the sun and earth, and m/, a' and n' belonging to another planet are introduced, the mean distance of the last is given by the equation m' ^^ M To complete the subject, it is necessary to notice a particular case of the problem, viz., when 4' = . Here the motion is in a right line, and from (6) it appears the velocity is infinite when the body arrives at 8. As the existence of another body here ought not to be considered, at least in a mathematical sense, as an obstacle to its further motion, it is plain the body will pass beyond and move in the same right line until its velocity is reduced to zero, when it will return on its path, which will thus be a por- tion of a right line of which S is the middle point. This cannot be consid- ered as a degenerate form of a conic section of which S is the focus. For when an ellipse is varied by augmenting the eccentricity but maintaining the greater axis constant, at the point the first has attained the limit unity, the ellipse has degenerated into two equal portions of right lines overlapping PROBLEM OF TWO BODIES jgj each other and having their extremities on. one side in the point S. Hence this case must be regarded as a singular solution. However, most of the properties of motion can be deduced from those of elliptic motion. Thus, if the length of the whole path denoted by 4a, the duration of an oscillation will be n/ M + fn' Whence we gather that the time, in which a planet, at rest at its mean dis- tance, would fall to the sun, is found by dividing its periodic time by 4\/2. 192 COLLECTED MATHEMATICAL WORKS OF G. W. HILL MEMOIE ISTo. 31. The Differential Equations of Dynamics. (The Analyst, Vol. I, pp. 200-303, 1874.) The general formula of dynamics is In the usual treatment of this equation, we have been asked to attribute to the symbols 8x, Sy, Sz, . . . . the signification they have in the calculus of variations. This, however, is unnecessary, except when we wish to deduce from it the principle of least action ; and the student unacquainted with this calculus may regard these symbols as multipliers, which, when all the points of the system are free, have any finite values we please, but when the coordinates are restricted to satisfy an equation U= 0, are subject to the condition 9a; cy dm an equation which, for brevity, we shall write 8U= 0. We shall confine our attention to those cases in which the equations of condition and the accelerating forces are functions of the coordinates and the time only, and in which the latter are equivalent to the partial differen- tial coefficients of a single function li taken with respect to the coordinates whose acceleration they express. Whenever a function as U involves, in addition to x, y, z, .... their differential coefficients with respect to the time, quantities which we shall denote by x', y', z' ,...., we shall suppose that 5 U involves, besides the terms written above, the following 3a; 92/ oz Moreover, as we shall have to differentiate such functions as 5 Z7 with respect to t, we shall meet with such quantities as -j— , and shall suppose that the order of the symbols d and S may be inverted, that is, we shall have equations such as ddx ^dx , , ^=5^= to'. DIFFERENTIAL EQUATIONS OF DYNAMICS 193 The reader will see in this only a notational assumption, without quantita- tive significance, serving merely as machinery of demonstration. It will be noted that tis a variable not subject to the operation S. We have liXdx + Y8y +Zdz) = 8a , and for convenience may put Then it will readily be perceived that the general formula can be written thus The coordinates x, y, z, . . . . , can be expressed as functions of the time and certain variables qi, independent of each other and whose number is equal to that of the variables x, y, z, . . ■ ■ , diminished by the number of equa- tions of condition. Substituting for x, y, z, . . ■ ■ , their values in terms of the new variables qt, it is plain that the last equation will take the follow- ing form: We can find the value of pi without actually making the substitution, from this consideration j since the original equation contains only the varia- tions 8x, 8y, Bz, ■ ■ ■ ■ , without the variations 8 -^, S ~, S -^, ■ ■ ■ ■ , it fol- Ctt Cvt (tt lows that, in its transformed state, it should contain only the variations 8qt without the variations 8 -~ . at Then writing ql for -^, the coefficient of ^q^ should vanish in the equation That is, since 11 does not contain ql, dT dqi Thus the general formula becomes 23 194 COLLECTED MATHEMATICAL WORKS OF G. W. HILL Because in this equation the variables qi are independent, we may equate the coefficient of each hqi to zero. Thus dt • dq'i dqt This is Lagrange's canonical form of the differential equations of motion. A simpler form may be obtained by substituting the variables ^^ for qi. By adding to and subtracting from the general formula, the term S . 2^ {pi, q't) , and writing it becomes Equating the coefficients of each variation Sqi and 8pi to zero gives the equations dpt^__dH_ dq, _ dH dt ~ ~ dqt ' dt ~~ dPi ' which are known as Hamilton's canonical form. The expression for H can take a simpler shape. From the value of T, it is evident that a certain part of it is independent of the variables ql , which may be denoted by Tq, another part Tj, involves the first powers, and a third Tg involves the squares and products of the same ; then T= To + 2\ + T^. By the theory of homogeneous functions I.(p,q,') = S,(^^q/)=T,+ 2T,. Hence, if we write we shall have CUBIC AND BIQUADRATIC EQUATIONS 195 MBMOIK No. 22. On the Solution of Cubic and Biquadratic Equations. (The Analyst, Vol. II, pp. i-8, 1875.) In nearly all treatises on algebra, the solution of these equations is pre- sented as accomplished by the aid of analytical artifices, which one seems, by some happy chance, to have stumbled upon. No doubt the processes were found in this manner by the original discoverers, Tartaglia, Cardan and Ferrari. But, for many reasons, it would be better to treat the sub- ject as one demanding invention rather than artifice. The equations can, as it were, be interrogated and compelled to yield up their secrets, if they have any. To say that an equation is solvable algebraically, is to say that an alge- braic expression can be found equivalent to the general root, that is, one involving a finite number of the operations of addition, subtraction, multi- plication, division and the extraction of roots of prime degree. If the expression does not involve the last mentioned operation, it is called rational, and if free from the two last, integral. However complex an algebraic expression involving radicals may be,- it is evident that there must be at least one radical which is involved in it rationally. Supposing this to be denoted by i2", n being a prime inte- ger, it is not difficult to convince one's self that, by the proper reductions, the expression can be exhibited thus : p, + PiR' + p^B" + . . . +p„-iB " , where ^01 Pii ■■ • ■ > ^^ 'lot involve the radical B". With no loss of gene- rality, we can suppose pi = l ; for if p^ is not zero, we can multiply the quantity under the radical sign by pi, and then take (pi i^)" as the radical ; and in the contrary case, if p,c is one of the quantities p which is not zero ; the simplification can be accomplished by putting R^ = plR''. Then 1_ 2_ « — 1 Po+ Ii^ + p,R-+ . . . + p„--,R " may be regarded as the most general form of an algebraic expression. 196 COLLECTED MATHEMATICAL WORKS OP G. W. HILL Here may be enunciated a general proposition, which, although I am not aware that it has ever been proved, is doubtless true and may be used for purposes of discovery. If an algebraic expression exists, equivalent to the general root of the equation a;'"4- aa;'"~'4- bx'"-''+ . . . + g = 0, it can be exhibited in the form given above, n being one of the prime fac- tors of m. Thus the algebraic expression of the root of the general equation of the S"* degree, if it existed, could be presented in the form p,+ m + p^R^ + PiRi + p^R^ , and that of the 6'*' degree in either of the two forms p,-ir Ri + p^Rl , p, + Ri. Solution of Cubic Equations. According to the foregoing proposition, the root of the general cubic equation sf+ ax'+ bx + c = 0, if it has an algebraic expression, must be presented in the form x=p + Ri +p'Ri. But, since we suppose that this is an irreducible expression involving radi- cals, it follows that it must satisfy the given equation, whichever of its three values is attributed to the radical \/B. Thus, calling either of the imagi- nary cube roots of unity a, the three roots of the cubic equation must be Xi=p+ Ri + p'Ri, Xi=p + aRi + a'p'Ri, X,=p + a'Ri+a*p'Ri. The first method that suggests itself for obtaining equations which shall give the values of p, p' and B, is to substitute these expressions in the sym- metric functions which are equivalent to the several coefficients a, b, c, viz., Xi+ Xi+ x,= — a, XiXi+ XiXi+ x^x^ = b, x^x^x^ =z — c. But a simpler proceeding is to employ the three symmetric functions 2. a;, 2 . a' and 2 . x^. Since any cube root, &s \/ B is a root of a;^ — B:=0, in which the coefficients denoted above by a and b are each zero, it follows that the sum of the three cube roots of any quantity, as well as the sum of CUBIC AND BIQUADRATIC EQUATIONS 197 their squares, is zero. Now, it is plain that if the value of x is raised to the n^^ power, a? = A + ^i2i+ ORl, where A, B and Care free from the radical \/-B, and are consequently the same whichever of the three roots x denotes. Thus, since 2.'v'i2 = 0, 2.^J?^ = 0, we have 2:. a;" = 3^. Thus, for computing the value of 2 . a;", we need only the part A which is free from the radical \^Ii. In this way we obtain and equate to their known values in terms of the coefficients a,h^ c, 2 .X =3p = — a, S.x^ = Z{p^+ 2p'R) = a'- 2b , i:.x'' = 3(p'+ R + Qpp'R + p'^R") = — a'+ 3aS - 3c . These equations afford the values of^, j?' and R; from the first two a , „ a^— 35 P = 3", p'R= 9 . and by substitution of these values in the last, a quadratic equation in B] thus the general cubic admits solution by radicals. For the sake of brevity, putting . a"- 35 _ 2a'— 9a5 + 27c we have R = B±i^B'-A\ and, as we may take at our option either of the two roots, we have choice of the two expressions for x, x= - :^a + IB + /^ B'— A'y + A[Ji + 1^ B'- A'}-K x=-^a + iB — ii/B'-A'-]i+AlB — >/B'-A']-i. The three values of x are obtained by attributing in succession to the single cube root appearing in either of these expressions its three values. I do not know why almost all algebraists prefer to put the root in the form x = —^a + y\.B + ^B'—A'}+''^\_B — >^/^—A']. 198 COLLECTED MATHEMATICAL WORKS OF G. W. HILL It is certainly easier in practice to make a division than an extraction of a cube root ; moreover, we are troubled, in the latter form, with the selection of the proper three values out of the nine of which it is susceptible, a diflS- culty which does not occur in the two former expressions. Solution of Biquadratic Eqtiations. An algebraic expression for the root of the general equation of the fourth degree ai'+ aa^+ bx''+ ex + d = , if it exists, can be presented in the form P + V Q. And if this denotes one of the roots, another will be P — V Qj but since x has four values, it is plain that P and Q must receive each two values. This condition will be fulfilled if we suppose that these quantities, in their turn, similarly to x, are rational functions of a second radical \/B. Thus we put P=pJr\IB, Q^q + q's/B. Then we have x=p + i^R + i^q + q'/^R. The four values of x are obtained by giving in succession to the radicals \/ Q and V R all the values they are, in combination, susceptible of. Thus Xi=p + ^/E + \'q + q'i^R, X2=p — i^R + ^q — q'^R, X3 =p + ^ R — ij q + q' si R, Xi-=p — nj R— >J q — q' hj R. By substituting these in the four symmetric functions 2 . a; , 2 . x^ X .7? and 2 . a;*, equations will be found determining p, q, §' and R. Here again, in computing 2 . a;", the radicals all disappear ; for, whenever a radical is present with one sign in any root, there is always another root in which it is present with the opposite sign ; thus these expressions in pairs cancel each other. Then, in deriving 2 . a;", it is necessary to preserve only the terms which are free from radicals. In this way we get J. a; =4p =—a, S.0(? = 4:ip^+mq + R) + 3g'^] --a?+^ab-Zc, S.y*z=4: f (/ +q + Rf + (4/ + l^pq' + q") R + ^q (/ + i?)] = a*- ^a?i + iac + W- id . CUBIC AND BIQUADRATIC EQUATIONS 199 From which we derive a _ 3a'— 8& , _ a'— 4aJ + 8c i'=— X' 1-^^ = — 16 — > §'-«= 32 , PS 3a'- 85 ™ 3a*- ICa'S + 16ac + 165'- ^U _ /3a'- 4a5 + 8c\' „ ^ 16-^+- — ^BT ^-t 64 J = «- The last is a cubic in i2, which, by the foregoing, is solvable by radi- cals; hence the general equation of the fourth degree is so solvable. In forming the value of £c, we may attribute to R as its value any one of the three roots of this equation. When a = , the case usually treated, the equations are simpler, viz., ^ = 0, q^R=-\l, q'R = —\c, If we should attempt to treat the general equation of the fifth degree in the preceding manner, we would be led to equations of higher degrees than the fifth, which must be regarded as a strong argument for the non-existence of an algebraic expression equivalent to the root of the general equation of this degree. 200 COLLECTED MATHEMATICAL WORKS OF G. W. HILL MEMOIE No. 33. On the Equilibrium of a Bar Fixed at One End Half Way between Two Centers of Force. (The Analyst, Vol. II, pp. 57-59, 1875.) "A very small bar of matter is movable about one extremity which is fixed half way between two centers of force attracting inversely as the square of the distance ; if Z be the length of the bar, and 2a the distance between the centers of force, prove that there will be two positions of equi- librium for the bar, or four, according as the ratio of the absolute intensity of the more powerful force to that of the less powerful is or is not greater than (a -f 21) -^ (a — 2l) : and distinguish between the stable and unstable positions."* Solution. Assume the fixed extremity of the bar as the origin of coordinates and the direction of the line joining the two centers of force as that of the axis of a;. Then x and y being the coordinates of a material point of the bar, and X and Y the forces acting on it, we have from the well-known equa- tions for the motion of a rigid body S '^'^-y^''' dm ^SixY-ylT). If ilf and M' denote the intensities of the forces at the unit of distance, we have _ M{a — x) dm M' (a + x) dm ^- ^^a-xf+f-\i-i{a + xf+f]V _ M ydm M'ydm Introduce polar coordinates, and put a; = rco8^j y = r&va.O, and since the mass of the bar may be supposed evenly distributed along its length, put dm^=dr, and take the integration with respect to r between the * Cambridge Problems for 1845. EQUILIBRIUM OF A BAR BETWEEN TWO CENTERS OP FORCE 201 limits and I. These substitutions made in the equations of motion, we get V cfe . „ PT Mrdr M'rdr 3" ^ = « «'^ /*r Mrdr M'rdr ~| i/o L [«'— 3a?- cos e + r'li + [a' + %ar cos d + r'YA ' Or, the integration performed , Z d^__ Msine 3 df ~ [a - ; cos ^ + V (a'- 3a? cos + P) a/ ](a^ ^ 2al cos « + ?') M' sin , y = r cos d ainu), z = r ain0, and thus is . c cos ff cos o) — a sin 9 r — 2ac & cos' e — a^ sin' 6 The element of volume of the mountain may be regarded as a rectan- gular solid whose sides are cZr, r cos 0(^cj, rcZ0, and p being its density, the element of mass is p r^ cos ddrddda. Its attraction on the unit of mass at the station is p cos 6 drdd da. From the symmetry of the cone it is plain that the component of the mountain's attraction in the direction of the axis of y is zero ; and the vertical component which diminishes the intensity of gravity at the station may be neglected. The component in the direc- tion of the axis of x is -/// cos' cos at dr de d . Integrating with respect to r, the limits are r = and r =. the value given by the equation of the surface. Thus ^ /* /*c cos (? cos o — « sin (9 _ . , X=2acpJ J c'cos'g-a'sin'g cios^ cos o> d0 da, . Next we integrate with respect to o. As r must be always positive, the limiting values of a are the two roots of the equation c cos o = a tan $ . Hence 204 COLLECTED MATHEMATICAL WORKS OF G. W. HILL /.r ccos^gcos-^[-^taiig] sing cos g 1 V L c' coo' e - a' Bin' e ~ ^c'coa'd-a'Bm'ej^^- The limits of integration are now from 6 =^0 to 6= the value given by the equation a tan = c. The second term within the brackets is integrable, and between the limits is „ " . . To simplify the first term, revert to a^ + (^ "^ the variable a , that is, put a tan d = c cos a . Then r /*rr (odio a' "1 L*'° sin o 1 + -^ cos' w J The expression within the brackets is a function of — , calling it F f — j , we have Now p' being the mean density and B the radius of the earth, the force of gravity is and S the deflection of the plumb-line is given by the equation tan 5 = •— = \a J P c g — ^t: y R The definite integral /* ludm it appears, must be computed by mechanical quadratures. As an example in illustration, suppose a = 5 miles, c = 2 miles, R = 3956 miles, p = 2.75 and p' = 5.67. For evaluating the definite inte- gral, divide the interval between and -— into 9 equal parts ; then h = 10° = 0.1745241. Compute the value of the function to be integrated mul- tiplied by h for the middle of each of these parts, that is, for 6) = 5°, 15° 115°. The three values beyond 90° are for the sake of the differences. We get GRAVITY AT THE FOOT OF A CONICAL, MOUNTAIN 205 tii. Ao. w. Ao. Ci>. Aq. 5° 0.1400956 45° 0.1737216 85" 0.3594408 15 0.1432880 55 0.1893800 95 0.3899633 35 0.1497300 65 0.3094393 105 0.3358781 35 0.1595134 75 0.3337701 115 0.3705385 As the function integrated remains the same when the sign of o is changed, all the odd orders of differences vanish for the argument w = 0. Then making A"^ = 0, for the argument 0=0, by summing and differencing, we get for the argument a =: 90°, J-^ = 1.6563687, A^= + 0.0305234, J' = + 0.0015408 , A' ^ + 0.0007833 . Thus the value of the definite integral is 1.6563687 + ^ (0.0305224) - ^^fj (0.0015408) + ^^^(0.0007833) = 1.6576363. Consequently i^(0.4) = 0.7955673, and the deflection 5 = 19".21174, 206 COLLECTED MATHEMATICAL WORKS OF G. W. HILL MBMOIE Ko. 25. On the Development of the Perturbative Function in Periodic Series. (The Analyst, Vol. 11, pp. 161-180. 1875.) 1. There are two modes of developing this function. In one, the numer- ical values of the elements involved are employed from the outset, and the results obtained belong only to the special case treated. This mode has been, almost exclusively, followed by Hansen, and is, perhaps, to be recom- mended when numerical results are chiefly desired. In the other, all the elements are left indeterminate, and thus is obtained a literal development possessing as much generality as possible. Certain investigations, arising from Jacobi's treatment of dynamical equations and Delaunay's method in the lunar theory, have invested the latter mode of development with addi- tional interest, and with it we shall be exclusively engaged in this article. In Liouville's Journal for 1860, M. Puiseux has given us two memoirs on this subject, in which appears the general term of this function, but his formulas seem susceptible of modifications which would render them much simpler. More recently, in the volume of the same journal for 18 73, M. Bourget has presented the development in a more concise form by employ- ing the Besselian functions, but as he discards the use of the functions 6^'\ his formulas on this account are more complex. It is hoped, that, even if the expressions, given hereafter, are deemed too cumbrous for practical use, they may still possess some interest from a theoretical point of view. 2. It is known that if we have a function yS' of a variable ^ , which is never infinite, and such that the relation function (C + 2in) — function (C) is satisfied for all integral values of i both positive and negative, it can be developed in a series of the form ^i. (Z;'"' cosiC -t- -ff;'" siniQ , in which i denotes a positive integer ; and that, in the cases where this series is infinite, it is convergent. THE PERTURBATIVE FUNCTION 207 In general, the handling of periodic series is easier if we introduce imaginary exponentials in the place of the circular functions. Thus, e denot- ing the base of natural logarithms, we shall put z = e^^~'^, whence 3 cos C = 2 + 2-\ 2 COS iZ = z'+ e-\ ^VC— 1) 8iDC = «-z-S 2V(— 1) sin *■£: = «'- 2-', « = cosC + V(— 1) sine, 2* = cosi? + V(— 1) siniC. The above theorem then comes to the same thing as to say that 8 is developable in a series of the form where the summation is extended to negative as well as positive values of i. The coeflBcients ^are given in terms of the coeflBcients (7 by the equations z;.w = c.H- c_,, ^/" = (C— c,,) V - 1 , except the case where i = 0, when ^'"^ =^G^. It will be seen that when 8 is real, (7; is a complex number a + &\/ — 1, and C_i, its conjugate a — hs/ — 1, which renders the coefficients ^real, as they should be. The integral A'fi?C - I (cos iC + V (— 1) sin iCj dZ, taken between the limits and 27t, vanishes in all cases except when i^=0, when its value is 271. Hence any function, capable of expansion in a series of positive and negative integral powers of z, integrated with respect to ^ between these limits, gives, as the result, 2n times the coefficient of z° in its expansion. And as the coefficient of z" in the function &~' is evidently (?<, we have 1 /*2ir G,^-=- I Sz-'dZ. This equation holds for all values of i, negative as well as positive, zero included. 3. Let us now suppose that t, denotes the mean anomaly of a planet, and let u be the eccentric anomaly, connected with the former by the equa- tion, e being the eccentricity, w — e sin M — C. In like manner as for ^, we introduce the imaginary exponential s = e"*'~^ Then the last equation can be written £(M— eslntt))/— 1 — eSy— 1, 208 COLLECTED MATHEMATICAL WORKS OF G. W. HILL and, by the introduction of the variables s and z, this becomes which is the transcendental equation connecting s and z . "We have dZ = (1 — e cosu)du=\ 1— -n[s + —] du. Substituting these values in the equation giving the value of Gi , and noticing that, as ^ and u both take the values and 27t together, the limits of integration, when u is the independent variable, are the same as for ^, we get ^.=ir*-*'-"[-i(-T)] du. But, from what precedes, we conclude that the coefficient of s* in the expan- sion of any function W, according to positive and negative powers of s , is 1 P^" -ij- / Ws~'du. Thus, from the foregoing expression for G^, we derive the following propo- sition : i being a positive or negative integer or zero, the coefficient of z', in the development of S, according to the powers of z, is equal to that of s^ in the devel- opment of *«-''[-l(-4)]. according to the powers of s . As most of the functions S, which are presented by astronomy for development in powers of z, are quite readily expanded in powers of s, this theorem is of much use. Another form can be given to it. For we have, integrating by parts Csz-'dZ = — V (— 1) fsz-^' + ^^dz Taking the integrals between the limits ^ = and ^ ^ 27t, we get ^'- 2w J ds^ "-^ Mt! Ja dS Whence we conclude this proposition : THE PERTURBATIVE FUNCTION 209 The coefficient of z' in the development of S according to the powers of z is equal to that of s'~^ in the development of i ds according to the powers of s . This theorem however is not applicable when i = . 4. We shall often have occasion for the expansion of the function in powers of s ; let us, for simplicity, put ;i = — , and We have whence we conclude that ■^^ — 1 . 2 . . . I L 1.(1 + 1)^ 1. 2(i + l){i + 2) — • • J • This series is not applicable when i is negative ; but if, in the function -- . 1 e"'. e ', we substitute — for a, and change the sign of /l, the function remains unchanged, hence and, consequently, by which the values of these functions for negative values of i can be derived from those in which i is positive. These functions are known as the Bes- selian. By putting ^< — 1 1 A' J. 1 ^ + i.(i + i)'^i.a(i + i)(i + 2) ••• one will have no difficulty in deducing the equation ^'-''=^'~i(i + 1) ^' + 1* 34 210 COLLECTED MATHEMATICAL WORKS OF G. W. HILL 5. We come now to the more complex function S of two variables ^ and ^' ; it is known that when this is never in6nite and is such that function (C + 2^, C'+ 2i'7r) = function (C, C) it can be developed in a series of the form 2,,„ [^1,1, cos (tC + i'C) + El:\, sin (iC + i'Z')] , where to one of the quantities i and i', we need assign only positive integral values, but to the other both positive and negative values. If we adopt another imaginary exponential z' =s^'^~^, this is the same as saying that where the summation is extended to all integral values positive and negative for i and i'. Since we have z'z''' = (COS iZ + V (— 1) sin iC)(cos i'Z' + V (— t) sin i'Z') = cos (iZ + CC) + V (— 1) sin (iC + i'C) , the relations, which connect the coefficients K with the coefficients G, are jrifi. = ((7,„-c_,,_,0V-i, unless i and i' are both zero, when -0-0,0 — ^0, • A course of reasoning, similar to that in the case of one variable, established that which holds for all integral values of i and i', positive, negative and zero. 6. Supposing that t,' denotes the mean anomaly of a second planet, whose eccentricity and eccentric anomaly are respectively e' and u', we have u' — e' sin u' = C', and by the adoption of the imaginary exponential s' = e"'''"^, this is trans- formed into It is not difficult to see that we have the following theorem : The coefficient of z^^' in the development of S , according to the powers of z and z', is equal to that of s's'*' in the development of according to the powers of s and s'. THE PEBTURBATIYE FUNCTION 211 7. After these preliminaries relative to the general development of func- tions in periodic series, we come to the matter more immediately engaging our attention. The perturbative function for the action of a planet, whose mass is m! , on another, whose mass is wi , is usually written T> r r 1 r cos ^n and that for the action of m. on ir^ n r 1 r' cos (l>~\ where A denotes their mutual distance, '^ their angular distance as seen from the sun, and r and r' their radii vectors. The problem proposed is then to develop these two functions in series whose general term is of the form (7j, i'zV*'. To this end it seems better to discuss the two portions of the gene- ral perturbative function, — and — — -j^ , separately, and not, as most investigators, attempt, by a particular notation, to combine, in a whole, these two parts. Thus, in developing — , we shall have the term common to both fimctions, and may suppose that r' denotes the radius vector which belongs to the planet more distant from the sun. But, in treating the sec- ond part, we shall suppose that r' belongs to the disturbing planet. The following equations are well known : j» — /2 — 2rr' cos + r', cos ip = cos {v + n) cos (v' + B') + cos / sin (v + 11) sin («' + H') , = cos(v — ^'4- 77 — 77') — 2 sin' I J sin (« + 77) sin(t)'+ 77'), where v and v' are the true anomalies, and n and 11' are the angular dis- tances of the perihelia from either point of intersection of the planes of the orbits, and /is their mutual inclination. 8. Attending then, in the first place, to the development of -r- , we have to notice what are the conditions under which this quantity can be devel- oped in powers of a and z!. In the case of two elliptic orbits, the only one we shall consider here, it is plain that -r- is always finite and continuous, provided the orbits have no point in common. Here we must make two cases according as the value of sin I is not or is zero. In the first case it is evident that the orbits can meet only on the line of intersection of their planes. Hence, p and p' denoting their semi-parameters, there will be 212 COLLECTED MATHEMATICAL WORKS OF G. W. HILL two, one or no points in common, according as two, one or none of the equations, / (1 + e' cos /7')-' =^ (1 + e cos n)-\ p' (1 — e' cos /?')"' =p (1 — e cos /7)-\ are satisfied. In the second case, where the orbits lie in the same plane, there will be two intersections or none, according as the equation p' [1 + e' cos (A — w')]- ' = ^ [1 + e cos (A — w)]- ', /I being the unknown quantity and a and o' the longitudes of the perihelia, admits real or imaginary roots. If we put pe' cos u>' — p'e cos ui^A cos a , pe' sin lu' — p'e sin u) =^A sin a , this equation takes the form A cos (A — 0-) =p' — p ■ The roots of this are imaginary when (p' — pY > ^V — 2pp'ee' cos (to — u>') + ^"e^ 9. If we put we have P — r'' — 2rr' cos {v — v'+n—- 11') + r\ Q =4: sin' |/. r sin (v + ll) . r' sin (v' + II') a series we shall denote thus 1 J^ ,_ ^., 1 . 3.. .(2^-1) p-H±i 10. In order that this development of —r- in a series of ascending pow- ers of Q, or, if one likes, of s,vd?\I, may be legitimate, it is necessary that the elements of the orbits should be such that the numerical value of-^ should be always less than unity. P is the square of the distance of the two planets after the plane of the orbit of one has been brought into coincidence with the plane of the other by revolving it about the line of THE PERTURBATrVE FUNCTION 213 intersection of the two planes. Taking then a system of rectangular axes passing through the center of the sun, and directing the axis of x along the line of intersection, it is plain the equations of the orbits may be written V(a;' + f) + ax + ^y =p, V(a;''+^") + a'x'+^'y' =p', a, /?, a', /3' being constants. And the variables x , y , x', y' satisfying these equations, the question depends on the finding of the values of them which render the expression D., yl_ (x-x'f+iy-y'y a maximum or a minimum. According to the known theory of maxima and minima, the equations, which, in combination with the equations of the orbits, give these values, are - 2i? C=« - ^') + Z' [t^s^^) + « ] = , %D {x-x') + / [^TC^^F) + «'] = 0, y'-2I){y-y')+^[jj/^rf)+P'\==^, 2/ + 2i)(y-2/')+M'[y(^l!p^) + /3']=0, where ^ and yl are the multipliers of the partial derivatives of the two equa- tions of condition. A complete investigation of this question would be con- ducted in the following manner. Eliminate from the seven equations last given the six quantities x, y, x', y', (i, fi' ; the result will be an algebraical equation determining the unknown D. Having derived the Sturmian functions of this, one will ascertain by the substitution of the values D = , — ■„.,, , Z) = + 00, and again of i) = — — . ,. ., ^ , Z) = — oo, whether any roots lie between these limits ; if none, -— can be expanded in a series of ascending powers of sin^ \ I, in the contrary case not. In this way we shall arrive at the condition or conditions necessary and sufficient for the legitimacy of this expansion. 11. This procedure would doubtless lead to very complicated formulas, hence we are obliged to pass over it. However, equations can be readily got, which, by a tentative process, afford the maximum and minimum values of D. Multiply the four equations last given respectively hy x,x',y, y' 214 COLLECTED MATHEMATICAL WOEKS OF G. W. HILL and add the resulting equations, having regard to the equations of the orbits and the value of D; we thus arrive at the simple relation p^i +p'/ji' = 0. Putting, for simplicity, x = r coad, x' =7^ cos6', the addition of the first and second of the same group of four equations gives /i (cos d + a) + ti' (cos e'+ a') = 0. By combining this with the preceding is obtained cos 0' + a' COS ^^ + a J' "" P " Again the addition of the same equations, multiplied severally by x, — x', 1/, — y', gives the equation 2I)(r" — r'')=p'fi'—p/i. Dividing the left member of this by 2D [x' — x) , and the terms of the right member by its equivalents derived from the first and second equations, we get r" — r^ p' p x' — X ~ cos &' + 0' "*" COS (? + a ' or r' COS 6' — r COS cos + a r'^ — r' 2p • This and the equation cos 0' + a' COS 6 + a p p determine the values of the variables Q and 0' which render D a maximum or minimum. When the orbits are nearly circular these values are in the neighborhood of 4 7t or !«. When both orbits are circles the solution is very simple, and we find, in order that the development may be legitimate, we must have . / a' — a a and a' being the mean distances of the planets from the sun. 12. Assuming that this development is legitimate, we have to develop _2S+_1 P 2 g* in terms of s and s'. We have THE PEE.TURBAT1VE FUNCTION 215 r cosv = a{cosu — e) = -g-fs-l 2ej, r sin z; = a V (l-e") smu= ^f_^ V (1 — e') (« — t) ' whence and by putting l + V(l-e') e 2 -''' 1 +VC1 — eO"*"' we get revV-l = arisil — —]• And the value of re""''"^ is evidently obtained by substituting in this -— for s, hence * s From these two equations may be derived r = ari(l — S Writing y for n — IT', we have (r'-'P)- ' =[l-2yC08iv-v'+r) + ^j-~- The right member of this is developable in a series of integral powers of the exponential gC-^'+v)!^-! when —is always less than unity. This con- dition is fulfilled when we have a(l+e) . ^»+i BJie-v'+y )V- 1 j^ — CO Bok + i is the same function of -r that Laplace's b^u+i is of ~= a . The -y- / -2- a' approximate value of — j- being a, any function of —j- can be expanded in a series of ascending powers of -^ — a by Taylor's Theorem. And as we have 216 consequently, COLLECTED MATHEMATICAL WORKS OE G. W. HILL [V(i--V)(i--f) J ■ V(l — a>V)|^l-^j n being an integer, and n ! denoting the product of all integers up to n inclu- sive, it being understood that ! = 1 . Expanding the last factor of this expression by the binomial theorem, and employing the notation [t,y] for the coefficient of a;'' in the expansion of (l + ^c)', we have, p being an inte- ger, 13. In the next place the development of Q in terms of s and s' must be formed. We have r sin (w + n) — 2J _i C*"^*" + ""'- 1 - rs- c + ^)V- 1] , r' sin {v' + /7') = ^ ,_^ [/£(«'+ n')V- 1 _ /e- («'+ nW- 1] , n + n' = o, h = e»v-i, and putting we find ^[''(^~t-^-t(i-'"'*')v]- Raising this expression to the k*-^ power, and multiplying by r'-'^'+" = [aV (1 - «''s')(l - -T-j] we find that the part of r'- '^'^ + " Q«^ which has h'"' as a factor is J_ „Y)?'- '* + " sin'' 4 2 (— l)""[^j w'][*, * — i'"— w'] („, \ai"' + aii' 1 — — ) C„/ \-2»'-l 1 ^j gl'"-l,+ l,'J^ill,^ THE PEETUEBATIVE FUNCTION 217 14. We are now in the possession of all the developments necessary for exhibiting the function -r- in terms of s and s'. In order to obtain the part of this function which has g^"h^"' for a factor, we must put, in the formulas of §12, j = i" — i"'+k — 2n', and the chief operation here is the addition of the exponents of the quan- tities s, 1 — as, 1 , and the similar functions of s' which are found s U + l (J) in the three formulas for (r'^^P) 2 , £?^and r'-'^* + "g\ For brevity we will write _ i.3...(a/fc-i) L*-"- 2.4. ...2^ • Then the part of — , which has g*"h^"' for a factor, is _1_2 ""5;"''' "2 V (- i)^-^"'+»-> W[^> ^'][^. ^ -i"'-n'Jn,p] j„7(i"— iV'+s_2,i') Cm' \—j!— 31— 1 — <" + <"' XV /-i- We observe that in this expression the summation with respect to n' affects only the integral coefficients \h, «'] , \h,'k — i"' — n'] and the upper index of the quantity 6, hence if a new function of a is assumed, which is a linear function of the 6's, and such that it will take the following simpler form : %a' jLl Li ^^ I n\ d^~ 2 k = i"' n = p = 1 — —] / B,' \-l-J.-l-4" + «"' X,'-«-P-V-'" + ""(l_a,V)-«:-p-i + <"-Wl _^j /'r". 15. In order to get the coefficient of zV*'- in the expansion of —^ , according to the foregoing investigation, we must multiply the preceding expression by 218 COLLECTED MATHEMATICAL WOKKS OF G. W. HILL Hence if for brevity we adopt the functional notation the coefficient of z'z'' g'"h^"' in —^ will be equal to the coefficient of s's''' in k — CP n = co p —n j_ y y V(_]y-<"+.-p W[w>j?] /vd ^^ ^J ^m fll k — i'" n = p=0 da" 2 \v y \ ^,' / If then the coefficient of s' in Lhe expansion of S is denoted by E fol- lowed by the same indices, and the coefficient of s'*' in the expansion of S^ by E' in like manner, E will be a function of e only, and E' a function of e' only ; and, it being understood that each argument is taken but once, that is, the negative of the argument is not considered, the coefficient of in the expansion of — is expressed thus a^n + l • T /ic-rp\ ,— \^K-rp + i-)^ As in this formula, h ought to be a positive integer, it will prevent embarrassment, if the arguments are so taken that i'" may not be negative. In the case where i, i', i" and i'" are all zero, the expression must be divided by 2.' 16. Thus we have arrived at an expression for the general coefficient involving only three signs of summation ; and it may be remarked that all the coefficients are exhibited in precisely similar forms. Thus, to pass from one argument to another, we have only to make the suitable changes in the two lower indices of the functions E and E' and in the upper indices of B , and commence the summation with reference to k with the new value of i'" instead of the old. Hence, from this expression, we can write out a scheme or blank form, which, when the indices proper to the argument are filled in, will be the coefficient of the cosine of it in the expansion of -r- . Such THE PERTURB ATIVE FUNCTION 219 a blank form is written below ; the indices i" and i'" are omitted from B , and the two lower indices from E and E', and the upper indices of these quantities, for the sake of facility in writing, are placed to the right and at the foot. The factor —j , common to the whole expression, is also omitted, so that the formula gives the coeflBcient in the expansion of — r- . In making use of it, one must commence at the portion which has sin^'"' h I for a factor, all the preceding parts being supposed to be suppressed. It is hoped that a sufficient number of terms have been written to render the law evident, so that they may be continued as far as desired. + 5^a"-§[E„E'_,-3E,E'_,+ E,E'_3] - i;^a'||[E.E'_,- 3E,E'_,+ 3E,E'_3- EsE'.,] + + ^sin4{ "B^E^E'- _ya^4!-^tE.E'_,-E,E'_3] + j^a'^[EiE'_,-2E,E'_,+ E3E'_,] — -12 3 " da? [EjE— J — 3E2C_3+ ot3t_4 — t4L_5j + +K-4{ ''^^^^^'- -4-'f'[E.E'.3-E3E'_J + l^a*^[E,E'_3-2E3E'_,+ E,E'_,] 1 .cm-. a 1.2.3" da' + . . . . [E,E'_3- 3E3E'_,+ 3E,E'_,- E,E'_e] + M^6'^'y{ «'B5E3E'_, _Ji.„4^rEE' — E E' 1 + j^«^^'[E3E'_,-2E,E'_,+ E,E'_c] 1 ,^'Bs |-E3E'_,_ 3E,E'_,+ 3E E'_e- £«£'_,] ~ 1.2.3 da" + . . . . 220 COLLECTED MATHEMATICAL WORKS OF G. W. HILL For illustration, let it be desired to obtain the coefficient of cos(2C-5r+2r), from which arises the larger part of the great inequality of Jupiter and Saturn ; we have only to imagine that the lower indices ( 2 I are everywhere applied to E, and the indices I _2 ) to E', the indices (2, 0) to B ; and as we have *'" = , we suppress nothing. 17. The quantities B are very simply expressed in terms of the h's. The following are all that are needed when terms of the eighth order with respect to the inclination of the orbits are neglected. B|''° ) 7,0 ^■' =*r'+*r'' Br 7,(0 -0,, B<'-» = Jf^>+45«+J|-»', B«-^ = 2J|'+"+25|*-», Bf ^6f, Bi''» = *!'+"+ 9*1'+''+ 9i|*-"+ 5<'-", ^r = 3J»+" + 95|« + 35i'-», Bp = 35i'+"+3M<-'>, Bi''" = iT. 18. In computing the factors of the preceding formula which depend on E and E', the following abbreviation can be used. M^ denoting the fac- tor which multiplies 1 a" Bm -t- 1 «! da" ' and A being the symbol of finite differences with respect to «, it is plain that J"ilf„ = (-l)"E,+„E'_„+„+„. Hence, if the products £* + „ E'_(i + „ + i) are computed for the various values of n, and are taken alternately with the positive and negative sign, and are written as if they were the successive differences of a function, we shall get the values of the factors ilf„ by filling out the scheme of differences. This abbreviation is applicable equally whether we are making a numerical THE PEETUEBATIVE FUNCTION 221 computation of the coeflBcient or a literal one. In the latter case the abbreviation can be applied separately to each term of the form CeV'' in the products £*£'_(», + i). 19. We proceed now to discuss the functions E. From their definition we have whence (^Jcosy. = i 5 [e (i) + E(-y)]cos^c, (-^Jsiny. = i 5 [e (|) - <-|)] '''''''■ From which we gather that the functions E can be computed by definite integrals, thus Let us now suppose that the coefficient of s*, in the expansion of /S 'v^" /■'"*" ^N in powers of s , is denoted by -E" | y i , then evidently By writing in the expression 1-^s for s and changing the sign ofy, it remains unaltered ; hence the relation By developing the factors of the expression by the binomial theorem, we get E0^ = i-iy-^ii -y, Jc -y] v<"'-^ V fi -^ (i±M=l) ,,., (^•+y)(^•+y-l)(^•-/fc)(^-/fc-l) , 1 222 COLLECTED MATHEMATICAL WORKS OP G. W. HILL This equation, as written, is correct only when h — j is not negative, but by the relation given above we can reduce the case of h — j negative to that where it is positive. The factor in the brackets is a case of the series treated by Gauss in a memoir entitled " Disquisiiiones generales circa seriem infinitam, cfcc." (See Gauss' Werke, Vol. Ill, p. 1 23, and especially the " Nachlass,'' p. 207.) According to Gauss' notation ^ (j) = (- 1)'-' [i - j, k -j-] iiJ--' F{- i -j,k — i, k-j + 1, o,^) . Whenever, of i -{- j and i — j, one is not negative, this series terminates after a certain number of terms, thus aflfording a finite expression for the function. But when these integers are both negative, the series is infinite. However, it can be easily transformed into another which like the former is finite. From Gauss' investigation of these series (see the volume just quoted, p. 209, equation [8 2]), we have F{a, /S, r, x) = (1 — a;)v-»-P F{y—a,r — P,r,x). Applying this to our expression, we get e(j\ = i—lf-'ii—j, k—j] ri'io'-^l — my + 'FCi + k+l, i—j + 1, k—j + 1, w"). This expression is evidently finite when i — j and i -\-j are negative. 20. The developments of the functions E in powers of e as far as e' have been tabulated by Prof. Cayley in the Memoirs of the Royal Astronomical Society, Vol. XXVII. It would conduce to the ready employment of the preceding formulas if we had the function E ( y i explicitly expanded in \h) ascending powers of e, but the attempts I have made to write such a series lead to extremely complex forms of the coefficients. Hence I shall give here only the coefficients of the lowest power of e in this function, which suffices for obtaining all the terms of the lowest order in any coefficient of the expan- sion of 1 -^ A. We have, when/ — h is positive, E (]" ) = [[t +j,j-k-] + ii +y,i-/fc-l]^ + [i +j,j-k-%] ^ THE PEETUKBATIVE FUNCTION 223 and when Tc — j is positive ^ (4 ) = [f *' -^ ' ^ -■^■] - 1^' -i ' -^ -i - 1] T + 1« -y > * -j - 2] o -... + [^-i,o]^^](-J 21. Thus in the example alluded to above, of the coefficient of cos (2^ — 5^' + 2y), we find that the terms of the lowest order in E and E' (omitting here, as in the scheme, the two lower indices), are E„=E,= E,= E3 = 1, E'-i= - [[-3,3]-[-3,2]| + [-3, 1]^ -[-^'0^li](T)=^^'"■ E'_,= - [[- 3, 3] _ [- 3, 3]f + [- 3, 1]^^ - [- 3, 0] jJi _ 5 9 -,'3 _8 4 5 p'i '-3=-[[-4,3]-[-4,3]f + [-4,l]f2-[-4,0]^3 '-*= - [[-5, 3] - [- 5,3]A + [_5,1]^^_[_5, 0]if3](4J=HF^" Bringing into use our method of abbreviation, we multiply each of the preceding numerical coefficients by 48 in order to avoid fractions, and then write them alternately with the positive and negative signs in a diagonal line, and from these, as successive orders of differences, derive the numbers standing in the vertical columns, thus : + 389 — 590 — 201 + 845 + 355 —1160 + 54 —315 — 60 + 381 — 6 +66 + 6 — 73 — 6 +6 and dividing the numbers of the first column respectively by 1, — 1, 1.2 — 1 . 2 . 3, we get the following as the terms of the lowest order in the coeffi- cient of cos (2^ — 5^' + 2y) in a' -i- A , 18" L * + ^"^ "1^ + ^^" ^S^ + " do? Y ' which agrees with that found in the books. The following additional terms 224 COLLECTED MATHEMATICAL WOBKS OF G. W. HILL of the same coefficient can be written from the second, third, &c., columns, viz., those which are multiplied by e'^ and the various powers of sin^ ^ /, -Us [S90a Br>+ 255a^%- + 30a3 %- + a^^] e'^ sin4 + gi[845«^Br"+ 315a3^ + 33a^^' + a^^j.^sin^ — &c 22. When we wish to obtain only the terms independent of ^ and ^', that is, those on which the secular perturbations depend, i =; and i' = , and the Besselian function / disappears from the expressions giving the values of E and E', and the coefficient of cos {i"'y + i"'d) in the expansion of -r- can be written A k =00 n = ca p'=~n v„* + " ~ d¥- «1^ J- .k+p + U .— {{h+p) . 23. In leaving the subject of the development of 1 -^A, it may be well to note that two other forms can be given to the expression of the general coefficient, by employing, instead of the expression given above, either of the following : — 7 ^— a ^^ OL 1 — z ^ — r ^-^V + 7) ,'(l-.Y)(l-4) But as they do not possess as much symmetry and brevity as the form given above, we will pass over them. 24. The second part of the Perturbative Function, omitting the factor m', is r r r I I ~1 — ^ji-coB4' = :;ji-\ cos'' Y COS (v — v'+r) + sin'-g cos (v + v'+ e) ^ COS ¥' = — -p? THE PEKTUEBATIVE FUNCTION 225 According to the first theorem of §3, the coefficient of 2" in — e'"^~^ is equal to that of s" in or it is equal to And, according to the second theorem, the coefficient of 2* in the same func- tion is equal to that of s* in ii['('-Tr]-«-"'-''=i('-^)-*<--*- Hence we have .■=+00 a .^^ * L ^ 2 J And by simply writing 1 -f- a for z, r . — e a I = + 00 The well-known differential equations of elliptic motion ^ J^ -n J + TT^/^o, supposing the axis of x to be directed towards the perihelion, give us the equation fi. f(i£i:). r"^ — dC and consequently these two * = + {» 4- e-v-i = ^ ^'J f'"''^'""- '^■■ + "1 «'■ By substituting these values in the expression given above for ^^ cos 4'. it is not difficult to see that, in it, the coefficient of cos(iC + i'C'-l-r) IS 25 a 4i I \_ -2 Xj L2 aj 226 COLLECTED MATHEMATICAL WORKS OF G. W. HILL and the coefl&cient of cos(iC + i'C'+fl) is — 4- sin»4 -^ U?-' - '"'^?+"l . i'-n' r^Sl'-"- "'V/i'.+ "l . In the special case of i = the middle factors of these expressions take the indeterminate form 0-^0, but then, in accordance with what has been shown above we should read — |e. Thus, by means of the Besselian functions, these coe£Scients take finite forms. DIFFERENTIAL EQUATIONS EMPLOYED BY DELAUNAY 227 MBMOIE Fo. 36 Demonstration of the Differential Equations Employed by Delaunay in the Lunar Theory. (The Analyst, Vol. Ill, pp. 65-70, 1876.) The method of treating the luaar theory adopted by Delaunay is so elegant that it cannot fail to become in the future the classic method of treating all the problems of celestial mechanics. The canonical system of equations employed by Delaunay is not demonstrated by him in his work, but he refers to a memoir of Binet inserted in the Journal de TEcole Poly- technique, Cahier XXVIII. Among the innumerable sets of canonical ele- ments it does not appear that a better can be selected. These equations can be established in a very elegant manner by using the properties of Lagrange's and Poisson's quantities (a, h) and [a, &]. But a demonstration founded on more direct and elementary considerations, is, on some accounts, to be preferred. Let a denote the mean distance, e the eccentricity, i the inclination of the orbit to a fixed plane, I the mean anomaly, g the angular distance of the lower apsis from the ascending node, h the longitude of the ascending node measured from a fixed line in the fixed plane, ^i the sura of the masses of the bodies whose relative motion is considered, and R the ordinary per- turbative function augmented by the term |p . Then if we put L = V^a, 6r=\/ [jLLa{l — e^)], H^s/ [}ta{l — e')] cosi, Delaunay's equations are m_ dR dH_ dR dt dg ' dt dh' dg _ _dR dh dR irt~ dO' dt dH' In terms of rectangular coordinates j._ fj? m[^ w! {xx' -\- yy' + z/) 2Z^ + iix' - xy + {y'- yj + iz'- zf]^ r" In this expression, for x,y,z, ought to be substituted their values deduced from the formulas of elliptic motion, and expressed in terms of L, G, H, 2 I, g, h. It should be noted that the term ^3= |^' of ^^^ ^^^^ ^^^^^ with respect to the disturbing force, has been added to B only to preserve dL dR dt er dl dR dt ~dL' 228 COLLECTED MATHEMATICAL WORKS OP G. W. HILL in the equations the canonical form : it is only by amplifying the significa- tion of the word that I can be called an element, as it is not constant in ellip- dl u? tic motion, but augments proportionally to the time and -5- = w := tr_ . dit Li It is chosen as a variable in preference to the element attached to it by addi- tion simply to prevent t from appearing in derivatives of B, outside of the functional signs sine and cosine. The equations d^x iLX _ 9J? d^y ny _ dR d^z ij.z _ dR d¥ '^7 -Qx' W ^ 7 -dy ' W^ 1^~ dz ' are well known ; here, however, B does not contain the term -^^ . By multiplying them severally by dx, dy, dz, adding and integrating, is obtained dx^+dy''Jrdz^ 11 m- PfdR . 9^, dR -, \ ¥dt^ T + W ^ J (d^'^'' + dy "^y + W^V- When the elements are made variable, this gives d I iJ. \ _ _ IdR dx dRdy dR dz\ dt U« / \dx dt dy dt '^ dz dtj ' dx _ dx_ dy _ dy dz dz But we have and hence dt -^^ dl ' di -^ di ■ dt -'^ dl ' d I iJ.\ IdR dx dR dy dR dz\ dR r) dt\2a) — ~"'\dx dl ^ dy dl '^ dz dl)~ ~ " dl Dividing both members of this equation by — n=- — \/fia~^, the left member is seen to be the differential of V^ = L. Consequently, dL _dR dt ~ dl' Denoting the true anomaly by v, the orthogonal projection of the radius vector on the line of nodes is r cos {v-\-g), and on a line perpendicular to it and in the plane of the orbit r sin {v ■\- g). And the latter projected on the plane of reference is r sin(« + g) cosi, and on a line perpendicular to this plane r sin {v + g) sin i. If the two projections lying in the plane of reference are again each projected on the axis of a;, their sum will be the value of the coordinate x, and the sum of their projections on the axis of y, the value of the coordinate y. Hence x = r cob{v + g) cos h — rsm{v + g) cos i sin ^, y = ?• cos (« + g) sin A + r sin {v + g) cos i cos h , z = r aiii(v + g) sin i , DIFFERENTIAL EQUATIONS EMPLOYED BY DELAUNAY 229 or, substituting for i its value in terms of G and H, x = r cos(v + g) coah — ^r sin (v + g) sin h , TT y = r coa (v + g) sin h + -^r sin (v + g) cos h, z = ^ r sin {v +g). As r and v are functions o£ L, G and I only, the preceding equations show the manner in which H, g and h are involved in JR. S' denotes double the areal velocity projected on the plane ay, or „ xdy — ydx ^= dt Consequently dH _ dB BR dt -^ dy ~y dx' But the foregoing values of », y, z show that we have and thus dx _ dy_ 9^ _n dff_dRdx dRdy dE dz _ dR dt -dx 9A + dy dh + dz dh- dh ' G denotes double the areal velocity, and evidently, if for the moment we suppose x and y to be drawn in the plane of the orbit, the axis of x towards the lower apsis, dG_ dR_ dR_dR dt ~ dy y dx ~ dv ' where, in the last B, for x, y, z must be substituted their values given above in terms o{r, v, G, H, g, h. Now, as the only way in which g is involved in these values, is by addition to «, it follows that dR _dB, dv ~ dg' and this equation is not affected when, for r and vin R, are substituted their values in terms of i, Gandl. Consequently dG_dR dt ~ dg ' In the elliptic theory '^''-^''^^ = V Q'- H' cos A, ydz — zdy dt :V^ioi-xf+y"^-h =0.* re'+x + h tan-' ^73-^ = , (1) With Lagrange's method of multipliers, if we denote these equations respectively by i = 0, ilf = 0, N-=0, and the multipliers of their difieren- tials by ^, j« , V, and take ^ to represent any one of the five variables x, x', y\ 6, 6', the general equation of the problem is dt ^d^~ d^ ~ d? d? '^ '^ d? '^'' dS * These equations subsist only as long as the spheres are in contact. 234 COLLECTED MATHEMATICAL WORKS OP G. W. HILL Applying this in succession to each of the five variables, and writing for simplicity ^ for tan' -1 y a/ — X , we get d'x in -jii = — A + ;u (1 + sin ?>) — p cos

+ i- sm y , ^ml?^= XB, (2) Adding the first and second of (2), d' (mz + m'af) dt' = ti-X. The two first of (1) and the two last of (2) give ■-m d^x df „ , d'e' , ,rd'x , ,«^V1 Substituting these values for % and ^ in the last equation, d'imx + m'od) „, , ,.d^x „ ,,d^v Integrating once and eliminating a/, ^ (wi + m') -^ + m'h (I - sin f) -J- = , (3) where the constant is zero because the spheres are supposed to set out together from a state of rest. As -^ , in general, is negative (4» can always be supposed in the first quadrant), it is evident from this equation, that if sin <^> I, the lower sphere will move horizontally towards the side on which the upper sphere is; but if sin^<|, in the opposite direction. Integrating (3) twice (7j» + %m') X + fim'x' + 3»i'Ay = a constant . Eliminating x and 4> from this by substituting their values in terms of x' and SOLUTION OF A PROBLEM IN THE MOTION OF ROLLING SPHERES. 235 y, we get as the equation of the path of the center of the upper sphere 7 (m + m')[a/- i^h^-y"'] + m' [^2h sin"' -|- + 5 = is equivalent to ^ cos ¥> + ( -^ + ^f 1 sin f = . And if we eliminate »' and y' from this by means of their values in terms of X and ^, we get cPx , . , dp' . ■^coa ^ + g amy — h ^ = , By eliminating second derivatives this becomes 49m \_j^- sin ?>] + 10m' (1 + sin vf [y ^' - l] = ' which, by substituting the value of-^, becomes {(3 is the initial value of ^) 70 (m + OT')[49m + 10m' + 20m' sin

][49m + 45m' + 20m' sin y — 25m' sin' ?>] = . 236 COLLECTED MATHEMATICAL WORKS OF G. W. HILL MEMOIE No. 28. Reduction of the Problem of Three Bodies. (The Analyst, Vol. Ill, pp. 179-185, 1876.) The object of this article is to find the three differential equations which virtually determine the sides of the triangle formed by the three bodies, bringing to our aid all the known finite integrals of the problem. Lagrange was the first to treat this question in his Essai sv/r le Problhme des Trois Corps (Oeuvres, Tome VI, p. 227) ; but the formulas lacking sym- metry, his editor, Serret, has, in a note, supplied this and pointed out an important error into which Otto Hesse, who had investigated this subject {Journal fur die Mathematik, Band LXXIV) had fallen. By adopting an orthogonal substitution, at the outset, for reducing the number of coordinates from nine to six, we can prevent the masses from entering the equations except through the potential function or its deriva- tives. In this way symmetry, indeed, appears to be lost, but there is so great a gain in condensation of the formulas, that we can carry out some of the eliminations which previous writers have been content only to indi- cate. Let I, »7, ^ ; ^', J?', ^'; ^", >/", ^" be the rectangular coordinates of the masses m , m', m", the expression for the living force will be d^'+dri'+dZ' ,d^"+dri"+di:" „d^"'+d7i"'+di:"' %df ^ "- Uf "^ "' 2df and A, A', A" being given by the equations J" = (r- 1")"+ W- >?"/+ (f'- <^")\ A" = (I"- ^y + (ri"- rif + (C"- C)^ the potential function + -77- + "■ — J ^ A' ^ A" Without lessening the generality, the origin of coordinates can be put REDUCTION OF THE PROBLEM OF THREE BODIES. 237 at the center of gravity, when the principle of the conservation of this cen- ter will furnish the equations mf + m'f ' + m"e" = , mt] + m'fj' + m">?" = , m? + m'C' + m"C" = , (1) By means of these relations three of the variables can be eliminated and the number thus reduced from nine to six. This transformation is most ele- gantly accomplished by putting ^ — ax + ^x', rj = ay + ^y' , i: — aZ + jSz', S' = a'x + /3V, rj' = a'y + ^'y', Z' = a'z + ^'z', ^"= a"x + ^"x', 7i"= a"y + [i"y', :"= a"z + /3"z', where a , a', a", /? , P', /3" are six constants which may be so taken that they satisfy the five equations ma + m'a' -^ in"a" = , mj5 -I- m'/J' + m"/3" = , map + m'a'p' + m"a"p" = , m,a'+m'a"+m"a"' =1, mP'+m'p''+m"p"' =1. (2) The first two are necessary in order that equations (l) may be satisfied ; the third is adopted in order that nothing but squares of diflferential coefiicients may occur in the transformed T; and, evidently, the last two may be adopted without thereby diminishing the generality of the transformation. These equations may be solved elegantly in the following manner: Put sj m^=Tc sin x cos £ , tj m' ^Tc sin y sin e , iij m" = Jc cos y ; and adopt the four quantities ^ , ^', u , a', such that sj ma — sin

' -I- sin ;- sin ^' cos ' = 2^ + rfw' + dyf" + dw'" df ■2(wv + wV + w'V), (5) an equation which is symmetrical. REDUCTION OF THE PROBLEM OF THREE BODIES. 241 It is evident now that, since the values of v, v', v" are known from the first three equations of (3), we shall have, as the equations determining w, w' and w", (4), (5) and the last of (3), provided we can find a relation connecting p with w, w', w", v, v', v" and the differentials of the first three. Such a relation can be found in the following manner : Assume the four indeterminates X, X', X", X'" so that the equations dx IF xX+ x'X'+ ^X"+ ^ X"'= 0, d£ dt §]L Y"^^ Y'"- yX+y'X'+^X"+--I^X"'=0, dz ~dt zX + z'X' + -^ X" + ^ X"'= , d£ dt are satisfied ; and treat the last as if they were equations of condition in the method of least squares, that is, multiply the first by x, the second by y, and the third by z, and take the sum for a first equation; and so on. In this way the normal equations formed from them are vX+v"X+^^X"+[^% + p)x"'=0, v"X+v'X+i[^-p)x"+i%X"' = 0, (i^ + p]x+i% X'+u"X"+ u'X"'=0. As the number of these equations exceeds that of those from which they are derived, they are not independent, and the determinant, formed from the coeflBcients, vanishes ; which is the condition determining p . This equation is dvdv'-dv"''-\\ . , „„, , „,. + 2 P'+ v'p-' + idt' v'dv"— v"dv' v'dv"' — 2v"dv'dv" + vdv'^ dt ■P + v"dv — vdtf' vdv"^— %v"dvdv"+ v'dv' V+ — ^, p + — dt idf V"p' + dt v'dv — vdv' . (v'dv -f vdv') dv"— v" (dvdv'+ dv'") -} „_r. T+ 4 (w-^ _ w'» - w"^)(v» - /» - /'») -P + 5S? or, expressed in terms of the new variables, -4 + 8 -4 —4 -4 P' + dvr'- d^ff"- dvf"'T dt' ~'w'd-w"—yf"dvf' vr"d'w — wd-w" , w%+ f m'a_.-.]g'^ 252 COLLECTED MATHEMATICAL WORKS OF G. W. HILL Let the last term of the second member of this equation be denoted by the series since r is a series of cosines, we must have, in consequence of the equations of condition which the a^ satisfy B^i= Bi, and the equations, which determine these coeflBcients, can be obtained from the formula. i^,. a,_ Bj = U(i + 1 + m) a< + f m^'a-,.!, when we attribute to i, in succession, all integral values from i^ to *= <», or which is preferable, from i = to i = — oo. The following are all the equations and terms which need be retained when it is proposed to neglect quantities of the same order of smallness as m^" ; a„i2„ + (ai + a_i) B^ -I- (aj + a_2) R^ = f m'a_i , a_ii?o + (a„ + a_2) A + a^Ri = — 4ma_i + f m'^a„ , a_272„ + (a_i + a_3) Bi + aoiJj + a^Rs = 8 (1 — m) a_2 + | m^a,, a_,jBi + a_ii?2 + S'oRs = 1^ (2 — m) a_3 + f 111%, a_3^i + a_A + a_ii23 + HoRi = 16 (3 — m) a_4 + f m'a3 . For the purpose of illustrating the present method, we content ourselves with giving the following approximate formula: — -^ + m' = l + 2m + fm — f m^aj + 4ma_i (&i + a_ 1) + [f m' — 4ma_0 (C' + r') + [8 (1 — m) a_, + f m^ (a, - a_.) + 4maL0 (^* + ^"O , where, for convenience in writing, it has been assumed that a^ ^ 1 , and consequently that a^ denotes here the ratio to ao, which, as has been mentioned above, is a function of m. The absolute term and the coefficient oi ^* + ^~* are affected with errors of the eighth order, while the coefficient of ^^+^~' is affected with one of the sixth order. We attend now to the remaining terms of 0. If we put Du ~ S,. (2i + 1) a,^" ~ ' • '^ ' it is plain that we shall have D's_ . 2-,.(2t + l)'a,C-" _ y ^._„ Ds ~ 2',. (ai + 1) a,e-" ~ *• •^•^ » and in consequence, ON THE MOTION OF THE LUNAR PERIGEE. 253 From this it will be seen that the development of -=^ will suffice for obtain- ing all the remaining terms of©. Let us put h = (2i + 1) a^ . The equations which determine the coefficients Ui are given by the formula but, in order to exhibit some of their properties, I write a few, m extenso, thus : + hU_, + h_^U_, + h_,{U„ — l) + h_,U^ + h_,U, + . . + hU^, + ho CLi + A_i(C/"o — 1) + h-,U, + h^sU, + . . + hU_, + h U_i + ?i, {Uo — V) + h U^ + h Ui + -- (19) When the subscripts of both the h and U in these equations are nega- tived, and the signs of the right-hand members reversed, the system of equations is the same as before. Hence, if we have found the value of Ui, which is a function of the A, the value of U_i will be got from it by simply negativing the subscripts of all the h involved in it and reversing the sign of the whole expression. When this operation is applied to the particular unknown Uq — 1 , we get the condition 27„-l = -(f7.-l); whence we have, rigorously, U,: This result can also be established by the aid of a definite integral, absolute term, in the definite integral The absolute term, in the development of -^^ — in powers of ^, is given by Z)"M '27tJo D'u ^ ^ 2tz»/ — iJo d'u d'u dz' The indefinite integral of the expression under the sign of integration is , d'u 1 rd'x , d'"ii -. — ^"1 254 COLLECTED MATHEMATICAL WORKS OF G. W. HILL and if, for the moment, we take p and ^ such that this integral takes the shape logp + fV — 1- The first term of this has the same value for -r = and t = 2n, and consequently contributes nothing to the value of the definite integral. Thus we have 1 1 /»2''D-+% , 1 r IT When T := , let ^ be assumed between and 27t : it will be found that ^ has the value or - or 7t or |7t according as v is of the form 4^ or 4|tf + 1 or 4(ti+ 2 or 4^ + 3. Moreover, when t augments, <^ also augments, and when ■r has passed over one circumference, 4> has also augmented by a circum- ference. Hence 'Sir X).'+1m 1 ni^ D-+ ^. ... ■^. = 1. It follows, therefore, that v denoting zero or a positive integer, the absolute term of the development of —jy^ — in integral powers of ij' is 1. And, in like manner, the absolute term of _^ - is — 1 . ix S Equations (19) are readily solved by successive approximations, and when terms of the tenth order are neglected, we can write ^ = 1 + 3 [ Ai — A_A + h h A_,]£;' + 2 [3^, — h h — 3A_A + 4Ai A_A —2Jh h h A_i] C* — 2[2A_j— A_iA_i — 2^1 A_, + 4A_,Ai h^., — 'ih_-Ji_.Ji_.,\ ]£;— + 2 [3A, — 3Ai h + h K h ] C' — 2 [3^._, — 3A_iA_, + h_Ji_Ji_i] Z-' + 2 [47*4 —iJ^k, + 4Ai Ai As — 2^, Aj — Ai Ai Aj A, ] C — 2 [4A_4 — 4:h_ih_3 + 4Ji_Ji_Ji_i — 2h_Ji_, — h_-Ji_.Ji_^h_^'\ ?-«, where we have supposed again that ^q = a^ = 1 . ON THE MOTION OF THE LUNAR PERIGEE. 255 With the same degree of approximation we have used for -y + m", © can be written e = l + 2m — |m» + fmX + 54a? + (12 — 4m)aia_i + (6 — 4m)ai., + [(6 + 12m) ai + (6 + 8m)a_i — f mT(:^ + C"') + [20maj + (16 + 20m) a_, — (9 + 40m) a? + 6aia_. + (7 + 4m) aL^ — fmVax — a_0](:' + r*). In the determination of the terms of the lunar coordinates which depend only on the parameter m, it has been found that, with errors of the sixth order, _ 3 6 + lam +^9m^ „ ^'-T^ 6-4m + m^ "" ' _ 3 38 + 2 8m + 9m'' , *-'- ^ 6-4m + m' °'' and, with errors of the eighth order, 07 2 + 4m + 3m' Fooo , An , n 2 00 !<59 — 35m -\^, Sj = -T^ ra A , 2iro» a i n ^38 + 40m + Qra' — 32 ^ j-— 5 m', 'ISO [6 — 4m + m''][30 — 4m + m'] L 6 — 4m + m'J a_, = 87 2 + 4m + 3m' F „„ ^^ _,_ 9^1 7 - m ~| ^. , "[6-4m + m'][30-4m + m'] [" ^^ " ^"^ + ^% - 4m + m'J "^ ' No use will be made of these formulas in the sequel of this memoir : they are given only that we may at need easily deduce an approximate literal expansion for the important function 0. III. In the preceding discussion it has been established that the determina- tion of the lunar inequalities, which have the simple power of the eccentricity as factor, depends on the integration of the linear differential equation to the treatment of which we accordingly proceed. We assume that the development of 0, in a series of the form has been obtained. Here we have the condition ©_< = ©£. If ©i, ©a, &c., are, to a considerable degree, smaller than ©o, an approximate statement of the equation is * These expressions will be established in another memoir. 256 COLLECTED MATHEMATICAL WORKS OF G. W. HILL the complete integral of which is K and K' being the arbitrary constants and c being written for \/©o- When the additional terms of are considered, the effect is to modify this value of c, and also to add to w new terms of the general form J.^±°+^'. It is plain, therefore, that we may suppose w = ^f(C,c) + ^'f(£:,-c), and may take, as a particular integral, w = 2'..b,:<'+»', hi being a constant coefl&cient. If this equivalent of w is substituted in the differential equation, we get the equation [c + a;Tb,-i'..^,-.b<=0, (20) which holds for all integral values for j, positive and negative. These con- ditions determine the ratios of all the coefficients b^ to one of them, as bo, which may then be regarded as the arbitrary constant. They also determine c, which is the ratio of the synodic to the anomalistic month. For the purpose of exhibiting more clearly the properties of the equations repre- sented generally by (20), I write a few of them in extenso: for convenience let [i} = ic + 2iy-e,; then (21) If, from this group of equations, infinite in number, and the number of terms in each equation also infinite, we eliminate all the b except one, we get a symmetrical determinant involving c, which, equated to zero, deter- mines this quantity. This equation we will denote thus: — S (c) = . (22) If, in (20), we put — c for c, — j for j, and suppose that b,- is now denoted by b_j, the equation is the same as at first; hence the determinant ... + [-2]b_ 8 - e.b_, - OA - ffA — ^M — . . = 0, . . . - e.b_, + [-l]b-,-«A -0A -SA -.. = 0, . . . - e,h_. - , - (F + 1) ^or.i-f'' Jo\ V »o (»o - l)(»o - ^)[^. - (* + !/][»„ - (A - 1)^] By attributing, in these equations, special integral values to Jc, will be obtained the values of all the single summations appearing in the preceding expression for n (0). With regard to the double summations, we may pro- ceed as follows : Substitute i + h for *', then resolve the expression under consideration into partial fractions with respect to i as variable, and sum between the limits — oo and -f <» ; the fractions occurring in the result thus obtained are next resolved into partial fractions with reference to k, and the summations, with reference to this integer, are taken between the limits 2 and + «> ; or, which is the same thing, between the limits and -\- oo • ON THE MOTION OF THE LUNAR PERIGEE. 267 and the terms corresponding to ^ = and k=l subtracted from the result, The single triple summation may be treated in an analogous manner. Thus we get ■^t, i' mi + 1 _ "^°K'i"^^°)pcoK^Vg.) 1,^1 9 1 TIFIF+T} 33V».(i-«./L V»o ^0 1-^0 a (4- 0o) J' ^Ui' 77 {i}{i+l}{i'}{i' + 2} _ -'^"Ha ^'°) pcot(W^o) __ 1,3,2,51 ^i.i' ■' {i}{i + l}{i'}{i' + l}{i' + 3} 3:rCOt(|-V0„) ■^i, i', i" 64V0„(l-6'„y(4-0„) 1_^ ' 7rCOt(7rVg„) _ J^ , 2 2 20 ~| . V«o 00 1-^0 4-^0 3(9- »„)J _ 2 ' V / f— J- + ^ , 9 -1 7rCOt(7rVg,) _ _25_ 138V«„(l-y„)' iL 00 1-^0 2C4-0„)J V^o 80, 4 9 9 4 _ tt" -I l-0„ ' (l-0„)^ 8(4-0„) + (4^::0J^ 9-«„ 30„r From which it follows that + + 4_« ' 9_« d + i^L V^o 00 1 — 00 2(4 32V0o(l-0o> 37rCOtf-|- ^00) ^]'' + ;:r-T7r 8V0.(l-0.)(4-0.) 0;0. ^cot(^V0.) ^r_ 1 2 9 1. + 128V"0;(l-0o)'tL 00 ^l-0o^2(4-0„)J ' +.^+ ^ -00 2(4- cot (tt V g„) _ 25 _ 1 + + + V0O 80, TTCOt^l-Vgoj p „cot(;r^g„) 1 2 2 20 -\^g 9;O--0,y(^-0,-)l V0. 0,^l-e„ + 4-0„^3(9-0„)J ^' cot^^yg.j p ^cot(7rVg,) 1 2 2 10 1^,^, 9-„(i_«„)(4-0„)L V0O «o l-0o'^4-e,^9-»„J ^ ^ .(24) (7 - 3e„) TT cot/-^ V 0.) 5'^ cot (-|- V 0o) 4V0.(l-0.X4-0oX9-0o) 010A + 16V0.(l-0o)(4-0o)(9-0o) 0^0. 268 COLLECTED MATHEMATICAL WORKS OF G. W. HILL This is the same result as would be obtained if, setting out with the equation 2) (c) = 0, and assuming that c = \/0o is an approximate value, we should expand the function sin^T— c) in ascending powers and pro- ducts of the coefficients ©j, Q^, &c. IV. In order to obtain a numerical result from the preceding investigation, we assume n = 17325594:".06085, n' = 1295977".41516, whence m = 0.08084 89338 08311.6. From an investigation (to be published hereafter) of the corresponding values of the a^, we have 2h, = + 0.00909 42448 77375.5 - 2h_,=- 0.01739 14939 23079.4 4A, = + 0.00011 75731 31569.1 - 4.h_, = + 0.00000 19654 85829.2 6A3 = + 0.00000 13613 28523.8 — 6A_a = + 0.00000 00738 11780.8 8h, = + 0.00000 00126 19314.9 - 8h_, = + 0.00000 00006 87885.7 lOAe = + 0.00000 00001 21722.9 -lQh_, = + 0.00000 00000 05777.1 12^6 = + 0.00000 00000 01147.9 -12A_e = + 0.00000 00000 00047.5 14A, = + 0.00000 00000 00010.6 - 14A_ , = + 0.00000 00000 00000.4 The values of the Ui derived from these are U^=z+ 0.00909 40932 76038.2 U_-,=- 01739 21860 78260.6 Z7, = + 0.00007 62192 02104.5 CL^ = + 0.00015 32094 08075.6 ^73 = + 0.00000 06474 24638.8 [!_,=- 0.00000 12670 56302.6 V^ = + 0.00000 00055 23086.8 U_, = + 0.00000 00115 67648.9 U, = + 0.00000 00000 47309.0 U_,=- 0.00000 00000 95049.5 Ue = + 0.00000 00000 00403.9 n_, = + 0.00000 00000 00867.3 Z7, = + 0.00000 00000 00003.4 CL, = - 0.00000 00000 00007.2 In combination with the values of Ei, which will be given elsewhere, these afford the following periodic series for : e = 1.15884 39395 96583 - 0.11408 80374 93807 cos 3t + 0.00076 64759 95109 cos 4t - 0.00001 83465 77790 cos 6r + 0.00000 01088 95009 cos 8r - 0.00000 00020 98671 cos IOt + 0.00000 00000 12103 cos 13t - 0.00000 00000 00211 cos 14r ON THE MOTION OF THE LUNAR PERIGEE. 269 The values of the coefficients ©p. ©i. ©a. &c., are the halves of these coefficients, except O^, which is equal to the first coefficient. On substituting the numerical values of these quantities in (24), and separating the sum of the terms into groups according to their order, for the sake of exhibiting the degree of convergence, we get Term of the zero order, 1.00000 00000 00000 Term of the 4*1 order, + 0.00180 46110 93432 7 Sum of the terms of the 8*'' order, + 0.00000 01808 63109 9 Sum of the terms of the 12tii order, + 0.00000 00000 64478 6 D (0) = 1,00180 47930 21011 2 As far as we can judge from induction, the value of n (O) would be affected, only in the 14*'' decimal, by the neglected remainder of the series, which is of the 16*'' order An error in n (0) is multiplied by 2.8 nearly in c. The value, which is derived thence for c , is = 1.07158 3277416016. In order that nothing may be wanting in the exact determination of this quantity, we will employ the value just obtained as an approximate value in the elimination between equations (21). The coefficients [i], as many of them as we have need for, have the following values : [- 4] = 46.8 , [1] = 8.27577 98905 1 , [- 3] = 23.13045 , [2] = 24.56211 3 , [-2]= 7.41678 05615 1, [3] = 48.85. [- 1] =-0.29688 63288 2300, If the quantities b^ are eliminated from equations (21) in the order b_i, b^, b_2, bj, b_3, bs, and b_4, it will be found that the coefficient of bp, in the principal equation, undergoes the following successive depressions : [0] = — 0.01055 32191 58933, [0](-i) = + 0.00040 72723 11650, [0](-i.i) = + 0.00001 50888 08423, [0](-2.-i.i) = + 0.00000 00253 21700, [0](-2. -1. 1. 8) = + 0.00000 00009 20430, [0](-3' -2' -1. !• 2) = + 0.00000 00000 03941 , [0](-3. -3, -1, 1, 3, 3) = + 0.00000 00000 00155 , [0](-i, -3, -2. -1, 1, 2, 3) = + 0.00000 00000 00008 . 270 COLLECTED MATHEMATICAL WORKS OF G. "W. HILL The last number is not sensibly changed by the elimination of any of the bj beyond h_^ on the one side, or bs on the other. This residual is so small that it will not be necessary to repeat the computation with another value of c : it will suflBce to subtract half of it from the assumed value of c. Thus we have as the final result : and, consequently, c =: 1.07158 33774 16012; 1 da dt = 0.00857 35730 04864. Let us compare this value with that obtained from Delaunay's literal expression,* 1 dw 3 , , 335 8 , 4071 « , 365493 ^ , 12833631 s ^-^ = T'"+ 32- *"+ W'"+ -204r"^+ -34576-"^ , 1373935965 , , 71038685589 , , 33145883707741 , ^ 589834 ^ 7077888 ^ 679477348 ' where m denotes the ratio of the mean motions of the sun and moon. On the substitution of the numerical values we have employed for these quan- tities, this series gives, term by term, -1-^ = 0.00419 6439 + 0.00394 3798 + 0.00099 5700 + 0.00030 3577 n dt + 0.00009 1395 + 0.00003 8300 + 0.00000 9836 + 0.00000 3468 = 0.00857 1503. From the comparison, it appears that the sum of the remainder of Delaunay's series is 0.00000 1070, somewhat less than would be inferred by induction from the terms of the series itself. And, although Delaunay has been at the gi'eat pains of computing 8 terms of this series, they do not suffice to give correctly the first 4 significant figures of the quantity sought. On the other hand, the terms of the highest order, computed in the expres- sion for n (0), were of the 12**^ order only ; and yet, as we have seen, they have sufficed for giving c exact nearly to the IS*'' decimal. As well as can be judged frotn induction, it would be necessary to prolong the series, in powers of »i, as far as m^', in order to obtain an equally precise result. Allowing that the two last figures of the foregoing value of - t- may be Th at vitiated by the accumulation of error arising from the very numerous opera- tions, we may, I think, assert that 13 decimals correctly correspond to the assumed value of m. It may be stated that all the computations have been made twice, and no inconsiderable portion of them three times. * Compte) Eendus de VAcademie des Sciences de Paris, Tom. LXXIV, p. 19. EMPIRICAL FORMULA FOR THE VOLUME OF ATMOSPHERIC AIR. 271 MEMOIK No. 30. Empirical Formula for the Volume of Atmospheric Air. (Analyst, Vol. IV, pp. 97-107, 1877.) The formula of Mariotte and Gay-Lussac is generally employed, in the laboratory, to reduce volumes, observed under one tension and temperature, to those which would have place under other tensions and temperatures. But Regnault, about 1845, made several series of experiments, which, if they may be relied upon, establish marked deviations from this formula. These experiments are detailed in the Memoires de I'Aeademie des Sciences de Paris, Tom. XXI. I propose to investigate a modification of the formula, the introduction of which makes it possible to satisfy nearly these experi- ments Let y denote the temperature, here always expressed in degrees of the centigrade scale ; P the tension or pressure, measured by the altitude, in meters, of a column of mercury, it is capable of supporting, the mercury being at the temperature O" and under the action of gravity which obtains at Regnault's laboratory ; and let V denote the volume. Then, for any given mass of air, these three quantities are so connected that, if any two of them are assigned, the remaining third is immediately determined. That is, we must have function (V, P, T) = 0, or, solved with respect to V, F= function (P, T). But the mode, in which T is to be measured, is arbitrary, and we may take atmospheric air as the thermometric substance, and assume that T increases, in direct proportion, as the volume, under constant pressure, increases. This gives V=F(F) +/(P). T. It is here taken for granted that, whatever may be the density of the air inclosed in the thermometer, its indications will be the same. It is true that the usual custom of experimenters has been to measure temperatures 272 COLLECTED MATHEMATICAL WORKS OF G. W. HILL by the augmentation of tensions under constant volume; but, when Ma- riotte's law holds, this gives results identical with those obtained by the former method. In this case we should have to write the equation F = Fcr)+f(r).T, but the first equation seems preferable. Now since, for any given constant temperature, the volume ought to be a function of the tension similar to what it is at any other temperature, it follows that, if ^^(P) is supposed to consist of a series of terms, each of the form KP'', where iT and k are constants, so that we may write then we ought to have f(P) = S.K,P'', where K^ denotes a constant, in general, different from K. Thus we should have The formula of Mariotte and Gay-Lussac assumes that F(P) and f{P) contain each only one term, in which ^^ — 1. But Regnault's experi- ments having shown the insufficiency of this, it is in order to see whether agreement between theory and observation cannot be brought about by annexing to V an additional term, for which k has a value different from — 1. Thus let us suppose that F= [K + E^T\ P-^ + {E' + K[T^ P-'-i _ E+K'P^ , K, + EiP^ rp - p i- p As F contains a factor, which is directly proportional to the mass of air considered, and inversely as the unit assumed for the measurement of volumes, we prefer to write the preceding equations thus : When the temperature is constant, the volumes are represented by the formula EMPIRICAL FORMULA FOR THE VOLUME OF ATMOSPHERIC AIR 273 that is, the result from Mariotte's law must be multiplied by the factor 1 + aP^, which differs but little from unity ; a is a small constant which measures the amplitude of the deviations from this law ; while |5 is a con- stant exponent so chosen that the more or less rapid variation of the devia- tions, in passing from one tension to another, may be represented as well as possible. It is evident that, in this manner, we get the utmost advantage that can be derived from the addition of a single term to V. The experiments of Regnault may be divided into two classes ; first, those where, the temperature remaining nearly constant, the volumes of the same mass of air, under different pressures, were observed ; second, those where, the volumes remaining nearly the same, the tensions were observed at the temperature of freezing and boiling water. It is obvious that experi- ments of these two kinds, extended over a suflEicient range of tension, would afford the data requisite for obtaining the values of the four constants a, a', a" and (i which enter into our adopted formula. The experiments of the first class are enumerated at pp. 374-379 of the volume quoted above. As the temperature is nearly the same for all, we assume that they have been made at the average of all the noted tempera- tures which is 4°. 747. To save labor, we may take the average of the observed volumes and tensions when they are nearly alike. In this way Regnault's 66 experi- ments are reduced to the 23 given in the following table. It may be noted that Fis here expressed by the number of grammes of mercury required to fill the volume. The column containing log (PF) exhibits the deviation from Mariotte's law ; did this law exactly hold, the numbers in this column would be identical for each series. It will be noted that, in general, they dimin- ish with increasing pressures. The volumes being supposed to be repre- sented by the equation a preliminary investigation has given the approximate values a=- 0.0024337 , /3 = 0.645 . With these have been computed the values of the expressions which stand at the head of the two last columns of the table, and which serve to obtain the coefficients of the equations of condition to be given presently. As the mass of air operated on was different in each series of experi- ments, K will have 9 different values ; it can, however, be eliminated. 28 274 Taking the COLLECTED MATHEMATICAL WORKS OF G. W. HILL common logarithms of each member of the equation last given, log E + log (1 + aPP) = log (P F) . Series. II. III. IV. V. VI. VII. VIII. IX. { Y . jr* i.^U, \JUQ, lug v-r r ). \ + aP? 1939.76 0.73899 4 3.156387 0.8244 969.65 1.47630 4 3.155790 1.2897 1939.37 2.11228 3 3.612412 1.6262 970.40 4.21020 3 3.611254 2.5430 642.82 6.35034 2 3.610886 3.3213 1939.72 2.06887 3 3.603472 1.6045 969.78 4.12663 6 3.602268 2.5102 1940.65 4.14235 2 3.905194 2.5164 979.78 8.17850 3 3.903803 3.9155 1939.85 4.21910 4 3.912988 2.5465 970.29 8.40648 4 3.911516 3.9863 626.91 12.98195 1 3.910545 5.2926 1940.23 6.77001 3 4.118444 3.4623 970.32 13.47353 4 4.116396 5.4226 685.11 19.00213 1 4.114562 6.7913 675.15 19.30191 2 4.115000 6.8612 1941.23 6.39003 2 4.093580 3.3347 969.98 12.72859 2 4.091543 5.2248 633.82 19.39954 1 4.089757 6.8842 1940.44 9.33401 3 4.257968 4.2676 970.53 18.54702 5 4.255283 6.6842 1945.06 11.47357 2 4.348632 4.8824 1053.78 21.05700 2 4.346146 7.2643 —0.1083 +0.2182 0.5281 1.5876 2.6664 0.5066 1.5452 1.5532 3.5737 1.5921 3.6857 5.8925 2.8758 6.1247 8.6846 8.8206 2.6861 5.7721 8.8654 4.1398 8.4774 5.1740 9.6137 To reduce the matter within the treatment of the method of least squares, it will be necessary to make some assumption regarding the prob- able errors of the observed P and F. We will, for convenience, suppose that they are such that the function log (P Y) has a probable error equal for all the observations ; an assumption somewhat precarious, it is true, but it seems that we cannot easily do better. Let the small corrections, which it is necessary to apply to the approx- imate values of log K, a and /3, be denoted by h log K, ha and ^/3, and let us put 8 log K= X , MSa = y, ad^ = z, where M denotes the modulus of common logarithms. 5 log (PF) being the excess of observed over calculated log {PV), we shall have the equation' of condition : P^ .. . P» x + 1 + aP y + l + aPf logP.z = a(PF). EMPIRICAL FORMULA FOR THE VOLUME OF ATMOSPHERIC AIR 275 A little consideration will show that x will be eliminated by taking the diflference of every two equations of condition arising from the same series, WW and attributing the weight - — to the resulting equation, w and w' denoting the weights of the equations whose difference is taken, and 2 . w the sura of the weights of all the equations in the series. Since the coeflBcients ofy and z, in the equations, are all positive and nearly proportional, it will be advantageous to adopt a new unknown u, such that Then the equations, with the weights that ought to be attributed to them, are Series. Weight. I. 0.4653i< , — 0.4490« = —0.000106 2 - 0.9168 —0.4685 = —0.000193 f II. i 1.6951 —0.6869 = +0.000255 % 1 0.7783 —0.2184 = +0.000448 % III. 0.9057 —0.4709 = —0.000252 2 IV. 1.3991 —0.3113 = +0.000076 1.2 r 1.4398 —0.3061 = +0.000038 Y V. 2.7461 —0.2764 = +0.000432 i 1.3063 +0.0296 = +0.000394 i 1.9603 —0.0183 = +0.000002 1.2 3.3290 +0.2605 = —0.000407 0.3 VI. 3.3989 +0.2800 = +0.000104 0.6 1.3687 +0.2787 = —0.000409 0.4 1.4386 +0.2982 = +0.000102 0.8 0.0699 +0.0195 = +0.000511 0.2 1.8901 -0.0642 = —0.000059 0.8 VII. ■ 3.5495 +0.2635 = —0.000117 0.4 1.6594 +0.3276 = —0.000058 0.4 VIII. 2.4166 +0.3099 = —0.000164 V IX. 2.3819 +0.4699 = —0.000005 1 The derived normal equations are 58.672 M — 0.0790 z = — 0.0005352, - 0.079 u + 2.4453 z = — 0.0000157. Whence M = - 0.000008962 , z = — 0.000006707 , y= + 0.000002216 , da= + 0.0000051, Sfi= + 0.00276. get Applying these corrections to the approximate values of a and (3 , we a = — 0.0034286, /3 = + 0.64776. 276 COLLECTED MATHEMATICAL WORKS OF G. W. HILL How well the experiments are represented by the formula, with these values of the constants, will best be seen from the following comparison of V P the values of ° ° given by Regnault and those computed from the for- ala: '' p;p^6— vjy o-uc Obs. Gal. Diff. 1.001414 1.001133 +281 1.001448 1.001132 +316 1.001224 1.001133 + 91 1.001421 1.001133 +288 1.002765 1.002233 +532 1.002759 1.002234 +525 1.002503 1.002236 +267 1.003539 1.004134 —595 1.003452 1.004133 —681 1.003309 1.004133 —824 1.002709 1.002209 +500 1.002724 1.002207 +517 1.002713 1.002206 +507 1.002528 1.002211 +317 1.002898 1.002203 +695 1.002762 1.002203 +559 1.003253 1.003417 —164 1.003090 1.003411 —321 1.003302 1.003407 —105 1.003336 1.003506 —170 1.003495 1.003508 — 13 1.003335 1.003508 —173 1.003448 1.003509 — 61 Obs. Cal. Difif. 1.005437 1.006694 —1257 1.005703 1.006694 — 991 1.004286 1.004777 — 491 1.004512 1.004770 — 258 1.004599 1.004779 — 180 1.004580 1.004771 — 191 1.008536 1.008106 + 430 1.008813 1.008108 + 705 1.008016 1.008286 — 270 1.008064 1.008269 — 205 1.007980 1.008288 — 308 1.004611 1.004601 + 10 1.004752 1.004601 + 151 1.008930 1.008648 + 282 1.008755 1.008642 + 113 1.006366 1.005876 + 490 1.006132 1.005880 + 252 1.006010 1.005869 + 141 1.006346 1.005878 + 468 1.005619 1.005738 — 121 1.005622 1.005736 — 114 1.005902 1.005832 + 70 It will be seen that the diflFerences, in the extreme cases, amount to a fourth part of the observed deviation from the law of Mariotte. Moreover, it is plain that some cause, which, varied from series to series, has operated to vitiate these experiments, since it is possible to determine a and /? so that any two series are well represented, but not possible when all the series are included in the investigation. It may be noted also that the experiments, in which the original volume was reduced to one-third, are not, in general, concordant with those where the reduction was to one-half. That these discrepancies are unavoidable will be evident from the fol- lowing exposition : Let us put com. log (PFJ = 2? (f). EMPIRICAL FORMULA FOR THE VOLUME OF ATMOSPHERIC AIR 277 The observations of Regnault may be condensed into the following nine results, all formed by combining tolerably concordant data : 1. F( L476)— F( 0.739)= 0.000598 2. F( 4.168)— F( 2.091) = 0.001181 3. F( 6.350)— F( 2.112)= 0.001526 4. F( 8.292)— F( 4.182) = 0.001437 5. F(12.982)— F( 4.219)= 0.002443 6. F(13.101)— F( 6.580) = 0.002042 7. F(19.276) — F( 6.580)= 0.003743 8. F(18.547)— F( 9.334) = 0.002685 9. P(21.057)—F(11.474)= 0.002486 These are the data actually furnished by Regnault for the determina- tion of the function F{P). Employing the graphical method, we endeavor to construct the curve whose equation is y=:F{x). One of the special values of F{x), as i^(0.739), may be taken arbitrarily, and then the value of -^'(1.476) becomes known. This premised, we see that each of the nine equations furnishes the length, direction and abscissae of the extremities of a chord, of the sought curve. Placing the chord, corresponding to the first equation, arbitrarily, and drawing the others on any part of the paper, but with the proper direction and abscissae of their extremities, we endeavor, by imparting a motion to all their points parallel to the axis of ?/, to make them fall into line as chords of the same continuous curve. We find that if we take 1, 2, 4, 6 and 7, they can be made to indicate a tolerably continuous curve ; but then 3, 5, 8 and 9 are not satisfied. Again, from this graphical process, we see that there cannot be much variation of curvature between the extremities of each chord, and hence the tangent to the curve, corresponding to the abscissa, which is the mean of the abscissae of the extremities, ought to be, very approximately, parallel to the chord ; or, in other terms, dx \ 2 j x^ — Xg This gives the following values of -^ : ■£ . dy dx ■ 1. 1.108 +0.0008113 2. 3.130 0.0005686 3. 4.231 0.0003770 4. 6.237 0.0003497 5. 8.600 0.0002788 6. 9.840 0.0003131 7. 12.428 0.0002948 8. 13.940 0.0002914 9. 16.265 +0.0002594 278 COLLECTED MATHEMATICAL WORKS OF G. W. HILL From the general course of these values of -^ , it may be gathered that this function, at first, diminishes rapidly, afterwards more slowly, and then tends, with higher values of cc, to become nearly constant. But while this is the conclusion from the tout ensemble, a comparison of some of the values contradicts it. Thus, from 1, 2 and 3, while -^ diminishes 0.0002427 ax in an interval 2.0 in x, it afterwards diminishes 0.0001916 in an interval 1.1 o{ X. All attempts then to represent these data by a curve, without singular points, must, evidently, show large errors. For the discussion of the second class of experiments, let us assume that a has the signification we have given it in the general formula for V. Then the volume remaining the same, if Pq ^^d P^ denote the tensions observed respectively at 0° and 100°, we have P, _ 1 + 100 «' + (« + 100 g") P? Po 1 + «PP ' p -^ is the quantity Regnault has designated by 1-f 100a, let us denote it by A ; then if, for convenience, we put r = 1 + 100a', / = a + 100a", each determined value of A will give the equation of condition r + Pl.r' = A + API.a. The following are Regnault's determinations of A augmented, in gene- ral, by 0.00018, for the reason we adopt the mean coeflScient 0.00018153 for the expansion of mercury between 0° and 100°, found by this experi- menter, instead of the value ^^Vir used by him (see Note, p. 31 of the volume) ; the last column contains the page of the volume, where the experi- ments may be found. f"" /.■ A. No. Obs. Ol)B.-Cal. Page. o!iio 0?149 1.36500 10 —0.00012 99 0.174 0.237 1.36531 3 —0.00004 99 0.266 0.362 1.36560 2 —0.00003 99 0.375 0.510 1.36598 4 +0.00005 99 0.548 0.746 1.36673 3 +0.00038 57 0.756 0.7535 1.36724 4 +0.00035 66 0.557 0.754 1.36651 18 +0.00014 43 0.656 0.757 1.36641 14 —0.00022 33 0.747 1.016 1.36663 3 —0.00014 68 0.771 1.049 1.36696 11 +0.00014 51 1.678 2.286 1.36778 2 —0.00059 109 1.693 2.306 1.36818 4 —0.00021 109 2.526 2.517 1.36962 2 —0.00018 114 2.622 2.614 1.36982 2 —0.00011 114 2.144 2.924 1.36912 2 +0.00007 109 3.656 4.992 1.37109 4 +0.00031 109 EMPIRICAL FORMULA FOR THE VOLUME OF ATMOSPHERIC AIR 279 Adopting, for convenience, as an unknown in the place of y , a; = ^ + / — 1.367, we have the following equations, to each of which we attribute a weight equal to a tenth of the number of experiments it is founded upon : z - 0.7086/ - 0.3268a = - 0.00300 X — 0.6064/ — 0.4398a = - 0.00169 X — O.482I7/ - 0.5790a = - 0.00140 X — 0.3534y' - 0.7a37a = - 0.00103 X - 0.1728/ - 0.93580 = — 0.00027 X — O.I6747' - 1.1410a = + 0.00034 X — 0.1670/ - 0.9354O = - 0.00049 X — 0.1650/ — 1.040 a = - 0.00059 X + 0.0105/ - 1.131 a = - 0.00037 X + 0.0317y' — 1.155 a = - 0.00004 X + 0.708 / — 1.913 a = + 0.00078 X + 0.718 7' - 1.934 a = + 0.00118 X + 0.818 7/ - 3.496 a = + 0.00263 X + 0.863 y/— 2.558 a = + 0.00383 X + 1.004 y' — 2.343 a = + 0.00212 X + 1.834 /- 3.175 a = +.0.00409 The derived normal equations, for determining x and y', are X - 0.0047/ - 1.162a = - 0.000144, - 0.0415* + 2.9547/ — 3.373a = + 0.007074, whence and x=— 0.000133 + 1.168a, / = + 0.002392 + 1.158a, a' = + 0.00364475 + 0.00010a, a" = + 0.00002392 + 0.00158a. The equation which determines a has already been obtained from the discussion of the first class of experiments ; it is °. "*" t'ly > = — 0.0024286 . 1 + 4.747a' The last three equations being solved, we gather that the volume of any mass of air is represented by the formula F= ^ [1 + aPP + (a' + a"P^) T] , in which a = - 0.002565, a'= + 0.0036445, a" =+ 0.00001987, /J= 0.64776. 280 COLLECTED MATHEMATICAL WORKS OF G. W. HILL How well the second class of experiments is satisfied by this formula may be seen from the numbers in the column headed Obs.-Cal. If we have T = ^-^^^^^^ = i29°.l, F takes the form 0.00001987 F Hence we have the noteworthy result that : About the temperature 130°, air follows quite exactly the law of Mariotte. For the following temperatures and pressures the volume vanishes : T. 0° P. 9995"49 — 50 4420.13 —100 2048.00 —150 896.26 —200 314.23 These numbers may be regarded as indications of the magnitude of pressure necessary for the condensation of air. The table is in accordance with the well-known fact that reduction of temperature facilitates conden- sation. A table is given below which will be found useful in the application of a! A- a."P^ of the formula. It contains the functions log (1 + aP^) and — — — =^ , * ^ ^ 1 + aP^ ' the latter being the coefficient of expansion under a constant pressure. As an example, let us suppose that the volume of a mass of air has been observed under the pressure 2". 5 and the temperature 20° ; it is required to find the factor necessary for reducing it to the pressure O^.TS and temperature 0°. From the table we get 3.07064. By employing the ordinary formula with the coefficient 0.003665 of expansion, there is obtained 3.06482, which differs from the preceding by about a 525**' part. Rigorously, observations of pressure made in localities having an inten- sity of gravity different from that which prevails at Regnault's laboratory ought to be multiplied by the ratio of the former to the latter. The latitude of Regnault's laboratory is stated at 48° 50' 14", the altitude above sea level at SO"", and the intensity of gravity at Q^.SOQe. EMPIRICAL FORMULA FOR THE VOLUME OF ATMOSPHERIC AIR 281 P. O^.O .1 .3 ,3 A .5 .6 .7 .8 .9 1 .0 1 .1 1 .2 1 .3 1 .4 1 .5 1 .6 1 .7 1 .8 1 .9 2 .0 3 .5 3 .0 3 .5 4 .0 4 .5 5 .0 5 .5 6 .0 6 .5 log(l + aPO. 0.000000 9.999749 9.999607 9.999489 9.999384 9.999288 9.999199 9.999115 9.999035 9.998958 9.998885 9.998814 9.998745 9.998678 9.998613 9.998549 9.998487 9.998436 9.998367 9.998308 9.998351 9.997979 9.997725 9.997485 9.997357 9.997039 9.996839 9.996636 9.996430 9.996339 351 143 118 105 96 89 84 80 77 73 71 69 67 65 64 63 61 59 59 57 273 354 340 328 318 310 203 196 191 186 CoefE.ofEip. 0.0036445 36511 36548 36579 36607 36683 36656 36678 36698 36719 36738 36756 36775 36793 36809 36836 36843 36858 36874 36889 36904 36977 37044 37106 37166 37334 37280 37333 37385 37436 66 37 31 28 35 34 33 30 21 19 18 19 18 16 17 16 16 16 15 15 73 67 63 60 58 56 53 52 51 49 P. 7"'.0 7 .5 8 .0 8 .5 9 .0 9 .5 10 .0 10 .5 11 .0 11 .5 12 .0 12 .5 13 .0 13 .5 14 .0 14 .5 15 .0 15 .5 16 .0 16 .5 17 .0 17 .5 18 .0 18 .5 19 .0 19 .5 20 .0 20 .5 21 .0 21 .5 log(l + aPP). 9.996053 9.995872 9.995695 9.995521 9.995352 9.995185 9.995021 9.994860 9.994702 9.994546 9.994393 9.994242 9.994093 9.993946 9.993800 9.993657 9.993515 9.993374 9.993336 9.993098 9.992962 9.992828 9.992695 9.993563 9.993433 9.993303 9.992174 9.993047 9.991931 9.991795 181 177 174 169 167 164 161 158 156 153 151 149 147 146 143 143 141 138 138 136 134 133 132 131 129 129 127 136 126 Coeff.ofExp. 0.0037485 37533 37580 37626 37671 37715 37758 37801 37843 37885 37935 37965 38005 38044 38083 38121 38159 38196 38233 38269 38306 38343 38377 38413 38447 38482 38516 38550 38584 38618 48 47 46 45 44 43 48 43 42 40 40 40 39 39 38 38 37 37 36 87 36 35 85 35 35 34 34 84 34 282 COLLECTED MATHEMATICAL WORKS OF G. W. HILL MEMOIE NO. 31. On Dr. Weiler's Secular Acceleration of the Moon's mean Motion. (Aatronomisehe Nachrichten, Vol. 91, pp. 251.254, 1878.) Dr. Weiler's conclusions are, in general, not admissible because the expressions he gives for the forces X, Y and Z* are incorrect. It is well known that the attraction of a body, whatever may be its bounding surface and law of interior density, always admits a potential function TT, such that "3F' 'W "35"" But if we form the expression Xdx + Ydy + Zdz from Dr. Weiler's values of X, Y and Z, it is found to be not an exact dif- ferential : hence these values are erroneous. They appear to have been derived by some illegitimate transformations from the formulas in the Micaniqiie Celeste, Tom. II, p. 22. After changing to Dr, Weiler's notation, Laplace's expressions for the attraction of a homo- geneous ellipsoid of revolution become i'du yr_ 3hx P"^ u*du yr_3hy P^^i^du__ ^_ 3Az /*' m' where k' is given by the equation 2Icf' = r' + X- y (r' - /)' + 4:Xz\ But Dr. Weiler seems to have put y = r. This cannot be done for the Jtf which is outside of the sign of integration, without losing some part of the attraction which is of the order of the small quantity A. Hansen {Fundamenta Nova, pp. 1-16) has elaborated this matter with great generality and much elegance. From this source we learn that the proper expression for the potential function of the action between the earth and moon is " r L ^ Mr' ^ "Mr' "J ' * Astronomlsche Nachrichten, Vol. 90, pp. 372-373. ON DR. WEILER'S SECULAR ACCELERATION OF MOON'S MEAN MOTION 283 where A , B and C are the moments about the axes oi x, y and a , supposed to coincide with the principal axes of rotation. In getting this expression, no assumption respecting the bounding surface or law of density of the earth is necessary ; it is only assumed that terms of the third and higher orders with respect to the ratio of the dimensions of the earth to the radius-vector of the moon may be neglected. Very nearly we have B = A, and, if this assumption is adopted, W takes the simpler form Q j^ If we put h=: X {M-\- m) , and a = f „ ^ — , a will be a constant inde- pendent of the linear and time units ; and measurements of arcs of the meri- dian, of the length of the second's pendulum, and the data afforded by the phenomena of precession and nutation, show that its value is very approxi- mately a= 0.0016395. The expressions of the forces, which ought to be substituted for those given by Dr. Weiler, are then X=- 284 COLLECTED MATHEMATICAL WORKS OF G. W. HILL MEMOIE N"o. 32. Besearches in the Lunar Theory.* (American Journal of Mathematics, Vol. I, pp. 5-26, 129-147, 245-260, 1878.) When we consider how we may best contribute to the advancement of this much-treated subject, we cannot fail to notice that the great majority of writers on it have had before them, as their ultimate aim, the construction of Tables ; that is, they have viewed the problem from the stand-point of practical astronomy rather than of mathematics. It is on this account that we find such a restricted choice of variables to express the position of the moon, and of parameters, in terms of which to express the coeflScients of the periodic terms. Again, their object compelling them to go over the whole field, they have neglected to notice many minor points of great interest to the mathematician, simply because the knowledge of them was unnecessary for the formation of Tables. But the developments having now been carried extremely far, without completely satisfying all desires, one is led to ask whether such modifications cannot be made in the processes of integration, and such coordinates and parameters adopted, that a much nearer approach may be had to the law of the series, and, at the same time, their convergence augmented. Now, as to choice of coordinates, it is known that, in the elliptic theory, the rectangular coordinates of a planet, relative to the central body, the axes being parallel to the axes of the ellipse described, can be developed, in terms of the time, in the following series : a; = a ^ — Ji.'"'' coeig, i ^ — oo i=+c ^=^ S 4^-^"''^^^^' t = 03 a and h being the semi-axes of the ellipse, e the eccentricity, g the mean anomaly and, for positive values of i , the Besselian function (in Hansen's notation) '^'^ ~ 1.2...il l.(i + l)^ 1.2.{i + l)ii + 2) J' » Communicated to the National Academy of Sciences at the session of April, 1877. RESEARCHES IN THE LUNAR THEORY 28 5 while, for negative values of i, we have and, for the special case of i = 0, we put the indeterminate 1 r(-i) — 8 /, Here the law of the series is manifest, and the approximation can easily be carried as far as we wish. But the longitude and latitude, variables employed by nearly all the lunar-theorists, are far from having such simple expressions ; in fact, their coefficients cannot be expressed finitely in terms of Besselian functions. And if this is true in the elliptic theory, how much more likely is a similar thing to be true when the complexity of the problem is increased by the consideration of disturbing forces ? We are then justified in thinking that the coefficients of the periodic terms in the development of rectangular or quasi-rectangular coordinates are less complex functions of their parameters than those of polar coordinates. There is also another advantage in employing coordinates of the former kind ; the differential equations are expressed in purely algebraic forms ; while, with the latter, circular functions immediately present themselves. For these reasons I have not hesitated to substitute rectangular for polar coordinates. Again, as to parameters, all those who have given literal developments, Laplace setting the example, have used the parameter m , the ratio of the sidereal month to the sidereal year. But a slight examination, even, of the results obtained, ought to convince any one that this is a most unfortunate selection in regard to convergence. Yet nothing seems to render this para- meter at all desirable, indeed, the ratio of the synodic month to the sidereal year would appear to be more naturally suggested as a parameter. Some instances of slow convergence with the parameter m may be given from Delaunay's Lunar Theory ; the development of the principal part of the 15 coefficient of the evection in longitude begins with the term — me, and ends .^, ^, ^ 413277465931033 , • • +u • • i ^ ^ *v, with the term — 15288238080 — "* ' ^Sam, in the principal part of the coefficient of the inequality whose argument is the difference of the mean 21 anomalies of the sun and moon, we find, at the beginning, the term — mee', n . .X. J iu ^ 1207454026843 , , ,, . u i.i +t, ^. i. .v and, at the end, the term — rrvee. It is probable that, by the adoption of some function of m as a parameter in place of this quantity, whose numerical value, in the case of our moon, should not greatly exceed 286 COLLECTED MATHEMATICAL WORKS OF G. W. HILL that of m, the foregoing large numerical coeflBcienta might be very much diminished. And nothing compels us to use the same parameter throughout ; one may be used in one class of inequalities, another in another, as may prove most advantageous. It is known what rapid convergence has been obtained in the series giving the values of logarithms, circular and elliptic functions, by simply adopting new parameters. Similar transformations, with like effects, are, perhaps, possible in the coeflBcients of the lunar ine- qualities. However, as far as my experience goes, no useful results are obtained by experimenting with the present known developments ; in every case it seems the proper parameter must be deduced from a priori consider- ations furnished in the course of the integration. With regard to the form of the differential equations to be employed, although Delaunay's method is very elegant, and, perhaps, as short as any, when one wishes to go over the whole ground of the lunar theory, it is sub- ject to some disadvantages when the attention is restricted to a certain class of lunar inequalities. Thus, do we wish to get only the inequalities whose coeflBcients depend solely on »i , we are yet compelled to develop the disturb- ing function B to all powers of e. Again, the method of integrating by undetermined coeflBcients is most likely to give us the nearest approach to the law of the series ; and, in this method, it is as easy to integrate a diflfer- ential equation of the second order as one of the first, while the labor is increased by augmenting the number of variables and equations. But Delaunay's method doubles the number of variables in order that the differ- ential equations may be all of the first order. Hence, in this disquisition, I have preferred to use the equations expressed in terms of the coordinates, rather than those in terms of the elements ; and, in general, always to dimin- ish the number of unknown quantities and equations by augmenting the order of the latter. In this way the labor of making a preliminary develop- ment of B in terms of the elliptic elements is avoided. In the present memoir I propose, dividing the periodic developments of the lunar coordinates into classes of terms, after the manner of Euler in his last Lunar Theory,* to treat the following five classes of inequalities: 1. Those which depend ouly on the ratio of the mean motions of the sun and moon. 2. Those which are proportional to the lunar eccentricity. 3. Those which are proportional to the sine of the lunar inclination. 4. Those which are proportional to the solar eccentricity. 5. Those which are proportional to the solar parallax. * Theoria Motuum Luna, nova methodo pertractaia. PetropoU, 1773. RESEARCHES IN THE LUNAR THEORY 287 A general method will also be given by which these investigations may be extended so as to cover the whole ground of the lunar theory. My methods have the advantage, which is not possessed by Delaunay's that they adapt themselves equally to a special numerical computation of the coeflQcients, or to a general literal development. The application of both procedures will be given in each of the just mentioned five classes of inequalities, so that it will be possible to compare them. I regret that, on account of the difficulty of the subject and the length of the investigation it seems to require, I have been obliged to pass over the important questions of the limits between which the series are con- vergent, and of the determination of superior limits to the errors committed in stopping short at definite points. There cannot be a reasonable doubt that, in all cases, where we are compelled to employ infinite series in the solution of a problem, analysis is capable of being prefected to the point of showing us within what limits our solution is legitimate, and also of giving us a limit which its error cannot surpass. When the coordinates are devel- oped in ascending powers of the time, or in ascending powers of a parameter attached as a multiplier to the disturbing forces, certain investigations of Cauchy afford us the means of replying to these questions. But when, for powers of the time, are substituted circular functions of it, and the coefficients of these are expanded in powers and products of certain parameters produced from the combination of the masses with certain of the arbitrary constants introduced by integration, it does not appear that anything in the writings of Cauchy will help us to the conditions of convergence. CHAPTER I. Differential Equations. — Properties of motion derived from Jacohi's integral. We set aside the action of the planets and the influence of the figures of the sun, earth and moon, together with the action of the last upon the sun, as also the product of the solar disturbing force on the moon by the small fraction obtained from dividing the mass of the earth by the mass of the sun. These are the same restrictions as those which Delaunay has imposed on his Lunar Theory contained in Vols, XXVIII and XXIX of the Memoirs of the Paris Academy of Sciences, The motion of the sun, about the earth, is then in accordance with the elliptic theory, and the ecliptic is a fixed plane. Let us take a system of rectangular axes, having its origin at the centre of gravity of the earth, the axis of x being constantly directed toward the centre of the sun, the axis of y toward a point in the ecliptic 90° in advance 288 COLLECTED MATHEMATICAL WORKS OF G. W. HILL of the sun, and the axis of z perpendicular to the ecliptic. In addition, we adopt the following notation : r' = the distance of the sun from the earth ; A' = the sun's longitude; fi = the sum of the masses of the earth and moon, measured by the Telocity these masses produce by their action, in a unit of time, and at the unit of distance; m' = the mass of the sun, measured in the same way; n' = the mean angular velocity of the sun about the earth; a' = the sun's mean distance from the earth . In accordance with one of the above-mentioned restrictions we have the equation: m' — n'^a'^ The axes of x and y having a velocity of rotation in their plane, equal to— -=— , it is evident that the square of the velocity of the moon, relative to cit the earth's centre, has for expression, in terms of the adopted coordinates, ^rp _ r dx _ dX' n" r dy dX' y d^ ^^-lir y~dr]^l~df^''~drj^^F dx^ + dy'^ -\r dz^ n dX' xdy--ydx dX''' ,, ., w '^^^t di — ^-^^^'^ +2/;- The potential function, in terms of the same coordinates, is V (a;' + «/' + 2") V[(»^ -«)' + «/' + 2'] ~7^** If the second radical in this expression is expanded in a series proceed- ing according to descending powers of r' and the first term — j- omitted, since it disappears in all differentiations with respect to the moon's coordi- nates, the following expression is obtained : a = V {x' + y' + z') + n"^lx'-^(f + z'-)-] + -^ -^ [^* - 3^ cf + «') + ! (f + m + RESEARCHES IN THE LUNAR THEORY 289 Since the diflferential equations of motion are of the form d dT _dT _ da dt ' -, d

' There does not seem to be any function of x, y and 2, which, adopted as a new variable to accompany u and s, would reduce this to a very simple form. However, when we are engaged in determining the inequalities independent of the inclination of the lunar orbit, this transformation will be useful to us. For, in this case, z = 0, and the values of u and s become u = x + y nj — 1, s =x-y>j—l, and T is given by the equation c.rp_duds dX' uds — sdu dX'^ ^^-~dF ~~di dt +~dF"*- Although il is expressed most simply by the systems of coordinates we have just employed, the integration of the differential equations will be easier, if we suppose that the axes of x and y have a constant instead of a variable velocity of rotation, the axis of x being made to pass through the RESEARCHES IN THE LUNAR THEORY 291 mean position of the sun instead of the true. To obtain the expression for T correspondent to this supposition, it is necessary only to write n' for — 5— in the former values. As for £i , it can be written thus "— r ^ [r'" — 2r'8 + r'Ji r" ' where r' = x' + y' + z' = the square of the moon's radius vector ; S = xcosu + y sino; o = the solar equation of the centre. This function being expanded in a series of descending powers of r', as before, we have S' f+i-' ''{7? + f) + w" a" lis^- ■i*^] a ■ i^S^- -fr-'^] a" /5 -[^„' dy - dy \^l^ + 3^-w-p^Y The problem is then reduced to the integration of two differential equations of the first order. Were this accomplished, and p eliminated from the two integral equations, we should have the equation of the orbit. If we put W=2x+ fA + Sa;^ — 2(7— /)»T, the differential equations can be written in the canonical form, dx __dW dy dp ' dp _ dW dy dx 294 COLLECTED MATHEMATICAL WORKS OF G. W. HILL It may be worth while to notice also the single partial differential equation, to the integration of which our problem can be reduced. Return- ing to the arbitrary linear and temporal units, and for convenience, reversing the sign of G, if a function of x and y can be found satisfying the partial differential equation and involving a single arbitrary constant h , distinct from that which can be joined to it by addition, the intermediate integrals of the problem will be dz dV , , dy dV , dt dx " dt dy ' and the final integrals a and c being two additional arbitrary constants. The truth of this will be evident if we differentiate the four integral equations with respect to t and compare severally the results with the partial differential coefiBcients of the partial differential equation with respect io x, y,h and G. Although, in this manner, the problem seems reduced to its briefest terms, yet, when we essay to solve it, setting out with this partial differen- tial equation, we are led to more complex expressions than would be expected. It would be advisable, in this method of proceeding, to substitute polar for rectangular coordinates, or to put a; = rcosv, y^raml denoting the longitude of the moon, we have IdX J dt dt ^ being a constant. Thus, after the longitude is determined in terms of t, the radius vector is obtained by a quadrature. But it can also be found, without the necessity of an integration, by dividing the integral by r^ and then eliminating the term —^ ^g- by means of its value derived from the second differential equation ; in this way we get *i3 M^ a -^ + |w'»8ia25. dt + iiS-fw''cosV. df As we desire to make constant numerical application of the general theory, established in what follows, to the particular case of the moon, we delay here, for a moment, to obtain the numerical values of the three con- stants /[/, n' and G. The value of |U may be derived either from the observed value of the constant of lunar parallax combined with the mean angular motion of the moon, or from the intensity of gravity at the earth's surface and the ratio of the moon's mass to that of the earth. We will adopt the latter procedure. The value of gravity at the equator, g'= 9.779741 metres, the unit of time being the mean solar second. We propose, however, to take the mean solar day as the unit of time, and the equatorial radius of the earth as the linear unit. This number must then be multiplied by w^ififooYT^ ' (6377397.15 metres is Bessel's value of the equatorial radius.) Moreover, 296 COLLECTED MATHEMATICAL WOEKS OF G. W. HILL the theory of the earth's figure shows that, in order to obtain the attractive force of the earth's mass, considered as concentrated at its centre of gravity, a second multiplication must be made by the factor 1.001818356. "With our units then this force is represented by the number 11468.338: and the moon's mass being taken at ^. ^^^,„ of the earth's, her attractive force is ° 81.52277 represented by the number 140.676. Consequently II = 11609.014. The sidereal mean motion of the sun in a Julian year is 1295977".41516, whence n' = 0.017202124. The value of G might be obtained from the observed values of the moon's coordinates and their rates of variation at any time. However, as the eccen- tricity of the earth's orbit is not zero, C obtained in this manner would be found to undergo slight variations. The mean of all the values obtained in a long series of observations might be adopted as the proper value of this quantity when regarded as constant. But it is much easier to derive it approximately from the series which will be established in the following chapter. Here n denotes the n! moon's sidereal mean motion, and m is put for , . In this formula the n — n' terms which involve the squares of the lunar eccentricity and inclination and of the solar parallax are neglected; this, however, is not of great moment, as, being multipled by at least m^ they are of the fourth order with respect to smallness. The observations give n = 0.22997085, hence C= 111.18883. If it is proposed to assume the units of time and length so that ^ and n' may both be unity, it will be found that the first is equal to 58.13236 mean solar days, and the second to 339.7898 equatorial radii of the earth. The corresponding value of Cis 3.254440. Let us now notice some of the properties of motion which can be derived from Jacobi's integral. This integral gives the square of the velocity relative to the moving axes of coordinates ; and, as this square is necessarily positive, the putting it equal to zero gives the equation of the surface which RESEARCHES IN THE LUNAR THEORY 297 separates those portions of space, in which the velocity is real, from those in which it is imaginary. This equation is, in its most general form, + ,.,„> :^^...^.n = ^ + \^'^'-^> [(«' - ^y + 2/1 , V (ar' + 2/' + «') V [(«' - «)' + 2/' + 2'] " ' 2 " "■ 3 which is seen to be of the 16th degree. As y and z enter it only in even powers, the surface is symmetrically situated with respect to the planes of xy and xz. The left member is necessarily positive, (the folds of the surface, for which either or both the radicals receive negative values, are excluded from consideration), hence the surface is inclosed within the cylinder whose axis passes through the centre of the sun perpendicularly to the ecliptic, and whose trace on this plane is a circle of the radius As, in general, the second term of the quantity, under the radical sign, is much smaller than the first, this radius is, quite approximately VSa'. Thus, in the case of our moon, assuming — j =sin8".848, we have this radius = V 3.00138 3a'. It is evident that, for all points without this cylinder, the velocity is real ; and as, for large values of 2, whether positive or negative, the left member of the equation becomes very small, it is plain that the cylinder is asymptotic to the surface. Every right line, perpendicular to the ecliptic, intersects the surface not more than twice, at equal distances from this plane, once above and once below. Let us now find the trace of the surface on the plane of xy. Putting p for the distance of a point on this trace from the centre of the sun, and it is evident that the cubic equation, will give the limits between which the values of p can oscillate. If C is negative, this equation has but one real root which is negative; consequently, in this case, the surface has no intersection with the plane of xy. But, in all the satellite systems we know, G is positive, and this condition is prob- ably necessary to insure stability. Hence we shall restrict our attention to the case where G is positive. Then all the roots of the equation are real, 298 COLLECTED MATHEMATICAL WORKS OP G. W. HILL and two are positive. It is between the latter roots that p must always be found. To compute them, we derive the auxiliary angle 6 from the formula or, since 6 differs but little from 90°, with more readiness from C08= = [^+*^?V^] or, as -72-72 is a small quantity, with suflBcient approximation from _ V^^^' COSS = 1 + The two roots are then The trace of the surface on the plane of xy is then wholly comprised in the annular space between the two circles described from the centre of the sun as centre with the radii pj and p^- Moreover, as in most satellite systems we ^*^® J^' ®*^"*^ *° * ^®^^ ^°^*^^ fraction, (for our moon -^3 = 322930.2 ) ' it is plain that, for points whose distance from the earth is comparable with their distance from the sun, the trace is approximately coincident with these circles. For the term -£^, in the equation, may then be neglected in com- parison with the other terms. In the case of our moon there is foumd e = 87° 52' 11".53 , and hence p^ =z 22815.15 , p, = 23816.09 , and, if r and p are regarded as the variables defining the position of a point in the plane xy, the following table gives some corresponding values of these RESEARCHES IN THE LUNAR THEORY 299 quantities, for each of the two branches of the trace approximating severally to the two circles. r. P- '/■. P- 433.3257 22878.69 439.7922 23751.81 450 22876.17 450 23753.37 500 22869.68 500 23760.04 600 22860.13 600 23769.85 1000 22841.59 1000 23788.87 10000 22817.70 10000 23813.43 46127.70 22815.68 47127.55 23815.53 The first and last values correspond to the four points where the curves intersect the axis of x on the hither and thither side of the sun. It will be seen that the approximation of the branches to the circles is quite close, except in the vicinity of the earth, where there is a slight protruding away from them. In addition to these two branches of the trace, there is, in the case where C exceeds a certain limit, a third closed one about the origin much smaller than the former. As the coordinates of points in this branch are small fractions of a', its equation may be written, quite approximately, r ^ It intersects the axis of j/ at a distance from the origin very nearly « — ^ y"- G ' and the axis of a; at points whose coordinates are the smallest (without regard to sign) roots of the equations jx_ _^ _nV^ =G + I w'V - \n" {a' — x)\ -JL+ -^^ = G + in"a" - i n" (a' - x)'. For the moon these quantities have the values 2/0 = 104.408, a^ = — 109.655, a;, = + 109.694 . This branch then does not differ much from a circle having its centre at the origin, more closely it approximates to the ellipse whose major axis = 0:2 — Xi, and minor axis ■=■ 2^o. 300 COLLECTED MATHEMATICAL WORKS OP G. W. HILL The value of the coordinate a, for the single intersection of the surface with the axis of 2 above the plane ofxy, is given by the single positive root of the equation z V (a" + z') For the moon the numerical value of this root is 00 = 102.956 . The intersection of the surface with the perpendicular to the plane of xy passing through the centre of the sun is, in like manner, given by the equation having but a single positive root, which is nearly 2 „i 1 + 1-^' or, with less exactitude, *.-l''' 2„ = la-. From this investigation it is possible to get a tolerably clear idea of the form of this surface. When G exceeds a certain limit, it consists of three separate folds. The first being quite small, relatively to the other two, is close, surrounds the earth and somewhat resembles an ellipsoid whose axes have been given above. The second is also closed, but surrounds the sun, and has approximately the form of an ellipsoid of revolution, the semiaxis in the plane of the ecliptic being somewhat less than a', and the semiaxis of revolution perpendicular to the ecliptic and passing through the sun being about two-thirds of this. This fold has a protuberance in the portion neigh- boring the earth. The third fold is not closed, but is asymptotic to the cylinder mentioned at the beginning of the investigation of the surface. Like the second, it also is nearly of revolution about an axis passing through the centre of the sun and perpendicular to the ecliptic. The radius of its trace on the ecliptic is about as much greater than a', as the radius of the trace of the second fold falls short of that quantity. The fold has a protuber- ance in the portion neighboring the earth, and which projects towards this RESEARCHES IN THE LUNAR THEORY 391 body. The whole fold resembles a cylinder bent inwards in a zone neigh- boring the ecliptic. What modifications take place in these folds when the constants involved in the equation of the surface are made to vary, will be clearly seen from the following exposition. Let us, for brevity, put and, for the moment, adopt a', the distance of the earth from the sun, as the linear unit, and transfer the origin to the centre of the sun, and moreover put At n"a" Then the intersections of the surface, with the axis of aj, will be given by the two roots of the equation x*-x'' — h3? + (A + 2 — 2r)a; — 3 = 0, which lie between the limits and 1 ; by the two roots of «* — ai'-Aa^ + (A + 2 + 2;')a; — 2 = 0, which lie between 1 and V^; and by the two roots of x^ — a^-'h3? + (h — 2 — 2r)x + 2 = 0, which lie between and — \/h. Hence, if G diminishes so much that the first of these three equations has the two roots, lying between the mentioned limits, equal, the first fold will have a contact with the second fold ; and if G fall below this limit, the roots become imaginary, and the two folds become one. Again, if G is diminished to the limit where the second equation has the mentioned pair of roots equal, the first fold will have a contact with the third; and when G is less than this, these two folds form but one. And when G is less than both these limits, there will be but one fold to the surface. In the spaces inclosed by the first and second folds the velocity, relative to the moving axes of coordinates, is real ; but, in the space lying between these folds and the third fold, it is imaginary; without the third fold it is again real. Thus, in those cases, where G and y have such values that the three folds exist, if the body, whose motion is considered, is found at any time within the first fold, it must forever remain within it, and its radius 302 COLLECTED MATHEMATICAL WORKS OF G. W. HILL vector will have a superior limit. If it be found within the second fold, the same thing is true, but the radius vector will have an inferior as well as a superior limit. And if it be found without the third fold, it must forever remain without, and its radius vector will have an inferior limit. Applying this theory to our satellite, we see that it is actually within the first fold, and consequently must always remain there, and its distance from the earth can never exceed 109.694 equatorial radii. Thus, the eccentricity of the earth^s orbit being neglected, we have a rigorous demon- stration of a superior limit to the radius vector of the moon. In the cases, where G and y have such values that the surface forms but one fold, Jacobi's integral does not afford any limits to the radius vector. When we neglect the solar parallax and the lunar inclination, the pre- ceding investigation is reduced to much simpler terms. The surface then degenerates into a plane curve, whose equation, of the sixth degree, is r ^ It is evidently symmetrical with respect to both axes of coordinates, and is contained between the two right lines, whose equations are , rw and which are asymptotic to it. It intersects the axis of y, at two points, whose coordinates are The cubic equation. gives the values of r, for which the curves intersect the axis of a;. If (2C)3 > 9fin', [20 this equation has two real roots between the limits and -|- y -^-fz' If (2(7)8 = Q^n', these roots become equal. And if (2(7)»<9/iw', RESEARCHES IN THE LUNAR THEORY 303 there are no real roots between these limits, and the curve has no intersec- tion with the axis of x. The figures below exhibit the three varieties of this curve. Fig. 1 represents the form of the curve in the case of our moon. In Fig. 2 we see that the small oval of Fig. 1 has enlarged and elongated itself so as to touch the two infinite branches; while, in Fig. 3, it has disappeared, the portions of the curve, lying on either side of the axis of x, having lifted themselves away from it, and the angles having become rounded off. In Fig. 1, the velocity is real within the oval, and also without the infinite branches, but it is imaginary in the portion of the plane lying between the oval and these branches. Hence, if the body be found, at any time, within the oval, it cannot escape thence, and its radius vector will have a superior limit; and, if it be found in one of the spaces on the concave side of the infinite branches, it cannot remove to the other, and its radius vector will have an inferior limit. In the case represented in Fig. 2, the same things are true, but it seems as if the body might escape from the oval to the infinite spaces, or vice versa, at the points where the curve intersects the axis of x. However, at these points, the force, no less than the velocity, is reduced to zero. For the distance of these points from the origin is the positive root of the equation 3^-^ = «' or V2g _ V9Atw' ~W~ 3n' 304 COLLECTED MATHEMATICAL WORKS OF G. W. HILL and this value is the same as that given by the equation 4- — 3w"= 0. r In consequence the forces vanish at these two points, and thus we have two particular solutions of our differential equations.* In the case represented in Pig. 3, the integral does not afford any superior or inferior limit to the radius vector. The surface, or, in the more simple case, the plane curve, we have discussed, is the locus of zero velocity ; and the surface or plane curve, upon which the velocity has a definite value, is precisely of the same character and has a similar equation. It is only necessary to suppose that the G of the preceding formulae is augmented by half the square of the value attrib- uted to the velocity. Thus, in the case of our moon, it is plain the curves of equal velocity will form a series of ovals surrounding the origin, and approaching it, and becoming more nearly circular as the velocity increases. Applying the simple formulae, where the solar parallax is neglected, to the moon, we find that the distance of the asymptotic lines, from the origin, is V |^, = 500.4992. The distance of the points on the axis of a;, at which the moon would remain stationary with respect to the sun, is ^3^, = 235.5971. If the auxiliary angle Q is derived from the equation we get e = 32° 49' 6".63 ; and the distances from the origin, at which the curve of zero velocity inter- sects the axis of as, are given by the two expressions y fll 3w' s:n(60°-|-), and the numbers are 109.6772 and 435.5623. These values differ but little from the previous more general determinations. * The corresponding Bolution, in the more general problem of three bodies, may be seen in the Mecanigue Celeste, Tom. IV, p. 310. RESEARCHES IN THE LUNAR THEORY 395 CHAPTER II. Determination of the inequalities which depend only on the ratio of the mean motions of the sun and moon. If the path of a body, whose motioa satisfies the difierential equations ^"'^-t-[f-3^']^ = «' dp intersect the axis of x at right angles, the circumstances of motion, before and after the intersection, are identical, but in reverse order with respect to the time. That is, if t be counted from the epoch when the body is on the axis of 33 , we shall have X — function (f ) , y = t. function (t') . For if, in the differential equations, the signs of y and t are reversed, but that of X left unchanged, the equations are the same as at first. A similar thing is true if the path intersect the axis of y at right angles; for if the signs of x and t are reversed, while that of y is not altered, the equations undergo no change. Now it is evident that the body may start from a given point on, and at right angles to, the axis of x, with different velocities; and that, within certain limits, it may reach the axis of y, and cross the same at correspond- ingly different angles. If the right angle lie between some of these, we judge, from the principle of continuity, that there is some intermediate velocity with which the body would arrive at and cross the axis of y at right angles. The difiiculty of this question does not permit its being treated by a literal analysis; but the tracing of the path of the body, in numerous special cases, by the application of mechanical quadratures to the differential equa- tions, enables us to state the following circumstances: If the body be projected at right angles to, and from a point on, the axis of a;, whose distances from the origin is less than 0.33 .... y -^, there is at least one (near the limit there are two) value of the initial velocity, with which the body, in arriving at the axis of y, will cross it at right angles. Beyond this limit it appears no initial velocity will serve to make the body reach the axis of y under the stated condition. If the body move from one axis to the other and cross both of them perpendicularly, it is plain, from the preceding developments, that its orbit 30 306 COLLECTED MATHEMATICAL WORKS OF G. W. HILL will be a closed curve symmetrical with respect to both axes. Thus is obtained a particular solution of the differential equations. While the general integrals involve four arbitrary constants, this solution, it is plain, has but two, which may be taken to be the distance from the origin at which the body crosses the the axis of x and the time of crossing. Certain considerations, connected with the employment of Fourier's Theorem and the possibility of developing functions in infinite series of periodic terms, show that, in this solution, the coordinates of the body can be represented, in a convergent manner, by series of the following form : x = A^GQi%\y(f, — t^'\ + A^ cos3[i'(^ — ^o)] + At cos5[i'(i — ^„)] + . . . , t/ = 5„Bin[.'(^ — O] + J?, sin 3 [v' (^ - g] + 5, sin 5 [^ (^ — i!,)] + . . ., where U denotes the time the body crosses the axis of as, and is the time V of a complete revolution of the body about the origin. We may regard v and the series, given above, may be written X = 2,. a.- cos (2i + 1) t , y = Sf.&t sin (2i + 1) r , the summation being extended to all integral values positive and negative zero included, for i. By adopting polar coordinates such that X =z r cos ¥> , y ^1" sin

. We may also remind the reader that they determine rigorously all the parts of the lunar coordinates which depend only on the ratio of the mean motions of the sun and moon and on the lunar eccentricity. The Jacobian integral, in the present notation, is Du.Ds + J^. +im'(u + sy=G. (us)* * ^ '' The most ready method of getting the values of the coefficients a^, is that of undetermined coefficients; the values of u and s, expressed by the pre- ceding summations with reference to i, being substituted in the differential equations, the resulting coefficient of each power of ^, in the left members, is equated to zero, which furnishes a series of equations of condition sufficient to determine all the quantities a;. For this purpose we may evidently employ any two independent combinations of the three equations last written, and it will be advisable to form these combinations in such a manner that the process of deriving the equations of condition may be facilitated in the largest degree. Now it will be recognized that the presence of the term 7-^, in one of the factors of the differential equations, is a hindrance to (lis)* their ready integration, being the single thing which prevents them from being linear with constant coefficients. Hence we avail ourselves of the possibility of eliminating it. Multiplying the first differential equation by 308 COLLECTED MATHEMATICAL WORKS OF G. W. HILL s, and the second by u, and taking in succession, the sum and difference, uD's + sD'u — 2m (uBs-sDu)— J— + iia'(u + sy =0, uD's — sD'u - 2m (uDs + sDu) + | m» (m' - s") = , then, adding to the first of these the integral equation, and retaining the second as it is, we have, as the final differential equations to be employed, D» {us) —Du.Ds — 2m {uDs — sDu) + | m= (w + s)' = C , D (uDs — sDu - 2mus) + f m^ (u' - s') = . It must be pointed out, however, that these equations are not, in all respects, a complete substitute for the original equations. It will be seen that [i ov X, an essential element in the problem, has disappeared from them, and that, in integration, an arbitrary constant, in excess of those admissible, will present itself This will be eliminated by substituting the integrals found in one of the original differential equations, in which fi or x is present ; the result being an equation of condition by which the superfluous constant can be expressed in terms oi (i and the remaining constants. We remark that the left members of our differential equations are homogeneous and of two dimensions with respect to u and s. If the first were differentiated, the constant C would disappear, and both equations would be homogeneous in all their terms. This property renders them exceedingly useful when equations of condition are to be obtained between the coefficients of the different periodic terms of the lunar coordinates, and it is for this purpose that we have given them their present form. From the signification of the symbol D , Bu =2,. (2i + 1) a,!:" + \ Ds =2,. (2t + 1) ■a_,_^C' + \ D'u = I,.(%i + lY3,j:"+\ Z)»s = 2', . (2t + 1)' a_ , _ iC" + ' ; also Du.Ds=- S, . [Z, . (2t + l)(2i — 2j + 1) a,a,_J C*, uDs — sDu=- 2Sj . [2, . {2i -j + 1) a 7; - ay 2 (4/— 1) - 4m + m' ' r,-|_(_/)^ 27 ^, 3 16/-3;-5-(37+ll)m L-^J *- ^' 8/ 2/ 2(4;-' — l)-4m + m' RESEARCHES IN THE LUNAR THEORY 311 In making a first approximation to the values of the coefiScients, one of the terms of the equation may be omitted; for, when / is positive, the term 2i . U) aia_4_^_i is a quantity four orders higher than that of the terms of the lowest order contained in the equation; and, when j is negative, the same thing is true of 2^. [/] aia_<+j_i. Hence, with this limitation, the equation may be written in the two forms ^i • [[;■ . *■] a.a *-, + if\ a,a_ , + , _ J = , ^i • [[— y. i'] aA+, + (— y) a,a_,+^_,] = . where / takes only positive values. From these two equations, by omitting all terms but those of the lowest order, we derive the following series of equations, determining the coefficients to the first degree of approximation : aoE^ ^ |_1 J Eoao , aoa_i = (— 1) a„a„ , a„aj = [3][aoai + aia„] + [2, 1] aia_i, aoa_2 = ( — 2) [ai,ai + aia„] + [— 2, — 1] aia_, , aoa, = [3][a„aj + &i&i + aja,,] +[3,1] aia_ j + [3,2] aja.i, ai,a_3 = (— 3) [a„aj + a^aj + ajao] + [— 3, — 1] a_ia2 + [— 3 , — 2] a_ jBi , aoa^ = [4][aoa3 + a^a., + aja, + a^ao] + [4,1] aia_s + [4,3] a^a. j + [4,3] aaa_i , a,a_4 = ( — 4) [a„a, + aiaj + a,ai + aaa,] + [— 4, — 1] a_ias + [ — 4,-2] a_2a3 + [— 4,— 3]a_3a,, The law of these equations is quite apparent, and they can easily be extended as far as desired. The first two give the values of aj and a_i, the following two the values of ag and a_2 by means of the values of aj and a._i already obtained, and so on, every two equations of the series giving the values of two coefficients by means of the values of all those which precede in the order of enumeration. A glance at the composition of these equations must convince us that all attempts to write explicity, even this approximate value of aj, would be unsuccessful on account of the excessive multiplicity of the terms. However, they may be regarded, in some sense, as giving the law of this approximate solution, since they exhibit clearly the mode in which each coefficient depends on all those which precede it. As to the degree of approximation aiforded by these equations, when the values are expanded in series of ascending powers of m, the first four terms are obtained correctly in the case of each coefficient. Thus a^ and a_i are afiected with errors of the 6th order, ag and a_2 with errors of the 8th order, ag and a_3 with errors of the 10th order, and so on. 312 COLLECTED MATHEMATICAL WORKS OF G. W. HILL The values of these quantities can be determined either in the literal form, where the parameter m is left indeterminate, as has been done by Plana and Delaunay, or as numbers, which mode has been followed by all the earlier lunar-theorists and Hansen. In the latter case, one will begin by computing the numerical values of the quantities [j, t], [y] and {J), corre- sponding to the assumed value of m, for all necessary values of the integers i andy. The great advantage of our equations consists in this, that we are able to extend the approximation as far as we wish, simply by writing explicitly the terms, our symbols giving the law of the coefficients. How rapid is the approximation in the terms of these equations will be apparent, when we say, that, after a certain number of terms are written, in order to carry this four orders higher, it is necessary to add to each of them only four new terms; and thereafter, every addition of four terms enables us to carry the approximation four orders farther. The process which may be followed to obtain the values of the a< with any desired degree of accuracy, is this: — the first approximate values will be got from the preceding group of equations until the a^ become of orders intended to be neglected; then one will recommence at the beginning, using the equations each augmented by the terms necessary to carry the approxi- mation four orders higher; substituting in the new terms the values obtained from the first approximation, and, in the old, ascertaining what changes are produced by employing the more exact values instead of the first approxi- mations. A second return to the beginning of the work will in like manner, push the degree of exactitude four orders higher. In this way any required degree of approximation may be attained. Whatever advantage the present process may have over those previously employed is plainly due to the use of the indeterminate integers i and j, which, although much used in the planetary theories, no one seems to have thought of introducing into the lunar theory. This enables us to perform a large mass of operations once for all. For the purpose of making evident the preceding assertions, and because we shall have occasion to use them, we write below the equations determin- ing the coefficients a; correct to quantities of the 13th order inclusive. aoai = [l][a5 + 3a_iai + Sa.ja^] + (1) [aLi + 3aoa_2 -|- 2aria_3] + [l,-2]a_2a_.-l-[l, — l]a_,a_j-|- [l,3]aja, + [l,3]a3a,, aoa_i = [— l][a!-i + 3aoa_j 4- Saja.s] -f- (— 1) [aj + 2a_ia, -l- 2a_ja,] + [— 1, — 3]a_3a_,-F [- 1, — 2]a_2a_, + [—1, 1] a^a, -h [— l,2]asa3, aoaj = [2][2a„ai + 2a_,aj + 2a_ja3] -t- (2) [2a_,a_j + 2aoa_3 + 2aia_,] + [2, 2] a_ja_. + [3, - 1] a_ia_, + [2, 1] a,a_x + [2, 3] a.a^ + [2, 4] a^, RESEARCHES IN THE LUNAR THEORY 313 aoa_, = [ — 2][2a_ia_, + 2aoa_5 + 2aia_ J + ( — 2) [Saoa, + 3a_iaj + 2a_2a3] + [ — 2, — 4]a_.a_2 + [— 2, — 3]a_3a_i + [—2, — l]a_iai + [—2, Ijaja,. + [— 2,2]aA, aoa, = [3] [a? + 2a„a2 + 2a_ia3j + (3) [aLj + 2a_ia_3 + 2a„a_4] + [3, — l]a_ia_,+ [3, l]aia_, + [3, 3]a2a_i + [3, 4]aA, aoa_3 = [— 3][aL2 + 2a_ia_3 + 2aoa_ J + (— 3) [a^ + 2a„a2 + 2a_ia3] + [- 3, - 4] a_,a_, + [— 3, — 2] a_ ^aj + [— 3, - 1] a_ a + [— 3, 1] a^a, , aja^ = [4][3aia3 + 2a„a3 + 2a_iaJ + (4) [3a_2a_3 + 2a_ia_j + 2a„a_5] + [4, — l]a_ia_, + [4, l]aia_3 + [4, 2]a2a_2 + [4, 3]a3a_i + [4, SJa^ai, aoa_j = [— 4][2a_2a_s + 2a_ia_i + 3aoa_s] + (— 4) [3aia2 + 2a„a3 + 2a_iaJ + [-4,- 5]a_5a_i + [— 4, — 3]a_3ai + [— 4, — 2]a_saj + [-4, — l]a_ia3 + [-4, l]aia5, a„a5 = [5][a^ + 3aia3 + 3a„aJ + [5, 1] aia_^ + [5, 2] aja_s + [5, 3] aaa., + [5, 4] a^a_i , aoa_j = (— 5) [a? + 3aia3 + 3a„aJ + [— 5, — 4] a_^ai + [ — 5j — 3] a_3a2 + [ — 5, — 2] a_ja3 + [ — 5, — 1] a_ia^, aoae = [6][2a2a3 + 3aiaj + 2a„a5j + [6, 1] aia_B + [6, 2] aja_4 + [6, 3] a3a_3 + [6, 4] a "''*»' and, after some reductions, »» = TftV fs A ; jTT^ri A ; n *38 + 40m + 9nv — 32 5 j = m'a„, 2 6 6 [-g _ 4.m ^ m='][dO — 4m + m''] \_ 6 — 4m + m" J »' 07 3 + 4m + dm" f oc r/^ , oa 7 — m 1 , *-' = ^ [6-4m + m'][30-4m+m'] [" ^^ " ^°^ + ^* W^T^^-^^j ^\ ■ It is evident that, however far the approximation may be carried, the only quantities, involved as divisors in the values of the a;, are the trino- mials, whose general expression is 2 (if - 1) — 4m + m', 314 COLLECTED MATHEMATICAL WORKS OP G. W. HILL or, particularizing, the series of divisors is 6 — 4nH- m', 30 — 4m + m.\ 70 - 4m + m^ It will be remarked that they diflfer only in their first terms, which are inde- pendent of m. Hence any expression, involving several divisors, can always be separated into several parts, each involving only one divisor, without any actual division by a trinomial in m. For instance, 1 _ 1 1 , 1 [6 - 4m + m'][30 — 4m + m=] ^ 6 — 4m + m' ^ 30 — 4m + 1 _ 1 [6 — 4m + m']'[30-4m + m'] ^[6 — 4m + m"]' " '^ fi _ Am J- m2 + ^' I 6-4m + m' ' ^ 30 — 4m + m»" Moreover when, after this transformation, any numerator contains more or other powers of m than two consecutive powers, it is clear it may be reduced so as to contain only these by eliminating the higher powers through sub- tracting certain multiples of the divisor which appears in the denominator, or, in other words, the fraction may be treated as if it were improper. From this we gather that the value of a< can be expressed thus ao ~ " ^ 6 — 4m + m» ^ [6 — 4m + m']' ^ [6 — 4m + m']' ^ ' " * 30-4m + m' [30-4m + m^]' [30-4m + m']' P P P ■^ 70-4m'+m' "*" [70-4mV m^ "*" [70-4m + m^ "*"••• + where Mg, M^ . . . ■ Ni, N^ . . . ■ P^, P^. . ■ ■ are entire functions of m each of the form Am" + ^m*+'. The advantage of this method of treatment consists in that nothing, which is given by the successive approximations, would be lost, as must be the case when the values are expanded in series of ascending powers of m . The preceding expressions, when put into this form, become a, -f- a_.i _ _ Q ^ + ra , -"-i;^ ^6-4m+m'°'' = 3 4- a_ \j 6-4m + mJ ' ViAA a. 323 + 109m _ q^ 23 — 11m 215 — 53m "1 , |_"TT^ ■'' 6 — 4m -i- m' [6 — 4m + m7 30 — 4m4-m'J ' _ « r,^« , 175 + 563m ,q 89 — 32m , ^ 361 - 10m l^, - A|_^^ + 6-4m + m' [6 — 4m -h m"]' "^ 30 - 4m + m" J Tf RBSBAHCHES IN THE LUNAR THEORY 315 The evident objection to this form for the coeflScients is that it makes the several terms very large, and of signs such that they nearly neutralize each other, the sum being very much smaller than any of the component terms. However it may be possible to remedy this imperfection by admitting three terms into the numerators, but, in this way, the problem is indeterminate, infinite variety being possible. It is remarkable that none of our system of divisors can vanish for any real value of m, since the quadratic equations, obtained by equating them to zero, have all imaginary roots. In this they differ from the binomial divisors met with when the integration is effected in approximations arranged according to ascending powers of the disturbing force. It is well known that the infinite series, obtained from the development, in ascending powers of m, of any fraction whose numerator is an entire function of m, and its denominator any integral power of a divisor of the previously mentioned series, is convergent, provided that m lies between the two square roots of the absolute term of the divisor. Hence any finite expression in m, involving these divisors, can be developed in such a series, provided that the numerical value of this parameter is less than \/6. The same, however, cannot be asserted when the expression really forms an infinite series, as it is in the equation just given for the value of -^ . Yet, on account of the simplicity with which these quantities can be expressed in this form, a,i and a_i containing each a single term, with an error of the sixth order only, this limit is worthy of attention. If the parameter m, hitherto employed by the lunar theorists, is taken as the quantity in powers of which to expand the value of a^, we shall have m = m And, substituting this value, the principal divisor 6 — 4m + m^ becomes 6 — 16m + llm*. Thus the limits, between which m must be con- tained, in order that convergent series may be obtained where this divisor intervenes, are ± Vtt- When we consider how little, in the case of our moon, m exceeds m, it will be plain that the series, in terms of m, are likely to be much more convergent than those in terms of m . If we inquire what function of m , of the form — , the quantity M [6 — 4m + m'']" can be expanded in powers of, with the greatest convergency, it is easily found that a should be — i. Then putting 1 + itn 316 COLLECTED MATHEMATICAL WOEKS OF G. W. HILL the divisor 6 — 4m + mMs changed into 6 + ^m', and there is introduced the additional divisor 1 + iwt. Here the series will be convergent provided m is less than 3. It is true the terms involving the succeeding divisors 30 — 4m + m^ &c., are not benefited by this change of parameter, but as they play an inferior role in this matter, I have chosen in as the parameter for the developments of the coefficients a; in series of ascending powers. To illustrate this matter, we have, in terms of the parameter nt, and with errors of the sixth order, »! + a-i _ _ r 2 + im _ 1 "1 J ao ~ U + iVm' 1 + itnJ'"' a„ U + iVm^ l + im^[l + imTj'"- Expanding these expressions in powers of m, we get ?i4^^ = -[m' + im'-|m* + ^m^ + . . . .], 3i] ■""■ 3i '1-^=1= i^m' + fm' + T^tn* — ■Hm' + .... Let these series be compared with those which correspond to them in the lunar theories of Plana or Delaunay, viz: nf + ^m' + ^m' + 4^m^ + , i^m' + H «i» + ^ra* + ^■H-m^ + The superiority of the former, in convergence and simplicity of numerial coefficients, is manifest. Much more might be said relative to possible modes of developing the coefficients aj in series, but we content ourselves with giving their values expanded in powers of m, the series being carried to terms of the ninth order inclusive. The denominators of the numerical fractions are written as pro- ducts of their prime factors, as, in this form, they can be more readily used, the principal labor in performing operations on these series being the reduc- RESEARCHES IN THE LUNAR THEORY 317 tion of the several fractional coeflBcients, to be added together, to a common denominator. a. _ 3 , , 1 3 , 7 . , 11 . 30749 , 1010531 , _ 18445871 8 _ 2114557853 , 2".3\5' 2".3''.5' a-.__i9.ni,_ 5 , ^Z , 14 . 7381 , 3574153 , 55218889^8 , 13620153029 "^ 2''.3^5' 2".3'.5 m' a. _ 35 . , 803 . , 6109 . , 897599 , 237203647 „. 44461407673 , "a^ W ^ W:3J '^ 2\3\5' 2\3'.5' ^ 2^3=.5* 2^3*.5\7 ^-»-0m*4. ^3 m»+ ^^^ „e 1 56339 , 79400351 8 , 8085846833 , ii:;~ ^'Wj ^FXP +2^3W + 2".3i5* ^ 2".3^5^7 a» _ 833 . , 27943 ^, , 12275527 ^8 , 27409853579 » ■57 ~ 2^ 3".5.7 2"'.3^5l7" ^ 2".3*.5'.7' a-3_ 1 . I ?1 ip, , 46951 8 , 14086643 , "a7 ~ "2^3" ^ 2'.3.5 ^ 2«.3\5^7 2'.3\5l7' a, _3537., 111809667 o a_.__23_., 1576553 „ 17 ~ 2".3 "^ 2".3^7'' ■ ■ ■ These values being substituted in the equations r cos u z= Sf.sif cos 2iT , r Bmo = 2 .a, sin 2ir , and the parameter changed to m, we get rcos. = a, {l+[-m'-i-m» + |-m'--i-m»-gf^m« + g|||m' 25239037 ._ 732931 . "l 3^ + 14929920'" 37324800 "^ • • • J "OS x!r r 25 . , 311 ^, , 9349 . _ 5831 , L'Me" "* "^ 960 "^ "^ 28800 216000 164645363 8 11321875589 . 1 „,^^ 552960000 ^ 19353600000 J r299 , , 30193 , , 379549 . , 181908179 , ■lcos6r [40% "^ +107520'" +1003520'" +1580544000"' •••J''°«^^ , r 1134^ ., 2350381 , 1poa8r+ I + L1966O8"' + 9031680"' •• -J ''°'^'^ + ---|' + + 318 COLLECTED MATHEMATICAL WOEKS OP G. W. HILL "-a„| |^_m +-^m +^m -3^m-33j^tn - r sin t> = a„ < \ ^ m' + -^m' + ^m* — ~ m" — ;^i;SS tn' - lUlH m' 269023019 ^3 _i5mm9^,;|^.^ 2^ 74649600 93312000 + [im^4-4|i-m» + Sm»-^^m' 256 480 28800 432000 3500287 ^8 _ 43885512859 j^,. _ 1 ^^^ ^^ 11520000 58060800000 + r ^69 e 24481 , 4419347 ^8 , 398314169 [_12288 107520 15052800 4741632000 , r 9875 8 , 32608451 , 1 • q , 1 + L1M6O8"' + 144506880 "' •••J^^°^^ + ---}- Our final differential equations are capable of furnishing only the ratios of the coefficients a.^, hence we must have recourse to one of the original equations if we wish to determine ao as a function of n and (i. By substi- tuting the values in the differential equation [i)» + 2mZ> + f m' - ^J M + |m»s = 0, we obtain (^ = ^'-^^^^^ + 1+ my + im»]a, + |in'a_._.} C"+'. Considering only the term of this, for which i^O, and supposing that the coefficient of ^ in the expansion of , ", 3 is denoted by J, we shall have ^J=l + 2m+fm' + fm'^. For brevity call the right member of this H; then, since (TO TO )^ to' ^ '^ we shall have The value of H is readily obtained from the value of -^ given above, and J must be found by substituting the values RESEARCHES IN THE LUNAR THEORY 319 in . °. 3 , and taking the coeflScient of ^. We get \US) s a«Q L Ho flifl J ^0 ^0 ^3 aja_^ +45?:i?^ + 3?2%_\ a,) a^ dif where the terms neglected are, at lowest, of the tenth order with respect to m. And, explicitly in terms of this parameter, r_i , ^Lm*--^m^ 53_ ,_ W7, j^, _ 4201313 . 14374939 , By means of which there is obtained ^'-LWAV- T'°+T°'+2304'^ 288"°' 41472 "" 8761 ^, __ 4967441 . , 14829273 ^^ 6912 7962624 "^ 39813120 or, in terms of the parameter m , M'[ 1 1 „, , 4 , 163 . 1147 , 79859 . 4811 „, , 9530295 . , 139240651 + i§H. m' + 9530295 . , 139240651 . "l "^ 10368 "^ 71663616 1074954240 " J " The quantity I -^^ I is usually designated a by the lunar-theorists; and, to make this appear as a factor of the expressions for r cos v and r sin i; , it would be necessary to multiply all the coefficients by the second factor of the preceding expression for ao- It seems simpler however to retain ao as the factor of linear magnitude ; for the astronomers have preferred to derive the constant of lunar parallax from direct observation of the moon, or, in other words, they have preferred to consider j« as a seventh element of the orbit; with this view of the matter, there is no incongruity in making ao everywhere replace jx. The expression for ao can be obtained in several other ways, which lead to more symmetrical formulae, and which also serve for verification of all the preceding developments. Tf, in the preceding equation giving the value of T^^ in terms of ^, we attribute to t the value 0, or, which is equiv- (usy alent, make (^ = 1 , we shall have m = s = 2i . a^, and, consequently 320 COLLECTED MATHEMATICAL WORKS OF G. W. HILL X , = r. . l(2i + 1 + m)' + 2mT a,. And thus, mindful of the value of x given above, we get ^° ~ ^^-' [2, . [(3t + 1 + m)» + 2m^] ^ ■ [^. • -^j\ ' Again the differential equation d-r dz r " gives ^.^^^..[(ai + l + my — m']a = a [1 — 0.08331972 cos 2t + 0.00114564 cos 4t + 0.00007409 cos 6r + 0.00000404 cos 8t], r sm o = a [ 0.12709553 sin 2t + 0.00098090 sin 4t + 0.00006099 sin 6t + 0.00000342 sin 8r]. log a = 9.5318013. For m = - r cos o = a [1 — 0.1622330 cos 2t r sin y = a [ 0.3542740 sia 2t + 0.0048920 cos 4t + 0.00059858 cos 6t + 0.000081198 cos 8t + 0.000011873 cos IOt + 0.000001849 cos 12t], log a = 9.5955815. + 0.0039840 sin 4t + 0.00049306 sin 6t + 0.000070196 sin 8t + 0.000010611 sin IOt + 0.0000016902 sin 12t], For moons of much longer lunations the methods hitherto used are not practicable, and, in consequence, we resort to mechanical quadratures. Here we shall have two cases. The satellite may be started at right angles to and from a point on the line of syzygies, and the motion traced across the first quadrant; or it may be started at right angles to and from a point on the line of quadratures, and the motion traced across the second quadrant; the prime object being to discover what value of the initial velocity will make the satellite intersect perpendicularly the axis at the farther side of the quadrant. 328 COLLECTED MATHEMATICAL WOKKS OP G. W. HILL The diflFerential equations give, as expressions of the values of the coordinates, in the first case, x = x, + 2 C'ydt - C f'\^- ^'\xdt\ y = ^f\^« - ^) dt -££t'^^" ' and, in the second case, ^ = - ^£{y. -y)dt- ££ [p - ^ ] ^*" ' Here the subscript (o) denotes values which belong to the beginning of motion, and d) will hereafter be used to denote those which belong to the end. Let V be the velocity, and a the angle, the direction of motion, relative to the rotating axes, makes with the moving line of syzygies. In the first case then Gq = 90°, and we wish to ascertain what value of Uq will make (Ti = 180°. Generally, for small values of v^, Oi will come out but little less than 270°; but, as Vo augments, a^ will be found to diminish, and, if xq does not exceed a certain limit, a value of Vq can be found which will make >7i i ^i ; fz - '72 > ^2 ; ^3 > >73 > ^3 • -^iid let the rectangular coordinates of the moon relative to the earth be denoted by x, y, z; those of the sun relative to the centre of c gravity of the earth and moon by x', y', and 2'; and those of the centre of gravity of the three bodies by X, F, and Z. Then from an attentive consideration of the subjoined figure, where S, E and M denote the posi- tions of the sun, earth and moon, G the centre of grav- ity of the last two bodies, and C the centre of gravity of all three, it will be seen that, if we put P- = nh + m^ + m^ we shall have ^2 = (i"' — l)x' — iix + X, f3 = (/-l)a;'+(l-,«)a;-(-X, MOTION OF THE CENTRE OF GRAVITY OF THE EARTH AND MOON 337 with two groups, of three equations each, for the r; and ^, obtained from these by writing, in the second members, for x and X, y and Y, and again z and Z. If we differentiate the equations just written, then square and add the results, after having multiplied them severally by Wj, m^, and m^, we shall get niid^l + m^d^l + m^d^l = ntifi'dx''' + m^ndx' + MdX''. Prom this equation it is evident that, if £l denote the potential func- tion, the differential equations, determining the variables x, y, z, x', y', z', are d'x SG d'y Sfl d'z Sfl '^''^-W^dx' '^''■^^W "'''-M^^Tz' Hence it may be gathered that the disturbing function for the motion of the sun relative to the centre of gravity of the earth and moon differs from the corresponding function for the motion of the moon relative to the earth only by a constant factor which depends on the masses. The expression for fl is where the A'sare given by the equations J?,, = (a;' + y.xf + (2/'+ tJ-yf + {%' + i>.%)\ ^l3 = [^'- (l-M)a;T + ly' - (1 - /^)«/T + [2' -(1-/^)^]% A\ 3 = x' + y^ + z^. Let us put r' = x' + y^ + z\ r" = x'^ + y'^ + z'\ rr' S = xx' + yy' + zz' . Then JJ ,, = r" + 2/ji.rr'8 + fi'V , Al ,^r"-20--fi) r/S + (1 - iJ-yr' . r 1 1 Since the ratio — ris only about — — ■, and u about-— , it is convenient to r' '' 400 "^ 80 expand, in D. , the reciprocals of Aj, 3 and Aj, 3 in infinite series proceeding according to ascending powers of — ^ . This, in both cases, evidently depends r on the development of 32 338 COLLECTED MATHEMATICAL WORKS OF G. W. HILL (1 - 'Hax + a") -i in powers of a. By the Theorem of Lagrange, in solving the equation y — a,F (y) = X with respect to «/ , we get , „. s , a" d.Fixf , , a" d''-\F(x)'' , whence ^ = 1 + a ^^JIS^ + «° d\F(xy ^ _ _ ^ al d'. Fjxy ^ da; c?a; 1.3 dx' ' ' ' n\ dx" Let us suppose that we have here F{y) ^= ^{'if — 1); the equation, on which y depends, becomes then?/ — ia(y — 1) = a^, and the resolution of this quadratic in y gives 1 — ay=-\/\ — 2aa;^-a^ and, by differenti- ation, -^ = (1 — 2 aa; + a^)~^. Consequently, 2 dx 2.4 dx' 2.. 2" dx" = 1 + ag-a; 3r6.5.4 3_3 4.3.2^-| 4r 8.7.6.5 ,_4 6.5.4.3 4.3 4.3. 2.1 "] "^" L2.4.6.8'^ 1-2.4.6.8^''^ 1.2'2. 4.6. Sj The law of the numerical coefficients in this series is so plain that we can set down as many terms as we have occasion for. In making the application to the reciprocals of Ai,2 and Ajg we must put, in the first case, a = — ^u — , , in the second, a — (1— /ct) -^ , and in both x = S. We obtain as the potential function proper for the relative motion of the moon about the earth. ]_^^«z,+ m3^ rj_ 1 ^ 1 1 -1 + r«, {[(1 -;.)-'• +M-']^ MOTION OP THE CENTEE OF GRAVITY OF THE EARTH AND MOON 339 .[a-,).,]^[t|.-|.fci] ,[a-.,.-„.]i^[|-i^-|.|-|.] - }• To get the similar function for the relative motion of the sun about the centre of gravity of the earth and moon, it is necessary to multiply the pre- ceding expression by The term of the potential function for the moon, factored by -^ , gives rise to inequalities in the lunar coordinates factored by — ^ . As this term has 1 — 2fias a factor, we see the correctness of the rule which directs to mul- tiply this class of inequalities by 1 — 2/a, in order to include the effect of the disturbance of the relative motion of the sun about the earth by the lunar mass. In treating the motion of the sun about the centre of gravity of the earth and moon, it will suffice to take two terms of the preceding expression and put Let the longitudes of the sun and moon be denoted respectively by TJ and /I, and neglect the latitudes; then _L,fl = !^+ \M,j.O~-t,) 4[3cos2(A-A') + 1]. The differential equations, determining / and %\ are where, it will be noticed, we have put r = a, and, after differentiation, in the final small terms, r'=a', -;o- = ft'l and ;i — ;i' = 'r the mean angular dis- a' tance of the moon from the sun. The integration of the second equation gives -L — + 3 „"a/^ (!_;,) __.|^_____ cos 2r + lJ =0. 340 COLLECTED MATHEMATICAL WORKS OF G. W. HILL -jj- = -5-7^ I r M (1 — /i) -Ts COS 3t , Oo being the arbitrary constant. We can now eliminate -j- from the first equation, and we get Let us suppose that this equation is satisfied by ^ = 00 + «'«! cos 2t , ai being a coefficient to be determined. Substituting this value of r' in the differential equation, we get the two equations of condition, (in' - 8»«' + 3 m'O «! - |to'' ^n — n!^ ^^ (1 — z^) -^^ = . Whence may be derived _ 3 w' (3 — m) ^^ X a" "•-3 (1— OTX4:-8m + 3m») '*'■ ''''^■ where, as is usually done in the lunar theory, we have put — = m. The value of r', thus obtained, being substituted in the expression for -=- , we get dV , , TO 4 — 3m + m" ,^ ^ a' „ a^ * 1— to4 — 8m + Snv ^ ' a" Integrating „ f , „// Q to'' 4 — 2m + to' ,, \ a' ■ o * (1 — nCf 4 — Sto + 3m' ^ -' d' The numerical values of the constant quantities, which enter into these formulas, are TO = 0.0748 , I. = g2^ , |-, = 0.002587 , n' = 1295977".4 . They give us r' = a' [1.00000 00200 + 0.00000 00003 cos 2r] , X' =e' + n't - 0". 0001 sin 2t . MOTION OF THE CENTRE OF GRAVITY OF THE EARTH AND MOON 341 The periodic terms of these equations are too small for consideration, r' but the constant term of —, may be noticed. If we should obtain the value of a' from measured values of / on the assumption that the value of the con- stant term is unity, it would be too large by the 0.00000 002 part, And I'm this value substituted in the equation n' = \ —j^ would give n' too small by the 0.00000 003 part, or n' would be too small by 0".03895 ; or the error in the mean longitude of the sun would amount to nearly 4" in a century, a quantity which could not, in the present state of astronomy, be neglected. However, it is only fair to state that astronomers proceed in a way the re- verse of this ; that is, they observe n' and thence deduce a', and in this case the term 0.00000 002 is without significance, since the logarithms of the radii vectores in the ephemerides are usually given to 7 decimals only. 342 COLLECTED MATHEMATICAL WORKS OP G. W. HILL MEMOIR Fo. 34. The Secular Acceleration of the Moon. (The Analyst, Vol. V, pp. 105-110, 1878.) In the Philosophical Transactions for 1853, Prof. J. C. Adams, of Cam- bridge University, England, showed that the values of the secular acceleration of the mean motion of the moon, obtained by Plana and Damoisean, were erroneous, for the simple reason that these authors had, inadvertently, made the solar eccentricity constant throughout a certain portion of the investiga- tion. This statement of Prof. Adams gave rise to an animated and pro- longed controversy, the history of which will, no doubt, always possess much interest. It is proposed to obtain here the coeflBcient of the term in the moon's mean motion involving the square of the solar eccentricity , supposed variable, to quantities of the order of the square of the sun's disturbing force, when the lunar eccentricity and inclination of orbit are neglected. The method employed has no novelty, having been used before by Mr. Donkin. But, at the end of the investigation, I have found that it is possible to do without an explicit development of iB in a periodic series, and thus the treatment is, to a considerable degree, abbreviated. Let ^ denote the mean longitude of the moon as affected by this secular inequality, and Wq the mean motion at a given epoch taken as the origin of time ; we propose to prove that, in the equation the true value of J? is 3/«'Y_37yi/w'Y ^U.i 64 [nj- ^ Employing the method of variation of the elements, we have, for deter- mining the four elements n, ^, e and u of the lunar orbit, these equations dn 3_9^ de Tutd-R dt ~ fia^ 9C ' dt~ fie 9-^ ^3 [1 + 3 cos 2 (/I— /)] , where il and X' are the true longitudes of the moon and sun. The constant part of i2 is evidently the same as that of ^n'^ a^—j^ , when we rejected, that is, it is equal to in'^a*(l + |e'^) . Considering first those terms in B which are independent of e (we need those multiplied by e only when taking account of the effects produced by the variations Se and 5u), we see that the only terms in B which produce terms in -=- , and, consequently, can give rise to terms independent of sines or (tt cosines of arguments in -^ , have arguments of the form 2^ + 4', where i^ de- notes an angle depending on the sun's mean motion. Hence, denoting any one of these terms of B by n'^a^ A cos (2 where A is independent of the lunar elements, but will generally contain e'^ and regard being had to this term alone, the equations determining the elements become ^ = 6n"A sin (2C + <*) , ^^ = n — i^ Acoa (3C + 0), where [i has been eliminated by using the equation fi = r^c^. Integrating these, and considering •^ as constant, since its variability affects only the terms in H multiplied by — g , but regard being had to the variability of a d e'^ through e', where we may consider—^ as constant, we obtain = -r-LAsin C2C + 0)-|^|f..^cos (2: + ^), This being only a first approximation in which we have had regard only to quantities of the order of n'^, we proceed to a second approximation. And first, in the expression for -=-, we substitute for ^, ^-\-S^ ; and we find, for 344 COLLECTED MATHEMATICAL WORKS OF G. W. HILL that part of the increment which is independent of the sines or cosines of arguments, the expression ~di~ 1? d.e"~df Integrating this and putting e'* — e'l =8. e^ , 2 n' d.e 8n and 8^ , Again, in the expression for -^ , increasing n and ^ by their variations dt ^ w' d.e'^ n n" Now the constant part of this value of 5 . -^ goes to form part of the constant Uq , hence, desiring to retain only the varying part, we may write ' ,g 8 . c'^ for A^ , and thus obtain ■ df- '^' n'd.e" ^^ • ^^^ In the next place let us consider the terms in B multiplied by e ; they are all of the form n" d'e A cos (w + kZ + (p') , where A and t^ possess the same quality as before, and k may be — 3 , — 1 or 1 . Representing a -\- Jc^ + '4' hy 6 , the equations determining the ele- ments are, regard being had to this term alone , di dt n dt ^ n dt n e In the last equation we have written only the term divided by e, since this alone can produce terms 5 . -^ of the kind we seek. Integrating the last two as we integrated in the former case, we obtain ;> 1 «'" 4 „..a /, _L 1 n"d.A'd.e'' . , se= l^Aame + ^^'^-i^i^^coso he n' ¥e n' d.e'^ dt THE SECULAR ACCELERATION OF THE MOON 345 Augmenting, in the expression for -5- , e and by these quantities, we ob- etc tain, regard being had only to the terms which are independent of sines or cosines of angles, . dn_ 3 n'* d.A'd.e" ' dt U rf d.^' dt ' Increasing, in the expression for -^ , the elements n, e and B b)' their vari- ations 5n, he and hQ , and preserving only the terms independent of the sines or cosines of angles, we get • dt~ U n' d.e" ' '^ 'Wl? ' In like manner as before, rejecting the constant part of this which coalesces with Wfl > we obtain , <^C 5 n"d.A' ■ dt ~ k n" d. e'" S.e" . (2) When formulas (1) and (2) are applied to all the terms of i2, to which each is applicable, and the results added, we shall have the complete value of 5 . ^ , since it is plain that the combination of two different terms in R will always produce terms in 5 . -^ involving the sines or cosines of angles. Denoting the mean anomalies of the moon and sun by ^ and ^', and the mean angular distance of the bodies by r, the part of a B, which is indepen- dent of e, may be written aR = A^+ Ai cos 2r + A^ cos f + A^ cos (2r— f) -f- A^ cos (2t + ^') . Formula (1) applied to this series gives dt ^^ d. e" We can obtain the terms in R multiplied by e from the series just given by using the equation dR_^ dRd.logr dR dX 9e 9r de d^ de = -2i2cose-|-2^sinf , 346 COLLECTED MATHEMATICAL WORKS OP G. W. HILL Whence ?^^ = - ZA, cos f — 3^1 cos (2t - e) + Ji cos (2t + f) — A, cos (e— r) - A^ cos (f + f) — 3 J, cos (3r _ f _ ^) + ^3 cos (2t — f + f) — 3^4 cos (2t + e' — f) + At cos (3t + ?' + ?). Applying formula (2) to this series gives 5 . ^ = w j^[- 5 (4^; + Al +Al)-i (Al + Al + Al)+5 (9A\ + 9Al-{- 9^J)] e . e' = ^ ^ [i|i {A\ + ^« + ^D - 10 (2^S + AX)-\ d.e'\ €b m 6 Adding this to the expression given by formula (1) , S.^= n^ [-V (^; + ^3 + Al) -10 i2Al + Al)} S . e'\ But, denoting the constant term of a^B^ by K, we have K=Al + ^iAl+Al + Al + Al). Or ^; + ^5 + Al = %K— (2^S + ^D , and But we have a^R = tV$p-I[¥ + 6cos 2(A - /) + • cos 4(A_A')] , and hence ^ is equal to the constant term of fj -j — ^ . In consequence, a" denoting the constant term of -jj by Z , we shall have Also we evidently have iir~ = ^. + ^.cose', and thus \:^L=%Al^-A\. Substituting this value THE SECULAR ACCELERATION OF THE MOON 347 But the constant term of — ^ is known to be 1 + Y*'^ 5 hence, in fine, dt 8* if To obtain -^ we must add to n both this term and that which arises, in at Inc^ dR the first approximation, from the term -^— in the differential equa- tion for -^ , which is therefore equal to the constant term of jo- , that dt n r^ is, to — — (1 + |e'2) . Thus n at n L(l+3e'2)_(B!^_3|Jl^;)5..". We could have added to the first two terms of this equation a term B^ e'l, where 5 is a numerical coeflScient, equal to the aggregate of the constants we have virtually neglected whenever we wrote h. e'^ for e'" , but it will be easily seen that this would not change the final result. We evidently have From which, to a suflBcient degree of approximation , Substituting this value of n, we get 348 COLLECTED MATHEMATICAL WORKS OF G. W. HILL MEMOIE No. 35. Note on Hansen's General Formulae for Perturbations. (American Journal of Mathematlce, Vol. IV, pp. 256-359, 1881.) The last form in which Hansen expressed the perturbations of the mean anomaly and equated radius vector is exhibited by the following equations : .( + ..+ /{f+-*^(-3^)"}«,«, n„z: (Equations 36 and 37, p. 97.)* It will be perceived that the right-hand member of the first of these in- volves three quantities, viz. W, v and -^ . But the last of these quantities has no share in defining the position of the body, and it is desirable to get rid of it, provided that can be done without complicating the equation. This is readily accomplished by means of the equation (33, p. 95) dz Aq ~dt~ h(l + v)' ' The result is w„« /W 4- if Why Hansen has not put the equation in this form I cannot imagine ; the advantage, not only as regards simplicity of expression, but also in point of ease of computation, is evident. Hansen develops W by Taylor's theorem, and, limiting ourselves to the second power of the disturbing force, we have W=W„ + [^)n,Sz=W,-2^Sz. When this value is substituted for W in the equation for n^^, we have a differential equation of the first order and degree for the determination of Sz, *See Auseinandersetzung einer zweckmdsaigen Methode zur Berechnung der absoluten Storungen der kleinen Planeten. Von P. A. Hansen. Erste Abhandlung. {Abhandlungen der Koniglichen Siichsischen Oesellsehaft der Wissenschaften. Band III.) The numbering of the equations and the paging are from this volume. NOTE ON HANSEN'S GENERAL FORMULAE FOR PERTURBATIONS. 349 the integral of which is well known. Terms of three dimensions with re- spect to disturbing forces being neglected, this procedure furnishes the equa- tion n,H = (1-— 3.) C [(1 + j}.)-r„ + "'] n,dt , which, however, is without interest other than analytical, as its use involves more labor than that of the equation given by Hansen. Hansen's equation for the determination of v has the disadvantage of not affording the constant term of this quantity, and is inconvenient in com- puting the portion, of the form At + Bt' + Gf + . . . ., which is independent of the arguments g , g' , &c., as the values of A, B, &c., must be determined to a degree of accuracy much beyond what is necessary in the case of the other terms. As all the arbitrary constants admissible have been introduced by the integrations which give z , it is evident there must exist an equation determining v without additional integrations. Hansen has virtually employed this in the place where he shows how the constant term of v is to be obtained, but has nowhere given it explicitly. This lacuna I propose to fill here. The equation 39, p. 97, smw may be employed to discover the value oi -j— . The known expressions for -£- cos 0) and J- sin u are i- cos ui p • -i- sin ■ ■' 1 _ X [r" — 'i/rrs + r"]*' If for fl are substituted only the first two terms of this expression, the differential equations are easily integrated, and the variables x, y,z and x', y' , z' represent the motion of two planets moving according to the laws of elliptic motion, whose mean motions are / Mm J / Mm' V 7^ ^""^ \-7^ In terms of symbols whose meaning is well known, we will put L= sj \Mmii.d\ , L' — 1^ [Mm'/i'a'] , f/G' — H'coah + i^G" — H"cosh' = 0, ifCP^^R^ sin A + V »" — H" sin A' = , H + H' = c, being an arbitrary constant. But , since i and i' are supposed contained between 0° and 180°, the radicals in these expressions must be taken posi- tively. Consequently the equations are equivalent to h' = h + 180°, H + H'=c, H—H'= <3" — <3"\ c These equations determine the values of the elements E, H' and h' in terms of the rest, and they may be used to eliminate them from R. Then it is plain, from the expression of s, given above, that h will also disappear from R, and we shall have ^ = function {L,'Q, L' , 0' , l,g, V,g'), and s takes the much simpler form s=-co6(v-v' +g-g')+ ^^ +^^^' ~ ^ sin (v + g) sin (w' + (/') . As to the partial derivatives of R with respect to L, L' , I, V , g , g' , they are evidently unchanged by this elimination of the elements H, E' , h, h' . But (^^^ and (^t^i) denoting the derivatives of R on the supposition of its containing the elements E, E' ,h,h' , we have (dR\_dR _dR dH dR dB' \mi~'dG dH dG dH' dO ' (dR\_ dR _dR^ dH_dR^ dH^ \W)~ dO' dH dO' dH' dG'' But we also have hence dR dR _ d(h' -h) _^ dH dH'~ dt ldR\_dR^_dR dCH+H') _ dR ['dG)~ dG dH dO dG ldR\-.dR _dR d{H+ H') _ dR^ \W) ~ dG' dH dO' dG' 356 COLLECTED MATHEMATICAL WORKS OP G. W. HILL Moreover dB^dB dH as 3^^ as 9 (g + H') _ dR dc dS dc dB! dc dH dc ~ dH' Thus the system of differential equations still retains its canonical form, and is dL _ dR dIJ^_ dR dG^_ dR dff _ dR dt ~ dl' dt ~ dV ' dt dg' dt dg' ' dl__dR dV___dR di^_dR dg' dR dt ~ dL' dt 327' dt dG' dt dO'' After this system of eight differential equations is integrated, the value of h is found by a quadrature from the equation dh__dR dt dc ' These integrations introduce nine arbitrary constants which, together with c, make ten. The reference of the coordinates to any arbitrary planes introduces three more, but one of these coalesces with the constant which completes the value of h. The time t does not explicitly enter R, hence the complete derivative of it with respect to t is dR_dR dL dRdl dt dL dt dl dt If, in this are substituted the values of -^ , -^- , . . . , from the equations at at just given, we shall find that it vanishes ; hence ^ = a constant is an integral of the system of differential equations. This integral may be employed to eliminate one of the elements, as L, from the equations. We can also take one of the elements, as I, for the independent variable in place of t. The system of equations, to be integrated, is then reduced to the ix dR dR dR dL' dl' da dg dG' W dl ~ ■ dR'' dl - ■ dR' dl ~ dR ' dL dL dL dR dR dR dl' dL' dg _ d& dg' _ dG' dl - dR ' dl dR' dl dR' dL dL dL NOTES ON THE THEORIES OP JUPITER AND SATURN 357 A simpler form can be given to them. If the solution of -B = a constant gives Zi= function {L' , G, G',l',g,g', I), and L is supposed to stand for the right member of this, the foregoing equa- tions can be written dl dl" dl ~ dg' dl ~ dg" dr___dL^ dg _ _dL M--^ dl ~ dL' ' dl " dO' dl~ dO' ■ When the values of L', G, G',V , g and g' in terms of I have been de rived from the integrals of these, they can be substituted in the equation -J- = — -^Y , which will then give t in terms of Z , by a quadrature. By inverting this we shall have I in terms of t ; and by substituting this in equa- tions previously obtained we shall have the values of all the other elements in terms of t . It will be noticed that ^ is a homogeneous function of Z , Z' , G , G' and c of the dimensions — 2 ; hence we shall have and, as a consequence of this, L ^ + L'^ + G% + Q' % + c^4= 2i2 = a constant. dt dt dt dt dt Thus, if the rate of motion of each angular element 1,1' . . . , be multiplied by the linear element which is conjugate to it, the sum of the products is invariable. The sines of half the inclinations of the orbits on the plane of maximum areas are Thus, in the special case where the two planets move in the same plane, we have G+G' =c. This equation may be employed to eliminate one of the elements G or G' from R , In the same case, the expression for s is reduced to s = — cos (« — v' + g — g'). 358 COLLECTED MATHEMATICAL WORKS OF G. W. HILL Then, if we put Q-Q' = r g-9' = r, R will be a function of L , L' , T ,1 , I' , y , and we shall have, for determi- ning these variables, the system of differential equations dL dR dt~ dl' dL' dR dt ~ dl" dr dR dt~ dr' dl dR dt~ dL' dl' dR dt ~ dL' ' dr dR dt~ dr' After these are integrated, the value of g + g' will be got by a quadra- ture from the equation d(g + g') _ dR dt Qc ■ If the value of L is obtained from the solution of -B = a constant, and we have L = function {L', F ,1' ,y ,T) , and I is adopted as the independent variable in place of t, the solution of this special case is reduced to the integration of the four equations dL' dL dl' dL dr dL dr dL dl ~ dV ' dl - dL' ' dl - dr' dl ~ di" The angle between the planes of the orbits of Jupiter and Saturn is about li°. This is small enough to make the terms, which are multiplied by the square of the sine of half of it, and which are besides of two or more dimensions with respect to disturbing forces, practically insignificant. Thus, while we are engaged in developing those terms of the coordinates which demand the highest degree of approximation relatively to disturbing forces, we shall assume that the planes coincide ; the determination of the effect of non-coincidence of these planes being reserved to the end, when it will be always sufficient to limit ourselves to the first power of the disturbing force. The coordinates usually preferred by astronomers are the logarithm of the radius vector, the longitude and the latitude. We suppose that the two last are referred to the plane of maximum areas. Let these coordinates be denoted by the symbols log . p , ;i and ^ ; and let the subscript (q) be applied to /I and /? when we wish to designate the similar coordinates corresponding to the variables x,y,z, a;',y,z'. Then we have P cos /J COS -i = r cos /S„ cos A„ + zr* cos /?'„ cos A'„ , P cos iS sin l = r cos /S, sin A„ + xr' cos /S',, sin A'„ , (0 sin /S = r sin /S, + z/ sin /S', . NOTES ON THE THEORIES OP JUPITER AND SATURN 359 From the first two equations are readily obtained the following two : — P cos (J cos (A — A„) = r cos /J„ + xr' cos /S',, cos (A'„ — /l„) , P cos /? sin (A — Ao) = zr' cos /S'o sin (A'o — Ao) . In the developments in infinite series which follow, the eccentricities of the orbits will be regarded as small quantities of the first order, the squares of the inclinations of the orbits on the plane of maximum areas as quantities of the third order, and x also as a quantity of the same order. Then all terms, whose order is higher than the sixth, will be neglected. This degree of approximation will be found amply sufficient for the most refined investigations. Under these conditions, we get log p = logr + I log [ 1 + 2x -^ s + x^^'J . = logr + .^s + ix^^' (l+2s'), ^ = ^0 + X ^ ^', - X -^ s;9„ . We will write >? for sin i* . Then, to the sufficient degree of approxi- mation, ^ ^ s = - X I- COS {v — if + g - g') + 2^ {y, + 7,'y ^ sin {I + g) sin (l' + g') . In like manner X £l?2L^o sin (A' _ A„) = X (1 + v'-v") 4r sill i^ - ^' +9-9') r cos /Jo r ^xri" -^ Sin {Zl—V + ^g—g') + xV'^- sin (I + i' + g + g') . The expressions for 7^^ and /3o in terms of elliptic elements are given by Delaunay.* Log r , as well as the following expressions + 3e-CO^«(3?+^')+ie"S(*^'+^'> ±i«'^s?n(^'-^')±^«"S(^^'-^')' * Theorie du Mouvement de la Lune. Tom. I, pp. 56-59. 360 COLLECTED MATHEMATICAL WORKS OF G. W. HILL are found in a memoir by Prof. Cayley* With these data we get logp = loga + ^e' + ^\^e* + ^e' + x' ^^ _(e_|fl3_^e=)co8Z-(|e»-^e* + ^e«)cos2? -(H^- t¥V ^) cos 3? - (II e^ - 129 e^) cos il — iU^ cos 5Z— 899 e« cos 61 ~'4{ [l-e'-ie"- (>? + v')l cos (?-?' + t/-^') + (|e - Je'-fee") cos (2i -V +g—g') + (_ |e' + |eV) cos (l + g—g') + (— 1 e — I e» + ^ ee") cos (J,'-g + g') + (^ e' _ | e"— i eV) cos Q -%l' + g —g') + ^e' cos (dl — l' +g-g'y- Iff" cos (I + V —g + g') + I ee' cos (5' — p') — » ee' cos (2i + ^r —g') — \ ee' cos (3? - 5' + 5^) + I «e' cos (2Z - %l' +g—g') + |e" cos (1—31' + g—g') + ^e" cos (l + V + g-g') + ^e' cos {H -I' + g -g') — ^\ e' cos (%l^l'-g + g') — 41 ^e' cos (3? + 5r - ^f') + ^eV cos (?— ^ + )V sin (4Z + 25f) — I vjV sin 2^ - 8| ,v sin (5? + 2(/) + ^ ,V sin (? - 2g-) + x^|(l-e'-ie" + ,'-,") sin (?-?+£r-5f') + (f e— ^e' — f ee") sin (21 - I' ^g—g") + (he+\e'-\ee")sm(l'-g+^) +( - f e' + |eV) sin (l + g-g') * Tables of the Development of Functions in the Theory of Elliptic Motion . Mem. Roy. Astr. Soc, Vol. XXIX, p. 191. NOTES ON THE THEORIES OF JUPITER AND SATURN 35 j + (i «' - I e'^- i eV) sin (I - 21' +g~g') ^ vr. ^a sin (3Z — Z' + ^ — /) + -I e' sin (Z + r - (/ + /) + f ee' sin (^r - ^') - f «e' sin {%i +g — g') + I ee' sin (2Z' — ^r + 5'') + I ««' sin (2? - 2Z' + ^ _ /) 4 f e'^ sin (; _ sr + ?«*) sin G - g) + I -qe^ sin (4Z + ^) + ^ 'ye' sin {21 — ^) + e 25 ^g* gjn (5; + ^) + /j r,e' sin (3Z —5-) - )?'e ein (U + 35^) + yf'e sin (3Z + 3^) + ;< -^ I ij sin (3?- Z' + 25- -/) + (ij + 2,') sin (?' + g') + ^-qe sin (8Z — Z' + 2g -g') - f ije sin (Z — Z' + 2g -g') — f ije' sin (2Z + 2g —g') + ^ );e' sin (2Z — 2Z' + 2g-g') + ^(v +2V)esin (I + I' + g') - ^(.v + 2^') e sin (I — l' — g') + i ('? + ^v') e' sin (3Z' +g')—B(y, + 2r,') e' sin g' \ . As written, these expressions give the coordinates of Jupiter. Those of Saturn are obtained by removing the accent from all the accented sym- bols, and applying it to those which are unaccented, x excepted, for which we have x' = x. Also it is to be remembered that we have A' = ^ + 180°. The coordinates of the two planets are obtained by employing in these formulas, for the quantities involved in them, the values they actually have at the time in question. The latter are determined by the differential equa- tions previously given ; but, instead of integrating these equations in one step, we may, as Delaunay has done in the lunar theory, divide the process into a series of transformations of the variables involved ; each of which must be made not only in the expressions for log p , /I , (3 , log p' , /I' , ^' , but also in R . As the introduction of I as the independent variable does not appear to be advantageous, we will suppose that the six variables L , L' ,T, I, I' , y are employed and that t is the independent variable. 34 362 COLLECTED MATHEMATICAL WORKS OF G. W. HILL Delaunay's method, somewhat amplified, amounts to this: — selecting the argument = i/ -j- i'V + *"/' suppose, for the moment, that B is lim- ited to the terms — B — A^ cos (t7 + i'V + i"Y) — ^j cos 2 {il + i'V + i"r) + . . . , where B , A^ . . . , are functions of i , Z' and T only. Then if it is found that the differential equations, corresponding to this limited R, are satisfied by the infinite series 0=0,{t + c) + dj sin [0„(^t + c)] + e^ sin 3[eo(^ + c)] + . . . , l-0) + /„ (i + c) + I, sin [»„ {t + c)] + I, sin 2 [So (i! + c)] + . . . , V = (V) + V„{t + c) + K sin [0„(t + c)] + n sin 2 [S„(if + c)] + . . . , ;-=:=(;-) + ^„ (if + C) + ^ sin [Oo (I + C)] + J-, sln 2 [So (< + c)] + . . . , i = jLo + ii cos [0„ (J, + c)] + Zj cos 2 [So (< f c)] + . . . , i' = i; + A' cos [e^{t + c)] + X; COS 2 [So (< + c)] + . . . , r = To + A COS [So (/ + c)] + r, COS 2 [S„(^ + c)] + . . . , where c, (?) , (Z') and {y) are arbitrary constants, the last three being equiv- alent to two independent constants, as we have the relation i{l)^i'{l') + i"{r) = o, and all the other coefficients are known functions of three other constants, a, a' and e , we can replace L hy L„ + A COS {il + i'V + i"r) + A cos 2 {il + i'V + i'V) + . . . , L' hy LI + L[ cos {il + i'V + i"r) + A cos 2 {il + i'V + i" y) + . . ., r by r„ + r, cos {U + i'V + i"r) + r; cos 2 {U + w + i"r) + . . ., I by I + I, sin {il + i'V + i"r) + h sin 2 {il + i'V + i"y) + . . . , V by V f l[ sin {il + i'V + V'f) + Z, sin 2 {il + i!V -I- i'V) -I- . . . , rbyr + n sin (i7 -1- i'? -1- i"r) + y, sin 2 {il + I'Z' + t"^) + . . ., and will have, for determining the new variables, I , V , y , a, a' . e, pre- cisely the same differential equations as we started with, provided we make all these substitutions in the function R , and regard the new variables L , L' , r as connected with a , a', e by the relations L ^L, + ^{o,L, + 20,L, +...), L' = L'„ +l{0,Li ^20,L[ + ...), r= n + ^(s, r, + 2s,z', + ...). It will be perceived that, as long as we are dealing with terms of R, whose arguments involve I or I' or both, the second members of the three equations, last written, have values which differ from the elliptic values of L, NOTES ON THE THEORIES OP JUPITER AND SATURN 3(53 L' and r only by quantities of the second order with respect to disturbing forces. Hence, if we propose to neglect third order terms, until we have reduced ^ to a function of the argument y only, we can assume that L, L' and r which are the elements conjugate to the arguments I, T and y, are expressed throughout in terms of a, a' and e, in the same way as in the ellip- tic theory. It may be added that these third order terms are found in experience to be much smaller than those which arise in other ways. 1