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l^**.-tA 3-.K5" -^j-^:^ j^.^^rM:'^ l\h\U^\ nt .ti*nriT uf XV 3/ O.?,' ^ 7 Va« w-\ y— «1^3», MECANIQUE CELESTE. MECANIQUE CELESTE. BY THE MARQUIS DE LA PLACE, PEER OF FRANCE; GRAND CROSS OF THE LEGION OF HONOR ; MEMBER OF THE FRENCH ACADEMY, OF THE ACADEMY OF SCIENCES OF PARIS, OF THE BOARD OF LONGITUDE OF FRANCE, OF THE ROYAL SOCIETIES OF LONOON AND GOTTINGEN, OF THE ACADEMIES OF SCIENCES OF RUSSIA, DENMARK. SWEDEN, PRUSSIA, HOLLAND, AND ITALY; MEMBER OF THE AMERICAN ACADEMY OF ARTS AND SCIENCES; ETC. TRANSLATED, WITH A COMMENTARY, NATHANIEL BOWDITCH, LL. D. TELLOW OF THE ROYAL SOCIETIES OF LONDON, EDINBURGH, AND DUBLIN; OF THE ASTRONOMICAL SOCIETY OF LONDON j OF THE PHILOSOPHICAL SOCIETY HELD AT PHILADELPHIA; OF THE AMERICAN ACADEMY OF ARTS AND SCIENCES J ETC. VOLUME III. BOSTON : FROM THE PRESS OF ISAAC R. BUTTS ; MILLIARD, GRAY, LITTLE, AND WILKINS, PUBLISHERS. M DCCC XXXIV. /i^i / fâb"^ Entered, according to Act of Congress, in the year 1829, By Nathaniel Bowditch, in the Clerk's Office of the District Court of Massachusetts. TO BONAPARTE MEMBER OF THE NATIONAL INSTITUTE. Citizen First Consul, You have permitted me to dedicate this work to you. It is gratifying and honorable to me to present it to the Hero, the Pacificator of Europe,* to whom France owes her prosperity, her greatness, and the most brilliant epoch of her glory ; to the enlightened Protector of the Sciences, who, himself distinguished in them, perceives, in their cultivation, the source of the most noble enjoyment, and, in their progress, the perfection of all useful arts and social institutions. May this work, consecrated to the most sublime of the natural sciences, be a durable monument of the gratitude inspired in those who cultivate them, by your kindness, and by the rewards of the government. Of all the truths which this work contains, the expression of this sentiment will ever be the most precious to me. Salutation and Respect, LA PLACE. [* This volume was published, by La Place, in 1802, soon after the peace of Amiens.] VOL. III. B ADVERTISEMENT. This volume contains the numerical values of the secular and periodical inequalities of the motions of the planets and moon ; the numbers, given in the original work, having been reduced from centesimal to sexagesimal seconds, to render them more convenient for reference. The Appendix contains many important formulas and tables, which are useful to astronomers in computing the motions of the planets and comets. Some of these tables are new, and the others have been varied in their forms, to render them more simple in their uses and applications : none of them have heretofore been published in this country. Several of the formulas have been introduced into the calculations of modern astronomy, since the commencement of the first part of the original work. The portrait of the author, accompanying this volume, was obtained in France, and is an impression from the original plate, which was engraved under his direction, for the Système du Monde. The fourth volume of the work will be put to press in the course of a few weeks. PREFACE. We have given, in the first part of this work, the general principles of the equilibrium and motion of bodies. The application of these principles to the motions of the heavenly bodies, has conducted us, by geometrical reasoning, without any hypothesis, to the law of universal attraction ; the action of gravity, and the motions of projectiles on the surface of the earth, being particular cases of this law. We have then taken into consideration, a system of bodies subjected to this great law of nature ; and have obtained, by a singular analysis, the general expressions of their motions, of their figures, and of the oscillations of the fluids which cover them. From these expressions, we have deduced all the known phenomena of the flow and ebb of the tide ; the variations of the degrees, and of the force of gravity at the surface of the earth ; the precession of the equinoxes ; the libration of the moon ; and the figure and rotation of Saturn's Rings. -We have also pointed out the cause, why these rings remain, permanently, in the plane of the equator of Saturn. Moreover, we have deduced, from the same theory of gravity, the principal equations of the motions of the planets ; particularly those of Jupiter and Saturn, whose great inequalities have a period of above nine hundred years. The inequalities in the motions of Jupiter and Saturn, presented, at first, to astronomers, nothing but anomalies, whose laws and causes were unknown; and, for a long time, these irregularities appeared to be inconsistent with the theory of gravity ; but a more thorough examination has shown, that they can be deduced from it ; and now, these motions are VOL. III. c PREFACE. one of the most striking proofs of the truth of this theory. We have developed the secular variations of the elements of the planetary system, which do not return to the same state till after the lapse of many centuries. In the midst of all these changes we have discovered the constancy of the mean motions, and of the mean distances of the bodies of this system ; which nature seems to have arranged, at its origin, for an eternal duration, upon the same principles as those which prevail, so admirably, upon the earth, for the preservation of individuals, and for the perpetuity of the species. From the single circumstance, that the motions are all in the same direction, and in planes but little inclined to each other, it follows, that the orbits of the planets and satellites must always be nearly circular, and but little inclined to each other. Thus, the variations of the obliquity of the ecliptic, which are always included within narrow limits, will never produce an eternal spring upon the earth. We have proved that the attraction of the terrestrial spheroid, by incessantly drawing towards its centre the hemisphere of the moon, which is directed towards the earth, transfers to the rotatory motion of this satellite, the great secular variations of its motion of revolution ; and, by this means, keeps always from our view, the other hemisphere. Lastly, we have demonstrated, in the motions of the three first satellites of Jupiter, the following remarkable law, namely, that, in consequence of their mutual attractions, the mean longitude of the first satellite, seen from the centre of Jupiter, minus three times that of the second satellite, plus twice that of the third satellite, is alivays exactly equal to two right angles ; so that they cannot all be eclipsed at the same time. It remains now to consider particularly the perturbations of the motions of the planets and comets about the sun ; of the moon about the earth ; and of the satellites about their primary planets. This is the object of the second part of this work, which is particularly devoted to the improvement of astronomical tables. PREFACE. xi The tables have followed the progress of the science, which serves as their basis ; and this progress was, at first, extremely slow. During a very long time, the apparent motions only of the planets were observed. This interval, which commenced in the most remote antiquity, may be considered as the infancy of Astronomy. It comprises the labors of Hipparchus and Ptolemy ; also, those of the Indians, the Arabs, and the Persians. The system of Ptolemy, which they successively adopted, is, in fact, nothing more than a method of representing the apparent motions ; and, on this account, it was useful to science. Such is the weakness of the human mind, that it often requires the aid of a theory, to connect together a series of observations. If we restrict the theory to this use, and take care not to attribute to it a reality which it does not possess, and afterwards frequently rectify it, by new observations, we may finally discover the true cause, or, at least, the laws of the phenomena. The history of Philosophy affords us more than one example, of the advantages which may be derived from an assumed theory ; and, of the errors to Avhich we are exposed, in considering it to be the true representation of nature. About the middle of the sixteenth century, Copernicus discovered, that the apparent motions of the heavenly bodies indicated a real motion of the earth about the sun, with a rotatory motion about its own axis : by this means, he showed to us the universe in a new point of view, and completely changed the face of Astronomy. A remarkable concurrence of discoveries will forever render memorable, in the history of science, the century immediately following this discovery ; a period which is also illustrious, by many master-pieces of literature and the fine arts. Kepler discovered the laws of the elliptical motion of the planets ; the telescope, which was invented by the most fortunate accident, and was immediately improved by Galileo, enabled him to see, in the heavens, new inequalities and new worlds. The application of the pendulum to clocks, by Huygens, and that xii PREFACE. of telescopes to the astronomical quadrant, gave more accurate measures of angles and times, and thus rendered sensible the least inequalities in the celestial motions. At the same time that observations presented to the human mind new phenomena, it created, to explain them, and to submit them to calculation, new instruments of thought. Napier invented logarithms : the analysis of curves, and the science of dynamics, were formed I)y the hands of Descartes and Galileo : Newton discovered the differential calculus, decomposed a ray of light, and penetrated into the general principle of gravity. In the century which has just passed, the successors of this great man have finished the superstructure, of which he laid the foundation. They have improved the analysis of infinitely small quantities, and have invented the calculus of partial differences, both infinitely small and finite : and have reduced the whole science of mechanics to formulas. In applying these discoveries to the law of gravity, they have deduced from it all the celestial phenomena ; and have given to the theories and to astronomical tables an unexpected degree of accuracy; which is to be attributed, in a great measure, to the labors of French mathematicians, and to the prizes proposed by the Academy of Sciences. To these discoveries in the last century, we must add those of Bradley, on the aberration of the stars, and on the nutation of the earth's axis : the numerous measures of the degrees of the meridian, and of the lengths of the pendulum ; of which operations, the first example was given by France, in sending academicians to the north, to the equator, and to the southern hemisphere, to observe the lengths of these degrees, and the intensity of gravity : the measure of the arc of the meridian, comprised between Dunkirk and Barcelona ; which has been determined by very accurate observation, and is used as the basis of the most simple and natural system of measures : the numerous voyages of discovery, undertaken to explore the different parts of the globe, and to observe the transits of PREFACE. xiii Venus over the sun's disc ; by which means, the exact determination of the dimensions of the sokir system has been obtained, as the fruit of these voyages : the discoveries, by Herschel, of the planet Uranus, its satellites, and two new satellites of Saturn : finally, if we add to all these discoveries, the admirable invention of the instrument of reflexion, so useful at sea ; that of the achromatic telescope ; also the repeating circle, and chronometer ; we must be satisfied, that the last century, considered with respect to the progress of the human mind, is worthy of that which preceded it. The century we have now entered upon, commenced under the most favorable auspices for Astronomy. Its first day was remarkable, by the discovery of the planet Ceres ; followed, almost immediately afterwards, by that of the planet Pallas, having nearly the same mean distance from the sun. The proximity of Jupiter to these two extremely small bodies ; the greatness of the excentricities and of the inclinations of their mutually intersecting orbits, must produce, in their motions, considerable inequalities, which will throw new light on the theory of the celestial attractions, and must give rise to farther improvements in Astronomy. It is chiefly in the application of analysis to the system of the world, that we perceive the power of this wonderful instrument ; without which, it would have been impossible to have discovered a mechanism which is so complicated in its effects, while it is so simple in its cause. The mathematician now includes in his formulas, the whole of the planetary system, and its successive variations ; he looks back, in imagination, to the several states, which the system has passed through, in the most remote ages ; and foretells what time will hereafter make known to observers. He sees this sublime spectacle, whose period includes several millions of years, repeated in a few centuries, in the system of the satellites of VOL. HI. D XIV PREFACE. Jupiter, by means of the rapidity of their revolutions ; which produce remarkable phenomena, similar to those which had been suspected, by astronomers, in the planetary motions ; but had not been determined, because they were either too complex, or too slow, for an accurate determination of their laws. The tlieory of gravity, which, by so many applications, has become a means of discovery, as certain as by observation itself, has made known to him several new inequalities, in the motions of the heavenly bodies, and enabled him to predict the return of the comet of 1 759, whose revolutions are rendered very unequal, by the attractions of Jupiter and Saturn. He has been enabled, by this means, to deduce, from observation, as from a rich mine, a great number of important and delicate elements, which, without the aid of analysis, would have been forever hidden from his view: such as the relative values of the masses of the sun, the planets and satellites, determined by the revolutions of these bodies, and by the development of their periodical and secular inequalities : the velocity of light, and the ellipticity of Jupiter ; which are given, by the eclipses of its satellites, with greater accuracy, than by direct observation: the rotation and oblateness of Uranus and Saturn; deduced from the consideration, that the different bodies which revolve about those two planets, are in the same plane, respectively : the parallaxes of the sun and moon : and, also, the figure of the earth, deduced from some lunar inequalities : for, we shall see hereafter, that the moon, by its motion, discloses to modern astronomy, the small ellipticity of the terrestrial spheroid, whose roundness was made known to the first observers by the eclipses of that luminary. Lastly, by a fortunate combination of analysis with observation, that body, which seems to have been given to the earth, to enlighten it, during the night, becomes also the most sure guide of the navigator ; who is protected by it from the dangers, to \vhich he was for a long time exposed, by the errors of his reckoning. PREFACE. XV The perfection of the theory, and of the lunar tables, to which he is indebted for this important object, and for that of determining, with accuracy, the position of the places he falls in with, is the fruit of the labors of mathematicians and astronomers, during the last fifty years: it unites all that can give value to a discovery ; the importance and usefulness of the object, its various applications, and the merit of the dififlculty which is overcome. It is thus, that the most abstract theories, diffused by numerous applications to nature and to the arts, have become inexhaustible sources of comfort and enjoyment, even to those who are wholly ignorant of the nature of these theories. CONTENTS OF THE THIRD VOLUME. PARTICULAR THEORIES OP THE MOTIONS OF THE HEAVENLY BODIES. SIXTH BOOK. THEORY OF THE PLANETARY MOTIONS. Object of this theory 1 CHAPTER I. FORMULAS FOR THE INEaUALITIES OF THE MOTIONS OP THE PLANETS, WHICH DEPEND ON THE SaUARES AND HIGHER POWERS OF THE EXCENTRICITIES AND INCLINATIONS OF THE ORBITS 4 ON TUE INEaUALITIES WHICH DEPEND UPON THE SaUARES AND PRODUCTS OP THE EXCENTRICITIES AND INCLINATIONS 4 Form of the terms which produce them [.3703,3704]. Influence of the ratio of the mean motions upon these terms, by reason of the small divisors, which are introduced by the integrations [3712]. Preparations of the diflerential equations for the different cases of these inequalities which occur in the solar system §L2 Considerations, by which we may distinguish the most important of these inequalities [3732-3735] §3 Development of the terms, which result in the expressions of the radius vector, of the longitude, and of the latitude of the disturbed planet [3736— 3800] §4,5,0 OX THE INEaUALITIES DEPENDING ON THE CUBES AND PRODUCTS OF THREE DIMENSIONS OF THE EXCENTRICITIES AND INCLINATIONS OF THE ORBITS, AND ON THEIE HIGHER POWERS 45 Form of the terms which produce them [3807—3807'] § 7 Examination of the cases where they become sensible. They depend on the circumstance, that the ratios of the mean motions are nearly commensurable. Application of these principles to the theory of Jupiter and Saturn, in terras of the third degree [3828, &c.] § 8 Inequalities depending on terms of the fifth degree [38.56']. They are sensible in the theory of Jupiter and Saturn. Calculation of them for these planets [3860, «Sic] § 9 VOL. MI. E XX CONTENTS OF THE THIRD VOLUME. longitude and the radius vector of the earth [4300', 4304]. The planets which produce them, are Venus, Mars, Jupiter and Saturn. Inequalities which are independent of the excentricities [4305,4306]. Inequalities depending on the first power of the excentricities [4307, 4308]. Inequalities depending on tlie second dimension of the excentricities and inclinations of the orbits [4309]. Inequalities depending on the third dimension of the same quantities [4311]. Inequalities of the motion of the earth in latitude [4312]. They are produced by the action of Venus and Jupiter ^ 2<j Inequalities of the motion of the Earth, produced by the action of the Moon [4324,4326]. §30 On the secular variations of the earth's orbit, of the equator, and of the length of the year [4329", &c.]. The action of the sun and moon has a considerable influence on these values. Determination of the epoch, when the greater axis of the earth's orbit coincided with the line of the equinoxes [4363"], and when these two lines were perpendicular to each other [4367'"] §31 CHAPTER XI. THEORY OF MAES 26S Examination of the limit to which the approximations must be carried, in the valuation of the radius vector [4371, &c.]. Numerical values of the sensible inequalities which affect the longitude and radius vector. The planets which produce them, are Venus, the Earth, Jupiter and Saturn. Inequalities which are independent of the excentricities [4.373, 4374]. Inequalities depending on the first power of the excentricities [4375, 4376]. Inequalities depending on the second dimension of the excentricities and inclinations of the orbits [4377—4380]. The inequalities in latitude are hardly sensible [4384]. The greatest of them arises from the action of Jupiter §32 CHAPTER Xll. THEORY OF JUPITER 275 Examination of the limit to which the approximations must be carried, in the valuation of the radius vector [4385, &c.]. Numerical values of the sensible inequalities afiecting the longitude and the radius vector. The planets which produce these inequalities, are the Earth, Saturn, and Uranus, but chiefly Saturn. Inequalities which are independent of the e.xcentricities [4388,4389]. Inequalities depending on the first power of the excentricities [4392,4393]. They are so large as to render it necessary to notice the variation of their coefiîcients. Inequalities depending on the squares and products of the excentricities and inclinations [4.394 — 4.397], They are produced only by the action of Saturn. CONTENTS OF THE THIRD VOLUME. Xxi Inequalities depending on the third and fifth dimensions of the excentricities and inclinations ; and also on the square of the disturbing force [4401, «fee.]. These last terms, which depend on the inequalities of a very long period, have considerable influence on the secular variations of the elliptical elements. Great inequality of the mean motions [4434]. It is produced by the action of Saturn. . §33 Inequalities in latitude [4457]. They are produced by the action of Saturn §34 CHAPTER XIII. THEOUV OF SATURi\ 299 Examination of the degree to which the approximations must be carried in the valuation of the radius vector [4460, &c.]. Numerical valuesof the sensible inequalities affecting the longitude and radius vector. The planets which produce them are Jupiter and Uranus. Inequalities which are independent of the excentricities [4463,446]. Inequalities depending on the first power of the excentricities [4466, 4467]. Inequalities depending on the squares and products of the excentricities and inclinations [4463—4471]. Inequalities depending on the third and fifth dimensions of the excentricities and inclinations, and also on the square of the disturbing force [4472', &c]. Great inequality of Saturn. It is the reaction of that of Jupiter § 35 Inequalities in latitude [4511]. They are produced by the action of Jupiter and Uranus. . §-36 CHAPTER XIV. THEORY OF UEANUS 314 Examination of the degree to which the approximations must be carried, in the valuation of the radius vector [4521, &c.]. Numerical values of the sensible inequalities affecting the longitude and radius vector. They are produced by the action of Jupiter and Saturn. Inequalities which are independent of the excentricities [4523, 4524]. Inequalities depending on the first power of the excentricities [4525, 4526]. Inequalities depending on the second dimension of the excentricities and inclinations [4.527—4529]. Inequalities depending on the third dimension of the excentricities and inclinations [4530]. There is only one of them produced by the action of Saturn § 37 Inequalities in latitude [4531]. They are produced by the action of Jupiter and Saturn. § 38 CHAPTER XV. O.N SOME EaUATIONS OF CONDITION, BETWEEN THE INEaUALlTlES OF THE PLANETS, WHICH MAY BE USED IN VEEIFn.NG THEIR NUMERICAL VALCES §39—43 318 CHAPTER XVI. ON THE MASSES OF THE PLANETS AND MOON 333 VOL. III. F XXII CONTENTS OF THE THIRD VOLUME. Reflections on the values given to those masses in § 21. New determination of those of Venus and Mars [4G05, 4608]. Discussion of that of the Moon, by the comparison of several phenomena which can determine it [4619 — 4637], such as the observation of the tides, the lunar equation in the tables of the Sun, the nutation of the Earth's axis, and the Moon's parallax. From these examinations, it appears, that this mass is rather less than that which is deduced from the tides observed at Brest [4037] §44 * CHAPTER XVII. ON THE FORMATION OF ASTRONOMICAL TABLES, AND ON THE INVARIABLE PLANE OF THE PLANETARY SVSTEM §45,46 341 CHAPTER XVIII. ON THE ACTION OF THE FIXED STARS UPON THF PLANETARY SYSTEM. . . . 343 The great distance of these bodies renders their action insensible [4673]. Reflections on the comparison of the preceding formulas with observations [4687, &c.] §47 SEVENTH BOOK. THEORY OF THE MOON. Explanation of this theory ; its particular difficulties [4692, &c.]. Considerations that must influence us in the approximations. How we may deduce from this theory, several important elements of the system of the world [4702, &c.], and among others, the oblateness of the Earlh [4709], which is thus obtained with greater accuracy than by direct observations 356 CHAPTER I. INTEGRATION OF THE DIFFERENTIAL EaUATIONS OF THE MOON'S MOTION 2QQ Difierential equations of this motion given in § 15 of the second book [4753 — 4756]. Method of noticing in the calculation, the non-sphericity of the Moon and Earth [4773]. ... § 1 Development of the quantities which occur in the differential equations, supposing these two bodies to be spherical [4780, &c.] §2 The ecliptic, in its secular motion, carries with it the moon's orbit, so that the mean inclination of this orbit to the ecliptic, remains always the same [4803]. This circumstance, indicated by analysis, simplifies the calculations, because it permits us to take the ecliptic for the fixed plane of projection [4804] §3 Investigation of the elliptical part of the motions of the Moon and Earth [4826, 4828, 4837,4838]. § 4 Principles relative to the degrees of smallness of the quantities which occur in the expressions of the co-ordinates of the moon [4841]. Examination of the influence of the successive integrations upon the different terms of these co-ordinates [4847, &c.]. Indication of the terms of the radius vector, wliich produce the evection [4850], and annual equation [4851]. §5 CONTENTS OF THE THIRD VOLUME. Xxiii Use to be made of these considerations. Development of the differential equation which produces the radius vector ; noticing only the first power of the disturbing force [4858 — 4903]. § 6, 7 Investigation of the terms of the order of the square and the higher powers of the disturbing masses, which acquire a sensible influence by integration [4904, &c.]. It is necessary to notice the perturbations of the Earth by the Moon [4909', 4948, &c.] §8 Connection of these terms with the preceding. Complete development of the differential equation which produces the radius vector [4961] § 9 Integration of tliis equation [4904, &c.]. Inequalities resulting from it. Expression of the motion of the lunar perigee [4982, &c.]. The variableness of the excentricity of the Earth's orbit produces a secular inequality in the constant term of the Moon's parallax ; but this inequality is insensible [4970]. The same cause produces a secular inequality in the motion of the Moon's perigee, which is conformable to observation. Analytical expression of this inequality [4985]. The excentricity of the Moon's orbit is subjected to a secular variation, which is analogous to that of the parallax, and like it, insensible [4987] § 10 Development of the differential equation which gives the latitude [501 8, &c.], noticing, in the first place, only the simple power of the disturbing forces §11 Investigation of tlie terms of the order of the square of those forces which acquire a sensible influence in the expression of the latitude [5039,&c.] §12 Connection of these terms with the preceding, and the complete development of the difierential equation which gives the latitude [5049] §13 Integration of this equation [5050, &c.]. Inequalities resulting from it. Expression of the retrograde motion of the nodes [5059]. The variableness of the excentricity of the Earth's orbit, produces in this motion a secular inequality. Analytical expression of this inequality [5059]. Its ratio to that of the perigee [5060]. The inclination of the lunar orbit to the true ecliptic, is likewise variable by means of the same cause ; but this variation is insensible [50G1] §14 Development of the differential equation which gives the time or the mean longitude in terras of the true longitude [5081, &c.] Integration of this equation. Inequalities which result from it [5095, &c.] The mean longitude also suffers a secular change, resulting from the variableness of the excentricity of the Earth's orbit; expression of this inequality. Analytical relations of the secular equations of the mean motions of the Moon, its perigee and nodes [5089, «Sic] § 15 Numerical determination of the several coefficients, occurring in the preceding formulas [51 17,&c.] and the numerical développent of the expression of the mean longitude [5220]. The perturbations of the Earth's orbit by the Moon, are reflected to the Moon by means of the Sun and are weakened by the transmission [.5225, 5226]. Numerical value of tlie motion of the perigee [5231], and of its secular equation [5232]. This equation has a contrary sign to that of the mean motion [.5232']. Numerical expression of the motion of the node [5233], and of Xxiv CONTENTS OF THE THIRD VOLUME. its secular equation [5234]. This equation has also a contrary sign to that of the mean motion [5234'] ; hence it follows, that the motions of the nodes and perigee decrease, while that of the Moon increases. Numerical ratios of these three secular equations [5235]. Secular equation of the mean anomaly [5238] § 16 The most sensible inequalities of the fourth order, which occur in the expression of tlie mean longitude [5240— 5305] §17 Numerical expression of the latitude [5308] §18 Numerical expression of the Moon's parallax [5331] § 19 CHAPTER II. ON THE LUÎJAR INEQUALITIES ARISING FROM THE OBLATENESS OF THE EARTH A.\D MOON 585 The oblateness of the Earth produces in the latitude of the Moon but one single inequality. We may represent this effect, by supposing that the orbit of the Moon, instead of moving on the plane of the ecliptic, with a constant inclination, to move with the same condition, upon a plane which always passes through the equinoxes between the ecliptic and equator [5352]. This inequality can be used for the determination of the oblateness of the Earth [.5358]. It is the reaction of the nutation of the Earth's axis upon the lunar spheroid [5398], and there would be an equilibrium about the centre of gravity of the Earth by means of the forces producing these two inequalities, if all the particles of the Earth and Moon were firmly connected with each other, the Moon compensating for the sraallness of the forces acting on it, by the length of the lever to which it is attached [5424]. The oblatenes of the Earth has no sensible influence on the radius vector of the Moon [.53G6] ; but it produces in the Moon's longitude one sensible inequality. The motions of the perigee and node are but very little augmented by it [5396, &c.] § 20 The non-sphericity of the Moon produces in its motion only insensible inequalities [5445, 5451, &c.] §21 CHAPTER III. ON THE INEaUALITIES OP TUE MOON DEPENDING ON THE ACTION OF THE PLANETS. G17 These inequalities are of two kinds, the first depends on the direct action of the planets on the motion of the Moon [5479, 5481] ; the second arises from the perturbations in the Earth's radius vector produced by the planets [5490]. These perturbations are reflected to the Moon by means of the Sun, and are augmented by the integrations which gives them small divisors. Determination of these inequalities for Venus, Mars, and Jupiter [5491, &c.]. The variableness of the excentricities of the orbits of the planets, introduces, in the mean longitude of the Moon, secular equations, analogous to that produced by the variation of the excentricity of the Earth's orbit, reflected to the Moon by means of the Sun ; but they are wholly insensible in comparison with this last. Thus the indirect action of the planets on the Moon, transmitted by means of the Sun, considerably exceeds their direct action, relative to this inequality [5539]. §22 CHAPTER IV. COMPARISON OF THE PRECEDING THEORY WITH OBSERVATION 642 Numerical values of the secular inequality of the mean motion of the Moon [5542, &c.], and those of the mean motions of the perigee and node of the Moon's orbit. Considerations which confirm their accuracy [5544, &c.] § 23 CONTENTS OF THE THIRD VOLUME. Periodical inequalities of the Moon's motion in longitude [5551, &c.]. Agreement of the coefficients given by the theory, with those of the lunar tables of Mason and Burg [5575, &c.]. One of these inequalities depends on the Sun's parallax [5581]. If we determine its coefficient by observation, we may deduce from it the same value of the Sun's paralla.ï,as that which is obtained by the transits of Venus [5589']. Another of these inequalities depends on the oblateness of the Earth [5590]. The value of its coefficient determined by the tables of Mason and Burg, indicates that the Earth is less flattened than in the hypothesis of homogeneity, and that the oblatenes is ^^-g. [5593] , S 34 Inequalities of the Moon's motion in latitude [5595, &c.]. Agreement of the coefficients given by the theory with those of the tables of Mason and Burg [5596]. One of these inequalities depends on the oblateness of the Earth [5598]. Its coefficient, determined by observation, gives tlie same oblateness [5602], as the inequality in longitude depending on the same element. So that these two results agree in proving, that the Earth is less flattened than in the hypothesis of homogeneity ^25 Numerical expression of the Moon's horizontal parallax [5C03]. Its agreement with the tables of Mason and Burg [5605] §26 CHAPTER V. ON AN INEaUALITV OF A LONG PERIOD, WHICH APPEARS TO EXIST IN THE MOON'S MOTION QQQ The action of the Sun on the Moon, produces in the motion of that satellite an inequality, whose argument is double the longitude of the node of the Moon's orbit, plus the longitude of its perigee, mimis three times the longitude of the Sun's perigee [5641, &c.]. The consideration of the non-spherical form of the Earth, may also introduce into the motion of the Moon, two other inequalities [5633, 5638'], with nearly the same period as that which we have just mentioned ; and in the present situation of the Sun's perigee, they are all three nearly confounded together. The coefficients of these three inequalities are very difficult to compute from the theory ; it appears that the two last must be wholly insensible [5637', 5639']. . § 27 The first is evidently indicated by observations. Determination of its coefficient [5665]. [This result was afterwards found to be incorrect, as is observed in the note, page 666, &c.]. § 28 CHAPTER VI. ON THE SECULAR VARIATIONS OF THE MOTIONS OF THE MOON AND EARTH, WHICH CAN BE PRODUCED BY THE RESISTANCE OF AN ETHEREAL FLUID SURROUNDING THE SUN. . . g-^g The resistance of the ether produces a secular equation in the Moon's mean motion [5715] ; but it does not produce any sensible inequality in the motions of the perigee and nodes [5713,5717] §29 The secular equation of the Earth's mean motion, produced by the resistance of the ether, is about one hundredth part of the corresponding equation of the Moon's mean motion [5740]. §30 VOL. III. ^ XXV Xxvi CONTENTS OF THE THIRD VOLUME. APPENDIX BY THE AUTHOR. The chief object of this appendix is to demonstrate a theorem, discovered by Mr. Poisson, that the mean motions of the planets are invariable, when we notice only the terms depending on the first and second powers of the disturbing forces [5744, &c.] This is done by giving new forms to some of the differential expressions of the elements of the orbits, as is observed in [.5743, &c]. Forms of these differentials, including all the terms depending upon the first power of the disturbing masses [5786 — 5791]. Expressions of the mean motion [5794] ; of the periodical inequalities in the elements [5873 — 5879] ; and of the secular inequalities of the elements [5882 — 58SS]. Investigation of the mutual action of two planets upon each other, referring their inequalities to an intermediate invariable plane [5905, &c]. New method of computing the lunar inequalities, depending upon the oblateness of the earth [5937—5973]. On the two great inequalities of Jupiter and Saturn ; correcting for the mistake in the signs of the functions JVC), JV(}) Sic. [5974—5981]. IN THE COMMENTARY Among the subjects treated of in the JVotes, we may mention the following : Correction to be made in the formula mfàR-{-m'rdR' = 0, [1202], in some of the terms of the order of the square of the disturbing masses [4004c, &c]. The necessity of this correction was first made known by Mr. Plana [400Gw, &c.]. Results of the discussion upon this subject, by Messrs. Plana, Pontecoulant, Poisson and La Place [40056'— 4008î]. New formula by La Place, relative to some of these terms [4008x]. This formula has been called " the last gift of La Place to Astronomy," being the last work he ever published. On the values of the constant quantities f,,f',g, &c. ; introduced into the integral expressions of or, ÔV, OS, by La Place [4058c, &c.] ; which were objected to by Mr. Plana. The results of La Place's calculation proved to be correct by him, and by Mr. Poisson, in [4058c — 40G0/i]. Corrected values of the masses of the planets, finally adopted by the author [40Gld]. Elements of the newly discovered planets Vesta, Juno, Pallas and Ceres ; corresponding to the 23d July, 1831, as given by Enckc [4079i]. Elements of the orbits of the comets of Halley, Olbers, Encke and Biela [4079»i]. Inequalities in the motions of Venus and the Earth, having a period of 2.39 years, and depending on terms of the fifth order of the excentricities and inclinations ; discovered and computed by Professor Airy [4296 a — q, 4310 c — /]. Mr. Ponteooulant's table of the part of the great inequality of the motion of Jupiter, depending on the square of the disturbing force [4431/]. Similar table for the inequalities of the motion of Saturn [4489c]. Results of the calculations of Professor Hansen [4489 n — p]. CONTENTS OF THE THIRD VOLUME. XXvii The action of the fixed stars affects the accuracy of the equation ta. m. \/ n -(- c's. ?)i'. y/a' -f- &c. = [46S5g-]. Results of the calculations of several authors relative to the sun's parallax, hy means of the parallactic inequality in the moon's longitude, and by the transits of Venus over the sun's disc [5589 a — m]. Inequality in the moon's longitude, whose period is about 179 years. It is found to be insensible [5611 a — g]; instead of being 15V39 at its maximum, as the author supposes in [5GG5]. The planets and comets move in a resisting medium, according to the observations of Encke's comet [5067 a — c]. Notice of the papers published by La Grange and Poisson, relative to the invaiiableness of the mean motions of the planets, which is treated of in the appendix to this volume [5741a — I]. It appears from the calculations of Nicolai, Encke and Airy, that the estimated value of the mass of Jupiter, adopted by La Place from Bouvard's calculations of its action on Saturn and Uranus, must be increased, to satisfy the observed perturbations of the planets Juno and "Vesta ; as well as those of Encke's comet, [5980 i — p]. APPENDIX BY THE TRANSLATOR. Formulas for the motion of a body in an elliptical orbit [.5985(1—19)] ; with their demonstrations [5984(3-25)]. Formulas for the motion of a body in a parabolic orbit [5986] ; with their demonstrations [5987]. Determination of the symbol log. A: = 8,2355814 .. . which is used in these calculations [5987(8)]. Formulas for the motions of a body in a hyperbolic orbit [5988] ; with their demonstrations [5989]. Kepler's problem for computing the true anomaly from the time, or the contrary, in an elliptic orbit. Indirect solution of this problem, according to Kepler's method, but arranged in formulas by Gauss [.5990]. Simpson's method for determining the true anomaly, in an ellipsis or hyperbola, where e is very nearly equal to unity, noticing only the first power of 1 — t, or e — J [5991(1— 12)]. Bessel's improved method for computing the terms depending on the second power of 1 -e or e-1 [.5991(1-40)] Gauss's method, in a very cxcentric ellipsis, noticing all the powers of e — 1 [5992]. Gauss's method of solution in a hyperbolic orbit, in which e — 1 is very small, noticing all tlie powers of this quantity [5993]. Olbers's method of computing the orbit of a comet [.5994, &c.]. Table of formulas which are used in this calculation [5994(.9— 45)]. Geometrical investigation of this method of calculation [5994(46—130")]. Remarks on the manner of determining the approximate values of the curtate distance p of the comet from the earth [5994(132—172)] Examples for illustrating these calculations [5994(173—242) ], using tables I, II, III. Remarks on the calculation of p by means of the equations (C), (/)) [.5994(136—103, 242', 242")]. Forms of the fundamental equations, adopted by Gauss for the determination of the curtate distance, or its equivalent expression it, by means of logarithms [5994(244, &c.)]. Solution of two examples, reduced to the form of Gauss [5994(247 — 250)]. Analytical investigation of the method of computing the orbit of a comet, [5994(263-403)]. Great advantage in having the intervals of times between the observations nearly equal to each other [5994(349)]. XXVÏn CONTENTS OF THE THIRD VOLUME. The method usually employed in this calculation requires some modification, when M appears under the form of a fraction, in which the numerators and denominators are both very small [5994(257)]. These methods are explained in [5994(387—392)]. Mr. Lubbock's method of computing the orbit of a comet [5994(405—458)]. Method of computing the elements of the orbit of a heavenly body; there being given the two radii r,r', the intermediate angle v' — v = 2f, and the time f — t of describing the angle 2/ [5995]. Collection of formulas for solving this problem, in an elliptical orbit [5995(4—67)] ; with their demonstrations [5995(08—174)]. Examples of the uses of these formulas [5995(175—193)]. Collection of formulas for solving this problem in a parabolic orbit [5996(2— :i;5)] ; with their demonstrations [.5996(26—50)] ; illustrated by an example in [5996(51-53)]. Collection of formulas for solving this problem in a hyperbolic orbit [5997(1 — 59)] ; with their demonstrations [5997(60—172)]. Example of the uses of these formulas [5997(173—183)]. Gauss's method of correcting for the efiect of the parallax and aberration of any newly discovered planet or comet, in computing its orbit by means of throe geocentric observations, with the intervals of time between them [5998]. Corrections in the places of the earth, on account of the planet's parallax [5998(47 — 50)]. Method of calculating the longitude and latitude of the zenith [5998(67 — 71)&,c.]; also the longitude and latitude of the planet from its right ascension and declination [5998 (97— 107) ], with examples. Method of correcting for the aberration of the planet [5998(108 — 117)]. Example for illustrating the calculations relative to the parallax and aberration [5998(118 — 126)]. Gauss's method of computing the orbit of a planet or comet, by means of three geocentric longitudes and latitudes, together with the times of observation [5999.] Table of the symbols and formulas which are used in this method [5999(9 — 54)]. Demonstrations of these formulas [5999(58, &c.)]. Example, containing the whole calculation of the elements of the orbit of Juno, from three observa- tions of Maskelyne [5999(274—650)]. CATALOGUE OF THE TABLES IN THE APPENDIX. Table i. Contains the square roots of the numbers from 0,001 to 10,1 ; to be used in Olbers's method of computing the orbit of a comet ; in finding r, r", c ; from j-9, r"% c^ ; which are given by three fundamental equations of this method [5994(31, 32, 33)]. Table II. To find the time T of describing a parabolic arc, by a comet ; there being given the sum of the radii r-\-r", and the chord c, connecting the two extreme parts of the arc. This table is computed by Lambert's formula [750], namely, 7 = 9"'''^', 688724. j (,- + r" + c)^ — (r + r" — c)* ( ; and the numbers are given to the nearest unit in the third decimal place, expressed in days and parts of a day. This degree of accuracy being abundantly sufficient for the purpose of computing the orbit of a comet, by Dr Olbers's method ; and the table serves to facilitate this part of the calculation. CONTENTS OF THE THIRD VOLUME. XXIX Table III. To find the anomaly U, corresponding to the time t' from the perihelion, expressed in days, in a parabolic orbit; where the perilielion distance is the same as the mean distance of the earth from the sun. The arguments of this table, as they were first arranged by days days Burckliardt, are the values of r, from I'^O ,0 to t' = G ,0; and the logarithm of t' from log.<'^0,700 to log. <'^5,00; the corresponding anomalies being given from 17=0'' to [/= 172''32"'09',9. We have also given Carlini's table for the first six days of the value of t'. This last table has for its argument log. of t' days ; and the corresponding numbers represent log. U in minutes, minus log. t' in days. 9S7 Table IV. To find the true anomaly v, in a very excentric ellipsis or hyperbola, from the corresponding anomaly U in a parabola; according to the method of Simpson, improved by Bessel. This table contains the coefficients of Simpson's correction, corresponding to the first power of (1 — e); and those of Bessel's correction, corresponding to the second power of (1 — c) ; for every degree of anomaly from 0** to 180''; as they were computed by Bessel 996 Table V. This table was computed by Gauss, for the purpose of finding the true anomaly v, corresponding to the time t from the perihelion, in a very excentric ellipsis, noticing all the powers of 1 — e 999 Table VI. This table is similar to Table V, and was computed by Gauss for finding the true anomaly r, corresponding to the time t from the perihelion, in a hyperbolic orbit, which approaches very nearly to the form of a parabola; noticing all the powers of (e— 1) 1002 Table VII. This was computed by Burckhardt, for the purpose of finding the time t, of describing an arc of a parabolic orbit ; there being given the radii r,r', and the described arc v'—v = 2f. 1005 Table VIII. This table was computed by Gauss, and is used with Table IX or Table X, in finding the elements of the orbit of a planet or comet, when there are given the two radii r, r', the included heliocentric arc v' — x) = 2/; and the time t' — t, of describing this arc, expressed in days. . 1006 Table IX. This table is used with Table VIII, in the computation of an elliptical orbit, by means of r,r',v' — V and t' — t 1012 Table X. This table is used with Table VIII, in the computation of a hyperbolic orbit, by means of r,r', v' — V, and t' — t 1013 Table XI. To convert centesimal degrees, minutes and seconds, into sexagesimals. ..... 1014 Table XII. To convert centesimal seconds into sexagesimals, and the contrary 1016 The Tables V — X, include all those which Gauss published in his Theoria Molus, etc. We have altered, in some respects, the arrangement and forms of these tables, to render them more convenient for use ; and upon comparison it will be found, that this appendix contains the most important of the methods which are given in that great work, as well as in that of Dr Olbers. The methods of Gauss being somewhat simplified, by reducing many of the processes to the common operations of spherical trigonometry, instead of using a great number of unusual auxiliary formulas, expressed in an analytical manner; and Olbers's calculations are abridged by the use of Tables I, II. VOL. III. ^ ERRATA. CORRECTIONS AND ADDITIONS IN VOLUME I. PagR. Line. 119 6 bot. For dw read dia. 120 13, 19, 21 J'or (zdx—xdy) read (zdx—xdz). 125 12 For dZ-j-dy,, read dZ-\-dz,. 7 \)o\.. For — l.m.Sn.ds, read —Xj.m.Sv.ds. 7 bot. For — y'ddx', read — yddx'. 4 bot. For Y read y. 3 bot. Insert dm in the last term. 7 Insert ( after xK 9 bot. For |. read 4.. 4 bot. For .2 read ^. 9 bot. For axis of z, read axis of x. 16 For dy read Sy. 10 bot. For {dp) read (J-p). 3 bet. For dr' read dr. 4 bot. For ag read a,g. 8 bot. For 0-.U read au'. 7 Change tlie accents in the denominator of I'. 1 bot. For /2a, read /2-2. 7 bot. For 2, read r2. 12 For shi.mt, read sin. ?nn<. 13 For cos.mt, read cos.mnt. 3 For sin.2»i<, read 2.sin.2ji(. 11 For [6S8a], read [66Su]. 1 For sin.4.(t),— 6), read sin.4.(i',— 6). 2 2 10 bot. For — , read — . r * r 3 For 0",5, read 0«,5. 1,2 bot. For logarithm, read logarithmic. 8 bot. For tang.(/g"— ;■), tang. {IS"'—j) ; read sin.(/3''— /), sin (/3"'— /). 7 bot. For éy, read d'y'. 6 For c, read c'. 4 For y',y', fcc, read y,y', &c. IS i^or iy',)/', &c., read y,y',&.c. 5 bot. For 6' reaa j'. 6 bot. For c=V', read c'=V'. 4 liot. For .111, read Jl^D. 8 bot. For [1034(;], read [1069a]. 1 bot. For the exponent — è, read 4. 134 147 147 147 159 182 183 209 215 220 230 234 235 280 281 301 371 371 378 378 381 398 413 455 464 475 478 480 487 495 499 542 581 585 Page. Line. 593 5 bot. for [109Sa], read [10976']. 608 Id For B, read B„. 618 15 For spherical angle, read spherical triangle. 679 5 bot. For m'p, read m'p' ; and/br m'q, read m'q'. 693 4 bot. For m, read m'. 715 15 bot. For andt, read an, in both formulas. IN VOLUME II. 370 16 For [1581a], read [1851a]. 510 11 bot. For >. read e. L' , L- 780 4 bot. For — , read — . r'3 T.'-i 781 5 bot. For read —- . IN VOLUME III. The same measures have been used for .correcting the mistakes ol the press in Volume III, as in printing the preceding volumes. The reader will also omit the third line from the bottom in page 501, which is unnecessarily repeated ; and at the end of the paragraph, page 556, line 16, will make the following addition of a paragraph which was accidentally omitted. " The function [5082s] contains also the terms depending on 120m2..4(8), 120m2../î(9) [5261c, e, line 1], which are derived from the part — Ja. funct. [4931;)] contained in [50825]. For by combining the term A&> ee' .e.os.{cio-\-e'mv) in [493 J^, col. 1] with — |.e.sin.(2i) — 2nit!— cr), in col. 2, we get the first of these terms; and by combining the term .4(9).e£'.cos.(CB — c'mtj), with — Je. sin. (2b — 2niii — ev), in col. 2, we get the second of these terms." Lastly, in page 458, line 3, we may add, that the function [4957] must be multiplied by the chief term of [4S90], or [4961 or 4960e]. to obtain the corresponding terms of SECOND PART. PARTICULAR THEORIES OP THE MOTIONS OF THE HEAVENLY BODIES. SIXTH BOOK. THEORY OF THE PLANETARY MOTIONS. The motions of the planets are sensibly disturbed by their mutual attractions, and it is important to determine accurately the inequalities which result from this cause ; for the purposes of verifying the law of universal gravitation, improving the accuracy of astronomical tables, and discovering whether any cause, foreign from the planetary system, produces a change in its constitution or its motions. The object of this book is to apply to the bodies of this system, the methods and general formula given in the first part of this work. We have developed in the second book, only those inequalities which are independent of the excentricities or inclinations of the orbits, and those which depend upon the first power of these quantities. But it is often indispensable to extend the approximation to the square and to the higher powers of these elements ; and sometimes it is also necessary to consider the terms depending on the square of the disturbmg force. We shall first give the formulas relative to these inequalities ; and shall then substitute in these formulas, and in those of the second book, the numbers or values of the elements corresponding to each planet. By this means we shall obtain the numerical expressions of the radius vector, and the motions of the ^jlanet in longitude and in latitude. Bouvard has willingly undertaken the calculation of these substitutions, and the zeal with which he has prosecuted this laborious work, deserves the acknowledgment of all astronomers. Several mathematicians had previously calculated the greater part of the planetary inequalities ; and their results have been useful in verifying those of Bouvard ; for when any difference has been found, he has examined into the source of VOL. HI. 1 PARTICULAR THEORIES OF THE the error, in order to satisfy himself of the accuracy of his own calculation. Lastly, he has reviewed with particular care, the calculation of those inequalities which had not been before computed ; and by means of several equations of condition, which obtain between these inequalities, I have been enabled to verify many of them. Notwithstanding all these precautions, there may possibly be found in the following results, some errors, which almost inevitably occur in such long calculations ; but there is reason to believe that they amount only to insensible quantities, and that they cannot be detrimental to the general accuracy of the tables founded upon them. These results, on account of their importance in the planetary astronomy, of which they are the basis, deserve to be verified with the same care that has been taken in the calculation of the tables of logarithms and of sines. The theories of Mercury, Venus, the Earth, and Mars, produce only periodical equations of small moment ; they are, however, very sensible, by modern observations, with which they agree in a remarkable manner. The development of the secular equations of the planets and of the moon will make known accurately the masses of these bodies, which is the only part of their theory that remains yet somewhat imperfect. It is chiefly in the motions of Jupiter and Saturn, the tAvo greatest bodies of the planetary system, that the mutual attraction of the planets is sensible. Their mean motions are nearly commensurable ; so that five times that of Saturn is nearly equal to twice that of Jupiter, and the great inequalities in the motions of these two bodies arise from this circumstance. When the laws and causes of these motions were unknown, they seemed, for a long time, to form an exception to the law of universal gravitation, and now they are one of the most striking proofs of its correctness. It is extremely curious to see with what precision the two principal equations of the motions of these planets, whose period includes more than nine hundred] years, satisfy ancient and modern observations. The development of these equations in future ages, will more and more prove this agreement of the theory and observation. To facilitate the comparison with distant observations, we have carried on the approximation to terms depending on the square of the disturbing force, and it is hoped that the values here assigned to these equations will vary but very little from those found by a long series of observations continued during an entire period. These equations have a great influence upon the secular variations of the orbits of Jupiter and Saturn, and we have developed the analytical and numerical expressions arising from this source. Lastly, the MOTIONS OF THE HEAVENLY BODIES. 3 planet Uranus is subjected to sensible inequalities, which we have determined, and which have been confirmed by observation. The first day of this century is remarkable for the discovery of a new planet, situated between the orbits of Jupiter and Mars,* and to which the name of Ceres has been given. It appears as a star of the eighth or ninth magnitude ; its excessive smallness renders its action insensible on the planetary system ; but it must suffer considerable perturbation from the attractions of the other planets, particularly Jupiter and Saturn, which ought to be ascertained. It is what we propose to do in the course of this work, after the elements of the orbit have been determined by observation to a sufficient degree of accuracy. It is hardly three centuries since Copernicus first introduced into astronomical tables the motion of the planets about the sun. A century afterwards, Kepler made known the laws of the elliptical motion, which he had discovered by observation ; and from these laws, Newton was led to the discovery of universal gravitation. Since these three memorable epochs in the history of the sciences, the progress of the infinitesimal analysis has enabled us to submit to calculation the numerous inequalities of the planets depending upon their reciprocal action ; and by this means the tables have acquired an unexpected degree of accuracy. It is believed that the following results will give to them a much greater degree of precision. * (2341) This volume was published by the author shortly after the discovery of Ceres, January 1, ISOl ; and before the discovery of the planets Pallas, Juno, and Vesta. He did [3698a] not compute the numerical values of the perturbations of their motions as he had intended. 4 PERTURBATIONS OF THE PLANETS. [Méc. Cél. rèr. Fi ret form. Radius. CHAPTER I. FORMULAS FOR THE INEQUALITIES OF THE MOTIONS OF THE PLANETS WHICH DEPEND UPON THE SQUARES AND HIGHER POWERS OF THE EXCENTRICITIES AND INCLINATIONS OF THE ORBITS. ON THE INEQUALITIES WHICH DEPEND UPON THE SQUARES AND PRODUCTS OF THE EXCENTRICITIES AND INCLINATIONS. Differen ^* '^^ determine these inequalities, we shall resume the formula [926],^ tial equa- [3699] = -jj^ + -^ + 2fdR + r.(j-y We have, as in [605', 669], f [3700] T = W^ [3701] r = a.{l +|e^ — e . cos. (nt-\-s — zi) — i el cos. 2. (n ï + s — w)} : hence the preceding differential equation becomes,} Differeu- tj^ rS tial eijua- -^ a . / u ;;^:,ir" 0=-^ + n\r6r+3rî'a.ôr.{e.cos.(nt+i—z,) + eKcos.2.(nt+e—z=)l [3702] "^ ^ sroo^nd ^2fdR + r.C^ form. -^ \ch- [3699a] * (2342) Substituting, in [926], the value of r R [928'], it becomes as in [3699]. r3700a] ^ {2343) The equation [3700] is easily deduced from [605'] ; and the value of r [3701] is the same as that in [669], neglecting tenns of the order e^. f (2344) If we use, for brevity, the same symbols as in [1018a], namely, [3702o] T=nt~nt-Jf-s'—s, W=nt+s—zi, b = ie^—e.cos.W—ié^.cos.2W, [3703i] we shall have r=a.(l+è) [3701]; hence r-^=a-^.{l-\-b)-3=a-^.{l—3b-\-6b^); neglecting 6' and the higher powers of b ; or, in other words, neglecting e^ e*, 8ic. Now, by VI. i. §1.] TERMS OF THE SECOND ORDER IN e, e', 7. 6 Now all the terms of the expression of R, depending on the squares and [3702] products of the excentricities and inclinations of the orbits, may be reduced .j,^^^^^,. to the one or the other of these two forms,* dopSdin- on anglea R = M. cos. { i . (n' t — nt + B' — s)-{-2nt-\-K\; (Pi-t form.j [3703] of two Rz=N. cos. { i . {11! t — n i + s' — + -^1 ; f®'"™'' '■"'"■i '^^'"■*] different forms. in which i includes all integral numbers, positive or negative, comprehending also i = [954"]. JVe shall, in the first place, consider the term [3703]. [3704'] It produces, in 2fàR-\-r.(^—\ the function f [3^04"] \ .^•^^~^);" . M-^a.(y^\ \ . cos. {i.(n't—nt+s'—{)+2nt+K\. \tn' -\-{2 — t).n ' \da J ), * ^ ' yi 1 4 [3705] [3702c] retaining ternis of the order e^, we get, successively, 66^=6e^.cos.^fF=3e^-|-3e^.cos.2 W; hence \ — ^h-{-Qb^=l + ie^-\-2e. cos. W+ f 1 œs. 2 W. Substituting this in r'^ [.37026], and then muhiplying by i^.rSr, we get [3Î02(?] ; which is easily reduced to the form [3702e], by the substitution of n^ [3700] and r^a. (1 — e.cos.^ [3701] in the last temi of the second member. Now we have — 3e^.cos.^ ?f = — f e^ — fe^.cos.2 fV; hence [3702e] becomes as in [3702/], '~^ = a3''^ ^ '''^a^' '^ ^ ^ ' ^^ "^ '^ "^ ^ ' ^°^' ^^+|fi^-C0S.2 TVl [3702a;] = 71^. rSr-\-n-.a5r.l<^e^+3e. cos. W-\-^e^. cos. 2W\.\\ — e. cos. W\ [3702e] =^n\r5r~\-n^.aSr .\Qe. COS. fF+ 3 e^. cos. 2 W\. [3702/] Substituting this in [3699], we get [3702]. * (2345) This will be e^^dent by the substitution of «,, v,, &,c. [1009,669] m [957]. It also appears from [957"", &c.] ; for in [3703], the coefficients of n' t, —nt, are i, i-2, [3704o] respectively ; their difference 2 expresses the order of the coefficient k [957'''', &c.], or that of M [3703] ; which must therefore be of the order 2 or e^. In like manner, the coefficients of ?i' t, — n t [3704] being both equal to i ; the coefficient JV may contain terms of the orders 0, 2, 4, he. [957"'S &;c.], which include those of the order e^ ; and [3704i] a very little attention to the remarks in [957^, &tc.] will show, that these are the only forms of this kind containing e^. t (2346) Substituting the expression r .(——j=a.( — J [962], in the function [3705a] [3704"], we get 2/dfl + 7- . (^^) =:2/di?+ a. ('^V In finding d R, we must [37056] suppose, as in [916'], the ordinaies of the body m to he the only variable quantities; or, in other words, we must consider nt as variable, and n't cojwton^, as is done in finding d/î [1012a — c]. Now in taking for R the form [3703] , R—M. cos. { i . (w' t — nt-}-s' — i)J^2nt-\-Kl, [3705cl VOL. HI. 2 6 PERTURBATIONS OF THE PLANETS. [Méc. Cél. We have seen, in the second book [1016], that the parts of — depending on the angles i.(n't — nt + e' — e) and i.(n't — nt-\-s — 5) + ni + £, °'^"" ° are of the following forms, [3706] — =F.cos.i.(n't — nt+e' — e)-^eG.cos.{i.(n't — 7it-\-s — s)-{-nt+e — ^ \ depending " o?Z'fiT.. -te'H.cos.{i.(n't—7it+i'—^) + nt+B—z>'\; hence the function [3707] 3n^.a&r.\e. cos. (7it + e — ra) + e\ cos. (2 n i + 2 s — 2 w) } will produce, in [3702], the following terms,* C{F+G).e\cos.{i.(7i't — nt + s'—i) + 2nt + 2s—2^] } [3708] |»-«-.^ -^H.€e'.cos.{i.(7i't—nt + e'—B) + 2nt + 2^—^—^'\l' Therefore, if we notice only the terms depending on the angle i.(n't — 7it + s—s) + 27it, [3709] and put (x = 1 ; which is equivalent to the siipposition that the sim^s mass is [3709'] equal to tmity, 7ieglecting the mass of the planet ; f 7ve shall have n^ «^ = 1 ; [3705d] we obtain d R = — {2 — i) .n. M .sm. {i. {n't — 7it -Jf- I'—s) + 2 7it -{- Kl-ctL Integrating this, and multiplying by 2, we get [3705e] 2fàR= .^,f~'':" .M.cos.{i . {n't-nt + ^-i) +2n t ^K\. The partial differential of R [3705c], relative to a, being multiplied by a, gives [3W] a.Ç^) = a.{'^).cos.ii.in't-r,t + s-s)+2nt + K}. Adding this to the expression [3705e], we get 2/d R-\-a . (j^j r as in [3705]. * (2347) The forms of the temis of -, assumed in [3706], are the same as those computed in [lOlG] ; the constant part corresponding to i = 0; and the secular [3708a] teiTOS being made to disappear, as in [1036", &c.]. Substituting these in [3707], and reducmg by formula [20] Int., retaining only the teims dependmg on the angle i . („' t — nt + s'—s)-\-2nt + K [3703], we get [3708]. t (2348) M being the mass of the sun, and m that of the planet, we have JU+m^M- [3709a] [914']. If we put Jkf=l, and neglect m on account of its smalhiess, we shall have fA=l ; and then fiom [3700], we shall get [3709']. VI. i. §1.] TER]\IS OF THE SECOND ORDER IN e, e', y. and then the differential equation [3702] will become* ^ rf2.(,6,) ^i{F+G).c\cos.\i.{n'l-nt^s'-z)+2nt + 2s-2v>\\ </«2 - ^ -\-H.ee'.co5.\i.{r^t—nt^^—s)-\-2nt^1i-zs-vi'\) (i)i+(2 — i).n \(/o/) Hence we get, by integration,! ^ ( (F+G).Ê2.cos.^ù(n'< — w< + £'— 5)+2ji< + 2s — 2-nf A «". < > I Values of ( +H.ee'.cos. fi.(ji'i — n<+E' — s) + 2?i <+2 s — «—• n'} )> f ^(îr V J depending [3710] rSr (in'-\- (2 — i) .n \da J '^ on angles of the first form. [3711] a2 {i.n'+(3 — i).Mf .{!'n'+(l — «).n| TOT If this expression of be considerable, and one of its divisors i n' + (3 — i) .n, i n' -\- (\ — i) . n, be very small, as is the case in the tlieory of Jupiter, disturbed by Saturn, when we suppose i = 5 ; 2n being [3712] nearly equal to 5n' ;% the variableness of the elements of the orbit will * (2349) Substituting, in [3702], the value of its third and fourth terms [3708], also the values of the fifth and sixth terms [3705], multiplied by n^ a^ = 1 , for the sake of [3710a] homogeneity; it becomes as in [3710]. [3711a] t (2350) If we put, in [865, 870'], y=rSr, a = n, a Q = 2 .a/f.™- (m,^ + 6,), the letters m, s bemg accented to prevent confusion in the notation, and 2 denoting the sign of finite integrals; we shall have the differential equation [3711&], whose integral [871] is as in [3711c], r 5 r = 2 . -4^, . f^- (m t + s)= "- ^ . [3711c] m^a_„a cos. ^ ' i i' m,^—n^ ^ Comparing the coefficient of < in the expressions [3710, 37116], we get OT,=i. (w'— M) + 2n; [3711d] hence m,^ — n-z={m,-{-n) .{m^ — ?i)= ^fn'_[-(3 — i).n\ .\in'-\-{l — i).n\; substituting [3711e] this in [3711c], and then dividmg by a^, we get [3711]. t (2351) We have, in [4077], for Saturn n'=43997''; and for Jupiter m=109256' r37n/-n nearly; hence 5?i' — 2 n = 1473^; which is quite small in comparison with n or n', being only y\ part of n. 8 PERTURBATIONS OF THE PLANETS. have a sensible influence on this expression ; it is important, therefore, to notice this circumstance. For this purpose we shall put the differential equation [37 1 0] under the following form,* 0= ^^^^-\-n-.rôr-\-7i-a^P.cos.{i.(n't — nt-\-s'—i)-{-2nt + 2s\ [3713] dt^ ' ^ / I I 3 + n'a^F. s'm.\i. (n't — nt + b'—b) J^2nt + 2s\. Integrating this, and neglecting the terms depending on the second and higher differentials of P, P', we shall obtain f r 7i" ^aTulof a^ {i7i'~\- (3 — i) . 71] . [ill' -\-{l — i) . ii] noticing / C _ \ d P' the secu- «/-.-. . . „ lar varia- tion of the _ elements. S / ' \in'-\-[^-i).nl . iin'-\-{l—i).n\ [3714] x< ^ > i i -ry > ) . v.(B) + \ p- 2.^KK-„)+2n|.— / ^.^ ^iÇn't-nt+s'-s)+2nt+2-:} ] [371%] * (2352) If we put, for brevîty, T,= i . {n't — nt -\- i' — s) -\-2n t -\-2s, the term depending on J*', in [3710], mil become [3711^] 3 „3 «2 2^ e^. COS. ( T; — 2 in) = f «2 «a Fe^. {cos. T, . cos. 2 « + sin. T, . sin. 2 ra | ; [3711i] if^veput |Fe2.cos.2zj=P; |Fe8,sb.2a=P', it becomes mV. {P. cos. T+P'. sin. TJ, as in [3713]. In like manner, the terms of [3710], depending on G, H, M, maybe reduced to the forms [371 li] ; P, P' being functions of the variable elements e, ra, Sic, ■■ ^ and r, T' mdependent of these variable elements ; observing, that n, a, s [1045', 1044"] are considered as constant, as well as the similar elements of the planet m'. ■f (2353) Using the abridged symbols m,. T, [3711(Z, g], and substitutmg, in [37116], r3714o] tlie flmction [3711/], instead of the temis under the sign 2, this differential equation becomes of the form [37146], and the integral [3711c], taken in the hypothesis that P, P' are constant, becomes as in [3714 c], [3714i] = ^l^Sdll j^n^,rôr + rv" é. {P. cos. T,+ P'. sin. T,\ ; 712 a2. 1 p. COS. r, +P'. sin, rj [3714c] r 5 r = . «1,2 — rfi We shall suppose r & r, to be increased by the quantity [r (5 r] , in consequence of the secular variation of P, F, so that mstead of [3714c], we shall have, generally, [3714d] ràr= ^^^^^^^ +['•5'-]. VI. i. §1.] TERMS OF THE SECOND ORDER IN e, e', y. 9 The formula [931] becomes, by putting (^ = 1,* — — î> • S i \ Values of a^ndt - ^ j^H.ee'.\i.{n't — nt-Jfi'—s)-^2nt-Jr2e—a—zi'l) [ Sv 1[^) depending on angles of the first è V = ; [3715] V/l — e2 and by giving to i all positive and negative values, including zero [3704'], [3715'] we shall obtain all the inequalities, in which the coefficient of n t differs from that of n't by two. Now as the value of r 5 r [37 1 4 c] satisfies the equation [3714J], supposing P,P' to be constant, and by hypothesis the value [3714(Z] satisfies the same equation [37146], when P, P' aie [37Ud'] variable by reason of the secular inequalities, we may substitute [3714d!] in [37 14 J], and then, from the resulting expression subtract the equation [37146], and we shall obtain an equation of the form [3714/], observing, that we must retain only the terms depending on the first and second differentials of P, F, namely, dP, dP', d^P, d^P', to the exclusion of P, F, [3714e] ^ d^.{ràr] o r t n I "a ^^.f p.cos. r, + P'.sin.rJ 0=-^^ + n-. ir&r-]+n~a^ (,„.-n^).rf<a ^^714/] Now we have, generally, d^.{P.co%.T)^d^P,cos. T,+ 2dP.d.{cos.T,) + Pd^.{cos.Ty, [37Ug] in wliich the term containing P is to be rejected [3714e] ; and if we neglect the term depending on d^P, on account of its smallness, we shall obtain d^. {P . COS. T,)=2dP.d. cos. r,= — 2dP.m^dt. sin. T, [371 1<Z, g]. [^W] In like manner we have d^. (F. sin. T,)=2d F. d . sin. T,= 2dF.m,dt. cos. T, ; [3714A] hence [3714/] becomes = ^^^ + n^[r<5r] + -^--^.j2m,.— .cos.T,-2m,.-.sm.r,|. [3714.] This is similar to the equation [37116], changing rSr into [riîr], representing by aQ the tenns depending on d P', d P. These ternis being divided by m^ — n^, give, as in [3711c], the following value of [?• 5 r] ; r-j,.-! S ^"^ iZ cos T- ^"'- — sb t\ [37144] Substituting this in [3714f/], connecting together the terms depending on cos. T^, also those depending on sin. T,, then substituting the value of m^ — n^ [3711c], and dividing by a^, we get [3714]. * (2354) We have 2 r . d5r -{- dr . ar = 2 d .{r^r) — dr . 5 r, as is easily [3715a] proved by developing the first temi of the second member, and reducing. Substitutmg VOL. III. 3 10 PERTURBATIONS OF THE PLANETS. [Méc. Cél. If the coefficient in'-\-(2 — i).n be very small, and this inequality be very sensible, as is the case in the theory of Uranus, disturbed [3715"] by Saturn [4527]; we must put the part of R depending on the this and [3705a] in [931], we obtain ^ -\- I \ 3 a In at . dli 4-2 an at . a . I — } [37156] <J w = ■ /(l-e^) The differential of [3701], being; multiplied by :; — — , becomes ■- a-ndt [3715c] — ^î^ "= ~ TT • ^^ • ^'"' (« ^ +^ — «) + «■• S'n-2 . (n < +-= — a)^ This is to be reduced, as in [370Sn], by substituting the value of — [3706], using tlie fonnula [IS] Int., and retaining only the terms depending on the angle T, [371l£-] ; hence we get [3715rf] -J^ = -^(F+G).e^.sin.(T-2«)-iifee'.sin.(T-.-.'). [3715e] Again, if we put, for brevity. To = i . {n't— n t + /— e) + 2 n( -\-K, tlie term of R [3703] will become R = M.cos.T^; hence the differential à R, found as in [916'], upon the [3715/] supposition that nt is the variable quantity, is dR = — (2 — i) .n d t . M .sm. T^i- Multiplying this by 3 a.ndt, integrating and using m, [37 lid], we get oafndt.dR = - . a M.cos. T„ . [3715g] To this we must add 2andt .a. (-j—j = 2andt .a . (-j—j • cos. T!, ; and then, by integrating the sum, we obtain [3715/.] f[saJndt.dR + 2andt.a.(^-^)]=\^^^^.aM+^Z^l^^ Substituting this and [o715£Z], in [3715i], we get [371.5]. [3715i] In the great inequalities of Jupiter and Saturn, the most important parts of Sv, Hv' [37 15 J, &c.] are those depending on the double integration of AR, d!RI, which introduces the divisor (5 m' — 2n)2. These paits are to be applied to the mean motions [3715fc] of the planets, as is shown in [1066", 1070"]. As we must frequently refer to these parts 6v, 5v', of the mean motions <^, ^' of the planets m, m', we shall here give their values, deduced from [1183, 1204, 3709a], or from the appendix [5794], representing the chief parts of & v, 5v' [37156, Sic.] ; [3715Z] Sv deduced from ^ = 3a n .fd t .fd R ; [3n5m] (5 v' deduced from ^' = 3 a' n'.fd i ./d' R. VI. i. §1.] TERMS OF THE SECOND ORDER IN e, e', y. 11 angle i.(n't — nt-\-s — s ) + 2 ?i ^ + 2 s [3703] , under the following form,* R= Q.cos. {i . (n' t — nt + s' — s) + 2nt + 2sl + Q'. sin. [i . (n' t — nt + s' — s) -^27it + 2sy^ and we shall have,t [3716] (6— 3i).«2a { ^--rr } , ^ ,^ la.ffndt.àR^- ji- I q + . ,, ''^ > .sin. \i.{>i't-nt+s-s)J^2nt+2 •'-' \in'+{2-i).n\^ I ^ in'-\-{2-î).n) [3717] (6-3»).n a_^ ) Q'___HL_ } .cos.{;.(M7-n<+s'-£)4-2n<+2s}. {in'+(2— i).nP / ni'+(2- * (-2355) Using, for brevity, K,= K — 2 s, and T, [.3711^], the expression of [3716a] R [3703] becomes iî = JIf .cos. (T,+Ar)=JM.cos./i:;.cos.2'— ^f . sin. Z;.sin.T, ; and by putting ^I.cos.Z,=::Q, — J/.sin.Z,= Q'; it changes into il=.Q.cos.r,4-Q'.sin.T, [37166] as in [3716] ; Q, Q' bemg like P, P' [3711À;], functions of the variable elements of the orbits, and T, independent of them. Now we have, in [4077], for Uranus n = 15425*'; for Saturn n'=: 43997'' nearly; hence 3m — ?i'=2278»'; which is much smaller than n [3716c] or n'; and by putting i^^ — 1, in the divisor in' -{-{2 — i).n, it becomes 3m — n'; [3716(/] therefore this small divisor must occur in computing the perturbations of Uranus by Saturn, as is observed in [3715'']. t (23.56) The difierential d R, deduced from [.3716i], considering nt as the variable quantity, as in [3715/], is dR = — {2 — i).7idt.q. sin. T, + (2 — i) .ndt.q'. cos. T, ; [3717a] hence we have 3 a .ffn dt.d R=ffa n^.di^.l{—6 + 3i).q.sm. T,-{- (6—3 î).q. cos. TJ. [37174] If the integral of the second member of this expression be taken, supposing Q, Q' to be constant, it ^\^ll produce the terms independent of d(^, c? Q' in [3717]. The terms depending on d(^, d(^ may be estimated by means of the general formula [1209è], which, by changing ^, B into Q, A, respectively, and neglecting d^Q, rf^Q, Sic, becomes ffAqdt^=qffAdt^-2.'^.fffAdt\ [3717c] From this formula, it appears, that the term depending on —, is easily deduced from that depending on Q , by changing Q into — 2 .—^.dt, and then integrating relatively to t, supposing — to be constant. In this way we easily deduce the term depending [3717*^] on dq [.3717] from that of q-, and in like manner we get the term depending on rfQ' from that of q. 12 PERTURBATIONS OF THE PLANETS. [Méc. Cél. Hence the formula [37156] will give* 2d.{r5r) UF-\-G).e^.smAi.(n't-7it4-s'-s)4-2nt + 2s-2ôil ) a^ndt - ( -\-H.ee'.sm.li.{n't—nt-\-e'—=)~{-2iii-\-2s—ss—zi']) bein;_ variable. [3719] ^ fin'-\-{2-i).n\^' [_" ^'' in'-\-(2-î).nJ in'-lf-[2—i).n -^{rr—, ~\ «QH — + }.sm.ii.{n't-nt+s'~i)+2nt + 0g\ Another form of tins value of ÔV, [3718] [3718'] For greater accuracy, we have neglected the divisor \/i — e^ in this expression of (5ï? ; because it does not affect the part of this expression which has the square of mi' +(2 — i) . n for a divisor, as we have seen in [1197]; and in the present case, this part is much greater than the others. Moreover, we must, as in [1197"", 1066", 1070"], apply this part [3719'] of &v to the mean motion of mf ; and as it is very nearly equal to the * (2357) Using the value of R [3716], or rather [37166,3711^]; taking its partial differential, relatively to «, which will aflect only Q, (^' ; multiplying by 2a^.ndt, and then integrating, we get [d/lBaj 2 and t . a. I —— = -r- ■ sin. 1 , . - — . cos. i ; ■^ \da / m, \da J ' m, \da J ' m^ being, as in [3711f/]. Substituting this in [3715J], also the values of the terms [3717, 3715rf], it becomes as m [3718] ; except that the divisor \/(l — c^) is neglected, [37186] for the reason mentioned in [3718'], namely, that the chief part of Sv or ^ [1195 or 1197] does not contain this divisor ; and as the other terms are very small, it may also be neglected in them. t (2353) The tenns of Sv [3718], having for divisor the square of j'w'-f- (2 — i) ■ n, [3719a] are those depending on 3 affndt .dR, computed in [3717]; and it is evident, that this part of S v much exceeds the other parts depending on F, G, H, he. Now, by [1066", 1070"], or by [1197>'"], the parts depending on 3affndt.dR, must be applied to the mean motion, and as the other parts, depending on the same angle, are much [37196] smaller, we may suppose that the whole of this equation is to be applied to the mean motion, as in [3720]. We may remark incidentally, that the expression of r [1066], as well as that of v [1070], contains the double integral ffndt.dR; hence, at the first view, it would seem that if v contain terms depending on this double integral with the small divisor [in' -{-{2 — ?') . «P, as in [3718], ?• would contain similar terms of the same order. But we must observe, that these terms of r, v [1066, 1070] are multiplied, [3719c] respectively, by ( — — - ) , f — — ) , or by their equivalent values a e . sin. (n i -}- s — a), l-t-2 e.cos. (n< -|- s — «) [669]. Hence these terms of v will be multiplied by I, VI. i. §1.] TERMS OF THE SECOND ORDER IN e, e', y. 13 whole term depending on the angle i.{n't — nt-\-i' — t) -\-2nt -^-2^, ive may apply this tvhole inequality to the mean motion of m. [3720] TIT 1. 11 1, • 1 1 r dP iJP' (JO dO' We shall obtani the values of — — , — — , — -^, —-i^, by takmg the (It (It ^ dt dt •' ^ differentials of the expressions P, P', Q, Q', relative to the excentricities and inclinations of the orbits, the positions of their perihelia and nodes, and then substituting the values of the differentials of these quantities. But we may obtain these values of — — , &c. more simply in the following manner. [3721] Find the value of P, for an epoch which is distant by two hundred years from the epoch taken for the origin of the time t ; then putting P^ for this value, and T for the interval of two hundred years, we shall have* Formula , ^ for the yj^ d ± J-. j-> détermina- 1 . j-^i'—f. [3723] " ' tion of dP, dP', T . £ J 1 1 c dP" do dq &c. In the same manner, we may nnd the values oi — — , — -^, — -^. •^ dt dt dt or TÔT To deduce the expression of — from that of — j-, we shall denote by -^, the part of — depending on the angle i.{n't—nt-\-i — 6)+2ni+2s, [3724] and we shall havef ràr r S^ — \-F.cos.i.{n't — nt-\-s — s)-{-Ge.cos.\i.{nt—nt-\-^—s)-\-nt-\-s — ra\ ( 5" — „ ' S '^ C ' [3/25] [3722] + i/e'. COS. ^i . {in—nt-\- s'— ;) -\-nt-\- s — jj'J S and those of r by the small quantity e, which will make it of a less order ; it will also be of [3719rf] a different form from those contained in this article, by reason of the factor sin.(n<-|-£— ra). dP (PP * (2359) From Taylor's theorem [617], we have P,=P + T.-— +| T^. — + &c.; and if we neglect the square and higher powers of T, on account of the smallness of the [3723a] terms, it becomes as in [3723]. t (2360) Adding — to the part of — [3706], we shall obtain all the terms of a 'a depending upon the angles i . [n' t — nt-\-i' — i), i.{n't — nt-\-i' — s) -^nt-\-i, [3725o] i.{n't — nt-\- s' — s) -{- 2 n t + 2 !. Multiplying this by - , we get [3725]. VOL. III. 4 14 PERTURBATIONS OF THE PLANETS. [Méc. Cél. Value of f), r, [3736] for the angles of the first form. [3726'] [3727] Computa- tion for angles of the second form. Hence we deduce* -^ = ^+i.(i^+2G).e-.cos.{?:.(n'i — »/ + /— + 2/1^ + 2;— 2^J > + lH.ee'.cos.li.(n't—nt-Jrs—i) + 2nt + 2i — z: — -/\ ) 2. JVe shall compute, in the same manner, the terms depending on the angle i.Çn't — nt-\-s' — e) ; and shall suppose, that, by carrying on the approximation to the first power only of the excentricities, we shall havef — = F.cos.i.(7i't—nt-\-s—i)-\-Ge.cos.\i.(7i't — nt + s — !) + nt + s — ^J + G'e .cos.j — i.(7i't — nt^s' — £) + nf-(-£ — to j + iïe'.cos.| i.(n't—nt+s'—s)-}-nt + s—^'\ -{'H'e'.cos.\—i.(n't—nt + s—s)-J^nt + i—^'\ ; [3726a] [37261] [3726c] [3726d] [3726e] [3726/] [3726g] [3726;i] * (2361) Using the symbols [3702«], namely, T=n't — nt-\-e'—s, TF=znt-\-s—a, W':= n't-{-s' — zs', the expressions [3725] will give, by transposing the terms depending on F, G, H ; f,..n,^ ca,,-~w^-^/yiXr+ V--UT' .F.cos.iT—-.Ge.cos.{iT+W) — -. He', cos. (i T-j- W/) ; r (5, )■ a ' a rSr and from [3701] we get - . :^ 1 -)- 4 «^ — ^ • cos. W — | e^. cos. 2 TV; which is to be substituted in [372Gè]. In making this substitution, we have, by hypothesis, only to notice terms of the order c^, ee', e'^, &,c. [3702', &c.], and of the same form as [3703]. Now (W* . /* the term -^ [3724] being already of the second order, we may substitute for the factor - , by which it is multiplied, the first term of its value [3726c], namely 1 ; in the coefficient of -F, we may use the term — i e^. cos. 2 TV ; and in the coefficients of G, H, the term — c . cos. TF ; by this means it will become as in [3726^]. Reducing this expression by means of [20] Int., and retaining only terms of the form [3703], it becomes as in [3726A], which is of the same form as in [3726]. S^^rjr^ e^ cos. 2 TV) . F. cos. iT-\-(e. cos. TV) .Ge. cos. (i T+TV) rSr + {e.cos.TV).He'.cos.{iT~\- TV,') = -^+ IFe^.cos. {i T+2?F) + JGe^.cos. {i r+2fr) -(-iffee'.cos. {i T+TV+T¥/). ■j- (2362) The expression of — [3727] is the same as [3706], making the alteration required by the supposition, that i is positive [3727']. If we use, for brevity, the symbols [3726a], this formula will become [3727a] [37276] — =F. cos.i T-\-Ge. cos.(i T-\-lV)+G'e-cos.(-i T+W)+He'. cos. (i T+ W/)+H'e'. cos.(- i T-\- W/). 5r The case of i^O, is separately considered in [3755'", Sjc.]. VI. i. §2.] TERMS OF THE SECOND ORDER IN e, e', y. 15 i being positive [3727rt, 6]. We shall then get* [3727] r ( G'+ G') . e2. COS. i . {n' t — nt + B' — e) ^ + He c' . COS. { i . {n't — nt^ s' — s) + -. (E) [3728] r r < -\- jn ee . cos. ^i.yn i — ni -^ s — jj -j- a — -n 5 > ( +H'ee'.cos.^;.(«'<-n? + 6'— £) — a + a'^) , , ^_^^_^^^^ * (2363) In finding the pai-t of rSr dependmg on the angle i.{n't — nt-{-s'—i), or iT, by means of the fomiula [3702], it is necessary to compute the part of 2/di? + r.( ) [3728a] depending upon the same angle, or upon R^^ JV . cos. {i T-\-L) [3704]. This [37286] gives for dR, similarly to [3705^7], the expression àR=nJ\r .i . sin. [i T-\-L) .dt • 2» hence 2/d R = ^— . JV . cos. {i T -\- L) ; also from [3705a], we obtain [3728c] Multiplying the sum of these two expressions by 1 = ji^ a^ [3709'], we get 2/dil+..(^) = «^«^5«^.(^)-^-^^.aA-^.cos.(.-T+Z). [3728.] Again, if we multiply [37276] by 3 n^ a^. {e . cos. PV-}- e^. cos. 2 W], we shall obtam [.3728/] the terms of [3702], which are multiphed by 3n^a.Sr; and as we have to notice only the terms depending on angles of the fonn i T [3726'], we may neglect the second rg^gg -, tenir of this factor e^.cos.2W [3728/], and then it will become 3 74^ a- . e . cos. W. In multiplying [3727è], by this last factor, and reducing by [20] Int., the term jP produces no term of the required form, and each of the other terms G, G', H H' produces one ; hence we finally obtain 3 n» a . 5 r. J e . cos. W-{- e^. cos. 2 ?F|:= f « V. { ( 6+ G') . e^ cos. i T+He e'. cos. {i T-\- W,'— TV) +11' eé. cos . {i T— W;+ W) ^^^^^''^ = |«V.{(G+G').e2.cQs.i T+Hee'.cos. (iT+a— «') ^^,„^., (3728*1 +ifée'.cos.(îT— a + tj'). The sum of the second members of the expressions [3728e, z], being represented by a Q rfs irSr) [3728ft] for brevity, the differential equation [3702] becomes = -^^^ — -\-n^.r()r-\-a.Q^; and we find by mspection, that a Q is equal to the numerator of the second member of [3728], [3728i] miltiplied by a^. This equation, being solved as in [37116, c], gives rôr = -^-^, usi^g^, [371k/]; hence we get '^ = „Tg;ê^ = „,(,„_" ^(^^^„) > as in [3728]. [3728.] 16 PERTURBATIONS OF THE PLANETS. [Méc. Cèl. ç{G—G').e\sm. i.{nt — nt + ;'—s) ■\ Value of 6v. — Hee.sm.\L{n't — 7it-Jrs' — s) — zs-\-zi'l') } .* (F) . , ■ .a^A-r-] — r-, — r-;; ■aJS >.sm.\i.{nt — nt-\-^—£)-{-L\ [in'— in \da/ [tn'—my ) / > > [3729] ^ V = ^1=1 [3730] r fi r* If we put -^ for the part of — , Avhich depends on angles of the rt [3729, form i,{n'l — n^ + s' — s),t and is also of the order of the square of * (2364) The value of 5 d [3729] is easily deduced from [37155]; since the denominator ^(1 — e^) is the same in both, also the first tenn of the numerator; and the other terms may be obtained by a calculation similar to that in [3728« — »]. For if we multiply the expression [3728c] by §andt, and [3728fZ] by 2 a n d t, and take the sum of the products, we shall get «] 3aj7idtAR + 2a.ncItui.(~)=\2n.a^('!^) — ^-.a:N'l.cos.{iT-{'L).dt. •' ' \da J i \da J n—n > \ ■ / Integrating this we get the two last terms of [37156], which are the same as the two last terms of the numerator of [3729], or those depending on JV, dJY. The only remaining term of [37156] is the second, which is found by multiplying the differential of ?• [3701] [3729i] by ^ ; whence we get — ^^^ = ^ . 5 e . sin. Jr+ e^ _ si„_ 2 Wi. ■' a-ndt a~n dt a Sr Substituting — [3727], we may neglect the term c^.sin. 2AF, and the term F, as in [-3728^, kc] ; the other terms being reduced as in [18, 19] Int., retaining only angles of the form i T; we get, in like manner, as in [3728A, Sic] ; —^±^==—-x.smW=l\{G-G'U^.smJT+Hc(^.s\n.{lT+W;-m-Hee\sm.(IT--l^^+W)\ a^n dt a [^''^^'l =h-\{G-G').e^sm.{T+Hce'.sm.(iT+:s - z/)-Hee'.s\n.(iT -in+^n')] ; [3729d] being the same as the terms depending on G, G', H, H\ [3729]. We may remark, that from the formulas [3728, 3729], we may deduce others similar to [3714, 3718], in which the secular variations of the elements c, -m, &,c. are noticed. I (2365) The second member of [3727] being denoted by F', it will include all the or . ■ ■ ■ r3731ol terms of — , depending on the angle i T, as far as the first power of the excentricities a (5 r [3726']. Adding to this the expression —, depending on the same angle, and on terms [37316] of the order e-, ee', Sic, we get — =ij"-f— , for the expression of —, containing VI. i. §3.] TERMS OF THE SECOND ORDER IN e, e', y. 17 the excentricities or inclinations, we shall have — = ^j^ + i .lGi-G'—F\.c~. COS. i.(n't — nt+ s' — V'''""'' + i. ATee'.cos.^î. (n'f — «^ + s' — s) + ^— ^'J [3731] for onglcB of the second form. + I . He e'. COS. \ i . (n' t — nt-\-^' — s) — ^ + ra' | . In these three expressions i must be supposed positive [3727']. [3731' 3. The great number of inecjualities depending on the squares of the excentricities, and of the inclinations, makes it troublesome to compute all of them ; and we must be guided in the selection of those which are of a sensible magnitude, by the following considerations. First. If the quantity in'-\-(2 — i).n differ but little from ±?i; then the one or the other of the divisors in' +(3 — i) . n, in'-\-(\ — i) . n, in the formula [3711], ^^^^^^ will be quite small, and by this means the expression may acquire a sensible value. Second. If the quantity in' -{-(2 — i).n be small, those terms of [3733] the formula [3715], having this quantity for a divisor, may become sensible, slîecîîtg^ Third. If the quantity i . (n' — n) differ but little from rh n, the one imp^tLt or the other of the divisors in' — (i-\-l).n, in' — (i — l).n, of the [3734] formula [3728], will be small, consequently this expression may acquire a sensible value. Fourth. If the quantity i . {n — n') be small, the terms terms as far as the order e^, ce', &ic. inclusively. Multiplying this by —, we r i,r rSr '' 7-1/ t i /- 1 c 1 • *■ get -. — ==; — g- . f . In the first member of this expression, we may put -=1, [37316'] as in [3726(Z], and in the factor of F', we may use the value [3726c] ; hence we shall get ^=:'^ + F'.{ — 1— ic^+e.cos. rr+ i e"". COS. 2 JV I [3731c] = ^— 5e^--F.cos.ir+F'.e.cos. W; [373W] the second of these expressions being easily deduced from the first, by observing, tliat of the four terms comprising the factor of F' [3731c], the first teim, — 1, produces nothing [3731e] of the order e^, when the value of F' [3727] is substituted ; the second tenn, — i e^, produces the term depending on F in [373 Ir/] ; the third produces the term depending on jP' [3731c/] ; and the fourth term, ^ t^. cos. 2 fV, produces nothing of the proposed form and order. Now substituting, in the term F'.e.cos.JV [313ld], the value of F', [3731/] or the second member of [3727], reducing the products by [20] Int., and retaining only angles of the form i T, it becomes as in [3731]. VOL. III. 5 18 PERTURBATIONS OF THE PLANETS. [Méc. Cél. [3735] •General value of R. [3736] First form. [3737] [3738] [3739] Values of ^■> y. 2, [3740] x', y', z'. [3740'] [3736a] of the formula [3729], which have this divisor, may become sensible. We must therefore estimate carefully all the inequalities subjected to either of these four conditions. 4. The quantities F, G, G', H, H', are determined by the approximative methods in the second book [1016, &c., 372T]. We shall now determine M, N ; and for this purpose we shall resume the value of R [913, &c.];* m'.[xx'-^yy'-^ zz') R^ m r' being the radius vector of m'. We shall take., for the fixed plane, the primitive orbit of m, and for the axis of x, the line of nodes of the orbit of m' upon this plane. If we put v for the angle formed by the radius r and the axis x ; v' for the angle formed by the same axis and by / ; also 7 for the tangent of the inclination of the two orbits to each other, we shall havef y = r . sin. v, 2^0; a; = r . cos. v X =^r . cos. V ; y- r . sm. V ^^ f .y . sin.w (2366) As there are only two bodies m, m', the value of R, X [913, 914] become n'.[xx'-\-yy-\-~z') X X R:= m' \ > M .■■'\ [37366] and by using r'^ = a'^ + if -{- z!^ [914'], we get [3736]. f (2367) In the annexed figure 72, C is the origin of the co-ordinates, or centre of the sun; C X, C Y, C Z, the axes of X, Y, Z, respectively ; M the place of the body to, supposing it to be situated nearly upon the [3740o] plane of xy [3737] ; M' the place of the body in. The co-ordinates of TO ai'e CA=x, AM^y, z=0 \ nearly ; those of to' are CA' = x', A'B'=y', B'M'=z'. Moreover angle.MC.^=«[924^],.lfC.^W, Then in the rectangular triangle CAM, we have C A — C M .cos. A C M, AM=C AI. sm. A CM, or in symbols, a: = r.cos. «, y = r .sin. i) [3740]. In the VI. i. §4.] TERMS OF THE SECOND ORDER IN e, é, 7. 19 Hence we get, by neglecting the fourth powers of 7,* [3741] R = -j^ . COS. {V I') J- . 3-3 . i COS. (V V) COS. (v^v)\ m r , , , m.y' r — - .COS. (V I') . — . !COS. (V' V) COS. (V'-\-V)\ Second .,.'2 V / ^^2i V -' V'/l fo,„ „f R. <i!.y~ rr'.{cos.(t)' — 1;) — cos.{v'-\-v)\ [3742] |r2— 2r;-'.cos.(D'— t>) + r'2}4 4 ' jr2_Orr'.cos.(D'— t') 4-r'2}2- We shall suppose, as m [954, 956], -^.cos.(n'i— n^ + s'— f) — Ja-— 2aa'.cos.(n'^— n^ + s'— s) + a'^î~^ [3743] = I x.A^'K COS. i.(n't — nt-\-s — ^t-',B<o. !«-— 2«a'.cos.(n'^— ni + s'— .^) + rt'2|-f=i2.jB»cos.ù(n7— n^ + £'— 0; [3^44] rectangular triangle CAM', we have C^'=CJ/'.cos.^'CJ»f', ^'Jf ^^CJU'.sin.^'CJ»/; or in symbols, x'=r'.cos.v' [.3740'], .4'J'/'=r'. sin. j;'. In the rectangular triangle A'B'M', we have, A' E=A'M'.cos.B'A' M', B' M'=A'M'.5m.B'A' M' ; substituting in these [3740c] 1 r the preceding value of ./2' ./»/', also cos.B' A! M' ^=-—-, , sm.B'A'M= , /(1 + 7-) v/(l + 7^) we get y', z' [3740']. * (2368) If we neglect 7^ as in [3741], we shall have (1 + 73)-*= l_ | y2 . hence we obtain from [3740'], y' = i-' . sm. v' — J 7^. r*. sin. u'; z'^='y^ .r'^.sm.^v' • [3742a] substitutmg these and the other values [3740, 3740'], in the first member of [37426], and then reducmg by [24, 17] Int., we get [3742c] ; ^ '^'+ i/ /+- ~'= '■'■'• (cos. u'. cos. ij-f-sin.u.sm.t;') — J 72.7-/. sin. jj. sin. o' [3742i] ^.irr'.cos.^!;'— î;) — i72.rr'.jcos.(t)'-i')— cos.(«'-fî))}. [3742c] Substituting this last expression in the first tenn of R [3736], we get the two first terms of [3742]. Again, if we develop the first member of [3742e], and substitute r^=x"-+y^+z^ r^=x^+y'^+^'^ [3740,3740'], also the expression [3742c], we get ^ (^-^)'+ {y'-yf+ (^-~)'=(-^"+2/'+~-)-2. (^^'+yy+zz')+ (x'==+2/'2+^'2) [3742e] = ^-2-2;y.cos.(«'-v)+r'2}-|-i72.rr'.{cos.(î,'-i,)-cos.(z>'+^)}. [3742/] Invc^lving this to the power — ^, we get \{^'-xf+{y'-yfJr{^'-zf\-i=\r'-'2rr^.cos.{v'-v)-^r'^]-^ _3 [3742^:] ^ Y^.rr. \coz.{v'—v)—co%.{v'\v) \ . {r^— 2r/.cos.(/— t;)-!-/^ j 2. substituting this in the last term of [3736], we get the two last terras of [3742]. 2» PERTURBATIONS OF THE PLANETS. [Méc. Cél. [3744'] and shall represent R = M.cos.\i. (n't —nt-i- s'—s) + 2nti-K\ [3703] , by the followmg function ; K™f]f ^= M'-'Ke'' .cos.\i.(n't—nt + s'—:) + 2nt + 2s—2z.\ [3^45'] -\-M^'Kee'.cos.\i.(n't—nt+s — s) + 2nt-\-2s—^ — ^'\ of the first '?^^^^«-, + M^~\ e'^cos. I i.(ii't—nt + i—s) + 2nt+2i—2z,'\ [3745'"] + M^'K y^ .cos.\i.(n't—nt+s'—s) + 2nt + 2s—2n \ ; n. n being the longitude of the ascending node of the orbit of m' tipon that [3746] fyj- j^^ counted from the line which is taken for the origin of the angle ni + f. We have, as in [669], [3747] -= 1 + ^e- — e. COS. (ni+£ — -a) — i e". cos. 2 . (nt-\-£ — ^i) : [3748] v = nt^i — n + 2e. sin. (n i + s — ^) + | ^'' sin. 2 . (n t + e — a). From these we get the values of —, v', by marking with one accent, the quantities n, e, s, &c. Then we have, as in [955], the product of 2 . A (". cos. {i . (7i't — nt + .='— s)|, by the sine or cosine of any angle ft-\-I\ which is equal to [3749] 2 • A'' ■ "^i li-(n't-n t + /— =) +ft +I\. Hence we easily obtain* [3750] M-^= f.J.-.(4^-5)..^(') + 2.(2.-l).«.('^) + «^('^)5; Values pondins 4^^-^ ^ ^ ^ \ da J \ da J \ dada / ^ to tlie first ^^, M-.= |.^,;_.,.(4;-3).^--..(.,--a,...(^')+...C^)^, [3750'"] M '31=— — . « a'. £''-'1 . 8 [3750a] * (2369) In [952, 953] we have r = o . ( 1 + wj ; v = vt -}-s — U-\- v,; the terni n being added to conform to the present notation. Comparing these witli [3747, .?748], [3750t] we get the following values of u^ , v^ , also the similar ones of w/, w/, using the abridged symbols [3726a] ; [3750c] «,= — e.cos.^F + 4 e^ — J e^. cos.2 ^F ; t), =2e .sin. ^F + f . e^siii.2 TF ; [3750rf] «; = — c'. COS. W'-\-^e"^—l e'- . cos. 2 W ; v;=2e'. sin. ?f '+ f . e'^ . sin. 2 W; M. ï. §4.] TERMS OF THE SECOND ORDER IN e, e', y. 21 and in the case of i=\ [3150y, ?/'], we have 4 a'* 8 Finding the squares and products of these quantities, then reducing them by [17 — 20] Int., retaining merely the temis of the second degree in e, e', y, which are the only terms now under consideration [3702'], we obtain the following system of equations. In these expressions we have substituted for JV its value W =^ T -\- IV -\- zs — -a' [3726a], [3750c] in order that the quantity n't-\- e' may not appear in the tenns of R, except in connexion with i, as in the assumed form of these terms of R, given in [3745, &c., 957]. The numbers prefixed to the formulas [3750/] express the order of the terms in the value of R [957]. = Je'2 — Je'2.cos.2.(r+ ?F+t3 — ^); = f Ê'2.sin.2.(r+ ?r+« — î/); = fe^.sin. 2^; = ie^-f le'-i.cos. 2 7F; '== è ee' . cos. (r+ w — ^') + A ce', cos. (7+ 2 W-\--a—Ta') ; = Je'2 4-Je'2.cos.2.(r + ?F+« — ^'); = — fe'.sin. (T^-^ — ^)— ee'.sin. {T +2 TV+zi—z,'); = — e^.sin. 2fF; ' = _ e'^.sm.2.{T-i-W+z, — z/); ,= ce'.sin. (T+w — •n') — ee'.sin. (r+2 W -}- ■a—zy') ; = 2e'2 — 2e'2.cos. 2.(T+ ÏV -^ zi — zi') ; = 2ee'.cos. (T+1^ — to')— 2e'2.cos. (T+2 ^F+a— to'); = 2e^ — 2e^cos.2W. Substituting these in [957], we shall obtain the terms of R depending upon M'-''\ M'-'^\ M'^', [3745, kc.]. The ternis of the fonn iVT^', arising from the terms of z, a/, in the two [3750g] lower lines of the value of R [957], will be considered hereafter in [3750m, &;c.]. In making these substitutions, we must use the following formulas, which are the same as those in [954f, 955a, 955/], changing TV into TV^, to prevent confusion in the notation. COS. JV^.iX.A <'> . cos. i T == i 2 . ^« . COS. ( J T + TV,) ; [3750^] sin. TV, . J 2 . iA^'l sin. {T= — ^2.iA « . cos. (i T + TV,) ; [3750i] COS. fT .is.i^^ra. COS. i r=i 2 . i^^w.cos. {i T+ W,). [3750A] VOL. III. 6 2 "/ 3 "/ 4 Î'/ 5 ^', 6 <- 7 v,u; 8 nr- 9 u,v; 10 uv. 11 w>; 12 ujv. 13 v',^ 14 v,v,' 15 r,2 [3750/] 22 PERTURBATIONS OF THE PLANETS. [Méc. Cèl. We shall represent R = N. cos. \i . (n' t — 7it + I'—s) + L\ [3704] We shall consider the terms depending on each of the factors M'-''\ iV/'", M'--'' [3745, &;c.] separately; and in the first place, shall take the tenns of the form M'^Ke^-cos. {iT-\-2W). These are evidently produced by the factors sin. 2 IV, cos. 2 TV, which occur in the terms of [3750/], marked 2, 5, 6, 10, 15; reducing the products by the formulas [3750A— A:]. These five terms, marked in the order in which they occur, without reduction, supposing them all to have the common factor ^ • c^- cos. {i T-(-2jF), and omitting 2 for brevity, are [37502] This expression is easily reduced to the form of the coefficient of —, in the value of M'-"'' [3750]. Proceeding iii the same manner with the parts of the terms 7, 9, 12, 14 [3750/], depending on the angle T+2?F-f-w — zs', we find that they produce in R [957] terms of the form iV/^'^e e'. cos. {{i + 1) . T+ 2 JV -{- zs — zs" } , which may [3750n] be represented by — ^ . e c'. cos. {(i + 1) . T+ 2 JF+ is — î/^, multiplied by the 4 following expression, which includes the terms as they occur, without any reduction ; [3750O] -««'(7a^) + 2*«-(^)-2-'(77-) + 4^'-^"- We may change in this i into i — 1 [.3715'], and then we get for the coefficient [3750;)] of — j.ce'.cos.(ir+2yF+ro— ra'), or — j . e e'. cos. {i T~{-2nt -{-2 s—zi— a'), an expression which is the same as the coefficient of — — , in the value of il/"' [37.50']. Again, the terms 3, 4, S, 11, 13 [3750/], depending on the angle 2. (r+ (F+ra— n'), produce in R [957], terms of the form M^-\ e'^ cos. { [i + 2) . r+ 2 fF+ 2 ro — 2^} ; [3750?] which may be expressed by ''^ .c'^.co5.\{i-\-2).T -{-2W -^2is — 2«'|, muhiplied by the following function, which includes all these terms as they occur, without reduction ; or, as it may be written, [3750,-'] i.(4i + 5).^'''-2.(2i+l).«'.('^) + a'^.(^). We may change in this i into %■ — 2 [3715'], and then we have for the coefficient [37505] of -.c'2.cos.(iT+2?F+2^— 2j:'), or ^. cos. (» T+2ni + 2s — 2^'), the 8 o m' [3750«] same quantity as the coefficient oî —, in the value of M^-' [3750"]. VI. i. §4.] TERMS OF THE SECOND ORDER IN e, e\ 7. 23 by the following terms ; * Terms of ^, -.-.-,«. . . - » depentlinff R= ^W, COS. I. (n't — 71 t+s'—e) [3752] on angles + N^'K ee'. cos. [i . (n't — n t + e'— s) + w — ^'| "'[ih^-] second + iV'^' . e e'. COS. ^ z . (n7 — ?i ^ + e' — s) — ra + ra' | ; 'farsa"] We shall now notice the terms depending on z, z , which were neglected in [3750^] ; these are the same as those depending on 7^, in the value of jR [3742]. As we neglect terms of a higher order than 7-, we may substitute, in these terms, the values r = a; r' = a'; v=^nt -{-s — Ii ; v =^n' t-\- ^ — U ; v' — v = n't-—nt'\-s — s=T; [3750u] v'-\-v = n't-[-nt-{-^-{-e — 2ll=T-{-2nt-{-2s — <iin; hence this part of R [374-2] becomes R = —'l^ .-^.{cos. T— cos.(jr+2n< + 2e — 2n)j ,^ , ■ [3750i>] * \a? — Saa.cos. T-|-« P Substituting, in the last term, the value of the denominator [3744], namely \ ^.B'^'K cos. iT, and reducing by means of the formula [3750^], it becomes m'ya f _-^.cos. T+4"-cos. (T+2n<4-2s— 2n) ) ^=~T~-< a- a~ V [3750k'1 (+|aa'.2.J5('\cos.(J+l).r-iaff'.2.5w.cos.{(i+l).r+2»U + 2s-2nn The last temi of this expression, changing i into i — 1 [3715'], becomes X.aa'. 2. B''-i'. COS. (ir+ 2 îii + 2£ — 2n); [3750i] which is of the same form as [3745'"], and is equal toit by putting M'-^'^ = . a a' . S . B^''~'\ 8 as in [3750'"]. In the case of i = l, the term [3750z] becomes — ^.^a a'. S^»^ COS. ( T + 2 n ^ + 2 £ — 2 n) ; [3750t/] connecting this with the second term of [3750f<'], namely, ^.^.cos.(T+2n< + 2j— 2n); [3750)/'] and putting the whole equal to this value [3745'"], we get, for this case, the same value of JW ">, as in [3751 ]. * (2370) By proceeding as in the last note, we shall find, that the substitution of the values [.3750/] in iî [957], produces terms depending on the angle i T, iT+zs — ra', [3752a] ïî"— « + «', as in [3752—3752"], without W, which occurs in the fomis [3745—3745'"]. 24 PERTURBATIONS OF THE PLANETS. [Méc. Cél. and we shall have CoefB- ;3753] JV(»)=-^.5(e«+e'=).r4i''.^«>-2a.f^')-a».f^)]-$.aa'.[B('-') + 5«^»]^; depending ^C. L \ da / \ da- /J 2 ) on angles , 4f^' ^ ' \ da / ^ ' \ da J ' \dada JS form. m.., ^».= -.^,.,+„,^..„+,.(..+,,..(^')+,.,+,).,.(^>)+„,(^)|. We shall calculate these terms separately, commencing with the angle i T, which is [37525] produced in R [957], by the substitution of the terms ^e^, Je'^, occurring in the terms of [3750/], marked 2, 3, 6, 8, 13, 15. These quantities produce in R, the expression — .cos. i T, multiplied by the following terms, WTitten down in the order in which they appear, without reduction, and omitting 2 for brevity ; Now if we multiply the first of the equations [1003] by — 1, and the third of these equations by — | ; the sum of their products will give , /dA^i)\ . ,„ /ddA<ii\ fd.»o\ AW^(.)\ substituting this in [3752f], we find, that the coefficient of e'^ is the same as that of c', and the whole expression becomes [3.5.,, _^.(..+ ..)4.i,^»_„.(l^)-i,...(^-^);.eo..-r. To this we must add the third term of [37502^], depending on cos. {i -\- I) . T, which, [3752/] by changing i into i—l, as in [3750c), becomes — . J aa'. 2. jB^'-i'. cos. i T. The expression [375'2e] is the same for —i, as for +i; because A'-'^ = A'''^ [954"]. Moreover, the term [3752/], by the same change of i, using J?(-i-i) == ^c.+D [956'], [3752g-] becomes '^ . i a a'. 2 . J5''+" . cos. i T. Hence, if we use only positive values of i, we must double the fonction [3752e], and add to it the two expressions [3752/ g] ; the sum of these three ftmctions, being put equal to N'-^'' . cos. i T [3752], gives the same value of iV^*", as in [3753]. In the case of i = \, this sum must be increased by the [3752/il first term of [3750w] ; by which means iV^°^ is increased by the quantity given in [3754]. The case of i=0, which is separately considered in [3755'''], produces, in R, the following expression, which is deduced from [3752c,/], by putting i=^0; VI. i. §4.] TERMS OF THE SECOND ORDER IN c, e', y. 25 In these three last expressiotis i is supposed to he positive and greater than [3753'"] zero. Incase i = \, we must add to iV"" the term — ^. ^^ [3752/i]. [3754] It is more convenient, for numerical calculations, to have the differentials relative to only one of the two quantities «, a', in these formulas.* Proceeding m the same manner with the angle iT-^-zi — w' [3752'], we find, that terms of this form are produced in R [957], by the substitution of the parts of the terms of [3750/] depending on the angle T-f-ra — i^, and marked 7, 9, 12, 14; reducing them by means of the formulas [954c, 955a,/]. Hence this part of R becomes equal to ~.ee'.cos.\{i -\-\) .T -\--a — -m'], multiplied by the following expression, retaining [37524] the terms according to the order of the numbers, without any reduction ; aa'.f -— — ) — 2ia. ( — - ) — 2ia'. — — ) + 4 i^. ./î*'^. [3752q \dada / \ da / \da / Changing i into i — 1, in [3752À:, Z], we find, that this part of R maybe represented by . ee'. JV^''.cos. {i T+ra — ra') [3753'] ; observing, that this change in the value [3752m] of i, reduces the expression [3752Z] to the same form as the factor of —, in the value of iV"' [3753']. We must retain only the positive values of i in [3752', 3753'] ; for if we ciiange the sign of Î, the expression cos. (i T-j-w— ■zs'), becomes cos.( — iT-\-vs—-a') [3752)1] or COS. {i T — ■n-j-'sj'), which is of tlie same form as [3752"]. Hence it appears, that we may deduce JV'-' [3752"] fi-om iV*'> [3752'], by changuig the sign of i. Performing [3752o] this operation on [3753'], we get [3753"], using ^^-'-i> = ^('+i> [3752/']. Finally, the case of i = 0, is found by putting i = in [3752ot], or in the similar terms depending [37520"] on JV^2i [.3752o]; observing, that when i = 0, the expressions JV'^', JV'2> [3753', 3753"] become equal to each other ; and this part of R becomes f . ... [ 4^...+ .„ . (1^) +.».. (•-■) +,y. (1^') I . e„. („-.,. ,,.„ * (2371) In making the reduction of M<" from [3750'] to [3755], it will be convenient to use the abridged symbols a™ . (-^^j = ^ï ; a"" . (Ç^') = .//'<::' ; and as the [3755o] index n is the same for all the terms depending on M''^\ we may neglect it, and put simply ' [37556] VOL. III. 7 26 PERTURBATIONS OF THE PLANETS. [Méc. Cél. This is obtained by means of [1003], from which we get [3755] JtfU,__^.^(2,-_2).(2,-_i).^(-«+2.(2;-l).«.(!^) + «^(î!^')^; [3755'] Jlf(^)= |\^(4i^_7i + 2).^^-> + 2.(2z-l).«.('-fp) + «^(''';;;^^^ Reduced values. [3755"] ./V-= ^.^(2._2).(2.--l).^<--2«.('^)-«^(^)5; [3755'"] JV^^>== -'.^(2^• + 2).(2^• + l).^u4-n_2«.(^'^^_„..(^''f^^^. [3755i>] 5, The case of i ^ deserves particular attention. We shall resume the expression [923], and shall consider, in the first place, the and the same symbols may be used in the reduction of Jlf' -' , .A'"'" , JV'-' . Then the coefficient of — ^m', in the value of M'-^'' [3750'], will become, by the substitution of the first and second formulas [1003], [3755c] = 2. (i — 1) . {2 . (i — 1) + 1 } . ^„+ ^4 i — 2^ . ^, + A., ^{2i — 2).(2i — l).A,-^2.{2i — l).A, + A,; which is the same as the coefficient of — J m' in [3755]. In hive manner, the coefficient of — , in [3750"], becomes, by using the first and third of the formulas [1003], 8 (;_2).(4i— 3).^o + 2.(2i — 3). lA, + A,l + {2Ao + 4.A, + A,\ f^^^^'^J =\{i-2).{ii-3)+4i — 4l.^, + 2.{2i-\).A,+A,; which is easily reduced to the form of the factor of —, in M^~'> [3755']. Again, the factor of i m, in the value of iV'" [3753'] becomes, by the substitution of the values in the first and second formulas [1003] ; A.{i — \f.A,—2.{l—\).A, + 2.{i — \).\Jl^+A,\ + \ — 2A,—A^\ ^2.{i—\).\2.{i—\) + \\.A, — 2Ay — A^; which is the same as the coefficient of \m, in tiie value of jV"' [3755"]. From this we may easily obtain A*'-', by merely changing the sign of i, as in [3752o]. * (2372) The terms of R depending on i = 0, are given in [3752», 3752p] ; they are independent of n t, n' t, and produce in ^ d a secular equation [3773] ; and on this account, they are carefully computed, though it is finally found, in [4446, 4505], that [3755e] VI. i. §5.] TERMS OF THE SECOND ORDER IN e, e', 7. 27 term — 5^ ^ ^, of the expression of d5v, given by this [3755"] On the secular '■dv formula. We have, as in [1037], by noticing only the terms affected with pi" of the arc of a circle n t* r a S r a — I m'. (h C + h' D).nt. cos. {n t + e)- ^ 1 _ /i . sin. (« t-\-i) — l. COS. (n t + i); [375G] = \m'.{lC + l' D).nt . sin. {n t + s) [3756'] they are insensible. To reduce these terms of R to the form [3764], we may use the following symbols, given in [1022, 1033] ; A:=c.sm.«; Z = e.cos.is; A'= e'. sm. w' ; r = e'.cos. ra' ; [3756a] e2=A2+/2. e'2 = A'2+Z'2; [37566] y.sin.n^y— p; y .cos.n = q' — q ; 7^=(p'—p)-+{q'—lf- t^^'^^'^l Now substituting, in [3752t], the values of c^, c'^, y^ [3756e, c], they will produce, respectively, the first, second, and fourth lines of the expression of R or ô R [3764] ; observing, that, by using the sign S, as in [917'], these terms of R may be represented [3756rf] by &R. The term [3752j7] produces the thii-d line of the same value of 5R; for we have, by using [3756a], e e'. cos. (a — -n') = e e'. (sin. ^s . sin. a' + cos. -m . cos. z^) = A h' + / Z' ; [3756e] substituting this in [3752p], it produces this term of &R [3764], having the factor hh'-\-ll'. This value of oR is to be used in the formula [923], to compute the part of 5v, which |-.,„„ is independent of the angles n t, n't; and of the second degree in A, A', /, V , &.C. * (2373) The object of the present computation is merely to ascertain the part of à v, mentioned in [.3756/], by means of the expression of dàv [923]. This may be reduced to the form [-3757^], by observing, that r i?' = r . ("^ W « . (^) [928', 962], [3757o] and that we have, identically, 2 r . 5 R' -{- R' . 6 r ^2 & . {r R!) — R' . 5 r. From the first [37574] of these equations, we see that R' is of the same order as R, or of the order m ; and by rejecting tenns of the order m'^, as in [-3768'], we may neglect the term — R'.Sr, and then this expression [37576], by the substitution of r R' [3757a], becomes 'd'>R\ [3757e] 2r.5R'-\-R'.ôr=2S.{rR') = 2a.( ^) . Substituting this in [923], also the value of r^dv [.3759], we get 'dûR d.{2r.d5r+drJr) + dt^.\^3fôàR+^a.(^)] ^^^^^^^ 1 r a'J.nrf<.v/(l— e9) 28 PERTURBATIONS OF THE PLANETS. [Méc. Cél. These give, by noticing only the terms depending on the squares and [3757] p^fjdy^cis qJ- ji^ i^ ]i'^ i'^ independent of the sines and cosines of nt + e, and its multiples * [3758] d.(2r.d5r + dr.&r) = —''^^^^^^.\(h'+n.C + (hh'+in.D\. In this we must substitute SR [3764], and those terms of dr, S r, which produce quantities of the form and order mentioned in [3756/]. Now these quantities will be obtained by selecting, from the general value [1037], the three terms contained in the r Ô V [3757e] second member of [3756], for -; and the terms in the second member of [3756'], for — . It is unnecessary to use any other tenns of a higher order in h, 1, &c. ; for if we retain, in -, any teim of the order h^, hi, 1% connected with sin.2.(?i^4-6) or cos.2. (m^ + s), a it must also be connected, in [3757rf], with terms of —, or of its differential, of the same [3757/] forms and order, producing terms of the fourth order in h, 1, and independent of the angles n t, n' t, which are neglected in this article. The same remarks will apply to other terms of - , depending on higher multiples of the angle nt-{-s. Having adopted [3757g-] this form of -, it will be unnecessary to retain any terms of — [1023, 1037], except a o Sr those in the second member of [3756] ; for, though other terms in — [1023], of the [3757/i] forms P, P'. sin. ()i / -(- s), P". cos. {nt -\- s), might produce, in 2r .d è r-\- dr .&r, quantities independent of the sine or cosine of the angle nt -\-e, or its multiples ; yet if we notice only terms of the order m', they will vanish in its differential, which occurs in [3757d, 3760] ; and this does not happen with the arcs of a circle retained in [3756'], as is shown in [3760]. * (2374) In finding the terms of 2 r . tZ i5 r -|- <7r . 5 r, of the order m', it is only [3758o] necessary to notice quantities of the form Q ■ « t.dt, containing the arc of a circle n i, Q being constant ; for if the function contain any constant term, or elements of tlie planet's orbit, it will either vanish from its differential ' [3760] or become of the order 7«'^, &;c. ; and terms depending on the sine and cosine of nt-\-s, ai-e neglected [3757]. Substituting r [3756], and its differential, in the first member of the following expression, we get [37585] 2 r . d 5 r -{- d r . 5 r = \2 a — 2 ah . sin. [nt -\- e) — 2al. cos. [ni -\-e)\ .d S r 4- I — ah . cos. {nt-\-s) -{- al . sin. [nt -\-e)] . n dt . or ; in which we must substitute the values of S r , d Sr. Now if, for a moment, we [3758c] put im'.a.{lC-Ji-rD) = L, im'. a .{h C-{-h'D) = H, we shall get, fiom [3756'] VI. i. ^5.] TERMS OF THE SECOND ORDER IN e, e', y. 29 We then have r" dv = a" n d t . \/ï^^ [1057]; hence we shall obtain [3759] d.{Or.dSr+dr.Sr) m'.ndt <(^j^,_^pyc+(hh' + ll').D\. [3760] r~ dv 4 We have, in [1071], (0, 1) = — 1 m' nC; [^^^rn'riD; [3761] therefore * d.{2r.dSr-{-dr.Sr) _,_^^,^Q^^y ^,^,j^p^ _ ^^ _ (hh'+ll')\. [3762] r^ dv We shall now consider the term — ^-^, , of the same formula [923] r-' dv [3758d] and from its difterentlal, the following expressions, retaining only the tenns which contain the ai-c of a circle, as in [3755'] ; S r =L .n t . sin. {n t -{- s) — H .nt . cos. (» t-\-s); dSr=L.7i^. tdt .COS. {71 t-j-s) -{- H . nP. t d t .sm. (jii + e). Substituting these values of &r, dSr, in the first members of the equations [3758e], reducing by [17 — 20] Int., retaining only the terms containing the arc of a circle, independent of the sine or cosine of nt-^ e, we get '2a.dSr = 0; — 2 a h. s'm. {nt -{- e) . dSr = — ahH .rfitdt ; — 2al.cos.{nt+s).d6r = — alL.n^tdt; [3758e] — ah . cos. (nt-\-s).ndt.5r^iahH.7i^tdt; -[- al . sin. [nt -{- s) .nd t .5r = ^ al L . n^t dt . The sum of the tenns in the first members of [3758e] is equal to the second member of [37586] ; consequently the first member of [3758J] is equal to the sum of the second members of [375Se] ; hence we get 2r.dSr + dr.5r= — iahH.n^tdt — ialL.n^tdt. [3758/] The differential of this expression becomes, by resubstituting [3758c], d.{2r.dSr + dr.Sr} = — in^a.dt^.{hH-{-lL) Dividing this by the expression of r'^dv [.3759], neglecting the divisor \/{l — e^), which only produces terms of the fourth degree in h, h', e, &c., it becomes as in [3760]. * (2-375) Substituting the values [3761] in [3760], we get [3762]. [3762a] VOL. III. 8 [3758g] 30 PERTURBATIONS OF THE PLANETS. [Méc. Cèl. or [3757 d'\. If we notice only the secular quantities depending on the [3763] squares and products of the excentricities and inclinations of the orbits, we shall have, by the analysis of the preceding article [37ô6d—f], Pan of V \ ^ \ 6R, ..™- +f(Mwj4.... + .„.('-)+2...(M^)+„..(J^), ponding to t=0. [3765] [3766] m p, p'l q, ([, denoting the same quantities as in [1032]. Hence we easily obtain, from Book II, ^ 55, 59,* aw.<5i2 = — 1.(0, l).{/r + r + /t'2 + Z'2| + [^].j/t/t'+n'i which gives f an.àùR = dh.\ — {Q,\).h+\^.h'\—dl. \(0,\).l—[^.l'\ -(0,1). dp. (p'-p)-(0,l)-dq.(q'-q). * (2376) If we multiply [3764] by an, we shall get the value of aii.ôR, which may be easily reduced to the form [3765] by the following considerations. The coefficient [3765a] of h^-^P is equal to '— [1073], and the coefficient of h'^-^-l'^ is of the same value ; as evidently appears by the substitution of the expression [3752rf]. The coefficient of {p'—p)^+ir/—fjf, in this product, is ^ m' ?i . «^ a'. B"'== J . (0, 1) [1130]. Lastly, the coefficient of h h'-\-Jl' in this product, is evidently equal to | m n, multiplied [37656] by the expression of D [1013], and this is shown iii [1071] to be equal lo [ôTÎ], as in [3765]. t (2377) In taking the differential of [3765], relatively to the characteristic d [37056], we must consider h, I, p, q as the variable quantities, and h', I', p, g' as constant ; and then we shall get an .do R = — {0,1) . {h d h -j-ld I) + [^] . {h'd h -^r d I) [3766a] ^ (0,1). \-{p'-p).dp-(q'-q).dri\; being the same as in [3766], with a slight alteration in the arrangement of the terms. VI. i. <^5.] TERMS OF THE SECOND ORDER IN e, e', y. 31 The second member of this equation becomes nothing, in virtue of the equations [1089, 1132] ; therefore we have* a ?t . d (5 K = ; [3767] hence we deduce, by observing that n" a^ = \ [3709'] ,t [37671 3dt.fdt.d&R_ 3m'.gdt m''g being the arbitrary constant quantity added to the integral fdôR [1012']. It now remams to consider the function „ ,— 1 , which r- d V occurs in the expression of d&v [923]. If we neglect the square of [3768'] 2 S . (r R') d t^ the disturbing force, this function will be reduced to — „ , — , r^ dv * (2378) Taking into consideration only two bodies, m, m', we get, as in [1072], ^=(0,l).?-[irr]./'; ^ = -(0,l).A+[ôZ].A'. [3767a] Multiplying the first of these equations by — R/, the second by dh, and adding the products, we find, that the sum of the terms of the first member vanishes ; consequently [37676] the sum of the terms in the second member, being the same as the terms depending on dh, dl, in [3766], must also vanish. Again, we have, in [11.31], ^=(0,1). (<?'-<?); ^'__(o,l).(/__p); [3767.] multiplying these, respectively, by — dq, dp, and taking the sum of the products; the first member becomes identically nothing, and the second member is the same as the terms [3767rf] depending on dp, dq [3766], which are therefore equal to nothing, as in [3767]. We may incidentally remark, that the quantities (0, 1), [Ôj]], &c. [3761] ; also dh, dl, &ic. [1102, 1102ff], are of the order m' ; consequently the second member of [3766] is of the order m'^ ; but its integration, in [3768], introduces divisors of the order g> gi> ,?î' ^c. [1102, 1102«], which are of the order m' [1097t] ; by this means, the [3767e] integral fdt.àèR [3768], is reduced to terms of the order m, like the other terms computed in this article. t (2.379) The integral of [-3767], using the constant g [1012'], is an.fàSR = an.m'g; lultiplying this we get [3768]. multiplying this by , and then dividing by r^ d v z^ a^ n d t . \/ {I — è') [3159}, [3768a] 32 PERTURBATIONS OF THE PLANETS. [Méc. Cél. ^ „ /dSR\ ,, [3769] or by [928, 962], to V^ .* This quantity produces, m.ndt.a-'.i— — I [3769'] in the first place, the term .^-—^ " ,t which is to be added ?iin' p'lJ t . , .3 m'. as: ndt [3769"] to , '; [3768], or to the equivalent expression /-;— ^ , deduced from ?r «^ = 1 [3767']; and the sum vanishes by the substitution of g = — -^a.l— — j [1017]. Resuming the expression of ôR [3764], we shall observe, that the function [3771] I' .aa'.B^'K{(p'-py+ (ç'-ç)^} + &c.î * (2380) We have, in [3757&, c], by neglecting tlie square of the disturbing [37690] force, 2 r 5 iî' + R ' 5 r = 2 5 . (r iî ' ) = 2 a . (jj^) • Multiplying this by d t"- and by l = n^tt^ [376T], and then dividing by r^ d v:^ a~ nd t .</{\—e^) [3759], we 2 a^. I —— ).ndt [37695] get ^ °" \ [3769], for the corresponding terra of d>]v. t (2381) The value of R [957], or rather [1011], gives, for the case of { = 0, [3770a] and for terms mdependent of nt, n't, S R = I i7i' . A'-''\ Substituting this in the term /djKOA of dSv [37696], it becomes as in [3769']. Now if we substitute g-=— Ja.(-^j [37706] . m'.ndt.a^ /rf.^(oi\ , ,. . [1017], in the term of dêv, [-3769"], it becomes — "; ,^_^o, ■\d^)' '^ destroyed by the equal and opposite term obtained in [3769'] ; so that this sum becomes [3770c] nothing, as in [3770]. The calculation [3767—3770] is in some respects a repetition of that in [1016", &ic.] ; and we see that the value of g, assumed in [1017], suffices even when we notice the parts of R contained in [3764]. % (2382) Taking into consideration only two bodies m, m, the differential of [3771] [37710] '^'" ''^ im'.aa.B''\\{p'—p).{dp'—dp) + {f/—q).{d(/—dq)l; observing that B^^^ [956] is a function of the constant quantities a, a' [1044"]. Now the first and second of the equations [1132] become as in [3767c], and the third and fourth of those [37716] equations give ^ =— (1,0). (q'—q) ; -^ =(1,0) .(/ — p). Hence the differential expression [3771a] becomes [3771c] ^ ^/ . „ „'.5(n . (p'—p) . (ç'_ î) . {— (1, 0) — (0, 1) + (1, 0) + (0, 1)( . ^ < ; VI. i. >§5.] TERMS OF THE SECOND ORDER IN e, e', y. SS is equal to a constant quantity independent of the time t, because its differential becomes nothing, in virtue of the equations [1132]; and if we consider only two planets, m, m', as we shall hereafter do, (p'—pf+{q'—qT [^TTr] will l)e a quantity independent of the time, in consequence of the same equations. Therefore the preceding function [3771] can produce in ^ n d t . a .{ ) '\'i^/ [3769], only a quantity independent of tdt, kc, which [3771] . v/l— e^ may therefore be neglected, since it may be supposed to be included in the value of ndt. Hence we shall have, by eliminating the partial differentials [3771"] of A^°^ and A^'\ relatively to a', by means of their values [1003],* [3772] in which the tenns between the braces mutually destroy each other, and render this quantity equal to nothing ; therefore the expression [3771] must be constant, and may be represented by G, and it will introduce into 5 R [3764] the constant quantity G. Now as this quantity, considered as a function of a, produces in [3771"], only a term wliich may be inckided in the expression of ndt, we may neglect it, and reject the tenn '■ ' depending on jB'* in [3764]. * (2-383) It appears from [3752(/], that the coefficients of }m'.{P-\-P), ^m'.{h'^+l'^), are equal in the value o{ S R [3764] ; these terms may therefore be connected together, as in [377*26]. Now if we put the two expressions of JV-^' [3753", 3755'"] equal to each other, then divide by | m', we shall have, for the case of i = 0, - ^,,, , ^ /dJim\ /(/.4(i)\ , , /dd.m)\ ^ ^,,, ^ /dA(.^i\ „ fddJiw\ 4^'-+2«.(-^) + 2a'.(— ) + «a'.(^-^,)=2^«>-2a.(-^)--«^(^; [3772«] substituting this in the coefficient of lm'.{hh'-\- II') [3764], it becomes as in [3772J] ; hence we get Taking the partial differential of this expression, relatively to a, and multiplying it by 2ndt .a^, we get [3772]. TOL. III. 9 [37726] 34 PERTURBATIONS OF THE PLANETS. [Méc. Ccl. Now if we collect together all these terms, we shall obtain,* irof" ,- vi'.ndt C /dAin\ _ fdd.m\ fdKm\~} epend- Expres- sion of [3773] [377:%] [3774] In this expression we may neglect the terms independent of the time t [3773e]. Hence it is easy to deduce the expression of (/ h v', by changing what relates to m into the corresponding terms of m' and the contrary ; and observing, [3775] that, though the value of J^'' [997], relative to the action of m' upon m, is different from its value relative to the action of m upon m', yet we may [3775] use, in the preceding expression, either of these values at pleasure.! But * (2384) The value of dSv [3773] is found, by adding together the several parts of the expression [3757(/], computed in this article ; and as tlie terms [3768' — 3771"] destroy [3773a] gj^jjjj Qfjjgj.^ there will remain only the terms [3762, 3772], to be connected together. The expi-ession [3762], by the substitution of the values of (0, 1), [""Til [1073] becomes and as the factors without the braces are the same as in [3772], the sum of the two expressions [3772, 37736] is easily found to be as in [3773] ; which is a function of the [3773c] elements of the orbits similar to that mentioned in [1345'"']. If all the terms of this function were constant, they might be included in the expression of the mean motion ndt. But e^ = h^ + P, e^ = h'^-^r^, he. [1108, 1109], are composed of con«to«^ quantities, and of others depending on the secular periodical variations of c, e, Stc. ; and it is evident, that the constant quantities produce in d 5 v terms of the same form as the mean motion ; they may therefore be neglected, as in [3771'", 3774]. [377.3rf] [3773e] t (2385) Substituting [964] in [963'], and then putting s:=i, we get [3775a] (a2_2 „ «'. cos. è + a'^)-i=a'-K\i bf + i'|>. cos. ê + if . cos. 2é -f- &c.^ Now the first member of this equation is symmetrical in a, a' ; tlierefore its second member must also be symmetrical ; so that we shall have, generally, a'~'.è'f equal to a synmretrical flinction of a, a'; and if we refer to the formulas [996, 997], we shall see, that for all ^ ' values of i, except i=l, the function ^<'' is likewise symmetrical. In the case of ? = 1, VI. i. §5.] TERMS OF THE SECOND ORDER IN e, e', y. 35 we may obtain dov' more easily by the following considerations. If we [3775"] add the value of d^iv, multiplied by w\/â, to the value of d!iv', multiplied by in'\/'a , Ave shall have, by substituting the partial differentials of A^^\ A'^\ relative to «, instead of those relative to a',* m\/a.(ISv-{-7nya'.cJov:^ ^ .lh^-{-PJf-h --{-I ^ . j^a . \^-j^ j + i a . {^-j^ J ^ [3776] corrresponding to the action of?»' upon m, we have ,/2''" = -- ;.& [997]; and in J " " [3775c] the action of m upon m', it becomes ./2'"=— — ^—-^i 5 hut we may neglect the parts —, — :^, because they produce nothing in dSv, dôv'. To prove this, we shall [3775rf] a observe, that by noticmg only the part .4''*= — , we shall get /rf./3Ui\ 1 /■ddAO)\ /ddJim\ 1-77-; = ^' [-1^)=^' l^I^j=^' [3775.] substituting these in [3773], the terms mutually destroy each other ; so that we may neglect this part of ./2'^*, and for similar reasons we may neglect the part ^'''=--, in [3775/-] computing the action of m upon m', and then the two expressions [3775c] become symmetrical in a, a', as in [3775']. * (2.386) Multiplying [3773] by tn^a, and dividing the second member by na^a=^l [3709'], we get, by reducing the factors without the braces to a symmetrical form, +.,„,....,.(*-+n. i^..(r)+^«'-c-^')+"'-r-^) \ Changing the elements m, a, v, h, I, he. into m', a, v', h', /', &c. and the contrary ; which does not, in the present case, alter the values of A^°^ or ^'" [3775], we obtain [3776t] the expression of m' \/a' .dSv'. The factors between the braces corresponding to the first, second, and third Imes of [3776a], become, respectively, as in the first members of [3776rf,/, h], and by means of the expressions [1003], they may be reduced to the 36 PERTURBATIONS OF THE PLANETS. [Méc. Cél. [3776] If we consider only two planets, m and m',* the differential of the second f„__„ , forms [3776e, ^, i]. In making these reductions, we may use the abridged symbols Ad, Ai, A.2, A3 [3755i], observing, that the index of A'^"'' or .4<" remains unchanged; ^^''^^ a'.(— ) + J«-.(^ + «'3.(-^):=|-.4o-^J + M2^„+4^,+^,i •\.\—QAu—\SA, — 9A^—A.:,\ [3776e] = — 5 .^j — ^A.2 — A^; l^nn ,,.(^)+,,'.{'^) + ,P. (^)=2.i-A-^.!+4.i2A+4^,+^,! + l — 6Ao—l8Ai—9A.,—A3\ [377%] =—4A^ — 5Aç,— As; /(/./3(iA /rf9.4<')\ /rf3./4''A [3776/.] -.^<"+«'.(— )-^V^.(^-2a'=.(^ = -.4„+|-.4„-^,|---.|2^,+4^,+^,| — 2 . 1 — 6.^0—18^1—9^2—^31 [3776i] = — 5.2o + 5A + -y-. ^2+9^^3 • Now substituting the values corresponding to [31~6c, g, i] in the value of m'^a. dSv, deduced from [3776«], by the change of the elements [37766], we get „V...«= i„„..„.(..+r=).|-.„.('/^)-,v...(^-^)-.=.('-^ +i„..,.(".-+"').S-5^-+-.C^')+V.'.(^)+..'.('5S')S- Adding together the two expressions [3776a, t], we obtain [3776], observing, that in this sum, the coefficient of h^ -\- P is found to be the same as that of A'^-j-/'^. We [3776i] may remark, that the factor ^ — , in the second line of [3776], is erroneously printed '■ in the original work. If we multiply the second member of [3776] by ?ia*^l [3709'], and substitute the expressions (0,1), [ôT\ [1073], we shall get, bstead of [3776], the following equation ; [3776m] m^a.d&v+mya'.dôv'= im\/a . dt . (0,1) .{h^ -\-P + h'^ + 1'^) — Zm\/a.dt. [KI] • {h li -\-ll'). * (2387) The differential of the equation [3776m], may be put under the following form; d.\m^a.dàv\-m!s/a'Mv']= Zm^a.dt.\{Q,\).{hdh.-\-ldl) — [^.{h'dh + rdl)\ ^^''^''"^ -\-Zms/a.dt.\{0,\).{Kdh:+l'dl') — [-^].{hdK^ldV)\. VI. i. ^5.] TERMS OF THE SECOND ORDER IN e, e', y. 37 member of this equation will be nothing, in virtue of the equations [1089] ; therefore %oe have, by noticing only secular periodical quantities, "for "■ = m s/a.dàv-]r m' \/7i ■ d <> v' ; [3777] which immediately gives d tS v\ when d o v is knoivn. The value of div is relative to the angle formed by the iivo radii veclores r and r + dr. To obtain its value relative to a fixed plane, we shall [3778] observe, that if we put dv^ for the projection of dv upon this plane, and neglect the fourth power of the inclination of the orbit, we shall find, as in [925],* dv=dv.\\^-ls''—\/^À. [3770] We have, as in [1051], s = q.sm.{nt + s) — J9.C0S. (n/ + j) + &C. ; [3780] which gives f rf5 = ^j2_^yji.cos.(7t^ + + (np + ^Yf/i.sin.(«^-f3) + Sic. ; [3781] Substituting, in the first line of the second member of this expression, the values d h, d I [3767a], it vanishes, because the terms mutually destroy each other. The second line of the second member becomes, by the substitution of the formulas [1093, 1094], equal to ^m^n'.dt.\{\,Q).{li'dh'-{-rdl') — ['ûV^.{hdh' + ldV)\, which vanishes also by [37776] the substitution of </A'= {(1,0) . /'— [To] J} . J<, dV=.\—{\,Q).h'-^[']^].h\.dt, [3777c] deduced from the third and fourth of the equations [1089]. This is also evident from the consideration, that the expression [3777i] may be derived from the first line of [3777a], by changing the elements relative to m into those corresponding to m', and the contrary ; [3777rf] and as that line is found to vanish by the substitution of the values of dh, dl [3767 «], the other will in like manner vanish by the substitution of the values of d A', d I [3777c]. Now the second member of [3777a] being equal to nothing, we have, by integration, m \/a.d&v-\- m' s/à .dhv'=Gdt \ G being a constant quantity independent of the secular periodical equations. This quantity Gdt may be supposed, as in [3771'"], [3777c] to be connected with ndt, ri'd t ; so that by noticing only the secular periodical equations, we may put tlie first member of the preceding equation equal to nothing, as in [3777]. • (2388) The equation [925] maybe put under the form di\=dv\/\l-\-s'^ — (T+TIilTrâv Developing this, and neglecting terms of the fourth degree in s or ds, we get [3779]. t (2389) The differential of .9 [.3780], considering p, q, t as variable, becomes as in [3781]. The squares of these expressions, which enter into the function [3779], are VOL. III. 10 38 PERTURBATIOiNS OF THE PLANETS. [Méc. Cél. [37811 hence we shall find, by neglecting the periodical quantities depending on n t , and observing that d v ■-= n d t, very nearly, [3782] dv^^dv-\-}2 .(q dp — p d q) ; therefore to obtain the value of d5v^, we must add the quantity [3783] \. (qdp — p dq) to the preceding value of dàv [3773]. If we only consider two planets m, m', we shall ha\r, by means of [1132, 1130],* [3784] {q('p—pdq) ■ m\/â+{f/dp'-p'dq').my^= — imm'.dt.aa'.B''\\{p'—p)^-lr{q'—qfl ; [3779a] of the order of the terms computed in this article [3702'], and by neglecting terms of a d s~ higher order, we may omit, in — [3779], the terms of dv [3748] depending on e, [37796] and put dvz^ndt, by which means we shall get d v^t=:z dv.\l-\-i s^ — i-ir — (» in which we must substitute s, ds [3780, 3781]. In making these substitutions, and noticing the terms independent of the sine and cosine of nt or its multiples, as is done in this article, where the secular periodical terms only are retained, we may, as in [3651a], put [3779c] sin.2 (n < + e) = è , cos.^ (n ^ + e) = i , sin. (« /■ + s) . cos. (n < -f s) =0 ; then the squares of [3780, 3781] will give, by neglecting dq^, dp^, which are of the order of the square of the disturbing forces, [3779rf] i,a_i.(ç2+_p2); Substituting these in [3779], we get [3782]. * (2390) Substituting the values of dp, dq, dp, dq' [.3767c, 3771 i] in the first members of [3784«, i>], and reducing the second expression by means of [109.3, 1094], we get the second members [3784o, c] ; [3784a] my'a.{qdp — pdq)^ms/a.{Q, \) .dt . \q ■ [q' — q) -^ P -(p' —p)\ ; [37846] w'v/«'- {q'fip'—p'dq') = m'y/a. {1,0) . d t .\— q'. {q'— q) —p'. {p' — l})\ [3784c] ==^m^a.{(),l).dt.\— q' . (7'— q) — p' . {p' — p)\- The sum of the two equations [3784f(, c] gives the value of [3784] under tlie form [3784c/] — m^a. [0,1). dt.\{q' — qf-j-{p'—p)^]; substituting (0,1) [11.30], and dividing 3 by na-==l [3709'], it becomes as in the second member of [3784]. VI. i. §6.] TERMS OF THE SECOND ORDER IN e, e', y. 39 and the second member of this equation is equal to dt, multiplied by a constant quantity ; * therefore by noticing only the secular periodical quantities, we shall have = m y'â .d&v^-^m' sjd . d^v^ ; & V and (5 v^ being relative to the fixed plane. 6. We shall now consider the inequalities in the motion in latitude, dependinrr on the products of the excentricities and inclinations of the orbits. For this purpose we shall resume the third of the equations [915] ; We shall take for the fixed plane the primitive orbit of m, in consequence of lohich we may put 2 = in the expression of (-i— )• We shall have, by [3736—3741], observing that z' = r's',\ dR rf7 [3784'] Tho Bame formula for reduced to the fixed plane. [3785] r"" 1^2— 2r/.cos.(î)' — i;)+r'2}*' [3785'] Differ- ential equation for the latitude. [3786] [3786'] [3787] [3788] * (2391) The differential of the second member of [3784], being divided by — 'imdt, becomes as in |'3771rt], and is therefore equal to nothing, as is shown in [3771c] ; hence [3785a] we find, as in [377 ItZ], that the first member of [3784] is equal to dt, multiphed by a constant quantity G, wliich may be neglected as in [377 le] ; so that by noticing only the secular periodical equations, we shall have {qdp—j)dq).m\/a-\-{q'dp' — p'dq').nJ\/a'^zO. [37854] Now we have found, in [3782], that by reducing « to a fixed plane, the value of dv or dàv must be augmented by ^.{qdp — pdq); and in like manner, the quantity d^v' must be increased by i.{q'dp' — pdq). Multiplying these by m\/a, m'\/a', respectively, [3785c] and adding the products, we get the increment of the function [3777], or the quantity to be added to it, to obtain the value of m \/a . dSv^-^-m' \/c! . d 5 v/. Now this increment vanishes by means of the equation [37856] ; consequently the function [3777], varied in this manner, becomes as in [3785]. t (2392) The latitude of the body ot', neglecting terms of the third order, being ^3737^-] represented by s , and the radius vector by r', we shall have, by the principles of orthographic projection, 2' = //, as in [3787]. Now / [37366] being independent of z, the partial differential of H [3736], relative to z, becomes (àR\ m' 2' m'.(z'- l(x'- ^f+iy'-yf+i^'-'-)]^ ' [3787a'] 40 PERTURBATIONS OF THE PLANETS. [Méc. Cèl. the differential equation in z, will by this means become* [3789] = ^ + n=2.{l + 3e.cos. (n^+£ — ^)| + m'. n^ a^. s \ r^ — 2 r r. cos. («' — ^)-\-'i'"~\~ We shall now putf ■clR [3790] ('l:^)^M.sin.{?:.(n7— n« + /— s)+2ni+^i+iV.sin.^'i'.(n7-nï+/— i)+L|, for the jiart of [3788] ffJR [3791] I77 r- — 2r r. cos. {v' — v)-{- r' -} ] '~ depending on the angles i.(n't — 7it-\-s' — s)-\-2nt and i.(7i't — nt-\-;' — s) ;* [-3792] and shall suppose, that by noticing only the inequalities of z, depending on and if we neglect quantities of the order s'^, we may reject tenns of the order z'~ or 7* in the denominator; then, as in [3742^], we shall have [3787i] {x'— xf 4- (i/' — i/f + {z' — zf = /-a _ 2 r r'. cos. {v — v) + r' 2 ; substituting this and z := 0, z'==r's' [3786', 3787] in [3787a'], we get [3788]. We may here remark, that the method used in this article, in finding the motion in [3787c] latitude, depending on terms of tlie order of the product of the excentricity hy the inchnatlon of the orbit, is difTerent from tliat proposed in [948], and used in [1025, &c.] in finding the terms independent of the excentricity. This last method may, however, be applied without any difficulty to terms depending on the excentricity, and we shall obtain the same [3787rf] result as in [3795 — 3797] ; as has been shown by Mr. Plana, in Vol. XII, page 449, &c. of Zach's Correspondance Astronomique, he. * (2393) We have, by means of [37026, c, 3700], [.3789o] (A ?--3= ,a a'^. { 1 + 3 e . cos. (n t + s—zi) -f- &c. | = 7i^. ^1+36. cos. (n t + 1— «) + &;c.| . Substituting this in [3786], also the expression [3788], multiphed by n'a^=l [3709'], we get [3789]. t (2394) The reasons for assuming these forms are evident from [3704a — 6], observing [3790«] that the object proposed at the commencement of this book, is to notice merely the terms depending on the squares and products of the excenlricities and inclinations. VI. i. §6] TERMS OF THE SECOND ORDER IN e, e', y. 41 the first power of the inclination of the orbits, the part of z, relative to the angle i . {n't — n i + s' — e) + nt, will be * z=^yaF. sin. \ i . («' t — n t + s' — s) + n t + s — n\. [3793] We then have, by retaining only the terms depending on the products of the excentricities and inclinations,! 0=^--— -\-nrz + %rf.€y.aFA ,.'.,, , , ^ , , [ + n''a\M.ûn.\i.{n't — nt-\-i—;)+27it + K\ + n'a\N.ûa.\i.(ri!t—nt + B'—c) + L\ ; [3794] * (2395) Putting, for brevity, we shall have, for the terms of s [1034] depending on iî''~", the expression F. \ (?' — q) . sin. Tg — (y — ^) . COS. Tal ; [3792i] substituting in this the values p' — p ^= y . sin. n, g' — q = y • cos. 11 [1033], it becomes [37926'] Fy . {sin. T3 . COS. n — cos. T3 . sin. n j = Fy . sin. ( T3 — n) = F7. sin. {i . {n' t—n t -J[- ^— s) -\- n t -{-E — n\. [3792e] Multiplying this by r, we get the corresponding part of z=^rs [3787,3796], to be [3792(/] substituted in the term 3 n^ e z . cos. {at -{- s — zs) [3789]. Now this term is of the [3792«] second order, or of the same order as the terms now under consideration [3702'] ; and by neglecting tliose of a higher order, we may substitute a for r, in the expression of z [3792(7], and we shall have z=a«; hence the term of s, computed in [3792c], produces in z [3792/"] the quantity [3T93]. Substituting this in [3792e], and reducing by means of [18] Int., we get the tenus depending on F in [3794]. In computing the value of the term [3792e], and neglecting quantities of the order m'^ or e^, it is not necessary to notice any other terms of s [1034], except those depending on B'-^'^^ or F, which we have used above. [3792g-] For the terms depending on the arc of a circle nt, in the second and third lines of [1034], vanish, as in [1051], in consequence of the secular variations of p, q. Again, having taken the primitive orbit of m for the fixed plane, we have z = or s = [3786'], at the commencement of the motion, corresponding to p = 0, q^O [1034, 1032] ; so that these terms may be neglected in computing [3792e]. Lastly, the terms of s depending on sin. {n't -]- e), cos. («' C -(- s'), in the fourth line of s [1034], may be considered as included in the term of 7*3 or of F [3792a], depending on i = I ; consequently the [3792i] function [3792é] is rightly expressed by the terms depending on F in [3794] ; the quantity F being of the order m' [3792a], as well as M, JV [3790, 3791]. [3792*] t (2396) The equation [3794] is easily deduced from [3789] ; for the two first terms are identically the same in each ; the third term depending on e, reduced as in [3792/, Sic], VOL. III. 11 [3792A] [3795] 4*2 PERTURBATIONS OF THE PLANETS. [Méc. Cél. hence we get, by integration,* iin^.ey.aF.sm.\i.(n't — nt + s'—s)-\-2nt-ir2î—7z—n\ ^ ._ ( +n''a\M.sm.\i.(n't—nt + i—s)-\-2nt-^K\ \ \ in'—(i— l).n\.\ in'—(i—3) . n \ i%n?.ey.aF.sm. [i.{n't — nt-\-^' — + '^ — n| ) \ +n?a^.N .ûn.\i.{n't—nt-]-s'—i)-\-L\ \ \in' — (i + \).n\.\in'—{i—\).7i\ We have the latitude s, by observing thatf s ^=~ =:--{-- .e . COS. (nt -\-i — ■a) : r a a ^ ^ ' therefore s may be obtained by dividing the preceding expression of z by a, and adding to it the quantity % ley.F.?.m.{i.{n't—nt-\'S—i) + 2nt+2t—^ — T\] + \ey.F.%m. \i.(n't — nt-\-s — £) + « — n|. [3796] [3797] [3794o] produces the terms depending on F [3794] ; the two remaining terms, comprised in the second line of the second member of [3789], are represented by the fonction [3791], or by the equivalent expression [3790] multiplied by 7t^a''=l, as in the two last lines of [3794]. * (2397) The equation [3794] is of the same forai as [865a], putting y = z, « = w; [3795o] then any term of [3794] depending on F, M, or JV, being represented by a.K.s\n. (m^t-\-s), [379561 the corresponding term of z will be represented by , ; , '"^ ', 'as in [871"] ; the ' X u i. -^ (m,-(-n).(m, — n) letters m,, e,, being accented to distinguish them from the similar letters of the present [3795c] article. Now putting m^:=zi.{^nl — n)-\-2n in the first and third of ^Aese ^erm^ of [3794], and m.i = i.{n' — n) in the second and fourth, we get, successively, the terms of z [3795] ; [3795d] all of which are of the order ?«' [3792Ar]. f (2398) We get, in like manner as in [3787], r5 = c; dividing this by r, or its [3796a] equivalent expression a.\\ — e. cos . {nt -\- e — ra)} [3701], we get the two values of s [3796], neglecting, in the last of them, the terms of the third order in e and z. J (2399) Substituting, in — .e.cos. (?i<-|-s — ra) [3796], the term of z of the first order y, assumed in [3793], and reducing the product by means of [18] Int., we obtain the corresponding values [3797]. Adding these to the term of - [3796], deduced from [3795], we get the terms of s. of the proposed forms and order. These terms are neglected VI. i. §6.] TERMS OF THE SECOND ORDER IN e, e', y. 45 Nothing more is required but to ascertain the values of M and iV; which may be easily found by the analysis in § 4. We shall, however, dispense with this calculation, because the inequalities of this order in latitude are insensible except in the orbits of Jupiter and Saturn, whose mean motions are nearly commensurable, and we shall give, in [3884 — 3888], a very simple method for the determination of these inequalities. If we refer the motion of m to a fixed plane, which is but very slightly inclined to that of its primitive orbit, putting tp for the inclination of the orbit to this plane, and a for the longitude of its ascending node ; we shall have the reduction of the motion in the orbit to this plane, by the method explained in Book II, ^22 [675, &c.],* — J- . tang."(p . sin. (2 v^ — 2 é) — tang. <?) . J 5 . cos. {v^ — ^) ; » being the motion v referred to the fixed plane. Hence the motion in latitude produces in the motion in longitude, upon the ecliptic, inequalities depending on the squares and higher powers of the excentricities and [3797'] [3798] [3799] [3800] [3600'] by the autlior in [3797'] on account of their smallness. The most important terms of the perturbation in latitude, of the second order, computed in [3885, 3886], are reduced to numbers [37976] in [4458, 4513], for Jupiter and Saturn, in whose orbits these terms have a sensible value. * (2400) In the annexed figure 73, AB \s tlie primitive orbit of the planet rn, A G the fixed plane, D the place of the planet, B D=^&s the perturbation in latitude now under consideration, which is perpendicular to A B ; lastly, the arcs B G, D EF are perpendicular to AG, and BE perpendicular to DF. Then by using the notation [669"], we have AB:=^v — 13, AG^=v^ — ê, BAG^ip; and in [676'], by neglecting tf^, ^B=^G' + tang.2iç,.sin.(2i;,— 2^) ; but on account of the smallness of cp, we may put tang.3 J 9 = ( ^ tang. <p )^ = J- tang.^ «j ; so that to reduce A B \o A G, we must apply the correction — ^tang.^ip.sin. (2t), — 26), as in the first term of [3800]. Again, since B D is perpendicular to AB, and BE perpendinijar to DF or B G, we have nearly, the angle ABG = angle D B E ; moreover, in the spherical triangle A B G, we have cos. ABG = sin. BAG . cos. A G [1345*-], or in symbols, cos. D5 jB== sin. ip . cos. («^ — d). Now in the right-angled triangle Bfil>, we have, very nearly, BE = BD.cos.DBE=iàs .sm.(p.cos.{v;—ê); and on account of the smallness of p, we may change sin.cp into tang.ç, also BE into FG; hence F f? = 5 s . tang. (?. cos. (j;,— Ô). Subtracting this from AG, we get AF; and in this way we obtain the second term of [3800]. i;-« [3800a] [38006] [3800c] [3800(/] [aSOOe] [3800/] [3800^] 44 PERTURBATIONS OF THE PLANETS. [Méc. Cél. [3800"] inclinations of the orbits ; but these inequalities are insensible except for Jupiter and Saturn. If we notice only the secular quantities, and put, as in [1032], [3801] tang. <p . sin. Ô =p ; tang, ip . cos. â = ç ; we shall have* [3802] is = t.~. sin. (n i + s) — t-'j^ • cos. (n t + s). [3803] The term — tang. (? . 5 5 . cos. (v^ — ê) produces the following expression, [3804] ^•^9'P~~P l! . so that we shall havef [3805] V-V + t. —^ , which agrees with what we have found in the preceding article [3782]. Î * (2401) If we suppose s to be a function of t, which becomes S, when t = 0, we [3801a] shall have, by the theorem [607, &c.], s = S + t . (^ — j + — . (^— j + &ic. If we neglect t- and the higlier powers of t, and notice only the secular inequalities, we shall get s — S=t.(-—-]. Now s — S, being the variation of s in the time t, is what is represented above by &s ; hence S s ^^ t . I j ; and by noticing only the secular inequalities depending on dp, d q, in [3781], we obtain /dS\ dq . , , dp , , > [3801c] [-j^) = -^ ■ sm. (« t + s) — ~. cos. (n t + s) ; consequently &s becomes as in [3802]. t (2402) Developing cos. (v— è) by [24] Int., and then substituting the values [3801], we get [3804a] — tang, (p . cos. (d, — ^)^ — tang. (p . \ cos. â . cos. i\-\- sin. Ô . sin . d J = — q .cos. i\ — p . sin. d, [38046] = — q. COS. {nt-\-s)— p. sm.{n(-\-s) ; observing, that as this quantity is of the order j}, g, and is to be multiplied by 5s, in [3800], which is also of the same order [3802, 3767c], we may put v^=^nt-\-s, neglecting, as usual, the terms of a higher order in p, q. Multiplying together the expressions [3802, 3804è], [3804c] and retaining only the quantities independent of the periodical angle 2nt-{-2s, we may use the values [3779c], and we shall get, for — tang, (p . i5 «. cos. (r, — è), the same [3804rf] expression as in [3304]. This represents the secular change of v, arising from the last term of [3800] ; and by adding it to v, it gives n,, as in [3805]. We may observe, that [3804e] the first term of [3800] produces no secular terms, or such as are independent of 2t', — 2^, and it is therefore neglected in this estimate of v, [3805]. [3805a] t (2403) Neglecting terms of the order t^ or m'^, we may suppose i.{qdp — pdq) VI. i. §7.] TERMS OF THE THIRD ORDER IN e, c', 7. 46 [3806] [3806'] ON THE INEaUALITIES DEPENDING ON THE CUBES AND PRODUCTS OF THREE DIMENSIONS OF THE EXCENTRICITIES AND INCLINATION'S OF THE ORBITS AND THEIR HIGHER POWERS. 7. The inequalities depending on the cubes and products of three dimensions of the excentricities and inclinations of the orbits, are susceptible .p„of„,„, r ^ r u. of /i of 01 two lorms, the third order. R = M. sin. {i . (n' t — nt-\- «' — s) + 3 n ^ + ^| ; [First form.] [3807] R = N. sin. {i . {n't — n / + s' — s) + nt + L]. [second fum,.] [3807'] We may determine them by the analysis employed in the preceding articles ; but as they become sensible only when they increase very slowly, we can make use of this circumstance to simplify the calculation. We shall resume 2d . (rS 7-) the expression [37156], and shall neglect the term — 3' , . > which is [3808] ci III (Xi z then insensible, t because of the smallness of the coefficient of t, in the inequalities now under consideration. Then this formula becomes àv= — '^^^^ + Sa.ffndt.AR+'2fndt..a\('^^^.X 13809] to be equal to Cdt, C being a constant quantity ; then [3782] becomes dvp=dv-\-Cdt, [3805J] whose integral is v^=v-\-C t, as in [3805]. * (2404) The reason for assuming these forms is evident from the principles used in [3704a — 6], For the coefficients of n't, — nt, in [3807], are i, i — 3, respectively; [3807o] their difference 3 expresses the order of the coefficient k [957"'", &c.], or that of M [3807], which must therefore be of the order e^. Again, the coefficients of n' t, — nt [3807'] are i, i — 1 ; their difference is I , consequently N may contain terms of the order [3807fc] 1, 3, 5, fee. [957'", fiic.] ; which include those of the order (? ; and it is evident from [957", &c.], that these forms embrace all these terms of the third order. t (2405) This remark applies exclusively to terms of the form [3807], like those in the three first lines of the second member of [3819], depending on the angles i.{r^ t—nt-\-s' — i) -\-Znt, whose differential introduces the very small factor i.(n' — n)+3n [3818(/]. But this small factor is not produced in the differential of the terms of the form [3807'], contained in the last line of the second member [3819] ; and then the term [3808] is not neglected, but is computed in [3822c]. X (2406) In the terms treated of in §7, and depending on the cubes of the excentricities, no quantities are finally retained except those which have the small divisor i.{n' — n)-\-3n, or its powers; and as the expression of 5v [37156] contains the function 2 d . {r5r), divided by a^. ndt ; we must examine whether this function contains the small divisor we have just mentioned. Now by the inspection of the value of rôr, or rather of Sr [1016], VOL. III. 12 [3808a] [38086] [3809a] [3809i] 46 PERTURBATIONS OF THE PLANETS. [Méc. Cél. The divisor y'l — e^ [37156] must be neglected for greater accuracy, as in Book II, ^54 or [3718']. We must also, by the same article, ayply these inequalities to the mean motion of the planet m, in computing its elliptical motion [3720]. This being premised, if we suppose R zzz: m! P. sin. \i . (n!t — nt -\-s'—^) + 3nf + 3-=! [3810] + m' F. cos. {i .{n't — nt-{-z'—i)-\-Snt + Si\; which comprises all the terms of R, where the coefficient of nt is greater [3811] or less than that of n't by the number 3 ; we shall get, as in [1209],* 3.(3 — i).m'n^.a [3809rf] [38126] 3a.ffndt.aR-. |i.(ra'— ?^)-j-3n}2 ;„, , 2(/P 3ddP' ).,.,, , , V , „ , „ , r»ûioi I -j r* + ( • ■ , , , I „ > ., — TT-r-, , , „ )^ , „ >.Bm.]i\nt—nt-\-s—s)-\rmt-\-3i\ \_mii\ 1 ^ \i.(n'—n)-\-3n\.dt \i.(n—n)->ç-3n\^.dt^S in 2dP' 3rfrfP ) ,. , , , , ,,„ ,„, [ {%.[n'—n)-{-Qn\.dt {i.(n'—n)-\-Qn\Kdt^S ■ we shall not find, in the preceding function, any term depending on the first power of e, [3809c] and having the divisor i.[n — »t)-j-3n. In quantities of the second order in e, c', given in [3711, 3714], we find such terms having the first power of that divisor ; but these terms depend upon angles of the form i . {n't — n t -\- ^ — s)-\-2nt , which are different from those under consideration in this article [3806' — 3807'] ; so that they may be neglected. To investigate the similar terms of the order e', which depend on the angle i.{n't — n t -\- e' — s)-\-3 nt, we may go through a calculation similar to that in [3703—3714], changing, however, the angle i . [n' t — nt-\-s' — t)-\-2nt into i.(n't — nt-\-s' — s)-\-3n't; which is the same as to increase the integral ntimher 2 — ;", connected with nt by unity; by which means the divisors in-{-{\ — i) .n, in'-\-{2 — i).n, in' -{-{3 — i).n, which occur in [3705,3710,3711,3714], are changed, respectively, into î»' + (2 — i).n, in'-\-{3 — i).n, iw'-j-(4 — i).n. Hence the quantity r^r, [3809e] similar to [3711], will contain a term of the order t^, depending on the form [3807], and having for divisor the first potver of the small quantity in'-\- {3 — i) .7i , as is hereafter found in [3819]; but this divisor will vanish from the dilTerential d.[r6r); therefore it may be neglected, as in [3809rt] ; and then the formula [37156] becomes as in [3809] ; omitting the divisor, \/{l — e^), for the reasons given in [3718']. * (2407) Substituting, in the first member of [1209], the assumed value of [:3812o] k.sm.{i'n't — int-\-^) [1208^'], it becomes ffan^.dt^.\q.sm.{i'n't — int-'ri's'—ie)-\-q.cos.{i'n't — int-{-i's'—is)\ = riia.a in.(i'n't — int+i's'—i£) (j ^ , 2dQ' , 3rf3Q Ad^q ^ ^ ^) ^_o (i'n' — inf 'i ^~'~ {i'n'—in).dt~'' (i'n'—inf.dt^ [i'n'—inf.dfi i.(i'n't — int + i'e'—{e) ^_ _ 2rfQ 3rf2Q- jd^q (i'n'—inf 'I ^ {i'n'—in).dt'{i'n'—inf.dVi'^(i'n'—inf.dt3~ VI. i. §7.] TERMS OF THE THIRD ORDER IN e, e', y. 47 Then we shall have * 2m'n ^ , „ /dR\ 2m 2.rndt.a-.('—) = — ^--. — , ,„ -. «2. f'^JLYcos.{i.{n't-nt+e-s)-^3nt-\-3sl -a". ('^^)-sin. {L{n't-nt+^-s)-\-3nt-\-3i] Lastly, we shall suppose, that by noticing only the angle i . (n't — nt + s — '.)-{-2nt + 2B, ' [3813] we havef r S r = H. COS. {i . (n't — nt + B'—s)-}-2nt + 2B-\-Al; [3814] Now if we take the difierential of [3810], relatively to d, then multiply it by 3 a .7idt, and prefix the double sign of integration, we shall get, by using for brevity, T=n't — nt + £'— 1 [3702a], [3812e] C — 3.{3 — {).m'.P'.sm.(iT+3nt-\-3i)) ^„^,„,, ■'■' •'•' ^+ 3. (3 — i) .m'.P.cos.(îT+3n!: + 3£)S The second member of this expression is of the same form as the first member of [38126], as is easily perceived by changing, in [3812J], i' into i, and i into i — 3; also putting Q = — 3 . ( 3 — z ) . m'. P', Q' = 3 . ( 3 — i) m'. P ; then making the same [3812<] changes in the second member of [3812è], we obtain for 3a .ffn dt . àR, the same expression as in [3812]. We may observe, as in [3714cZ'], that the secular variations of [3812/] the elements are noticed by the introduction of the differentials dP, dP', ddP, ddP', which are computed in [4415, &c., 4484, &c.]. * (2408) The partial differential of R [3810], taken relatively to a, being multiplied, [3813a] by 2/1 dt . a^, and then integrated, gives [3813]. t (2409) The expression [3814] is equivalent to that in [3711]; if being taken for the coefficient of any one of the terms of this formula, and A representing that one of the [3814a] quantities — 2 in, — « — ra', K — 2 s, which is connected with this coefficient H ; ^001411 observing that H is of the second dimension in e, e'. The differential of [3701], is dr=^ae.ndt.sm.{nt-^E — «) -|- &:c. ; multiplying this by [3814], and neglecting [3814c] terms of the fourth order, we get, by using T [3812c], ^-^ .dr^ Hae .ndt .cos.{i T-\-2nt -\-2 s -{- A) .sin. {nt-{-B — ts) = lHae.ndt.sin.{iT-\-3nt-{-3s—-a-{-A) [3814d] — i Ha e.nd t .s'm. {iT-}- nt-{- s -\- zs -\- A). As this is of the third order [38146], we may, in the first member, put r= a, and then dividing by — andt, we get '^'•^' -|ffe.sin.(iT+3n<+3E-^ + ^) [3814,] (findt + iiîe.sin. (iT-f ««-(- t-^-a + A). 48 PERTURBATIONS OF THE PLANETS. [Méc Ctl. [3814] ^ ^*'^"S determined as in [3814a], and having the very small divisor r. (n'— w) +3n ; then the first term of ôv [3809] gives the following expression ; [3815] __^Ll_! ^_i/fe.sin.|z. (ra'i — «f + s' — f) + 3»i + 3s — a + Jj. Hence we shall find, by noticing only terms which have the divisor [3816] i . {n' — w) + 3w,* SC _, Sa.rfP 3a.ddP' ■) . C{.{n't-nt+i'-s)') i {t.{n'-,i)i-3n].dt \i.(n'-n)-+-3n]-2.dPS l+3nt~\-3s S 5 p 2a. dP' 3a. ddP } C{.{n't-nt+;'-s)-) '—UP- •COS. Ter mi of \i.{n'-n)-l-3n].dt \i.{n'-n)-\-3nl'i.df^S (+3n«+3e <5« ( a^. (~). COS. U. (n't — 7it + s' — s) + Snt-\-3s\ heihird 2m'n y \da/ ' ^ ' ' ' ' * i.(n' — n)-|-3n ] /dP'\ / — a'^.f^j.sin. \i.{n't — nt-\-s' — E) + 3n<+3£} — 3 Ue.sin. \i.{n't—nt^^—s)-\-Znt-{-Ze—vi-[-A\. The differential equation [3699] [3818] o = ^' + '-^ + Vaie+..Q.t The first term of the second member is the same as in [3815] ; the second term is noticed r'î8l4/'l *" [3822rf]. We may observe, that it is not necessaiy to notice terms of the order t^ in dr [3814c], because they depend on the elliptical motion, and have no divisor of the form i . (n' — ?i) -j- 3 ?! ; moreover they must be multiplied by terms of the order e, [3814^] which occur in — [1023], to produce terms of the third order now under consideration ; and these terms of [1023] do not contain the small divisor just mentioned. * (2410) Substituting, in the expression of i5i; [3809], the values of the terms in its [3816a] ^^^^^^^ member, given in [3815, 3812, 3813], we get [3817]. f (2411) The expression [3818] is the same as [3699], from which we have deduced [3702], and by using [3705a], it becomes [3818a] 0==^^^' + n^r5?-4-^3«2a.5r.[e.cos.(n<+j-rt)+e2.cos.2.(«)'4-E-^)]+2rd/î+a.('^y^. This is solved as in [3711&, c], and if any term of the expression between the braces be [3818o'] represented, as in [37116], by aif. sin. (?»,< -}- «,)> or o.K .cos. {mt -\- s^), the corresponding terms of rhr [3711c] will contain the divisor m^ — n^, or rather the two divisors (m.-\-n), {m^ — n). To find the values of m^ producing the divisor i.{n' — n)-|-3n [3818'], VI. i. §7.] TERMS OF THE THIRD ORDER IN e, e\ y. 49 gives, by noticing only the terms which have tlie divisor i.(n' — n)-{-3ii, [3818'] 2.{{~3).mn ( aP .sm. \i.{n't—nt i-s' — i)^3nt-\-Ss\ ■) J- (5 )• a i . («'—«) +3 11 ' I -JraP'.cos.\i.{nt — nt-\-!'—s)-\-3nt + 3;l ^ — ^He. COS. \i. (n't— nt + s—s)-^3nt + 3s—^ + A} + iHe.cos.{i.(n't—7it + s—s) + nt + s + ^ + Al. [3819] Terms of ri)r. we shall put it successively equal to ???, + n and m, — n ; and we shall get m,= 2.((i' — «)-)-2n, 'm^=i.[i\! — n)-\-An', but we may neglect the last, because the coefficients of n, n dilfer by 4, and the terms depending on it must be of the fourth dimension in c, e' [3704n', &lc.], which are here neglected. Therefore, in finding r5r, we need notice only the following terms. First. Where m.^ = i . [n' — n) -(- 2 n . Secoii'l. Where the quantity R, or rather fdR, contains the divisor i . [n' — ?i) -j- 3 w [3813']. Hence it is evident, that we may neglect a . (— — j, which produces no such terms. The part of R, given in [3810], produces in 2/d R, the following terms, -2.(i — 3).m'.?i ( 77(7i'— n) + 3V ■ ^ [3818fc] [3818c] and [3818rfl P.sm.{i.{nt — nt-\-s'—e)-{-3nt-\-3s] ) + P'.cos.f/.(n'<— ni + s'— s)4-3 7i< + 3£| V These come under the second form [38186], in which o-K has the divisor i.{n' — n)-\-3n. The part of rSr [3818a"], depending on these terms, is found by dividing them by jn/ — )i^ ; ?H, being in this case equal to i . {ii — n) -j- 3 ?i ; and by hypothesis it is very small in comparison with n. Thus, for Jupiter and Saturn, where i=5, it becomes m^=i.(n' — «) -l-3?( = .5 w' — 2n^=j\n [3711/]; so that m,^ is less than ~7^, for the divisor m^ — n^, we may write simply — 71^=^ — a~^ [3709']. Therefore, by multiplying [3818c], by — a^, we get the part of rSr corresponding to these terms of 2/d R ; and then dividing this result by a^, we obtain the corresponding terms of The terms thus computed agree with those in [3819], depending on P, P'. necessary to notice the terms of 2/d R, like those depending on [3703, 3704], because terms depending on different angles from those proposed in [3807, 3807'], or else such as have not the small divisor mentioned in [3818']. The next term of ajBT [3818a'J, which we shall notice, is that depending on the quantity Sn^a.ôr .t^. cos. 2 . (?i i -f s — ra) [3818rt] ; and as we retain merely the terms of the [3818/] third dimension in e, e', &:c., it will only be necessary to notice terms of the first dimension in 6r. Now if we examine [1023], we shall find, that none of its terms, of that order, have the small divisor [3818']; therefore we may neglect this part, and then the only remaining quantity in [3818a], producing terms of a ^, is Sn^a.Ar .e .cos. {nt -\- e — ts). As this contains the factor e, we may notice in o r only terms of the second dimension, in [3818^] VOL. III. 13 rSr It is not [3818f] they will produce in — 50 PERTURBATIONS OF THE PLANETS. [Méc. Cél. Adding this expression to that in [3814], [3620] '-Jl^ff^cos.{i.(n't — nt + i'—^) + 2nt + 2s + A], or we obtain* To^r^sof sr^ H .cos.li.(n't — nt-^e'—s)-i-2nt + 2^ + Al [3821] — He.cos.{i.(n't—nt + i'—^) + Snt + 3s—^ + Al -irHe.cos.{i.(7i't—nt + s—B)-{-nti-s + z: + A\ 2.(1— S). m n ( aP.sm.\i.{n't — nt+s'—i)-\-3nt-\-3sl '^ i.{7i'—n)-i-3n' i + aP'. COS. {i. {n't — iit-^-s'—sJ-^-Snt-^Qs] order to procure those of the third dimension, which are the only ones investigated in this article. The terms of the second dimension, which can produce the angles proposed in [.3807, 3807'], are evidently included in the form [3814] or [3820] ; multiplying this by 3n^o^. e . cos. (?i< + ^ — «)> and reducing by [20] Int., it becomes r „ „ „Ç cosAi.{n't—nt-{-s'—s)-\-3nt-{-3s-zs-\-A\} [3818A] 3n^a.ôr.e.cos.{nt + e-^) X a=^^''-''''-l + cos.li.{n't-nt + s'-s) + nt+s+^+A\ V Now He [3814t] is of the third dimension in c, e', he, and by neglecting higher dimensions, we may put - = 1 [3701], and then we shall have for the remaining terms of o.K.cos.{m,t + e,) [3818n], &He.7i^a'^.cos.ii.(nt—nt-\-s — s)-\-3nt-\-3e—Ts+Al [3818i] z ( -Jf-îHe.n^a^.cos.{i.{n't—7it + ^—c) + nti-B-J^zi-\-.^. Dividing this by m^ — n^ [3Sl8a"], we get the corresponding terms of rSr. Now for the first of these angles i.{n't—nt-{-s — £)^3nt-lrSs—zi-\-A, we have OT,=i.(n'— n) + 3M, and as this is very small [3818rf], it may be neglected ; and then the divisor becomes — n^- [3818A:] In the second angle [3S18i], the value of 7«, is i.{n' — n)+n or \i.{n' — ?(.)+3n} — 2ra, which is nearly equal to — 2n ; hence m^ — n^ is nearly Sn^; consequently this divisor is nearly equal to 3n^- Therefore if we divide these terms of [3818i] by — Ji^ and 3n^, respectively, we shall obtain the corresponding terms of r S r ; lastly, dividing these result» by a^, we get the terms of —5- depending on He, as in [3819]. [38180 * (2412) None of the terms of ^ or — , of the order m'e, contain the small a- a divisor [3818'], as is evident from the inspection of the formula [1016] ; so that the terms of — , containing this divisor, and which must be noticed, are Included in the functions of the second members of [3819, 3320]. Adding these quantities together, and multiplying VI. i. §7.] TERMS OF THE THIRD ORDER IN e, e', y. 51 This value of — produces in 2 v, an inequality depending on the angle [3822] i . (7i' t — nt-\-;' — e) + n f + f, which has i.(n' — n)-\-Sn for a divisor. [3822^ To determine it, we shall resume the expression of (5 v, given by the formula [931].* The part — \ , ' — of this expression produces [3822"] It m 1i Or V in (5 V the term 6v = ^ He . s'm.[i . (n't — nt-\- s' — e) + ^ î + ^ +« + -<4 } ; [3823] which is the only one of this kind having the divisor i . (n' — n)-}-3n. The inequality of i5î) depending on the angle i.(n't — nt-\-s' — B)~\-2nt-\-2s, [3824] noticing only the terms having the divisor i . (n' — «)-}-3n, is, by [3715, 3814], very nearly equal to 2H.s\n.{i.(n't—nt + B'—B)+2nt + 2e + Al. [3825] their sum by -, which, by [3701], is equal to l-\- e .cos. {nt -\- e—zs) -j-kc., we [38216] Of get the coiTesponding temis of — . The quantities produced by this multiplication are equal to the sum of the terms [3819, 3820], with the additional term produced by multiplying the Rinction [3820J by e . cos. {nt-\- s — «), and this term is He. COS. {nt-{-e — w) .cos. \i.{n' t — nt-{-s' — s) -f 2 n i + 2 e -f .4| , [3821c] which, by [23] Int., becomes iHe. cos. \i .{lit — nt-^s' — s) + 3 n i + 3 s — ts + A] -\-^ He. COS. {i.{nt — nt-\-s'—2) -{- 7i t -^ s -{- vs -}- jl] . Connecting this with the other terms [3819, 3820], we obtain, by reduction, the function — [3821]. [3821d] * (2413) This formula, by the substitution of [3715a, 3705a], becomes as in [37156], the part mentioned in [3822"] being represented by — '— . Now the last [3822o] a^. ndt <fi. ndt term of the second member of [3819] depends on the angle i T -\- n i -{- s -\- is -{- A [3702a], mentioned in [3822'], and if we substitute it in the first term of the preceding 2d.{rûr) . , , expression „ -~— , it produces the tenn * a-. ndt ^ — \i. {n — n)+7i].-^. sin. \i T-|- n < + e + w + ^ f ; [382261 and as we have, very nearly, — \i.{n' — n)-\-n] =2n [3818Ar] ; it becomes 2He . sin. \i T -\-nt -\- -a -\- A). Again, the second term of [3822a] has already been computed in [.3814e], and contains the quantity i He .sm.{i T-\~nl-\-s-{-zi -\-A) ; [3822rf] connecting this with the preceding [3822c], the sum becomes as in [3823]. 52 PERTURBATIONS OF THE PLANETS. [Méc. Cél. Therefore, if we denote this inequality by [3826] K.sm.{i.(n't—nt + i'—s)-\-27it+2B-{-B}* Terras of weshallhave, in 6v, the following expression, ÔV. [3827] i<v = ^Jï^e.sia.{i.(n't—nt-JrB'—s)-j-nt + B-Jr^ + B\. 8. // is chiefly in the theory of Jupiter and Saturn that these different inequalities are sensible. If we suppose i =^ 5, the function [3828] i . («'— ?i) + 3 n = 5 n' — 2 n , becomes very small [381 8rf], in consequence of the nearly commensurable ratio which obtains between the mean motions of these planets ; and from this cause the corresponding terms of or, ov acquire great values. To determine them, we shall resume the expression of R [3742]. The partf [3829] — .cos.(y— f) -j-\cos.{v'—v) — cos.{v'+v)\+-^.— ^ ^, ^ 4 r^ 4 J,-2-2r)-'.cos.(y'-«;)-fr'2p * (2414) The parts of R [957, 1011], represented by M, JV [3703, 3704], do not contain the small divisor i . {71 — n)-\-3n, as is evident from inspection. Moreover, [3826a] F, G, H [3706], being the parts of — [1016], depending on terms of the first degree in e, e, do not contain this divisor, as appears by the inspection of [1016]. Therefore no part 2rf.(?"(5r) of ÔV [3715], except the first term —^^ — 7—, contains this divisor ; and if we substitute a^.ndt in this term the value of r (5 ?• [3814], we shall obtain, in ô v, tlie terra 2 [38266] .li.{n'—7i)-{-2nl.H.sm.li.{n'i—7it-\-e'—s)-J[-2nt-{'S-j-A} ; substituting — \i.{7i' — ?))-(- 2 «}=?i [3S22c], it becomes as in [3825]. If we now compare the expressions [3825, 3823], we find, that [3823] may be derived from [3825] [3826c] by multiplying its coefficient by |e, and decreasing the argument by nt-^B — sj. The same process of derivation being used upon the assumed form [3826], produces the expression [3827] ; which is computed in [4439] for Jupiter, by tliis very simple process. t (2415) We shall suppose, as in [1009, 956c, 963'', 1018a], for tlie sake of brevity, [3829a] r =a(l + Mj; r' =a'.(l+j«;); v =^nt + s-\- v, ; v'=n't + s'-^- v,' ; [-38296] a^ = a m, ; a' = a' w/ ; a"=: v,' — r, ; a. ^ - ; [38296'] T ^n't-nt-\-^—e; dT^{n'—n).dt; [3829c) W=nt-\-B — -ro ; W'=^n't-\-B' — -a; [38S9e'] M,, «/ v' — » are of the order of the excentrlcities, and a is changed into a^, to VI. i. §8.] TERMS OF THE THIRD ORDER IN e, e', y. 63 produces no term of the third order of the excentricities and inclinations, [3830] distinguish it from a [963'^]. If we represent the function [3829] by m, and suppose U to be the part of this value independent of u,, «/, f,, vj, we shall have U as in [3829/]; [3829d] observing that the last term of [3829J becomes in this case, by using [3744, 3749], im'.'j^.aci'.cos.T.la^-2ad.cos.T+a'^l~^ = ^^m'.f. ad. cos.T .is . B''\cos.iT = im'.f.aa'.iS.B"Kcos.{i-\-l).T = 1 m'. y^. a a'. -2. B^'-il. cos. i T ; t3829e] [3829/] [3829g:] U=L — ^g- . cos. T — T ™'' 7*- "7^ ■ cos. T -j- i '«'• y®- -^ • cos. [n t -\- n t -\- ^ -{- 1) + I m. f. aa'.S.. B^-^\ cos. i T ; i being as in [3715']. The development of u, as far as the second powers of aj, a', a" being found as in [957e], is «= -■+-(^) +-• O +^"- © +H.'. (^)+^..^'. G^.) the tenns of the third order, obtained in the same manner, are +i-^"'G-S^)+--^"-G-i^)H-*-'-(^)- We have given this full development of îi, because it will hereafter be of use in the notes on this article ; and for the same purpose, we shall also insert the following expressions, deduced [3829t] from the comparison of the values of ao, a', a" [3829J, a] with [659, 668, 669] ; ao = rt .|i e^ — (e — f e') . cos. fV — i e^.cos. 2W — i e^. cos. 3Wl=au/, [3829*] a'=a'.|Je'2— (e'— tc'3).cos. fP— ^ e'2.cos.2 JF' — § e'^. cos. 3 ^'} = «'m/ ; [3829i] "■~^-(2e-ie3).sin. fF-Je2.sin.2?r-ife3.sin.3^FS^"''~'''' ^^^^^ From these values it appears, by a slight exammation, that none of the terms of U [3829/] produce quantities of the third order, depending on the angle 5 7i'i — 2nt, now under consideration. For the terms of [3829/], multiplied by y% of the second order, depend [3829«] on the angles T, n' < -f n < + s' -f s , i T ; and when we combine these with terms of the VOL. III. 14 64 PERTURBATIONS OF THE PLANETS. [Méc. Cél. Value of R for tliis case. [3831] [3832] depending on the angle 5 n't — 2nt; such terms can therefore only arise from the remaining part* m'.y^ rr'.cos.{v'-\-v) R ^j.2 — ,2;-/. COS. (j;' v) -\- JSfS I r^ — 2 r r'. cos. (ti' — v)-\-r'^l- and then the expressions of P and P' [3810] will be the same, whether we consider the action of m' on m, or that of m on m'. We shall now investigate these values of P, P'. [3829o] [3831a] [3831o'] [38315] [38316'] [3S31c] [3831d] [3831e] [3831e'] [3831/] ^w< order in a,,, a', a" [38297i: — ?»], they will not produce the angle bn't — '^nt. The only remaining term of C/ [3329/] is the fii'st, depending on cos. T or cos.(?i'i — nt-\-^ — s); and if this were multiplied by a term depending on the angle An't—nt, it would produce a quantity of the required form ; but none of the powers and products of cio , a', a" [3829Ar — ?»]; retamed in [3829^, K] contain terms of the third order depending on this angle ; therefore we may also reject this temi, as in [3830]. * (2416) If we reject the terras of R [3742], mentioned in [3829], which we have proved, in the last note, not to contain terms of the required form and order, we shall obtain for R the âmction [3831]. This expression is not altered by changing r, v into r', «', respectively, and the contrary ; so that it w'Ul be of the same form, whether we compute the action of ?«' upon m, or that of m upon to' ; but in the first case it will be multiplied by in, in the second by m. Supposing, as in [3829fZ], that the general value of the fonction R [3831] is represented by u, and that it becomes equal to U, by putting r = «, ?•'=(/ v^^nt-\-s, v'^n't-{-e', v — » = ?t'i — n ^ + s' — s=T, we shall get the first of the following expressions of U [3831c]. The second expression [3831<?] is deduced fi-om the first by the substitution of the values [.3743, 3744], neglecting, however, the first term of [3743], which makes an exception in the value of A'-''', in the case of i = 1 ; because this term produces no effect in the present calculation, as we have seen in [3829o] ; lJ=—m'. \(v^-2 a a. cos. T+a'^-^—{m'.y^.aa'.cos. {n't+nt-\-s'+2) .{(?- 2 aa'. cos. T-\-a'^\-i = im'.S.A^'\cos.iT—im'.f.aa'.cos.{n't-{-7it-\-s'-j-s).S.B'-'\cos.{T = i m. 2 . A^'\ COS. i T— I iri. y^. a a'. 2 . B'^ ". cos. [i T-\- 'int + 2 s— 2 n) . We may remark, that, in reducing [3831(/] to the form [3831e], we obtain, in the first place, from [3749], cos.{n't-\-nt + s'-Jrs).:s.B^'\cos.iT=X.B^'\cos.{iT-\-n't-j-nt-{-s'-\-s) = 2.B''\cos. {(i+l).T+2«^ + 2£} ; and by changing i into i — 1, it becomes X. B'-'~^\ cos. \iT-\- 2 n t -\- 2 b\ ; but as this quantity is to be multiplied by y^, we must change 2n<-|-2s into 2nt-{-2s — 2n, as in [.3745'" — 3748], and then the value of U becomes as in [383 le]. VI. i.^,^ 8.] TERNIS OF THE THIRD ORDER IN e, C, 7. 55 We have, in Book II, ^22, by carrying on the approximation to terms of the third order of the excentricities [659, 668, 669],* — ^e\cos.(3nt+3B—3^) ; v = ni4-e+(2e — ie=').sin. (w^ + s— tï)+ f e-.sin. (2nï + 2s— 2x^) +11 e^sin. (3w< + 3s— 3a). Values of r, 1'. [3834] * (2417) We shall now commence the investigation of the part of R depending upon the first term of [3S31e], namely, U^=^ m'. 2 . ^''\ cos.i T; the other terms depending on B''"", being computed in [3840a, Sic.]. Substituting this value of U, in the terms [3829^, A], we get the following value of R, [3834o] 1 R = 2, 3 4 5, 6 7, 8 9, 10 11, 12 13, 14 15, 16 17, 18 19, 20 im'.2.^<''.cos.^T + + * m' . ag . 2 . f — — \ . cos. I r+ J m' . a'. 2 . ( -7-^ j -cos. i T , — im'. 0.". Si. A^'\ sin. iT , „ /ddA(i)\ .^, 1 , , /dd.m\ .^ + im'.ao2. 2 . -r-7- ) . cos. i T+Jm'.ao a' . 2 . ( -— — ) .cos.i T \ da^ J " \dada J + l+i m'. a' ^ . 2 . i^-^) .cos.iT-i m'. ^a". 2 i . (1^) . sin. i T —im'.oJa/'.Xi. (-j^) ■ sin. i T— ^rn'.a''^. 2 Î^A^'K cos. i T + A-'.S^2.('-^).cos.iT+.™'.ao^a'.2.(^).cos.ir^ + ^-'-'^-(£^.)---^+^V..'.a'3.2.(^^).cos..-T + < ->'.a„V'.2;.('i^) .sin.zr-i».'.aoa"^.2P.(^).cos.tr -im'.a'2a".2i. (^) . sb. i T-im'.a'a"^. 2^^. (^^Vcos.i T — àm'.aoa'a".2i/^^,') . sin. i r+^ . a"^. 2 i'. ^». sm.i T \dada/ ' 12 Terras of R depend- ing on [38346] We must substitute, in this expression, the values of a^, a', a" [3829A: — ni], and retain only the terms of the third dimension, and of the form 5n't—2nt [3834"], in which the coefficients of n't, nt differ by 3. Now as these coefficients are equal in the angle i T, which occurs m [38346], this difference in the coefficients of ri't, nt must arise from the [3834c] 56 PERTURBATIONS OF THE PLANETS. [Méc. Cél. [3834'] This being premised, if we develop R [3831] according to the order of the powers and products of a^, a', a" ; and it is evident, from [957^'", Sic], that such terms [3834d] must have for a factor, some one of the four quantities e'*, e'^e, e'e^, c^. If we take the powers and products of tlie quantities a,,, a', a" [3929/!: — to], of the tliird dimension, and reduce them by means of [17 — 20] Int., we shall find, that the greatest angles connected [.3834e] with these factors e'\ e'^e, c' <?, ê, are, respectively, 3 7P, 2W'^W, W'-{-2Jr, 3fV; it is not necessary to notice the smaller angles TV, W , 2 W — W, Sic, because they do not produce terms of the form bn't — 2n t [3834c] ; substituting ?F'= T-\-nt-\-B—-a', W= nt-{-s — -m [3829c] ; they become, respectively. [3834e'] [38.34/] [3834êr] [3834^] [3834i] [3834ft] 3 T+3Mi + 3£ — 3«'; 2T+3n<-f-3e — 2i3'— ra; r_|_3,j;_j-3£_,3'— 2i3 ; 3m; + 3£ — 3îï. Now we perceive, by inspection, that the cosine of any one of these angles is multiplied, in [38346], by a tenn of the form ^/''. cos. z T ; and its sine by a term of the form ^/''. sin. I T; the products reduced by the formula [3749], are found to depend, respectively, upon the angles (i + 3).T-}-3n< + 3£ — 3«'; (« + 2). r+ 3 n < + 3 e — 2^^' — ts ; (i_|_l). T-\-Znt-{-3s — z^—2-a; i r+ 3 n < + 3 s — 3«. In order to reduce all the angles to the form i T, we must change, in the first, i into i—3; in the second, i into i — 2 ; in the third, i into i — 1 ; and make the same changes in the index of ^/'' ; by this means the terms in question become of the forms e'3. 2 . ^i''-=>. COS. (i T+ 3 Ji < + 3 £ — 3 ^) ; e'=e . 2 .A^^'-^K COS. (i T + 3 n « + 3 s — 2 73'— i^) ; e 62.2.^''-». COS. (i T+ 3 B ^ + 3 £ — TO — 2 13 ) ; e» . 2 . ^ w. COS. {i T+ 3 ÎI i + 3 £ — 3 w) . Putting i=:5, as in [3828], these expressions become of the same forms as the four first terms of R [3835], depending on M'°^ M'", Jf'^', M<^\ respectively. The two remaining terms M''^\ M^^\ depend on JS""", which was neglected in [3834a], and will be computed in [3840a, &c.]. We may remark, that the exponent of e, in any one of the terms [3834A], being increased by i — 3, gives the corresponding index of ^, , and when i = 5, we have for this increment i — 3 :^ 2 . We shall now proceed to the computation of the values of the powers and products of a,, a', a", which occur in the expression of R [3834J], retaining only the tenns [38342] depending on e'' e^, which are wanted in finding the values of M'-°\ VI. i.§8.] TERMS OF THE THIRD ORDER IN e, e', 7. 67 terms depending on the angle 5 n't — 2nt, we shall obtain an expression [3834"] of the following form, J\P^\ M'--\ M'-^K These quantities are arranged in the following table, in the order in wliich they occur in [38346], noticing only the greatest angles mentioned in [3834e] ; [3834m] 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0-0 a' a" tt,' = — frt.c». cos. 3 TV; = J-l e' 3. sin. 3 W— if e^. sin. 3 W ; = i 0^ f3. COS. 3 W ; aott a' 2 i a' « . e'2 e . cos. (2 W'^ W) -\- ^a' a .e' ê. cos. ( ?F'+ 2 7F) ; irt'2.e'3.cos.3fF'; a^a" = fa.e3.sb.3»^— |«e'2e.sin.(2?F+?F) — Jrt.e'e2.sin.(JF'+2ff) a' a" = — f a', e' 3. sin. 3 ?F'+ 1 a', e'e^. sin. ( ?F'+2 ?r) -f i a', e'^ e . sin. (2 W'^ W) i c\ C0S.3 W+ye^cosX W'-{-2 7F)+f e'2e.cos.(2 fF'^- W^)— f e'3.cos.3 W = ftf3 ao' = — i fl^. e'. cos. 3 W ; a/ a' = — i a' a", e' f?. cos. ( ?F'+ 2 ?F) ; a.oa'2 == — ia'2a.e"2e.cos. (2?F'+ W'); a'3 = — |«'3.e'3.cos.3l'F'; a;-ia" = — \ «2. c^. sin. 3 W + * «^ g' ^a, gin. ( /F'+ 2 W) ; aoa"2 = a.e3.cos.3?F— 2a.e'e2.cos.(?F'+2/^) + «.e'2e.cos.(2^'+rr) ; a'^a" = i«'2.e'3.sin.3 JF'— ia'2.e'2e.sin. (2R^'+ ?F); a'a"2 = a'.e'3.cos.3 TF'— 2a'. e'^e .cos.(2ff' +?F)4-«'.e'e2.cos. (?F'+2(F) ; ao a' a" = — 1 a a', e e^. sin. ( W'-\- 2 W') + J a a', e' ^^ e . sin. (2 fF'+ W) ; a"3 = 2e3.sin.3fF— 6e'e2.sin.(fF'+2?'F)4-6e'2e.sin.(2?F'+fF)-2e'3.sin.3W'' . We shall use these expressions in the following notes, in computing Jlf", JV/<", fiic. ; and we shall also make use of the following formulas, which are deduced from [95.5e — A], by taking the differentials relative to T, and dividing by àzdT, changing also W into ?F^ , as in [3T50A, &tc.] ; sin. W^.is. P. A^'-> . sin. i T= — ^ 2 . P. ^^'l cos. {i T-\-TV); COS. fF, . 1 2 . P. A'^\ sin. i T=: i^.P. A^'\ sin. {i T+ IV,) ; sin. W,.\-s..P. A''\ cos.i T^ ^ 2 . P. A^'\ sin. (/ T-\- W,) ; cos. fF, . I 2 . i3. ^'0. cos.i T= X V . î3. ^(0. cos.(i r+ fFJ, VOL. III. 15 [3835a] [38356] [3835c] [3835(f] [3835el 58 PERTURBATIONS OF THE PLANETS. [Méc. Cél. General form of for termaof the third order- R= M^''\e'\cos.(5 7i't — 2nt + 5s' — 2s — 3z^') ~R +ai^'Ke'-e.cos. (5n'i — 2 n i + 5 s' — 2 s — 2«' — ^) + M(^>. e'e-. COS. (5 h' i — 2 jU + 5 s' — 2 e — ^' — 2 ^) [3835] +M(^'.e^cos. (5 n' t — 2 n i -^ 5 e' — 2 s — 3 ^) + M'^' . e'f. COS. (5n't — 2nt + ôs' — 2s — zi'-~2n) + M'-'Key-. COS. (5 n' t — 2n t + 5 e' — 2 s — z^ — 2 n) ; and we shall find, after all the reductions,* !(2) (3) (3) 389 6';' + 201 a . ^-^+ 27 a^ ^ + a3. Ç^ •4 a a da-' rf a^ [3836a] * (2418) The pait of R [3835], depending on e'*, may be put under the form M'-°K e'\ COS. {iT-\- 3 W) or iH'»\ e'^. cos. (2T+3?F'), using T, ÏV, &lc. [38296', c] ; the coefficient of T being i=2. Terms of this kind are produced in i?, by multiplying the quantities which are connected with e'^ in [3835aJ, by the corresponding terms with which they are combined in [38346], and then reducing the products by means of the formulas [955, 955a — h, 33356]. The terms depending on ^® and its differentials, are [38366] giygjj ;,^ ^jjg value of Jli"" [3S36c/], in the order in which they occur, without any reduction, and omitting 2 for brevity ; so that the terms of [3835a], marked 4, 10, 20, are connected /dA'''i\ /rf2^(i)\ /rfS^Wx With ^«; 3, 9, 18 with (^); 7, 17 with (-^-^j ; 14 with (—-) . Substituting i^^2 [3S36o] in this first value of M^°\ we get the second value of [3836e] ; [3836c] and this, by using the values [1003], becomes as in [3836/], or by reduction, as in [3836^]. Lastly, substituting in this the values [996 — 1001], we get [3836/j], which is easily reduce d to the form [3836] ; [3836d] /(/./?(3)\ /d-A^~^\ „ /rf3./3f3)\ [3836e] =W-^^--W-'.«^('-;^)+il-«-.(^)-.V-'--•(^ + -.^6^'^'+18a.(— j+9a^(-^) + a^.(^( [3836g] =^s^W.^«,^.^^^^«.(^__j + |^^^«3.^__j + _.«3.(-^-^j /- (3) (2) (21 ■ 7,1' S 12) dbk c. d^bi , d3Ji ( [3836/.] =:i^,.)— 389 6,— 201a. -^ — 27 a2.^--a3.—f , VI. i. §8.] TERMS OF THE THIRD ORDER IN e, e', y. 59 16 / * a a, dix.^ ria-* > * (2419) Proceeding as in the last note, we find, that the part of R [3835] depending on e'=e, may be put under the form M'-'\ t'^ e .cQs.{iT -^'2 W'-\- W) [3829è', c], [383ra] in which the coefficient of T is i = 3. Substituting the values [3835a] in [3834&], we obtain the first of the following values of JV/'" ; observing, that the terms of [3835a] depending on e'^e, marked 10, 20, are connected with ^<'^ ; the terms 8, 16, with i— — j ; the terms 9, 18, vnû\ ( , , ) ; the terms 6, 19, with ( , ', , ) ; the term 17 \ da' J \dadaj with C^-^^); and the term 13 with (^^^^^r^A- Substituting i = 3 in [3837(-], [38376] we get [3837f?] ; and this, by using the values [1003], becomes as in [3837e], or by reduction, as in [3837/]. Lastly, substituting in this the values [996 — 1001], we get [3837 0-], which is equivalent to [3837] ; [3837c] , , /dd^\ ,,,,., ; , ,a /' d--A(0 \ ^ , ,2 /rf3^U)>, = - \\«- m'. A ^3>- 1^ m'. a . (^-j^j + f |- m'. a'. (^ j [3837rf] , ,, , , /ddJ10)\ ^ , ,„ /rf2^(3K /d3jia)\ 60 PERTURBATIONS OF THE PLANETS. [Méc. Cél. ( W (4) (4) ■) [3838] „'iJf(.= _^.- 3966';+ 184a. ^ + 25a^^ + a^^;* 16 ( ^ da. do.'' da.-^ ) * (2420) We may compute [3838, 3839] as in the two last notes, but it is rather less laborious to derive them from M'-''\ M'-'-\ by changing the symbols as below, namely, [3838o] For i, n't, nt, e', i, zî , •n, e', e, «', a; a', aj, T; [38386] Write — i, nt, n't, i, s', zi , -a', e, e' , a, a; a„, a', — T. The changes in these three last values of a', o.^ , T, evidently follow from those proposed in the other symbols, using [3829À:, /]. The value a" [3829m] is not altered, except in its sign, because e.sin. W^ changes into e'. sin.?F', and e'.sin. ?F' into e.sin. fF, &ic. ; moreover, A^'^ is not altered, because we have A'-~''' ^^^ A^''' [954"]; we also have, as [3838c] j^ [3831c, rf], —\a^— 2a a', cos. T-\- a'^l-i=i X .A^^\co5.iT ; and as the first member is symmetrical in a, a', the second, or A''\ must also be symmetrical, and wll [3838rf] not be varied by putting a, a' for a', a, respectively; lastly, the expression of iî [38346] is not altered by making these changes ; observing, that the quantities i a", i T remain unchanged. Now the part of R [3835] depending on c'e^, may be put under the form [3S38e] J»f (2). e' e^. cos. {i T+ 2 W-{- W), in which the coefficient of T is i = 4 . Comparing this with [3837a], we find, that by making the changes [3838a, 6], the expression [3837a], corresponding to i = — 4, will become like [3838e], and M'^^ will change into M'--^ ; we may therefore obtain the values of M^"'^ [3838/1], by changing a, a', i into «', a, — i, respectively ; then putting i = 4, we get [3838A']. This value may be reduced to the form [3838J], by the substitution of the values [1003], and also the partial differential of the second of this system of equations, taken relatively to a, which gives [3838/] Reducing the expression [.3S38i], we get [3838^] ; and by the substitution of the values [996—1001], it becomes as in [3838?], being the same as [3838] ; M'-''-> = m'.A<~^{—^i^^^i^+iii:.n'.(^^yy^i-li^]+m'.a.(^^^y\ — [3838/i'] [3838i] dada' J ' /f/.-î'-DX /(/.^HA /ddJl'^^\ =W»'.-«»-tt..'.»'.(-^)+W-«'...(^)-iS.«'..«'.(,-^,) VI. i. §8.] TERMS OF THE THIRD ORDER IN e, e', 7. 61 a' M'^' = in 48 !(5) (5) (5) -\ 2 ri a d iS? do? ) C (3) ^ a'ilf W _ _ ^ . ^ 10 63 +a . lï^ > ;t Id / ^ f/a > [3839] [3840] f (J) (-1) '41 -J = 7-r-, . J — 396 i — 184 a . — 25 a-', -—^r a'. — — - ( . lb a f - a a da~ aa'^ ) * (2421) The part of R [3835] depending on e^, may be put under the form M^^''.e^.cos.{iT-\-3 TF), in which the coefficient of T is x=5. Comparing this with [3836a], we find, that by making the changes a, a', i, &ic. into a, a, — i, &c., respectively, as in [3838a, 6], the expression [3836(/] will become as in [3839i]. This represents the value of Jf^"", or the coefficient of c^ in [3835]; and by putting i^5, it becomes as in [3839i'] ; which, by means of [996—1001], is easily reduced to the form [3839] ; 48a C * ' do. ' rfa2 ' do. . t (2422) The values of iV/C", M*^) [3840, 3841] depend on the second term of [3831e] ; and by retaining only this term, we shall have JJ ^ — | m'. 7^. aa'.'S. . B^'~^K cos. T, , supposing, for a moment, that T^z=i . (n' t — nt -\- s' — i) -\-2nt -{-2s — 2 n . As this expression is multiplied by 7^, of the second order, we need only notice terms of the first order in ao, a', a", in the development of u or R, and we shall get for this part of jR, the following expression [3829^], «--C^)+'^'-(7 ""U.-.f^ ft' / dT, obser\ing, that we notice in this article only terms of the third dimension. The values of aQ, a', to be substituted in this expression, are the same as in [3829A:, Z] ; and by retaining terms of the first order, w-e have ao = — ae . cos. fV, a == — a'e'. cos. M'". The angle T, represents the mean value of i . {v — v) -\-2v ; its increment, depending VOL. III. 16 [3838^] [3838Z] [3839a] [38394] [38396'] [3839c] [3840o] [3840a'] [3840A] [.3840c] [3840(/] 62 PERTURBATIONS OF THE PLANETS. [Méc. Cél. [3841] «'M<^' = ^.<7 6l + a.^^. r3840rf1 °" ^'" ^'' [^^~^"]' '•'' o-"=i.{vJ — ■dJ -j- 2 !;,= «!)/ — {i — 2).i;,, and by substituting v;=2e'.sm.TV', v,= 2e.sin.fV [669], we get a", and then [38406] becomes R^-e'.[a'.oos.W'.(^)-2i.sin.W'.(-^^\ [3840e] _e.^«.cos.^F.(^)+C2.--4).sin.rF.(j^)|; and by substituting the partial differentials of U [3840a], we obtain, without any reduction, R= ^- m'. c' f. COS. TV. J a' a . 2 . B^'-^K cos. T^ + a' ^ « . 2 . C^^^) ■ cos. T, I + i m'. c' 72. sin. W .a'a.Xi. £('-i>. sin. T. [3840/-] + J m' . c 72 . COS. TF. \a'a.-s:.B <'-». cos. T^ + a^ «'. 2 . (^-^^) . cos. T^ I — i m' . c 7^ . sin. W .a'a.X. (2 i — 4) . S''-», sin. T^ . The terms of this expression, depending on c' 7^, contain the factors cos. tV'.cos. T^, [3840g-] and sin.fF'.sin.T^, both of which, as in [17, 20] Int., produce the terms icos. (T^-^W), which, by putting i = 4, becomes icos.{5n'( — 2nt-\-5s' — 2s — •ra' — 2n) [3840»']. Comparing this with the term depending on Jfef '''' in [3835], we get the first of the following expressions, omitting 2 for brevity, and then by successive reductions, using [963''', 1006—1008], we finally obtain [3840/], which is easily reduced to the form [3840] ; [3840/1] M'-^^=-i^m'. i a' a .B^'-^^-{- a'^a. (^fj^) \ — ^ m'. a'a.i. S^'-" [3840i] = J^ m'. a' a . \— 1 B'^^ + a'. (^-^^ j= J, m'. a'n . J- 7 5«' + [-3B'^>-a . (^')] I (3) ,:««, =,,.,..4-,OB™-..(i^')|=,,„..„.„.)_i£,4'_^..^ I [3840i] ICa C 2 ' rfa In like manner, the terms of [3840/"], depending on e 7^, contain the factors COS. ^F. cos. T4, sin. ?r. sin. T4, producing the term | cos. (T4+ ^F), which, [:3840»i] by putting i = 5, becomes ^cos.(5n't — 27it-{-5s — 26 — « — 2n) [3840o']. Comparing this with the term depending on M^^^ [3835], we get the first of the following VI. i. §8.] TERMS OF THE THIRD ORDER IN e, e', y. 63 Hence we deduce* General + a'M(^). e'e~. sin. (^+ 2 w) + a'M^^'. el sin. 3 « [3842] + a'M<^'. e'7'. sin. (2 n + ^') + a'M'^'. e y". sin. (2 n + ..). and of We shall get in', a' P', by changing the sines into cosines, in this expression °^^,' of in!, a' P ; and it will be easy to deduce the values of a P, a P, by [3843] expressions, in which we must put i = 5, and then, by reducing as above, it becomes as in [3840p] ; whence we easily deduce [3841], M^^^=j'^m'. I «'rt.B<'-" + a2rt'Y!^^") I -j-j\m'.a'a.{2i—4).B^'-'^ [3840n] [3840o] (4)- 16 o' C t "T d a ) * (2423) In the case of i = 5, if we use, for a moment, the abridged symbol [3842a] T5=5n't — 2nt-\-5e' — 2s, the value of R [3810] becomes R = m'. P. sin. Tg -f 7n'. P. cos. T^ . [3842o'] Now each tenn of R [3835] may be easily reduced to the form [3842»'] ; since, if we take, for example, the fii-st ^<'". e'lcos. (T^ — 3ra'), and develop it by [24] Int., it [38426] becomes J/"'>.e'3.sin.3w'.sin.T5+^/<®.e'3.cos.'n'.cos.r5. Comparing this witli [3842a'], we get for the parts of m'.P, m'.P', the following expressions, m'.P = Jlf(°i.e'3.sin. 3^3', m'. F=M^°\ e'^ cos-Sz/, [38426'] as in [3842, 3843]. In like manner, we obtain the other terms of [3842] from [3835]. The values of P, P', deduced from [3842, 3843], may be put under the following fonns, which will be of use hereafter. Expres- sions of P=S.M'.e"'. e". f\ sin. (6' îi'-f 6 ts -f 2 c n), ^' P- [3842c] P'= 1 . M'. e"'. e\ f. COS. (6' î3'+ 6 rt + 2 c n) ; 2 being the characteristic of finite intégrais, and h\ b, c, integral numbers, including zero, satisfying the equation è' + 6 + 2c=:3. 64 PERTURBATIONS OF THE PLANETS. [Méc. Cél. [3843'] multiplying a' P, a' F, by ~ or a. We shall then find, by putting i — 5, in the expressions of ôv and — [3817,3827,3821],* Ex près - sioiiofthe /' ( ^n HP ?t« rltfP' "^ \ ^-P'+T^^Vt.-,-/ o P ^,o l -sin. {5n't-2nt+5s'-2s) ÔV —6m'. n^ ) i. (5n—2n).dt {5n'—'2nf.dt^) ^ ' ' of the °" third order. [3844] (5n'_2„)9 \ r 'ia.dP' Sa.ddP ( „ 2a. dP' Sa.ddP ) , , — ^ ciF— , ,, — , g , o V, j.o ^ •cos.(5?i'/-2n<+5s'— 2s) C (5n'— 2>i).d< (5n'—2nf.dt~ ) ^ ' ' 2 / \ — „ cc'v n.. V r r J o cA 5 n' — 2 n' ' — a^f—ysm.{rjn't—2nti-5s'—2s) — ^He.s'm. {5n't — 2nt-l-5E'—2s—zj-\-A) -]-^Ke.s\n.{5n't—4nt-J^5s'—4s-\-7S-\-B) ; Exprès- ^ J, terTo.-'" — = H .cos.(5nV— 3/U + 5s'— 3e+^) — iîe.cos.(5n'^— 2ra« + 5£'— 2s— «+^) of.the _|_ He.cos.{bn't — Ant-\-bs' — As-\--ui-\-A) order. [3845] +^^^.\aP.sm.{bn't — 2nt-^bs'—2i)-YaP'.co5.{^n't — 2nt + bt'—2s)\. [=3845'] If we suppose i = — 2,t and change the elements of m into [3844a] * (2424) Adding the terms of .5^ [3817,3827], and putting i=b, we get [3844]. Putting i = 5, in [3821], we obtain [3845]. t (2425) By restricting ourselves to terms of the first order of the masses, and of the [3846a] third dimension In e, e', y, the expression of — [3831] becomes symmetrical in the elements of m, m', so that these elements may be Interchanged without altering this value R R of — [3831 «, «']. The same symmetry obtains in the expression of — [3810] ; for [384(36] if we put, for a moment, T^ = 5n' t — 2jit ^5s'—2s, T,.= 5nt — 2 ?f'C+ 5 s — 2e', and retain, in [3810], only the two terms arising from the successive substitution of the values » = 5, i = — 2, It becomes [3846c] ^=P- sin. T, + P'. cos. T, + P^ . sin. Tg + P'o- cos. T, ; Py, P'o, Tq, being, respectively, the values of P, P', T^, when the elements a, n, e, &c. are changed into a', n', e, Sic, and the contrary, this being necessary to preserve the [3846(/'] symmetry [3846»]. In computing the action of tn upon m, it Is not necessary to notice VI. i. §8.] TERMS OF THE THIRD ORDER IN e, c, y. &b the conesponding ones, relative to Ht', and the contrary, we shall obtain C ,„, , '2a'. (IP Sa'.ddP' ) . ,, , ^ , ^ , r^ ^ \ I '{5n'—2n).dt {57i'— inf. dt^ y ^ ' ^f ( ,„ 2«'.rfP' Sa'.ddP } /r '. f> V 1 C ' O \ ( Expros- I (5n'-2n).rfi i5n'-2nf.dt^ S ^ 'J tcrmsof _ . 15m.)i'- he terms of iv' of the third 2m. n' ) vaa / • ■ - r order. «'-.r^j.cos.(5?i'^— 2n^ + 5e'— 2s) 5n'— 2n ) .„ /dP — a'^.(—\.sm.{on't—2nt-\-b^—2s) [3846] — IH'e'.sin. (57!'< — 2)U + 5s'— Qe— t3'+^') + îi:V.sin. (3n7 — 2;U + 3£'— 2ê-[-w'+B') ; t / Exprès- ^= if'.cos.(47i'<— 2n< + 4£'— 26+^')— iîV.cos.(5n'!!— 2n!;-f5£'— 2e— w'+^O ,';°"„^"ffi- + i/V.cos.(3«V— 2n< + 3£'— 25 + w' + ^') «nhe order. ^°™'"' |«'P.sin.(5n'< — 2ni + 5E'— 2£)+a'P'.cos.(5n7 — 2n< + 5£'— 2£)} ; [3847] 5n'— 2n if', cos. (4 n' i — 2 n ^ + 4 e' — 2 £ + J' ) being the part of — r^- depending [3848] onthe angle An't — 2nt* and ^'. sin. (4n'i — 2n i + 4=' — 2£+-B') the angle T^, because it does not produce terms having the small divisor 5 n' — 2n. [.3846rf"] In making the change of the elements of m into those of m', according to the directions [3845'], the value of — , corresponding to the action of m upon 7n', becomes - = Po . sin. T. + P'o . COS. To + P. sin. T, + P'. cos. T, . [3846e] m The second members of [3846c, e], are evidently identical ; but in this last expression the terms depending on the angle Tg, are derived from those of [3846c], which depend r„o,^^^ on i^ — 2 ; by changing the elements ?«, «, e, Sic. into those of m', a, e', &c., as in [3845']. Lastly, we may observe, that the quantities P, P', connected, respectively, with sin. Tj, cos. Tj, are the same in [.3846c, e]. Hence we may derive ôv from Sv, by taking the sum of the two parts of 5 » [3817,3827], putting i = — 2, then changing m, a, n, e, H, K, k.c. into m, a', n', e, H', K', &.C., respectively ; by which means we get [3846]. In like manner, we may derive [3847] from [3821]. * (2426) These terms correspond to [3814, 3826], putting i= — 2, and changing the elements as in [3845']. A'OL. III. 17 66 PERTURBATIONS OF THE PLANETS. [Méc. Cél. being the part of 6 v' relative to the same angle. In these various inequalities, we shall, for greater simplicity, refer the origin of the angles to [3840] the common intersection of the orbits of Jupiter and Saturn ; as Ave have already done in the development of the expression of R [3736 — 3738], and shall continue to do in the following article. For the sake of symmetry, we shall retain the angle n, which must be supposed equal to nothing. lioZf * We shall determine the differentials - — , , ' , - — , — — , in ,P,dir, iW dt^ ' dt ' dt^ ' ^^- the following manner. We shall compute, for the two epochs of [3849] 1750 and 1950, which embrace an interval of 200 Julian years, the , - (7e (/« de' d-ul d y du i i n values or --, —, —, —— , —, -— ; and shall represent these [3850] dt' dt ' dt' dt ' dt' dt quantities, at the second of these epochs, by — ', — -^, -~-, &c. ; we shall then have, by supposing t to be expressed in Julian years,* [3851] ^'^^ + 200.^% dt dt dt'' in which the differentials de, dde, in the second member, correspond to the epoch 1750. The value of e,t for any time t, neglecting the cube * (2427) We have, as in [607, &c.], [3850a] ,^f7+,.(^) + .,..(^J_^) + &e., [38506] d € (i€ u beins; a function of ^, which becomes U, when t=0. Now puttine m=— -', C7:= — , ,- T , ••11/' r '^^, ^^ 1 dde as in [3850], we get, by retaining only the farst power of r, 77 =" 77 + ^ • jTs > which, by putting ^=200, the interval mentioned in [3849'], becomes as in [3851]. From this roo,-/. -, ddi 1 \de, de} de de, . [38.50c] we get ^^ =§00 ' Jdl " rfIS ' ^^^^ ^^"'"'^ °^ rf"i ' ^ ' being computed, as m [4238, he, 4330a, &c.], for the epochs 1750, 1950 ; we obtain, by substitution, in [3850c], dde the value of -t^j corresponding to the epoch 1750. t (2428) Putting U=e, M=e,, in [3850«], we get [•3852a] e=e + t/^^+it^.'^ [3852]; in which we must substitute the values of e, ^^ , ^ [3850, 3850c], for the epoch 1750^ VI. i. §8.] TERMS OF THE THIRD ORDER IN e, e', 7. 67 of t and its higher powers, is de (1 d I clt' dt-' being supposed to correspond to the year 1 750 ; this expression maij be used for ten or tioelve centuries before or after that epoch* [3853] In like manner, we may determine the values of ^, e', ^', 7, and n ; Til [OOOO J thence we may compute the values of P, corresponding to the three epochs 1750, 2250, and 2750. If we represent these values by P, P,, P„, and the general expression of P byf P4./ 'll^'l ^i^. [3854] ^+^•77 + 2 ' dt^ ' ^ we shall have, by putting successively, t = 500, t = 1000, dP. 950000 1 — ^ — + 250000.2- ^^, P =P+ 500.^+ 250000 . 1 . -r;^ ; [3855] Values of dP, ddP. [3856] p^ = p + 1000 . ^ + 1000000 . è • ^ ; ^^855'] Ct Z (t I hence we obtain! dP 4P — .3P — P, ddP P„—2P, + P d t ÏÔÔÔ ' ~dl^ 250000 * (2429) To give some idea of the rapidity with which the terms of the series [3852] decrease, we may take the value of e'" [4407] for the case of < = 1000, and we shall find t .-^=329% — i<2 «=8^; so that the second is about ^V P^rt of the [3853a] (13 e fii-st ; and with the same rate of decrease, the third tenu it^-:r^ will be insensible; [38534] similar remarks may be made relative to tlie other terms of [4407, Stc.]. t (2430) Tlie expression [3854] is similar to [3850a], and by putting, successively, < = 500, f=1000, we get p, P„ [3855,3855']. Î (2431) Multiplying [3855] by 4, [3855'] by —1, adding the products, and then dividmg by 1000, we get — [3856]. Again, multiplying [3855] by — 2, adding [38560] the product to [3855'], and then dividing by 250000, we get jjy [3856]. 68 PERTURBATIONS OF THE PLANETS. [Méc. Cél. 9. The terms depending on the ffth powers of the excentriciiies may have [3856'! " sensible influence on the great inequalities of Jupiter and Saturn ; but the calculation is very troublesome on account of its excessive length. The importance of the subject has, however, induced that very skilful astronomer Burckhardt, to undertake the computation. He has discussed, with scrupulous .ggg„ attention, all the terms of this order depending on the angle bn't — 2n^, neglecting merely those terms which depend on the products of the excentricities by the fourth power of the mutual inclinations of the orbits ; which produce only insensible quantities. The expression of R [3742] [3857'] corresponds to the action of in' upon m ; and the part of the expression which has the most influence on this inequality, is the product of m' by the following factor,* „ J ~.ri-'.\cos.{v'—v) — cos.{v'-\-v)\ [3858] -= — -—==== + 3-. m Vr'^—'2rr'.cos.{v'—v)+r'~ [,■''— 2rr'.cos.{v'—v)^r'^^ [3858'] This factor is the same for both planets ;\ by developing it, and noticing * (2432) If we proceed by a method similar to that used in [3d29«, &.C.], we may prove, as in [3829?i, &c.], that the second and third terms of R [3742], namely, [3858o] J- . — -{cos. (d' — v) — cos.{v-\-v)], do not have any influence in producing terms of the order now under consideration, depending on the angle bnt — 2nt, and by neglecting them, and also the first term of [3742], which is noticed in [3S61, 3868], we obtain the value of -r [3858]. t (2433) As 7 enters into R [3858] only in the even powers, and the quantities [3859a] multiplied by y^ are neglected [3857], the terms of R of the fifth order, must contain factors of the following forms, [38596] e'^ c'^e, e'^e^, t'^c\ e' e^ e" ; y^e'^, y^e'^e, y'^ e' e\ y^ e^ ; of which the six first terms compose all the combinations of e, t', of the fifth dimension, and the remaining terms all the combinations of e, e, of the third dimension, multiplied by 7^ of the second dimension. Now we see, as in [957"", 957''^], that if R contain a series of terms of the form ?;*'. Ar. cos. (5?i'/ — 2nt-\-A), the first term of the series [3859c] will be of the order i' — i = 5 — 2^3, or of the third order ; the second term will be of the order i' — i-\-2, or of the ffth order ; and by noticing only terms of the fifth order, the angles will become, respectively, of the forms [3859]. For in the elliptical [.3859d] motion the angle nf-\-s is always connected with — w, 7i't-\-^ with — «' [669, 957'^'] ; VI. i. §9.] TERMS OF THE FIFTH ORDER IN e, e', 7. 69 only the products of the excentricities and inclinations corresponding to the angle 5 n't — 2 w ^, we shall have a function of this form, R VI ~= N "". COS. (5 «' i — 2 ?U + 5 s' — 2 £ — 4 ^' + -.:) + iV (' ) . COS. (5 n' i — 2 n Ï + 5 a' — 2 £ — 3 ^') -{-N^''\cos.{5n't—2nt + ôs' — 2s — 2^' — ^) + N'-^K COS. (5 n't — 2nt + 5 =' — 2 s — ^' — 2^) + N '^'. COS. (5nt — 2nt + 5^' — 2s—3z^) + iV(^>. COS. (5 7i' t — 2n t + Ô s' — 2 s + z^' — 4>^) + TV (**'. COS. (5 71' t — 2n t + 5 s' — 2s — 2^' + ^ — 2u) + N '-'K COS. (5 n' t — 2 n t + 5 B —2 s —^' —2n) + iV(^'. COS. (5 n' t — 2 71 1 + 5 s' — 2 s — ^ — 2 n) + ^<^cos. (5n'i — 27if + 5s' — 2£ + ^' — 2^ — 2n). and we find* [3858" Forms of the terms in R uf the fifth dimen- sion in (0) [3859] and in tlie terms depending on 7^, the angle 2n't~\-2s' is connected with — 2n; so that if the coefficients of w, -n', n, be represented by g, g", g", respectively, we shall always have, by noticing the signs g -\- g' -\- g" :^ — 3; which is similar to [959], [3859e] changing the signs of the coefficients. Moreover, the sum of the coefficients g, g', g", considering them all as positive, must not exceed 5 [957'"], because the present calculation is restricted to terms of the fifth order. Thus, for example, a term depending on the angle 5 n't — 2nt-\-5^ — 2 s — 5to'+2«, must be rejected, because the sum of [3859/] the coefficients of -n', «, taking them positively, is 7, corresponding to terms of the seventh order. Now a slight examination will show, that the values of g, g, g" , which satisfy the equation g -^ g^ -\- ^' ^ — 3 [3S59e], with the prescribed condition, are as in the [3859g-] following table ; the corresponding numbers being placed in the same vertical lines. These numbers agree with [3859] ; Values of g', _4, — 3, — 2, — 1, 0, 1; Values of g, 1, 0,-1, —2, —.3, —4; Values of g", 0, 0, 0, 0, 0, ; ■2, —1, 0, 1; 1, 0, —1, —2; ■2, —2, —2, —2. [3859;i] * (24.34) The signs of ah these values of a' N^'>\ a! N^'^', &c. [.3860— .3860'"], have been changed from the original so as to correct the error mentioned by the author in [5974, Sic.]. Before the discover)' of this mistake, he had computed and used these [3860a] erroneous values in ascertaining the inequalities of Jupiter and Saturn [4431, 4487] ; hence it becomes necessary to apply the corrections of the mean longitudes, given in [5976, 5977, &ic.]. We have given [3860—3860''] as they were printed by the author, VOL. III. 18 70 PERTURBATIONS OF THE PLANETS. [Méc. Cél. [3860] a'iV('" = — 768" (') (1) (1) 3138 b , — 13 a. — -^ — 1556 o?. —— — 438 a». 1 à — 38a^ (1) cW d a. f/as do.^ Terms of the fînh dimen- siou in r, e', y. [3860'] «'iV(»=_ + e'3y2 384"" (2) — (20267 e' 2+ 24896 ê) .h''— (7223 e'=+ 8 1 44 e^) . a . (2) (2) f/2i, (2) ,3 z^ . < + ( 1 094 €'-+ 3692 e^) . a=. Vf + (482 e'^+ 1 436 e^) . a^ '-^^ ' "° ) a a"' ^ c/ a' (2) (2) + (41 e'"~+ 140 e^) . a\ ^ + (e'2 + 4e^) .«-^^ (3) s 590a.(6^ + 6j + 255a^(^l^ + ^) 0) (3), , (1) (3) 2~ + w.,2 / ~i "" • V~T7ir I f/a2 f/ tt' d a' [38606] [3660c] [:3860d] [36(!0f] [3860/] correcting the signs as above ; but without pretending to verify more tlian one or two terms of each of the coefficients. Tiie calculations of Burckliardt, on this subject, are given in the Mémoires de FInstttnt, T. IX, 1808, p. 59, supp., but generally with wrong signs. From what has been said in the preceding notes [3809a — 38.56rt], concerning the terms of the third order, we may form some idea of the great labor of computing and reducing the terms of the fifdi order [3860-3860''^]. The series [3829^—?», 38346] must be very much increased by the introduction of terms of the fourth and fifth orders ; a table similar to [38350] must be formed, containing terms of the fifth order, depending on the proposed angles and on the powers and products of a^ , a', a", as far as the fifth order inclusively. Then we obtain, as in [38.36f/, 3837c, &ic.], values of iY'»>, JV"', &c., depending on ^''' and its differentials relatively to a, n' ; which may be reduced to the differentials relative to a only, by extending the table [1003] to differentials of the fifth order; finally, by the substitution of the values ./2'*', B'-'\ and then- differentials, in terms of ftj^, èj, and their differentials [996—1003], we get the required values of JV"", JV"', &ic. This short sketch of the method of computing the terms of the fifth and higher orders, must suffice ; more minuteness would be inconsistent with the prescribed limits to the notes on this work ; in which we have proposed to point out and illustrate the methods of computing the various inequalities, by occasional examples, without attempting to verify the immense number of numerical calculations with which the work abounds. VI. i. ^9.] TERMS OF THE FIFTH ORDER IN e, e', y. 71 (3) —(109392e'2+53064e=).&'"— (42368 e'^+23436.e^).a.^ '''^''~ '76^*\ +(1064e'^+2088e^).a^Ç^+(1572e'^+1710e^).a='.'^ (3) (3) + (152 e-+192e==) .a^ i^ + (4e-+6e^).a^^ e'^e^s / (2) (4)\ „ fdb^ dbi. (2) (4) 128 , (2) (4). / (2) (4) 1 da? ^ da? y ' \ (/a» (4) —(42912c'^+199848e'-2).6'"— (21728 e2+82032e'^).a.'" è da. (4) (4) + (116 e'-+210 e'^) . a^ ^ + (4 e^+ 6 e'^) . a^. ^ (3) (5) 580a. (63 + 63) +234a^(^+^ \ z s ' \ da. da. / (3) (5)v / (3) (5), I d'^hs. d^h^\ fd^h^ d^b"\ (51 — (11840e=+152000e'^).6®— (6560e-+65168e'2).a.^ 4 </a g3 ) „ (5) (5) "'^'' ~ 768 • \ — (592 e^+ 4720 e'^) . a^ ^ + ( 1 52 e^ 920 e'^) . a^. ^- (5) (5) + (26 e" + 128 e'2) . a^ Ç^ + (e"- + 4 e'^) . a^. ^ 554a.(6:+6")+222a^(^ + ^-^) ' / (4) (6)v , (4) (6), 384 ' [3860" (4) ,^,«^« o , r^^/^oo «X db^ Terms of , g 1 (4) (4) I the fifth a'iV(^)=__./ _(640e2+2970e'=).a=. ^ + (864e^+1854e'2).al 1_^ sioo in c, e', 7. [3860" [38601- 72 PERTURBATIONS OF THE PLANETS. [Méc. Cél. (6) „ (O [3860^] «'A^W^ 1_^ 768 41448. 6/+ 18392 a.. ^ + 1780 a=^. , , 4 do. «a^ (6) (6) (fi) 156 a^ î-^ — 29 a^ lif — a^ ^ C ,o, (2) (2) (3) [3860V.] a'iV<^)= ^^. < — 85 a . 6 3 + 85a^^+ 21 a^. ^ + a^ 1-1^ 1-28 ( ^ a a «/a-' rfa^ (3) (3) f] 7, 3 3 2 Terms of the fiflh dimen- sion in 1 e, e', r. \ (56 e^ + 842 e'=) . a . 6 3 + (4 e^ + 87 e'^) . a^ -ip [3860vii] d' ]\fO)^ <^^ ■ 128 1 (3) (3) -(16e^+20.a3.^-(2e^+e-).a^Çi| (4) _ _ (174 e^ 196 e'^) . a. 6*^' + (50 e-+ 180 e'~) . a=, '^ [3860vi.i] ^('^^(S)— IZ!. 128 \ W „ W 1 .3îÇ^ + (2e-+e^).a^îÇM + (14 e'^— e^) . «.3. — f + (2 e'^ + e"^) . a^ e'e^ yS ^ (6) (5) (r.) ■\ [3860U] a'iV<«)=:l^.<580a.èr+86a^^— 8a3.1!ij_a^^i •. ( ? eta da- «/a-*) When we consider the action of m! upon m, we must augment «W"' [3860], by increasing h with the term r? or — a [3743], which increases 5 (t 3125 a. c'*e a'N'-"'' by —^ .* When we consider the action of m upon m', 768 '■ [3861] * (2435) In [996], we have, generally, — . è ' = — ^''' ; but in the particular case [3861a] of 1=1, this becomes, as in [997], -.Z."' — -^ = — A^^K The part -^^ being introduced by the tenn -^^ .cos. {n't — nt -\- s' — s) [954], which does not occur in the terms noticed in the value of R [3858], so that wherever the quantity — , •^'', occurs, [38616] we ought to add ; or in other words, b ought to be increased by the term , , fit * u Ct or — a . To notice this circumstance, we must apply a correction to the vakie VI. i. §9.] TERMS OF THE FIFTH ORDER IN e, e,y. 13 , , jti) , 1 , . , . , ,T,n^ 1 SOOe'^e we must add to b, the term = ; which increases a N'-"' by -;.^ „ • [3862] •i a-* 763 a-^ This behig premised, we shall multiply the preceding values of a' N'-°\ a'iV''', &c. by m', and shall reduce each of the cosines by which they are multiplied in the function [3859], into sines and cosines of [38(32'] 5 n't — 2nt-{-5s — 2s; Avhich gives to this function the following form,* Value of ..'/?= m'.a'P,.sm.(ôn't — 2nt + 5s'—2s) «• [Action of m' on ml. [38631 + m'. a' P;. cos. (5n't — 2nt + 5 s'— 2 e). We shall likewise multiply by m the values of «'iV<% rt'iV*'', &c. relative to the action of m upon m' ; and shall reduce the sines and cosines of a' N'-'^'' [3860], which may be computed by supposing II = — a, whicli (11 U) (Z 6 1 (id h jL gives "^ =^ — 1) "d'^'^^' ^'^' Substituting these in [3860], It becomes _ '^ . |_3138 a + 13 a? = ^J^Ë±iîll , [3861c] /Uo /Do as in [3361]. When we are computing the action of m on m', the fonnula [3861a] becomes « * a- a' I i cfiS a' I i S' SO that the correction of è^j' is — a-^, and the correction of a'./V"'* for this case, will be found by putting &j = — a.-- in the expression [3860]. Now this value of 6 j gives ,,(1) (1) (1) (1, substituting these in that expression of a' N ^^\ it becomes ~ïml^-^~^^^^ — ^ X 13 + 6 X1556 — 24 X 438 + 120 X 38 — 720}='|^, as in [3862]. * (2436) The reduction here used is the same as that in [3842J, &c.], by which tiie fonction [3835] is reduced to the form of [3842n'], and were it not for the terms [3861,3862], the values of P,, P/ [3863] would be identical with P„, P,,' [3865], respectively ; for the factor [3358] is the same for both planets ; and the reasoning made [3864a] use of in [3846a— ^] will serve to prove, in [3863, 3865], that P,, P/ will be respectively equal to P„, P/, if we neglect the ternis [3361,3862], and we shall show, in [3866i], [38646] that these ternis do not affect the result. VOL. III. 19 74 PERTURBATIONS OF THE PLANETS. [Méc. Cél. [3864] of the function [3859] to sines and cosines of brilt — 2/1^ + 5=' — 2;; which will give to it the following form, Value of R. 'R = m . «' P^, . sin. (ôn't — 2nt + ôs'—2s) [38651 [Action of «I on m']. ^ +m.a'PJ.cos.(5n't — 2nt + ôs'—2s). We shall then substitute these values successively, in the expressions of 6v, 6v', of the preceding article [3844,3846], neglecting their second [3865'] differences, because of the smallness of these quantities ; and in this way we shall obtain the parts of the inequalities of Jupiter and Saturn, corresponding to the angle 5 n't — 2nt, and depending on the powers and products of the excentricities and inclinations of the orbits of the fifth order. We may here observe, that in consequence of the ratio which obtains [3806] between the mean motions of Jupiter and Saturn, we have 3125 a^^ 500;* ji' 2 n' ** 4 [3867] for a^==— and on' is very nearly equal to 2n; consequently ■- ^ = — . 71" 11" /Co Hence it follows, that the value of a' N *°^ is the same, ivhether ice consider the action of m' upon in, or that of m upon m'. Hence we may deduce the preceding part of 6 v' from the corresponding part of 6 v, by multiplying [3868] the latter by — ^J'~ . -.f [3806a] [38666] [3868a] [38686] * (24.37) We have nearly l=7v^a^ = n"' a' ^ [3109']; hence iL.=-^^=:a3 [,38296] ; n' 2 /n'\2 4 but by [3318f/], we have nearly 5n' — 271^=0, or - = -; therefore a^=(-)=— , 31 O \7l / fit) as in [3867], and 3125a^ = 500, or 3125 a = '—j- ; substitutmg this in the increment of a'JV^"' [3861], correspondmg to the action of rn! upon m, it changes into the expression [3862], representing the increment of «'JV'"' in the action of m upon m', as we have remarked in [38646]. ■j- (2438) If we multiply the factor — '— — - , connected with the chief term [an' — 2 rap of ^t; [-3844], by tlie quantity — ^ ' „ . - [3868], the product becomes im.n- a ^ \5m.n"^ a' 15m.n'~ 1 (5n'— 2>!)3 ■ a ~ (5n'— 2n)3 ' a ' j the same as the corresponding fac the other part, -, being multiplied into the terms aP, aP', adP, adP', kc. [-3844], in which the part — — is the same as the corresponding: factor of the terms of i5 y' [3846] ; (on — 2n)3 X o VI. i. §10.] TERMS OF THE THIRD ORDER IN MERCURY. 75 10. In the theory of Mercury disturbed by the Earth, we must notice the ine([uulity depending on the angle nt — 4 n7 ; because the mean motion [3869] of Mercury is very nearly four times that of the Earth [4077a]. Supposing inequaiuv m to be Mercury and m' the Earth, we shall obtain the proposed inequality Siércuô" by putting i = 4, in the expression oi àv [3817]. Considering the [3870] extreme minuteness of this inequality, we may neglect all the terms dP (IP' depending 071 -r-, ^— , and retain only those having the divisor (n— 4?i'/. [3871] Hence we shall get* iv = , "'•"" .iaP'.sin.fa^— 4rt'^ + s— 40+«P-cos.r?z^— 4n'^ + -— 40i- [3872] (« — Any ' ^ ^ ' ' We can easily determine P and P' in the following manner. We may T S T calculate, by formula [3711], the value of — g-, corresponding to the angle -I n't — 2nt, by substituting in it i := 4. Hence we obtain a [3873] value of —5- of the form,t «- ^-: L . e^cos. (4n'i — 2nf + 4s' — 2e — 2w) + L"'.ee'. cos. (4n'« — 2w^ + 4s' — 2s — ^ — ^') + U^K e'K cos. (4 n't — 2 n ^ + 4 s' — 2 s— 2 ^') + U^\ y"~ . cos. (4 n't — 2n ^ + 4s' — 2 £ — 2 n). We shall then observe, that this value of ^ results from the variations (r of the excentricity and perihelion, depending on nt — 4<n't, in the elliptical [3874] produces the corresponding expressions a P, a P, a' d P, a d P', he. [3846] ; the values P, P' of S v', having been proved in the two last notes to be respectively equal to those of P, P', in S v. [3868c] * (2439) Neglecting dP, dP', ddP, ddP, and H, in [3817], and putting z'=4, [3872a] we obtain the expression [3872]. t (2440) The two first of the angles [3874], connected with e^, e e', are explicitly contained in [3711] ; the others, as well as these two, are included in the form cos. \i .{n't — nt-[- s— s)'\-2nt-\- K\, [3873o] which occurs in [3711], and is developed in [3745 — 3745'"]. 76 PERTURBATIONS OF THE PLANETS. [Méc. Cél. [3875] expression of ^.— . This expression contains the term — c.cos. (n/+;— cî), whose variation is* T Ô T [3876] — ^ = — èe . cos. (n Ï + s — ra) — eôzs . sin. (n t -\- s — ûî) ; 6e and (5w being the variations of e and ^3, depending on 7i t — i'li't. [3876c] * (2441) If we square the value of r [3701], and substitute cos.^ {7it -\- e — ■ui)=zi -\-^ COS. '2.{nt -\-s — k), we shall get r3=a2.{l + f c^— 2c.cos.(ji?+s— to) — ie-.cos.a.Cnf + e — ro) + &;c.|. [3876rt] In the troubled orbit the elements r, a, e, s, ss, n t, are increased by the variations [38766] 'J''; <5 «) Se, Sis, Sv, respectively; and if we neglect the squares and products of these variations, the increment of the preceding expression will be found by taking its differential relatively to the characteristic S ; hence we get 2riir='iaôa.\l-lr§ c^— &,c. \ -j- a-. \3 c 5 c — 2(Se. cos. («<-)-£ — ûj) — 2 c ô a . sin. {n t -\- s — zs) — &c. } . Dividing this by 9 a^, it becomes of the form r 1' [3876rf] — 2 = — Se. cos. {nt-\- s — to) — e f5 to . sin. {n t -{- s — to) -(- X ; representing, for brevity, by the symbol X, all the terms of the second member, excepting the two parts explicitly retained by the author in [38T6]. If we neglect X, and substitute '-■ '*''•' in the remaining terms the values of Se, e o a [3877, 3878], we sbal] get the expression of — [3879], which the author supposes to be identical with [3874], and thence by integration obtains Sv [3882]. In the Memoirs of the Astronomical Society of London, Vol. II, page 358, Sic, Mr. Plana has pointed out some defects in this method, and ha? shown, that the terms depending on X materially alter the result. To prove this, he has computed directly the terms of Sv depending on the divisor [n — 4 n')^, using formulas similar to those in [3335 — 3841] ; which we shall give in [3881r — w'] ; after going over [.3876?] the calculation by the method of the author. From the comparison made in [3883w, y], it appears, that this method of La Place cannot be considered, in an analytical point of view, as a very near approximation to the truth ; though he seems rather unwilling [3876/t] to admit the fact, in a note he published on the subject in the Connaisance des Terns, for 1829, page 249. [3876/] VI. i. §10.] TERMS OF THE THIRD ORDER IN MERCURY. 77 We shall have, by [1288, 1297],* ôe= __ , ■< i j^j.sm. {4nt—nt -{-As — i)-\-l--- ). COS. {in' t — nt-\-4£— s)Ç; [3877] y COS. (4 n't — nt+4 s'— s) + ('^^ . sin . {4n't — ni -{-é^—s)l; [3878] yn'.an C /d P e a = j— , . < — I -^ n — 4 H ^ \ a e hence the variation of — e.cos. (ni + ' — ^) becomesf r^r m'.a?i Ç/dP :£;^,.j('^).sin.(2n<-4n'i+2£-4£'-«) — ('^Vcos.(3n<-4n'<-f2s-4£'-î:r)l. [3879] This function is identical with the preceding expression of -y [3874] ; therefore if we change, in both of them, 2nt-\-2s into » ^ + s + ^ + - , [3880] V being the semi-circumference, we shall obtain J T-;- ^ -T- ).cos.rwi — 4w'^+£ — 4/)+ ^— ) . sm. (îi^ — 4n'i + £ — 4s')^ M — 4w i\de/ ^ \de J ^ ') = L . e^sin. (4w'i — wi + 4E' — 5 — 3ot) + L" '. e e'. sin. (4 m' i — n i + 4 a' — s — î^' — 2 ^) [3881] + L(=>.e'^sin. (4?j'^ — n^+4a' — s — 2a' — w) + L'^>.7-.sin. (4n7— >i^ + 4s' — E — 73 — 2n). [3877o] [3879o] * (2442) The expression of R [12S7] is the same as in [.3810] ; so that P, P' have the same values in both formulas. Now putting t' = 4, » = 1, (j.= 1 [-3709], in the expression of à -a [1297], and then multiplying it by e, we get the value of e5a [3878]. The variation i5e [1288] becomes, by similar substitutions, of the same form as in [3S77]. t (2443) Putting, for a moment, Ati! t — nt -\- A^ — s^rT^, nt-\-e — -^s^zW; then multiplying [387'i] by — cos. ?r, also [3878] by — sin. ?r, and adding the J* 6 J* products, we get for the second member of [3876], or the value of —^ , the expression [3879e] ; reducing this by means of [22, 24] Int., it becomes as in [3879c], which is equivalent to [3879] ; '■^='^^,-\(^)-i-sm.T,.cos.W+cos.T,.sm.JV}-(^\{^^^^ [38796] t (2444) We have two expressions of — [3874, 3879], depending upon the angle 2nt — Aii't, and it is evident, that if it were not for the terms produced by the VOL. Ill, 20 78 PERTURBATIONS OF THE PLANETS. [Méc. Cél. [38S1'] If we integrate this equation relatively to e,* and then multiply it by 11 — An we shall obtain I\L . eKsin.(4^n't — nt-\-4,£' — s — 3^) + 1LW. e^e'. sin. (4n'ï — ni + 4s' — ; — ^' — 2^) + L(^'. ee'~. sin. (4 n'^ — Ji ^ + 4 s' — s — 2 ^' — ^) + L<^'. ey-.sin. (4n'/ — wi +4£' — s — ^ — 2 n) function X [3876e], they would be identical ; therefore they will still be equal to each other, if we change the angle 2nt-\-2e into 7it-{-e-\--s-\-^ir. Now if we make this change in [38741, we shall find, that a term of the form cos.(4n't—2?ii-\-'ie' — 2s4-A), becomes [38806] L J' cos.{An't—nt-\-4£' — i-{-A — zi — i •r) =^ sin. (4 7i' t — n t -\- 4 (' — s-\-A — ■si); and the second member of the expression [3874] changes into the second member of [3881]. In like manner, sin. (2 ?i ( — 4 n't -{-2 s — 4 s' — w) becomes [.3880c] sin. {nt — 4 n' t -\- s —■ 4 s' -\- ^v) — cos. {nt — 47i't-{-i — 4s); and COS. {2nt — 4n't -\-2s — 4 s' — «) becomes [3880d] COS. {n t—4n't-\-s — 4 s'+ è *) = —sin. {nt — 4 n't -\-s — 4 s') ; hence the second member of [3879] becomes as in the first member of [3881]. * (2445) Multiplying the equation [3881] by de, and then integrating it relatively to e, in order to obtain the values of P, P', we get J!!Î:^Ap.co5.(nt — 47i't-\-B — 4s')-\-P'.sm.{nt — 47i't^s — 4s')\ n— 4n' i > = iL . e^ .sm.{4n't — nt-{-4s — s — 3-a) [38816] + i L^^'. e^ e'. sin. {4n't — nt-\-4 s'— s — u'— 2 ra) + L<2'. ee'2. sin. (4 7i't — n < + 4 e'— £ — 2^— tn) -}- L'-^1.ey^.sm.{4n't — nt-\-4e — s — -us — 2n). 3n The first member of this expression being multiplied by _ , , becomes equal to the value 0Î Sv [3872] ; therefore 5 v will be obtained by multiplying the second member [3881c] of [38816] by — — ; and in this way we obtain [3882]. In the integration relative to e [3881a, 6], we may add terms depending on e'^, and e 'f, which are considered as constant in the integrations ; but the excentricity of the Earth's orbit e', being only [3881rf] about -rV of e [4080], the term depending on e'^, must be much smaller tlian the VI. i. §10.] TERMS OF THE THIRD ORDER IN MERCURY. 79 In this integration, we neglect the terms of P and P' depending on [3882] [3881e] [3881/] others ; and the same remark will apply to the term depending on e' 7^. The author has neglected these terms, because they are so much less than those which are included in the expression [3882]. Having followed the author in this indirect method of computing the value of <^v [3882], we shall now proceed to the direct investigation of the same inequality. For this purpose we must have an expression of R, similar to [3835], depending on the angle 4 ti! t — nt. This expression is evidently of the following form, R = M«» . e" . COS. (4 71 1 — 71 1 + 4 e'— s — 3 z>') + Jtf "' . c'^e . COS. (4 ?i'< — ?i < -f- 4 e'— s — 2 ra'— -55) + M'-> . e' c2 . COS. {An't — 71 1-{- 4 s'— s _ ra'— 2 ts) + JU"'. e» .COS. (4?i'< — ?8/ + 4£'— £ — 3«) + JJf («. e' f. COS. (4 n'< — ?U + 4 e'— s — a'— 2 n) -\-M^^\ey^. COS. {A7i't — 7it-\-4^—s — zs—2n); but the factors JW«>, M'-^\ he. are different from those in [3836, Sic] ; we shall give their values in [3S8lr — !«']. If we suppose, for a moment, the preceding expression of R to be put under the form R= 2 M .cos. {4 n't— Tit -{- K), we shall have d JÎ = )! 2 M . sin. {4 n't — nt + K) [916']. Substituting this in the expression of the ^^ mean longitude ^ [3715/], we shall get the corresponding term, Sv = 3rrandt.àR= — -^^„.:sM.sm.(4n't — nt-i-K); [388U] •''' [4n'—nf therefore the value of 5v may be easily derived from R [38S1/], by multiplying it by ; , and changing the cosi7ies into sines. The terms of jR may be very [3881il easily obtained from the values of Jkf'"', M'-^\ he, computed in [3836(7— 3840o], by merely decreasing the value of i by unity ; so as to change the angle 5 n't — 2 7it into 4 n't — nt. In this way of computing M'-°\ we must use the decreased value i=\ [3836a], and then [3836(Z] becomes as in [3881r]. In computing .M"' from [3881/] [3837c], we have the decreased value i^2 [3837a]; hence we get [3881s]. [.3881jn] From [3838e, A], we get the decreased value i=3, and JF-' [38810- From [3839«, b], [3881n] we get the decreased value i = 4, and M'^' [3881m]. These expressions are reduced, in rgggj ^ the fii-st place, by means of the formulas [1003], and then by [996—1001] ; so that we finally obtain the values [3881r', s', t', u']. Observing, that in computing JIf ' [3881/], we must (1) notice the increments of 6, and , represented by — a and — 1, respectively, [3881o] ' da. as in [38616 — c], by which means we shall obtain the first term, — —,.\—256o.l, [S88U] 80 PERTURBATIONS OF THE PLANETS. [Méc. Cél. [3883] e'^ and e' y- \ but as the excentricity of the orbit of the Earth is quite small, in the expression [388 b'J, which is omitted by Mr. Plana by mistake. In like manner, [3881o'] from [3840Â] and the decreased value z — 3 [3840^], we obtain Jl/™ [.3881 «] ; also from [3881;>] [3840rt] and the decreased value i=A [3840m], we obtain M'^) [388 1 w] ; which, by similar substitutions [1008, Sic], are reduced to the forms [3831t)', w']. In making these [3881?] successive reductions, we have used the abridged expression [3755a], ./2 '"=«'" '3 rfa'3 [3881.] JIf .0,^ ^ . 564^<"-48«'. C-f^) + 12«-. f'-^Ua' 48 ^ \da' J ^ V f'a'- / ^^, C64^'i> + 48.[^'» + ^/')] + 12.[2^(" + 4^/» + ^3W] "^^'1 + [6.^W+18^i(')+9.^a<"+^3("] = J . ^ 14-2 ^'"+114^/"+ 21 ^3<'> +^3(1' I ,0) (1) (IV [3881,-'] = -^ 5_256a+142è!' + 114a.'^ + 21a^ '"' ^ ^^-^ '''' ' 104^^-' + 26a.(^)-40«'.('if!^Ul0a'«.r^^ \ da J \da / \dadaj 104 ^'2) _^ 26 ^^(2) _j_ 40 . |-^(2) _|. ^^(3)-j _^ 10 . [2 ^/2) ^ ^ ® ■] + 4 . [2^«>+ 4 ^p + ^2^2,-] _|. |-6 ^^(2)+ 6^3(2)+ ^3« 152^«)_)_ 108 ^ ®_|_ 20 ^3'2i_^ _^^c3)^ [3881«] J/(i> = — -. 16 ÏG (S) <2) .„,(2) __(2) [3881(] 3f(2)^ ^ [3881,'] = J!-,.jl526r+108a.liL + 20a2.^+a3.^ii '-•«"'--«■•C^')+-«-(^')--«-G^) C 126^"^+ 21 . [^(3)+ ^ ffl-| ^ 60 ^/='+ 10 . [2 ^/3' + ^2i3)j (+6^3^=1+ [3 ^2"'+^3'3>] I 147 ^'■»+ 101 ^ o)_|. 19 ^^(3) _^ _^^C3,| ÏG m' 16 , C „v (3) (3) (3) [3881f ] = _ -^, . j 147 6, + 101 a . !iil- + 19 a=2. ^ + a^. ^^ ; VI. i. §10.] TERMS OF THE THIRD ORDER IN MERCURY. 81 in comparison with that of Mercury, and the inequality in question is very [388!?] f (4) (4) (4) -J ..».= |....^_5B».+..(i^)^='^...,.^-5««+[-3i,...-„(';£:>)] IG t ' \ da J) W o' C f ' da. ^' [3881t)] [3881w'] JIf (5)= -^. a'a . ^ 5 B® + « . ( — ) (■ [3881i<;] I 5 6 3 4- a . -—^ C • L'^ooiw) j 16 " a' ■ C ^ '/■^ (1) rfSfe. If we substitute in these the numerical vahies [409.5 — 4102'!, also ^ = 5,340815, da? (2) a.-^-^^ 1,96112, given by Mr. Plana, in Vol. II, page 366, of the Memoirs of the Astronomical Society of London, we shall obtain, by supposing «'=1, a' JH'0'= — m'. 0,-3411; a'JW*'>= m'.3,3192; a' M'2'= _m'. 1,4808 ; o'J»f(3i== m'.0,2181; a'JJf ^4' = — >«'. 0,1921 ; a'^®^ m'. 0,0690. [3883a] [38836] The last four of these numbers agree nearly with those given by Mr. Plana; but he finds «']»/<«' = — m'. 2,40567, a'.y»f(» = m'. 2,94.30 ; so that he makes .M'"' seven [3883c] times too great, and .Af <" about a seventh part too small. The first of these mistakes arises from the omission of the term — 256 a [388 lo] ; the second is an error in the numerical calculations. We must observe, that the indices of M in La Place's notation, namely, 0, 1, 2, 3, 4, 5, correspond, respectively, to 3, 2, 1, 0, 5, 4, in [3883d] the notation used by Mr. Plana. In computing the value of 5 v, Mr. Plana uses the elements coiTesponding to the year 1800, namely, e'= 0,0163.5.32; e = 0,2056163; 7 = tang. 7' 0" 6' ; w' = 99'^ 30™ 5' ; [3883e] «=74'' 21™ 47^; n = 4.5''.57"'3P ; «'=1; « = 0,.38709; and ?i', w [4077] ; .329630 t^^^^3 '"^ 35^36' he also reduces the mass m' from .JoqV^ [4061] to ^, which makes [3883/] VOL. Ill, 21 82 PERTURBATIONS OF THE PLANETS. [Méc. Céî. [3883"] small, we may neglect these terms without any sensible error [388 Id]. H."=— 0,0713 [42.30']; then by the method [3881 î], he finally obtains [3883g-] 6v = 0^5596 . sin. {4:n't — nt-\-4s — i—l6'' 59" 20"). If we correct the errors mentioned in [3883c] ; also another error, in his substitution of the value of 2 n, which is taken too small by 40'', in [3881/] ; it will become [3883^] Sv = 0',61 . sin. {An't — nt-^-A s — s — 21'' 19"=). This differs but very little from the computation of La Place in [4283], namely, ôv = (1 +|x") . 0',69 . sin. {4n"t—nt-\-4^'—s~19^2'^ 1.3') [3883t] = 0-',64 . sin. (4 n" < — 71 f + 4 e"— s — Id'' 2'" 13') [3883/]. Notwithstanding this near agreement in the numerical results, the method of La Place is essentially defective, as may be seen by comparing the term depending on e^ in the expression [3881i,/], namely, [3883i] Sv = —~~ . JJf (31. c». sin. (4 n't — nt-}-4s'—s — 3 o), with that given by La Place in [3882], t-'^'^^^'] ê V == ". , . L . c^.sin. (4n't — n t + 41'— s — 3z:). [38S3n To compute the value of L, we may observe, that L.ë'.cos.{4n't—2nt-{-4e'—2B — 2zs) is the term of ~, depending on e~, in [38741. Now the term of — [3711], [3883m] ai ^ ^ a'^ <- ■> corresponding to i=4, and having the divisor 4n — n, is [388.3n] 4'l?3^-«-^^+«-(77' -r- . a Jli 4- a^.[—-) -rpL , \?±J . n^. COS. (4 n'< — 2m < + 4 s'— 2 5 — 2 s) (4n — î!.).(4n— 3n) ^ ' ' ■ and as we retain here only the terms depending on c^, we may put M^=M^^'^ e^ [.3703,3745]; moreover, we have, in the present case, very nearly 4 ?i' — 2n = — n, 4n' — 3n = — 2n [3869] ; hence this term of ^ becomes \4aM-^+a^.('-^\\ [38830] _(__ \ da /) ,nc'.cos.{4 7H — 2jit + 4i'—2s — 2^). 2.(4J^'— n) ^ ' ' Now we may obtain the expression of M'"' [3S8.3p], by putting i=^4 [3883m], in [3750], The partial differential, relative to a, is as in [38837]. Substituting these two values VI. i. §11.] TERMS OF THE SECOND ORDER IN THE LATITUDE. 83 11. It follows, from [1337'— 1342], that the two terms of R [3835], represented by R= M^*K e' r. COS. (5 n't — 2n t + 5 s' — 2s — zs'—2n) [3884] + M^'\e f- cos.(5 n't — 2ni + 5 s' — 2s — z: — 2n), in the first member of [3883;-], and making the same reductions as in [999, &c.]. we get [3S83«], by putting «'=1, r (4) (4) (4) \ = — .)176è +n4a.-— 4-20a^. -j-^+a^.-— --( . [3883*] Substituting this in [3883o], and putting the resuk equal to L . e^ COS. (4 n' i; — 2 » ^ -f- 4 £'— 2 c- — 2 tn) [3883Z'], vre get r (4) (4) (4) -s L^^:^ 176è';+114a.^ + 20a^Çf + a3.^% ; [3883^ 16.(4)1 — n) C 2^ da. ' dofi da-' ) "■ J consequently the part of 5v [3883?], computed by La Place, is C (4^ (4) W 5 16. (-In'— nf ( 5- ' da. ' daS ^ rfaS V whereas the real value, obtained by the direct method [3881i, m'], is 3, = _ "•"";•-" „ .^1.366l' + 93a.^+18a^.^+a3.^^ [3883.] , , ,. , r5 . ) 1.36 6 1 + 93 a . -^ + 18 a^. -7-^- + «-'• -7—0 10.(4,i'— n)2 ( 4 ' da ' da^ ' rf a^ If we substitute in these expressions the values given in [4095, &c.], we shall find, that the coefficient of — — ^ '■ , in the first is 12, 54, and in the second 10, 50; so [3883jc] 16. (4 n' — n)2 that La Place's method makes this term too great by about one fifth part ; and the same [.3883i] discrepancy occurs in the coefficients of most of the terms of these two formulas. 84 PERTURBATIONS OF THE PLANETS. [Méc. Cél. produce in the value of s, or in the motion of m in latitude, the inequality,* [3885] 6 5 = — — -; — - • < > • 5n — 2?i ^ _j_ 3j(5) _ f ^ _ 5;„_ (5 ,^/^ _3nt + 5s'—3s — -a—n) ) Moreover the same terms produce in the value of s', or in the motion of m' in latitude, the inequality f 2a'»' m S ^''^•e'7-sin.(4n'<-2«^ + 4a'_2s-^'-n)^ [3886] i s = ■^, — . - . < > ; 5n'— 2?J m ( j^ jyj^^K ey . sm. {47i' t—2n t -\- 4 s'— 2 s — zs — n) ) There is a small inequality in the motion of the Earth, depending on the same angle nt — 4.n"t, given by the author in [4311]. He seems to have computed it from [3883y] the term for Mercury [4283], hy means of the formula [1208], ôv" = — 5v.~^,, using (5« = — 0',690412 [4283], and the other elements [4061, 4079]. This method will answer, as the inequality is extremely small. * (2446) Putting;, in the term of iî [1337"], tang. 9/ = 7, it becomes [.3885o] R = m k . y^. cos. [i' n' t — int -\- Jl — g ()/) ; [3885a'] comparing this with [3884], we get 5" =2, d/ = n, j':=5, i = 2; also in the first term, m'k = M^'^'^ . e', ^ =5 s' — 2 £ — s/ ; and in the second term, mk = M^^'^. e, [388561 •' i^ = 5 s' — 2 s — ro. Substituting these in [1342], which is obtained from the integrals [1341«, 1341], we obtain in s, from the first term, the quantity [3885c] _-l^.J»fW).e'y.sin.(5rt'/-2«^-« + 5 3'-2.-^'-n); and from the second term, the quantity [3885d] --^^^^.M^'\cy.^m.{bnt-2nt-v-\-b^'-2^-^-U); observing, that (j,=:l [3709]. Putting, in these, for v, its mean value nt-\-e [3834], and connecting the two preceding terms, they become as in [3885]. t (2447) The terms of R [3884], used in computing s [3885], are deduced from the fonction [3831], which is multiplied by the factor or mass m. In computing the [3886o] value of s' , corresponding to the planet m', and to the same angles, we must use the factor m, instead of in ; therefore the value of R to be used in computing s', is equal to the function [3884], multiplied by — ; which amounts to the same thing as to change [38866] M^^\ M^^\ into — ,.JI/^", and —,.M^^\ respectively. VI. i. §11.] TERMS OF THE SECOND ORDER IN THE LATITUDE. 85 n being, as in the preceding inequality of s, the longitude of the ascending rgggg,, node of the orbit of in! upon that of m. These are the only sensible inequalities in latitude, in the planetary system, depending on the product of the excentricities and inclinations of the orbits. We have seen, in [3800], that the value of 5 s produces in the motion of m, reduced to the fixed plane, the term — tang. ? . 6 s . cos. {i\ — '') ; [3887] by substituting the preceding inequality of s [3885] in this term, we shall obtain a term depending on bn't — 2 n ^ , which must be added to the If we now compare the value of s [-3885] with the vakie of R [3884], we shall find, that s may be derived from R, by multiplying it by '■ — ; then integrating relatively [388(;c] to t, as in [.38S5è, &c., 1341»], and after integration, decreasing the angles by the quantity i' — n [3885c], or by its mean value nt-\-s — 11. In like manner, we may derive s' [3886rf] from R [3884], after multiplying it by the factor ^ [38866]. This value of ^,.il is to be multiplied by — '- — , to correspond with [3886c], and it will become „ ( JW<«.e'y.cos. (.5?i'^ — 2?U + 5£'— 2 e — ra' — 2n)) -2a'ri'.f/^.-,.^ >; [3886e] -(- JIf ^5)_ e y _ COS. (5 ?t'^ — 2 Ji < + 5 e'— 2 £ — « — 2 n) ) m and then by integration, we get 2a' n' m <» iV/«' . e'"/ . sin. (5 n'^ — 2 71 i! + 5 s'— 2 £ — ^' — 2 n) 5ji' — 2îi ' m' ' -\- iW^s). g y . sin. (5 „'^ _ 3 „ i _|_ 5 £'_ 2 c _ js — 2 n) [388(;c'l The angles bnt — 2nt + 5s' — ^2£ — is — 2n, Sic, must now be decreased by v — n'=nt-\-s' — n', corresponding to the planet m', as in [.3886c/]; the angle n' being the longitude of the ascending node of the orbit of 7n upon that of m' ; in the same rsgsry] manner as n [3746] is the ascendirig node of m' upon that of m ; and it is evident, that n' = 180'' -f n ; hence v'—U'=n't-{-s'—n— 180"'. Subtractmg this from the angles which occur in [3886e'], it becomes 9«'„' ™ C M^'^\e'y.sm. (4n't — 2 nt-{- As — 2 s — zj'—n + ISO")) 5«'_2n m' ^ _j_ J/(5)_ g ^ , ^j,,, (4 ^'^ _ 2 „ ^ + 4 a' _ 2 £_ « _ n + 180") which is easily reduced to the form [3886]. VOL. III. 22 86 PERTURBATIONS OF THE PLANETS, [Méc. Ce). g great inequality of the motion of m; but this term is insensible for Jupiter and Saturn.* * (2448) The functions 5 s, ôs' [3885,3886], reduced to numbers in [4458,4513], [3887o] j^j.g Qf jjjg Qj.£jej. 3i or 9' ; these are muhiphed by tang. 9 in [3887], and as this tangent is very small [4082], these terms may be neglected. VI. ii. sU2.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 87 CHAPTER II. INEaUALITIES DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 12. The great inequalities which toe have just investigated, prochice other sensible ones, depending on the square of the disturbing force. We have given the analytical expressions in [1213, 1214, 1306 — 1309] ; and it follows, from [1197, 1213], that if we put the great inequality of Jupiter ^= H. sin. (5 n't — 2nt + ÔB — 2s+ Â), [38891 we shall have e^^juaiu!"; of Jupiter. ^^^_^. (^'»V«;+^"'V«) .sin.2.(5u'^-2n^ + 5a'-2s + J), [3890] 8 m ya for the corresponding inequality of Jupiter, depending on the square of the disturbing force* This inequality, like that from ivhich it is derived, is to [3890] be added to the mean motion of Jupiter. In like manner, if we put the great inequality of Saturn ^'= — ïï'. sin. (5 n't — 2nt-\-ôs' — 2s+Z')» [3891] we shall have ^Sties of Saturn. 6v'=--. ^- !^-—L )LJ . sm. 2.(5n't — 2nt-i-5s'—2;+A), [3891'] 8 m y a ^ * (2449) The great inequality of Jupiter is found, by substituting, in ^ [1197], ,A=1 [.3709], also i = 2, i'=5; and if we put 6m'.an^k ^ = 5.^— 2e + :4, T,= 5n't — 2nt + 5^ — 2s, ^=— (5n'-2n)g ' ^^^^^''^ we get ^ = H.sln. (jr5 + ;i), as in [3889]. Making the same substitutions in the [3890c] terms of the second order [121.3J, it becomes as in [3890]. 88 PERTURBATIONS OF THE PLANETS, [Méc. Cél. [3891"] for the correspondmg inequality of Saturn,* which must be added to the mean motion of Saturn. The variations of the excentricities and perihelion may introduce similar inequalities in the mean motions of the tivo planets. To determine them, [3891"'] we shall ohserve, that if we notice only the cubes and products of three dimensions, of the excentricities and inclinations of the orbits, we shall havef [3892] 3a.ffndt.dR = — Ga m'.ff^i- dt". P.cos.{5n't—2nt + 5s'—2s) — P'. sin. (5n'<~9n t + 5s'—2 s) [3891a] [3891&] [3891d] [3891e] [3892ol [38925] [3892c] * (2150) Substitutmg ^ [3S90c] in [1208], we get the great inequality of Saturn ^' = mi/a — . ,„ , _, -,.H.sm.{T.^-JrJl); m' j/a' putting tliis equal to the assumed value [3891], we obtain — , mv/o _ _ H =-—-,. H, and jI = J1l. m\/a Now by comparing the two formulas [1213, 1214], we find, that the part of the great inequality of Saturn, depending on the square of the disturliing force, is equal to the m\/a [3891c] corresponding part of the great inequality of Jupiter, multiplied by — , , , using the expression of this inequality of Jupiter [3S90], that of Saturn becomes and by _g TOv/a (2m'\/a'-\-5ms/a) m' y/a! m't/a' sin.2.(7;+:5)^i^'-. '^-'^;^-^°U in.2.(7; + :^); the second of these formulas being deduced from the first, by the substitution of H [38916]. This last expression agrees with that in [3891'], except that Â is changed into JÎ', so as to make both the expressions [3S91, 3891'] depend on the same argument; observing, that these quantities are very nearly equal to each other, since, in the year 17.50, we have ^ = 4''22'"2P [44.34], and .1'= 4'' 21'" 20" [4492]. f (2451) The part of R depending on the angle Zn't — 2nt, and terms of the third degree in e, c', y, h.c., is given in [3842«']. Its differential, relatively to the characteristic d [916'], is àR = — 2nJ.ndt.\P. cos. % — P' . sin. %]. Multiplying this by Za.ndi, and prefixing the double sign of integration, we get [3892], which represents the part of 5v [3715i], depending on diî, the divisor \/(l — e®) being neglected, as in [3718']. The quantities P, P', which occur in this expression, are, given in [3842, 3843], in terms of the elements of the orbits of m, m'. VI. il. §1-2.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 89 which gives, in 3 a .ffn d t . d R, the quantity* àe-\(j-)-C03.[57it—2nt+5s'—2B)—{'—-\sm.{5n'l-2ntj-5s'—'is)l + 75. J (-—^ . cos.(Mt—2nt-^5£'—2s)-r—-Ymn.(5n'l-Qnt+5s'—2 s)l + de'. }('~ycos.{Wt-'2nti-5e'-2s)~Çj^\sm.{5n't--'2nt+5s'—2s). -Qam'.ffn\W.( ); [3893] y+6ra'. j ('y-Vcos.lSn'f— 2jie+5c-'— 2e)— f— Vsin.(5n'<— 2?U+5s'— 25); / -^5 y. j r_-yco3.(5)i'<— 2;i<+5£'— 2=)— ^j- Vsin.(5ra'«— 2(!<-(-5s'— 2;)^ ^+5n. \{j^ • '=°^- (ô«'«-2n/+5s'-2;)-(^^) .sin.(5n'<-2n<+5='-2.=)|^ 6e, 6 a, 6e', 6^', 6), 6U, being the parts of e, ra, e', ra', 7, n, respectively, depending upon the angle 5n't — 2nt. We have, by means of [3842c], f /dP\ fdP'\ /dP'\ /'dP\ [j^)-'-[j7)' W;=-^-(rfrj' [3894] /dP\ , /dP'\ /dP'\ , fdP\ * (2452) We have already noticed the effect of the secular variations of P, P', in the terms of 3a.ffndt.dR [3812,3812/], depending on sin.Tj, cos.Tj; using, for brevity, T5 [38906]. The object of the present investigation is to ascertain whether the periodical variations of e, e', to, -as', r, n, depending on the angle T5, which are computed in [3893a] [1288, 1297, Sic], produce, in the function 3 a .ffndt . dR, any secular or periodical inequaUties. Now if we suppose the elements e, e', w, -ra', /, II, to be increased by the variations 5e, &e', S-a, ôts, Sy, Su, respectively, the corresponding increments of P, P', will be obtained, by means of [607 — 612], in the following forms, these parts of the general values of P, P', being substituted in [3892], produce the expression [3893]. t (2453) The equations [3894 — 3894"], are easily deduced from the general values VOL. III. 23 90 PERTURBATIONS OF THE PLANETS, [Méc. Cél. Moreover we have, as in [1297, 1288],* [3895] (— Vco3.(5«'i-2n^ + 5£'-26)-('^Vsin.(5n'^-2n/+5£'-26)= '^"^]~^^"l erî^; [3895'] (^).cos.(5«V-2?ii + 5s'~2£)+('^Vsin.(5n7-2ni+5£'-2e) = -'^^7^.<5e; we likewise havef [3896] (~\.cos.{^n't~2nt + bi-2s) — (^\.Bm.(bn't-2nt+b^-^i)= ''^"'^"l e'cî^^: \dc' J ' \de: J ^ ' ^ m.a'n' ' [3896'] (^).cos.(5n't-2nt + 5s'-2s)4-('^).s[n.(57i't-2}it4-5B'—2e)=-^^^^^^^^ \"6/ ' \de / ^ ' m.an of P, P' [.3S42c], which give [38940] (^\ = 2 è . JJi'. e'". e*. y^'. cos. (6V+ & ra + 2 c n) ; [38946] (^— 'j=:2&.iVf'.e"''.e''-^y'^^cos.(i'îï' + 6w4-2cn). These expressions satisfy the first of the equations [3894] ; and in hke manner, we may prove the others to be accurate, by the substitution of the partial differentials of P, P' [3842c]. * (2454) The value of R [3842a'], is the same as that assumed in [1287], [3S95o] supposing (*=!, i' ■=b, i = 2, as in [3890«]. Making the same substitutions in ÔC, i5a [1288, 1297], we get, by using the abridged symbols [3846^,(1], the following expressions, which are easily reduced to the forms [3895', 3895] ; [3895i] 6e = — -r-, — — . i -r- • COS. i 5 + ( -7— . sm. ^5 > [3B95C] s^= .^!!l^5(l£yeos.T,-f^).sin.rJ. f (2455) The values Se', e'S'm', depending on the angle Tr,, noticing only terms of the third order in e, e', 7 [3891'"], are easily deduced from those of ôe, cSts [3895,3895'], by a process similar to that employed in [3846« — gl ; using also the same abridged symbols [.3895e] Ts, Tg, Po, Pq, kc. For if we substitute, in [1288], the values i'= — 2, i = —5, we get the following term of Se, which may be added to [38956], to obtain a symmetrical form of Se, similar to [3S46i, Sic], This last temi may, however, be neglected in computing the value of Se ; because it has not the small divisor 5 71' — 2n. Now changing the elements m, a, n, e, &ic. into [SSgSg-] ^,^ ^^1^ ^^^/^ ^t^ ^ç,_^ ^j^j jj^g contrary, as in [3846a, d\, we find, that the part of S e', arising VI. ii. §12.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 91 To obtain the values of 5 y and Sn, we shall observe, that the latitude of m, [3696"] above the primitive orhit of m', is 5= — y . s'la. (v — n),* which gives [3897] as = — 5y . sin. (v — n) + 7 . a n , cos. (v — n), [3898] Now we have, in [1342],t ( ('iPl.cos.(ôn't — 2nt + 52'—2s — v + n) 7)i.a?i ) \dy/ "^ ' ^ ''•'-^^^ZJ^rS ,.p,, . _ . „ ^ [3899] dP' rf7 sin. (5 n't — 2nt + ôs' — 2 s — v + n) [3895h] [38975] from [3895i], has the divisor 5n — 2n', which is large ; therefore this part is small and may be neglected. The other part, derived from [3895/], becomes „ , m.a'n' i/dP\ ^ , fdP\ . ^ ■) 5n—Zn i\de / \de'/ ^^ whlcli is easily reduced to the form [3896']. In the same manner, we may derive S ■a' [3896] from Sz, [3895c]. ^^^^^'1 * (2456) It may not be amiss to remark, that the object of the calculation in [3896"— 3902], is to ascertain the parts of ôy, 7 5 n [3900, 3901], arising from the [3897a] perturbation of m in latitude, by the action of m' ; supposing the fixed ■plane io le the primitive orbit of m! [3897]; these parts are denoted by 5^/, y5„n, respectively, in [3899']. In like manner, the action of m upon m' affects the values of ôy, y Su by terms which are represented by «, 7, 7 5^ n, respectively, [3904]. The sum of these l35J7c] two paits of i5 7 gives the complete value of 5 7, as in the first equation [3905] ; and the sum of the two parts of <5 n gives the complete value of S n, as in the second of the equations [.3905]. Having made these preliminary observations, we shall now remark, that L^*'-'''] the expression [3897] is similar to [679], changing v, into v, tang. 9 into 7 [669", 3739] ; and à into n+180'' [669", 3746] ; observing, that as n [3746 or -3902] is the longitude [3897«] of the ascending node of m' upon the orbit of m, we shall have n 4- 180'', for that of the [3897/^1 ascending node of m upon the orbit of m, taken for the fixed plane [3896"]. Hence [679] becomes 5 = 7.sin.(i' — n— 180'')=:— 7.sin.(j; — n), as in [.3897]. Supposing [3897^-] now 7, n to vaiy ; the corresponding variation of 4- will be as in [3898]. t (24.57) Using the values [.3895«], also ^ = 2, tang.ç); = 7, é;=n [3902, 1.3-37'] ; also, for brevity Ts=^5n't — 2nt-\-5s'—2e, Ts = 5n' i — 2n t + A — 2n; [3899a] the expressions of R [13.37"], and « or us [1342], become Ii = m k.y-. COS. T^, ûs=—^~y~^^.y.sm.[Ts — v-{-n). [38996] [3900] 92 PERTURBATIONS OF THE PLANETS, [Méc. Cél. Comparing this expression with the preceding [3898], we shall obtain, [3899'] for the parts of 6y, y an, depending upon the action of m! upon m, which ô„. we shall represent bj s^^y, 7^,,^^ 6,7==— "';"" ■5fl^Vsin.(5n'^-2ftt+5£'-2s) + ('^Vcos.(5n7-2n<4-5s'-2c-) j; "' 5?i'— 2ra i\dy / ^ ' \dy J ^ ' ^3 [3901] yS, n= -4^-5f^Vcos.(5«7-2w<+5s'-2.0-('^Vsin.(5?i'^-2u<+5a'-2;)?; ' " 5?l — 271 i\dy J \<iy/ ) 7' n- in which y is the mutual inclination of the tivo orbits to each other, and n the longitude of the ascending node of in' upon the orbit of m [37461. These [3903] ^ . . *^, , f . ^ ' , , T g_ quantities also vary by the action oi m upon m ; so that it we put these [3904] last variations equal to 6^ y, 5^ n ; the whole variations being 6y, iu ; we shall have* [3905] 5y = S,y +ô^^y ■ 6 H = ^, H + 6„ H ; m.a'n' ms/a m.a'n' m\/a [3906] ^,7='^, • '5,/ '/ = -TT^ • «5,, /- ; ^,^=^, • '5„n = -^-— .6 n. •■ J ' m!. a n ' m\/a m.aii " m \/a " If we compare this value oî 5 s with that of R, we shall find, that &$ = —-, — ^;r •{ ~, — ), [3899c] 5,1—2/1 \dyj' provided we increase the angle 5>i'/ — 2nt by the quantity 90'' — v-\-'n., by which [3899</] means cos. Tg will change into cos. ( Ts + 90'^ — v-\-Jl) = — sin. ( Tg — v -\-n); and if we use R [3842a'], the expression of 8s [3899c], becomes as in [3899/,^, or 3399]. [3899c] Now if we put, for brevity, v — n = v, , and develop the terms of [3899^], by means of [22, 24] Int., it becomes, as in [3899A], [3899/] 5s=g-^^^.»/.^Q.sin.(T,+90^-r + n) + (^).cos.(r5 + 90"-. + n)^ ^'""'^^ =5-£k--'K^)-'^°^-(^^-^')-(i^)-^*"-^^'-^')^ ^--^ -^B-n ■ { [(?).sln.T.+Q.cos.Tj.si„.,+ [Q.cos.T,-(^).sin.T.].cos.^. Comparing this with 5«^ — (5y. sin.D,-)-7i5n.cos. t), [3898], and putting the coefficients of sin. v^, cos. V, , separately equal to each other in both expressions, we get [3900, 3901]. If we compare the value of 7<5„n [3901] with that of R [3842a'], we easily perceive [3899A;] that it may be put under the form 7(S,,n = — an .fdt .i-—\; and having found 7(5„n [3899/] by this formula, we get from it the value of S^y, by changing the angle T^ into T^-\-9(y^, as is evident by comparing the two expressions [3901, 3900]. * (24.58) From the expression of 7i5„n [3899A:], we may obtain the value of y <î; n, [3906a] corresponding to the action of m upon m' ; by observing that the values of P, P' , which VI. ii. ^^ 12.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 93 This beins; premised, if we substitute these different quantities in the c,,,mi'ityof ■- tlie mean function [3893], Ave shall find that it vanishes.* Therefore the variations ^,''11^1' of the excentricities, of the perihelia, of the nodes and of the inclinations of [3906'] the orbits, corresponding to the two great inequalities of Jupiter and Saturn, téms here do not introduce into the mean motion of Jupiter, or into the greater axis of its l^^^^fj vanishes. occur in R [3842(}'], are the same in both cases, as is remarked in [3832 or 3846/', &c.] ; so that it is only necessary to cliange R [3831] into —-R, and an into a'nf, to obtain from [3899^-], the expression yô^n = — —,. a' n'.fd t .U-—\. Dividing this [390C6] by 7Ô„n [3899t], we get the first form of '5,n [3906]; and by applying the principle [3906c] of derivation [3899Z] to this value of y^'^, we obtain that of 5/ [3906]. The second forms [3906] are derived from the first, by putting an=a-^, a'n'=a-i [3709']. Substituting the values [3906] in [3905], we get m'.an-\~7n.a'n' , . m'.anA-m.a'n' . Sy= T .5„y; y^n= -f~- .yS,,n; 3906e in which we must substitute for 5,,y, 7 5„ n, their values [3900, 3901]. Therefore, to obtain the complete values of 5y, y 5n, we must change the factor m'. a n into [3906/"] m'. an -\- m . a n', in the formulas [3900, 3901]. * (2459) If we substitute the values [3S94— 3901] in [3393], we shall find, that the terms of this expression mutually destroy each other. In proving this, we shall neglect the factor — 6 am'.ffn^dt^, which affects all the terms; and shall use the symbol Tj [38906], also, for brevity, 5n^^ nn'-^n ^ 5n'-2n ^3907»] m .an m.an m .an--\-m.a n Then the expressions [3895, 3895'] may be put under the following forms [39076] ; the similar values [3896, 3896'] become as in [3907c] ; and if we change, in [3900, 3901], the factor m'.an into m'. an -\-m . a'n', in order to obtain the complete values of Sy, yon [3906/], they will become as in [3907(/] ; (^-£-)-^os.T, + (^).sln.T,=-M.Je; (^) . cos.r,-(^^) .sin.T,=./lf,.e..; [3907.] {'£)-^os.T, + ('l^yn.T,^^M.,.Se', (If ) . cos.T.-('iÇ).sin.r,=M,.e'.-.'; [3907, (^).cos.7,+ (^).sin.T, = _J>/3.57; (^) . cos.r,-(^').sin.r,=./If3.7<Sn. [3907d] VOL. III. 24 94 PERTURBATIONS OF THE PLANETS, [Méc. Cél. [3907] [3907^-] [3907A] [3907i] [3907fc] [S9071] [3907m] [3907n] [3907o] [3907p] [3907j] oi'bit, considered as a variable ellipsis, any sensible inequality, depending on the square of the disturbing force ; and it is evident, that the same result holds good in the mean ^notion of Saturn and in the greater axis of its orbit. Substituting tliese values in the first members of the following equations [3907e — ff], then reducing, by the neglect of the terms which mutually destroy each other and putting sin.^ 7^5 -(- cos.^ Tr,= I, we get [3907f] —Mi.5e.cos.T,—M,.e5v5.sm.T,= (^^y, M,.5e.sm.T,—M,.eô,z.cos.T,= —(^); [3907/] —M^.Se'.cos.T,—J\1^.e'&^'.s[n.T,= (j£^ ; M,.S e'.sm.T,—M^.e'&^.cos.T,=—(^^') ; —J\l^.&'y.cos.T,—M;.y&n.s\n.T,= (~^^ ; M,.Sy .sm.T,—M,.ySu.cos.T,= -('^) ; Now the first line of [3893] becomes, by the substitution of M^ . e f5 « [3907//] equal to Se. {Ml . e <5 w) ^ Mi . e 5 e . (5 ra . The second line of [3893] becomes, by the substitution of [3894], equal to e 5 ra . j (-i~) ■ cos. T^ -f ( — ) • sin. T^ I , and by using — ^j.^e [3907i], it is reduced to C(5«.( — Mi.&c) = — Mi-eS e.S-a ; adding this to the first line [3907 /i], the sum becomes zero. In lilce manner, the third line of [3893], by the substitution of M.^ . e' 5 ra' [3907c], is equal to S e'. (Jk/^ . e'Szs') ^M^.e'S e'. d -n' ; and the fourth line, by the successive substitutions of [3894'] and — M^.Se' [3907c], is e'&z/.^ — Mg . 5 e') ^ — 31^. e S e .6zs' ; the sum of these two lines is therefore equal to zero. Substituting M^-yàlî [3907<^] in the fifth line of [3893], it becomes S y . {M3 .yôJl)Tz= M3 .yôy.SH; and by successively using the equations [3894"], also the value of — M^Sy [3907rf], we shall find, that the sixth line of [3893] is ySu.( — ^3.157)= — M3.ySy.oll; therefore the sum of the fifth and sixth lines is equal to zero. Hence we see that all the terms of [3893], included between the braces, mutually destroy each other, as is observed in [3906'] ; consequently the values of èe, ÔZS, 5e', Sz/, 5 y, Su [3895—3901], do not produce in 3a.ffndt.AR [3892 or 3715i] any term of the order of the square of the disturbing forces. The function 3 a .ffndt .dR, represents the mean motion of the planet m [1183]; therefore the variation of the mean motion, arising from these values of i5 e, S -a, S e', &c. is nothing. Again, from [3709'], we have 2a = 2n '" , and as the mean motion nt or n, is not affected by these values of Se, ^ ra, Sic, it follows, that the transverse axis of the ellipsis 2 a is not affected by the variations Se, S -a, &c. now under consideration, as is observed in [3906"]. The same result holds good when we notice the variations of the motions of the body m', disturbed by m, as in [3907]. VI. il. §13.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 95 13. TVe shall now consider the variations of the excentricities and of the perihelia. We have given, in [1287 — 1309], the expressions of the ^ (1 C (/ CT d c' d Ts' , increments ot —-, —, —, 77"' dependnig on the two great [3i)08] inequalities of Jupiter and Saturn, and we have observed, in [1309", &c.], that the variations of e, ro, e', ra', relative to the angle 5 n' t — 2>t^,t * (-2460) Tlie expression de [1284], is integrated in [1286], and put under another form in [1283]. Now as this last expression is used in this article, we shall take its differential relatively to t, and then change the angles n't, nt into ^', ^, respectively, [39080! as in [1194'"] ; for the purpose of noticing the inequalities of the mean motion. If we put fA=l, i'^o, i^2, as in [.389.5rt], we shall get from [1288] the following value of de; and in like manner, from [1297], we get d-m [.3908f/] ; de=-m.andt.^ (^^).cos.(.5^'-2^+5s'-2s)_ (^^).sin.(5^'-2^+5s'-2a)^; [390Sc] rf^=-,«'.««rf<.^^.(lÇ).sin.(5>'-2^+5s'-2£)+i.('^').cos.(.5f-2^+5s'_23)^. [3908d] t (2461) If we put the values of §, ^', under the forms ^=nt-}-N, ^'^n't-{-JV', [.3909o] we shall find, by comparison with [1304, 1305], and using the symbols [3890a, b], ^= (IÉS^-^^-^°^-^^-^'-^'"-ï^5l ; [39095] ,,, 6m'. a n^ m^a ,„ „ ^ = -W^;ji:2^^-^a' ■ ^^--- ^^-^'- ^■'"- î'^l- [39095'] Substituting the values [3909«] in the first member of the following expression, we get 5 ^'— 2 ^ + 5 £'— 2 E = 5 «7— 2 M < + 5 e'— 2 £ + (5 JV'— 2 JV) ^T^ + (5iV'— 2iV), [3909c] and by neglecting the square and higher powers of 5N'—2JV, using also [60,61] Int., we obtain sin. (5 ^— 2 ^ + 5 E-— 2 = sin. T, + (5 N'— 2 N) . cos. T, ; cos. (5 ^'— 2 I + 5 £'— 2 = COS. T, — {bN'— 2 N) . sin. T^. ^^^^^'^^ Substituting these in the value of de [3908c], or, as it may be called, dàe, we get dhe= m!.andt.\-(;^-£).cos.T, + (^-£).^n.T,\ [3909e] + rr^.andt.{b N'- 2 iV) . ^ (^^-^) . cos. T, + (—") . sin. tJ . 96 PERTURBATIONS OF THE PLANETS, [Méc. Cél. may introduce in these expressions some variations similar to those produced The part of this expression, depending on the factor 5 JV' — 2 JV, is of the order itJ^ ; and as the other terms are of the order m', we must notice, in them, the variations of {- — 1, (— — ), arising from the variations of Se, (5 ra, &tc. The additional [3909/] terms of the vahies of f— — j, {— — j, from this source, may be found by changing P, P' into [ - — ) , ( -T— ) , respectively, in [3393c, rf] ; and as the former quantity is multiplied by — m'.and t .cos-T^, in [3909e], and the latter by m'. andt .sin.T^, the complete expression of doe will be d5e^= rn'.andt .< — (~T~) • cos. Ts + ( -;— j . sin. Tg ^ + m'. andt.{5 JY'— 2JV)A (^^ . cos. T5 + (^) ■ sin. tJ ( + ('^).Se + (^).S.+ (:^).Se' \'\de^ J \dedTSj ' \dedc'J [3909;»] — m'.andt. cos. Ts . < \.^\ded^'J ^\dedy) '^\dednj \~\de^J ^\ded-a) '\dede'J f -)- m'. audi . sin. Î5 . <^ > • \'\ded-!a'J ^\dedyj '^~\deduj ) Now if we take the partial differentials of [3894—3894"], relatively to e, we get /ddP\ (dP'\, f''''P!\ (i±^'\ — _('!JL\_, fill-'] \d7d^) = \d7)'^''-\ de-i J' \ded^)— \de J -Kdc^) ^^^^■3 {d^)='\-d^)' \:d7d^j=-'\iûd7)' \d7dn)~~'^''\dedy)'' \dedn ) '''\dedy)' Substituting these in [3909/t], and retaining only the terms of the order m!^ ; or in other words, neglecting those terms of the first line of [3909A], which are independent VI. ii. § 13.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 97 by the two great inequalities. If we apply this method to the elements of of the factor 5 A'' — 2 JV and the second differentials ddP, ddP', we get dÔ6= m'. a ndt. (5 JV'— 2 A") . | (^\ . cos. T, + Cj^) ■ sin- T, I ddP\ /dP'\ , fddP'\ fddP\,, , /ddP'\ ,, ,, /ddP\ , , /ddP'\ (ddp\ fddP\ ^ ,- — ?«'. a lid t. COS. T^. . de^ J \ de -\-m' . and t .sin. T^ ddP'\ , / dP oe — [.3909fr] /ddP\ ,,,,/'ddP'\ , /ddP\ ,^ [-d7d7)-'^^+[-dId^)-^^^-[d7di)-^^^ We must substitute in this the vahies [3895 — 3896', 3906/], and tlien by integration, we shall obtain 6 e [3910], as will appear by the following calculations, using the abridged symbols to denote the factors of the three difterent groups of terms which occur in [3910]. If ^ve compare these expressions with those in [3907a], we shall obtain the following values of m'.an, which will be used hereafter ; these equations are easily proved to be identical, by the substitution of [3907a, 3909/] and reducing. m'.an=:MiK:,=zJ\'LJV2—M^.{J^^ + J^:,). [3909ml First. We have, by means of [3909&, U], m'.andt .{i)N'—2JV)=:^— —-:-—— . ^ — ~ ^, , ' .{P. cos. Tr,— P'. sm.T^l.d t [3909rt] = — 2 A*! .f P. cos. T^ — P'. sin. Ti\.dt. Multiplying this by (-v— ) • cos. Tj -f f— — j .sin.Tj, we obtain the value of the first line of [3909Â-], as in the first member of the following expression, which, by means of [1, 6, 31] Int., is reduced to the form [3909o] ; -<iN,.dt.\P.cos.T,-P. sin.Ts^.J (^) .cos.Ts + ^^Vsin.Ts^ > • [3909ol Its integral gives the terms of i5e [3910], depending on the factor (5mv/a + 3m'v/«')- VOL. III. 25 98 PERTURBATIONS OF THE PLANETS, [Méc. Cél. [3909] the orbits of Jupiter and Saturn, and put àe, «î ts, for the variations arising Second. The term of [3909A"], connected with the factor ( — ~j . dt, is as in the first member of [3909p] ; wliich, by the successive substitutions of [3909»!, 3907e], becomes as in [39095], whose integral gives the corresponding term in the fourth hne of S e [3910] ; [3909p] m'.an . I— Se .cos.T^—eûzi .sm.Ts] = MijX^. { — Ô e .COS. T^— e Ù 7Ô . sm.T^l [3909g] =JV,J—Mi.Se.cos.7\—J\I^.e5z:.s\n.T,l=Mj'^\. ITiird. The term of [3909A:], connected with [-j-tt )-dt, is as in [3909r], and by reduction, using [3909m, 3907e], it becomes as in [3909s] ; whose integral gives the corresponding term of the fourth line of [3910] ; [3909r] ni'.an.{Se.sin.T^ — eS-a.cos.T5l=^M^J\'^.\5e .s'm. T^ — e^^.cos.Tgl [3909«] =zJV2.{Mi.ôe.sm.T5—M^.eôzs.cos.Ts\ = —M.{^\ Fourth. We may proceed in the same manner with tlie terms of [3909Ar] . connected with the factors ( . , , ] .dt, { - — — r r"; which will be found to be represented, \dede / \dede / ^ |3909«1 respectively, by the first members of [3909/j, r], accenting the symbols e, 8e, S-!^. If we also put 7n'. anz^MjJV^ [3909m], and reduce the formulas as in [3909c, «] by [3909m] using the expressions [3907/], they will become, respectively, JV^.i — ], ~"^3-(^j- Multiplying these by the factors [3909;:], and integrating relatively to t, they become as in the last line of the expression [3910]. /ddP\ , Fifth. In like manner, the terms of [3909t], connected with the factors , , ).dt, [m9v] , \dedyj ( -—].dt, will be represented by the first members of [3909p, j-], changing e, êe, ôzs, XdedyJ into 7, (5 7, 5n, respectively. Then substituting in', an = M^. {Nç,-{- JV^) [3909m], and reducing the formulas, as in [3909(7, *]> using [3907^-], they become respectively, [.3909«>] (A'a + Ns) . (^) , — (JVo + A',) . (''^^ . Multiplying these by the factors [3909y], and integrating relatively to t, we get the corresponding terms of oe [3910] ; the terms depending on JVo being in the fourth line, and those on JV^ in the last line of [3910]. Sixth. The two remaining terms of [3909A:] are as in the first member of [3909x] ; which is reduced to the form in the second member, by the substitution of m'. an^ M^ JV^ [3909m], and M^.Su [39076]. Reducing the products by means of [31, 32] Int., VI.ii.§13.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 99 from the square of the disturbing force, we shall find [ \de) \de {5n'—2nf mV«' \ 2.(5ji'— 2n ^ \ de J '\de Inequality of the ex- centricitv dP'\ ^ /dP\y \ ofJupitei. 2.(5)i'— 2îi) . C0S.2 . (5 n't—2nt-\-5;'—'2s) rfF \ f<idP\ JdP\ { ddP' \ . /dP'\ fi^\f'[f\ f ddP' \y • I^J-yde"- ) \de)\dei J'^Kdy j\dedyj \dYj\ded y )ç^ [3910] WiP\~_fdP'Yl m'g.aans ) , iMlZ_AfiZ_l.cos.2.(5n'«-2n<.+5E'_2 5) "5n'— 27i*\ '^ 4.(5n'— 2n).e - lliZ_AlijL.sin.2.(5ji'f— 2n<+5e'— 2j) 2.(5n'— 2îi).e ,m'.aa'.nn'.t WdP'\ /ddP\/dP\ / ddP' \ /dP'\ / ddP \ /dP\ / ddP' \] 5n'—^n 'i\d^ )\dede' ) \de')'\dede')'~\ dj )\dedy) \dy)'\dedy)\ .* 5 [3909x] it becomes as in [3909(/] ; then integrating relatively to t, it produces the terms depending on COS. 2 T, sin. 2 T, in the fifth or sixth lines of [3910] ; m'.andtA — \~T~) •5'"- ^s — \~r~) '^°^' "^ A ' ^'^ - •^.^'.l-©.s-.n.n-('^),cos.r,|. J(^^).».n-(-).sin.r.| * (2462) If we compare the expressions of de, dis [3908c, r/], we shall find, that dvi may be derived from de, by subtracting 90"^ from the angle 5^'— 2 ^ + 5 e'— 2e, [3910a] and connectbg the factor - with each of the quantities (-7- )) (-7—) 5 '^y th's means [39104] the angle Ts is also changed mto T5 — 90'', in all the terms of [3909e, h, A;], in which 100 PERTURBATIONS OF THE PLANETS, [Méc. Cél. a e [3911] Inequality 3m'2.aan3 (5mv/a+2mVa') j i '- Vrfe / \d<"/> . „ ,£. „ o ^ic- o< perigee of (5n — 2ref.e jnVa' \ 2.(5n' 2 ni Jupiter. 1 ^ ' + i ^ 1 ^^ i-^.co3.2. 5;i'<— 2n(+5;'-2s) 2.(5re'— 2ji) dP\ /ddP\ /'dP'\ /ddP'\ /dP-\ f'l±^\,/<l£\ f'^'^^'M d7)\de^ )'^\de )\de^ J^Uyj\dedj)'^\d^)\did:^)y^ M'^.a^n^ j+ ^V^^"^^^ > ..in.2(5„'/-2n< + 5.c'-2.) ^(5?i'— 2ra).e ' "j 2 . (5 ?i' — 2 n) . e + \£L-L-h£— L. COS. 2. [5 n't — in t-j- 5 s'— 2 s) (5ra' — 2 re), e I?! m "(5 re'.gg'nn'.< 5/'ijP\ fddP\ /dP'\ / ddP' \ /dP\ /ddP\ /dP'\ /ddP'\ n'—2n).e'l\d7)\dede')'\de')\dede')'^\(r^)\dedy)'\dy)\dedyl [3910c] T5 explicitly occms ; observing that no change must be made m the factor 5 JV' — 2N. Hence it appears, that if we change in [.3909A] the angle T5 into T5 — 90"^, without ahering 5 JV' — 2 JV, and then muhiply tlie resuUing expression by - , we shall obtain [3910rf] all the terms of d Szi, except those arising from the variation of the factor - , connected with the quantities (— — J, (—. — ) [39106]. These last quantities depend upon the two following terms of dôzs, namely, [3910e] ,«'. an dt.ll -("£). sin. T.- ('i^') . cos. T, ] , corresponding to the two first terms of [.3909e] ; and as the variation of - is 13910/] _^_f^_l_.^(l^).sin.T,+ (^).cos.r,} [39076]; also m'.an=M^JVn [3909m], this part of dozs will be represented by JVo , ( /dP\ . „ /dP"' [3910g] VI. ii. §13.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 101 The parts of these expressions, proportional to the time t, give the secular variations of the exceutricity and of the perihelion, depending on the square of the disturbine; forces. To obtain the periodical terms of v depending [3912] on this sqxiare, we shall consider the term 2e.sin. (n^ + s — -in) [3748], [3012'] in the elliptical expression of the true longitude. If we put 6 e, (5 ra, for the variations of e, ■^, depending upon the angle bn't — ^.nt-^-bi' — 2;, Tills is to be connected with the terms mentioned In [3910(ZJ, to obtain the complete value of do -a; and then by integration, we shall get 5-ui [.3911], as will appear by the following investigation, taking the terms in the same order as in the preceding note [3909« — y]. [3910A] [3910î] In the first place, the terms depending on .5 A'"' — 2 JV, are multiplied by the factor {— — j . cos. Tj-j-f —— j . sin.Tg, in the expression of d5e [3909A], which becomes -.(— — J.sln.Ts . ( — — j .C0S.T5, in dô-a [3910f/]. Now it is evident, by inspection, that this last expression may be derived from the first, by changing ( — -- 1 into - . ( — ) , /dP'\ . 1 /dP\ « \ e/ [39ioi] and (-7-] iiito .( — j, without varying the angle T5, or the factor 5 JV'— SA"; therefore v. e may use the same process of derivation in obtaining the part of dozs, depending on oN' — 2.¥, from the similar part of due [3909A:] ; or in other words, the part of us [3911], connected with the factor bm,y'a-\-'im s/a', from the similar part of &e, [3910]. We shall now apply the principle of derivation mentioned in [3910f/], to the terms [3909p — «■], and we shall find, that the factor of -.{— — ].dt, in do-,, deduced e Kde'i J ' [3910mJ from [3909^], is N^.\—M,.5e.ûn. Ts-f- itfj . e 5 « . cos. Ts | = A', . (^\ [3907e] , producing the term — • ("3~) • (-7~r) • ''^- '" dàzi, whose integral is as in the first [3910ml term of the fourth line of 5 a [3911]. The term [3909s], by similar reductions, A% /dP'\ /ddP'\ gives — • ( 7^] • ( "TT" ) • ^ ' ^'^^ *^™^ [3909^] give [3910n] A3 /rfPN (ddP_^ A-3 (dJF\ (ddP'\ [3910,] e \dt)-\dtde')-'' e \de' J ' [dede' J ' ^ ' the terms [3909ic] give as in the fourth and seventh lines of 5 a [3911]. VOL. III. 26 102 PERTURBATIONS OF THE PLANETS, [Méc. Cél. and upon the first power of the disturbing force,* also 6' e, 6' w, for the [3913] preceding variations of e, w, depending upon the double of this angle ; f [3913'] moreover, if we denote by à s the sum of the two inequalities of m, the Lastly, the terms of dSzi, deduced from those of doc, in the first member of [3909r], by the principle of derivation [3910f7], are [3910p] m'.ajirf<.| — -.(^— j.sin.T5+-.(^— J.C0S.T5J .Sis; which, by the substitution of 7n'.a7i = M^JV„ [3909m], and 5 a [3907i], becomes [39105] Adding these terms to those in [3910o-], and putting cos.^ Tj — sin.- T3 = cos. 2 Ts, 2 sin. Tj . COS. Tg = sin. 2 T^ , we get and by integration, it produces the terms of i5 -si, depending on sin. 2 T5 , cos. 2 Tj , in the fifth and sixth lines of [3911]. [3912a] * (2463) These values of 5 c, 5^, are given by the formulas [3907i]. t (2464) The formulas [3910—3912'] give, by using T5 [38906], 3m'2.a3n3 ( 5»n/a-t-amVa') A L V f<e / \dej_i 3.(5n'— 2n)3" my a' '\ r^,/dP'\ ^/dP\ [3913a] [39136] 3m'3. f)3»,3 ( ^m^a-\-^mVa') 1 L \de/ \dt/A VLii.§13.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 103 one depending on the angle 5n't — '2.nt-{-Bi — 2s, the other upon the double of this angle,* the term 2e . sin. (nt^s — ^), will become [3913"] (2 e + 2 5 e + 2 <5' e) . sin. (« ^ + = -f- d s — ^ — 5 =i — 5' ^) . [39i4i If we neglect the cube of the disturbing force, the preceding expression may [3i)U'| be developed in the following form,t 2 e . sin. (nt-\-s~\- 6 s — ra) + 2 (5 e . sin. (n t 4- s — ra) — 2e6is . cos. (nt -\-s — ra) ^ [3915] + \2ô' e + 265^ .5s — e. Ç6v>)-\. sin. (n t + s — w) — [2ed'îJ + 2oe.da — 26s .ôe}. cos. (nt + s — zi). The term 2 e . s'm. (nt -\- s -{- 5; — i^) is that obtained by increasing the [3915] * (2465) The great inequalities [1197, 1213, Sic.], are to be applied to the mean motion of the planet [1070"]. If we notice only the chief terms of â s, having the divisor [3914a] (5?i'— 2n)^ they will become, by putting i^^5 in [3817], and using Tj [3S90è] ; ^ ' = (sl'-atp • ^ ^- ^"'- ^= - ^'- ''"• ^= ' • P^^^''^ We may remark, tliat the terms of v [3748], depending on e^, e^, &c., are here neglected [3914c] by the author, on account of their smallness ; they are, however, noticed by him in the fourth volume [9062, &c.]. I (2466) Putting a = nt + i-[-5s — TS, b^ô-^^ôt^', in [22] Int., we get [3915a] the second member of [3915&], which is successively reduced to the form [3915c], by usmg [43, 44] Int., neglecting terms of the order m'^, and finally putting [3915a'] cos. o = cos. [nt-\- S' — to) — us. sin. (w t -}- s — ra) in the term multiplied by 5 w ; sin.(n<-|-£-|-5£— ^— 5i3— ô'î3) = sin.«.cos.((533-{-5'xn) — cos. a.sin.(r5x3-|-i5'-5j) [39156] = {1 — i . (Sts)^] .sin.w — (6z!-\-S'zi) .cos. a =sin.a — J.((Jûj)~.sin.(7i^-f"^~^)~("''+'^'®) •cos.(w^-|-£— ro) -\- o-us. 5 1. sin. {7it-\-s — ra). Multiplying this by 2e4-2i5e-r2i5'e, and neglecting terms of the order m' ^, it becomes as in [3915] ; observing, that in the term multiplied by 2 i5 e, we may put sin. a = sin. (n < -}" ^ — w) + i5 s . cos. (lit -\- i — w). [3915rf] 104 PERTURBATIONS OF THE PLANETS, [Méc. Cél. [3915 "] mean motion n t, by & j, in the elliptical part, according to the directions in [1070"]. The two terms [3916] 2ie . sin. (nt -\-s — n) — 2 e (î ^ . cos. {nt-\-s — cr) , form the inequality depending on the angle Qnt — Sn't^Ss — 5e', given by the formula [3718].* If we then substitute in the other terms, the * (2467) If we put ?' = 5 in [3814,3825], where only the terms having the divisor 5 ft'— 2 m are retained [3818', 3824], we get [391Go] '-^==i/.cos.(5n'i— 3n« + 5£'— 3s + .^); (5j; = 2H.sin.(5n'i— 3«« + 5e'— 3e+^) ; and we may observe, that this value of ^ v is easily obtained from that of r(ir, by means [3916!>1 of the formula [3718]; retaining only its first term 5v=^ ^ , which contains the small divisor 5 n' — 2;i [3814, &ic.]. If we substitute, in this last expression of (5 v, the value of r (5 r [3876f/], neglecting the small terms depending on X, it becomes [3916e] (St) = 2 ue. sin. (n C + s — w) — 2 e (5« . cos. [nt -\-s---ui). Comparing these two values of 5d [3916a, c], we find, that the two terms in the second fine of [3915], depend on the angle bn't — Qnt-\-bi' — 3e, or ^nt — bn't-\-'è3 — 5s', as in [3916']. The same result may be obtained by the substitution of the values of <3 e, c 5 w [39076] in [3916], and using the symbols Tg == 5 w'ï! — 2 « < + 5 s'— 2 e, W=nt -\-s — Î3 ; since it becomes, by successive reductions, as in [3916^]; being of the form mentioned in [3916'] ; 25e .sin.?^— 2e5«.cos.?f =— ^ 'KS) ' ''°'' ^^ "^ (Ï) ' "'"" ^' \ ' ™' ^ [3916e] ' = — ^ ■ (^) • Icos. Ts . cos. TV-\- sin. T, . sin. W\ Ml \de/ ' [3916V] -I- J- . (^\ . |sin. Ts'. COS. fV— cos.Ts . sin. ?F} [3016/1 =--. {-) • COS. in - TV)+^. f^') . sin. (T.-W) =—■—. (--] .cos. (5 7i' t — 3 n t 4- 5 t'—3 s + zs) [3016g] 2 + — . f — ] .sin. (5n'< — 3?ii-f-5s'— 3e + -5î). VI. ii. «^^ 13.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 105 values of he, ôt^ [39076], and for ô' e, b' -., their preceding values [391 3f/, 6] ; the sum will give, by neglecting terms depending on the sine and cosine of «ï + 5, because they are comprised in the equation [3917] of the centre," (on— 2»)3' mVa' f \(/e/ \de/S _ 3m-s.«^r.3 [5mM±ivW^) ^j,,(dJ-\_^_ f^V..sin. (5. ^-10,/^ + 5.-10.-'-^). [3918'] {5n'— 2n)3 m'y/»' C \de J \dejS ^ ir we put, in [3916-], (^) = — 7»/jIJ.sin.(^ — ^), (^-£^ = M,H.co5.{A — ^), [3916A] and reduce the result by means of [21] Int., it becomes equal to 2 H. sin. (5 Ji7 — 3 n t + b^ — 3^+Jl). [3916il This is of the same form as [382.5], which represents the most important term of this form and order, having the small divisor 5?i'— 2 k [3824]. The factor ZJ' is of the second .gg^g-,! fdP\ fdP'\ dimension in c, e' [3314i], being of the same order as the quantities (77)' ITT")' For the values of P, P [12S7], which correspond to the angle T5, are of the third dimension in e, e', &,c. [957'"', &c.], and their differential coefficients, which occur [3916t] in [391fiir], are of a lower order by one degree. * (2468) The first and second lines of the expression [3915] are accounted for in [3915", 3916] ; the remaining terms become, by using the abridged symbols W, T^ [3916rf], \2à'e-{-2e5cz.6s~e.{S^f\.sm.Tr-\-\ — 2c.o'is — 2Se.ii^-{-2ôe.Se\.cosJV; [3917a] in which we must substitute the values of Se, 5 -a [39076], (V e, 5' ■a [3913a, b], Si [39146]. In making these substitutions, the terms Szi.Ss, [S-a)^, Se. 5 -a, Ss.Se, will produce factors of the forms ^.cos.^Tj, ^'. sin.^Tg, »4". sin. Tj . cos. T, , or à^ + ^^.cos.2r5, i.(]'—hA'.cos.2T^, ^ ^ . sin. 2 T., [1,6,31] Int. Substituting these in [3917«], we find that the parts ^ A, ^Jl', independent of 2 T^ , produce terms depending on sin. W, cos. W, of the form a . sin. W -\-h .sm. (V ; which, by putting a=k .sir., p, h^Tc . cos. p, and reducing by [21] Int., becomes fc . sin. (/f'-j- (3) = Ar. sin. (n ^ + ; — « + (3)- This maybe connected with the equation [3917c'] of the centre [3915'], as is observed in [3917] ; therefore these terms may be neglected, and we may substitute in [3917«] the following values, C0S.3 Tj = 1 COS. 2 T5 ; sin.^ T,= — ic^.2T^; sin. T5 . cos. T, = | sin. 2 1\ . [3917d] Substituting these in the square of Svs, multiplied by — e, deduced from [39076, a], we get VOL. III. 27 [39176] [3917c] 106 PERTURBATIONS OF THE PLANETS, [Méc. Cél. This inequality may be put under the form [3921] ; for if we represent by [3919] 6v = K. sin. (5 7i' t — 3 ni + 5 s' — 3 e + B), This term is destroyed by the corresponding terms of 2à' e, deduced from the third hne of [3913a], so that the sum becomes [3917/] oye_e.Mn)2==— ,-- 3 m''^. a^n^ (5 m \/a -j- 2 m V«') :^-©+^-©]---^' 15»'— 2n)3 m'^/af + :''■■©-^•(^^)] .2T, Multiplying the value of e&zs [3901 b, a], by as [3914a], and reducing the product by means of the expressions [3917(7], we get, by putting the factor 6, in this last 1 1 /• r. 2m'i/i' expression, under the lorm 3 . , , , , mya [39l7g] 2eSzs,(is: 3»i'2.a2n3 2mVa' (5n' — 2n)3' my a' [39177i] Adding this to [3917/], and putting, for brevity, ^£, = - 3m'3.aan3 (5mv/fl + 4 m'y/a') (5n'-2n)3' 711' y^a' we get [39l7i] 25'e + 2e5«.â£ — e.(^î3)2= Again, multiplying together the two equations [39076], and dividing by ^M^^.e [3907a], we get, by substitutmg the values [3917»^/], [3917fe] — 25c J« = m'a.a3 7i2 de J \de (5n'— 2n)2.e J r/dP\ /dP' Adding this to the expression à' tz [3913?»], multiplied by — 2e, it is destroyed by the terra depending on the third line of [3913i], and the sum becomes 3m'2.a2n3 (5mv/a + 2?nVa') \o»l.{l\ (5„' — 2n)3 m'\/a vJ+[m^)+-q: .cos.2Ts -[-m-^m-- VI. ii. §13.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 107 the inequality of m, depending on 3 7it — 5n't-\-3i — 5 s';* and as in [3889], the great inequality ^ = H. sin. (5 n'i — 2 « ï + 5 e' — 2 s + Â), [392oj Multiplying —My.Se [3907i] by — jj^, also by <5 e [3914J], and then reducing by means of [3907a, 3917(/], we get +[--f^)+'--(^: .cos.Srs The sum of [3917/, m], using JI:/4 [3917A], is 5 — Seô'îs— 25e.5t3 + 2i56,5e}=<( >. [3917n] Multiplying [3917i] by sin. W^, and [39177i] by cos. fF, then adding the products, we find that the first member is equal to the expression [3917a] ; and the second member, by the substitution of sin. 2 Ts . sin. W -f cos. 2 T5 . cos. ?F = cos. (W—2T5), cos. 2 T, . sin. TV— sin. 2 T5 . cos. fV= sin. ( W— 2 T5 ), becomes [39iro] and by resubstituting the values of M^ , T5 , ?F [3917A, 3916J], it becomes as in [3918,3918']. * (2469) The expression [3919] is of the same form as that assumed in [3826], or that computed in [3916^], assuming 1 = 5; moreover [3920] is the same as [3889]. [3920o] Hence if we put, for brevity, T5 = 5 ?i' < — 2 71 < + 5 e' — 2 s, fV^ = nt -\- s, and [3920o'] then make the two expressions [3919,3916^] equal to each other; also [3920, 39096, a], «sing M [3907a] ; we shall obtain the two following equations ; K.ûn.{n-W,+B)=^^^^^.\-(^i^ycos.{T,-W,+^)+(^-^ysm^^^^ [392061 H.sin.CTs-f- 1) = (5 Jlyjja •{P-c0s.r5-F.sb.T5}. [3930c] 108 PEUTUllEATiOiVS OF THE PLANETS, [Môc. Cél. the preceding inequality will be, by ^69, of the second book,* [3921] ^^_ (5>V«+4>»V«') ^ff^-,3i^^.(5,,,_10^,,^5,_10,_^_^j, m\/a ^ ' In like manner, we shall find, by noticing only the secular variations,! * (2470) Multiplying together the equations [39206, c], and reducing the products by [17 — 20] Int., we find that the first member becomes equal to [3931a] I ÛK . COS. {W^J^Â — B) — lTl K. COS. ( fF» — 2 T-.— B — A) ; and the product, in the second member, depends on similar angles JV», W^ — 2T5. Now as these expressions must be equal to each other, whatever be the value of t, we may put the terms depending on the angle /Fg — 2 T5 in both members, separately equal to each other, and v/e shall get _ _ 6m'2.(i2n3 ) L \de. J ' \de/_\ [.•39216] — è HK. cos. { W..—'iTi,—B-A) = — ,r„>_g„a . < ^D7l — ^raj-' \ |- //p/v /,!P\-\ This equation being identical, we may change W., — 2T5, into !-V„ — 2 T5 -{- OO"* ; by which means, the expressions cos. (TF. — 2 75 — B — 7]), sm. {IV.2 — 2 T5 — ûj), [3921c] COS. (H'., — 2T5— ®), become, respectively, —ejn. (FF3 — 2 '/'s — iî— J), cos. ( JV^ — 2 Ts — tn), — sin. ( TFg — 2 Tj — -) 5 substituting these in [3921^-], and multiplymg the result by '^,—^ , t'le first member of the product becomes as in the second member of [3921] ; and the second member of this product includes the terms [392W] [391S, 3918'] ; observing, that fF.j — 2 1\=: but — 10 «' < + .5 ; — 10 s' [3920rt'] ; therefore the inequality [3921] is equal to the sum of the two expressions [3918, 391S']. t (2471) Using the abridged symbols P„, P^, T,,, Sic. [38466— f7] ; also [3922a] Z = b?J — 22,-\-b^ — 2s. Z^=b2, — 2^' + 5£ — 2s'; we find, that the expression of de [3908c] may be rendered symmetrical by the introduction of the two terms depending on the angle Z^ , or T^ , in the value of R [3S46c] ; so that we may put , <.fdP\ „ fdP'\ . _, /dP^\ _ fdP'n\ . „? [3Î326] de = —m'.andt.j^[-jjycos.Z—i^-j-ysm.Z+^~ycos.Z, — l^^j^ysm.Z,^. In computing S e from this expression, it is not necessary to notice the angle Zg , because [S922C] it does not produce terms which are so essentially increased by the small divisor 5n' — 2n, as has been already observed in [3846f/"]. From this expression of de. we may derive VI.ii.§13.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE, depending on the square of the disturbing force, ôe'=- 3 irfi. a3 n3. t (ôm\/a-\-2m'\/a') [5n — 2n)-.a'' m\/a 5)i' a'g n'3 ■ t ^/(/P'N / ddP \ /dP\ fddj^ ,i'—2n 'i\de' )\ilc"i J\d7)\d7^ dP 77 dP' dy + mm'.aa'.nn'.t (.^dP'\ /ddP\ A/j 5n' — 2?^ l\de. Jydedc'J \d dP e ddP' dede'J xdy ddP de'dj ddP_ de'dy dP\ / ddP' \ J^,}\de'dy) ■dP\ /ddP' dy J \dt'dy^ 109 Secular inenuality of ilie ex- centricity of Saturn. [3922] iliat of de, by changing the elements of tlie body m into those of ?«.', and tlie contrary ; by which means P changes into Pq [3846(/, &c.], P' into P'o , Z into Zo, « into «', e into c', &:c. ; hence we have (/e':=: — m.a'n' (lt.\ [ -r-ri -cos. Z, U'/e'/' ,sIn.Zo + (^). cos Z — rfP' 77 .sin.Z^ Neglecting the terms of this expression depending on the angle Z,,, because they do not produce by integration the small divisor bn' — 2k; then substituting the values of sin. Z, cos.Z [.3909fZ, .3922rt], we get the following value of dc, or as it may be written d&e', being similar to [3909e], d <i c == m .a n d t .< — — — . cos. i s + — r-r • sm. 1 r, > I \de J ^ ' \de' J '' ) + m . a' n' d t . {5 JV'— 2 A") . ^ (''|^') . cos. T, +Çj^) ■ si"- T, I . The part of this expression depending on 5 JV' — 2 JV, is easil)- deduced from that in the tirst line of [3909fc], or from its development in [3909o] ; by multiplymg it by —, , and changing the partial differentials of P, P', relative to e, into those relative to e. Hence we obtain the following expression of the part of d5e', depending on the factor (5JV'— 2JV) [3922/], '■r^)---©+[''-©+-'-(S) -., m. an , — J\ , . — .d t. Now by successive reductions, using an = a [3709'], an'=a' we get [3922d] [3922e] [39226'] [3922/] [3922^] [3922A] m.a ' a hence from [3909/], we obtain m'.an VOL. in. 3 m'~. «2 }i3 (.5 m \/a -\- 2 j^V"') "^ • " (on'— 2n)2 28 3mS. a3?i3 (.5nn/a-f-2mVa') {5îi' — 2nf.a'' m^a [3922^1'] [3922i-] 110 PERTURBATIONS OF THE PLANETS, [Méc. Cùl. of the perigeo of Saturn. [3922/] [3922?»] [3923c] {5n'—2nf.a'e'' m\/a 'X \de'J~ '\de'J^ dP\ /ddP\ /dP'\ fddP'\ fdP\ fddP\ fdP'\ /ddP'\) nAa"-inKt (,/dP\ /ddP\ , fdP'\ fddP'\ , fdP\ fddP\ , yr\dy )\U'dyJS mm ■(5^ .aa'.7in. t WdP\ / ddP \ /dP'\ /ddP'^. /dP\ / dJP \ ^ /dP"\ / d d P' \) ^. —2n).e''i\dej\dede')'\de )\dede')^\dy )'\dedj) ' \d^ )\d7I^) ^ ' Substituting this in [3S22A], and integrating, we find, that the terms multiphed by i, become as in tlie first line of [3922] ; the otlier part depending on 2 T5 , produce in (5 e' tlie terms [39224] ~2.(3n'-2n)W" '^Wa j r- /.^p^x /,;p . sin. 2 T, . cos. 2 75 If we compare the terms of d rS e, whicli are independent of (5 .V — 2 7V) [SOODe], with those of dSe [3922/*], we find, that the latter maybe derived from the former by changing the elements m, a, n, e, w, &,c. into m', a', n', c', ■cr', &ic., respectively, without altering P, P', T5 ; and as the divisor 5// — 2« is introduced merely by the integration of terms depending on the sine or cosine of the angle T^ and its multiples, this divisor xvill also be unchanged. Now making these changes in the secular terms, in the fourth and seventh lines of 5 e [3910], we obtain the similar terms in the second and third lines of S e [3922] ; moreover the periodical terms, depending on 2 T5 , in the fifth and sixth lines of oe [3910], produce the following terms of oc', t3^22n] u-E^;r^^,- \ K77) -[-17) J-^°^-'^^^-^- W) • (77) -^'"-^^^^ 3 • [3922o] The sum of ihn expressions [3922Àr, rî\ may be represented by o'e', to conform to the notation in [3913], the characteristic &' being used to include the terms depending on [3922p] the angle 2T5. These terms are used in [3924c]. * (2472) In the same manner as we have deduced the expressions [3922'^, e,/] from [3903c], we may obtain the following expressions of d -,, d 3', d & t^' from [3903(Z] ; [3923«1 ^.=-.'.««^..51.(^).sm.Z+^.('^).cos.Z+J.(^).sin.Z,+ ^(^).cos.Z„>; [3923.] d.'=-m.a'n'dt^^,.(^ysn.Z,+l('^^^^^ d5zi'=. m.a'n'dt}—\. f^") . sin.7',— -,. f'-^Vcos. T, \ ( e \de / e \de' J ) + m.aVrf^(5JV'-2A').^-i.(^).cos.T5 + ^,.(^).sin.T,|. VI. ii. § 13.] DEPEiXDING ON THE SQUARE OF THE DISTURBING FORCE. ] ] I We also find, that the motion of wi' in longitude, is affected with the inc(|na]itj* 3iiAa3n3 (3 mv/a-|-2niVV) (an'— 2n)3.a'' m^a .cos.(4 nt—0 n't+i 5—!) e'— tj') + ■ , /dP'\ „ /dP\l [3924] This last expression being developed, as in [392r2o-, &,c.], and integrated, gives this part of ^33'. It may also be derived from die' [39:2:2/], in the following manner. We perceive, by inspection, that the part of [3923c], depending on the factor 5 JY' — 2 JV, can be derived from the corresponding terms of doe [3922/"], by changing same /dP\ . 1 /dP'\ , /dP'\ . 1 /dP\ ,^ , , ( — 1 into -, . -r-r , and -rr mto î-\~r~, )■ Ii we make the \de'/ e' Kde'J' \de'J e \de' J changes in the first line of S e' [3922], which was derived from the factor 5 JV' — 2 JV, [3922 j, &.C.], we get the first line of the expression of ù'bj' [3923] ; and the periodical terms of e'tJ^', corresponding to [3922A:], become equal to the following function, which is used in [3924n] ; 3m-.a?n^ {5m^a-{-^m'^a) rfc' 2.(5»i'— 2)1)3. a' m\/a -[-(f)--(S)] . cos. 2 Ts 1. 2 7; [;3933f/] [392:iÉ] [3923/1 [3923g-] The part of (/ 0" to' [3923c], which is independent of 5 JV' — 2 N, may be derived from the corresponding part of (ZtSts [3908f/, 3910a — e, or 3911 J, by the principle of derivation mentioned in [3922/, &c.] ; that is, by changing m, a, n, e, to, &z;c. into to', a', n', c', to', S:c., i-esjjectively, ivithout (dtering P, P', T~,, or the divisor on' — 2?i. In this way, we find that the fourth and seventh lines of [3911] give the second and third lines of [3923] ; and the periodical terms, corresponding to the fifth and sixth lines of [3911], produce in c' d to' the following terms, 7n2.o'2n'2 Cr/£/P\2 fdP'V^^ . ^™ , ^ /dP\ /dP'\ ^) The sum of the expressions [392.3/, h] depending on the angle 2 T5, represents the value of c' 0' to', [3913] ; which is used in the next note. [3923i] * (2473) The expression [3924] represents, for the planet m', the terms similar to those in [3918, 3918'], which correspond to the planet tn, and are derived from the function [3917a]. The similar function, relative to the planet to', using the symbols T, = 5n't — 2nt-\-5^—2s, JV'=n't -\- ^—-^^ is |2 Ô' e' + 2 e' 6 to' . 5 / — e'. (f3 to')2| . sin. TV — \2 e 0' zi' -{- 2 5 e'. à to'— 2 .5 s'. S e'\ . cos. W. [3924«] [39246] 112 PERTURBATIONS OF THE PLANETS, [Méc. Cél. If we denote the inequality of m', depending on the angle 2nt — ■in't-{-2; — 4s', [3925] 6 v' = K'. sin. (4 n'^ — 2 M i + 4 s' — 2 E + 5'), [3924c] By the inspection of [39076, c, a], we perceive, that o c, 5zi, become equal to <) ë , '3ra', respectively, by changing the elements m, a, e, he. into m', a', ë, he, iviihout altering P, P', T5, or the divisor 5n' — 2n ; upon the principles of derivation used in [3923^]. By this method of derivation, we may obtain — ë.{5zi')^ from [3917c], and we find, that it is equal to, and of an opposite sign to the part of 2 5' c' [3922n] ; so that these terms destroy each other, in the value of 2 0' e' — c'. (0 ro')^ ; and then the other part of 2'5'e' [3922/:;], spoken of in [3922o], produces the following expression ; [3924rf] -e'.(<5ra')^ 3m3.a3n3 (5m/a + 2HtVa' ) (5)i' — 2n)3. a'' m\/'a + dp de' .cos.2T^ Now if we represent, as in [3913'], by 5 s', the part of 5 v' [3846, &c.], depending on the angles Tg, 2T^, and notice, as in [3914a, Sic], only the chief terms of 5s' depending on Tj, we shall get the following value, which is similar to [39146], [3924e] 6e'-. 15 m . a'n'^ S-P.cos.T, + P'.sin.T,}. ■(5n'— 2îî)a Multiplying this by 2 e' 5 to' [3907c, n], and substituting the values [3917c?], we get /dP'-^ [3924/] [3924e'] [3924/1] [3994i] [3924*] dP t' ' à ûî . f) s := Ti — . < + [ (5?i'— 2îi)3 ■^■■(^ dP de' 2T, .C0S.2T, We have very nearly 5?i'=2w [38 18c/], and n^(P^n~c? [3709']; multiplying these two ecjuations together, and the product by 3 m^, we get 15 m^. a'^ k'^= 6 vr. a^ n^ ; substituting this in the first factor of the second member of [3924/], it becomes 15to2. a'2?i'3 3 m9. a3 n3 2 711 \/a {5n'— 2/1)3 {5n'— 2îi)3.a' m /a ' and then the sum of [3924c/,/] becomes, by \vi'iting, for brevity, M^ 3 nfi. (t3 »3 {3mv/a + 2mVV)_ (5ji' — 2n)3.a' my' a i.2ï; 2 <5' e' + 2 e' <5 i;i'. <S£' — e'. ((5 îs' )2= +^.[P'.©-p.O].cos..r. Vf. ii. § 13.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 113 and the great inequality of m' [3891] by* ^'= — H'. sin. (5 n't — 27it + 5s' — 2s + Â'), [3926] Again, if we multiply together the two equations [-SOOTc], and divide the product by iJ\I\.e' [3907n], we shall get an expression of — 2ôe'.ôzi', similar to [3917À;], »k', a, n, e, being changed into m, a', ii! , e', respectively, without altering the divisor 5n' — 2n. Adding this to the part of — '^t'h'-ui, deduced from [3923A], we [3924?] find that the sum becomes nothing ; and the term of e' i5 ' ra' [3923/] produces the following expression, ^p.(^^^+P^(l^^1.eos.2T, — 2e'ô'a'— 2(5e'.5w'= — — T-;— — .' -—^, ^^^< >. [3924m -M-£)--m--^' 2 Multiplying —M-.-oe' [3907c] by —j^, and by (5 s' [3924e], and reducing, using [3907a, 3917(/], we get rp.(^)+P'.(^)1.cos.2r, ^^^•^'= ron'-2nf ■< - ..^.. ..- - >; [3924n] H-m-^-m---^ in which we must substitute the factor [3924/t] ; then the resulting expression being added to [3924w], using M^ [3924*], the sum becomes ^'•■['■■('S)+^'-(S): . cos. 2 7\ -|2e'a"V + 25e'.ow' — 2(5s'.5e'}=<( \. [39240 -'=■[-■■©--• (^: .sin.STs Multiplying the equation [3924^'] by sin. W, and [3924o] by cos. W, then adding [3924p] the products, we find that the first member of the sum is equal to the flmction [39246] • the second member, reduced by formulas similar to [3917o], is which, by resubstituting the values [3924i, «], becomes as in [3924]. * (2474) If we interchange the elements of the bodies m, m', in [3826], and suppose B to become B', and i = — 2, we shall obtain an inequality of the body m', of the form [392.5]. Substituting % = bnt—2nt-\-bz -2e, ?F3=n7 + £', W'=nt-\-î—u/, t^^^*^ we find that the expressions [3925, 3926] become, respectively, àv'=K'.5m.{Ts—W^-{-B'); ^'= — H'.sin. (T5 + J'). [3926t] VOL. III. 29 114 PERTURBATIONS OF THE PLANETS, [Méc. Ce}. we shall find, that the inequality of m', depending on the angle ^nt — 9n'i + 4s — 9/, is represented by [3927] 6v'=\ S^''''^''^^"^^'''\ h' K'.ûxi.ant — 9n't + ^B—9^'—B' — 7i'). m\/a ^ These may be reduced to forms similar to [39206, c], respectively, by observing, tliat the term 2e'.sm.{n' t-^-s — zi), in the motion of m' , similar to that of m [3913''], may be developed as in [3915], and will contain the terms 2 (5 c'. sin. W — 2 e' ^ w' . cos. W, which may be reduced, as in [3916/"], to the form n , and by the usual process, as in [3916A, i], it may be reduced to the form K'.^m.{T^—W'-\-B^). Now if we put B^=B'—zi', and W=W3 — -^' [3926a], it becomes, as in [39266], iT'. sin. (Tg — fFg-j- J5') ; so that by substituting the value of Al^ [3907a], we shall have identically, in like manner as in [39206], [39-26e] K\àn.{T,-W,+B') = ^^l;^^^^^.^-(^^^ Putting the two expressions of the chief terms of the great inequality [3924e, 39266] equal to each other, we get, by changing the signs, [3926/] E'. sin. {T, + .5') = '^^:^,-\P- cos. T,- P'. sin. T,^ The identical equations [3926e,/] are similar to [39206, c], and may be derived from them [.3926g-] by changing m!, a, n, e, to, J3, B, K, H, fV^, into m, a', n, e, -a', A', B', K', H', W^, respectively; also multiplying the second member of [3920c] by if-, without altering the angle T^ , or the divisor ( 5 n' — 2 n ). Making the same changes in the product of these two equations, and in [39216], we get from this last the following equation ; 15m9.o'3n3 3 L \de J \de [3926.] -àH'jr'.cos.(^3-22'.-5'-..';=-^^;^;^^.^_ ^ sin.(^3— 275— ra') , This equation being identical, we may change fV^ — 2T5, into IV^ — 2 T5 -\- 90'' ; then multiplyine; by 7p~, , we find, that the second member of the product ^ •' ° -^ 2 ni v/o ^ becomes equal to the expression [3924] ; and the first member becomes equal to [3927] ; [.3926t] observing that W^-2T^ = 'int-9n' t'j-4e-9s' and 15^^. a'^M'^^Gm^ a^n^ [3924§-] ; therefore the expression [3927] is equivalent to [3924]. VI. ii. § 14.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 115 14. The nodes and inclinations of the orbits of Jupiter and Saturn are subjected to variations analogous to the preceding. To determine them, toe shall observe, that c, ç', being the inclinations of the orbits to a fixed plane, and ^ 6' the longitude of their ascending nodes, we shall have, as in [1338], [3928] bj reason of the smallness of <p, 9',* 7 . sin. n = (?'. sin. è' — ç . sin. è ; [3029] '/ , COS. n = (p. cos. 6' — ip . cos. é. [3929'] Moreover, from [3906], we havef m'\/d in'\/a i . (?'. sin. 0') = — 5:^ .5.(9. sin. ; ^3930] à . ((?'. COS. ù') = — '^^^^ .5.((p. COS. Ô). [3930'] The subject of the small inequalities, treated of in this article, is resumed by the author [3926A] in the fourth volume [9062, Sic] ; where he notices terms of the order m'^. e^, &c., which are omitted in [.3914c]. His object in using the indirect methods, adopted in this [392(5q article, is to avoid the great labor of a direct calculation ; assuming as a principle, that these very small inequalities may be determined in this manner to a sufficient degree of exactness, for all the purposes of practical astronomy ; as will appear from the minute examination f3926»i] of the terms of this kind in [9041 — 9114]. *i (2475) Comparing the notation in [1337', 3902], we get â/ = n; tang.i|)/=:tang.7=y [3929a] nearly; hence the equations [1.338] become p' — p=y.sm.n, q — q = y .cq?,.TI. [39296] Now on account of the smallness of 9, we have very nearly |? = 9 . sin. â, q= o . cos. [1334]; and in like manner, for the orbit of m', p' = 9' . sin. â', q' = ip' . cos. 6 Substituting these in [3929J], we get [3929, 3929']. [3929c] t (2476) The variation of the second member of [3929], arising from the action of the body m' upon m, is represented by — (5 . ( 9 . sin. ê ) , because ç', é', do not [3030a] vary by the action of m'. The variation of the first member of the same equation, usbg the characteristics &,, <S„, as in [-3399', -3904], is <5,,. (7 . sin. n) ; hence by development, we have — S.{((>. sin. = ^7- sin. n + 7 . J,, n . cos. n. [39306] In like manner, the variation of the second member of [3929], relative to the action of the body m, which does not affect 9, ê, is ô . {tp. sin. ^) ; and that of the first member is 116 PERTURBATIONS OF THE PLANETS, [Méc. Cél. From these four equations, we deduce the following,* in! \/a' . ^ . • ^ X ) [3931] &,==-^^^_ç-^^^,.{6y.cos.(n-é)-y.6n.sm.(u-ê)]; [3931'] cp5è :z= jn^ a c, _ gjj^_ ^^ — o) + y.^n . COS. (n — ê)\ ; [3932'] [3930c] 0) h^ = — . "', , , , .Uy . sin. (n — â') + 7.6n.cos. (n — ô')L m \/a -\- m \/a ' ^ / ■ ' v ^ > 5^.(7sin. n); hence we get [.3930fZ]. Substituting successively in this the values [3906, 39306], we finally obtain [3930/], as in [3930], [3930rf] (5 . ( ip' . sin. â' ) z= 5, y . sin. n + 7 • <5, n . cos. n [.3930e] = £;^.|<5„7.sln.n + 7.<S,,n.cos.n^ [3906]; [3930/] =-£^.5.(9.sin.â) [3930è]. [3930iir] In the same way, we may deduce [3930'] from [3929']. [3931ff] * (2477) We shall put, for brevity, M= — ,'"^", , , , Jf, = — 7 ^'', -, ; then taking the variation of [3929], relative to the characteristic 5, we get, by the substitution of [3930], the following equation, 5.(7. sin. n ) ^ <5 • ( <?'• sin. Ô') — (5 . ( p . sin. Ô ) [.39316] = — ^^.5.(?.sin.é)— 5.(ç,.sin.â) = — ITT • 5 . (9 -sin. â), («V» •'"7 or [39316'] 5 . ( (p . sin. () ) = — M7 . 5 . ( y . sin. n ) . [39316"] In like manner, from [3929', 3930'], we get 5 . ( <p . cos. ^ ) = — Jlf7 . . ( 7 . cos. n ) . Developing these two equations, we obtain [3931c] i5 (p . sin. Ô -f (p (5 Ô . cos. & == — M~ . (^ 7 • sin. n + 7 . 5 n . cos. n ) ; [3931d] ^ (p . cos. â — (p (5 â . sin. = — .M- . (^ 7 . cos. 11 — 7 . 5 n . sin. n ) . Multiplying [3931c, t/] by sin. Ô, cos. ^, respectively; adding the products, and substituting sin.- ^ -f cos.^ â ^ 1 , sin. n . sm. â + cos. n . cos. â = cos. ( n — Ô ) , [3931e] cos . n.sin. é — sin. n .cos. ô = — sin. (n — é), we get [3931]. Again, multiplying [3931c, rf] by cos. d, — sin..", respectively; adding the products, and making similar substitutions, we get [3931']. t (2478) We may compute the equations [3932, 3932'] from [3929—3930'], in like [3932a] manner as in the last note ; or more simply by derivation, in the following manner. VI. ii. >^ 14.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 117 Therefore the variations of ç, è, <?', o', depend on the variations of 7 and n. We have, by ^12,* ( (—].cos.(57i't—2nt + ôs'—2B) ,.,,.. , (—).sm.(57i't—2nt + 5e'—20 (/ r m' \/a' + (^ycos.(ôn't—2nt + 5^'—2s) If we change m, a, (p, ê, 7, into ?»', «', ç»', «', — 7, and the contrary respectively, in [39326] the equations [.3929 — 3930'], they will remain unaltered, as will be evident by changing the signs of the two first of these equations, and multiplying those which are derived from the two last by the factor —'^- Making the changes [3932&] in [3931,3931'], which [39.32c] are deduced from [3929—3930'], we get [3932, 3932']. * (2479) Substituting the values <5„ 7, 3„ n [3900, 3901], in [3906e], and using, for brevity, the symbols T5 [3890i], also an = a~^, a n' :i=^ a' ~ ^ , M, = "''■ ° "+"^ • "' "' = (!!LV^+J!^Vg:) ^ ^j M _,„,,„ „_(i!H^+_Z!^vV).„,-.„„^ [3933«] m'.an m' \/a' m ^/a we get The divisor 5n' — 2n is introduced in 5 s, &c. [1342,3899 — 3901], by the integration relative to t, spoken of in [1341a. &c.], in finding p, q, s [1341, 1342] ; where the angle T5 is considered as the only variable quantity ; the very small terms, of a different [.39.33c'] form or order, depending on the variations of the elements, which enter into the second members of [1342, he, 39336, c], being neglected. If we again resume the differentials of the expressions [.3933è, c], upon the same principles, we shall get d [.3933rf] ^^-M3..'.an.^(4^).cos.T,-Q.sin.T,^; ^) . sin. T, + (^) . cos. n^ . [393:3.] [3933/] rf.{6n) ,, m'.an ( /d P — ; — = — Mg. . dt 7 These equations are equivalent to [3933, 3933'], omitting the characteristic S, which merely signifies, that the calculation is restricted to terms depending on the angle Ts [3893']. VOL. III. 30 118 PERTURBATIONS OF THE PLANETS, [Méc. Cél. r3933"l Hence we deduce, by neglecting periodical quantities* ivhose effect is insensible, and observing thatf * (2479a) If we compare the expressions [3842, 4401] with the numerical vahies [SPSSg'] of e'% e", y, or tang. 7 [4080, 4409], we shall easily perceive, that the terms of P [3842], depending on 7, are not a thirtieth part so great as some of the tenns depending on e", e" ; therefore the periodical inequalities depending on the variation of 7, will evidently be much less than those arising from the variations of c'", e". Now from the computation made in [4438, 4496], it appears, that these last inequalities are nearly 4' and 9' ; hence it is evident, that we may neglect the periodical quantities spoken of in [3933"]. 13933/t [3933i] [39346] [3934c] [3935a] [39355] t (2480) Dividing [3842] by a', and taking the partial differentials relatively to 7, we get [3934a] ^'- (^) = 2 M^'\ e' 7 . sin. ( 2 n + «') + 2 M^^\ e 7 . sin. ( 2 n + ^ ) ; m'. ('^] = 2 M^». c'. sin. (2 n + ^j') + 2.¥'5). e . sin. (2 n + -^ï). \0 7^ / Multiplying the second of these equations by 7, it becomes equal to the first ; hence we get, by dividing by m', 7./-— -j = (- — j. In like manner, from the values of m'.ct'P' [3843], we obtain y .( ., j:^(- — j; dividing the first of these expressions by the second, we get an equation, which is easily reduced to the form [3934]. t (2481) To obtain the effect of the variations of P, P', ,?, ^', in dy [393-3], we may proceed in the same manner as we have done in notes 2461, 2462 [3909a, &ic.], in finding the variations of de, d^. In the first place, we must substitute, as in [3908a], ^, 1^' for nt, n't, in [3933], and use the symbols [3933»]; hence we get rf7 = _.;^f8.m'.anrf^^(^).cos.(.5^'-2^^-5s'-2.)-(î^).sin.(5^'-2^+5.'-2£)^. Substituting in this the values [3909f?], we get the following expression, which is nearly '" similar to [3909e], changing e into 7, &:c., and writing, as usual, dSy for Sy, Ç /dP\ /dP'\ ') d5yz= Ma ■ m'. andt . < — ( -— ) . cos. 7^ + ( —r~ ) . sin. T5 f C \dyj \dy J > [3935c] + M^.m'.andt.{?>^'—2]V) Âi^-^Vcos. r5+('~) .sin. 7^^ . The variation of this expression, arising from i5 e, '5 w, (5 e', (5 13', 5y, 5 n, in the two first terms, may be found as in [3909e — fc] ; or more simply by derivation, in the following VI. il. § 11.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 119 3 »i'2. g g ?i3 (my/a-f wtVa') (5my'a+2mVa') ( /'<'^'\_p' {iJL\ \ '{ân'—^nf' mVa' ' ?nV«' ' ( Xd-y) Kdj/S <5r = - + + 5 n' — 2 )i m V"' mm'.aa'.nn' [mi/a-\-m'\^a') 'Sn'— 2^" f/rfP'N /ddP\ {dP\ /rfrfP'N) + mV a') ^ f''P'\ f 'ldP \_fdP\ (ddP'\ } _ ^-V^^ "^^ i Vrf«^/ 'Ue'rf"// \de')'\de'dy)\' Inefiuiilily ill tlin inclinutioii of the orbits of Jupiter mid Siituni. [3935] manner. If we change, in d <) c [.3909e], e into y, zs into n, and the contrary ; also [3935;;] m' into Ms-m', without aUering the values of P, P\ JY, JV', T5, e, z/, he; we shall find, that this expression of d5 e becomes equal to that of do y [3935c] ; and by making the same changes in the other expressions of d5e [3909A, /c], we shall get the similar values of day. After making these changes in [3909/c], and putting, for brevity) M^=^Mg.m'.an [-39330], we may alter the arrangement of the quantities, so that the [-39.356] ternis depending on the same differential coefficient may be connected together, and we shall get dh= ^/9.rf^(5JV'-2JV).[('^^).cos.r,+ (^).sin.T5j + JI/g.^/^f^Y(-^e.cos.T5-cfe.sin.T,)+J»f9.(/^(^^y(r5e.sin.7;-efc.cos.r5) + ^ig.rft.['^).(-5e^cos.^5-cW.sin.^3)+^/o.r/^('^^V(^e'.sin.T5-e'^^'.cos.^5) de'd dedy \dedy [3935/] -^M,.dt. (^^).(-57.cos.2'5_76n.sin.r5)+M9.(/^-(^).('5r.sin.T5-y5n.cos.r5) _JI/.../^5n.^(^i^).sin.T.+ ('^).cos.T4. We may neglect tlie fourth and fifth lines of this expression, values [3907O-] in the fourth line, it becomes equal to — , For if we substitute the multiplied by the terms in [3935e-] the first member of [3934], and is therefore equal to nothing. IMoreover, by using the value of ^ n [3907 fZ], we find that the lower line of the expression [3935/] becomes of a similar form to tiiat in the second member of [3909a;] ; the partial difl^erentials of P, P' being taken relative to y, instead of e. Hence we find, as in [3909y], that this line of [3935/"] depends upon xhe periodical quantities sin. 2 T5 , cos. 2T5, which are neglected in the present calculation [3933"] . The three remaining lines of the expression [3935/] being reduced, and integrated relatively to t, produce respectively the three lines of the expresson of ^7 [3935]. For if we compare the first line of [3909/t], multiplied Mg=-^ [3933«], with the first line of [3935/], we shall find that they become [-393.5A] by [3935i] identical, by changing the partial difierentials relative to e into those relative to y ; hence 120 PERTURBATIONS OF THE PLANETS, [Méc. Cél Inequality in the place of the node. [393G] <5n: 3m'2.a2ji3 [nn/a-^m'y/a') [5m\/a-\-2m'^/a'] (5n' — 2n)2.y 7n'\/a' m'^a' + 73 «2 {nn/a~\-m'^a') (5n'— 2n).y m'^a' mm' aa' nn' {m\/a-\-in'y^a') (5n' — 2n).y m'\/a' \de J' \dedy ) ~^ \d7 ) ' \d7d edy J /ddP \d e dP' 7 ddP' 7 ddP 7 ,/dP\ 'ddP\ , fdP'\ /ddP'\ /dP\ fddP\ fdP'\ (ddP \dt' )' \de'dy) ' \d7) ' \de'd ' \de'dyj ' . (dP\ /ddP\ /dP'\ /ddP'\ [3935it] [3935/] [3935j?i] [3936a] [39366] [3936c] [3936rf] [3936e] we obtain the coefficient of t, in the term of 5 y, depending on the first line of [3935/"], by multiplying the first line of [3910J, which is derived from the first of [3909A:], by M^ [3935i], and changing the differential divisor de into dy, as in the first line of [3935]. Again, substituting the values [3907e] in the second line of [3935/], and using Mg m' 2. «2 jtS {m\/a-\- my a') [3933a, 3907o], we get the second line of [3935]. Mg m' \/a! Lastly, substituting [3907/], and mm. aa tin [m\/a-\-m!\/a!^ m' \/a' [3933a, 3907a], Mç^ 5 n'— 2 II in the third line of [3935/], we get the third line of [3935]. * (2482) We may compute (5n from [3933e], in the same manner as we have found h y [3935] from [3933f/] in the last note ; or we may use the principle of derivation; observing that the expressions of dy, ydïl [3933(Z, e] have a relation to each other, which is similar to that of de, ed-m [3908c, (/]. Moreover the former values may be derived from the latter, by changing e, «, &tc., into y, n, Sic, respectively, as in [3935f/] ; therefore we may derive the expression of (5 n from that of 5 y, in the same manner as we have derived «îw from 8 e, in note 2462 [3910cf, &ic.]. Proceeding now as in that note, we shall find, by changing e into 7, &:c. in the terms [3910p, q], and reducing as in [3910?-], that these terms depend on the periodical quantities sin. STs, cos. 2r5, which are neglected in [3933"] and in [3935/(]. In the terms depending on the factor 5JV' — 2 JV, we find, by proceeding as in [39107i:], that we must change \dy) nito - . 7 — — ) ; and by making these (dP'\ , fdP'\ . 1 /dP' changes in the first line oî S y [3935], we get the corresponding terms of ^n in the first line of [3936]. The remaining terms corresponding to those which are computed in [3910m — 0], depend on the second differentials ddP, ddP', and maybe computed from the second, third, and fourth lines of [3935/]; changing T5 into T5 — 90'', as VI. ii. {s 15.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 121 15. If we wish to determine, for any time whatever, the elements of the planetiuy orbits, we must integrate the differential equations [1089, 1132], by the method explained in [1096, Sic.] ; but in our present ignorance of the exact values of the masses of several of the planets, this calculation '■ " '' would be of no practical use in astronomy ; and it becomes indis])ensable to notice the secular variations, depending on the square of the disturbing force, which we have just determined ; since they are very sensible in the orbits of Jupiter and Saturn. These variations increase the values of , , -=— , -f-, — — , &c., relative to these two planets, by the at at at at d t . . ^ h'\Se"' I'^iTu'" l''\Se'" h".ôu" p^.&ca'" , q''\Sè'"' „ quantities* __+-^; ~; ii___ + i__; &c., [3938] in [3910a — d], and substituting the values [3907e — gl ; by this means we shall obtain the corresponding terms, which are to be multiplied by — in d 5u ; or by — in Su, namely, A/P\ /ddP\ , /dP'\ /ddP'\-) , Ma (,/'dP\ /ddP\ , /dP'\ (ddP [393G/] \ /ddP\ /dP^\ /ddP'\~) Ms_ WdP\ / ddP \ •Ml ■ l\de J' \dedy ) ^~ \d7 ) ' \dedy )\~^ M2 i\de')' \de'dy) ' \de' J ' \de'dy Mg WdP\ /ddP\ /dP'\ /ddP'\ + .¥3 ■ l\dy ) ■ yi^J + Wr / ' \ dy^ ) Substituting in this the values [3935/, m], also 17-= 5 K ■ o + '~^> — S — C • T-r, [3933a, 3907oJ, [3936g-] we get, by a slight reduction, the second and third lines of [3936]. * (2483) The equations [1022], corresponding to Jupiter and Saturn, are Ai" = e'\ sin. ra'" ; I" = é\ zo%. v>" ; A"=e\ sin. to" ; /"= e". cos. ■b". [3938o] Taking the variations of these quantities, relatively to the characteristic i5, used as in [3938'], and then substituting the values of sin. ra'", cos. ra'", &ic., deduced from [3938^], we get <5 Ai'= 5 e'". sin. ra'" + e'". <5 w'\ cos. to'"= rîe'" . -|^ + e'\ cS to-". -^ ; [.39386] J /*" = 5 e'". cos. TO — t" . 5 to'", sin. to'" = ô e'" . ^ — e'". h to'". — , &c. [3938c] e'" e'" The secular part of any one of the quantities (5e'", (Îto'", he, 5ra" [3910, 3911, 3922, 3923], may be put under the form ht'^^At; A being a function of the elements of the orbits, of the order m'^. Its differential, divided by at, gives —— =^A== — ; observing, [39,'3S(/] that the variations of A may be neglected, because they are of the order m', and are VOL. III. 31 122 PERTURBATIONS OF THE PLANETS, [Méc. Cél. considering only in 6 é" , 6 n'% the quantities proportional to the time t, ^'"' J determined in the preceding articles. We must substitute, in these last quantities, the values of e", sin. ~", cos. ra'', &c., expressed in terms of h", /'", &c.* The diiferential equations [1089] will then cease to [.?939] be linear ; but it will be easj to integrate them by known methods of approximation, when, after the lapse of many centuries, the exact values of the planetary masses shall be known. In the present state of astronomy, it is sufficiently accurate to have the secular variations of the elements of the orbits, expressed in a series ascending according to the powers of the time, carrying on the approximation no farther than to include the second poAver. We have seen, in [1114", 1139'"], that the state of the planetary system is stable, or in other words, that the excentricities of the orbits are small, and their planes but little inclined to each other. We have deduced this important result of the system of the world from the equation [1153],t [3940] [3940'] [3941] constant = (e- + o"") . m \/a + (e'' + <?'-) . m' \/a + &c. ; for the second member of this equation being small in the present state of the system, it must always remain so ; consequently the excentricities r394'>l ^^^ inclinations of the orbits Avill always be quite small. J We shall now prove that the differential of the preceding equation [3941], [3943] (cde + ^dv) .m^a-\- (e' d e' + <f>' d v') . m' \/a' + &c. = , multiplied by He'", which is of the order m'^, producing terms of the order m'^. For a similar reason, we may nesrlect the variations of ^ , — , Sic. in findine the differentials of [39386, Sic.]. Hence the differential of the last expression in [3938e], divided by dt, is ,.^^t,o -, ^^f^'" doe'" h'" , . rf(5wv Jiv ^jiv 7jiv iT^iv liy r„„„„-, . . [3938e] --— ^ — — . — + e" — -— .- = —-.-_ -f e'' . -, as m [3938], omitting the dt dt e" ' di e'" t v ^ t e'" l J' t> characteristic 6 in the first member. In a similar way, we may obtain the other values ~dT' 'dJ [3938/] [3938] from [3938c, fee] ; also the variations of "^ , '^ , &c. from [1132,1032]. * (2484) The equations [3938a] give e'^' = ^{h'^^ + l'^^), e" =^^{Jc'' + l"""), as [3939a] in [1108]; which are to be substituted in [3938]; and when the resulting quantities are added, respectively, to the second members of [1089, 1132], they cease to be linear in A"', l", he, as is observed in [3939]. t (2485) Neglecting terms of the order (p*, we may put tang.^<p=fp^ and then [1153] r3940a] becomes as in [3941]. [.3941a] t (2486) This must be understood with the restrictions mentioned in note 762 [1 1 14a, &c.]. VI. ii. v^l5.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 123 [3943'1 [3943"] obtains even when we notice the secular variations of the elements of the orbits determined in the preceding articles [3910, 3922, 3935, &lc.]. Hence it will follow, that these variations do not affect the stability of the planetary system. To render this evident, it is only necessary to prove, that if we represent the mass of Jupiter by m, that of Saturn by ??j', and put ie, âe', Ô!?, à if, respectively, for the secular variations of e, é, <p, ip', which were found by the preceding calculations, we shall have (e 6 e + ? â ?) . m \/« + (e' 6 e' + 9' ô ?') . m' /a' = . [3944] If we substitute, in the function (pa© . m^/a + 'P'^'P'- w*'v/^<') the values of «, 6ç, 9', 6ip', given in the preceding article, it becomes* m m' y/a a [3944'] yày ; [3945] [394Gn m s/ a -j- m' y/n' which changes the equation [3944] into , , , , / / , , mm'\/aa' . eôe.mi/a-Jreôe.mi/a'-\ ; — , , . , . y ôy = 0. [3946] ^ m \/a -\- m \/a ■' We shall now commence with the consideration of the first line of the expression of ie [3910], which becomes, by the substitution of a^n^=\ [3709'] ,t -5 6=- r^, ; ,7 , ; \ nt.]P. (---]— P'. (--)[. [3947] * (2487) Multiplying [3931,3932] by <f.7n^a, (p'.mYa', respectively, and adding the products, we get />** ■( , , , ,//'' mmVaa' ^ -^r • 1 — P • cos. ( n — ^ + o'. cos. ( n — é')L ^/..<p^^ + q)^9= _^ .<^ V. [3944a] m^a + nWa' ( ^ y S H . { ^ . sin. ( H — â)- 9'. sin. ( n-é')\ ) Now multiplying [3929, 3929'] by sin. n, cos. 11, respectively, adding the products, and putting sin.^n + cos.^n^l, sin. n.sin.â'-f cos n.cos.â' = cos.( 11 — Ô'), 8ic. [24] Int., [39446] we get [3944c]. In like manner, multiplying [3929] by — cos. n, and [3929'] by sin. n, and reducing the sum of the products, it becomes as in [3944f?] ; (?'. COS. ( n — è') — 9 . COS. ( n ^ é ) = y ; [3944c] (?'. sin. {U — é') — cp. sin. ( n — â ) = . [3944(i] Substituting these in [3944a], it becomes as in [3945] ; and by this means [3944] changes into [3946]. t (2488) Substituting a^ n^ = - [3946'] in the first line of 5 e [3910], it becomes as in [3947]. Again, substituting a^ n^ = n [3946'], in the first hne of oe' [3922], [394eo] we get [3943] ; in like manner, the first line of [3935] becomes as in [3949]. 124 PERTURBATIONS OF THE PLANETS, [Méc. Cél. In the second place, we shall consider the first line of the expression of àé [3922], [3948] , V _ .3m.(5,V«+2mV«') C /rf^N /dP\ ) Lastly, we shall notice the first line of the expression of 5 y [3935], [3949] 3m'.(5mv/« + 2mV«') (>V« + mV»') ^^ ^ (^ /^^^ p' /l^\? (5n'— 2h)2. «^/f/ ■ mV«' ' ( ' \'h J 'vhj\' If we notice only these terms, we shall find* , , , , / , / , mm'\/na' e à e . m\/a + e à e . m u a -\ — — — , , , , .y&y * * m \/a ~\- m \/a' [3950] 3mm'.(5m\/f> + 2m'\/a') \ ' L '\d7 J '^ ^ ' \de' ) ~^"'" \J^ )j [3950] Now P, F', being homogeneous functions of e, e', y, of the third dimension, we shall havef therefore the equation [3950] will become ,,,,,,, mm'\/aa' ^ [3952] ede .m\/a + e àe.m i/a -i ; — . , , , .y6y = 0. * (2489) Substituting the terms of 5e, 6c, 6 y [3947,3948,3949], in the first member of the expression [3946], it becomes as in the second member of [3950]. f (2490) The expressions of P, P' [3842, 3843], are evidently homogeneous in e, e', y, and of the third dimension. Now the theorem in homogeneous functions [3950a] [1001a], by putting n = 3, a = e, a'^e', a"=y, A"^=P, becomes as in [3951]; and if we put ^''':= P', we get [3951']. Substituting these in [3950], we get [3952]. VI. ii. §15.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 125 We shall, in tlie next place, consider the followhig terms in the fourth line of 6e [3910],* ^ {5n'-'in).a'l\de )\de'i ) \de ) \ dt"~ )^\dy )'\dedj ) [d y ) '\d e dyJS ' ^^ and the terms in the third line of 6e' [3922], ,, mm'.t ^/rfP'\ /ddP\ fdP\ /ddP'\ fdP'\ / ddP \ /dP\/ddP'\-i "^ — [5n'—%n)Vaa''l\d c )\jid^j~\J7 J\J777)^\1^ jXdTd^r^JTy )'\^^ ^1 also the terms in the second line of & y [3935], . m'2.< ( mt/a+)«Va') i/dP'\ /ddP\ (dP\ (ddl ['yii—'in).a' m'^a' C\de J ' \dedyj \de J '\ded m'^.mt j \ de J fdP\ C /ddP'-^ ddP'\ , (ddP'\ tddP'\~i dti )+^- Vrferfe'J+^" \dtdy)\ (bn!—2n).\/a'\ , /dP'\ C /ddP\ , , /ddP\ , /ddP\-) I -^[-djji'idûyJ+'-UVdVj+^i-d^n fdP\ i /ddP'\ , , fddP'\ , /ddP'\} -w}-r\iTj^)'^'-Wd^)+^-[ih^)s [3955] we shall have, by noticing these terms only, and observing that we have, as in [3934], /'dP'\ /ddP\ /dP\ /ddP'\ / , , , , / , , m m'\/a a' eôe .m\/a + e 6e. m i/a -j ; — ; , , , .y6y '!^^ 5, (^-\+c- r^^^Uy i'^-^\l\ de )'l \de^ )^^-\dede')^''-\dedy)S \ ;t [3957] * (-2491) The part of (5 e in the fourth line of [3910], by tlie substitution of «2n2^- [,3946'], becomes as in [.395.3]. Again, we have an = ^ , a'ti=— [.3946'], a a', n «'= — — - ; substituting this in the tiiird h'ne à e' [3922], it becomes as in [3954]. Lastly, substituting a^ n^ = - [3746'], in the second line of f5 y [3935], it becomes [39526] as in [3955]. t (2492) Adding the two terms [3956] to the two terms hettveen the braces, in tlie last factor of the expression of '5 y [3955] ; it becomes of a symmetrical form with the [3957o] values of 5e, a e' [3953,3954]. Substituting these values of & e, he', &y, in the first member of [3957], and connecting togetlier the terms depending on the same factors of the [3957i] first order, it becomes as in the second member of [3957]. VOL. III. 32 126 PERTURBATIONS OF THE PLANETS, [Méc. Cél. [3957'] ( ) and ( ) are homogeneous in e, e', 7, and of the second \d e J \de J dimension ; hence we have* <^ dP_ in fcJdP\ . , fddP\ . fddP\ /ddP'\ , , /ddP'\ , /ddP'\ _ /dP'\ [3958"] Moreover ( - — ■ ) , ( ^ ) are homogeneous in e, c', 7, of the second \«7/ \'h / dimension ; therefore we have hence we find, by noticing these terms only,! [3960] eôe.m^a + e'ô e'. m' k/o! + -^p''^y ,.76-^ = 0. ^ ^ m \/a -f- m \/a Lastly, we shall consider the following terms of f>e,X included in [3958a] [39586] * (2493) It evidently appears from tlie values of P, P' [3842, 3843], that /dP\ fdP'\ /dP\ /dP'\ , ...,-, ( -— j , ( — — j , ( 7~ ) ) ( ";; — ) îire homogeneous tunc.tions m e, e , 7, of the second degree, corresponding to the formula [1001a,], supposing « = e, n'=e, a" = y, m=2. If we put, in this formula, ^(''=^-— j, we get [3958]; and ^"i=r-_j gives [3958']. In like manner, by putting successively, ^"i = ( — j, ^'■' = f-—j [1001a], we get [3959, 3959']. ^ t (2494) Substituting the values [3958, 3958'] in the first and second lines of the [3960a] second member of [3957], we find that these terms mutually destroy each other. In like [.30(106] manner, the terms in the third and fourth lines of [3957], are destroyed by the substitution of [3959, 3959'] ; and the whole expression becomes as in [3960]. t (2495) Substituting aa'.nn'= [3952«], in the last lines of the values [3061a] of Sc, 5 y [3910,3935], we get [3961,3963], respectively. Putting a'^n'~=-, [3952o], in the second line of & t [3922], we get [3962]. VI. il. § 15.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 127 the seventh line of [3910], m m'. I WdP'\ fddP /dP\ ( 'i'^P' \_,(dP'\ fddP \ /dJl\fddP'\) {r}n'—'în]Vaa''l\,le'J\dede'J \de' j\de de' )'T'\d y )\,ledy j\,ïi )\77d^) )' and the terms of 6e', in the second line of [3922], namely, g .^ »A< W'^P"\ fddP\ /dP\ / ddP' \ /dP^\ / ddP \ /dP\ /ddP'\-) also those terms of o;. , in the third line of [3935], 5y = (my/a + mVn' ) W'^fl\ (ddP\ (dP\ I'ddP' (.5?i' — 2n).^/aa' m'\/a' ' ' ' \\dt' J ' \de' dy Hence we shall have, by noticing these terms only,* dP \ /ddP'\ -) de' J\de'dyJ ) ' , , , , , , , , mm' Wa a' e ie . m i/a 4- e ôe. m wa -\ —- — , . -. 5 7 = . Therefore the equations [3946, 3941] hold good, even ichen ive notice the terms depending on the square of the disturbing force [3910, 3922, 3935]. * (2496) Substituting the values of à e, Sc', 5 y [.3961—3963], in tlie first member of [3964], and reducing, as in the preceding notes, by means of formulas similar to [3958 — 3959'], we shall find, that the terms mutually destroy each other. But without taking the trouble of writing down these formulas at full length, we may abridge the calculation, by the principle of derivation, in the following manner. If we multiply the values of 6e, 5e', 5 y [-3953, .3954, 3955], by the factor m\/a III' y/a' and in the terms whicli are connected with the two differential coefiîcients ( — — ) , ( -—- ] , change the partial differentials of P, P', of tlie first order relative to de, into those relative to de; and in the differentials of the «ccowfZ on/cr, d e^ into de de', de de' into de'^, d e d y into d e' d y, the other differentials being unchanged ; we shall obtain the three expressions [3961, 3962, 3963], respectively. The same changes in the partial differentials may be made in [3958 3958']; as is evident by putting, in [1001«], a = e, a'=^t', a"^y; and then .7'" ^ f — j , to obtain the equation corresponding to [39.58] ; also ./2®==(-T-r j , to obtain the equation corresponding to [3958']. To render the expression [3963] symmetrical, we may, as in [3957a], add the two terms [3956] to those between the braces in [3963]. Hence it is evident, tliat if we substitute these values of oe, 5e', Sy [3961, 3962, 3963, 3964/], in the first member of [3957], liie result will be equal to the second member of [3957], multiplied by the factor [3964i], changing also the partial f39(;il [.3002] Tlie sta- hility of the orbit of a planet 19 not (lis- lurbed by [3904] lerrns of the order of the [30tJ4'] ptjuare of the dis- turbing forcen. [39<34a] [39046] [.39G4c] [39(>4(/] [3964e] [:39C4/] [3964g:] 1-28 PERTURBATIONS OF THE PLANETS, [Mtc. Cél. The determination of the invariable plane, given in ^62, Book II, is founded on the three equations,* [3965] c =m \/a.{l-t^) • COS. (p + m' ^f77(l^^'2) . cos. y' + &c. ; [3965'] c' = m i/a ."(1— e^) • sin. (? . COS. â -j- ?/«' y/«'.(i — e'"-^) . sin. <?'. cos. o' + &c. ; [3963"] c"= 7rt \/aT{l^^) . sin. (? . sin. ô + m' \/«'.(l— e''-^) . sin. a', sin. ;)' + &c. ; « and a' being constant, having regard even to the terms [3906' — 3907], [3965'"] depending on the square of the disturbing force. The first of these equations gives, by neglecting the products of four dimensions in e, e', &c., W, ({>', &c.,t [3966] constant = ( c" + if" ) . hj \/rt + ( e' - + o' - ) . m' \/a' + &c. ; and we have just seen, in [3964'], that the terms depending on the square [3966] of the disturbing force, do not- affect the accuracy of this equation. The [3964/i] differentials, as in [3964f]. Now the third and fourth lines of the terms between the braces, in the second member of [3957], remain unchanged [3964(/] ; they must therefore vanish, as in [39605], by the substitution of the expressions [3959, 3959'J. In hke [3964il manner, the first and second lines vanish, as in [3960a], by the substitution of the two equations found in [3964e], corresponding to [3958, 3958']. Hence the second member wholly vanishes, and the result becomes as in [3964]. We may remark, that this [3964/t] demonstration is restricted to terms having the small divisor (5n' — 2rt); but it is extended to other terms in [5935, Sic.]. * (2497) Substituting ( 1 -f- tang.-.p)~' =cos. 9 ; ( 1 -)- la'ig-^ <?')~*=^ cos. p', &c. [3965a.] j^ [1151], it becomes as in [3965]. Making the same substitutions in d , d' [1158,1159], and putting also, as in [1156], ?; . cos.ffl = sin. 9 . sin. ^ ; (jr. cos. (p= sin. 9. cos. ^ ; y. cos. 9':^ sin. 9'. sin. d', &jc., we get [3965', 3965"] It may be remarked, that the quantities c', c", are in the original work called c", c', respectively ; tliey are here altered so as to conform to the notation in [1158, 1159]. t (2498) If we neglect terms of the order t"*, ©'', we shall have [3966a] /a.(i_e2) = (l — ie2)./«, cos.(p=l — Iv^ [44] Int. ; hence m \/a . (1— e^j . cos. o == m \/a — J . («^ -(- 9-) . ?« \/o ; substituting this and the similar terms of a', c', 9', Sic, in [3965], it becomes [39666] c = m /« -f- ?«'/«' + &.C. — I .\{t^-\- (f) .m\/a-\- (c'^ + 9'^) . m' \/a' -f- &:c.|. Multiplying this by — 2, and transposing the constant terms — 2m\/a, — 2in'\/a' — &«;. to the first member, we get [3966]. [39656] [3965c] VI ii. À 16.j DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 129 equation [3965"] gives, by neglecting the products of three dimensions in c, e', &;c., (.?, ?', &.c.,* 6.(3. sin. (1) . m v/fl + 6 . (.;'. sin. c') . m\/a' + &c. = 0. [3967] Now if we notice only the terms depending on the square of the disturbing [3967'] force [3931 — 3936], f this equation will hold good ; therefore the expression f"= 7)1 ^/^(i^.2) . sin. V sin. <-' + m' \/a'.{[^^^) . sin. ç. sin. l,' + &e. [3961:'] [3965"], ^vill not be affected by these terms. In like manner, we find, [3968] that a similar result is obtained from the equation [3965'], c'= m y/a7(l— 6^) . sin. o . cos. J + m'/a'.(l— e'^) . sin. ©'. cos. è' -f &c. [3969] Hence the invariable plane, determined in § 62, of the second book [3969'] [1162, 1162'], remains unchanged, even when toe notice these terms depending on the square of the disturbing force. 16. The terms depending on the square of the disturbing force, have a sensible influence on the two great inequalities of Jupiter and Saturn ; t we [3969"] * (2499) Neglecting terms of tlie order y^ ^'^ we may put siii.(p=(p; sin.o'=:(p', &c. [4.3] Int. If we also neglect terms of tlie order e^?, e'^ç)', &c., the equation [3965"] may be put under the form c"=: ( 9 . sin. 6) . m s/ a -\- ( ç'. sin. a') . ni' \/a' -\- &c. ; and [3967a] if we take the variation relatively to the characteristic (3, it becomes as in [3967]. t (2500) The terms here referred to, are those mentioned in [3943'], and computed for two planets in [3929— 393-3']. The equations [3930, 3930'] may be put under the [3968a] following forms, 0.(9. sin. é) .m s/ a -f- ^ • ( ç'. sin. ^ ) . m' i/(/'^ ; [39686] .((p .cos. é) .m\/a -f- 0" . (ç'. cos. H) .m'^a=^0. In the same manner, other planets produce similar expressions, and the sum of all the equations, corresponding to the first, forms the equation [3967] ; a similar equation may [3968c] also be obtained from the sum of the equations of the second form. % (2501) Substituting the expressions [37566, c, e], in SR [3764], it becomes as in [3970J ; observing, that the coefficients of h^ + P, h'^+l'^ [3764], are equal to [3969a] each other, as appears by multiplying [3752i] by — 4. VOL. III. S3 130 PERTURBATIONS OF THE PLANETS, [Méc. Ctl. shall proceed to determine the most considerable of these terms. We have seen, in [3764], that the expression of R or iR contains the function <«= -^.(.H.-).i2„.(^)+„..(''-:)( ,™, +.'.....os.(.-.).|4.».+.„.(:^) + 2..('->„..(^) [3970] ^f ^^'*^ increase the quantities e, fi', w, ra', r, in this expression, by their variations, depending on the angle b n t — 2nt,* we shall obtain in R some terms depending on the same angle ; and it would seem, on account of the divisor on' — 2n, connected with these variations, that these terms mioht become sensible. But we must observe, that this divisor [3970"] disappears in d R, because the differential characteristic d, refers only to the co-ordinates of m, or to the variations of e, ^ [916'] ; so that it introduces the multiplicator on' — 2 n. Now we have seen, that the great [3970'"] inequality of m depends chiefly on the term 3 affn dt . dR [1070"]. The inecjualities of the radius vector and the longitude, Avhich depend on the variations of the exccntricities and perihelion, relative to the angle [3971] 5 n't — 27it, have therefore very little influence on the two great inequalities of Jupiter and Saturn. We shall see hereafter [4392, &c., 4466, &c.], that the most sensible inequalities of these two planets, depending on the simple exccntricities * (2502) The variation of c, «', ■ro, 8ic., here referred to, are tliose represented ro970a] '^y ^ ^' '^*'' ''^' ^''" [3907 J, c, c/] ; all of which have the divisor 5 «' — 2 îi [3907«] ; but the divisor is destroyed in finding their differentials (/ e, d -a, Stc, as is evident from [3908c, &ic.]. Hence it follows, that the differential of the expression [3970] gives, [39704] in d H R or d R, terms depending on ede, e e' d w, &c., wliich do not contain this divisor ; and if we substitute them in the chief term of the great inequality [3970'"], they will produce terms which are of the order ?h'^. or of the order m', in comparison [3970c] with the chief terms computed in [3844, 4418, 4474] ; but as these terms of the order w'^, [3970(i] have the same divisor (5 n — 2 ?i)^, a* the chief terra, it seems proper to examine carefully into their exact values, instead of neglecting them, as the author has done. We shall also see, in [4006^, &ic., 4431/"], that several terms, omitted by the author, similar [39/ Oc] ■- . . -11 -1 to those treated of in this article, are quite as important as those which he has retauied. VI. il. § 16.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 131 of the orbits, are relative to the angle nt — 2 n't. We shall put* -= F. COS. (7ii — 2n't + s — 2i' + A), [397^] -5, a 5r for the term of — , depending on this angle ; and 6v :=E.sm. (nt — 2n't-\-s — 2s'-JrB), [3973] lor the term of 6 v , depending on the same angle ; also for the correspondmg terms oi -7- and & v , ^ = F'. COS. (nt — 2tt't + s — 2s' + A'); [3974] ôv' = E'.sm.(nt—2n't + s — 2s'-^B'). [3^4'] If we suppose that R corresponds to Saturn, disturbed bij Jupiter, and [3974"] then develop it relatively to the squares and products of the excentricities and inclinations of the orbits, noticing only the angle Sn' t — n t, we shall [3074'"] obtain, as in [3745, Stc], a function of this form,t R = il/(»'. é~ . cos. {Qn't — nt + 3^' — i — 2 z-I) + il/fi' . e e'. cos. (3n'i — M i + 3 i' — s — ^ — ^') + M^-Ke'. cos. (3m'< — n ^ + 3;' — 5 — 2«) + M'^'.7=. cos. {Sn't — nt + Qi'—s — 2n). * (2503) The terms of 5« [4392], depending on the angle nt — 2 n't, or rather on 2»i''i — 71'^/, are of the order 136' or 56', and may be reduced to the form [3973]; [.3973a] those of v' [4466] are of the order 182% 417% and may be reduced to the form [3974'] ; they are the largest terms of the expressions [4392, 4666]. In like manner, tlie parts of —, 4 [4393, 4467], may be reduced to the forms [3972, 3974]; the last of [.3973i] " CI which is the greatest term of [4467] . t (2504) This value of R is similar to that assumed in [3745 — 3745'"], changing reciprocally the elements of m' into those of m ; also M'--'> into M'-°\ M^"^ into M^-'' ; [3975a] and afterwards putting i = — 1. This form of the angles in the value of R, is selected because it produces, in connexion with the variations [3972—3974'], terms in dR, d' R, [39756] of the order m^, depending on the same angle 5 n't — 2 n t, as the great inequality, as is seen, in [3979, 3982, 3985, 3989, 3991]. We may remark incidentally, that in this article [3975] 132 PERTURBATIONS OF THE PLANETS, [Méc. Cél. [3976] The quantity 3I^''\ e'". cos. [3 n't — ?i< + 3s' — s — 2y>') arises from the development of the term of R, denoted by yi'''. cos. («' — v) ;* in which we must increase ?• by S r, r' by i r', v by 6v, v' by i v'. This is the same as to increase, in the development of this term, a by or. a by 6 /■', and n t — nt by &v' — -5 1; ; by which means it produces the following expression,! iJ = _ 3/(»). e'=. {iv' — i v) . sin. (3 m' ^ — n i + 3 / — J — 2 ^') [3976'] [39' [3978] + a. (—,--) . e'-. ~ . COS. (3n't—7it + 3^:'—s — 2zi') d a a + a'. (^^^ .e'-.~. COS. (3n/t — nt+3/—s — 2 ^'). \ da J a [3975c] [3975rf] [3975e] [3976a] [3976t] the values B, i?, , difler from those in other parts of the work ; since B, B, [3974", 4005'] take the place of jf?', B [1199'], respectively; m being the mass of Jupiter, »«' that of Saturn. The object of the author, in making this change in the value of B, is to obtain express formulas for the direct computation of the inequalities of Saturn, which are much larger than those of Jupiter ; and then to deduce the corresponding smaller ones of Jupiter, by means of the formula [1208] ; it being evident, that this method of deduction, in the cases where it can be applied, must be more accurate in finding the small inequalities of Jupiter from the large ones of Saturn, than in an inverse process. * (2505) The part of B, independent of j'^, corresponding to the action of Jupiter iipon Saturn, is found by changing, in [3742], ?»', r, r', v, v', into m, r . r, v', v, respectively ; and if we suppose, that when a, a, nt -\- s, ii t -\- s', are changed into r, r, v', v, respectively, the quantity .4''' [3743] becomes .^/'', we shall get, from [3742, 3743], for this part of B, the following expression, :- . 5; . .^/''. cos. ?" . («' — i;). [3976c] jR = — .cos. («' — v) — ■ ,, o — h — ; 7~~i rT~^^w'' '■ ,.a \ '' v/i' — 2rr.cos.[v — «) + ? jj Substituting in this the values of r, r', v, v' [952, 953], we obtain an expression of B, [.3976(/] of the same form as [957], and possessing the properties mentioned in [957 — 963] ; moreover, the term multiplied by the factor e'^, being represented by [3976c] M^''\e'^.cos. \i . {71' t — nt -^ s' — i) + 2n' t -{- 2s' — 2 z:'} [9.57—959'], becomes of the form [3976], by putting i=l ; then the corresponding term of B [3976r] is of the same form as in [3976']. t (2506) The term Jf '"'. c'^. cos. (3 ?i'/— ?i < + 3 s'— £— 2 to') [3975], is produced in the function B, by a development similar to that which is used in [957], that is, by [3977a] the substitution of the cV/p^icoZ values of u^, v,, &c., without noticing the perturbations [3972 — 3974']. If we wish also to notice these terms, we may suppose a, a', v, v', to be VI.ii.^16.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. \S3 This produces in R, the terms* R^—i, il/"". E'. e'-. COS. ( 5 »' / — 2 n ^ -f 5 /— 2 .■ — 2 ^'— B') + h M"". E . e'-. COS. (5n't — 2nf + 5s'—2i — 2^' — B) [3979] + 1 a' . (^^]. F'. e\ COS. (5 I,.' t-~2nt + o^'—2i-^2^'—A') \ a a J + 1 a . ('^'*^) . F. e'\ COS. (5 n' ï — 2 « « + 5 .-'— 2 -= — 2 ^' — ^ ). increased, respectively, by Sr, (ir, 5v, 5 v' ; liy which means A'Kcos.i.{v' — r) [SQ^ya'] will be augmented by the three terms in the second member of the following expression, in which we have retained the factor i=l, for the purpose of more easy derivation hereafter; [39776] />.\A''\ cos. i.{v' — v)] = — A''\ i .{Sv'—5v). sin. i . ( v'~v) and in the same manner as we have derived from .^''^ cos. i .{v' — v) the term .¥<">. f'^ cos. \i . {n t — nt -\' s — s) -i^ 2 n' t -\- 2 s'—2tz'] [3976e], [.3977f] we may derive the three terms [.3978] from those in [39776]. Thus the first term of the second member of [39776] is the variation of ^'\cos.i.{v' — v) or of J">.cos.(t)' — v), [3977rf] supposing the angle i . {v — u) to increase by i.((5j)' — àv); in like manner, the first line of [3978] is the variation of the term iH'"'. e'2. COS. \i . {n't — nt + i' — e) -f 2 u'< + 2 e'— 2 zi'], [3977e] supposing the angle i .{n't — nt -\- s — s ) , corresponding to i . {v' — v), to increase by the same quantity 6 v' — 5 v . The second line of [3978] is deduced from the second [.3977e'] term in the second member of [39776], by supposing a to be increased by S r in ^'" and .W". Lastly, the third line of [3978] is derived from the third term of the second [3977/] member of [39776], by supposing a to be increased by 5 r , in ^"' and JW'"'. * (2507) The expression [3979] is deduced from [3978] by the substitution of [3972 — 3974], and reducing by [17 — 20] Int., retaining only the angles which are similar to that of the great inequality, depending on bn't — 2nt = {Zn't—nt) — {nt — '2n't) ; [39796] or the difference between the angles contained in [3978] and those in [3972 — 3974']. VOL. in. 34 134 PERTURBATIONS OF THE PLANETS, [Méc. Cél. ^^- We shall pxit à' R for the differential of R, supposing the co-ordinates of m' to be the only variable quantities. In the terms multiplied by E' [3981] and F', the part 5 7i't — ni, of the angle 5 n'i — 2n/,* is relative to these co-ordinates. In the terms multiplied by E and F, the part 3 n t^ of the same angle 5 n't — 2nt, is relative to the same co-ordinates; therefore we shall have, by noticing only the preceding terms of R [3979], rt'd'i?z= i.{5n' — n).di.a'M(°KE'.e'^.sm.{5n't — 2nt-{-às' — 2; — 2m' — B') [3982] [3983] — i.(5?i' — ?0-'/^.«'^-(^^)-F'.e'2.sin.(5n'<— 2»(« + 5.='— 2.-— 2^'— ./î') — ^.n'dt.a'J\I'°\E.e'^.sm.{5n't~2nt-{-5s' — 2e — 2zi' — B) — ^ .n'tl ( .aa'. (■^~-\ . F.e'^.sm.{5 n' t — 2nt + 5 ! —2s —2-:^' — A). The term ilf ". e e'. cos. (3 ?i'ï — n ï + 3 /— .= — ^ — ^') [3975], results from the development of A'^-\ cos. 2 . (v' — v), in the expression * (2508) The difterential relative to d' [-3980], does not affect nt in the angle [.3989«] 3ii'f — nt, which occurs explicitly in [3975], so that d'.{3n't — nl) = 3 7i'cl t ; but 6 v' [39836] this cliaracterlstic d' affects the w/io/e of the values of —, i5d' [3974,3974'], connected with F', E', consequently the whole of the angle nt — 2 7i't, which occurs in these values, must be considered as variable, and its differential is n (t i — 2n'dt. The difference of these two expressions gives [393'25'] [3982f] à'.{ron't — 2nl)=à'.[3n'l—n() — à'.{nt — 2n'i) = {5n'—n).dt; which must be taken for the differential of tlie angle b n' t — 2nt [3979J], depending on E', F', in the first and third lines of [3979] ; hence we obtain the first and second (3982(/] lines of [3932]. In like manner, the differential relative to d' does not affect the [398ae] expressions of —, Sv [3972, 3973], connected with the factors F, E ; or in other words, the differential of the angle nt — 2n' t, connected with these factors, must vanish : and we shall have A'.{nt — 2?i'<)=0; subtracting this from [3932a], we get, in this case, for the differential of [3979i], [398%] d'. (5 7!'< — 2h<) =d'. (3?i'<— M^— d'. (?i t — 2^^ t)=3 n' dt . Substituting this in the differential of the second and fourtli lines of [39791, we get, [3983/!] ^ L J' 6 ' respectively, the third and fourth lines of [3982]. [39841 [3985] VI. ii. ^ 16.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 135 of R* Therefore we must vary, in this term, a by ir, a' by ir', ^^ also 2 n't — 2nt by 26v' — 2iv; and by this means we obtain the following terms of R, R = —2 M^'\ e e'. (d v' — 6v). sin. (3n' t — ti t + 3e —s — z^ — zi') + a. ) . ee. — . cos. (3n r — nt + 3e' — s — ra — ra') \ d a J a + a'. f^^!L_) .ee'.^.cos. (3 n'/ — n r + 3e' — s — ^ — ^'). \ an y a Hence the part of a'd'R, relative to this expression, is a'à'R= {5n'—n).dt.a'J\P^\E'.cc'.sm.{5n't—2nt-lr5s—2s — 7Z—zi' — B') — i.{57i' — n).dt.a'-^.('^-^^\F'.ee'.sm.{5n't — '2nt + bs-22—z^-zs'-A') — 37idt .a' M^'lE .ce'.sm.{5n' t — 2nt-j-5i' — 2i — ^—^' —B) — in'dt.aa'.(^-^^\F.ee'.sm.{57i't — 2nt + 5s' — 2s—z!—'u/—A). The term M<^>. e". cos. (3 n' t — nt + 3^— ^ — 2^) [3975], arises [.3986] from the development of J'^'. cos. (3 y' — 3i'), in the expression * (2509) Proceeding witli the term depending on M^^K [3975], in the same manner as we have done with that multiphed by AI"^\ in tlie tliree preceding notes, we find, that it may be put under the form M^'Kee.cos.\i. {n' t — n t -^ s — s) -J^- ,1' t -j- n t -j- s" -{- e — -ui' — tz], [3984a] supposing i = 2 ; by whicli means it becomes as in the second Une of [3975], and tlie corresponding term of [39~6c], is of the form à?ft.^/*^. cos. t . {v'— v)~A'''''.cos.2 . {v'—v). [.39844] The variations of this term, depending on or, fir', ou, f5 «', are as in [3977è], supposing i = 2; and from these we may deduce the functions [3984, 3985], by a computation similar to that used in finding [3978, 3982]. We may, however, obtain the former by derivation in a more simple manner; for if we change M''^\ c'^, — 2 -a', into rr^,.u. M'", ee, — « — to', respectively, we shall find, that the first term of [-3975] becomes like the second ; and the doubling the values of '5 v', & v, in [.397761, on account of the ^ ' ' L J' [3984rfl factor r = 2, make it necessary that we should double the values of E, E' [3973,3974']. Making these changes in [3978, 3982], they become, respectively, as in [3984, 3985). 136 PERTURBATIONS OF THE PLANETS, [Méc. Cél. of R* Therefore we must vary, in this term, a by 6 r, a' by 6 r', and 13987] 3n't — 3nt by Sôv' — 3iv ; hence we get the following terms of B. R = — 3 M''~\ e\ (ôv' — iv) . sïn. (3 n' t — n t + 3 s'—s — 2zs) [3988] + a . f'^^\ . e". - . COS. (3n't — nt + 3s' — s — 2^) \ d a J a + «'. y^^) • è\ ~ . COS. (3n'i — n ï + 3 -=' — s — 2^). Therefore the part of a' d'R, relative to this expression, is a'à'R= §.{5n'—ii).cIt.aM^-^''.E'.e^.sm.{5n't—2nt + 5s' — 2e—2vs — B') — i.{5n'— n). dt.a"^.(-^^^].F'. e^.sm. {5 n't — 2nt + 5 s'— 2 s — 2zi — A') L3989] —^.n'dt.a'J\'r-\E.e''.sm.{rj7i't — 2nt + 5s'—2e—2z^—B) — %.n'dt.aa'.f-^\F.e''.sm.(5n't — 2nt-{-5s'—2i—2z^—A). [3989'] Lastly, the term M'=' . -/. cos. (3n' t — 7it + 3 s' — s — 2n) [3976], [3989"] arises from the term multiplied by -/.cos. (3r' — v), in the expression of i2;t * (2510) Proceeding as in the last note, we may put the term [.3975], depending on M'-'^-, under the form ^3988o] M<^\ e^. cos. \i . {n' < — n Ï + s'— s) + 2 n < + 2 e— 2 ^f , supposing i = 3 ; and then the corresponding term of [3976f] is of the form pjjjggj, i to'. A}'K cos. i . {v — v) = A^^'. cos. 3 .{v'—v). The variations of this term are as in [3977 J], supposing i = 3; from which we may get [3988, 3989], in the same manner as [3978, 3982] were found. The same result may be obtained more easily by derivation, as in the last note ; by changing, in [3975, &,c.], [3988c] M^"', e'^, A'-'\ 2zi', into M'-'', e^, A'-^\ 2 s, respectively; by which means the first term of [3975], changes into the third; and tlie trebling of the values of ôv', ôv, in [3988(n [3977i], on account of the factor /:=:3, makes it necessary to change E, E' [3973,3974'] into 3E, 3 E', respectively. Making these changes in [3978,3982], they become as in [3988, 3989], respectively. f (2511) We must now compute the terms arising from the introduction of the increments [3990a] i5 r, Si-', &v, 5v', in the expressions of J-, r', v, v , connected with the factor 7®, in the value of R [3742] ; which were neglected in [3976«]. These terms of R may be VI. il. § 16.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 137 we must therefore vary a by 6r, a' by àr', S n't by Siv', and n t bv à V ; hence we obtain the following terms, R =. — M<''. y^.(3ôv'—&v). sin. (Sti't — nt-\-3/ — s — 2u) + a . ('^_:^\ . "-, iZ'. COS. (3n't — n t + 3 /— e — 2 n) \ da J a + «'. ( ^/V)./'. 4-cos.('3n'« — n< + 3/ — E — 2n). \ (I a J a ^ [3990] deduced from those depending on y^, in [3742], by changing the elements as in [3976r(]. These four terms of R [3742] are ah'eady muUiphed by the factor y^, of the second dimension, and as none of a higher order are noticed in [397.5], we may substitute in these terms, r=a, r' = a' , v ^=nt -{- b — n, v'=n(-\-s' — IT; and retain only angles of the form 3n'< — nt, assumed in [3975]. Now it is evident, that the two first of these terms of R [3742], depending on the angles cos. (i;' — v), cos. {v' -\- v), produce the angles n't — nt, n't-\-nt, which are not included in the proposed form. The third of these terms [3742] contains v' — v in its numerator and denominator, and when the denominator is developed, as in [3744], the whole term will depend on quantities of the form cos. ?*.(«' — v) or cos. i.(n'^ — nt), which are not comprised in the form ^n!t — nt, now under consideration ; so that we need only retain the last term of [3742], which, by making the changes indicated in [3976a], may be put under Ttt 'V T T COS I y' ~l~ v ^ the form R = {- . '- ' j. Now if in the formula [3744], 4 {,-2 — 2 rr'. COS. (w'—î))+»-'2 1 ^ we change a, a', nt-\-s, n't-\-s', B'-'\ into r, r, v, v', J5/", we shall get [3990i] [3!)90c] [3990</] [3990(/'] \r^— 2rr'. cos. {v'— v ) -f r'~l ^ = ^S. 5». cos. i . ( v'—v). Substituting this in R [3990e], and reducing by means of formula [3749], it becomes iî = — :i m . f. r r. * 2 . B'p. cos. \i . (v'—v) -f «' + jj j. If we change î into i — 1, and put — |- ?« . r r*. i?/'-" = JW''', we get R = f.S.M'-'^.cos. {i.{v'—v)'}-2v]; which in the case of i^3, produces a term of the form R = M'-^\'^^. cos. {3 v' — v). Taking the variations of this term, as in [3977a', &c.], we get the following expression, similar to [3977è], &.{M'^\f.cQs.{Sv'-v) \ = —.¥"1.^2. {3Sv'— S «) .sin. (3 v'—v) [3990/] [3990/'] [3990g-] [3f>90A] [3990i] Substituting in this the values [3990è], we obtain [3990]. VOL. III. 35 [3991] 138 PERTURBATIONS OF THE PLANETS, [Méc. Cél. Hence we obtain in a' d' R, the following terms,* aà'R= ^.{5n~7i).dt.a'M'-^\E'.f.sm.{5n't — 27^-^-5;'— 2! — ^n — B') — i.{5n'—n).dt.a'~. [~j^) F'. f. sin. (5 n't — 2n t + 5 s'— 2 s — 2 n— ^') — ^n'dt. a'M^^'lE. f. sin. (5 wV — 2 ?i ^ + 5 s' — 2 s — 2 n—B) — ^n'di.aa'.l -j^j ■ F.f. sm. {5n't — 2 nt -{-5i'— 2c — 2 n—^). The most sensible inequalities, arising from the squares and products of the [3991] excentricities and inclinations of the orbits, which neither have 5 n' — 2 nf for a divisor, nor depend upon the variations of the elements relative io the * (2512) Tlie expression [3991] is deduced from [3990], in the same manner as [3982] is from [3978] ; or more easily by tlie principle of derivation. For if we cliange [3991a] M'^°\ e'2, 5v', — 2 «', into Jl/"\ y^, Si'iv', — 2n, respectively, the function [3978] will become as in [3990] ; consequently E' [3974'] must be changed, as in [3984f/], [.39916] jj^^^^ g^,_ Making the same changes in [3982], which was deduced from [3978], we get [3991]. t (2.513) The divisors in [3714, 3715], which may be small, in the theory of the perturbations of Jupiter and Saturn, are i>i'-\-{3 — i)-n, in' -{-{I — î).n, în'-{-[2 — i).n; "•^ and since n'^fn nearly [38 18fZ], they become (3 — %i)-n, (1 — f?).H, (2 — |-î).«. If we put / = 5, the first divisor becomes 0, the others being large. If i = 4, the • last divisor becomes — f ?i, and the others are larger. If / = 3, the last divisor becomes ^ n, and the others are greater then this quantity ; and it is evident, that next to i^5, this value of i gives the least value to the divisors [3992a] ; therefore the terms of 7-0 r, ÔV [3714,3715], of the second order, relative to the quantities e, e', y, and depending on the angle 3?*'/ — nt, maybe increased by this divisor, so as to become greater than other terms of the same order, relative to e, e', y, which have not a small divisor. This reasoning is confirmed a posteriori by the inspection of the numerical values of 5r"', Sr", Hv", û v" [4397,4470,4394,4468], in which the terms depending on the angle 3 n't — n t, are generally greater than any of those that are noticed in [3991'], [.3992^] excepting 4n't—2nt. This last angle is here neglected, because the terms or, ôv,hc., depending upon it, do not produce in [3995], functions of the form [3998], depending on the angle 5 n' t — 2 n t, which are the only ones under consideration at the present moment. Now if we notice only the temis depending on the angle 3 7i't — 7it, in [39926] [3992c] or [3992c] [3714, 3715], we shall obtain for —, Sv, quantities of the forms [3992, 3993], 6r' and in like manner, in —, Sv', terms of the forms [3994, 3994']. VI. il. §16.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 139 angle bn't — Int, are those corresponding to the angle 5 n't — nt. We shall put — = G . COS. (3 n'i — n < + 3 e'— Î + C) , [3999] (5 r for the part of -, depending on this angle ; also ÔV = H. sin. (3 n't — n t + 3 s' — s + D), [3993] for the part of ô v, depending on the same angle ; in like manner, ^-4 =^ G'. COS. (3 71' t — nt + 3 i' — z + C') , [3994] a ^ Ô v = H'. sin. (3n't — nt-ir 3s'— s+D'), [3994'] 5 r' for the parts of -7-, èv', depending on the same angle. The expression of R, developed relative to the first power of the excentricities, contains the two following terms,* R= iV'O'.e.cos. (nt — 2n't + s — 2i'-\-zs) [3995] + N^'Ke'. cos. (nt — 2n't + e—2s' + ^'). * (2514) In the same manner as we have deduced, from R [3976c], the three terms [3916e, 3984», 3988a], of the second order in e, e', we may obtain two of the [3995a] first order in e, e', of the following forms, R= :^-i^\e.cos.\i.{nt-nt^e'-s)-{-7it-i-s-z,\ ^^^^^^ + JV('>.e'.cos. li.(«'<— ?i<4-s'— 6) + ?i7 + e'-ra'}. If we put i = 2, in the first of these terms, it becomes of the same form as the first [3995c] term of [3995] ; and by proceeding in like manner as in note 2506, we easily perceive [3995(/] that this term arises from the development of A'-'^^.cos.i . {v' — v), supposing i = 2, [3995e] as in [3995c]. Moreover the second term of R [3995è], becomes of the same form as the second term of [3995], by putting i=l; and then the term Jl'^'\cos. i . {v'—v), [3995/] upon which it depends, becomes .a'", cos. [v — v), as in [3998']. We have already computed, in the case of i = 2, the effect of the substitution of the variations 5r, or', ôv, Sv', in the development of .^^-'.cos. 2. ( y' — v) [3984i], and [3995g-] we have found that this substitution, in [3984i(»], produces the function [3984]. A similar method may be followed with the first line of R [39956] ; but it is more simple to derive ^ ' it from [3984a, 3984]. This is done by changing, in [3984a], the factor M^^Kee' into JVC", e, and decreasing the angle, which is contained under the sign cos., by the [3995i] [3996] [3997] 140 PERTURBATIONS OF THE PLANETS, [Méc. Cél. [3995'] The first of these terms arises from the development of J'^.cos. (2«' — 2v), in the expression of R ; and in this development we must increase a hy 6r, [3995"] a' by or', 2 n't — 2nt by 2ôv' — 26v; from which we obtain the following expression, R= 2 iV'"'. e . (6 v'— &v). sin. (^nt — 2n!t + s — 2i'-\--:z) + a. (~ — ) .e.— .COS. (îit — 2n't + s — 2s' i-^) \ a a y a ^ ^ /,7JV(0)\ xj + «'. -7^).e.— .cos. (71/ — 2n7 + £ — 2£'+^). \ da J d ^ ^ Hence we get in iî, the following terms,* /?= iV'^i/'.e.cos. (5n'/ — 2n/ + 5E'_2£ — ^ + Z)') — m\H.e.(io%.{b'n:t — 2nt^bi—2i — ^ + D) + ia'.f-^j.G'. e.cos. (5w'/ — 2 7i/ + 5f'— 2.^ — ^ + C") + i«. (— — j .G.e.cos. (5 7t'/ — 2ni + 5/— 2s — ra + C). To obtain the corresponding part of d'7?, we must vary the angle [3997'] 2jj/^ — jj^^ jj^ j^l^g terms multiplied by H' and G';t but in the terms [39954] quantity ?i'i-|-£' — ra' ; by which means it becomes as in the first line of [39956]; then putting ! = 2, it becomes as in the first term of [3995]. The same changes being made in [3984], which was derived from [3984n], it becomes as in [3996] ; observing that when the angle 3 n' i — nt-\-Z^ — i — « — ■s/ [3984] is decreased by the quantity nV -\- s' — ra' [3995fc], its sine becomes [3995i] sin. (2 w' ^ — n < + 2 s' — s— ^ ) = — sin. (n < - 2 «'< + s - 2 £'+ ra) , as in the first line of [3996], and its cosine is as in the second and third lines of the same expression. * (2515) Substituting, in [3996], the values of hr, hv, 5/, <5 y' [3992-3994'], [3997a] reducing the products by [17—20] Int., and retaining only the terms depending on the angle 5 ii! t — 2?i<, it becomes as in [3997]. t (2516) The characteristic d' [3980] affects only the angle 2 m'!', in [3995], so [3998o] that in these terms we shall have à'.^nt — 'i.n t) = — '2,n dt ; but d' aflects the the whole values of ~7 , ^ f', consequently also the whole of the angle 3 n t — nt. VI. il. §16.] DEPENDING ON THE SCJUARE OF THE DISTURBING FORCE. 141 multiplied by H and G, we must only vary 2 n't; hence we obtain [3997"] a'd'B= — (ôn—n).clt.a'j\'^'>\H'.e.sm.{:in't~2nt^5s—2s — r.~\-D') _j.(5„'_„).rf^„'aY''^VG'.e.sin.(5«7 — 2H^ + 5a'-2£— ra+C) 4--2n'tlt.a'JY^'>\H.e.sm.(5n'i — 2nt4-5.' — 2s — a^D) [3998] -~-^].G.e.sm.(5n't — 2nt-{-5e' — 2E—zi-\-C). The term N^'K e'. cos. Çti t — 2 n' t + s — 2 s' + z,') , arises from the r3998'l development of the term of iî, represented by ^''. cos. («' — v)* [3d95f'\ ; ' which occurs in the terms [-3994, .3994'], which are multiplied hy G', II'; so that in these terms we shall have d'.[3n't — n l) =^3 n' c1 1 — ndt. Subtracting [3998a] [39986] from this, we get d'. {5n't—2nt) = d'.(3ji't—n <) — d'. {nt — 2n' t) = {5 n'—n) . d t, [3998e] for the dlTerential of the angle 5 n' t — 2?!.^, which occurs in the terms of R [3997], depending on G', 11' ; it being evident, that the angle 5 ft' t — 2 n t is produced in these ^ terms by combining the angles 3 n' t — nt, ni — 2 n't, as in [3998c]. Substituting [3998ci] this in the differential of the first and third lines of [3997], taken relatively to d', we get the first and second lines of [3998], containing the flictors G', H', as in [3997']. or Again, the characteristic d' [3930] does not affect —, îi v, so that in their values [3992, .3993], which contain the factors G, H, we have d'.{3n't — nt) = 0; subtracting from tliis the expression [.3998n], we get [3998e] d'. ( .5 7i't — 2ni) = d'.{3n't—nt) — d'. {nt — 2n't)=2 n' d i ; [3998e'] which is to be substituted in the differenlial of tlie second and fourth lines of [3997], taken relatively to d', to obtain the third and fourth lines of [3998], containing tlie factors G, H, as in [3997"]. The whole value of d'^ is to be mukiplied by a', to '"^^^^•^^ obtain ddR [3998]. * (2517) We have seen, in [3995/], that the second term of [3995], ./V'". e'. COS. ( /i ^ — 2 n't-\-s—2 i' + to'), [3999a] is derived from a term of i?, of the form .,4^". cos (i;' — v), corresponding to i=\; being of the same form as [3977(/]. Now tlie effect of the substitution of the variations of or, (5/, Ô (', dv', in tlie development of this quantity, having been computed in [3978], we may deduce from it the terms of R [3999], corresponding to the present case, by a similar method of derivation to that made use of in [3995/i— /]. Thus, instead of the ^^^^^^^ VOL. III. 36 142 PERTURBATIONS OF THE PLANETS, [Mtc. Cél. r3998"l ^^ must therefore vary, in this term, a hy 6 r, a' by ô r', n' t — nt by i v — 6v, and we get the following expression, R=. N^'K e'. (6 v' — 6v) . sin. {n t — 2n' t + s — 2s' + v>') [3999] + a. ( -— — ) . e'.— . COS. (n t — 2 n'?; +s— 2 e' + jj') (/ a a [4000] + «'. [~T-r .e'. — .COS. (n^— 2n'^ + f — 2s' + ï5'). \ ail J a Therefore the part of a'd'B, relative to these terms, is* a'à'R = — i.{5n'—n). lit. a JV-^'.H'.e'. sm.(5n't—2ni + 5^—2 s — z^'-]-D') ///jV(i)\ — i.(5?i'— nj.f/i-.a's. f-— -j.G'.e'.sin. (5M'i — 2m< + 5s'— 2s— to'+C) + n'(Z<.aW'.H.c'.sin.(5n'^ — 2n< + .5='— 2s— ûj' + D) — ?i'rf^.o«'. ^-^\G.e'.sin.(5?).'/ — 2Ki'-)-5s'— 2j— îj'+C). The values of M<% iV/'^', M'*', M' =>, are determined in the formulas [4000'] Qf ^^^ jjy changing the quantities relative to m into those relative to m', and the contrary [3975a, 6].t The values of A'^"* and N'-^^ are determined operations mentioned in [3995?], we must, in the present case, change the factor M'-"''. e'" r3977el into A'"', e' ; and decrease the ani^le which is contained under the sign cos., [3999cl •- J ' o c- J '■ by n't-\-e' — to'; by which means [3977e] becomes as in the second line of [39956], [3999(i] or tlie second line of [3995], supposing ?'=1. Now making the same changes in [3978], which is derived from [3977e], it becomes as in [3999] ; observing that when the angle 3n'i — nt-{-3^—s—2-a' [3978], is decreased by n't-\-i — ia' [3999c], it becomes 2 )/< — n <-(- 2 s'— s — to'= — ( ji < — 2«'< + e — 2 s'+ra') . * (2518) The function [4000] may be deduced from [3999], by the method we have used in computing [3997] from [3996]. It may, however, be deduced more easily from [3999/] [-3995^ 3997J . by changing JV*»', e, ra, 6v, ôv', into .V<'>, e', to', iSv, i&v', respectively. For by this means, [3996] changes into [3999]; and H, H' [3993, 3994'] become L ^■' i H, I H', respectively. These changes being made in [3998], it becomes as in [4000]. f (2519) If we put i = — 1, in the terms of R [1011], depending on e, e', and [4000a] retain only these two terms, putting also .4'~'> = .4<'' [954"], we get, for this part of R, relative to the action of Saturn on Jupiter, R=^ — -~.]a.[- — — 2^<')^.e.cos. {2nt — ntA-2s — s — zi) i ( \ da / S ^ ' ' [40005] — ^ ■\<''- ('-TT^W4^^='^.e'. cos. (2nt — v!t-\-2i — s — z^'). VI. il. § 16.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 143 by the equations, a'A'^(»)=— 2m.rtVi'-'— im. ««'. (-^) ; [400i] a' iV<" = m . «' . J('> — i m . a'\ C^^ • [4001'] Connecting together all these partial expressions of a'A'R, we obtain a term of this form,* a'à'R = m n'. I.dt . sin. {5n' t — 2nt + 5^' — '2,^ — 0) . [4002] Hence the term 3 a ffn d t . d' R, of the expression of <5 v', givesf [4002] iv'= — ,^"'''''''\ .sm. (5n't — 2nt + ôs'—2s — 0). [4003] [5 II — 2 II)- ^ This is the most sensible term of the great inequality of Saturn, depending on the square of the disturbing force. [4000c] Changing, reciprocally, ibe elements of m' into those of m, we get the corresponding part of -R, relative to the action of Jupiter on Saturn. Comparing this with the assumed form [-S^QS], after having changed the signs of all the terms contained under the sign cos., in [3995], we get the expressions of JV'», ^'" [4001, 4001']. * (2520) Adding together the parts of a d' R [3982, 3985, 3989, 3991, 3998, 4000], and putting, for brevity, T^ = 5nt — 27it-{-5s' — 2e, we get a series of terms [4002a] of the first form [4002f] ; /' being used for brevity, for the coefficients, and O' for the quantity connected with Tj. Developing this by [23] Int., we get the second form [4002c or 4002fi?J ; in which we may substitute 2./'.cos. 0'=mM'. /.cos. O, 2./'. sin. 0'= — mn'. /. sin. O, [4002J] and we obtain the first form [4002e], which by means of [22] Int., becomes as in the second form of [4002e], agreeing with [4002], a'd'R = dt.-Z.r. sin. (Ts -\- 0')=^dt .S. . F. {sin. T^ . cos. O'+cos. Tj . sin. 0'\ [4002e] = (/ ^ . sin. Tj . 2 . /'. cos. O'-^d t . cos. T^ . 2 . /'. sin. O' [4002(f| = mn'.I.dt.\sm. T, . cos. O — cos. T5 . sin 0\ = mn'.l. dt .sm. (T. — O). [4002e] t (2521) Multiplying [4002] by S n' d i , and then integrating it twice, relatively to t, we get, for 3 a'ffn'd t . à'R, the expression [4003] : and this quantity is evidently [4003o] the most important one in the value of u v, depending on the term now under consideration, included in the expression [3715m]. 144 PERTURBATIONS OF THE PLANETS, [Méc. Céî. [4003'] [4004] [4005] If the expression of R, divided by the disturbing mass, be the same for Jupiter and Saturn, we shall have, as in [1208], the coiresjjonding inequality of Jupiter 6 v, by substituting the preceding value &v' [4003] in the formula m' \/tt' 6V = m\/a .6V\ but the value of ^4'" [3775c] is not the same for the two planets, therefore the terms* ilf C). e'\ COS. (3 m' ^ — /U + 3 .-'— s — 2 ^J) ; iV">. e'. COS. (nt — 2n't + s — 2 e + ^') ; divided by the disturbing mass, are different for each of them. But it follows, from [1202], that by noticing only the terms having the divisor (5 n' — 2 n)", we shall have in this case,t m.fdR^+in'.fd'R^O ; [4004a] [4004ft] [4004c] [4004i] * (2522) The terms mentioned in [4004] are derived from «3'^\ cos. (îj' — v), as it appears in [3976', .3998'] ; but the value of A'-''' is not the same, in computing the action of m upon m' ; as it is in computing the action of m' upon m [377.5c]. Now we have already remarked, in Vol. I, page 651, that the method of finding the inequality of Jupiter from that of Saturn, by means of the formula [1208 or 4003'], is not applicable, without some restriction, to the computation of terms of the order of the square of the disturbing force. This is evident from the consideration, tliat in the equation ni.fdR^m'.fd'R' = [1 202] , from which the formula [1208] is derived, terms of the third order in m, m' are neglected, which is equivalent to the neglect of terms of the second order in R, R' ; being of the same order as the terms computed in [3982 — 4002]. t (2523) This formula is corrected for a typographical mistake in the original work, [4005a] and is the same as in [4004c] ; terms of the third order in m, m being neglected. We have already spoken of the different meanings of the symbol R, and it may not be amiss again to repeat, that ?» is the mass of Jupiter, w' that of Saturn ; also in formula [40056] [4004c], the value of R corresponds to the action of rd on m [913], and R' to the action of m on ?«' [1199']. These are changed in the present article to R^ [4005'] and R [3974''], respectively. VI. il. §16] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 146 R^ being tvhat R becoiries relatively to the action of Saturn on Jupiter, and ,Af.,.r,^ the differential characteristic d referring to the co-ordinates of Jupiter.* * (2524) Substituting &v' [4003] in the formula [4003'], we get the corresponding inequality of &v [4006]. This method of deriving 5v from <)v', would be sufficiently accurate, were it not for the terms of the third order in m, m', omitted in [4004c, 4003']. These neglected terms make it necessary either to correct the result obtained in [4006], or to compute, tn a direct manner, the value of 5v from the formula ôv^Saffndt .dR [3715Z]. Thus, for the terms of R,, which are similar to those of R [3978, 3984, 3988, 3990, 3996, 3999], we must compute the corresponding values of adR^, similar to [3982, 3985, &c. — 4000], and by combining all of them together, we get the value of adR^, corresponding to [4002]. This is to be substituted in [4005f], to obtain the required inequality 5v, which is to be used instead of [4006]. It will not, however, be necessary to repeat the whole of these calculations, since we shall soon show that the terms of R, of the form and order in the development [3742], combined with those of a similar development of R^, satisfy the equation [4005], when we except the terms depending on A'-^\ and notice only such quantities as have been under consideration in this article, namely, those which are of the order of the square of the disturbing force, and depend on the angle 5 n' t — 2nt. For if we put A = cos. ( v' — v) — ^7®. cos. {v' — I' ) ~t~ 4 7^' ^^^- ( '^'~H * ) > X B =^ — {r^ — 2 r r'. cos. {v' — i' ) 4~ '"' '^ ^ ^ 3. . 4" ^7^-\cos.{v' — v) — COS. ( !)'-[- 1' ) } • \r^ — 2rr'. cos. (d' — v)-{-r'^\ ^ ' we shall obtain the value of R [4005/], corresponding, as in [3974''], to the disturbing force of Jupiter upon Saturn ; the expression is derived from [3742], by changing m, r, v into m', /, v', and the contrary. Moreover R^ [4005/', 4005'] corresponds to the action of Saturn upon Jupiter, being the same as in [3742], R=m.^ .--if-mB; [Action of Jupiter on Saturn.] R/=^ m'A . \-m B ; [Action of Saturn on Jupiter.] respectively, in [3975—3991]; also JV'^ JV»>, into -.A*'»', - .JV'" [3995— 4001'] ; or in other words, we may compute the parts of R^ , depending on B, by multiplying the VOL. III. 37 [4005i'] [40056"] [4005c] [4005rf] [4005e] [4005/] [4005^-] [4005/i] [4005i] [4005*] [4005/] [4005/'] If we neglect, for a moment, the term A, we shall have R^mB, R, = m' B ; I whence R,^ — .R; so that the terms of R^, corresponding to R [3975], maybe [4005m] found by changing M^'>\ M^'\ M'^\ M^^\ into -.J/«\ - .M^", -'.J/<2', -.JfO), [4005»!] 146 PERTURBATIONS OF THE PLANETS, [Méc. Cél. Hence it follows, that the inequality of Jupiter, corresponding to the corresponding terms of it [3978, 3984, &jc.] by — . In finding tlie differentials relative to d, we shall proceed in the same order as we have done in finding those relative to d' [4005o] [39S2ff, 8ic.], observing that d does not affect Sti't, in the angle 3 n't — nt, which [4005;/] occurs explicitly in [3975]. Hence we shall have d . {3 n't — ni)^^ — n dt, similar 6r' to [3982rt] ; moreover, as the sign d does not affect the values of —, Sv', the differential of the angle nt — ■2n't, which occurs in these values, or in the terms connected with [4005î] £', i^' [3974', 3974], is d . {nt~2n'i) = 0. The difference of these two expressions, corresponding to the equation [3982c], is [4005r] d . {5 n' t — 2 n t) = d . {3 n t — 71 1) — d . {71 1 — 2 n' t) = — )i d t ; [4005r'] now we have very nearly 5 71' — 2 ?i= [3818rf] ; and the inequalities S v, iv', under consideration, are very small, as we shall see in [4431/] ; therefore we may put — J! = — ( 5 71' — n), and the preceding expression becomes [4005s] d.{57it—2nt)^ — ( 5 n' — 71) . d t ; which is equal to that of d'. ( 5 m' t — 2 7it) [3982c], but has a different sign. Hence, by noticing only the part of R, depending on B, and connected with the factors E', F', we have d/? = — d'iî ; substituting this in the differential of R^ [4005»j], taken relatively to d, we get dR=~.dR^ .A'R; which is easily reduced to the [4005u] fo'™ ;« .diî, -[-?«'. d'/{ = [4005]. In like manner, the differential d affects the whole of the values —, &v [3972, 3973], depending on the factors E, F ; so that the differential d, of the angle 71 1 — 2iH, connected with these terms, is [4005i'] d .{7it — 2 7i't) =: ndt — 2n'd t . Subtracting this from [4005j:>], we get d.{5n't — 27it) = d.{3n't — ni) — d.{7it — 27i'i)=^27i'dt — 27idt: and by substituting 2 m' — 2 }i = — 3 ?i' [4005/-'], it becomes d.{5 7it — 2 71 1)= — 3 71' dt = — d'. {5 n't— 2 nt) [3982^] ; r4005rl hence, for these terms, we also get, as in [4005^], dR^ — d'R and ?«.di?,-}-m'.d'-R = 0. The same result holds good when the terms of R, instead of depending on the angle [4005yJ 3 n't — 71 1 [3975], have other forms, as for example, nt — 2 7i' t [3995] ; which are to be combined with the corresponding terms of S 7-, ôv, (5 /, 6 v', so as to produce the angle 5 7i'i — 2 7it. Thus, if instead of the particular values of R, — [3975, 3974], we assume the following general values, [4005î" R = M.cos.{ i\ n't — i^nt + Jli), ~ = F'. cos. ( it n t — i'., n' t -\- A^^) ; [400.5«P [4005w'i VI. ii. ^S 16.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 147 preceding expression [4003], is ^ 3m. «ay- ^ ^■^^^.Q^,^_2nt-\-5s—2e — 0). [4006] (5ji' — 2n)2 ^ in which i\ -j- i'^ = 5 ; {^ -(- i, = 2 ; we shall find that the products of these two [4005i'] expressions, contained in a function similar to [3978], will produce a term depending on the angle 5n't—2nt, as in [-3979]. In this case, the equations [3982c, 4005r] become, respectively, by suhstituting i\-{-i'„ = 5 [400.5-'], [400(Ja] d'.(5ii't — 2nt)^zà'.{i'i n' t — iiiit) — à'.{^nt — i'^ n' t ) [4006i] = i\ n (It — {^i^ndt — i'o n' d () = 5 n d t — i.^ ndt ; à..{bn't — 2nt)=^à.{i\nt — i\?U) — d. (îoni — i'.^n't)^^ — i^ndt. [4006<;] The sum of these two equations, substituting iy -|- to= 2 ; 5 ?j' — 2 ?« = [400.5cr', /], is ù'.{bnt—2nt)-i^A.{biît — 'int)^bn'dt—2ndt = Q, or à'R^àR^Q, [4006rf] as in [400.5^] ; and from this we get, generally, as in [400.5x, 4005] , m.àR^-\-m' .à'R=zO. [4006e] Hence it follows, that if we put àvy, i5z).,, for the parts of èv, of this form and order, dependuig on Jl, B, respectively; also &v\, ôv'ç^, for the similar parts of ôv', we shall have 5 D =: 5 Di + 5 1>2 ; ôv' = Sv\-{-Sv'„; [4006/] and the formula [4006e] gives, as in [1202, Sic], the following expression, similar to [4003'], [4006e'] Sv^= — 5v'2."^. [meg] my a From this formula we may compute 5t)o, after having found or'j, by a direct process similar to that used in [3975 — 4003]. In computing the terms of avy, àv\, depending on A [4005A], we may neglect the two terms containing y^, for the same reasons as in [3990ff— c]. Then we shall have simply ^ = cos. (î;' — v) ; hence the corresponding parts of R, R, [400.5/,/'], become [4006/i] R = m. ^ .cos.{v'—v); R^=m'.-^^.cos. .{v'—v). [400fo-] These quantities evidently depend on the term connected with the coefficient A <'', in the development of — [954, 957], as is evident by the substitution of the values [952, 953]. Hence we have, by noticing only this part of A'-^\ A'^'> = m • -J ; in computing êv\, arising from the action of Jupiter on Saturn ; [4006^] ^'i>=m'. — ; in computing Sv, arising from the action of Saturn on Jupiter. [4006/] Now A^^' occurs only in the development of the term .^'". cos. ( r' — v); and it is [4006m] 148 PERTURBATIONS OF THE PLANETS, [Méc. Cél. 17. In the inequalities of Jupiter and Saturn, in which the coefficient [4006'] of t is neither 5n'- nor differs from it by the quantity n, in [4006n] therefore found in JJf*"' [3976,3976'], also in JV'(" [4001'] ; but not in M^'\ M'--\ M"', [400(3o] JV'"' [3983, 3986, 3989", 3995'] ; so that in these last terms we shall have (5 Uj = 0, [400G/)] 5^'j = 0, à'Vç,=^iv, (5 î)'g = (5 1)' ; consequently the value of 5v may be correctly obtained from i5 v', in these cases, by means of the formula [4003']. A different process [4006?] must be used with the terms depending on M'-^^, JV*'\ which contain A^^\ For we must compute (5 ti'j in a direct manner, by means of the value of ^'" [4006Zr] ; also dv-^, from [4006r] [4006Z] ; by a process similar to that used in computing &v' or &v'^, in [3982,4002']. [400C«] Having thus obtained i5 Dj , ùv\, iv'.^, we get àv^, by means of the formula [4006^], and then by substitution in [4006/"], we obtain the values of 5v, Sv', corresponding to r.««^ „ these terms. These remarks are not restricted to the two forms of R, treated of by the [4006s ] author in [3975, .3995], but apply generally to others of a similar nature, contained in the general table, which we shall give in [4006zt]. In addition to the terms of R, depending on the angles 3 n't — ni, ni — 2n'i ; [4006<] treated of by the author in [3975, 3995] ; there is an infinite number of a similar nature ; some of which are deserving of peculiar notice, on account of their magnitudes ; and one of them is of nearly the same order as those we have already noticed. The 20 forms of R, S 7-, 5v, êr, ôv', Sic, producing the angle 5 n't — 2 n i , are contained in the annexed table. Thus the form which is marked with the number 6, includes the terms of R, depending on the angle 3 n't — nt, as in ; the first form assumed by the author in [3975] ; and when this is combined with 6r, 5v, &:c., of the form 2n't—nt, it produces terms depending on 5n'i — 2nt, as in [3979]. We may also take these angles in an inverse order, corresponding to the accented numbers, supposing, as in the number 6', that R depends on the angle 2n i — n t , corresponding to the second form of the author, in [3995], and ér, 5v, he. depend on the angle 3 n't — nt . The numerical values of these terms of ^i', 5v', are given inaccurately in [4432,4488]; as was first observed by Mr. Plana, in the second volume of the Memoirs of the Astronomical Society of London ; in which he has given the calculations of the [4006d] separate terms at full length ; and has also noticed the terms of R, of the forms 5', 3, 4 ; observing, however, that they have hardly any sensible effect in the complete values of &v, 5 v'. The final values of ôv, ô v', computed by Mr. Plana, by a direct process, and independently of each other, did not satisfy the equation [400-3'] ; and this numerical result, he considered as a demonstration a posteriori, that this formula could not be applied [4006!^] ^^ ^j^ ^1^^^^ jg^.^^^^ ^j. jj^g ^^,jg|. ^f ^jjg square of the disturbing masses. In consequence [4006«] No. Coefficienl3 of ( in the terms of R. Coefficients of t in tlie terms of Sr, év, or', iv'. V 2! 3' 4' 5' 6' 1 2 3 4 5 6 n' 2ji' 3n' n' — n 3 n' — n 5 n'- 2 n An'—2n 3n' — 2n 2n'— 2n An' — n 2n' — n v=n'—n; { = any positive integer. 7 8 9 10 5n'—2n-\-i\i 5n' — 3 ?i -|-iv 5n' — 4?i-|-iv 5n' — 5n-\-iy I V t V — n iv — 2n iv — ■ 3 n r 8' 9' 10' No. Coefficients of £ in tlie terms i>f Sr, iv, 6r', 6v'. Coefficients of ( in the terms of R. VI. ii. §17.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 149 Jupiter, or n' in Saturn; we must increase nt and n't by their great r^^Qg,,, inequalities depending on bn't — Int. For we have seen [1070"], of tliese remarks, La Place resumed the subject in a memoir published in the Connaissance des Terns for the year 1829 ; in which he tacitly admits the inaccuracy of the application of the formula [4003'] to all these terms of the order of the square of the disturbing forces ; and gives a new formula [400Si], expressing the relation between the complete values of the terms of 5v, 5 v', like tJiose computed in this article, and others of a similar form and order, calculated by Mr. Plana [4006v]. This new formula has been called the last gift of La Place to astronomy. Upon applying the numerical values of ôv, 5 v', given by Mr. Plana, to this formula, it was not satisfied ; whence La Place inferred, that these numerical calculations of Mr. Plana were incomplete or inaccurate. Some strictures having been made on this formula by Mr. Plana, in the Memorie dclla Reale Accademia delle Scienze di Torino, Tom. XXXI ; it was followed by two other demonstrations of this new formula ; the first by Mr. Poisson in a memoir published in the Connaissance des Terns for 1831 ; the second by Mr. Pontécoulant, in the same work, for 1833. In the memoir of Mr. Poisson, he notices the term of the form 1, in the table [4006m], and shows, that it is of sufficient importance to be introduced into the calculation. Under these circumstances, he recommends a revision of the whole calculation, by taking into consideration all the forms comprised in the table [4006it], which produce terms of i5 v, Sv'. of any sensible magnitude. This extremely laborious task has been performed by Mr. Pontécoulant, who has given the abridged results of his investigation in the Connaissance des Terns for the year 1833, from which we shall make some extracts, in the notes upon the twelfth and thirteenth chapters of this book, in treating of the orbits of Jupiter and Saturn. These results, so far as they relate to terms of the forms 6, 6' [4006?;], computed in this article, differ but very little from those of La Place [4432, 4488], except in the signs ; and upon referring to the original manuscript, in which these last calculations were made, a mistake in the signs was discovered. Finally, Mr. Pontécoulant suggested to Mr. Plana, some corrections which were necessary in his work ; and upon the revision of his calculation, it was found, that the results were almost identical with those of Mr. Pontécoulant ; these corrected values, combined with the other terms of this kind computed by Mr. Pontécoulant, are found to satisfy very nearly the new formula of La Place [4008x]. We shall now give the demonstration of this formula. For this purpose, we shall use the same notation as in [1198], in which M represents the sun's mass, m the mass of Jupiter, in' the mass of Saturn ; x, ij, z, the rectangular co-ordinates of Jupiter, referred to the sun's centre ; r its radius vector, &c. ; and the same letters accented correspond to the orbit of Saturn. Then putting, for brevity. x^'+yy'+-'^' VOL. III. w xx'+yy'+-' ^lf^x'-xf+{y'-yf+{z'~zfl [4006x] [4006y] [4006z] [4007a] [40076] [4007c] [4007 (i] [4007e] [4007/] [4007e-] 38 150 PERTURBATIONS OF THE PLANETS, [Méc. Cél. [4006'"] that these great inequalities must be added to the mean motion, in the [4007/1] we get, as in [949,1200], by observing that r^^x^-V-y^-^-z^ r'^=x'^-^y"^-^z'~ [ÇiW], [4007il R = in • (iv' -\- X) ; [For tho action of Saturn upon Jupiter.] [4007/c] R':= m . (it) -\- X) ; [For the action of Jupiter upon Saturn.] Now if we multiply the formula [1198] by M-\-m-{-m', it will become of the form [4007o] ; for the two first terms of the second member of the product, or those in the first line of [1198], may be put under the form, [4007i] ^{dx^ + dy^ + dz"~) , ,3 {dx'^+dy-!i+dz"- dt^ ' dt^ ' of which the first line is the same as in the first line of [4007o]. Connecting the terms in the second line of [4007?] with those produced by the second line of [1198], namely, {mdx4-m,'dx')^ {mdy-j-m'dy')- (mrfz-j- m'rfz'p f*°°'"l dt^ dT^ ■ dV^ ' it produces the second line of [4007o] ; observing, that ??i^ d x^ -{- m' - dx'~ — ( m d x -\-m' dx'Y = — 2 m m'. dxd x', he. The first and second terms of the third line of [1198] produce, without any reduction, the [4007;i] third line of [4007o], and the last term of [1198] gives the last of [4007o], using X [4007^] ; hence we have constant = ( M+ m!).m. ■ ^ -* J^ ' -\- {M-\-m) .m'. ^ ^ ^ [4007o] [4007p] _ , Crfxrfx' , dvdy' , dzdz'") -2mm'.^-^^ + -^ + ^ + 2 . ( JIf + m + 7n') . m m'. X. Tr 1-11 1 r dx'i + dif- + dz^ rfa:'2 + rfy2 + (/i'2 , ^^ ^ . If we multiply the values of /^^ — , f,o [1199,1200], by [M -\- m') . m, {M-{'m).m', respectively; and add the products, we shall get, for the first line of the second member of [4007o], the following expression, (^i+^')..,.^^J^,2/diï^ + (J^f+>»)•m^ f•^•7^'"'^ -2/d'i^| VI. ii. ^^ IT.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 151 formulas of the elliptical motion ; they must therefore be added to the same If we substitute this in [4007o], we shall find, that the term having the divisor r, is 2 m — . \ {M + m') .{M+m) — {M+m+ m') . 31], [4007p'] which, by reduction, is ; and in like manner, the term depending on r , is — ;; — ; so that if after this substitution is made, we divide the whole expression by 2, and transpose the tenns depending on d -R, d'R', we shall obtain the following equation, in which nothing is omitted, the constant quantity being included in the signs /, ( M + m') .m..fàR + {M-\- m ) . m'.fd' R'= m m. + ^) , /dxdx'-\-dydy'-\-dzdz'\ - '" '" • V dt' — ) ^^^^^^^ -{- {M + m -\- m') .mm'.y-.. We must now consider the terms of this equation affected with the small divisor 5n' — 2n, and ha\Tng 5 n't — 2nt for the argument ; these temis being the only ones which can acquire the di\Tsor (5»' — 2n)^ by another integration in J'fdR, ffd'R', or in [4007)-] the expression of the longitudes of the two planets [3715/, »*] ; and in making this investigation, we shall reject all terms of the order in'*. In the first place, we shall observe, that the expression in the second line of the second member of [40075'] ^'^^^ ^'^^ contain such tenns of the order ??i^, as is evident from the reasoning in note 819 [1201'], [4007«] where it is sho^^Ti, that these terms of the order ?«^, arise fi'om the substitution of the elhptical values of x, x', y, ij , &c. ; and to obtam terms of the order »i', we must augment these elhptical values of x, x, Sic. by the terms depending on the perturbations. These terms may be easily obtained by considering the orbits as variable ellipses, in which we may suppose X, x', to be of the forms, x = ^1 -f- 5i . cos. (n / + Ci) + &:c. ; [4007<] x' =: Ay + B.2 . COS. ( n't -)- Co) -j- &c. ; [4007k] Ai, B^, Ci, &c., c/^2, Bo, Cj, &.C. being functions of the elements of the orbits. These elements for the planet Jupiter are ; the mean longitude of this planet nt -\- e; E the mean longitude of the epoch ; a the semi-transverse axis of the ellipsis ; e the excentricity ; « the longitude of the perihelion ; y the inclination of the ellipsis to a fixed plane ; and è the longitude of the ascending node. The same letters being accented, [4007u"] represent the corresponding elements of the orbit of Saturn. In the values of all these elements, the secular inequalities are supposed to be included. The differential of the expression [4007/, u], bemg found as in [1168'], become dx = — B,.ndt. sin. (n / + C^) — &c. ; [4007t.] dx'= — Br,.n'dt. sin. {nt-{- Co) — &c. [4007w] [4007u'] 152 PERTURBATIONS OF THE PLANETS, [Méc. Cél. quantities in the development of R. Let [4007] R^H. COS. (i' n' t — int + A), [4007a-] [40086] [4008i'] [4008c] [4008rf] [4008e] [4008/] Tlie product dx dx', will therefore contain only periodical quantities of the form, H . cos. {in't — int-\-E); H, E, being functions of the elements of the orbits ; and i', i, integral numbers, positive or negative ; moreover n't, nt, in the planetary system, are incommensurable quantities [1197"]. Now if we consider the elements as variable, their variations, corresponding to the great inequalities of Jupiter and Saturn, will have the same argument as these inequalities, [i007y] namely, 5 n t — 2nt, and they have 5 ?i' — 2 n for a divisor, as is evident from what we have seen in [1197, 1286, 1294, 1341, 1345'], or more completely in the appendix to this volume [5872 — 5879]. Substituting these variations in [4007x], and reducing by [17 — 20] Int., we shall obtain terms having this divisor; but it is evident, that they will [4007z] not have the same argument, except z' = 10 and i = 4; in which case /J" will be of the order e^ [957^''', &,c.], which is neglected, because we notice only terms of the third order relative to the excentricities e, e', and of the same order relative to the masses in, mf. [4008a] The same remarks may be made with regard to the products d y dy', d z d z' ; hence we conclude, that the fonction included in the second line of [4007^] does not contain terms of the order n? or it?, which has for its argument 5h7 — 2)i<, and for divisor 5/i' — 2?i; so that we may substitute, in [40075], ^^^ following expression. ■mm. dxdx'-\-dy dy'-\-dz dz' 0. In the fonction comprised in the third line of [4007 (^], namely, (./li-f-w + ?»') . mm'. X, we may change the factor M -{- m-\-7n' into ./If ; it being evident, that the neglected quantities do not comprise terms of the order m^, having the argument 5 n' t — 2nt and the divisor 5n' — 2n. Then substituting, in X [4007^], the elliptical values of x, x' [4007<, u], and the similar values of y, y', z, z' ; it becomes, by development, of the form, ■k = A-\-K.cos. {5n't — 2nt-J[- I) + Q., in which A represents the part depending on the argument zero, and Q all the terms depending on angles of the form i'n't-\-int, i', i, being integral numbers, positive or negative, excluding those pi-oducing the argument 5 n't — 2nt, which is connected with K, and the argument zero connected with A ; hence we have (.W + w + m) .mm! .\ — M . m-rri .\A-\- K . CQ's,.{'ô'){ t —2nt ^ I) A^ q}. The quantity mm'.— [4007<7], is of the third order in 7n, in, and as the value of r [4008g-] contains no term having the divisor 5 ?i' — 2n, except it be of the order ??/, we may neglect this term, because it produces nothing except of the order m"" ; and the same is to VI. ii. §1~] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 163 be any term of this development ; and 6v = L. sin. (i'n't—int-\-B), [4008] VI be observed relatively to m m'.—. Substituting these and [40086,/] in [4007^], we get M. { mfdR + m'fd'R' } + m m'. \fdR +/d'jR' \r:^M. m m'. \ A+K. cos. (5 7t7 — 2 n ï +/ ) + Q ^ [4008A] We shall represent by (R), {R'), the parts of R, R', respectively, of the order m ; [4008t] then using the characteristic (S of variations, we shall put àR, ôR', for the remaining parts of the same quantities of the order mP, &lc., and we shall have R={R)-\-5R, R'={R')~\-6R'. [4008i] If we also put [(.2) + (^) .cos.(5?i'<-2n^-|-/)] for the part of .^+Z:cos.(5n'i;— 2n<+/), [4008J] which is independent of m, m ; and prefix the sign <5 before the same quantity, to denote the remaining part, we shall have ^ + Z.cos.(5 7i'< — 2«ï' + /)+ Q=[(.^) + (^).cos.(.5îi'^ — 2n<+/)] -\-&.\A-^K.cos.{biït — ^nt + I]+q. Substituting [4008fc, m] in [4008/(], and neglecting the terms mtn'.fd5R, nim'.fd' 5R', which are of the order m^ ; also the terms M.mm. Q, because the integration does not introduce the divisor 5 ?i' — 2 7i, we get M.\mfd{R)-^m'.fd'{R')\-i-mm'.\fd{R)-^fd'{R')]-JrM.\mfd5R-Jrm'fd'ôR'l =M.mm'.[{A) + {K).cos.{5n't—2nt+I)]-JrM.mm'.S.{A-\-K.cos.{5n't—2nt-^l)]. Now equating separately the parts of this equation, which are of the order m^, and those of the order m?; putting also M=^l [-3709], in terms of the order m^ we get M. \m. f d{R) +m'.f d' {R)\ = M. mm'. [{A) -{-{K). COS. {57i't — 2nt-\-l)']; [4008p] mm'.{fd{R) +/d' (R) ] + m ./d 6 R + m'./d' ÔR'=mm'.5.\A -^K. cos. (.5 n' ( — 2 n t+I) | . [4008?] 14008m] [4008?»] [4006o] [4008r] If we neglect the terms of the second member of [400S»/], or in other words, the terms of the elliptical value of X, depending on the two arguments zero and bn't — 2nt, we shall have the following expression [4008s], which includes all the arguments except these two ; and is accurate both as it regards terms of the third order of the masses m, m', and of the third order relative to the excentricities and inclinations, m m'.\fd {R) +/d' {R) \ + m.fd8R-{- m'.fd'ôR r= 0. [4008s] Substituting M=^ 1 [4008o] in the product of [4008p], by the quantity m', we get, by neglecting terms of the two forms and 5n't — 2nt [4008;-], mm'fdR-^m'-.fd'R'^^O. Subtracting this from [4008s], we obtain m.fdiR-{- m'./d' Ô R' -{- {m — m') . m'./d' R' = 0. [4008«] VOL. III. 39 [4008<] 154 PERTURBATIONS OF THE PLANETS, [Méc. Cél. the corresponding inequality of Jupiter.* If we increase 7it, n't, by their great inequalities in the expression [4007], there will result in ii a term of the form,t [4009] R = ±qH. COS. {i' n't — i7it±{5 n't — 2nt) + A±E). and since a'~ n = a' '~ n' = 1 [3866»] , neglecting terms of the order m, this may be put under the following form, terms of the order in* being neglected, [4008t;] m «* n.fdSR^ m'. a' ^ n'.fd' 5 i?' + ( m — m') . m', a' ^ n'.fd'R' = . Now if we put ^, ^', for the great inequalities of Jupiter and Saturn; S^^, S^^', for the [4008t)'] parts of i^, ^', depending on dôR, d'SR'; or in other words, those which depend on the combinations [4006m], excluding the angles zero and 5 n't — 2nt, we sliall have, as in [.371 5Z, m], [i008w] S^^ = 3an.ffdt.d&R; S^ ^'=3 a' n'.ffd t .d'5 R' ; ?,'^3 a' n'.ffdt .d'R lastfoT*^" Now multiplying [4008?;], by 3dt, integrating and substituting [4008 w], we get mula, which [4008.r] m /a • 5, ? + '«' /«'• 'I ■? ' + ( »* — '»') • ™'- /«'• ■? ' ^ 5 inctudoa terms of wMch IS the last formula of La Place, proposed to be demonstrated in [4007^^ ; and the the order trfi. complete values of (S, ^ , (5^ ^ ought to satisfy it ; so that if one of these quantities be rifioR 1 accurately computed, the other may be deduced from it ; but the usefulness of the theorem is restricted by the circumstance, that it can only be applied to the results obtained from all [4008z] the sensible terms of this kind, taken collectively; or to all the terms corresponding ic each of the six factors e', e^ «', e e'^, e'*, ey^, e' y^. * (2525) The relation between R and 5v is expressed by the equation [.37155]. A particular case of this formula is considered in [3703, 3715], in wliich [4009a] R = M. cos. ( m,t + K) [3703, 371 Irf] ; [40095] and we find, by mere inspection, that the third and fourth terms of uv [37155] have, as in [3715A], the divisors m^, m^ ; also by comparing [3702, 371 If], we find, that the terms of hv [37155], depending on hr, have the divisor mf — ?t^, or ?;?, ±h- It is [4009c] easy to generalize this result, as in [4010], where lUi^i' n' — in. t (2526) If we increase n't by the great inequality of Saturn [3891], and nt by that of Jupiter [3889], the angle i'n!t — int, which occurs in [4007, 4008], will be increased by a quantity, which we shall represent by p ; then putting, for brevity, [4012a] Ts=bn!i — 2nt-\-b e'—2i; — i'H'. cos.A'—iU. cos.^= 2f/ .cos.c ; — i'H'.sin.J'— 2'il.sin.^=2^.sin. c; 5 e' — 2 s -f c = £. [4011] [4012] [4012'] VI. H. § 17.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 155 Now tlic series of operations, which connects H and L, gives to the parts of // the divisors (/'n' — inf, i' n' — in, i' n' — in±n [40096, c] ; [4010] and the same series of operations gives to the inequalities corresponding to the parts of R [4009], the divisors* [i' n' — in±:(5n' — 2n)}-, i'n'—inzt(5)i' — 2n), i' n'—in± {bn' — 2n) ±n. If i'n'—i7i or i' n' — inztn be not small quantities of the order 5n' — 2n, we may neglect 5 n' — 2n in these divisors,! and then the inequality, corresponding to R= ±qH. cos. [i' n' t — int±(5n't — 2nt) + A^E}, [4013] will be ÔV = ±qL . sin. { i' n' t — i n t ± (5 n' t — 2 7it) -^ B ±El ; [4014] we get, successively, p = —i'H'. sin. ( n + ^') — i H- sin. {T, + ^) [40126] = — i' H'.\sm. Tj . COS. J'+ COS. T^ . sm.Â'1—i H. |sin. T^ . cos. 7l + cos. Tg . sln.^| = 2«j'.{sin.T5.cos.c-[-cos.T5.sin.c^^25'.sin.(T5-j-<^) = 2q.sm.[bn't — 'ilnt-\-E). [4012c] If we increase the angle i' n' t — int-{-Jl [4007] by the quantity p ; then develop the expression by means of [61] Int., we shall obtain an additional term of the order p, and represented by — p H .s\n. {i' 7i't — int-\-A). Substituting in this the value of [4012«i] p [4012c], and then reducing by [17] Int., it becomes, as in [4009], qH.cos.{in't — int-{-{57i't—2nt)-j-A-\-E]—qH.cos.\i'n't—int — {5n't—27ii)-\-A—E\. [4012e] * (2.527) The coefficient of t, in [4007], is i' 7i' — in, and from this arise the divisors [4010] ; but in the term [4009], this coefficient is augmented by the quantity ±(5 7i' — 2)t); which requires a corresponding increase in the resulting divisors [4010]; [4014o] by this means the divisors [4010], depending upon the term [4007], change into those given in [4012]. If we suppose 5 7i' — 2 ?i to be very small, in comparison with [40146] i' 7i' — t?t or i' 7i' — in ±71, we may neglect it ; and then the chain of operations connecting H, L [4007,4003], will have the same divisors as that connecting q H, q L [4014c] [4013, 4014]. Now [4007] is changed into [4013], by multiplying by ± ?, and augmenting the angle i'n't — int by ±{57i't — 2 7it)zizE. Applying the same [4014rf] process of derivation to [4008], we get the corresponding inequality of Jupiter, as in [4014]. t (2528) In restricting the formula [4014] to the terms mentioned in [4006'], we 5,j' 271 may consider the part which is neglected in [4012'], as of an order , or j\ of that retained [3818fr] ; so that the error of the terms ôv [4014] is of the order ^^qL; 156 PERTURBATIONS OF THE PLANETS, [Méc. Cél. which is the same as to increase nt, n't, by the great inequalities in the term of àv [4008].* We must also increase, in the terms depending on the first power of the excentricities, the quantities e, e', -us, ra', by their variations, depending [4016] upon the angle bii!t — 2?i i ; but it is evident, that this will not produce any sensible inequalities.! 18. The coefficients of the inequalities of the planets vary on account of the Manner of ^ ^ n i * t ' • i • • i ihflffirA secular variations of the elements of their orbits : we may notice this in the secular followinq manner. We must first put the inequality relative to any angle variations *■ ^lemans. *' *^' t— i^t, undcr the form t [4017] P. sin. (i' n't — int + i't — is)-\-P'. cos. (i' 7i' t — in t + i' s' — is). and as rj is of the order ^p [4012c], it becomes of the order ^l^p L. Now the great [40156] inequalities of Jupiter and Saturn being nearly 1265', — 2957', [44.34, 4474], the quantity 2) [4012ff] becomes — 5 X 2957'— -3 X 1265' = — 18580% or about y^- of the radius ; r4015c] consequently the quantity -j-^^pL is less than tïs ^ tV -^' °'' ^^^^ than y J^ij L ; and the error of this computation of i5 y [4014], arising from this source, will generally be less than ■j^jjjy of the inequality [4008], which is under consideration. * (2528«) If we increase n'i, nt, by the great inequalities, using j; [4012J], the expression 6 v [4008] will become S v z= L . sin. [i' n't — i ni -{-B ~\- p). Developing [40]5(/] this as in [60] Int., we get ôv = L.sm.{{'n't — i nt ~{- B) -j-jiL. cos. [i' n't — int-j-B). Substituting j} [4012c], and reducing by [19] Int., it becomes equal to the sum of the two expressions [4008, 4014]. t (2529) The smallness of these terms may be seen, by a rough examination of the increment of the value of R [1011], arising from the introduction of the part of c oi ô e [4016a] [1286], when we put ?:'==5, z = 2, a=l, " := 74 [3818f/], m'=^-J^^, e = 0,05 [4061rf, 4080] ; observing that as i' — i==: 3, ^- [1281'], may be considered as of the order e^ and (~) of the order e^ ; so that 5e [1286] may be considered as of [40106] [4017a] the order 74 m'. e^. cos. (5 n'< — 2nt-\-A}, or ^i^ c . cos. (5 ?i'i — 2nt-\-J) nearly. Consequently this increment of e produces terms of the order y^i^, in comparison with those depending on e, in [4392], none of which amount to 200'; hence it is evidenti that these terms are insensible. X (2530) The form assumed in [4017] has been frequently used, as, for example, in [371 li]. Vl.ii.-^IS.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 157 We must determine the values of P, P\ for the epoch 1750, and then put tang. A = ^ ■■, L = ^/W+pr^ ; [4018] the sign of sin. A is the same as that of P', and its cosine is the same [4018] sign as that of P [401 9f?] ; then the proposed inequality will be* L . sin. {i'n' t — int^ i' /— i s + J) . [40i9] We must determine the values of P, P', for 1950, noticing the secular variations of the elements of the orbits ; and we shall have for this inequality, in 1950, (L + <5 L ) . sin. (i'n't—i nt + i'e — is-\-A + 6A). [4020] If we denote by t the number of Julian years elapsed since 1750, the preceding inequality relative to the time t will assume the following form,t Çl + ^-^^ . sin. U'n't — int + i' s'— i t -\-A + 200 [4021] Under this form it may be used for several centuries before and after 1 750. But this calculation is not necessary except with those inequalities which are quite large. In the two great inequalities of Jupiter and Saturn, it will be useful to continue the approximation as far as the square of the time, in the part [4021'] * (2531) Using, for brevity, i'71't — int-\-i's' — ie=^Tg; then developing [4019] r^Q^g^-, by means of [21] Int., and putting the expressions [4017, 4019] equal to each other, we get, identically, P. sm. Tg + P'. COS. Tg = L. sin. {Tg-^A)=L. cos. A . sin. Tg + L. sm.A. cos. Tg. [40196] Comparing the coefficients of sin. Tg, cos. Tg, separately, in both members, we get P = L.cos.Jl, P'=Z,. sin. ^. Dividing the second by the first, also taking the sum [4019c] of their squares, we get [4018]. The quantity L being considered as positive, we [4019(f) get, from [4019c], the signs of sin. A, cos. A, as in [4018]. t (2532) If 5L, &A, represent the variations of L, A, in 200 years, between i.bl. t.SA 1750 and 1950; then their variations in t years will be represented by "^^j '2ÔÔ ' '■ ^^^ respectively. Substituting these in [4020], it becomes as in [4021]. VOL. III. 40 158 PERTURBATIONS OF THE PLANETS, [Méc. Cél. [4022] [4022'] Great in- equality of Jupiter, reduced to a tabular form. which has the divisor {bn! — 2nf. This part of the expression of àv is as in [3844], \aP'- 2a. dP Sa.ddP' ûv = 6 m'. n~ [5n'-2n).dt (5n'— 2n)2.rf(2 '.sm.{5>i't—2ntJ^5i — 2s) {5n'—27if — ^aP- (OJI' 2a. dP' 3a.ddP } :—2n).dt (an—2n)-.dl^^ ^ ' the values of P, P', and of their differentials, being relative to any time whatever /. By developing them in series, ascending according to the powers of the time, and retaining only the second power, and the first and second differentials of P, P', the preceding quantity will become* [4023] 5i, = - 6 ml, rP' (5n'-2n)2 2a. dP Sa.ddP' (5n'—2n).dt {5n'—2nf.dt^ ( dP' , Oa.ddP 7 , , , aap,>.^in.{5n't-2nt+5^-2s) I dt ^ (5n' 2a. dP' ;-2H).dri\'- i).dl-i) Sa.ddP {5n'—27i).dt (5n'—2n)KdtZ , C dP 2a.ddP' ) , , , ' I dt {5n'~2n).dt^^~ - dfi ddP\ dt^ >.cos.(5re'<— 2n<+5s'— 2s) * (2533) The values of P, P', and their difFerentials [4022], must be computed for the particular time t, for which the value o( 5v is wanted ; but this is an inconvenient method; therefore the functions by which sin.Tj, cos.Ts [3842a], are multipHed in [4022], [4022al ^''^ developed in [4023] in series, ascending according to the powers of t. This is done by means of the formula [oS50«], neglecting i^, and the higher powers of t. Thus, if we put the factor of sin. Tj, included between the braces in the first line of [4022], equal to u, and take its first and second differentials, neglecting the differentials of the third and higher orders ; we shall get the following values of U, and its differentials ; in which the terms in the second members correspond to the epoch < = ; [40326] [4022c] [4022d] U = aP' 2a. dP Sa.ddP' /dt \d dU T (5n'-2n).dt (5n'-2n)2.rf<a ' dP' 2a.ddP /ddU\ a.ddP' dt ' (5n'-2 7i).dV2' /ddU\ dfi Substituting these in [3850a], we get for u, the same expression as the factor of sin. Tj, in the first and second lines of [4023]. In the same manner, the factor of cos. T^, in the second line of [4022], produces the corresponding factor, in the third and fourth lines of [4023]. VI. il. § IS.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 159 The values of P, P', and their differaitials, correspond to the epoch of 1750, and are determined by the method in [3850, &c.] ; the other parts of the great inequality of m being rather small, it will be sufficient, by what has aheady been shown, to notice the first power of the time. This great inequality will then have the folloAving form, [4024] ôv= (A +5 t + Ct^). sin. (5n't — 2nt + 5 s'— 20 + (A'+B't + C't") . cos. (5n't — 2nt + 5 /— 2 . We may also put the great inequality of m' under the same form, by which means it will be easy to reduce these inequalities into tables. If ice wish to reduce the preceding inequality to one term, loe must calculate it for the three epochs 1750, 2250, 2750. Let f3 . sin. (5n't — 2nt + 5 b'— 2s + a) [4025] be this inequality in the year 1750; and |3^, a,; (3,, a„, the values of p, a [4025] at the epochs 2250, 2750 ; then the inequality corresponding to any fquaiuyof time whatever t, will be* reduced / ds , -, „ ddP;\ . ^_ , _ , ,~ , ^ . . dA , , „ ddA ) the differentials p and a correspond to the epoch in 1750; and we shall have, by [3854— 3856], f to one term. [4026] d^ 4 3,— .3(3 — p,,^ dt 1000 ' dd^ p„— 2(3,+ 3_ dt^ 250000 ' [4027] d\ 4 a,— 3 a — A„ , dt ~ 1000 dd\ A„ — 2a, + a dt^ ~ 250000 [4027'] * (2534) p and A being functions of t, we shall have, as in [3850«], e4-t '^^^^t^ '^'^^ and A _i_y ^^ J^ii2 '^'^^ [4025a] for their values ; using for p, A, and their differentials, the values corresponding to the epoch in 1750. Substituting these in [4025], it becomes as in [4026]. t (2535) If in the general formulas [3854—3856], we change P, P,, P„, into |8, 3,, p„, the expression [3854] will become like the first of the functions [4025a] ; [4027o] J , , . , d 13 ddp and by making the same changes in [3856], we shall get the values of — , — 160 PERTURBATIONS OF THE PLANETS, [Méc. Cél. In conformity to the remark we have made in [3720], these two great [4027"] inequalities of Jupiter and Saturn must be applied respectively to their mean motions. [4027]. In like manner, by changing, in [3854—3856], P, P,, P„, into A, A,, A„, the formula [3854] will become as in the second of the functions [4025a], and [3856] [4027c] will give the values of —, — - [4027']. VI.iii.§18'] DEPENDING ON THE OBLATENESS OF THE SUN. 161 CHAPTER III. PERTURBATIONS DEPENDING ON THE ELLIPTICITY OF THE SUN. 18'. Since the sun is endowed with a rotatory motion, its figure will not be perfectly spherical. We shall now investigate the effect of its ellipticity on the motions of the planets ; putting p = the ellipticity of the sun, expressed in parts of its radius ; q = the ratio of the centrifugal force to the gravity at the sun's equator ; (X = the sine of the planet's declination relative to the sun's equator ; D = the sun's semi-diameter ; 1 = the sun's mass, usually called M ; R = {?-\fi)-^'i^'-\)' Symbols. [4028] then it will follow, from [1812], that the sun's ellipticity adds to the vaiuoof function R [913], the quantity* dependine on the ellipticity. [4029] * (2536) We shall suppose m', m", ??i"', &c. to represent the particles of the sun's mass ; considering it as being composed of concentrical elliptical strata of variable densities, symmetrically arranged about its centre of gravity, taken as the origin of the co-ordinates of these particles x', y' , £ ; x", y" &ic. The co-ordinates of the attracted planet m being represented by x, y, z, and its distance from the sun 7-=\/(.r^-j- )/^-f-c^). In this case, the expression of R [91.3] will be reduced to its last temi 7?= — — ; {xx'+yy'+zz') any term of the form because depending on the particle m', whose co-ordinates are x', y', s^, is destroyed by a similar term, depending on an equal particle m', whose co-ordinates are — x', — y', — 2'. Substituting, in [4029è], the value of X [914], VOL. III. 41 [4029a] [40296] [4029c] 162 PERTURBATIONS OF THE PLANETS, [Méc. Cél. [4030] If we notice only this part of R, and put fdR = g-^R; g being a constant quantity ; we shall find, that the differential equation in r 6r [926, 928'] becomes, by neglecting* the square of tJ., [4031] 0=-^^^ + -^^— + 2^+^ ^ .t neglecting terms of the order m' m", and using the sign / to represent the sum of the [4029dl terms depending on all the particles, we get iî = — C'TTr', ,„ , , , ,„ , , , —. This expression of R corresponds to that of — V in [1385"', 1386], m' being the attracting particle, and \/ \ {x' — a,)^ -\- {y — y)^ -{- {z' — z)^ } its distance from the attracted planet ; hence iî = — V; and by substituting the value of V [1812], we get [4029e] -« — —,.— ra •JU.[iJ. s). The last term being multiplied by D^, to render it homogeneous with the first, because in [1812, 1795"], the semi-diameter of the body M is put equal to unity, and here it is [4029/] supposed to be D. Again, by comparing [1670', 4028], we get a.(p = q; also by comparing [1801, &lc., 4028], we get o.h = p. Substituting these in [40296], we obtam [4029^] -^7 r3 -M-ii^—s). Now if the sun were of a spherical form, with no rotatory motion, we should have M [4029A] P = 0, 7^0, and then J? = — — [4209^]. Subtracting this from the general value of R [4029^'-], we get the part of it depending on the sun's ellipticity, namely, [4029i] R^_^Illl^t^.M.ii^^—^), and by putting, as in [4028], the sun's mass Jlf = 1, it becomes as in [4029]. * (2537) The inclination of the sun's equator to the ecliptic is less than 8'^, and its sine [4030a] j^ ^g^^jy ^^ g^ ^j^^j ^2 „-j„st be less than {if, or ^3-; which may be neglected in [4030t] comparison with ^ 5 and then [4029] becomes R = — ^.(P — s?)-"^- t (2538) Substituting, in [926], the value of rPt'=r. (— ) [928'], also i^=n^a^ ,-100], we get d2.(,v5r) , ,fia?.rSr , „^,„ t /dR\ [40316] 0=-±^ + ^^~ + 2fàR + r.(jy). [4031a] ^ ^ [3700], we get Now the value of R [4030&], depending on the sun's ellipticity, gives [4031c] fdR^-i.{?-hq)-D'-fà.'^- = -h{p-ii)-^+g; '••C^)=(''-*?)-Tr^ VI.iii.§lS'.] DEPENDING ON THE OBLATENESS OF THE SUN. 163 To determine the constant quantity g, we shall observe, that the formula [931] gives, in àv, the quantity* 3a.ngt + {^ — h(]) ' — .lit; [4032] a' n t denoting the mean motion of the planet ; this quantity must be equal to zero ; therefore we have ST = ^ ^ . ^ 3 «3 Hence the differential equation in r&r becomes, by neglecting the square of c, and observing that n-a'^^=\ [3709'] ,t + ^l^Mï , n~. Z)-. { 1 + 3 e . COS. (n t + i — ^)]. but from [4031c], we get a?' 3a/diî + 2ar.(^)==3«i^ + (p-Aî).^ = 3«^ + (p-H)' [4032'] [4033] [4033'] [4034] substituting tliese in [40316J, we get [4031]. We may observe, that the symbol (J- [4031a] is entirely different from that in [4028]. * (2539) The constant quantity g is to he found, as in note 699, Vol. I, page 550, by putting the terms of [931], multiplied by t, or rather by ^oZT^)' equal to nothing. These terms are evidently produced by the two last terms of [931], 3 afn dt.fdR + 2afndt.r. 0^^ ; [4032a] [40326] noticing merely the term a of the value of r, which is evidently the only part which affects the coefficient of t, now under consideration. Multiplying this last expression by ndt, and integrating, it becomes as in [4032], which represents the part of ô v, connected with [4032c] the factor t. Putting this equal to nothing, we get [4033]. t (2540) We have r = a.\l—e. cos. ( n < + s — w ) } [3747], neglecting e^ ; ^^^g^^,^ hence we get, by using [4033'], i = 1 .n + 3 e . cos. (ni4-s — zi)\=^nm + 3e. cos. (n t + t—z,)]; [4034i] substituting this, and g [4033], in [4031], we get [4034]. 164 PERTURBATIONS OF THE PLANETS, [Méc. Cél. This gives, by integration,* [4035] ^ = i.(p — 19).^.^ — 3e.«^.sin.(«< + f — «)}. The elliptical part of - is 1 — 2 e . cos. (nt + s — zi) [3876a] ; and if we suppose w to vary by 6^^, we shall have [3876f/],t [4036] -g- =r — eô'a . sin. (nt-{- s — w). * (2541) This integration is made as in [865 — 871"], putting rSr^^y'; hence [4034] becomes, by connecting together the terms depending on e, [4035a] o = ^+n^.y'—i.{p—hq)-n^-D^+\n^y'-\-i.{p—iq).7v'-D^.3e.cos.{nt-\-e—:;!). [40356] Putting y'^y-]-^.(p — ^q).D'^, and neglecting the term of the order ye, or e^, we get [4035c] = -j^l^ n^ y + 2 . {f ~i q) . 71^. D^. e .COS. {n t -\- s — zi); [4035rf] which is of the same form as [865a, 870', 871'], changing a or m into n, s into s — w, a -fir into 2.(p — ^q)n^. D~.e, and then [871"] becomes y = — ^ — .sin.(?t < + £ — «) = — (p — iq) -711 .D^.e .sin. Çnt-\-e — to) ; substituting this in y' or rSr [4035i], we get [4035e] r5r = i.(p — i<?).D2 — (p — iy).ni.I32.e.sin. (n^-|-£ — to); dividing this by «^, we obtain [4035]. We may remark, that the term of the form air. cos. {nf-{-i — to) [871'J is included in the elliptical motion, and it is not necessary to notice this term in the present calculation. V Ô r ■f (2542) Comparing together the expressions of — j- [3876^, 4035], we find, that if the coefficients of sin. (7it-\-s — to) be put equal to each other, we shall get D- [4036a] — e ^5 TO = i . ( p— i Ç ) . — . ( — 3e .7it); whence we obtain 'îis, as in the first equation [4037]. The second expression [4037] is deduced from the first by the substitution of n = a ^ [3709']. Again, since the formula [4035] does not contain a term depending on n t . cos. {71 i -\- e — to), and [4036c] in [3876] this cosine is connected with the factor ôe, we shall have (îe = 0. The VI.iii.§18'.] DEPENDING ON THE OBLATENESS OF THE SUN. 166 If we compare this expression of %- with the preceding, we shall obtain ^^^ ,^^ "" of the perihelion f rjo T)- f arising 6^=.(p_i9).^.nï = (p-è7).-^ [4036«,6]; [4037] W ^ from the Ct oblateneai of Ihe 3UU, is therefore the most sensible effect of the ellipticity of the sun, upon the motion '"«nsibie. of a planet in its orbit, is a direct motion in its perihelion ; but this motion [4037'] being in the inverse ratio of the square root of the seventh power of the greater axis of the planetary ellipsis, îve see that it cannot be sensible except [4038] in Mercury [4036/], To find the effect of the sun^s ellipticity upon the position of the orbit, we shall resume the third of the equations [915]. This equation may be put under the following form,* d,lz n^a^.z , f(lR\ ^^dr-^-^ + yiû)' f4039] 2:2 We shall take the solar equator for the fixed plane, which gives n^= — ^ [4039'] [4040fl] ; then by observing that r = x^ + ^/^+z', we shall havef — j = 3.(p — i9).-^^5— .3; [4040] constant part of —3- , which is nearly equal to that of — , is represented in the present case by the first term of the second member of [4035] ; so that we shall have 'i^i.{9-iq).^, [4036^] as in [4042]. Now we shall see, in [4262 — 4265'], that if the sun be homogeneous, we shall have, for the orbit of the planet Mercury, 5j3 = (p — ^q) . — .<=0',012.? nearly [4036c] [4265] ; and this expression is much smaller for the other planets, on account of 4he divisor a^ ; so that it produces only 12°" in a thousand years for Mercury, and is much less for the other [4036/"] planets. The quantity 5 r [4036'/, 4260 — 4263] is evidendy insensible. * (2543) Substituting i>.^n^ a^ [3700] in the third equation [915], it becomes [4039a] as in [4039]. t (2544) In [4028], (a is put for the sine of the planet's declination above the plane [403951 of the sun's equator, its perpendicular distance above this plane being z, and its distance VOL. III. 42 166 PERTURBATIONS OF THE PLANETS, [Méc. Cél. [4041] hence the preceding differential equation becomes* dch now by what precedes [4036f/], we have [4042] ^=i.(p-ly).^^; hence we obtain [4043] = ^" + n^..|l+2.(p-ic).^'|. This gives, by integration, f [4044] z = ip .sin. <nt . (1 + (p — I q) • ~^ ) — 4? [4045] - being the inclination of the orbit to the solar equator,! and d an arbitrary z [4040o] from the sun's centre r; hence we evidently have (a^-; also r = y/(x^ -j- y^ 4" ~^) [914']. Substituting this value of fA in [4029], we get [4040i] ij_(p_iç).D2.^E:__L^. / d r\ z Taking its partial differential relatively to z, neglecting z^, and observing that i—]=-, we get [4040c] (^) = (p-i<?).i3^.^?| + i^ = 3.(p-i<?).f.z. 1 1 _ 7l2 î"5 «5 o2 1 1 n~ Retaining only the constant part of r, we may put - = — = — - [3709'], and then the preceding expression [4040c] becomes as in [4040]. * (2545) Noticing only the terms of r, depending on the sun's ellipticity, we may put, [4041a] as in [4036c/] , r=za-j-5r, whence -==-.n '^j. Substituting this and [4040] in [4039], we get [4041] ; and if we use [4036(^], it becomes as in [4043]. t (2546) Comparing [865', 4043], we get y = z, a = n .U -j- {p — ^q) . — ^, " by neglecting (p — hlT- Substituting these in the first value of y [864a]; changing also b into <p, and (p into — ê, we get [4044]. X (2547) The sine of the declination is equal to - [4040a], and its greatest value [4045a] is equal to - [4044] or - nearly ; which evidently represents the sine of the inclination of the orbit to the solar equator. VI. iii. § 18'.] DEPENDING ON THE OBLATENESS OF THE SUN. 167 [4045'] [4046] constant quantity. Tims the nodes of the orbit on this equator have a retrograde motion equal to the direct motion of the perihelion, and which cannot therefore be sensible, except in the orbit of Mercury* At the same time ive see that the sun'' s ellipticity has no influence on the excentricity of the planeCs orbit [4046f ], or on the inclination of this orbit to the solar equator ; it cannot therefore alter the stability of the planetary system. * (2548) It is evident from the form of the angle, which occurs in [4044], that the D- retrograde motion of the node in tJie time t is represented by nt . {^ — J?)--t7) [4046a] because the body is in the node wjien c = 0, and it completes its revolution, to the same node, while the angle nt -{-nt .{if — è <p) . -5- increases by 360''; the mean [40466] periodical revolution being performed in the time t, which makes nt = 360'' [4032']. Hence it is evident, that the retrograde motion of the node in the time t is nearly equal to the difference of these quantities, as in [4046a], being the same as the direct motion of the perihelion [4037]. As (5e = [4036c], the excentricity is not affected by the sun's [4046c] ellipticity, neither does it affect the inclination - of the planet's orbit to the sun's equator [4045a], which is constant, because ç is one of the constant quantities obtained by integration. The results found in this chapter agree with those found by Mr. Plana in the Memoirs of the Royal Society of London, Vol. II, page 344, &c., noticing the term neglected by [4046rfl La Place in [4030] ; makmg also the computation directly from the formulas [5788—5791], and carrying on the approximation to a rather greater degree of accuracy. 168 PERTURBATIONS OF THE PLANETS, [Méc. Cél. [4047] CHAPTER IV. PERTURBATIONS OF THE MOTIONS OF THE PLANETS, ARISING FROM THE ACTION OF THEIR SATELLITES. 19. The theorems of ^10, Book II [442", &c.], afford a simple and accurate method of ascertaining the perturbations of the planets from tlie action of their satellites. We have seen, in [451', &;c.], that the common centre of gravity of the planet and its satellites, describes very nearly an elliptical orbit about the sun. If we consider this common orbit as the ellipsis of the planet ; the relative position of the satellites, compared with each other and with the sun, will give the position of the planet, relative to this common centre of gravity, consequently also the perturbations which the planet suffers from its satellites. Let M^ the mass of the planet ; Symbol.. Ji :^ ^j^g radlus vector of the common orbit, or the orbit of the centre of gravity of the planet and satellites, the origin being the sun's centre ; V == the angle formed by the radius R, and the invariable line, taken in the comm07i orbit, as the origin of the longitudes ; m, ml, &c. the masses of the satellites ; [4048] r, /■', &c. the radii vectores of the satellites, the origin being the common centre of gravity of the planet and its satellites ; V, v', &c. the longitudes of the satellites, referred to this common centre ; s, s', &c. the latitudes of the satellites above the common orbit, and viewed from the common centre ; X, Y, Z the rectangular co-ordinates of the planet ; taking the common centre of gravity of the planet and its satellites for their origin ; the radius R for the axis of X ; and for the axis of Z the line perpendicular to the plane of the common orbit. VI.iv.§19.] ARISING FROM THE ACTION OF THEIR SATELLITES. 169 We shall have very nearly, from the properties of the centre of gravity, and by observing that the masses of the satellites are very small, in comparison, with that of the planet,* = MX + mr. cos. ( v — f/) + m' r'. cos. (v' —U) + &c. ; 0=^MY-i-mr. sin. ( i' — C/) + m' r'. sin. {v'—U) + hc.\ [4050] = M Z + m . r s + m', r s + &c. The perturbation of the radius vector is nearly equal to X; consequently it is equal to Perturba- tions. .r. cos. Ct; — U) .r'.cosJv' — U) — &c.=: Perturbation of radius vector. [4051] The perturbation of the motion of the planet in longitude, as seen from the r R — ^•-B-sin,(t; — U) — - . — .sin.(v' — U) — &c. = Perturbation in longitude. [4052] wU -cCr Jim. JAj Y sun, is very nearly — ; therefore it is equal to m r * (2549) If we let fall from the points where the bodies M, in, m', &IC. are situated, perpendiculars upon the axes of X, Y, Z, the distances of these perpendiculars from [4050a] the common centre of gravity of the planet and its satellites, taken as the origin, will be, respectively, as follows; On the axis of X ; X; r . cos. {v — U) ; ?•'. cos. {v' — U), &,c. ; [40506] On the axis of F; Y ; r . sin. {v— U) ; r' . sin. {v — U), &c. ; [4050c] On the axis of Z ; Z ; r s ; r's',hc. nearly. [4050d] Multiplying the distances [4050/^] by the masses M, m, m, &c. ; and taking the sum of [4050c] these products, it will become equal to nothing, by means of the first of the equations [124] ; hence we get the first of the equations [4050]. In like manner, by multiplying the distances, measured on the axis of Y, by M, m, m!, &ic., respectively, and putting the sum [4050/"] of the products equal to nothing, we get the second of the equations [4050]. The third of these equations is formed by a similar sum, corresponding to the axis of Z. From Y Z these three equations, we may find the values of X, —, —, as in [4051,4052, 4053]; and as the radius R, or axis X, passes through the place of the common centre of gravity, Y Z it is evident that these quantities X, —, — will represent, respectively, the perturbations [4050^] of the radius vector, of the longitude and of the latitude, conformably to what is said above. VOL. III. 43 170 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. Lastly, the perturbation of the motion of the planet in latitude, as seen from the sun, is very nearly -^ ; hence it is nearly equal to m rs m! r' s' c -r» i • • i • i [4053] ■ — IT; • p^ M ' ~R "^ Perturbation m latitude. These different perturbations are sensible only in the earth, disturbed by the moon. The masses of Jujnter^s satellites are very small in comparison with that of the planet, and their elongations, seen from the sun, are so very [4054] small, that these perturbations of Jupiter are insensible. There is every reason to believe that this is also the case for Saturn and Uranus. VI. v.§-20.] ELLIPTICAL PART OF THE RADIUS VECTOR. 171 CHAPTER V. CONSIDERATIONS ON THE ELLIPTICAI. PART OF THE RADIUS VECTOR, AND ON THE MOTION OF A PLANET. 20. We have determined, in [1017, &c.], the arbitrary constant quantities, so that the mean motion and the equation of the centre may not be changed by the mutual action of the planets. Now we have, in the elliptical hypothesis,* — y— == **"5 ^^^^ ''^('-^^ of the sun being put equal [4055] to unity. Hence we obtain 2. a = n ' . (1 + X»i) ; [4056] for the semi-transverse axis, which must be used in the elliptical part of the radius vector. If we suppose, in conformity to the principles assumed in [4078 — 4079, &c.], that «==n~*; a'=n'~'\ &c. ; [4057] we must increase a, a', &c. in the calculation of the elliptical part of the • (2550) This is the same as [3700], putting, as in [3709a], iJ. = M-\-m, and M^l, as in [4055]. From this we get a = 7i'~^.(l-f m)^ = n~"^.(l + im — T^V^^+^O; [4056o] which, by neglectmg terms of the order m^, becomes as in [4056]. 172 PERTURBATIONS OF THE PLANETS; [Méc. Cél. [4058] radius vector by the quantities ^m a, \ m a. Stc. respectively ; but this Increment of the radius augmentation is only sensible in the orbits of Jupiter and Saturn. * (2551) The values of a", ce', for Jupiter and Saturn [4079], are respectively augmented by the correction [4058], in the expressions [4451, 4510]. The similar augmentation, corresponding to the other great planet Uranus, is |^m"o", which, by using [4058a] [40586] ?»'' [4061], becomes If this quantity were an arc of the planet's orbit, [4058c] [4058rf] 58512 perpe7i(Ucu1ar to the radius vector, it would subtend only an angle of 3'~,6, when viewed from the sun ; but being in the direction of the radius vector, it produces no change in the longitude, seen from the sun ; or from the earth, when the planet is in conjunction or in opposition. The most favorable situation for augmenting the effect of this correction, in the geocentric longitude of the planet, is when the earth is nearly at its greatest angle of elongation from the sun, as seen from the planet. This angle for the planet Uranus is quite small, its sine being represented by — ; = j^ nearly [4079] ; and as the above correction 3°",6 is to be diminished in the same ratio, it produces only 0'',2 for the greatest possible effect of this augmentation of the radius, in changing the place of the planet Uranus, as seen from the earth ; consequently this correction is wholly insensible. [4058e] We have already observed in the commentary in Vol. I, page 561, that Mr. Plana makes some objections to the introduction of the constant quantity^, in the integral [1012'], and he has also urged similar remarks against the use of the constant quantities _/), f^ [1015'], in finding the integral i5m [1015] ; but a little consideration will show, that these objections do not apply to the accuracy of the results, or to the astronomical tables founded upon them ; but merely to the most convenient way of ascertaining, as a mere matter of curiosity, the orbit a body would describe if it were not acted upon by the disturbing force, or of computing the whole effect of the disturbing force in a given time. This subject has been discussed very ably by Mr. Poisson, in the Connaissance des Terns for the year 1831, [4058/"] pag. 23 — 33 ; and we shall, in the remaining part of this note, avail ourselves of his remarks. The complete integrals of the three differential equations [545], which determine the co-ordinates x, y, z, of the planet referred to the sun's centre as their origin, contain six arbitrary constant quantities [571«], which we shall denote by a, h, c, Sec. ; and the same is true in using the polar co-ordinates r, v, s; as we have already seen, in [602"], in the Jirst ajrproximation, where the disturbing forces are neglected, and the simple elliptical motion obtained. In a second approximation, in which we notice only the first power of the disturbing forces, we may put &r, Sv, 5 s for the increments of r, v, s ; and then the integrations being made, as in [1015, &:c., 1021, 1030], will introduce six new arbitrary constant quantities, a', b', d, &;c. ; these accented letters being taken for symmetry, instead 0Î g, fi, fl, &.C., used by La Place. A third approximation includes terms of the second order of the disturbing forces, and by similar integrations, produces six other constant quantities o", h" , c", Sic, and so on successively. If ive restrict ourselves to the second [4058g [4058;i] [4058i] VI.v.s^20.] ELLIPTICAL PART OF THE RADIUS VECTOR. 173 We must then apply to the radius vector the corrections given by the approximation, neglecting terms of the order of the square of the disturbing forces, tlie [40584] polar co-ordinates will be r-\-6r, v-\-5v, s-\-&s, containing the twelve constant quantities a, h, c, &,c. ; «', 6', d, &ic., which must, by the nature of the question, be reduced to six only, or to six distinct functions Jl, B, C, D, E, F, of these twelve quantities. The [4058i] values of A, B, C, &;c. may be determined by the position, velocity, and direction of the planet at a given moment ; or by the comparison of the values of r-\-ô r, v-\-Sv, s -\- ôs, with those deduced from observation ; in each case the result will be fixed and determined. On the contrary, we may assume at pleasure any values of a, b', c', &c. ; and the values [4058m] thus assigned to these terms, will determine absolutely the quantities a, b, c, fee, which differ but little from A, B, C, he. on account of the smallness of the disturbing forces. If we wish that or, Sv, &s should express the effects produced by the disturbing forces 011 the radius vector, the longitude and the latitude of the disturbed planet ; we must determine a, b, c, &.c. so that the elliptical co-ordinates r, v, s, and their differential coefficients —, —, — , may represent the position, the velocity, and the direction of dt dt dt the planet at the commencement of this interval of time ; and afterwards determine a, I', (,•', &ic., so that we may have at the same epoch 0, &v = 0, (is=^0; d.ir ~dt = 0, lit = 0, d.Ss ~di = 0. At the end of the time t, counted from the same epoch, r will be the distance of the planet from the sun, wliich will obtain, if the disturbing force cease to act from the commencement, and r will be the augmentation of distance produced by this force. Similar remarks may be made relative to the quantities v, Sv ; or s, Ss. If we determine a', b', d by other conditions, the perturbations of the troubled orbit will no longer be loholly expressed by the quantities 5 r, S v, S s ; because the elliptical parts r, v, s, are also affected by means of the constant quantities a, b. c, Sic, tvhich partake of the disturbing forces, and are different from what they would be if these forces were suppressed. But this is not attended with any inconvenience ; since it does not prevent these complete values of r -[- 5 r, v-\-Sv, s -{-5 s, from representing, at every instant, the true position of the planet, wliich is the object of the tables of its motion, into which tliese values are finally reduced. Instead of considering directly the increments ô r, 6 v, ô s, of the elliptical orbit, we may use the method depending on the variation of the arbitrary constant quantities ; supposing Sa, Sb, S c, &c. to be the increments of the constant quantities a, b, c, he., contained in r, v, s. These six variable quantities S a, Sb, S c, &c. will be given by direct integration of formulas similar to [1177], or like those collected together in the appendix [5786 — 5791], supposing that we neglect the second and higher powers of the disturbing forces. These values will then be of the forms, (5a=^o, + a; Sb = b^-\- fi; Sc^c,-{-y, he. VOL. III. 44 [4058n] [4058o] [4058p] [40589] [4058r] [4058*] [4058*'] [4058f| 174 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. formulas of Book II, ^50 [1020, &c.], and bj the preceding articles [4058u] rt, , b^, c, being new arbitrary constant quantities, and a, p, y, &c. functions of t, and of a, h, c, &.C. Substituting a-\-ôa, b-{-Sb, c-\-Sc, &.C. for a, b, c, he. in the values of r, V, s, we shall obtain for the co-ordinates of the disturbed planet, expressions which are equivalent to the preceding values of r -]- 5 r, v -\-ôv, s -{- 8 s. The constant [4058t)] quantities «^, f>^, c^, &c., as well as a, p, y, Sic, are of the order of the disturbing forces; therefore, by neglecting terms of the second order, as in [4053s'J, we may put, in the values of a, p, y, &,c.; a + », for a, ^ + 6, for p, c+c, for c, Sic; by which means a-f-a,! ^-f"^,j c+c,, he. will be the six arbitrary constant quantities, which occur in the values of r-\-5r, v -\- S v, s-\-Ss. This shows how the arbitrary constant quantities, contained in the co-ordinates of the disturbed planet, as found by the two first approximations, are reduced to the number corresponding to the system of dilTerential equations upon which they depend. [4058u»] If ive ivish to chtermint the total effect of the disturbing forces upon each of the elliptical elements, during a given time, we must find, as above, the constant quantities [4058r] a, b, c, &c.; by means of the position, the velocity, and the direction of the planet at the commencement of this interval of time; and tlien the constant quantities a^, b^, c, , by means of the equations [i058y] a -fa =3 0, 6^-|-p = 0, c^-{.y=zO, kc, corresponding to the same instant. The effect of the disturbing force at the end of any proposed time t, will be expressed by means of the quantities 5 a, Sb, So, &c., which will then contain nothing arbitrary. This is practised in the theory of comets, In wiilch the [40582] values of Sa, 5b, 5 c, he. are calculated, by quadratures, for the interval of time between the two successive appearances of a comet. These general considerations agree with the method used by La Place in the second book of this work. In the abovementioned paper of the Connaissance des Terns for the year 1831, page 29, he., Mr. Poisson has applied these principles to the investigation of the effect of the whole disturbing force of a planet m', upon another planet m, moving f4059al in the same plane. The radius vector and the longitude of the planet m being affected by this action, but not its latitude, because the bodies m, m' move in the same plane. In this case, the six arbitrary constant quantities mentioned in [4058/], are reduced to four. If we neglect terms of the order e^ in the elliptical motion of the body rn, the expressions of the radius vector and longitude [669, 605'], become [40596] [4059c] r=^a — ae .cos. {n t -\- s — -n) ; [4059d] t) = n < + s + 2 e . sin. {ni-\-e — ra ) ; [4059c] n^a^ = M^m = !x. If we suppose the body m to begin to disturb the motion of m at the commencement VI. v.§20.] ELLIPTICAL PART OF THE RADIUS VECTOR. 175 [3706 — 4058]. The expression of àr [1020] contains these two terms, ir = — m' a .fe . cos. (7it~\- s — ra) — ni' a ./' e'. cos. (nt-{- ; — ra') ; [4059] of the time t, we may determine the effect of the perturbation of the radius vector by means of the value of Sr [1016], in whicii the arbitrary constant quantities are retained. [4059/"] The expression o{ &v [I0-21] would give the perturbations in longitude, if particular values had not been assigned to the arbitrary constant quantities g, f, f. To obviate this objection, we must retain these arbitrary quantities as they are found in the functions \\0-2\b, c, f/, e], whose sum is assumed in the first line of the note in page 556, Vol. I [4059g-J [1021e — -/], for the value of S v. In order to simplify this calculation, it will be convenient to change the form of the terms depending on /, /' ; by developing the sines and cosines of the angles nt-\-s — «, 7it-^s — ra', into terms depending on sin. ?i ^, cos. jj f, by the method used in [1023((] ; and changing the values of the arbitrary constant quantities /, /', so that the part of the expression of — [1016], depending upon them, [4059A] may be put under the form /. cos. n t +./ '• sin. n t. The corresponding terms of the value [40.59t] of 5 r may be found by multiplying this expression by 2, and changing the angle n t into n ( -j- 90'' ; as is evident, by comparing the terms of — [1016], depending on f, f, with those of ay [1021i]; hence these terms of àv become — 2f .s\n. nt-\-2f'. cos. nt. [405941 We may also add an arbitrary constant quantity h, to the part of <S v, computed in either of the integrations [1021 rf, f], and retain the terms m'.ant.j3g-\-a. f — — j ^ [1021t?, e], [4059J] which were put equal to nothing in [102iy]. Making these changes in the expressions or of —, Sv [1016, 1021] ; neglecting the other terms of the order c or e', because this degree of accuracy is sufficient in our present calculation, which is only designed for the purpose of illustration ; and supposing also, for brevity, as in [1018a], [4059m] v = n — n'; T=7i't — nt + s' — s; G = a\ (^^) ~{- ^ . a jî% we get — = — 2m'.ag—im'. a^. ( — — ^ + J m'. n^. 2 . — . cos. i T4-f. cos.nt4- f. sin. n t ; [4059n] o \ da J f^v-^ — n^ &v = h — 'if.%m.nt-\-2f'.cos.nt->rm'.nt.'X3ag + a^.(^^\i Cna .,., 2n3. G ? . . _, which are substantially the same as the equations (5), (6), of Mr. Poisson, in the paper [4059o] 176 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. f and f being determined by the two following equations, given in [1018], [4059p] [4059c] [4059;] abovementioned ; observing, that i includes all integral numbers, positive and negative, except i= [1012'] ; whereas he only uses the positive values of i. Now if we use the expression of g [1017], the terms depending on nt will vanish from &v, and then & r [1020] will contain the constant part | m . a'' da but this is not the whole [40595] [4059<] effect of the disturbing force upon the radius vector ; because a part of this perturbation is introduced in the value of 7!, which is affected by the value of g, assumed in [1017], and n is connected with a by means of the equation [4059c]. We shall, for greater simphcity, take, as the epoch, the instant of the mean conjunction of the planets m, m' ; so that we shall then have < = 0, r=0 ; also s' = s. We shall also suppose that the body m', at that instant, commences its action upon the radius vector, and upon the longitude of the body m. Now we may find, from the tables of the planet's motion, the numerical values of r, v, —, —, when t = Q; and these are to be put equal to the values deduced from [4059c, f/]. These four equations, being combined with [4059e], determine the constant quantities n, a, e, s, w ; and then the formulas [4059c, t1] determine the elliptical motion, which obtains, if the disturbing force cease to act at the epoch ^ = 0. This being premised, we must put t^O, T=^0 [4059r], in the four equations [4058o], d.èr „ d.ôv dt [4059m] [4059t)] [4059«i] iSr^O; , 6v^0; -^=0; ^=0; and by substituting in them the values [4059», o], we may obtain the values of the four arbitrary constant quantities g, f, f, h, introduced by the second approximation. If we substitute these values of g,f,f', h, in 5r, Sv [4059n,o], they will express, at the end of the time t, the effect of the disturbing force during that time. Now the differential of 5r [4059?i], relative to t, being found, and substituted in the third equation [4059<], gives /'=0, when t = 0, T^O [4059/-]. With this value of /', and those of 5v [4059o, «], together with < = 0, T= 0, we get A = 0. Substituting these values d.6v of t, h, f, in the equations ^j- = 0, we obtain the follow ng equations, ■d.û<-0) dt = [4059<], using also the values [4059n, o], = — 2 m'. a g — J m', ft^. da + i7}i'.n^.S. !v9- -/; 0== — 2/n + ?«'.«. ^Saj+fl^.r^^^^^—Jm'. 2.^^. «^o- 2 «3. G [4059:c] Multiplying the equation [4059i;] by 2 n, and adding the product to [4059io] we find that the terms depending on /, G, ( —-: J , vanish from the sum, which becomes = — m. nag — J m'. 2 . — . a A^''' ; VI.v.§20.] ELLIPTICAL PART OF THE RADIUS VECTOR. 177 [4060] [m9y] whence s; = — — . 2 . .^ '\ Substituting this in [4059y], we get /=- — • 2 . A-'^ + i m'. «2. ( — — — I m'. n^ 2 . :^^ — - . By means of the values of /', /(, g, f [4059it, y], the expressions [4059m, o] become — = .2.^^''.(1 — COS.ÎtO irn.rt^. .(1 COS.?in a V \ d a / f. _ [40592] + i m. 11^. 2 .— —; ; • (cos. i T — cos. n ; C San .,., , „ /d.mx} 2\ [4059z'] [40600] [40606] If we retain merely the non-periodical parts of r, v, 5r, Sv [4059f, d, z, s'], and resubstitute the value of v [4059m], we shall get , ^ , m'.a^n .... . , „ fd.m\ ' ' »i— n \da / v4-5v = nt-\-i-^m'.nt.]^- . 2. ^'^-fa^. (— — )C ; ' ' ' t 2.()i— 7i') \ da / !) ' for the expressions of the mean distance and mean longitude of the planet m- The expressions of the same mean distance and mean longitude, according to La Place's calculation [1020, 1021], are r-\-àr = a-\-\m:.a^.(^-^\, v + àv = nt. [4060c] The differences between these values, and those in [4060a, &], are merely apparent, and arise from using different values of n, a, in [4060c] from those in [4060a, i]. To render this evident, we shall suppose, for a moment, that n, t represents the mean motion of the planet m, derived from observation ; then, by putting the coefficient of t, in the equation [40606], equal to n^t, we shall have , / <^ 3an ^ am i 2 f dA^ ^ ^ [4060d] ^ i 2.(n— 7i') ' \ da J <) VOL. III. 45 178 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. The preceding part of the radius vector [4059] may be united in the same table with the elliptical part of the radius.* Let a, be the value of a, deduced from the equation a = fA^.?i '•' [4059f], when '• *■' n, is substituted for ?t ; so that this equation holds good for a, n, and also for «^ , n, ; we shall have successively, by development, neglecting the square of n, — n, Substituting in this the value of n^ — n [4060cZ], we get, by transposition, [4000^] «=-«'+«»•«• 1-2-:i;;=;ô-'--^+"-V-^) I • This value of a being substituted in [4060«], we find, that the parts depending on A'-''' destroy each other, and we have / d A'-O'i \ [4060/i] '• + (5 ?• = a, + ^ ?«'. a^. \~J^) • Now as we neglect terms of the oi'der m ^, we may change a into «, , in the part depending on ^"" ; and then the expression [4060/t] becomes of the same form as in [4060f] ; being equivalent to that found by La Place. This calculation serves to illustrate and confirm his [40(j0i] method of calculation ; and shows, at the same time, how we can dispose of the additional arbitrary constant quantities, which are introduced by the integrations of 6 ?•, Sv; so as to conform to the actual situations and motions of the attracting bodies ; and to investigate the part of the effect of the disturbing forces, that we have particularl}' considered in this note. * (2552) We have here omitted a clause, in which the author directs, that the sign [4060ft] of the term of f, depending on cldA^^'', should be changed; because we have previously corrected the mistake, and given the accurate expression of /' in [1021g-], which agrees with that in [4060]. VI. vi.<^21.] NUMERICAL ELEMENTS. 179 CHAPTER VI. NUMERICAL VALUES OF THE DIFFERENT QUANTITIES WHICH ENTER INTO THE EXPRESSIONS OF Till; PLANETARY INEaUALITIES. 21. To reduce to numbers, the formulas contained in the second book and in the preceding chapters, we shall use the following data ; Masses of the Sun and Planets* Sun, M = 1 ; Mercury, m = ^^ ; log. m = 93,6934013 ; Venus, m' =: j^ ; log. m' = 94,4166538 ; The Earth, m" = j^^ ; log. m" = 94,4819733 ; Mars, m"'== \^~ \ log. w"'= 93,7337490 ; Jupiter, ^"=1^^ ; log- m" = 96,9717990 ; 1067,09 ° Saturn, vf = ^±^ ; log. m" = 96,4737383 ; Uranus, m"= "^^ ; log- m"= 95,7098763. * (2553) The factors l+fx, 1+(ji', Uc. in the values of m, m', &c. [4061], are not inserted in the original work ; but as they are introduced in [4230'], and frequently Masses of the planets, the masi of the sun beiiii? unity. [4061 J [4061o| 180 PERTURBATIONS OF THE PLANETS; [Méc. Cél. Of all these masses, that of Jupiter is the most accurately determined ; it is obtained by means of the formula [709]. If we put T for the time [40615] [4061c] Masses finally adopieil by the author. [406 W] used in computing the perturbations of the motions of the planets, it was thought best, for the sake of convenient reference, to insert them in this place. When the author printed this part of the work, he supposed, in conformity with the best observations, which could then be procured, that the masses of the planets were as in the table [4061], putting each of the quantities (x, (a', &ic. equal to zero. Since tliat time, he has been induced, by other observations, to make successive corrections in these masses, as in [4605, 4608, 9161, &c.]. In his last edition of the Système du Monde, he adopts the following Corrected Masses of the Planets. ^ (A =0; log. ??i =93,6934013; ^' =z — 0,0.56030 ; log. m = 94,3916120 ; (a" = — 0,0T1297 ; log. m" = 94,4498499 ; (;/"= — 0,275000 ; log. m'" = 93,5940870 ; |xi-= — 0,003186 ; log. m'':= 96,9704133 ; f;.^ = — 0,043451 ; log. m' == 96,4544455 ; ij:'= 0,088514 ; log. m"'' = 95,7467105. Saturn, nf = Uranus, m'' = The alterations here made in tlie values of »«', ?»"', are in conformity with the results of the calculations of Burckhardt, in his late solar tables, by comparing the observed perturbations [4061e] of the earth's orbit with the theory. The change in the value of m", arises from the supposition, that the sun's horizontal parallax is nearly equal to 8',6 [5589], instead of 8^,8, assumed in [4073]. Lastly, the values of nt", m'', m'", are obtained, by Mr. Bouvard, from the observations used in constructing his new tables of Jupiter, Saturn, and Uranus, by comparing the theory with the actual perturbations depending upon their mutual attractions. [4061/] Putting the values in [4061] equal to those in [4061fZ], respectively, we get the corresponding values of (a, f.'/, he. [4061f/]. Lindeneau, in his tables of Mercury, printed r4061ffl ill 1813, supposes that the mass of Venus ought to be increased to a-jaVioJ making |j,'= 0,09643 nearly; to satisfy the perturbations of Mercury, by the action of Venus. Encke, in his Astronomisches Jahrbuch for 1831, states, that the mass of Jupiter tû5 j.fls^ > deduced by Nicolai, from the perturbations of Juno, agrees better with the observations [40G1/I.] of Pallas and Vesta, than the mass adopted by La Place [4061, 4065], and that it probably VI. vi.^21.] NUMERICAL ELEMENTS. 181 of the sidéral revolution of the planet m' ; T for that of one of its satellites ; q for the sine of the greatest angle, under which the mean radius of the orbit of this satellite appears, when viewed from the centre of the sun, [40G2] at the mean distance of the planet from that centre ; then the mass of the sun being taken for unity, that of the planet will be expressed by * T .,r! \-q\ 7=— i; = mass of the planet. [4063] T [4061&] agrees also better for Vesta. Comparing this with [4061], we get (a''' =0,012492. When [406lt] we take into consideration that \he first value of fi''==0 [4061, 4065] is obtained from the observed elongations of the sateUites of Jupiter; the secondvdXue, (a'= — 0,003186 [4061«/], from the perturbations of Saturn and Uranus ; the third value, (^'=0,012492 [4061z], from the perturbations of the newly discovered planets ; we shall not be surprised in finding these small diflerences in the results of methods, which are so wholly independent of each other. Nothing is known relatively to the masses of these new planets or the masses of the [4061ot] comets, except that they are all very small ; so that their action on the other bodies of the system is wholly insensible. * (2554) This is deduced from [709], —^^ — .i—\, in which we must write [4062a] I* for M, as is evident from [706'] ; and as m' represents the mass of the planet, in the present notation, we have n = M + ?»'. Moreover p is the mass of the satellite [Î07'], and M that of the sun [706'] ; h the mean distance of the satellite from the planet ; a the mean distance of the planet from the sun ; so that — represents the quantity we [40626] [4062c] q [4062] ; hence the preceding equation [4062a] becomes - J^ , = ^^ / \ if / T \ 2 1 neglect p in comparison with m', and put JW= 1 ; also, for brevity, cf. (—\ =~ , we a 1 get, as in [4063], m'=^ =- — - . If we put r, p" for the mean densities of the [4062rf] 1_- ft-i bodies m'', m"; also R'% R" for the radii ; we shall have nearly, as in [2106], ?«>'■= 4 * . piv. (/3iv^3 . ^v ^ I ^ ^ pv_ ^ji-y^ [4062e] Hence we easily obtain the relative densities of these two bodies, ^~ = — .(-—] • [4062/1 pv m^ yR'" / This may be used for ascertaining the densities of all the bodies, whose masses are known, and whose apparent diameters have been well observed. VOL. III. 46 182 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. We have, relatively to the fourth satellite,* q = sin. 1530",38 = sin. 495',84 ; [4064] T = 4332'")%602208 = 433244* 27"- 10',8 ; r= 16'^'",6890 = 16"16''32"'09-,6. From [4063, 4064], we obtain [4065] m" [4064a] 1067,09 The mass of Saturn is found by the same method ; supposing the sidéral revolution of its sixth satellite to be 15"'''%9453 = 15"22"41'" 13',9, and the greatest angle, under which the mean radius of the orbit of this satellite appears, when viewed from the sun, in the mean distances of Saturn, [4066] 552'',47=179'. The mass of Uranus has, in like manner, been obtained, by supposing, conformably to the observations of Herschel, that the duration of the sidéral revolution of its fourth satellite, is 13'''ys4559 = 13'10''56'"29',8; [4067] jji^j ^]^Q mean radius of the orbit of tliis satellite, viewed from the sun, at the mean distance of Uranus, 13S",512 = 44',23. But the greatest elongations of the satellites of Saturn and Uranus have not been so accurately ascertained as that of the fourth satellite of Jupiter. Observations of these elongations deserve the careful attention of astronomers. The mass of the earth is found in the following manner. If we take the mean distance of the earth from the sun for unity, the arc described by the earth, in a centesimal second of time, will be obtained by dividing the circumference of a circle, whose radius is unity, by the number of [4068] seconds in a sidéral year, 36525638"''-,4. Dividing the square of this arc [4068] by the diameter, we obtain its versed sine = — r;^»^ jf which is the space the earth falls towards t!ie sun in a centesimal second, by means of its relative motion about the sun. On the parallel of latitude, whose sine is * (2555) The values of c/, T [4064], are nearly the same as those used in the theory of this satellite [6781,6785] ; the value of T corresponds to the mean motion n'" [4077]. t (2556) The radius of the orbit being 1, its circumference is 6,28.318 nearly; if we [4068a] divide this by 36525638,4, and take half the square of the product, we get the expression of the versed sine, corresponding to this arc, as in [4068']. VI. vi. }21.] NUMERICAL ELEMENTS. 183 equal to \/}, the attraction of the earth causes a body to fall through 3""", 66553* ill one centesimal second. To deduce from tiiis the earth's attraction at the mean distance of the earth from the sun, we must multiply it by the square of the sine of the sun's parallax, and divide the product by the number of metres contained in that distance. Now the earth's radius on tlie proposed parallel, isf 6369374'™'-; therefore, by dividing this number by the sine of the sun's parallax, supposing it to be 2T',2 — S%8, we obtain the mean radius of the earth's orbit, expressed in metres. Hence it follows, that the effect of the attraction of the earth, at a distance equal to that of the mean distance of the earth from the sun, is equal to the product of ihe fraction i'^'^^., by the cube of the sine of 27",2 ; ^ bSby-i 1 4 consequently it is equal to J 10- Subtractins this fraction from 1479565 10-33-' we obtain — 1479560,5 10- for the effect of the attraction of the sun, [4069] [4069'] [4070] [4071] [4071'] * (2557) This computation varies a little from that in [388"] or in [3SSf/] ; probably owing to a small difference in the ellipticity, used in reducing the observations. t (25.58) LT^sing the polar and equatorial semi-axes of the earth, 6356677™'-, 6375709"'"'' [2035i], whose difference is 19032""^'-, we find the radius corresponding to the latitude, whose sine is /-L, to be 6375709""='- — i X l9032'"'='-= 6369365""^'-, agreeing nearly with [4069']. J (2559) Gravity decreases, in proceeding from the earth's surface, inversely, as the square of the distance of the attracted point ; or as the square of the sine of the horizontal parallax of that point nearly. Hence the earth's attraction, at the distance of the sun, will cause a body to fall through a space represented by 3"""-,66553 X (sin. O's par.)"^, in one centesimal second of time. To reduce this from metres to parts of the mean distance of the earth from the sun, we must divide it by that distance, which is evidently equal earth's radius 6-369374 '"o'- r n i i • so that the space lallen through m a second, becomes to sin.27",2 \3. sin. Os' par. ^ô^\~ ■ ■ (sin- Q's par.)^ = , as in [4071'!. Now in [4063'], we have found, that Doo9.!}/4 10-0 ■- -" ■- -• the earth falls towards the sun, in the same time, by the combined action of the sua and 1479565 — 4,488.5 1479560,5 earth 1479565 10^0 ' nearly ; and as that of the earth is hence the effect of the sun alone is 4,4885 low ' 4,4885 1479560,5 „^„„, , ■ . r.-, to — ,^.,„ , or 1 to 329630 nearly, as in [4072]. 1020 1020 the mass of the earth is to that of tiie sun [4069a] [4070a] [4071a] [40716] [4071c] [4071rf] lOio 10-0 184 PERTURBATIONS OF THE PLACETS; [Méc. Cél. at the same distance. Hence the masses of the sun and earth are in the ratio of the numbers 1479560,5 to 4,4885; consequentlj the mass of [4072] the earth is . If the sun's parallax differ a little from the quantity we have assumed in [4070], the value of the earth's mass will varv as [4073] the cube of that parallax, compared with the cube of 21", 2 = 8",8 [4071c]. We have computed the mass of Venus from the formulas [4251, 4332, &c.], which express the secular diminution of the obliquity of the ecliptic to the [4074] equator; supposing it, by observation, to be 154',30^50'. This diminution is obtained from those observations which appear the most to be relied upon.* With respect to the masses of Mercury and Mars, we have supposed, according to observation, that the mean diameters of Mercury, Mars, and Jupiter, viewed at the mean distance of the earth from the sun, are, respectively, [4075] 21",60 = 7-; 35",19 = 11%4; 626",04 = 202-,84. Now Jupiter's mass being ascertained, we could, by means of these diameters, obtain the masses of Mercury and Mars, if the relative densities of these three planets were known. It we compare the masses of the Earth, Jupiter, and Saturn, with their magnitudes, respectively, we find, that the densities of these planets are very nearly in the inverse ratio of their mean distances from the * (2560) K we change 7, A [310-2f] into 0", «", respectively, to confonn to the [4074a] notation used in [4082. 4083] ; we shall find, that the arc F G^y . cos. A [3109c], which represents the difference between the inclinations of the equator to the fixed echptic of 1750 and to the variable ecliptic of 1750 -|- ^j is equal to o". cos. é", or q" [4249]. [40746] The value of q" is found by integrating the second equation [4251]. In this expression of q", the coefficients of fi, fif", (1% fi", are small, and the value of i^'^ [4061 J] is small and tolerably well ascertained ; therefore we need only retain /. so that the intesn^ [4074c] becomes q" = — ( 0". 500955 -p 0',309951 . ,u.') .t. If we suppose ,a' =^ 0, the annual [4074dl decrement becomes 0*..500955, being nearly as in [4074]. The action of the planet Venus has more effect in producing this change of obliquity, than that of all the other planets taken together; as is evident fcom the inspection of the value of d q'' [4251]; in which [4074e] we find, that the coefficient of ,a' exceeds the sum of the coefficients of the other quantities, ji, (i'", 11'", (Ji\ fi". We have already remarked, in [3380/! — q], that the author increased the annual variation to 0'\521154 [4613] ; on the other hand, Mr. Poisson uses 45692 [4074/1 [33Sqp], and Mr. Bessel 0-',48368 [3380j] ; each of them varying the values of ,a, ^', &c., so as to conform to their assumed decrements. VI. vi.§21] NUMERICAL ELEMENTS. 186 1 / O \3 a" ,„ 1 /D"'\3 a'" ' "' ine-r nn ' \ 7)iv ^ • „w J 1067,09 V-D'V « 1067,09 V^ and by substituting the values [4076c, 4079], we get, for m, m", rather greater values than those in [4061]. These diflerences probably arise from having used different values of D, D", D\ which cannot be obtained, by observation, to a great degree of accuracy. In some of the subsequent calculations, it will be sufficiently accurate to use the values of n, n, Sic. to the nearest degree; and for convenience of reference we have here inserted these approximate values ; 71=1661°; 7i'=650°, n"=400^, n"'=212=',7, w» = 330,7, n- = 13^,6, ■nr' = 4P,Q. VOL. III. 47 [4076] sun ;* we shall therefore adopt the same hypothesis, relatively to the three planets IMercury, Mars, and Jupiter ; whence we obtain the preceding values of the masses of IVIercury and Mars [4061]. The irradiation and the other difficulties attending the measures of the diameters of the planets, taken in connexion with the uncertainty of the hypothesis adopted on the law of their densities, render these estimated values somewhat doubtful, and this uncertainty seems to be increased from the circumstance, that the hypothesis is not correct relative to the masses of Venus and Uranus. Fortunately, Mercury and Mars have only a very small [4076] influence on the planetary system ; and it will be easy to correct the following results, so far as they are affected by this cause, whenever the development of the secular inequalities shall make known exactly the values of these masses. * (2561) The densities of the Earth, Jupiter, and Saturn, given by the author in the Système du Monde, are 3,93 ; 0,99 ; 0,55 ; respectively, being found as in [4062/, Sic.]. [4076a] These densities of Jupiter and Saturn are nearly in the inverse ratio of the distances a", a" [4079] ; but the density of the earth differs considerably from this rule. If we suppose this ratio of the densities to hold good for the three planets Mercury, Mars, Jupiter, and represent their apparent diameters [4075], by D=21",60, I>"'=35",19, -D'^=:626",04; [4076c] [40766] #13 n"'3 71' ^3 the corresponding masses will be m = b . — ; ?»'"= b . —^ ; m''= J . — — ; i being [4076(f] a constant quantity, to be found by means of the value of m'" [4061] ; which gives [4076/] [4076^] [4076A] 186 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. Mean motions of the planets. [4077] [4078] The time t 18 expressed in Julian years. Mean distances of the planets from the flun. [4079] 22. Mean sidéral motions of the Planets in a Julian year of 365| days, or the values of n, n', &c. Sexagesimals. Mercury, .n = 16608076",50 = 5381016%786 ; log. n =6,7308643 Venus, . . . n' = 6501980",00 = 210664r,520 ; log. n' =6,3235906 The Earth, n" = 3999930",09 = 1295977%349 ; log. n" = 6,1 125974 Mars, n"'= 2126701",00= 689051%124 ; log. /i"' = 5,8382514 71'"= 337210",78= 109256-,293; log. ?i'^ = 5,0384465 n^ = 135792",34= 43996%718; log. n^ = 4,6434203 n''= 47606",62= 15424^545, log. n" = 4, 1882124 Jupiter, Saturn, Uranus, If we use these values of n, n', &c., the time t ivill be represented in Julian years ; hence if we put the mean distance of the earth from the sun equal to xmity, we shall obtain, from Kepler's law [385'"], the following mean distances of the planets from the sun. Mean distances of the Planets from the Sun, or the semi-major axes of their orbits.* Mercury, a = 0,38709812 Venus, «' = 0,72333230 The Earth, a" = 1,00000000 Mars, «'"= 1,52369352 Jupiter, a'^ = 5,20116636 Saturn, a" = 9,53787090 Uranus, a" 19,18330500 log. a log. a' 9,5878211 9,8593379 log. a" = 0,0000000 log. «'"=0,1828976 log. rr = 0,7161007 log. a" = 0,9794514 log. «^' = 1,2829234. * (2562) These values of «, a', &C. are deduced from [4077], by putting them, [4079a] respectively, equal to I — V, ( — r, (— :F, &c. The elements of the orbits of the newly discovered planets, Ceres, Pallas, Vesta, and Juno, were first computed by Gauss, and have since been repeatedly corrected by him, VI.vi.§â-2.] NUMERICAL ELEMENTS. 187 The mutual action of the planets alters a little their mean distances ; we shall, in [4451, 4510], determine these alterations. aiul by other astronomers ; taking notice of the most important perturbations, from the [40794] action of the nearest phinets ; so that we can now compute the places of these bodies with a considerable degree of accuracy. The usual methods of finding the perturbations can be applied to these small planets ; but the great excentricities and inclinations of some [4079eJ of their orbits, will make it necessary, when great accuracy is required, to notice the terms depending on the powers and products of these two elements, of a higher order than is generally used with the other planets. The laborious task of ascertaining all the inequalities of these four planets, was not performed by the author of this work ; and it will probably be [4079<i] a long while before it can be done completely, on account of the small imperfections in the present estimated values of the elements, which have not yet been determined with perfect accuracy in the short period since the bodies have been observed. It is evident, also, that [4079e] until these elements have been found very nearly, it will not be of much use to compute several of the very small inequalities, with tiie extreme minuteness which is used relatively to the other planets. In computing the Jahrluch, it has been found most convenient by Encke to apply the corrections directly to the elements of the orbit, rather than to the elliptical places of [4070/'] the bodies ; in a manner similar to that which is used in finding the elements of a comet, in two successive returns. He finds, when the elements are thus adjusted to any particular [407yg'] moment of time, that they will give, tolerably well, the places of the planet for a considerable period, on each side of this epoch. The elements of the orbits obtained by him, for these four planets, about the time of the opposition of Pallas, in the year 1831, are as in the [4079A] following table ; which will serve to give an idea of the relative positions of the orbits at that time ; remarking, that these elements must not be confounded with the memi values. Epoch 1831, July 23d, 0'', mean time at Berlin. I Vesta. Mean longitude, 84'' 47" 03' Mean anomaly, 195 35 26 Longitude of the perihelion, .... 249 11 37 Longitude of the ascending node, . 103 20 28 Inclination, 7 07 57 Excentricity, 0,0885601 Mean daily sidéral motion, 97775540 Semi-major axis, 2,.361484 Periodic revolution corresponding, . 1325,5 days 74''39"'44' 20 22 31 54 17 13 170 52 34 13 02 10 0,2555592 813',525.33 2,669464 1593,1 days 290'' . 38"' 12»- 169 33 11 121 05 01 172 38 30 34 35 49 0,2419986 768%54421 2,772631 1686,3 days Cereg. 307'' 03'" 26'- 159 22 02 147 41 23 SO 53 50 10 .36 56 0,0767379 769-26059 2,770907 1684,7 days Elements of Veatu, Juno, Pallas, and Ceres. [4079i] 188 PERTURBATIONS OF THE PLANETS; [Méc. Cél. Eicen- tricities of the orbits of the planets. [4080] Ratios of the excentricities to the mean distances, or the values of e, e', ^c. for the year 1750. Mercury, e = 0,20551320 Venus, e' = 0,00688405 The Earth, e" =0,01681395 Mars, e"= 0,09308767 Jupiter, e'' = 0,04807670 Saturn, e'' = 0,05622460 Uranus, e" = 0,04669950 log. e =9,3128397; log. e' = 7,8378440 ; log. e" = 8,2256698 ; log. e"'= 8,9688922; log. e"= 8,6819346; log. &■ = 8,7499264 ; log. e''= 8,6693122. [407M] [4079Z] Elements of the orbits of the four known periodical comets. [4079m] [4079n] The distances of tlie planets Pallas and Ceres from the sun, are so nearly equal to each other, that it may sometimes happen, in finding the apparent orbits, in the precedins; manner, that the order of the bodies will be inverted, relative their distances from the sun, by means of the perturbations. Besides these planets, there are four comets, whose periodical revolutions have been discovered by Halley, Gibers, Encke, and Biela. They have been usually called by the names of the discoverers i-espectively. That of Olbers has been observed only once, at the time of its return to the perihelion in 1815 ; the others have been observed in several successive revolutions. Periodic revolution, Time of perihelion, Longitude of perihelion on the orbit, Longitude of the ascending node, Inclination, Excentricity, Semi-major axis, Of the seven periodical bodies, which have been made known to astronomers since the commencement of the present century, three were discovered by Dr. Olbers of Bremen ; namely, Vesta, Pallas, and the comet of 1S15. His great success in the discovery of these remarkable bodies, which had silently performed their revolutions in the heavens for ages, unperceived by astronomers, induced an eminent German writer to style him» the fortunate Columbus of the planetary ivorld. Halley's. Olbeis's. Encke's. Biela's. 7G years 74 years 1204 days 6,7 years Nov. 7, 1835 April 26,1815 Jan. 10, 1829 Nov. 27,1832 304' 31 "'43' 149"^ 2-" 157'^18'"35' 109'' 56™ 45' 55 ,30 83 29 .334 24 15 248 12 24 17 44 24 44 30 13 22 34 13 13 13 0,9675212 0,9313 0,8446862 0,751748 17,98705 17,7 2,224346 3,-53683 VI.vi.§22.] NUMERICAL ELEMENTS. 189 Longitudes of the perihelia in the year 1 750, or the values of ^, ts', ^c. Mercury, « = 8P,7401 = 13'33^5S' Venus, • ^' = 142°,1241 = 127 54 42 The Earth, ^" -= 109^,5790 = 98 37 16 Mars, ^"' = 368°,3037 = 331 28 24 Jupiter, ^'"^ 11°,5012= 10 21 04 Saturn, zy" = 97°,9466 = 88 09 07 Uranus, ^"= 185°,1262 = 166 36 49. Inclinations of the orbits to the ecliptic in the year 1750, or the values of f, <p', ^c. Loagitudes of the perihelia in 1750. [4081] Mercury, Venus, The Earth, cp" = Mars, ^'"^ Jupiter, tp"' = Saturn, <?' = 2°,7762 Uranus, <p" = 0^,8596 9 = 7°,7778= 7''00™00'; 9' = 3°,7701 = 3 23 35 ; ?" = 0° ; 2°,0556 = 1 51 00 1°,4636= 1 19 02 2 29 55 46 25 Inclina- tions of the orbits to the fixed ecliptic of 1750. [4082] Longitudes of the ascending nodes on the ecliptic of the year 1750, or the values of ô, 6', ^c. Mercury, . . Venus, . . . . The Earth, Mars, . . . . = 50^,3836= 45''20™43^; == 82°,7093= 74 26 18 ; as in [4249—4251]; '=: 52°,9376= 47 38 38 Jupiter, «'"= 108°,7846 = 97 54 22 Saturn, ô' = 123°,8960 = 111 30 23 Uranus, r = 80^,7015 = 72 37 53 VOL. III. 48 Longitudo;^ of tlie ascending nodes of the orbits on the fixed ecliptic of 1750. [4083] 190 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. Epoch. All these longitudes are counted from the mean vernal equinox, at the epoch [4084] of December olst, 1749, mid-daij, mean time at Paris. ^Ve may here Lpnguude observe, that hij the longitude of the perihelion, is to be understood, the rlôJiT <^'^fc>^c^ of the perihelion from the ascending node, counted on the orbit, increased by the longitude of that node. 23. We have obtained the following results, by the formulas of §49, Book II. MERCURY AND VENUS, [4085] [4086] hence we deduce Then we obtain* 0. = - = 0,53516076; 6_;. = 2,145969210; rin 6 \ = — 0,515245873. 6^ = 2,1721751 ; b''^ = 0,6057052 ; 6^^' = 0,2465877 ; ,(3) ^^L. 6, = 0,1107665; ,w 6^ = 0,0520855; ,(5) 6V= 0,0251378; ,(6) [4087] 5^ = 0,0123166 ; ,P) 6\'= 0,0060633 (S) 6^;'== 0,0029287 ; 6^^'= 0,0012758. * (25G3) From a, a [4079], we have a=-, as in [4085]. Then from [989]. [40S6a] we find, 6,, b_^, as in [4086]; from these we get b<, b, [40S7], by means of the formulas [990, 991]. Then putting, in [966], s^i, and successively, ?:=-2. /=3, I = 4, Sec. we obtain the remaining terms of [4037]. From these last, we get those [40866] in [40S8], by putting, successively, 2 = 0, j* = l, Sec, and s = i, in [981]. The same values, being substituted in [98-2], give [4089] ; also [983] gives [4090] _ Lastly, by taking the partial differential of [983], relative to a, we shall get an expression U) d*b s ■ [4086c] of ; in which we must put s = i ; then j'^0; /=1, &;c. ; and we (0) a) shall get [4091]. Again, the formulas [99-2] give ba. , bs. , [409:2]; from these two Vl.vi.S^a.] VALUES OF b'^, AND ITS DIFFERENTIALS FOR MERCURY. 191 (0) dbl, da. = 0,780206 ; do. 1,457891 ; dbi do. = 0,691487 ; (41 dbi do. 0,423818; (6) dbi rfa = 0,147708 ; a) db^ do. 0,085953 ; (0) dH^ da? = 2,756285 ; (11 dHi da? = 2,426165; (3) d"-bi do? = 3,381072 ; (41 dHk do? = 2,826559 ; (6) dHi da? = 1,511016; (71 dH>, da? = 1,014134; (01 dHk do? = 11,308703; m dHk da? - 12,064245 ; (31 da? = 14,584366 ; (41 dH^ do? = 16,067040; (61 dHk da? = 13,720218. (2) d*bK do.* = 69,60594 ; (?) d^b^ da.* = 82,36773 ; (51 d*bk_ da.* = 105,33962. r-1 db.i do. - 1,070071 ; j/^ dbi do. = 0,252376 ; [4088] 77"' db^ do. = 0,050726. (21 dHi do? = 3,395022 ; d^fl do? = 2,137906 ; [4089] Mercury and Venue . (21 dHi, (51 dH^ da? 11,983424; 15,617274; (41 d*b), 92,72610 ; [4090] [4091] terms, we may obtain the others of [4092], by means of the formula [966] ; putting s = |, and, successively, i^2, i=3, &;c. The values [4093] are found from [981], by putting s = f, and i = 2, i = 3, he. Those in [4094] are deduced from [982], by [4086d] using similar values of s, i ; observing to substitute, in any of these formulas, the values of b, or its differentials, which occur, and have been found in the preceding parts of the calculation. All the other terms of this article, §23, are found in the same manner, except those in [4113, 4119, 4124, Sec], where a is very small ; and there is no difliculty in the [4086e] calculation, except the ennui, arising from a long and uninteresting numerical calculation. 192 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. (!) (2) 4,214154 ; 6^ = 3,035376 ; 63 = 1,950536 ; [4092] Mercury and Venus. (3) b^= 1,192372; 2- 0,238807. (4) 6 , = 0,708667 ; (5) 63 =0,413762; [4093] (2) = 12,50630; (3) ^ = 9,76666 ; da. (1) dbî 7,08399 (5) db^ da. = 4,88781. [4094] (3) dHi = 78,09476 ; (4) dHi da.^ 67,14764. [4095] MERCURY AND THE EARTH. hence we deduce a = - = 0,38709812 ; [4096] , (0) 6^; =2,07565247; ~2 ,(I) 6 , = —0,37970591. Then we get Mercurjr and the Earth. 6*^"' = 2,081980; (1) 6, =0,411140; 2 è'f = 0,120178 ; [4097] (3) 6^ = 0,038900 ; (4) 6 J = 0,013202; ô'J = 0,004603 ; (6) 6 = 0,001629 ; (V) 6^ =0,000573; (8) b, =0,000177. VI. vi.§:2;3.] VALUES OF T' AND ITS DIFFERENTIALS FOR MERCURY. 193 (0) da. (3) dbj do. (6) dbi = 0,464378 ; =. 0,316756 ; = 0,026130 ; (0) cPbj da.' (3) dH^ da.^ (2) dHj ,(0) = 1,672199; = 1,852364; = 6,49232; 6 =2,871833; 6*'' = 0,334212 ; (I) db), (4) dbj do. m dbj da. (1) d^bi (4) dHj_ do.^ (3) dHj da.^ .(') 1,199633; 0,141792; 0,011153. 1,220775; 1,197245; : 5,45663; 63 =1,576062; ,w 63 =0,153779. (2) dbj do. (5) dbis da. 0,665739 ; 0,061433 ; [4098] (2) '^l^^ = 2,235935 ; do.^ (5) '^^J^ = 0,670874. [4099 Mercury and the Earth. (4) ^'^^ _ 6,51373. [410C 6^!' = 0,747619; (3) dbi = 3,05535. [4101] [4102] MERCURY AND MARS. hence we deduce a = ^, = 0,25405312 ; ,(0) b'^ = 2,03240384 ; , (1) r: = — 0,25198657. [4103] Mercury and Mare. [4104] VOL. III. 49 194 PERTURBATIONS OF THE PLANETS; [Méc. Cél. Then we have [4105] Mercury and Mars. [4106] [4107] b^'^ = 2,033500 ; ,(3) (0) ,(1) dbl do. do. (0) (3) d^bj doJ" = 1,050458. b =0,260462; ,(2) 6 J ==0,049765; (5) = 0,010546; '.= = 0,002331 ; b ^ = 0,000538. = 0,273829 ; (1) dbi d a = 1,077839; ir^ = 0,402980 ; do. = 0,127139 ; db^ do. = 0,037781. = 1,244725; do? = 0,656780 ; (3) ^= 1,778641 ; [4108] è^J^= 2,322536; ,(1) rJ = 0,863876 ; 6''' = 0,272085. [4109] Mercury hcncc WB deduce and Jupiter. [4110] MERCURY AND JUPITER. a = - = 0,07442555 ; a" f^ = 2,00277053 ; 2 &"; = — 0,07437397. ,(0) ,(1) In computing the values of 6 , 6 , &c., by means of the formulas [966 — 983], it is found, that the successive terms of the series become more inaccurate, particularly if o. be rather small ; because these values Vl.vi. §23.] VALUES OF b^^ AND ITS DIFFERENTIALS FOR MERCURY. 195 are the differences of numbers, Avhicli vary but little from each other ; so that we are under the necessity of computing them to an extreme degree [4lll] of exactness, to enable us to determine correctly their differences,* and this requires the use of tables of logarithms to ten or twelve places of decimals. To obviate this inconvenience, we may have recourse to the value of b '\ developed in a series, by means of the formulas [976, 984— 985],t 'ill (i±i' a2J_*-(*+l) (^+')-(^-H+l ) ^4 *-=^-— 172737^::^^ — •"-'•< ^ [4112] 1.2.3 • (i+l).(i+2).(i+3) • "^ This value of 6'"' is, in the present case, very converging, on account of the smallness of a. We shall hereafter use it, in finding the values of b , b \ &ic.; 6'°\ &c., in ail cases where a is rather small. i h ^ By this method we have computed, for Mercury and Jupiter, the following values ; (0) (1) (2) 6 = 2,002778 ; b,= 0,074581 ; 6, = 0,004164 ; [4113] Mercury (3) (4) and b^ = 0,000258 ; b^ = 0,000017. '"•'''"• * (2564) Thus, if we put s = i and i = 2, in [966], it becomes <i) (0) (2) (l+a').6a— ia.è , [4111a] ** = f^ —^ • Now ht, is much smaller than h. or h. [4105], and the preceding value of b' is divided by the small quantity J a. Hence it necessarily follows, that the terms (1 + a^) .b, and — ^ a . è , , in the numerator of this expression, must be very nearly equal to each other; and their difference, which is to be divided by a quantity of the r^mii order a, must therefore be very accurately computed. The same takes place in b\, &.c. t (2565) The quantity h is the coefficient of cos. i ê, in a-^ [976] ; and X-* is the product of the two factors [985]. If we multiply these factors, and retain only terms of the form 0=*='^*^, putting c'"^-' +c-'^»^"' = 2.cos.i é [12] Int., it becomes [4n2„j as in [4112]. 196 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. [4114] Mercury and .lupiler. [4115] (0) dbs d CL da. m dUi 0,074891 ; : 0,010428; 1,018876; Cl) dbj_ da. dHj da.^ 1,006269; = 0,171781 (2) dbi, d a (2) dHj_ do.^ 0,111380; = 1,499780; [4116] 6^ = 2,025143 ; b'l = 0,225613 ; ,(2) 0,020984. MERCURY AND SATURN. hence we deduce [4117] [4118] ^Zr Then we find Saturn. ,(0) [4119] 6 j'= 2,000823: 6^^^ = 0,000042 ; 0. = - = 0,04058547 ; 6^°' = 2,00082368 ; 6^'' =—0,04057711. —2 6^^ = 0,040610; 2 6'^ = 0,000001. ,(2) 6^^^=0,001236; [4120] [4121] rf&è da. = 0,040662 ; (1) dbi da. 1,001841 ; (3) dhh da. = 0,003085. (0) dH^ da.^ = 1,003904; d< da.^ :^ 0,091840; (2) dbj^ da. = 0,060919 ; (2) dHi 1,469188. VI.vi.'§23.] VALUES OF b'^ AND ITS DIFFERENTIALS FOR VENUS. 197 MERCURY AND URANUS. hence we deduce a= 4- = 0,02017895; a" 6_\ = 2,00020360; 6!1', = — 0,02017792, Then we find 6^^*= 2,000182; 6^''= 0,020183; (0) (1) 020196: ^^^ =1 (2) è^'= 0,000306; do. do. = 1,000913. [4122] [4123] Mercury and Utanuf. [4124] [4125] VENUS AND THE EARTH. hence we deduce «L = ^, = 0,72333230 ; a 6 , = 2,27159162; — 3 Then we obtain 6 ''! = — 0,672263] (0) b^ = 2,386343 ; b'^ = 0,942413 ; (3) b. =0,323359; 6*" = 0,206811 ; 6 J = 0,090412 ; ▼OL. III. .cn 60 6^ =0,527589; 2 6^ = 0,135616; (8) 6i = 0,061101 ; 6^ = 0,041731. [4126] [4127] Venus and the Earth. [4128] 198 [4129] [4130] Venus and the Earth. [4131] [4133] [4133] (0) db^ do. (3) djb^ do. (6) dbj d a (0) dH^ do.^ (3) dHj do.^ (6) do.^ (0) do.^ (3) d^ do.^ , (0) PERTURBATIONS OF THE PLANETS ; : 1,643709; 1,738781;" 0,867147 ; 7,719923 ; 9,112527; : 7,842733. : 66,55335 ; : 62,87646 ; [Méc. Cél. 63 = 9,992539 ; ,(3) b, = 6,953940 ; K3) d_H_ do. (I) dbi do. = 2,272414; (2) dbi do. : 2,069770 ; (4) do. : 1,407491 ; (5) db^ do. = 1,113704; df^ do. : 0,668830. do.^ = 7,531096; (2) d^fii do? = 8,558595 ; (4) d^i : 9,107400 ; (5) dn^ _ /7«2 = 8,634030; d^èl do.^ (4) dH^ do.^ = 57,35721 ; 66,32409 ; ,(i) Ô; = 8,871894; rt^) b\ = 4,704321 ; = 56,65440 ; (4) dbi do. (2) dHi = 58,19633; dH (5) da» i = 70,54326. ,(2) 6 y = 7,386580 ; 6 ; = 3,652052. 50,90290. VENUS AND MARS. [4134] Venus and Mars. [4135] hence we deduce a = 4; = 0,47472320 ; a 6^°J= 2,11436649; 6"j = — 0,46094390. VI. vd.^SS.] VALUES OF i^;' AND ITS DIFFERENTIALS FOR VENUS. 199 Then we find 67=2,129668; S 5^ = 0,521624; 3 fe'f = 0,187726; 5 6*^' = 0,074675 ; 6*^'* = 0,031127; 6'f = 0,013337; [4136] 2 (6) 6 , = 0,005829. dh (0) do. (3) 1 = 0,631752; ^ = 0,510976; do. (0) ^ = 2,192778; do.- (3) dHh do? = 2,628516 ; (0) dH^ do? (3) dHi, da? 7,65440 ; = 10,66513. (1) db^ do. 1,330781 ; do. 0,279002 ; (1) dH^^ do? -- 1,815836; do? = 2,004429. J .3 = 8,45655 ; do? (2) dh^ do. (5) db^ da. (2) d^i do? 0,884106 ; 0,147606. = 2,795574 ; i^ = 8,17676 ; [4137] [4138] VonuB and Mars. [4139] ,(0) 6 = 3,523572 ; 2 .<3) 6, =0,722687. 6*3^ = 2,304481 ; (2) dH da. 8,47521. .(2) &3 = 1,325959; [4140] [4141] VENUS AND JUPITER. a=- = 0,13907116; VenuB and Jupiter. [4142] 200 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. hence we deduce [4143] b^'l = 2,00968215 ; 6''' = — 0,13873412. Then we have 6^ = 2,009778; 6'^'= 0,140092; 6'J'= 0,014623; [4144] 6^^=0,001695; f4) 6^ = 0,000206; (5) 6^ = 0,000026. Venus and Jupiter. 2 [4145] \^ =0,142160; a a ^=1,022206; da -^ = 0,212046 ; da ' dh '^ "* = 0,036783 ; d 0. "/* =0,006111. da rdi /irfîi (0) ^=1,067532; (1) Ç^= 0,325869; (2) '^^*_ 1,575190; [4146] (3) '^l^^ -0,533951. [4147] C) 63 =2,089736; (1) b^ = 0,432801 ; (2) 63= 0,075054. VENUS AND SATURN. [4148] Vennsand hcHCe WC dcduCC Saturn. [4149] a = - = 0,07583790 ; b^^ = 2,00287673 ; b^'\ = — 0,07578334. ,w Vr.vi.§23.] VALUES OF b'J AND ITS DIFFERENTIALS FOR VENUS. 201 riieii we obtain (0) 6j = 2,002886 ; (1) = 0,076002 ; (2) -. 0,004323 ; [4150] 6^' = 0,000273 ; 2 '*: = 0,000018. [4151] (0) ''f * = 0,076331 ; da. (1) dbi, do. = 1,006490; (2) dbi _ da. 0,114267; [4152] 7/'" ,* -0,011085. da Venus anil Saturn. (0) ''^=1,019629; ft a.-' d^i da? = 0,172510; (2) dHi do? 1,419950. [41.53] (0) b\ =2,026116; !>":- ^ = 0,229988 ; = 0,021791. [4154] VENUS AND URANUS. hence we deduce Then we find a = — = 0,03770634 ; r\ = 2,00071095; -'2 — 0,03769964. [4155] [41.56] Venu3 and Uranus. ,(0) 0^=2,000712; ,(1) 6^=0,037725; ,(2) b\= 0,001067 ; [4157] ,0) 4 67 = 0,000034. (0) dbi da. VOL. Ill = 0,716690 ; (1) dbj da. = 1,000829; 51 (2) db i do. = 0,056634. [41.58] 202 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. THE EARTH AND MARS. [4159] [4160] a =. — = 0,65630030 ; hence we deduce Then [4161] The Earth and Mnrs. [4169] [4163] [4164] .(0) (3) (6) 6, = (0) da. (3) (6) da. (0) (3) dHj do.^ (6) dH^ do? (0) dHh do.^ (3) dH^ 2,291132; 0,224598 ; 0,046595 ; = 1,228078; : 1,240990; : 0,473942 ; 4,985108 ; 6,057860 ; : 4,388001. 29,03400 ; 33,29381 ; , (0) 6_j = 2,22192172; ,(") 6_^ = _ 0,61874262. = 0,804563 , a'^' h = 0,129973 ; 0) = 0,028480 ; (1) dbh do. = 1,871211; (4) d b ^ do. ^ 0,920710 ; 7 7^" db^ do. = 0,333444. (1) dn^ do.^ = 4,744671 ; (4) d^bi .1 .2 = 5,776483 ; (I) ^ = 29,78930 ; (4) d^bi Vf- = 36,32093 ; (SJ h- = 0,405584 ; (5) = 0,077170; 2 : 0,0175565. 1,601236; (5) ,7 „ = 0,666207 ; (2) dH^ do? (?) dHk = 5,731111 5,141993; (2) îÇii. = 30,18848; (5) dH), = 37,23908. VI. v-i. §-23.] VALUES OF ù^'^ AND ITS DIFFERENTIALS FOR THE EARTH. 203 (0) 6 y = 6,856336 ; 6''' = 3,255964 ; 6^'' = 1,174650. (2) ^ = 31,80897; do. ,(') J 3 = 5,727893 ; fi'I' = 2,351254 ; f^ =^ 4,404530 ; 6'^' = 1,671668; [4165] (3) (5) '^ = 32,26285 ; .... 'Ill ^ 18,25867. [4166] a a «a THE EARTH AND JUPITER. hence we deduce Then 6^ = 2,018885; 6^^'= 0,004516; ,(6) a= - = 0,19226461 ; &'"[= 2,01852593; 6^'! = — 0,19137205. 6^ = 0,195003; 2 6'^'= 0,000779 ; ,(2) 6^=0,028195; (5) 6^ = 0,000132; [4167] [4168] The Earth and Jupiter. [4169] 6, =0,000023. (3) 0,200586 ; = 0,070932 ; d a da. 1 ,043204 ; 0,016369; (2) da. (5) 0,297995 ; = 0,003448 ; [4170] 204 PERTURBATIONS OF THE PLANETS ; [Méc. Ct [4171] (0) dHi, 1,132355; = 0,746681. The Earth d 0? and Jupiter. (0) [4172] 1!^ = 1,4727 14; (1) d^bi = 0,466165; Vf = 2,874986 ; (2) dH^ do? 1,628667; (2) ^ = 1,418830. [4173] ,(0) 6 ' = 2,176460; (3) h 3 = 0,032493. 6*'' = 0,619063; 5 ,(2) b =0,148198; [4174] THE EARTH AND SATURN. hence we deduce a = - = 0,10484520 ; (0) b_^ = 2,00550004 : [4175] (1) 6_ J = — 0,10470094. Then The Earth and Saturn ^'I' = 2,005535 ; [4176] 6? 'S (0) dbi, d<x. = 0,000724 ; = 0,106155; [4177] (3) dbi do. = 0,020779. fe'l'^ 0,105283; s (4) b, = 0,000066. dJl d Ol = 1,012536; ,(2) b =0,008282; (2) dbi 0,158723; Vl.vi.§-23.] VALUES OF 6*;' AND ITS DIFFERENTIALS FOR MARS. 205 (0) (0) 1,037816; b = 2,050321 ; ^ 0,246193 ; do? : 1,526303, (1) (2) b : 1 = 0,321144; *,= = 0,041977 [4178] [4179] THE EARTH AND URANUS. hence we deduce a = -=0,05212866; [4180] , (0) 6 ; = 2,00135893; -i .0) 6_^ = — 0,05211095. [4181] Then we find . (0) b =2,001355; .0) 6- = 0,000089. 2 (1) 6 =0,052182; 6^=0,002040; 2 The Earth and Uranus. [4182] (0) dbi ~ = 0,052288 ; -(0 da. ' - 1,003060; (2) '^ = 0,078449. o a [4183] MARS AND JUPITER. a = — =0,29295212. [4184] VOL. III. 52 206 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. hence we deduce [4185] Then (0) b i= 2,04314576; (1) & , = — 0,28977479. 'r = •2,045112; b'^= 0,302922 ; 2 <- : 0,066812; [4186] 2 = 0,016357; (4) 6^=0,004192; "h 0,001109; "> = 0,000297 ; (7) 6^ = 0,000081. (0) dbi da. = 0,324004 ; 0) ,^ -1,105998; do. C2) do. = 0,473717; [4187] (3) db^ da. = 0,172096; (4) '^Z* =0,058420; a OL db'l - 019258: do. Mara and Jupiter. (6) dbh do. = 0,006173. (0) dHi do.^ = 1,338759; 3. m ^ = 0,794557 ; (2) do.^ ' = 1,871538 ; [4188] (3) do.'' = 1,258858; (4) '^'\*- 0,623184. (0) d^bi do.^ = 2,69358 ; ^ = 3,77722 ; (2) = 2,91068; [4189] (3) dHi do.^ = 5,47068. (0) *3 = = 2,444762 ; 6 '=1,040206; 3 : 0,376693; [4190] 'b% i =0,127942. ^ VI. vi.§23.] VALUES OF 6"' AND ITS DIFFERENTIALS FOR MARS. 207 (0) db^ da. = 3,48815 ; (I) db^ do. = 4,80540 ; (2) ^ = 2,99684. a a [4191] MARS AND SATURN. a = — = 0,15975187; [4192] hence we deduce Then we find (0) (3) 6_ 1 = 2,01278081 ; 6^'^ =—0,15924060. (0) (1) (3) 6, =2,012945; 6, =0,161305, 6, = 0,019347 ; - 2 2 w h- = 0,002577 ; = 0,000360 ; (0) dbi do. = 0,164463; (1) dhi . da. = 1,029493; dbi do. ~ = 0,048740 ; do. = 0,009065. (0) dHi do? = 1,090095; (1) dHi da? - 0,379322 ; (3) - n fi9nfi.S9 (5) b^ = 0,000052. (2) dbi 0,244843 ; (2) Ç^ = 1,596248; da.-' [4193] [4194] Mars and Saturn. [4195] [4196] da? , !0) 6; = 2,119585; b'l' = 0,503071 ; ,(2) 6y = 0,100136; [4197] 2 208 PERTURBATIONS OF THE PLANETS ; [Méc. Cél MARS AND URANUS. hence we deduce [4198] [4199] Mars Then we find aod Uranus. ,(0) [4200] 6^ = 2,003167; , C3) a = - = 0,07942807 ; a"' 5'°^ = 2,00315565 ; f\ = — 0,07936538. —5 6'" = 0,079617; (4) 6V = 0,000314 ; b.= 0,000022 2 2 ,(2) 6^=0,004746; [4201] (0) da. (3) d_H d a. 0,079995 ; 0,011982. (1) dbk = 1,007144<; (2) dH da. = 0,119822: JUPITER AND SATURN. [4202] Jupiter and Saturn. [4203] a = - = 0,54531725; a" hence we deduce (0) 6_j =2,15168241; felj= — 0,52421272, Then we have 6^^' = 2,1802348; 6*'' = 0,6206406 ; •3 b'^ = 0,2576379 ; VI. vi.§^3.j VALUES OF i'j'AND ITS DIFFERENTIALS FOR JUPITER. 209 (3) bi =0,1179750; (6) 5, = 0,0139345 ; ,(9) èy= 0,0018056; 2 b^ =0,0565522; (7) b . = 0,0070481 ; 6*J"L 0,0008632 ; 6, =0,0278360; (8) 6^ = 0,0035837 ; bl'L 0,0003223. [4204] 77'"' db i da. = 0,808789 ; (3) db^ da. = 0,726550 ; (6) dbi da. = 0,163506; (9) = 0,033083 ; (0) - 2.875229 : da.^ (3) '^^ = 3,533622 ; dHi do? = 1,664586; (9) ^ = 0,485135. da? da. (4) (7) da. (10) db i (1) dHj do? (4) dH^ da? = 1,483154; = 0,453285 ; 0,096019 ; 0,020265. 2,552788 (7) dH^ 1^ = 2,995647 ; = 1,144377; (2) db^ = 1,105160; (5) db^ (8) dbi, da. 0,274717; = 0,056171 ; (2) da? (55 dH}, da? (8) d^b^ d a' = 3,521040; = 2,302428 ; i = 0,760603 ; [420.5] Jupiter and Saturn. [4206] (0) da.^ 12,128630 ; (3) dH,, = 15,454850 ; d^bk ~~ = 14,958762 ; VOL. III. (I) dHj da? = 12,878804; (4) ^1^ = 17,058155; Vf= 12,234874; 53 (9) (Z^èj = 12,832050; (5) dHk (8) = 16,655445; [4207] , , = 9,566420. 210 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. [4208] (0) d^hi (?) d*bi^ 84,40159 ; = 89,8615; (6) dHi 118,6607; (1) d*bi do} (4) d'bi, da.* (7) da." = 83,94825 ; 101,3809; = 115,9588. (2) d'bk^ — 87,3027 • da.* (5) dH^ da." = 113,5238; [4209] Jupiter and Saturn. [4210] d'b^ da.^ (1) = 747,480 ; dH (3) d a: I = 785,884 ; (6) ^—^ = 912,301. da."- f^ = 4,358387 ; 6''' = 1,295672; 6^" = 0,273629 ; b'^ = 0,053922. 3 ' d^bi da.' (4) dHi (3) da.^ = 753,417; = 819,180; 6^^^ = 3,185493; z f^ = 0,784084 ; 67 = 0,158799; 2 d' H d a5 d^ (5) 761,843: ■■ 884,505 ; (2) b = 2,082131 ; ,(5) 63 =0,466047; (8) b , = 0,092290 ; [4211] (0) ^ = 14,681324; da. ^=10,598611 ; d a db i = 3,710043 ; (1) db$ da. (.1) dbj da. (7) db§ d a 15,239657 ; 7,802247 ; : 2,426079 ; (2) ^=13,416026; (5) db§ da. = 5,470398 ; (8) 1^^=1,563695. a a. (0) dHj da.^ = 96,68536 ; (I) ^2/,;t VI- = 94.91 701; d a.-' (2) Vf = 93,19282; VI. vi.§'23.] VALUES OF li^" AND ITS DIFFERENTIALS FOR JUPITER. 211 (3) dHè d<x? («) dH^ da? (0) d^i do? (3) dH§ do? (6) dHi da? = 86,90215 ; = 47,48185; 830,0586 ; 785,5855 ; = 574,9115. (4) ll/f= 75,08115; dHi do? 35,74355. do? = 830,1580 ; (4) d^b3- Vf = 740,6775 ; do." (5) d^b§ do? = 61,10115; dH§ do? = 810,1045; (5) d^b^ Vf = 666,4080 ; do? [4212] Jupiter and Saturn. [4213] JUPITER AND URANUS. hence we deduce Then we get 6^°' = 2,038359 ; S 6^ = 0,012879 ; 2 (6) 6, =0,000185. do 4 a = — = 0,27112980; a" 6'"! = 2,03692776 ; 6l'! = — 0,26861497. 6''' = 0,278966 ; ft'"' = 0,003058 ; da. = 0,295410; dix. = 1,089551; 6 , = 0,056906 ; 6^^^ = 0,000745 ; (2) da. [4214] [4215] Jopiter and Uranus. [4216] = 0,433630 ; [4217] 212 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. Jupiter and UranCB. (3) ^,^ - 0,145398 ; a a Ç^= 1,283434; da'' [4218] (3) '^'\* =1,133359. da'' (0) b 3 = 2,372983 ; [4219] 2" 6 , =0,099260. (4) da rf^tT = 0,045930 ; = 0,714932 ; (1) b =-0,938794; (5) dbi da. (2) dHi da? 0,015410. 1,815451 ; 6 "'=0,315186; 3 [4220] hence we deduce SATURN AND URANUS. a = — =0,49719638; a"' (0) b 1=2,12564287; [4221] (1) b i = — 0,48131675. Saturn and Uranus. Then we get (0) 6, = 2,144440; (') 6^ = 0,552007 ; (2) 6 =0,208313; 2 (3) [4222] 6^=0,086834; (4) 6 =0,037909; i (5) 0^ = 0,016990; (6) 6 , = 0,007728 ; (V) 6 =0,003522; i .<8) 6 =0,001547. 2 (0) dh 1 ^ = 0,683055; da db. (1) = 1,373806; (2) 1^=0,949128; da ,(i) Vl.vi. ^v,>3.] VALUES OF bj AND ITS DIFFERENTIALS FOR SATURN. 213 (3) i^A = 0,572896 ; do. (6) dbl = 0,098799 ; dbj da. (7) dbi do. = 0,327198; 0,053642. (5) dbj da. 0,181370; [4223] (0) " 1^ = 2,377102; = 2,881218; d a.-2 d°~ (3) 'b^ d 0.^ d^ (6) H d 0.^ rf3 (0) d a? J3 (?) bi do? rC) 1,067430. = 8,798999 ; = 11,904140; 6; = 3,750905; (3) b =0,872105; d^bj do!^ dn\ = 2,017767; = 2,278077 ; (1) d 0? 9,578267 ; d^b^ --^= 12,988670; do."* f^ = 2,547992 ; 6^'J = 0,482564 ; (2) d^bi = 2,992245 ; (5) d^J. = 1,616470; (2) do? 9,425450 ; (5) 12,135721. "> = 1,530452; (5) b = 0,262146. [4224] Saturn and Uranu». [4225] [4226] (2) ^=9,75656; do. (3) db^ -P = 7,24097 ; da. (4) 'iH ^ 4^95062. do. [4227] YOL. III. 54 214 PERTURBATIONS OF THE PLANETS ; [Méc. Ctl CHAPTER VU. NUMERICAL EXPRESSIONS OF THE SECULAR VARIATIONS OF THE ELEMENTS OF THE PLANETARV ORBITS. 24. We shall now give the numerical values of the secular variations of the elements of the planetary orbits. For this purpose we shall resume the differential variations of the excentricities, perihelia and inclinations of the îbVthe"' orbits ril22, 1126, 1142, 1143, 11461. To reduce these formulas to num- computa- L ' ' ' ' J (oln^Lc. bers, we must previously determine the numerical values of the quantities (0,1), [^J], &c. These are obtained by computing, in the first place, the values of (0,1), [ÔJ] ; by means of the formulas [1076, 1082],* (1) . _ 1 , 3 m. n a? . b_i [4^8] CQ'^)=- 4.(l-a^) ^ I'onnHlas 14228'] [JM ] = o n_.2^a (1) (D) 3m'. no.. I (l+a^).6_^ + Aa.6_è \ 2.(l-a2)2 From these we have deduced the values of (1,0) ["iT|, by means of the equations [1093, 1094]. (0) U) * 2566. The values of mf, n, a, b_i, 6_j to be used in tliese formulas are given in IA09R [4061 — 4222]. By the formula [4228] we must compute the values corresponding to the exterior planets, namely ; (0,1), (0,2), (0,-3), (0,4), (0,5), (0,6); (1,2), (1,3), (1,4), (1,5), (1,6); (2,3), (2,4), (2.5), (2,6); (.3,4), (3^), (3,6); (4,-5), (4,6); (5,6); and the similar ones of [4228'], namely; [J^] Sic; [wi] &ic. ; [W] fee; [J±] &lc.; [JJ] he; [Jfi^j. The remaining terms corresponding to interior planets are to be deduced from these by the formulas [4229]. Thus, if it be required to compute (4,5), [^] cor- responding to the action of Saturn upon Jupiter. The value of m' to be used in [4228], [42286] VI.vii.^^24.] SECULAR VARIATIONS OF THE ELEMENTS. 215 (1,0) m .\/a 7-(0,i); m.\/a ^ 7)1. y a Bv this means we have obtained the following results, in seconds, supposing tlie numerical characters 0, 1,2, 3, 4, 5, 6 to refer respectively to Mercury, Venus, the Earth, Mars, Jupiter, Saturn, and Uranus. The preceding masses of the planets [AOG], AOGl d], hove been multiplied by 1 + f^j 1 + f^', 1 + /') &.C. respectively, in order that these results may be immediately corrected, for any change in the values of the masses, tohich may hereafter be found ne- cessary. (0,1) = (1 +,a').3",052453 (0,2) = (1 +|/').0%963818 (0,3) = (1 + /") . 0',040631 (0,4) =- (1 +(^'0.1',575473 (0,5) = (1 + f' ) • 0'>080560 (0,6) = (1 +,^'')-0',001702 mi = (1 +f^').l'',961407 Ul] = (1 +(^").0%457195 [m] = (1 +/-"')-0',012797 [ÎZ] = +(^'0•0^ 146329 UKl = (1 +H-O-0~',004086 ["îZJ "= (1 +(^'')-0''000042. [42291 [42,30] [4230'] [4231] Mercury. (1,0) = (1 +,a ).0^422318 (1.2) = (1 +/').7\416280 (1.3) = (1 +|j."').0',148161 (1.4) = (1 +M-'^). 4', 131 166 (1.5) = (1 +f^O-0',207370 (1.6) = (1 + ,a-) . 0%004354 (2.0) = (1 +,., ).0',097574 (2.1) = (1 +a').5',426695 (2,3) = (1 +;^-"').0',432999 rvi = (!+/-■■ .0,271367; i,,.i = (1 +P- / \ 6-, 174974; [-^1 = (1 +(- ///\ .0',085252; iHi == (1+,- v\ 0',7 16427; L^J = (1+'^- V \ ,019641 ; [ÎZ] = (1+1^ \'i\ . 0',000205. [^] ; (1 + 1^' 0',046285 ; [IZ] = (1 +f^ \ 4',5 18397; [H] = (1+f^' tl\ 0',332961 ; [4232] VcTiua. The Ea rill. [4233] is that of Saturn, ?n'- = ^l^ '"•." ^ [40611, the value of n is that of n'" = 109256'29.3 3339,40 ^^^ [4^8c] [4077]; the value of a is 0,54.531725, [4202]; then we have è_à = 2.15168241, b_h = -0,52421272 [4203]. Substituting these in [4228, 4228'] we get the values of (4,5), [i£] as in [4235]. Lastly the formulas [4229] give (5,4) = •(4,5); [m;] = ™'^V^.[T£]; hence we obtain (5,4), \jr\ as in [4236], using the factor [4228(/] 1 +(*'" instead of 1 + /J.^ In like manner the other formulas [4231 — 4237] are to be computed. 216 PERTURBATIONS OF THE PLANETS ; [Méc. Ctl- (2,4) = (i+(^'0- 6',947861 ; [m] = = (l+^-'o- 1 ',662036; TheEaiih. (2,5) = (l + p-n- 0,340441 ; [iï] = = (1 + ^^). 0%044514; (2,6) = (i+H-^O- 0',007095 ; [ISJ = = (1 + ^") . 0',000463. (3,0) = (i+O- OSO 18662; [m] = = (1+f^ ). 0',005878 ; - (3,1) = (1 + ^')- 0',491880; [s^] -. == (M-.'). 0',283029 ; [4234] (3,2) = (1+^-"). r, 964546 ; r^] = = (1+^"). 1 ',510657; MlTB. (3,4) = (i+i^-O- 14',411136; Lm] = (i+^-'O- 5',2 19092; (3,5) = (1+^')- 0%658341 ; [^] = (1+.-). 0,131041 ; (3,6) = (1+^-')- 0%013436; [3,6] = (ï+f"). 0,001333. (4,0) = (l+f^ )• 0%000226 lii] = (i+M- ). 0',000021 ; (4,1) = (1 + M-'). 0%004291 [ÏZ] = (i+,v). 0',000744 ; [4235] (4,2) = (1+f^"). O',009862 [4,.] = (i+fx"). 0',002359 ; Jupiter. (4,3) = (i+n- 0,004509 ; [i^] = (i+O- 0',001633; (4,5) = (1+pO- 7%701937 w^ = (1+,-^. 5%034195; (4,6) = (i + i^^O- 0',096647, [ja] = (1+H.^'O- 0',032446. (5,0) = (i + (^ )• 0',000027 ; [m] = (i+(0- 0^000001 ; (5,1) = (i+f^'). 0',000501 ; [ ^'M = (1+.'). 0',000047 ; [4236] (5,2) = (1 + 1^."). 0,001123 ; [^.^J ^ (1 +..")• 0^000147; (5,3) = - (i+O- 0',000479 ; [m] == (i+O- 0',000095 ; (5,4) = = (l+f^-) 17%90ô446 ; Ua\ = (l+(^'0- 11%703495; (5,6) = = +!'■') • 0,355214 ■ [.6] = (l+r'.^0- 0',2 13356. (6,0) = - (i+O 0',000002 ; [M] = (]+..)• 0,000000 , (6,1) = = (1 + p-') . O',000043 ; [^] = (1+.-') 0',000002 ; 14237] (6,2) = = (1+f^") . 0',000096 ; [^] =^ (1 +(■.") O',000006 ; L '■'"-" J 1 1 rciti us. (6,3) = = (i+O 0\000040 ; [Mj = (i+O O',000004 ; (6,4) = = (l+(^") 0%919814 U±\ = (l+(^-0 O',308803 ; (6,5) = = (1+^0 1 ',454 176 ; un = (1+f^') 0^873434. [4237 25. By means of these values and the formulas [1122, 1126, 1142, 1143, 1146] the following results have been obtained; ivhich exhibit, at the epoch of 1750, the annual variations of the elements^ during a year of 3651 days, namely, VI. vii.§25] SECULAR VARIATIONS OF THE ELEMENTS. 217 dl' •2de the annual sidéral motion of the perihelion in longitude in 1 750 ;* [42381 [4238'] = the annual variation of the equation of the centre, or that of double the excentricity in 1750 ;t -— = the annual variation of the inclination of the orbit to the fixed ecliptic r^omn d t ^ [4239] of 1750 Symbol! . — -!^ the annual variation of the inclination of the orbit to the apparent ,.^„^, d t ^^ [4239] ecliptic ; d è -— = the annual sidéral motion of the ascending node of the orbit upon the d t fixed ecliptic of 1750 ; de — ' := the annual sidéral motion of the same node upon the apparent d t ecliptic. Î [4240] [4241] * (2567) Neglecting terms of the order i^, we get u=^U-\-t.— — , by Taylor's [4238a] theorem [.38-50a]. The time t is counted in Julian years [4078] and the values of n, n', n' kx,. [4077] are taken to conform to this unit of time, so that n"i, which represents generally the motion of the earth in the time t, will become simply n", in one year, or when t=\. Now U being the value of m when < = 0, if we subtract it from the value for dU [42386] the case of ^=1, which by [42.38a] is U -\- —, we shall get the annual variation of It equal to — . Therefore if we write successively «, 2e, ip, 9,, è, 6^, for u, we shall obtain the annual variations of these quantities respectively, namely, -— , °' [4238c] '^-T,' Ti' Ti' 77' Tr ^°^ '" [4080 — 4083] « represents the longitude of the perihelion, e the excentricity of the orbit, 9 the inclination of the orbit, and è the longitude moqqji of the ascending node of m, upon the ^retZ ecliptic. Moreover, 9, is, as in [1143'"], the inclination, and ê^ the longitude of the node counted upon the apparent ecliptic. With one ^ .poo accent above these quantities, they correspond to the body m'; and with iivo accents to the body ni' , &«;. t (2568) Neglecting terms of the order e^, in the equation of the centre [3748], it becomes 2 c . sin. {nt-\- t — ra) ; the maximum value being 2 e, whose annual variation is [4239a] ^.~ [4238c]. X (2569) The formulas used for computing the values [4242 — 4248] are as follows. [4242a] VOL. in. 65 218 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. MERCURY. ^ = 5',627032 + 3',014032 . (^'+ 0',929932 . (x"+ 0%041846 . f^'" at + P,560043 . f^''+ 0',079478 . i>y+ 0S001702. f."'. 2 .j^ = 0',013690 + 0', 021948 . |x'+ 0',00651 1 . m-"— 0', 002330 . t^'" — 0',012560 . i>>'+ 0',000116 . f^^+ ,000004 . t^^ Mercury. ^ ^ ~ 0',1 19993 — 0',087951 . f^'— 0%000052 . ix"'— 0^028764 . f^^" — ,003215 . f^"— 0',00001 1 . /j-". [4242] jf^-^ 0'' ^ ^'^^^^ + 0%068409 . H-'+ 0^000508 . f^"'+ 0^098085 . fx'^ + 0%0 10373 . i>y+ 0%000033 . f^"'. — = — 4",224994— P,764590. m-'— 0',963817 . p-"— 0',029951 .(x'" dt — P,396112 . M--— 0%068989 . (x^— 0^001535 . fx". — ' = — 7%566802 — 0',097574 . f^ — 4',054426 . t^'— 0^963817 . (>■" d t — 0',143774 . fx'"— 2%1 87093 . i^^''— 0%1 17889 . m-" — 0%002228 . fi.^'. The values of — , —- , &ic. are given in [1126] ; 2.—, 2.—-, &c. are derived from d t (t t (t I u t [««, f"'-'^ '-?■ '^■■'^■'•"' 'iï- s.*"-'™" [.142, .143]. L..,y ^S f.8.. General ^nd -r-^ , — ^, &ic. are obtained from [1 146]. If we put i for the number of accents o«er exprès- " ' " ' the'annuai <p; «) &•€. SO that (p*'', -sj^'^ Sic. represent the values of (p, «, 8ic. corresponding to the planet of the eie- vvhicli is numbered i, according to the notation adopted in [4230] ; and suppose the sign 2 ments of,, 7.7 • 1 • i • /^ the orbits, of finite integrals to include all the values of k, contained m the series of numbers, 0, 1,2, 3, [4242c] 4, 5, 6 [4230], excepting i =^ ]c ; then the four first of the preceding equations, may be put under the following forms, as is evident by mere inspection, [4242d] ^^=i;.J(i,Ar) — [Tr].^\cos.(:=<'>— «W)5; [1126] [4242e] 2.^ = — 22.[TF].e^'i.sin.(ra«— b"'1); [1122] _ _2%343127 — 4%315] 77 . (^ — 5',754638 . ^."+ 1 ',203777 . t^" (It VI. vii. §25.] SECULAR VARIATIONS OF THE ELEMENTS. 219 VENUS. + 6',435827 . (j."+ 0^083814. (x'+0',003269 . ,j.''. 2 .^ = — 0%260567 — 0',090479 . fx — 0%101170. fx"— 0',006378 . >'■"' a t — 0%061 143 . <— O',001409 . f'+ 0',000012 . m-". ^ = — 0%015950 + 0S025200 . ^ + 0%002I57 . f^'"— 0%037854 . (x- ^^^^^ — 0',005455 . fx'' -I- 0%000002 . /x"'. ^' = 0%044538 + 0',019377 . f^ — 0',004148 . fx"'+ 0%025810 . t^- [4243] + 0',003500 . fx'— 0^000001 . t^-. — = _ 9,900996 + 0%342053 . fx _ 7',416280 . f."— 0%0761 12 . m-'" — 2',66 1705 . fx-— o%085589 . K— 0',003363 . ,.''. ^ = — 18%387762 + 0', 165450 . fx — 5^426693 . m-'_ 7^416280 . v^' — 0',286675 . fx'"— 5',133067 . jx"— 0',285519 . v" — 0',004978 . fx". ^ = 2.[ a_] .tang. ^(«. sin. (ôO-ô^'i); ^242/] [1142,1143] In like manner the expressions [1146] may be reduced to the forms [4242i, fc], supposing the orbits of all the other planets to be referred to that which is numbered I [4230] ; ?/'' [4242ft] bebg the indination, and â|'' the longitude of the node of the orbit denoted by i referred to that which is denoted by I; conformably to the notation [1 143''] ; the fixed plane being the orbit of /, at the epoch 1750, -^ = 2.{ (i,Ar)_(/,A') \ . tang. <p'*'. sin. (âio_^(B); [4242i] -^ = -{l,i)-Mi,^) + ^A{i,^-{l,^\-~^yCO,.{è'~^-è^''^). [42424] 220 PERTURBATIONS OF THE PLANETS; [Méc. Céi. THE EARTH. -- = 11* ,949588 — 0',414923 . f^ + 3',813276 . (^'+ r,546163 . ^^"' at The Earth 4- 6^804392 . i>^"+ 0', 194066 . 1^."+ 0',006614 M-' [4244] 2 /-^ = — 0',1 87638 — 0',008057 . (^ + 0',030435 . i^'— 0',049410 . f/-" — 0', 159738 . i>'"— 0S000909 . f^''^- 0',000040 . f.^'. Instead of excepting Ic = i [4242c] , we may suppose the sign 2 to include all the numbers [4242Z] 0, 1,2, 3,4,5,6 [4230]; putting {i,i)=iO, [TTJ = 0, in all the formulas [4242«i — A:] ; observing also that the first term of [4242Ar], namely — (^j ^) is that which arises from the tano^. &^''^ [4242m] value A; = i, under the sign 2 ; because then ° — — = 1; cos.(é''' — Ô(''>)=:1. We may moreover remark, that as the orbit of the planet /, in 1750, is taken for the fixed plane [4232/t], tang. <p"' must be of the order m, and since this is multiplied, in [4242/], by quanti- [4242n] ties of the same order, the product will be of the order m^, which is neglected ; likewise the term depending on tang. 9''' vanishes, because it is multiplied by sin. {&'■''' — â*'') = 0. If we now substitute in [4242f/— t] the values [40S0— 4083, 4231—4237], we shall [4242o] obtain the expressions [4242 — 4248] For the sake of illustration, we shall give a few examples of the numerical calculations in the following notes. * (2570) As an example of the formula [4242(/J, we shall compute the action of Mercury on the Earth, in which case i^ 2, A: = 0, and the corresponding terms of this formula [4244a] are (2,0) — [Ml • -• cos. (n"— w). Substituting the values of (2,0), [aiô"], e, e", to, ra" [4233, 4080, 4081], it becomes. [42446] (1 + p.) • ^^ 0-,097574 - 0',046285 . °^^lf^l- cos. (98^ 37" 1 6-73" 33'» 58') ^ = (1 + fJ^) . { 0',097574 — 0',512497|= — 0^414923 — 0^414923 . ij. ; in which the part depending on fx is the same as in — — [4244], the other part — 0',414923 is included in the constant term 1 1',949588, which is the sum of all the coefficients of /a, ja'd, 14244c] .... dzi" &ic. noticing their signs. This constant quantity represents the value of -r-r-, supposing (a, /J.', &.C. to vanish, or the numerical values of the masses [4061] to be correct. VI.vii.<^25.] SECULAR VARIATIONS OF THE ELEMENTS. 221 MARS. ^" ^ 15',677160 + 0',015944 . \>. + 0^511046 . f.'+ 2%129320 . ^' a t + 12%312891 . (/-"+ 0^693878 . f^^+ 0',014082 . ^''K 2. ^"= 0%372537 + 0^002363 . (x + 0',001566 . ,x'-f 0',040492 . /' + 0',314982.,j."+ 0',013167 . p."— 0^000032 . f^". 1^ = — 0',293800 + 0^,000092 . ^ — 0',013146 . f^'— 0^254879 . m-'" — 0%025790 . vy— 0',000076 . ^^K *'"" 1^ = — 0%012984 — 0',000388 . (/. + 0',131893 . f^'— 0S131999 . f." dt — 0',0 12454 . V- — 0',000036 . p". [4245] — = — 9%728234 + 0%052224 . \^ + 0',3 14067 . (.'— P,964546 . ^' a t — 7%855103 . ^i'— 0',266532 . f^"— 0%008345 . ,x^'. ^ = — 22%789674 — 0',31 8395 . fx — 8%577599 . f.'— 1 ',964546 . fx" — 0^,432999 . fx'"— 1 P,015955 . i>'r— 0',469146 . fx" — 0',011033.fx-. de" In like manner the terms of 2 . — [4242cj, depending on Mercury, become by using [4244ti] tiie same values as above, — (1 + fx) • [111] -26. sin. (ra"— -a) = — (I + (x) • 0,046285 X 2 X 0,20551320 . sin. (98'' 3T" 16'- 73'' aS" 58*) [4244e] = — ( 1 + ^) . 0',00805T = — OS008057 — 0',008057 . fx, in which the coefficient of (x is the same as in [4244], and the quantity — 0',00S057 forms part of the constant quantity — 0", 187638 [4244], as in the dTS" r T T •! rfwCO case of — — [4244c]. In like manner we may compute any other values —rrt d_f> dt ' VOL. III. 56 222 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. JUPITER. '^— = 6%599770 + 0',000186 . f^^ + 0',004330 . (^'+ 0',009837 . m" dt Jupiter. + 0^002047 . f^"'+ 6%457871 . (j.''+ 0% 125498 . ix"': 0%55441 8 — 0',000008 . m- + 0',000009 . f^'+ 0',00( — 0',000191 . /x"'+ 0',553308 . f>-''+ 0',001220 . i^", * ^ = _0, 078140 + 0',000022.(x+0',0001 01 . |x'+0^000112.f^"' at — 0',078933 . M-' + O',000557 . f^". ^ = — 0^223178 — 0%009491 . m. — 0%128114 . (^'— 0',010646 . (x'" [4346] _ 0^075444 . (^^"4- 0%0005 1 6 . f^^'. — = 6^456281 + 0',000509 . (^ + 0%005857 . i^'— 0%009862 .(." dt — 0%000461 . f^"'+ 6%505571 . f^"— 0%045332 . j."'. ^^ _ 14%663377 — 0',316227 . t>- — 12S828736 . i^'— 0^009862 . t^" dt — œ,389153 . (^"'— 6%947861 . f^'^+ 5%877561 . t^" — 0,049 100 . H-^'. * (2571) As an example of the use of the formula [4242/], we shall compute the [4946a] part of — — depending on the action of Mars. In this case i ^ 4, A: = 3, and the corresponding terms of the formula become, by using the values [4080 — 4083. 4231 — 4237] ; (4,3) . tang. (?'". sin. {&"— é'") [42466] = (1 + 1^'") ■ 0',004509 X tang. I'' 51™ X sin. (97'' 54™ 22'— 47'' SS" 38') = (1 +|i,"').0',000112 =0',000112 + 0%000112.fA"' of which the part depending on ;*'" is the same as in -— [4246], and the other term 0' ,000112 forms part of the constant quantity — 0',078140 of this formula. [4246c] In like manner by putting i = 4, A = 3, Z = 2 in [4242 J], and using the same data, VI. vii. ^S25.] SECULAR VARIATIONS OF THE ELEMENTS. 223 SATURN + 0%000550.f^"'+ 15',790810.(x-+0%3]9768. f.' , = 16%1 12726 + 0',000022 . t^ + 0',000496 . fx'+ 0S001080 . i^" (It 2 . ^ = — I ',080409 — OSOOOOOO . f. + O',000000 . f.'+ 0^000001 . /-•" a t — 0%000016 . (^"'— P,099919 . fj^"+ 0%019524 . (x-. 1^ = 0%099740 + O',000003 . fx + 0^00001 8 . m-'+ 0%000014 . i>!" ^^^^^^^ + 0^,096696 . H-" -f 0^003010 . fx". ^ = — 0-,155290 — 0^,010955 . fx_ o-,1939] 8 . ^— 0%012542 . f^'" [4247] + 0%059175 . fx-+ 0', 002950 . m-". * 'jj =— 9%005292 + 0%000004 . ,x + 0',000042 . ^^— 0%001 123 . ^' — 0',000323 . (x'"— 8%734249 . f^-— 0%269642 . iCK ^ = — 19^041499 — 0',1 10961 . fx — 5',883249.f^'— 0',001123 . ^." at — 0', 141 41 4 . p.'"— 12',292960 . t^"— 0',340441 . (x" — 0',271351 .fx". we get the part of —7^, or as it is called -7^- [4246], depending on Mars, equal to \ (4,3) - (2,3) I . tang. 9'" . sin. (é-— é'") = (1 + fx"').|0' ,004.509 — 0',432999| X tang. 1'' 51"' X sin. (9T'54"'22'— 47"' 33™ 38') [4246rf] = _ (1 _f- ^"') . 0',010643 = — 0',010643 - 0',010643 . ix'", which agree very nearly with the corresponding terms of -77 [4246]. * (2572) Putting i = 5 in [4242g-], we get the expression of —, and the terms corresponding to the action of any one of the planets, is found by using the value of k corresponding to it ; thus for Mars k = 3, and the terms depending on this planet become, by using the data [4080— 4083, 4231 — 4237], 224 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. URANUS. ^' = 2',454851 + 0',000003 . (^ + 0',000043 . ^'+ 0',000095 . fx" at + 0',000048 . ^^"'+ r,2 10830 . i>>''+ 1 ',243833 . i^\ 2 . — = — 0', 1081 84 — 0',000000 . (^ — O',000000 . f^'— 0',000000 . /x" dt + 0',000000 . fj.'"— OSOl 1952 . yi"— 0',096232 . i>y. ^ = _ 0',048861 + 0',000000 . i^ + 0',000000 . f/.'+ 0%000000 . f^'" """"'• — 0',009036.(x-— 0^039826.,x^ 4^'- = — 0',027460 — 0',005492.pi + ^,010145./— 0',005907 . f^^'" a t [4248] + 0',05921 7 . 9^" — 0%030502 . i>.\ — = 2^,700876 +0',00001 7. f^+0S000146.fx'—0',000096.(^" dt + 0',000047 . M-"'+ 0',496382 . i>:"-\- 2^204381 . i>.\ l!il' = _ 34\403396 — 0',78851 7 . \>- — 23^81 5885 . i>!— 0^000096 . f^" dt — 0',938767 .M-'"— 10',200902 . ,x-+ 1 ',347 866. j^" — 0',007096.fi.". ■(5'3)+(5'3)-tïï!|^---(^'-^"') = (I + fx."'). 5 — 0-,000479 + 0'.000479 /^"^' ^,„^^'"° '.cos.(lll''30"'23'-47''3S'"38^) ] \ ^ '^ ' I ' ' - tang. 2'^ 29™ 55' ^ ' ) [4247i] = (1 + fJ^'") .{ — 0',000479 + 0^000156 } = _0',000323 — 0^000323.fA"', as in ^[4247]. d é^ • Putting 1 = 5, Z = 2, in [4242^], we obtain the expression of — j-, in the notation [4247c] °f [4247]. The term of this expression corresponding to Mars, is found by putting k=3, and using tlie above data, by which means it becomes, VI. vil. § ^'5.] SECULAR VARIATIONS OF THE ELEMENTS. 226 The variations of the earth's orbit are not included in the preceding formulas ; they may be determined by the equations * tang. <?". sin. f = p" ; tang. /. cos. o" = q". [4349] With resjiect to the values of p", q", we may determine them by the formulas [1132, &:c.], and we have, by taking the ecliptic of \1 50 for the [4249] fixed plane,i d p" in which t is the number of Julian years elapsed since 1750, and -rr^ [4250] r/q" rldp' , , &.C. are taken to correspond to that epoch. It is only necessary to notice the first power of t in these formulas, if t be less than 300. If t do not exceed 1000 or 1200, we may reject the third and higher powers of t ; and we may do the same even with the most ancient observations, [4250'] [4250"] - (5,3)+ <(5,3)-(2,3)}.^^'.cos.(év_r) = (!-]_ |j,"'). ) _ 0^000479 + (0',000179 — 0',432999). ^'.^^ ^ ■. cos. 63-' 51" 45'^ [4247rfJ \ ^ '^ ' I IV. I / tang. S"* 29"' 55» 3 = (1 +fj-"').{— 0',000179 — 0',141035| = — 0,141514 — 0%141514.(a"', which differs 0',0001 from that given by the author. We have thus given an example of the numerical calculations of each of the formulas [4212(/ — k'\. * (2573) The formulas [4249] are similar to [1032], accenting p, q, Sic. with tioo r4249„] accents, in order to conform to the case now under consideration. t (2574) Putting successively m =p"; U = p" ; or u = 5", TJ = q", in the formula [3850«], we get the following expressions of p" , q", in which the quantities p", q", and their differentials, in the second members, correspond to the epoch of 1750. Now at that epoch we have 9" = [4249'] ; substituting this in [42506] [4249], we get p"= 0, q"= ; hence the formulas [4250a] become as in [4250]. VOL. III. 57 226 PERTURBATIONS OF THE PLANETS; [Méc. Cél. taking into view their imperfections. We obtain from the formulas [4250], Value. the following results.* CO r re 3- po.ndiag rrrlil '^ = 0%076721 + 0',008420 . m- + 0%0863 16.^'+ 0%009423 . ,a"' odiit. a t [4251] — 0»,022021 .M.'"— 0^005446 . i^--+ 0s000029.|x^'. ^ = — 0%500955 — 0%008522./x — 0',309951 .,/- 0%010335.f/^" — 0',1 58234. /x'"— 0',013821 .F-^— 0%000091 .f^^'. theperi" 26. Wc havB seen, in [4037], that the oblateness of the sun produces, in helion de- the"t'n,p°" the perihelia of the planetary orbits, a small motion, which is represented by, cily of the sun. y-j2 [4252] 5«=.(p_X^).-:.„^. * (2575) If we substitute tlie values p", q" [4250J, in the terms of -j-, — d p" [1132], depending upon p", or ç", they produce terms of the order {(2,0) +(2,l)+&ic. }• — ; [4251a] . , , . . ., dp" dq" ,. , • , c u <• u or 01 the order m m comparison with — , — , which occur in the first members ol these ^ dt' dt' equations ; therefore these terms may be neglected, and then the values of — — , — • [1132], become, Ji" = (2,0) . q + (2,1). 2' + (2,3) . q"'+ &c. ; [42516] . „ ^ ^ _ (2,0) . p - (2,1) ./- (2,3) .p"'~ he. [4251c] Substituting p = tang. cp. sin. é, p' =tang. 9'. sin.ô', &,c.; q ^tang. q> .cos.^, &;c. we get [4251rf] ''■JT ^ (2.0) • tang. <p . cos. ^ + (2,1 ) . tang. 9'. cos. 6' + (2,3) . tang. 9'". cos. ()'"+ Sic. ; [4251e] rfl" ="~ (^'^^ • tang.9-sin.â — (2,1) . tang.9'. sin. â'— (2,3) .tang. 9'". sin.ô'"— he ; and by using the values [4082, 4083, 4233], they become as in [4251] nearly. Thus the (l p" term of —- , depending on Mars, is [4251/] (2,3). tang. 9'". cos. r= (1 + |x"') .0',4.32999 X tang. l-'Sl™ X cos.47''38'" 38' = (l+H."')-0''009423, Vl.vii.§26.J SECULAR VARIATIONS OF THE ELEMENTS. 227 We shall consider the motion relatively to Mercury. Now q is the ratio of [43531 the centrifugal force to gravity at the solar equator [4028] ; and if mt be the sun's angular rotary motion, the centrifugal force at the solar equator will ^oss'i bo ni'D* Puttine; the mass of the sun enual to S, we havef -^,.l=^»"^ or ,,„,„ * 1 a^ [4254] .S = ti"-. a"', which gives the gravity at the solar equator, S n"2.a"3 2)3 DP ' therefore we have % m' D^ /ot\2 /D\3 [4255] The time of the sun's revolution about its axis, according to observations, is nearly equal to 25'*°y%417. The duration of the earth's sidéral revolution is [4257] 365'""^S256 ; hence we obtain, TO 365,256 n" 25,417 The apparent semidiameter of the sun, at its mean distance, is 96P,632; which gives [4258] [4259] dp" in which the coefficient of iu-"' is the same as in the value of — - [4251]. In like manner at dp" dq" we find the Other terms of ——, — — [42511. dt dt ^ ■■ * (2576) The angular rotary velocity being to, and the equatorial radius D ; the actual velocity of a point of the surface of the equator will be represented by to D. The square [4253o] of this, divided by the radius D, gives the centrifugal force [54'], equal to m^D, as in [4253]. t (2577) We have n^ = ^3 = "^^^ [3700, .3709a] ; and in like manner n"'^=^^ . ^^254a] Now changing M into S to conform to the notation [4254], neglecting also to" in comparison Ç /t"3 with S, we obtain -^ = n"^ [4254]; multiplying by — we get [4255]. [42546] jt"2.a"3 . t (2578) The centrifugal force ni'D [4253'], divided by the gravity -p— , gives q [4253], as in [4256] ; substituting the values [4258, 4260] it becomes r4255o] q = (^54^^)^. (sin. 961',632)=' = 0,000020926, as in [4261]. 228 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. [4260] ^ = sin. 961%632; a therefore we have [4261] y == 0,0000209268. If the sun be homogeneous, we have p = l;q [1590', 1592'], m which case the motion of Mercury's perihelion [4252], produced by the ellipticity of the sun, is* or the equivalent expression, [4264] <5 « = ^7 . (sin. 961 ',632)-. T- Y. nt. [4262] If we substitute in this formula the values of n, «, a" [4077, 4079], it [4265] becomes f>-r^ = ,012250.^ ; so that it increases '-^ [4242] by the quantity 0',012250, which is nearly insensible. This must be still farther decreased if the sun be formed of strata whose densities increase from the surface to the centre, as there is reason to believe is the case.f Hence we may neglect this [4266] expression for Mercury, and much more so for the other planets. The variations of the nodes and inclinations of the orbits, depending on the same cause, may also be rejected on account of their smallness [4045'] * (2579) The density of the sun being supposed uniform, we have 'h? =: ^q nearly [4262a] [1590']. Moreover by [1592'] the polar semiaxis being 1, the equatorial semiaxis is \/{\ -f- 'i^) = 1 + J^^ =^ 1 4~ i*/ nearly ; so that the ellipticity p is nearly equal to fç, as in [4262] ; substituting this in [4252J we get [4263]. Now we have [42621] V^-7,- 5 = ( -"• 961',632)^ g)' [4260] ; hence [4263J becomes as in [4264] ; and by using the values of q, a, a", n [4261, 4079, 4077], it becomes as in [4265], namely, [4262c] 5^^ I X (0,0000209268) X (sin. 961',632f X (0,.38709812)-2 X 538101 6^ < = 0',01 250. <. t (2580) The effect of increasing the density towards the centre is seen, in the extreme r4266a] case, when the whole mass is collected in the centre, and p = io-tp [1732'"]; or in the present notation f^hq [1726', 4253]. Substituting this in [4252], we get ira=Oj so that in this case the ellipticity has no effect on the motion of the perihelion ; hence it [42665] appears that this increase of density, towards the centre, decreases the motion of the perihelion. We have supposed, in this example, that I) remains unaltered, the density being considered as infinitely rare, from the suiface towards the centre. VI. viii. V-î"?-] THEORY OF MERCURY. 229 CHAPTER VIII. THEORY OF MERCURY. 27. The inequalities of the planets which are independent of the excentricities, and those which depend on the first power of the excentricities, were computed by means of the formulas [1020, 1021, 1030], having previously ascertained the values of ^"'*, ^^'' &c. and their differences, by the formulas [963'^ — 1008]. The results of these calculations are contained in this, and in the following chapters, neglecting the perturbations of the radius vector, whose effect on the geocentric longitude of the planet is less than one centesimal second. To determine * [4267] TertiiB whieh may be neglected on account of their ainallnc". • (2581) Let S be the sun, E the earth, M Mercury, supposing it to move in the plane of the ecliptic ; S T the line drawn from the sun towards the first point of Aries in the heavens, being the hne from which the longitude v, v" are counted. Then S E =t" F 74 (4Q(Jf''al Hence the longitude of the sun, as it appears from the earth, is 180''-[-f"; and if from this we subtract the angle of elongation SEM =^ E, we shall obtain the geocentric longitude of Mercury V=lSO''+i'"— £. Now if SM=r be increased by the quantity MJ\1! = or, the angle E will increase by the quantity MEM'z^SE,'^' while V, v" remain unaltered ; therefore the variation of the preceding value of V will be 5V^ — f5£. If we draw .M'.Y, EF, perpendicular to EM, .S./V/ respectively, w-e shall have in the -similar triangles J\1JYM', MFE; ME : EF :: MM : M'N; [4268rf] EF ME' (5 E = — 5 V : [42(3S6] [4268c] hence iV/'JV=dr. angle MEM' Dividing this by M' E, or ME, we obtain very nearly the EF or substituting EF = S E.sm.ESM ME'i' =r".sm.{v—v"), and ME^=r"^—2r"r .cos.{v—v")-}-t^=r''^.\l—2<x.cos.{v — v")■j-oJ [6■2 Int. 4268], we get [1269]. VOL. III. 68 [4'i(>er] 230 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. the limit, which an inequality in the radius vector must attain, to produce one second in the geocentric longitude of Mercury, we shall observe that if [4a68] we put this longitude equal to V, and r = ?"a-, we shall have for the variation ôV corresponding to Sr, ,. Sr sin (v — v") [4270] The maximum of the function s'in.(« — v'') 1 — 2a.cos.(j;— i;")-l-a'' corresponds to^ [4271] cos.(z^-t?) =j-p- ;J [4271'] which gives ^ [4270e] for this maximum ; therefore we shall then have,t * (25S2) The maximum of [4270] is found, by taking the differential, supposing v to 270 b^ t'l^ variable quantity; putting it equal to zero, and dividing by d.{v — v"). This differential expression being multiplied by |1 — 2a. cos. (y — ii")-|-a-p becomes, without [4270i] reduction, as in the first of the following expressions, and this is easily reduced to the last form [4270(/] ; [4970c] = cos.(«— v") . \ 1 — 2o..cos.(« — v")-\-a?l — 2a. sin.^. (t) — v") = (1 +o.^).cos.{v — v")—2a..{cos.^.{v—v") + sm.^.{v — v")] [4370dJ = (1 +a2).cos.(i'— î>") — 2 a. From this we easily obtain [4271] ; thence . / 4a2 U_l-a2 [4270e] V {l + c>-2)V l+a2 l-2a.cos.(.-.") +a2= 1 - .^-p^, + a=== \-:^^ . Dividing the first of these expressions by the second, we get the value of the maximum of the function [4270], as in [4271']. t (2583) Substituting in [4269] the value of the function [4270], at its maximum t4271a] 6r 1 [4271'], we find &V = ; . :; :,; hence we get or [4272]. n. viii.^27.] THEORY OF MERCURY. 231 <S,- __,•".(! —a'^).. 5 V. [4272] If we suppose iV = ± 1"= ± 0,324, and take for r, r", the mean distances of Mercury and the earth from the sun [4079], we shall have by ^^^^^^ \vliat precedes r" = 1 ; a = 0,38709812 [4095] ; hence we obtain* 6r=^ 0,000001335 ; [4274] therefore ive may neglect all the inequalities of the radius vector of Mercury, in which the coefjicient is less than, rt 0,000001. Among the inequalities of [4275] the motion in longitude, we shall retain generally only those whose coefficients i„equaii- e\ceed a quarter of a centesimal second [0,081]; but as the inequalities ^^^^J^» depending on the simple angular distances of the planets can be introduced ^«JTbe into the same table with those of greater magnitude, they are retained. Inequalities of Mercury, independent of the excentriciiies. iv = (1+0 + (l+0- 0%662353 . sin. (n't — nt + e— s) \ — r,457111 . %\n.2{n't — nt + ^' — ^) j — 0', 128075 . s\n.3(n't — n < + s' — ' — 0',029264 . sin.4(n'« — nt + s—^) — 0%008905 . sin.5(n'« — nt + i —() 0',201 688 . sin. (n"t — nt + ^"- s) — 0', 165645 . sm.2(n"t — n i + «" — — 0',016901 . sin.3(n"« - nt + s"- — O',003127 . sm A(n"t - ni + s"- [4276] Inequali- ties inde- pendent of the ec- centrici- ties. 0',569336 . sin. {n'H — nt + i"- _|- (1 4- ^iv) . I _ 0-,l 1 8384 . sin.2(n'H — nt + ^'— — O',003 118. sin.3(M''i — nt + s'^— * (2584) Using the mean values r = a, r" = a" [4079], we get a [4095], substituting these and 6V = ± 1", or o V = dz sin. 1" = ± 0,324 . sin. 1% we obtEÙn [4274] [4274o] 232 PERTURBATIONS OF THE PLANETS ; [Méc. C^l. / 0,0000000376* \ \ — 0,0000004094 . cos. (n't-nt + i'-s) 1 [4277] i r = — (1 + /) . + 0,0000015545 . cos.2(n't — n i + e' — i) + 0,0000001702 . cos.3(n't — 7i i + s' — s) + 0,0000000437 . cos.4(n' t — 7it + i—s) * (2585) The parts of or, 5v [1023,1024] independent of the excentricities are, by using T [.3702a], [4277a] ^. = _^ . a3.(^_ j+ _ . V. I -^^ ,,^,,^_^ ^■cos.^ i'; [42774] 5 u = — . 2 . ) :- . a ^" -\ r-. ,. , ' -5 -r, ( • sin. i 1 ; [4277rf] [4277c] in which m, «, w, s correspond to the disturbed planet, and m', a n', s', to the disturbing planet. These expressions must be accented so as to conform to the notation [4061, 4077 — 4083], taking for i all integral numbers from i= — œ to z'^co. For example, if we wish to calculate the action of Mars on the earth, we must, in the formulas [4277«, &], change m, a, n, s into vi", a", n", s", &c. corresponding to the [4277 e] disturbed planet; and m', a', n', s'. &c. into ?;/", «'", n'", b", kc. respectively, for the disturbing planet. As an example of the use of these formulas we shall apply them to the computation of the perturbations of Mercury by the action of Venus. The constant part of 5r deduced r4Q77/'l from the first term of [4277a] is as in the first expression [4277ri]. This is successively reduced, by the substitution of the values •0) fdA^o)^ 1 dbi ^g^^g^^ „ = 0,.38709812 [4079], da ) (i''2 do- (0) [4277g-] -= a = 0,53516076 [4085], -^ = 0,780206 [4088], ?«'== ii^ [4061]: a' d CL oboloU III) m' „ /'dA('>)\ m' a" db^ ùr =^ -—.cr. -~ — 1 = ■ a. G ' \ da J 6 a'2 da. [4277/t] ^ (0, = - - . a a2. — : = — (1 +,a') . 0,0000000.370, as in [4277]. Again by putting successively i=!, i=— 1, .^'i'=^(-" [954"J, in [4277o], and connecting the two terms, we obtain the part of &r depending on cos. T, namely. VI. viii.§27.] THEORY OF MERCURY. 233 Iiicqual'.ties depending on the first power of the excentricities.* 0-,295201.sin.(H.'^ + .-'_^) -4-,030852.sm.(2?i'«— nt + 2-:'— £ — ^) - ]%686n4>.s\n.{3nt~2nt + 3e' — 2s — z^) 6-i, = (1 + ,/) . ^ + 0'/J93989 . sin. (3 n't — 2nt + 3s—2 s — ^') ) [4278] + (»%'293992. sin.(4 n't — 3 n < + 4 .' — 3 £ — ^) — 0%17682;).sin.(2M/;_ nf + 2e— i' — ^) + 0%394 1?,6 . sin. (3 n t—2n't + 3 s —2e' — ^) Ô r = m' n^a.) ^ '^ " ^ , "~" ( . cos. T ; [4277i] [42774] in which we must substitute a.^"'==a^ — a.OA, o-'. ( == a^ — a'^. \ da J do. [997,1000,963''], and use tlie values [4277f] coriesponding to the disturbing and disturbed planets. Tiius in computing tiie action of Venus upon Mercury, we must use the values o, a, m'[4-277o-], ji = 538101 6',736, ?i'-^ 210664 1',520 [4077], ^i, [4087], (1) j^ [4088], and we shall get (5 r = 0,0000004094 . cos. T, as in the second line of [4277]. The terms depending on cos. 2 T, cos. 3 T, cos.4 T, &c. are found from [4277n], by using successively, i = =p2, i==\^3. i = ^4, &c. [4277to1 In like manner, the part of '5 v [42776], depending on sin. T, is found by using i = ^l ; hence we have < \ da / ' n — n > ^'^-n^'.l j£^,^, . aA'-r^ -f - t^^^,;;]:^,; _ J }.~;^ ^ • sin. T. ^^^..^^ Substituting the values of the elements given in [4277^,/], it becomes 0^6623. sin. T, as in the first line of [4276] ; the other terms depending on sin. 2 T, sin. 3 T, he. are found [4277o] in like manner, from [4277/./], by using successively « = zt 2, « = ±3) he. The similar terms, corresponding to the other planets, are com[)uted by means of the same formulas [4277a, 6], altering the accents as in [4277t]. The results of these calculations are given in [4289, 4290 ; 4305,4306 ; 4.373, 4374; 4388, 4389; 4463,4464; 4523,4524]. ^^^'^'^^ * (2586) The terms depending on the first power of the excentricities are those parts of ir, ÔV, [1020. 1021], containing e and e. The calculation of these terms is made as [4278a] in the preceding note ; using for e the excentricity [4C80], corresponding to the disturbed VOL. III. 59 234 PERTURBATIONS OF THE PLANETS; [Mée. Cél- Inequali- ties de- pending on tbetirst power of the excen- tricities. [4279] 0',09541 8 . sin.(n"i + e"— ^) + (1 +f^")-<| — 0',461708.sin.(2n"i_ wi + 2s"_ £_™) + ,244148.8111.(3 n"t — 2nt + 3/'— 2 e — ™) 0',236346. sin.(n'''i + s" _ ra) + (1 + f^'^) . { — (r,572172. s\n.(7i'H + ^'^ — ^'0 - 3 ,278687 . sin. (2 n'H _ n < + 2 1" — e (1+^^). O',084]67 . sm.{n'i + s" —z^") + 0',395493 . sin.(2n"ï — rU +2 5" -.) ') 3r = — (1 +p.').0,0000013482.cos.(3n'i — 2n^+3£' — 26— z;j) — (1 + 1^"). 0,0000029625 . cos.(2 n"i— nt+2 s'"— s — ^). Inequalities depending on the scjuares and products of the excentricities and inclinations of the orbits. [4280] [4281] These inequalities have been calculated by the formulas of [3711 — 3755]. Now twice the motion of Mercury differs but very little from five times that of Venus ;* so that 5(n' — n) + 2n is very nearly equal to — n; we must therefore, as in [3732], notice the inequality depending on 3nt — 5 n't. The angle 37i't — 7it varies quite slowly, therefore it is necessary to notice the inequality depending on it [3733]. Moreover the motion of Mercury is very nearly equal to four times that of the earth, so that 4.(n" — n) + 2w differs but little from — n; therefore, we must, as in [3732], notice the inequality depending on 2nt — 4<n"t. Hence we obtain, [4282a] planet; and for e the value [4080] corresponding to the disturbing planet; these symbols being accented so as to conform to these two bodies. * (2587) Using the values [4076^] we have very nearly 2 n .5n'=z 72° = 23" 3n' — n=:289'^ = ^, and 71 — 4 ?j" = 61° = — ; so that these three quantities are small in comparison with 71, as is observed above. Hence 5 («' — n) -j- 2»t is very nearly r4282tl equal to — «, and must be noticed as in [3732] ; also 3 (ji' — 74) -(- 2 n is very small, and must be noticed as in [3733] ; lastly 4 (?i" — n) -j- 2 n is very nearly equal to — n, and must be noticed as in [3732]. The ;enns ofiî[3745-3745"'jdependingon these angles VI.viii.§27.] THEORY OF MERCURY. 235 ^ l',690443.sin.(3n< — 5»'<+3î— 5e'— 43^18'"32')) i r = _ ( 1 + f^ ). s o-,597664 . sin. (3 n t— n / + 3 /— £ + 4O"36™350 ( M^'Sl V ■' second — ( 1 + f.") . 0',263474 . sin. (2 n / — 4 n'7 + 2 s — 4 s"— 41 M 1 "" 46^ ir = (1 -f ,j.').0,0000016056.cos.(3n^ — 5n'i + 3£ — 5e' — 42^58"'04'). the order. [4282] are found by pulling in the first case ?'^ 5 ; in tlie second i^3, and in the third i = 4. The values of ^W"', iW'", M^'^^, .'V/'^', corresponding to these values of i, are successively [4282c] obtained from [3750, 3755, 3755', 3750'"] ; and they may be reduced to terms of U'\ - — , &:c. by means of the formulas [996 — 1001]. These values are to be substituted separately for Jfcf in the expressions of -^, àv, [3711,3715], and we shall [4282d] obtain very nearly the terras of — , 5 r, having the small divisors 5 n' — 2 n, 3 n' — n, 4 n' — n, which are the only ones necessary to be noticed in this place. Now [4282e] if we use, for a moment, the abridged symbol, T.^iJn't — n t 4- ^ — e)-\-2nt4-2e i^ [4282/] [371 lij-], the resulting terms of — or 5r [3711, &ic.] will be of the form [4282/t]. Developing this by [24], Int. it becomes as in [4282?]; substituting .^jsin. Z?, for the coefficient of sin. 7', also ,/2,cos. -Bj, for the coefficient of cos. T^, it changes into [4282t], and is finally reduced to the form [4282/], by means of [24], Int. [428%] dr = J»f/<".cos.(T— 2;n)+J/;".cos(T— a— ^')+M/2\cos.(r— 2îi') + i>7/3>.cos.(T— 2n) [4232^] = { M}^'. cos. 2 Î3 + AJ^'K cos. (îi + -/) + Jl/'^'. cos. 25/ + M/". cos. 2 n | . cos. T, + { ./U;»'. sin. 2 w + .W ">. sin.(i^ + z,') + iVif \ sin. 2 ^' + M,'^\ sin. 2 n| . sin. T, ^*^^^'^ = ^1.5 cos. S, . COS. T, + sin. 5, . sin. 1] } [42824] = A, . cos.{T-B,), as in [4282]. [4282^] In like manner the several terms of i5 v may be reduced to the form A-3. sin.(T, — B.-,) ; there is no other difficulty than the tediousness of the numerical calculation, arising from its [4282m] length. We may observe that the quantities 7^, 2 IT, which occur in [3745'"], are not explicitly included among the data [4077 — 4083], but must be computed from the formulas [4282n] [10.32, 103.3]. 7 .sin.n =z tang. 9'. sin. â' — tang. 9. sin. é; 7. cos. n = tang. 9'. cos. â'— tang. 9. cos. Ô; [4282o] supposing 9, é to correspond to the disturhed planet, and 9', è' to the cUsiurbing jilanet ; these symbols being accented so as to conform to the notation [4230] ; then using the values [4082, 4083] we get the required values of 7, n. 236 PERTURBATIONS OF THE PLANETS ; [Méc. Ct Inequalities depending on the cubes and jirodncts of th-ee dimensions of the excentricities and inclinations of the orbits. The first of these inequalities, depending on the angle 2nt — 5 n't, is [4282'] computed by means of the formula [3844] ;* the second, depending on the angle nt — 4 /<% is found by means of [3882] ;t hence we obtain, ÔV = —{l +f^')-8',483765.sin.(2ni — 5w'^ + 2s— 5.='+30''13'"36') ineq^.ii. — (1 + O • 0',690612 . sin.( n i — 4 n"t + £ — 4="+ 19^02'" 13'). ties of Ihe etder. The inequalities of Mercury's motion in latitude, may be calculated by- means of the formula [1030] ; but as they are insensible, being less than [4283] ^ quarter of a centesimal second, it was thought unnecessary to insert them. r4283ol * (2538) The first line of [4283] is obtained from the formula [3844], connecting all the terms into one, as in \_4282h — ?]. [42836] t (2589) The second line of [4283] is obtained from [3882], reducing all the terms into one, as in [4282/i — Z]. We have already seen in [3883/(], that the correction, as it is given by the author, in [4283], is rather too great ; his method of computation [3882] being i J merely an approximation. The direct method of computation has already been explained in the previous notes [3876a— 3833io] ; and it is unnecessary to say more upon the subject [4283rf] ji^ jijjg place. There is a similar equation in the earth's motion [4311, 3S83i/]. VI. ix. ^^28.] THEORY OF VENUS. 237 CHAPTER IX. THEORY OF VENUS. 28. If we put - = a, and V equal to the geocentric longitude ^4284] of Venus, we shall find that the equation [4272], 6r = — r". (1 — a=) . (5 V, [4285] will become, relatively to Venus, ^r'=. — r".(l— a'=).6V'. ["286] Taking for r', »", tlie mean distances of Venus and the earth from the sun [4079J, we shall have, as in [4126], a = 0,72333230 ; therefore by [4287] putting 6 V = ± 1"— ± 0',324, we shall obtain, 6 r' = :f 0,0000007489. [4288] Therefore we shall neglect those inequalities of the radius vector whose ''"'"» coefficients are less than 0,0000007. We shall also neglect the inequalities .'"gLfeci on account of the motion in longitude, which are less than a quarter of a centesimal "f">, ^ ^ sinullnes*. second, or 0',081. Inequalities of Venus, independent of the excentricities. '+ 5',015931 . sin. {n"t — n' t + s"— s'Y +11',424392 . sm.2(7i"t — n' t + b"— s') - 7%253867 . sm.S(ti"t — n' t -\- e"— s') — p-,056720 . smA(n"t — n! t + /'— /) iV = {\+ O . ( _ Q, 345898 . sin.5(n"^ - n' t + ."- > ^'''""^ — 0% 145382 . sin.6(w" t - n' t + a"— £') — 0',069726 . sin.7(n"« - n' t + s"— s') — 0%036207 . sln.^n"t — n' t -\- e"- i') ^ VOL. III. 60 238 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. + (1 + n 0%079908 . sin. (n'"t . 0% 105987 . sm.2(n"'t ■ 0^010853.sin.3(n"'/ 0',002332 . sm.4>(n"'t . n't + £'" — £') n't + s"'-s') n't + i"' — [4289] Inequali- ties inde- pendent of the ex- centrici- tias. + (l+f^'0- 2% 891 136 . sin. (n'^t — n' t + >— =') 0%877624 . sin.2(n'^i — n't + s'^— s') 0',040034 . sin.3(?i'^i — n't + s"— .') 0',002754 . sin.4(n'^i —n't + e'"— s') 0% 190473. sin. (n't- + (1 + 'O • <! — 0',039859 . sin.2(«^^ - 0',00 1306. sin. 3(n^r [4290] ir' ^ (1+f.'"). — 0,0000003145 + 0,0000038362 . cos. (n"t - n't + s"- s') I + 0,0000165050 . cos.2(n"^ — n't + .-"— i') — 0,0000140155 . cos.3(n"t — n't + s"— i') \ — 0,0000024255 . cos.4(n" t — n't + s"— i') 1 — 0,0000008873 . cos.ô{n"t - n't + /'- e') i - 0,0000004021 . cos.6(n"^ — n't + s"_ e') I — 0,0000002033 .cos.7(n"^ — n't + ;"_ s') — 0,0000001094 .cos.8(n'7 - n't + s"— =') '—0,0000003106 - 0,0000048903 . cos. (n'^t — n't + s"— i') + (1 + \^") . / — 0,0000021911 .cos.2(n''^ — n't + ^'—s') 1 _ 0,0000001 155 . cos.3(?i'^^ — n't + s'"— e^ _ 0,0000000098 . cosA{nH — n't + b"-— ,') * (2590) The values 5v', &r' [4289,4290], were computed from the formulas "' [4277a, 6], accenting the symbols as in [4277c], so as to conform to the present case. the excen- tricitiei. VI. ix.§28.] THEORY OF VENUS. 239 InequttlUies depending on the first potoer of the excentricities* i r' = (1 + ,a) . 0% 800933 . sin.(2 n't —nt + 2s'—s — ^) 0',073206 . sin. {n"t + ;" — ^') — OM 27720 . sin. (7ft + s" — ^") -I- 0^1631 15 . sin. (2 n"t — n't + 2 s" — s' — ^') — 0', 1 1 3443 . sin. (2 n" ? — n'f + 2 s" — a' — ^,") / Inequalt — 1 ',549550 . sin. (3 n"i — 2 w'< + 3 e" — 2 s' — z^') "'',<'•>■ ^ \ I / I pending + (1 + (..") . / + 4',766332 . sin. (3 n"t — 2 n't + 3 s" — 2 .=' — ^") ) pp'-'r' \ ' ' V ' ' / the excen — 0^299478 . sin. (4>n"t — 3n't + 4^ b" — 3 s' — ^') + 0',947648 . sin. (4 n"t — 3 n'i + 4 e" — 3 =' — t.") — 0',69 1 744 . sin. (5 n" f — 4 n' i + 5 a" — 4 -=' — ^') + 2', 196527 . sin. (5 n"t — 4 n'i + 5 s" — 4 / — ^") / [4991] + 0% 106435 . sin. (3 n' t — 2n"t +3^ — 2 b"— ^') — (1 + P-'") . P,092755 . sin. (3 71'" t —2n't + 3 a'"— 2/— ^"') — P,503893 . sin. {n'H + 3'"—^'^) 0%32n08 . sin. (2 n'^t — n't + 2 b'"— s' — a') ' ^ ' '^ ^ \ ^ 0',232430 . sin. (2 n'-'i — n'^ + 2 a''— / — a-) — 0',163470 . sin. (3 n'^t — 2 n't + 3 b"— 2 b'—^'^) — (1 + ,a') . 0%218743 . sin. (n" t + b^ — z-^) ; 6 r' = (1 + (.) . 0,0000008831 . cos. (2 n'i — n ^ + 2 /— a — ^) r 0,00000 1 6482 . cos. (3 /t" « — 2 n' < + 3 a" — 2 / — îj") y + (1 4-^") .<^_ 0,00000 11406 . cos. (5n"t — An't + 5a" — 4s' — «')> ^^^^^^ (+ 0,0000036421 . cos. (5n!'t — A n't -{-5^' — 4a' — ^"); — ( 1 _^ ijJ" ) . 0,0000019404 . cos. (3 n'" t— 2 n't -^3 a'" — 2 £' — ^"'). * (2591) The terras of &v', Sr' [4291,4292] are computed from the parts o( S v, or [1021, 1020] depending upon the excentricities e, e'; in the same manner as the [4291o] calculation is made for Mercury in [4278a]. 240 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. Inequalities depending on the squares and products of tivo dimensions of the excentricities and inclinations of the orbits. .5 v' = — ( 1 + ,;.) . 0',333596 . sin. (4^n' t — 2nt -{- 4,s'—2 £ —39" 30" 30') ( 1%505036 . sin. (5 ?i"t — 3n't+ôs"—3 s' + 20'' 54" 260 > [4993] — (1 + •/) .{ } - ^ ^ ^ ^ l^ 0',089351 . sin. (4 n"t — 2 n' i + 4 s"— 2 s' + 26" 66'" 32') ) i„c^uai,- + ( 1 + //" ) . 2^009677 . sin. (3 n"'t— n' t + 3 /"— s' + 65" 53'" 09'). tie3 of the ' V ' ' ^ ' V ' ' / secund order. rj,j^^ mean motions of Mercury, Venus, the earth and Mars, bear such proportions to each other that the quantities 2n — 5n', 5 n" — 3n' and [4393] n' — 5 n'" are very small in comparison with n';* hence it follows from the remarks made in [3732, &c.], that the preceding inequalities [4293] are the only ones of the order of the square of the excentricities which can become sensible. Inequalities depending on terms of the third order, relative to the powers and products of the excentricities and inclinations of the orbits. [4294] ,5 Î)' = (1 + M-) . r,184842 . sin. (2nt — 5 n't + 2s — 5e' + 30" 13"' 36'). t Inequali- ties of Ilic , n TT • 1 • 1 "'''•' Inequalities of the motion of Venus m latitude. order. The formulas of ^ 51. Book I. giv^ej n * (259-2) The values [4076A] give, very nearly, 2 » — 5 7i'= 72^ = - ; [4293a] 5n"— 3 7/ = 50^= - : «' — 3 «'"= 12=' = -?^ : all of which are small. The 13 ' 54 ' first of these gives 4?i' — 2n nearly equal to — ti', and corresponds to tlie first form mentioned in [3732]. The second quantity 5 n" — 3 )i', and the third n' — 3 »'", being nearly equal to zero, correspond to the second form [3733]. The [4293fc] terms of àv' [4293] corresponding to these quantities are to be computed from [3715], and reduced as in [4282/i— Z]. The term depending on An" t — 2 n' = 300° == Jn' nearly, is computed for the same reasons as that in [4310']. t (2593) This is obtained from [3817], reducing the several terms to one, as t^^^^^l in [4282A-Z]. [4295a] X [2594) If we change, in [1030], n, a, e, n', a', i', into n', a, s, n", a", i VI.ix428.] THEORY OF VENUS. 241 0%124804.sin.(n"< + £"_0 6s'=—(l+t^"). 4- 0',090932 . sin. (2 n"t — n't +2 s"- + 0',073443 . sin.(3 n"t — 2 n't + 3 s"— 2 e' + 0S081481 . sin. (4 n"t — 3 nV + 4 /'— 3 s + 0',312535 . sin.(5 n"t —A n't + 5 s"— 4 s — 0',078119.sin.(2n'i— n"t + 2i'— s"— è') Ineqoali- tioi in tbe latitude. -0 -0 — '•') \ [4295] [42956] — (1 + ix"') . 0%148701 . sm.(3n"'t—2n't + 3 ê'" — 2 s — n'") + (1 +(x'"').0%161414.sin.(2n'7-n'^+2«'^ — f'-n"). respectively, we shall obtain the value of 5 s' corresponding to Venus disturbed by the earth ; and by neglecting the term containing the arc of a circle n t without the signs of sine and cosine, as is done in [1051] ; also excluding i = [1028, &ic.] from the sign 2, we get. In this formula, y [1026'] represents the inclination, and n the longitude of tlie ascending node of the orbit of the disturbing planet, above that of the disturbed planet. These quantities for the earth's action upon Venus are, nearly y = tang. q>', and n= \ëO''-{-è' • (p' being the inclination of the orbit of Venus to the fixed orbit of the earth ; and è' the longitude of the ascending node of the orbit of Venus upon that of the [4295d] earth [4082,408.3]. For Mars they become /", n'"; for Jupiter y'% U", he. In the expression [42956] we must include all positive and negative integral values of i, [4295c] except 1=0 [1028, &;c.]. The values of y, /, &c. II, n', &ic. are deduced from those of cp, <p', kc. é, è', Sic. [4082, 4083] ; by means of formulas similar to those in [4282o]. Thus if we wish to find the part of 5 s' depending on the angle [4295/"] 2n"t — n't. we must put i=2, in [42956], and the term in question becomes, Now the factor n'^— (2n"— n')2 = 4 n".(n' — n") ; also B'''^ = ~ . b'''.^ [1006]; a •> <j substituting these and y, n [4295c], in [4295^], it becomes, [4295c] -m'.n'Ka'^a" 6f.tang. m' m.{2n"i—n't + 2^'-^—6') 2 4rt".(n'— n").a"3 (.-, [4295;.] VOL. III. 61 242 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. [4295'] n'" being here the longitude of the ascending node of the orbit of Mars upon that of Venus,* and n'" the longitude of the ascending node of the If ill this we substitute m" ^l^' [4061], n"=: 1295977', ?»'= 2106641' [4077], [42951"] (1) „' . , , . 63=8,871894 [4132], — = 0,72333230 [4126], <p' = 3'^ 23"" 35' [4082]; it is J a [4295;t] reduced to — 0',090932.(1 + fx"). sin.(2n"< — n'^ + 2 s"— s' - â'), as in [4295]. In the same way we may compute other terms. If we suppose i= 1, there will be found two corresponding terms in [4295i] ; namely, [4995/] -^J^l., • ^ • tang- ?'• 1 1 — ^«" ^- ^'^^ ■ sin- («" < + s"— «'}• to) But by changing a' into a", in [1006], to conform to this case, we have a" ^ B''°'' ^ b ^i [4295m] j^gjjpg thg preceding expression becomes -^ — ^ • (~) • ^''ng- <p' • ( 1 — 2-^3)- If a' W) use the values of m", n', n", - [4295i"] ; also 6 3^ = 9,992539 [4132]; we get to) 3' 2 we [4295n] (0) 0',031231, for the part independent of b ^ ; and — 0',156035, for the part (0) depending on b ^ ; the sum is — 0^,124804 . sin. (71" < -{"'" — ^') > ^^ '" '''^ ^'^^ line of [4295]. * (2595) A small inequality in the mean motion of Venus, depending on terms of the fifth order of the powers and products of the excentricities, has lately been discovered by rAvjQni Mr. Airy, arising from the action of the earth upon that planet. This inequality affects the mean motion, the radius vector, tiie perihelion, the excentricity, and the latitude ; its period [429661 is nearly 239 years ; being the time required for the argument 8 7it — 13 «"^ to increase from 0' to 360''. This appears from the values of n', n" [4077] ; from which we [429Gf] get 8 m' — 1 3 7i" = 5427' = — — nearly; and as this quantity is very small, it follows that tlie mean motions of Venus and the earth must be affected by inequalities, depending upon the argument 8n'< — 13?i"^; in like manner as the mutual attraction of Jupiter and Saturn produces the great inequalities of these planets in [1 196, 1204] ; supposing the accents on the letters a, n, &ic. to be increased to conform to the present notation, and putting i' = 8, i" = 13. The variations in the excentricities and in the motions of the perihelia, similar to tiiose of Jupiter and Saturn [1298 — 1302], are in the present case nearly insensible. The inequalities of the mean motions of Venus and the earth, ^', ^" depending ' on the argument 8n'i — 13 n"^, are of the order 13 — 8 := 5 [957^'", &c.], or of the fifth order nlative to the powers and products of the excentricities. Now e, e" are [4296/] both quite small, so that the largest of them e" gives e"* = . .r nearly ; but this VI.ix.§28.] THEORY OF VENUS. 243 orbit of Jupiter upon that of Veaus. very minute fraction is multiplied, in [1 1 97] , by .^J,'^'^ =3 X 13 X (239)''= 2200000 [4296^] nearly, in finding the value of ^" ; and by this means the correction is very much increased. The theory and numerical computation of this inequality are given by Mr. Air)',in an elaborate paper on this subject, in the Philosophical Transactions of the Royal Society of London for ^ '«'"*] 1832; using the data [4061—4083]; and putting (x' = — 0,045, (ji"=0, so that [4296i] m'= . He finds the correction 2^ of the mean motion of Venus, to be represented by [4296ft] ^r=: {2',946-r.OS0002970|.sin.{8«^ — 13n'7 4-8s'— 13£"+220''44'»34'-M0%76|. [429«] He also obtains the following equations, depending on the same cause, and similar to those Siven in [1298-1302] ; 5 b' = _ 5',70 . cos.(8 n't—\2n"t + Ss— 13 s") ; [4296i»] W= — 0,000000190 , sin.(8 ?i'< — 13n"i-\- Be' — 13 s") ; [4296n] 5s=0',0151 .sin.(9n'i — 13n"< + 9£' — 13£" + 140''31"'). ^4296^, These corrections of 5 ro', Se, S s, may be generally neglected, as insensible; as also that in the radius vector, similar to [1197]. We shall give, in [4310c—/], the corresponding corrections of the earth's motion. The expressions of 8,', ^" [4296Z, 43 1 Oc] , are subject t^'^^^p] to the noted equation [1208], which in the present case becomes 7n'./a'.^'+mV«"-l"=0. ^**^^ 244 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. CHAPTER X. THEOKY OP THE EARTH'S MOTION. [4296] [4297] [4298] 29. If we suppose the geocentric longitude of Venus to be represented T by V, and -, = <>•; V'* will be a function of a and v' — v". Then we shall have, by [4269], &\'= — à ft . sin. («' — v'') 1— 2 a. COS. («'—«") 4- a2' which gives, as in [4272], where ôV is at its maximum, 5 a. 6V'=r — ■ l-a^» (4297ol * (2596) In strictness it is not the angle V which is to be considered as a function of a and v — v'' exclusively, but the angle of elongation E of Venus, as seen from the earth. This will appear by referring to fig- 74, page 229 ; supposing M to represent the place of Venus ; S M = /, °fSM = v'. For it is evident that the angle of elongation E=^SEM will remain the same, if the angle ESM = v' — v" and the -^— , = — do not vary, whatever changes may be made in the absolute lengths of the lines SM, SE. This inadvertence of the author, in using V for E does not however affect the result of his calculation [4297. &c.] ; because the differentials only of these quantities are used ; and we have, as in [4268c] (5 V' = — 6 E. Now in [4268, 4269] [42976] ratio a = -- [4297c] we have supposed r" to be invai-iable, so that the variation of — = a or — =: ô a ; [4297rf] substituting this in [4269], and accenting the letters r', v', so as to correspond to the planet Venus, we get the expression [4297]. This Is reduced to the form [4298], by the substitution of the maximum value of the coefficient of — Sa [4271'], in the second member of [4297]. VI. X. ^^29.] THEORY OF THE EARTH. 245 a.5r" Supposing r" only to vary in 6 a, we have 5 a. =^ ~ ;* therefore, [42t>y] èr"=r"MlZ^ .6Y'. [4300] a If we put 6V' = ±1"= ± 0',324, and take for r' and r", the mean [43001 distances of Venus and the earth from the sun [4079], we shall get. 6r"= ±0,000001035. [4301] r [4301'] If we put V" for the geocentric longitude of Mars, and — =^ a, we shall have, by [4272],t 6f'=— r"'. (1 — a-) . 6 V". [4^02] If we take for r", ?'", the mean distances of the earth and Mars from the sun, we shall have, a = 0,65630030 [4159] ; r'" = 1 ,52369352 [4079] ; [4303] Terms and if we put 6\"' ^ ± 1" = ±0'.324, we shall obtain, which ^ ' may be i /•" = =F 0,000001363 ; [4304] neglected therefore, we may neglect ilie inequalities of sr", whose coefficients are ofîhciT * (2597) If we suppose / to be invariable in the value of a [4296], we shall get Ja = — '^= — °^ [4299]; substituting this in [4298], we obtain [4300] ; which is reduced to the form [4301], by the substitution of âV' = ±l" [4300'], r" ^ I [40r9] and a == 0,7233323 [4126]. t (2598) Venus, being an inferior planet to the earth, is situated in the same relative position as the earth is to Mars ; therefore the equation [4286], which obtains relatively to [4301»] Venus and the earth, may be applied to the earth and Mars, by substituting in [4286] the value of a [4284], and then adding one more accent to each of the symbols r', r" , V ; by which means we shall obtain 3r" = — 7-'" . A-^V .5 V" [4286]. In this ^^g^j^^ case V" is the change of the longitude of the earth, as seen from Mars, arising from the t/^^Qi^^ increment 5 ?•" ; and is evidently equal to the increment of the geocentric longitude of Mars, depending upon the same cause, which is represented by 5\"'; hence we get r43Qjj| ^ r"==-r"'. (\ -~\ . V", as in [4302]. VOL. III. 62 246 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. less than ±0,000001.* JVe shall also neglect those inequalities of the [4304'] earth^s ^notion in longitude, lohich are less than a quarter of a centesimal second, or 0',081. Inequalities of the Earth, independent of the excentricities.f 5%290878 . sin. (n't — n" t + s'— z") — 6',015891 . sm.2(n'i — n" t + e'— î") — 0%743445 . sin. 2{n't — n" t + i'— /') .,,,,/— 0%225439 . sin.4rn'i — n" t + s'— n ' ^ — 0S091210. sin.5(w'^ — n" t + s'— s") — 0%042805 . sin. Q{n' t - n" t + s'— .") - 0',022027 . ûn.l{n't - n" t + s'— e") — 0%0 12053 . sin. 8(n'i — n" i + i'- =") [4305} .„.n.aii- / 0',427214 . sin. (n"7 _ n" t + s'" — /') ties inde- / p™t"-' "'■ f 3^483037 . sin. 2(n"'ï — n" t + ^"' — e") ""'""" \ — 0',21 5249 . sin. 3(n"'t — n" t + s'" - ,") + (1 + f^'") • ( — 0',047022 . sin.4(n"7 — n" t + s'" — /') — 0^015871 . sin.5(M"7 — n" t + b'" _ s") — 0%006458 . sin. 6(ti"'t — n"t + £'" — £") — 0^002923 . sin. 7(n"'t — n"t + s'" — s") + (1 + f^'') 7^059053 . sin. {n"t — n" t + £'"— e") - 2'-,674257 . sin.2(n'^'i — n" t + ."—.") — 0', 167770 . ûn.S{n'''t — n" t + ."— s") — 0',016549 . sin. 4(?i'"/ — n" t + 5'^— s") ( 0S439410 . sin. {n't — n"t + s'— s") + ( 1 4- f^.'') . I — 0', 1 1 1 1 . sin. 2{n''t — n"t + £^— s") / — 0»,004]45 . sin. 3(n't — n"t + s'— /') * (2599) This quantity, independent of its sign, is less than either of the values [4301,4304], corresponding to the 7îea?-cs< inferior and superior planets ; and for the more [4304a] distant planets this degree of accuracy is more than is absolutely requisite, in the present state of astronomy. t (2600) The quantities [4305, 4306] are deduced from [4277a, b] ; accenting the [4305o] symbols so as to correspond to the present case, and using the data [4061, 8:c.]. Vl.x.^^-29.] THEORY OF THE EARTH. 247 , 0,0000015553 ^ — 0,0000060012 . COS. {n't — n"t + s'— i") \ 4- 0,0000171431 . COS. 2(n't - n"t + i'— s") Ô r" = (1 + f^') • ^ + 0,0000027072 . cos. 3(n't — n"i + s'— e") + 0,0000009358 . cos. A(n't — n"t + s'— e") + 0,0000004086 . cos. ô(n' t — n"t + b'— b") . + 0,0000002008 . cos. 6{n't — n"t + s'— /') ^ ,— 0,0000000478 + 0,0000005487 . cos. (n"'t — n"t + /"_ s") + (1 4_ f.'") . ) + 0,0000080620 . cos. 2(n"7 — n"t + e'"— s") V i„e,„,u. — 0,0000006475 . cos. S(n"'t — n"t + «"'— ^") \ 'ifEt — 0,0000001643 . cos. 4(n"7 — n"t + £'"— £") tricities. — 0,0000011581 \ [4306] + 0,00001 59384 . cos. (n^'t — n"t + é"— e") + (1 + f^") . <f — 0,0000090986 . cos. ^.{n^t — n"t + s'"— f") ' _ 0,0000006550 . cos. S{n'H — n"t + 1"—^') - 0,0000000704 . cos. 4(«'7 — n"t + s'"— s") / -0,0000000580 ^ + (1 -ff;^'). <[+ 0,00000 10337. cos. («7 — m"^4-£^— z")\. — 0,0000003859 . cos. 2{n't —7i"t + £'— s")) In the solar tables of La Caille, Mayer, La Lande, Delambre and Zach, published before the year 1803, the chief correction of the radius vector of the earth's orbit, arising from the action of Jupiter, is given with a wrong sign ; in consequence of taking, for n"t-\-s'', the ' sun's longitude, instead of that of the earth, in finding the argument corresponding to the terms which were used, namely, + 0,0000 1 59384 . cos . {n'^i — n"t-\- £'"— e") — 0,0000090986 . cos . 2 (n-i — n't -f- e^' — e") . [4305c ] This mistake was first made known in a letter communicated by me to La Lande, and u^did} noticed in vol. 8, p. 449, of the Moiiatliche Correspond enz for 1803. 248 PERTURBATIONS OF THE PLANETS; [Méc. Cél. Inequalities depending on the first power of the excetitricities. iric<iuali- ties de- pending on the first power of the excen- tricitiea. [4307] 0^075910 — 0', 129675 - 0', 145 179 -0%168981 + 1', 186390 — 2',342956 + O"-, 722424 V+ 0S2 16368 — r,095603 + 2, 137658 — 0",087400 , + 0%661950 — 0', 103758 , + 0',807 1 1 1 — 0-, 1349 15 + (i+0 sin. (n' t + ^' — z,") sin, {2 n't — n"t + 2 e' — =-" — ^") sin. (2 n"t — n'i + 2 /' — i' — z,") sin. (2 n"t — n't + 2 /' — s' — ^') sin. (3 )i" t — 2n't + Ss" — 2.' — ~/') sin. (3 n" t — 2 n' t-^3e" — 2^ — ^') sin. (4 n" t — 3n't + 4>i" — 3 ^' — z^") sin. (4 n"t — 3 n't + 4 s" —3 / — ^') sin. (5 n"t — 4 n' i + 5 ;" — 4 / — ^") sin. (2 n"'ï — n" t + 2 /" — /' — ^") sin. (2 n"'t — n" i + 2 c-'"— :-" — ^"') sin. (3 n"'t —2n"t + 3 i'"— 2/'— v/') sin. (3 n"'t — 2n"t + 3 .'"— 2 ."_ ^"') sin. (4 n"'t — 3 n" i + 4 --'"— 3 ;"_ w" ) sin. (4 H"'t — 3 n" t + 4 s'"— 3 ^"—^"') sin. (5 /t"7 — 4 7i" t -1- 5 s'"— 4 ="— r/") + (!+(-"') + (1 + O 0',302092 . sin. (n'-'ï -1- e'"— ^") — 2%539884 . sin. {n'H + .=" — j.") — 1%492044 . sin. (? w'7 — «'7 + 2 .>— /'— ^") + 0',606399 . sin. (2 n'7 — n"t + 2 «'"— /'— ^'0 — 0',543364 . sin. (3 n"t — 2n"t-\-3 s"— 2 1"— z^"") — 0', 148925 . sin. (2n"« — n'^'i + 2 3"— e-—^") \— 0^093643 . sin. {2n"t — n"t +2£"— j'"— ^'') J — 0',359921 . sin. {n't + e^ — ra') > ( — 0',151752 . sin. (2 n7 — ?i"i + 2 s'— e"— ^") ^ ' * (2601) The terms of àv", or" [4307, 4308] are computed as in the theory of '^^"'"^ Mercury [4278«]. VI.x.§29.] THEORY OF THE EARTH. 249 r_ 0,0000030439 . cos. (3 ift — 2 n' ^ + 3 s" — 2 / — ^") y àr"= (1 +.a').^ — 0,0000049815 . cos. (4»"/ — Sn't + 4s" — 3 e' —^")} (+ 0,0000015895 . cos. (4ïi"/ — 3n'^ 4- 4 s" _ 3e' — ^')) 4- (1 +,x"') . 0,0000017707 . cos. {^n"'t—Sn"t + 4 a'"— 3 s" — ^"') [4308] — 0,0000030410 . cos. (2 n'^ï— n"^ + 2£'''— £" — ra")" 4-(l 4- f^'0.<| + 0,0000012652 . cos. (27rt— n"t + 2e''— e" — ^'O! -0,0000018101 . COS. (3n''f — 2n"^+ 3s"—2s" — ^''')'^ Inequalities depending on the squares and products of the excentricities and inclinations of the orbits,* Inequali- 6 v" = (1 + I'.') . r-,125575 . sill. (5 n'7 — 3n't + 5-="— 3/+ 21''02"' 1 8^ iVJi''' order. C + 0^993935 . sin. (4 n"'t — 2 n"t + 4 s'"— 2 s"+ 67H8"560 ) [4309] ■^ ^ "^ ^ ^ ■ ^ + 0^351 796 . sin. (5 n"'t — 3 n"t + 5 s"'- 3 a" + 68'' 25'" 09^ ) ' The mean motions of Venus, the earth and Mars bear such proportions to each other, that the quantities 5 n" — 3 n', 4 n'" — 2 n" are small ^^gj^-j in comparison with n" ; hence it follows, from [3733], that the two first of these inequalities are the only ones of this order which are deserving of notice. However we have calculated the third ; because 3h"— 5?r, being very nearly equal to ^n", it is satisfactory to show, by •[4310] a direct calculation, that this inequality acquires by integration only a very insensible value. t n [4309a] * (2602) From [4076A] we get, very nearly, 5n" — 3 n' = 50° = - ; 4 n"' — 2 n" r= 50= = ^ ; .3 ?i"— 5 ?i"' = J 37° = ~. These angles ought therefore 3 to be noticed, as in [3733] ; and by making the computation, as for Mercury [4282a— jp], we may reduce the terras, depending on each angle, to one single term, as in [42S2/t — /]. t (2603) We have already mentioned, in [4296/;], that Mr. Airy had discovered an inequality in the earth's motion, depending on terms of the fifth order of the excentricities [4310a] and inclinations, connected with the angle 8 n't — 13 n"i. He has given in the paper mentioned in [4296A] the numerical values of the inequalities of the mean motion |", [43106] of the perihelion ozi", of the excentricity ôe", and of the latitude &s", namely, VOL. III. 63 250 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. Inequali- ties of ihe third order. [4311] hiequali- tio3 ill the latitude. [4312] Symbols. [4313] Inequalities depending on the powers and prodiccts of three dimensions of the excentricities and inclinations of the orbits. iv" = (1 +1^.) .0',069915 . sm.(ni— 4n"i + s— 4=-"+19''O2'"130.* Periodical inequalities of the Earth's motion in latitude. We find, by formula [1030],t ( 0%0991 80 . sin. (2 n't —n't + 2 /'— ;'— è') > \ 0',234256 . sin. (4 n!'t — 3 n't + 4 /'— 3 ='— é') $ + (1 + (J-") . 0',164703 . sin. (2 n"t — n't + 2 .-- — s" _ é-). Inequalities of the Earth depending upon the Moon. 30. If we jHit U = the longitude of the moon, as viewed from the centre of the earth ; v" = the longitude of the earth, as viewed from the centre of the sun ; R = the radius vector of the moon ; its origin being the earth's centre ; r" = the radius vector of the earth ; its origin being the sun's centre ; m = the mass of the moon ; M = the mass of the earth ; s = the latitude of the moon, as viewed from the earth's centre. [4310c] [4310rf] [431 Of] [4310/] [4311a] [4312a] ^"= (2',059 — ^.0',0002076).sin.(8n7— 13 «"<+ 8 s— 13 /'+ 40''44'"34'— MO',76); 5 ra" = 2',268 . sin. (8 n't— 13 n" / + 8 s' — 13 s" + 60'' 16"') ; <5e"= — 0,0000001849. cos.(8n'<— 13 m" <+ S s' — 13 £" + 60''16"') ; 5s" =. 0',0105 . sin. (8 n't — 12 n"t-\- 8 ; — 12 s" — 39'' 29'"). * (2604) The direct calculation of this inequality can be made, by a process like that which is used for Mercury, in [3881f, &c.] ; but it is probable that the author deduced it from the similar inequality of Mercury [4283], by the method given in [3883y]. f (2605) The terms of [4312] are computed by means of the formula [4295/!»] ; changing, in the first place, n, «', e', into n", i", respectively. Then changing m , n , a , s mto earth ; or into m'", n^" the earth. a', s', in computing the action of Venus on the , respectively, in computing the action of Jupiter on VI.x.^30.] THEORY OF THE EARTH. 251 we shall have, for the inequality of the earth's motion in longitude [4052], produced by the action of the moon,* ôv"^ - -.-.sin.(U—v). The inequality of the radius vector of the earth [4051] is ôi-"^ — jj.R.cos.(U—v"); and the inequality of the earth's motion in latitude [4053] is „ m R 711 The moon's action produces a perturba- tion in the longitude ; [4314] in the radius ; [4315] latitude. [4316] in the For greater accuracy, we must substitute f —— for —, expressions of these three inequalities. We shall suppose conformably to the phenomena of the tides [2706,2768], m R^ 3S_ ^"3 ' [4317] * (-2606) The moon's action upon the eartli produces, in the radius vector, the longitude and the latitude of the earth, the ineciualilics given in [4051, 4052, 4053] ; namely, m . r . cos.(v — U) ; m r MR .{v-U); m Jl rs [4314a] and by comparing the notation used in [4047, 404S], with that in [4313], it appears r^^Ub] that we must change R, r, v, U, into ?•", R, U, v", respectively, to conform nearly to the notation of this article. By this means the preceding expressions become, m R . ,-rT „, ™ Rs — ^.R.cos.{U—v"); M M r- corresponding respectively to the formulas [4315, 4314, 4316]. In the original work the divisor r", by mistake, is omitted in [4314], and inserted in [4315] ; we have rectified this mistake. f (2607) The radius r [4048] has for its origin the common centre of gravity of the earth and moon. This is changed into R, in [4314&], to conform to the present notation ; but as the origin of R [4313] is in the centre of the earth, the value of the radius ■■ "^ is too great, and must be decreased in the ratio of M to M -\-m; which is equivalent M to the multiplication of the perturbations [4314 — 4316] by ; or in other words [43164] to change the divisor M into M-\-7n, in all three of these formulas. 252 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. S being the sun's mass. Now, bj the theory of central forces [3700],* we have, [4318] -^ = n/ ; - = n - ; n,t behig the moon's mean motion; hence we obtain, [4319] m 3n"2 M+i [4319] We have by observation - = 0,0748013 [5117, 4835] ; hence we get, 71, [4320] Mass of consequently. m 1 the moon. M+m 59,6 ' m 1 If we suppose the sun's horizontal parallax to be 27",2 = 8',8, and [43221 *^^® moon's mean horizontal parallax 10661" = 3454' = 57'" 34',t we [4323] shall have, R sun's lior. par. ?• moon's hor. par. 3454,0 ' * (2608) Substituting fi^Ji-j-m [3709«] in [.3700], then changing a, n, into [4318a] Jl^ n^, respectively, we get the first of the equations [4318], corresponding to the moon's motion about the earth. Changing in this, iW, m, B, n, into S, M, r", n", [43186] and neglecting M in comparison with .S', we get the second of the equations [4318] ; corresponding to the earth's motion about the sun. MuUiplying the first of the equations r4318cl [■^■^■^®]' hy „ , and the second by 3; then substituting the products in [4317] we set ir?-i n2=3?j"^: dividing this by n^, we obtain [43191; substituting in this [4318rf] ^I-\--"' a J / ' L J J 6 the value [4319'], we finally get the expression of the mass of the moon [4321]. This was afterwards found to be too great [4631, 1190i, &c.], as we have already observed in [.3380 J, &ic.]. Instead of supposing, as in [2706], tliat the ratio of the mean force of the moon on the r.fj.g -, tides, is to that of the sun as .3 to 1, we may express it more generally by 3(1 — |3)to 1 ; by which means the second members of the equations [4317, 4319, 4320], will be [4318/] multiplied by 1 — (3 ; and the last of these expressions will become - = -r^rê '■> [4318g-] whence we get the following expression, which will be used hereafter, — = — . [4322a] t (2609) This parallax, taken for the mean between the greatest and least values, VI. X. §30.] THEORY OF THE EARTH. 253 consequently,* iv" = _ 27",2524 . sin. (C/— v") = — 8%8298 . sin. {U—v") ; pptturba- tionn in the longiliide, [43'24] or" = — 0,000042808 . cos. (U— v"). a„d in .h.. radius. Then taking for s the greatest inequality of the moon in latitude, which ^43251 wo shall suppose to be 18543'. sin. (f/ — f) [5308]; U — being the pcnurba tion of moon's distance from her ascending node; we shall obtain t inlau'"' tude- 6 s" ^ — 0%7938 . sin. (U—è), [432e'] for the inequality of the earth's motion in latitude. We must add it to the terms of is" [4312], to obtain the complete value of 6s"; and by taking this sitm, with a contrary sign, we have the inequalities of the sun^s utïtâde. apparent motion in latitude. These inequalities in the latitude have an influence on the obliquity of the ecliptic, deduced from the observations of [43'27] the meridian altitudes of the sun near the solstices. They have also an influence upon the time of the equinox, deduced from observations of the sun, when near the equinoxes, as well as upon the right-ascensions and declinations of the stars, determined by comparing directly their places in exceeds, by .33% the constant quantity in Burg's tables [5603], and is nearly conformable to the resuh given by La Lande in *5' 1698 of tlie third edition of his astronomy. For the purpose of illustration, we may neglect all the inequalities of the moon's pai-allax, except [43'22?i] those depending on the moon's mean anomaly ; then taking the coefficients to the nearest second, we have, from Burg's tables [5603], J) 's hor. par. = 342P' + 187" . cos. (mean anom.) -)- 10'. cos. (2 mean anom.). r4322cl The greatest value of this expression, corresponding to the perigee, or the mean anom. = 0, is .3421>--f 18T"+10'; and the least value, in the apogee is 342P — 187^+10'. The ?nert« of these two values 342P+ 10% exceeds hy 10% the constant term 3421"; and it is from causes similar to this, that the difference above-mentioned depends. [4322rfJ * (2610) The inequalities [4324] are deduced from [4314, 4315], by using the values [4321,4323], and multiplying the value of S v" by the expression of the radius in [4324«J seconds 206264%8. t (2611) Substituting the values [4321, 432-3], and s [4.325], in [4316J, we get Û »" [4-326] ; changing M into M-fm, in all these calculations, as in [4316e]. VOL. III. 64 [432fi<»J 254 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. Perturba- tion of the sun in declina- tion, [4328] and in right- aacenston. [4329] [4329'] Increment of O's declination = — the heavens with that of the sun. On account of the great accuracy of modern observations, it is necessary to notice these inequalities. It is evident that this correction increases the apparent declination of the sun, by the quantity,* Ss" . COS. (obliquity of the ecliptic) _ COS. (sun's declination) ' and its ajiparent right-ascension is also increased, by the following expression, r c ^1 • 1 5s". sin. (obliquity of the ecliptic) . cos. (sun's riffht-ascension) Inc. of O's nght-ascen. = ^^ — -, — , / ,. — ~ \ ° COS. (sun s declination) fVe must therefore decrease, hy these quantities, the observed declinations and right-ascensions of the sun, to obtain those lohich ivould be observed, if the earth did not quit the plane of the ecliptic. * (2612) Let ECC be the ecliptic, -EQQ' the equator, P the north equator; then if the earth's latitude, north of the ecliptic, be 5s", that of the sun will be south, and may be represented by CU = &s" perpendicular to the ecliptic. P CL Q, r4328al PC'L'Q^, are circles of declination, perpendicular to the equator, and L L' is parallel to the equator. The small differential triangle CLU, may be supposed rectangular in L, and angle LC L'= 90' — angle E CQ^. Then in the spherical triangle E C(^, we have, by [1345^2], cos.i:CQ = sin.L CL'=^s\n.CE q.cos.Eq-, COS. CE q [43386] sm.ECq=cos.LCL'= COS. C Q Now the declination is decreased by the quantity C L ; the right-ascension is LL' LL' lEqaator [4328c] increased by the quantity QQ' = sin. PL COS. dec. and we have [4328rf] LL'=^CL'. sin. LCL' = 5s" . sin. CEq. cos. E q ; hence we get. [4328e] Increm. dec.= — CL =^—CU. cos. L C L-- , „ COS. CEQ OS . COS. CQ , as in [4328] ; and I . , y-^ ^, L L' sin. C E Q .cos.E Q [4328/] Increm. nght-ascen. Q Q' = -—g- ==Ss' -^ ^, as in [4329] COS. dec. VI. x.§31.] THEORY OF THE EARTH. 255 [4329" On the secular variations in the Earthh orbit, in its equator, and in the length of the year. 31. We have given, in [4244, 4249, &c.], the secular variations of the elements of the earth's orbit ; but the influence of these variations on the most important phenomena of astronomy has been an inducement to compute them with greater accuracy, noticing the square of the time t;* supposing t to denote the number of Julian years elapsed since 1750. We have found by the methods given in [1096 — 1126], and using the values of the masses of [4329" the planets [4061], that the coefficient of the equation of the centre of the earth's orbit is represented by,t * (261.3) The values of e^, tang. « [1109, 1110], give those of e"^, tang, a"; by changing the quantities corresponding to m, into those relative to m", and the contrary. The formulas, thus found, may be developed in series, ascending according to the powers of t, by Taylor's theorem [.3850a] ; hence we easily deduce the values of c", •zs", in similar forms. The calculation may also be made by the method pointed out in the following note. t (2614) We have, by Taylor's theorem, as in [1126'"], 2e' 2 de" = 2E+~.t + dde" ~dfi neglecting the higher powers of t ; the values of — , to the epoch 1750. The differential of -— , -— - , being taken to correspond de" — [1122], taken according to the directions dde" [4329a] [43296] [4330a] [43306] [4330c] in [1126"], or as in note 768, vol. I. p. 612, and divided by dt, gives -— , m terms of e, e', 8ic. w, -n', &.c. and of their first differentials. Substituting in this expression, the values of these first differentials, given in [1122, 1126], it changes into a function of the finite quantities e, c', fee. tn, -s/, &,c. ; and by substituting the values of these quantities, [4330(/] dde" for the year 1750, given in [4030,4081], we obtain the expression of dfi Moreover, de" by similar substitutions, we get the value of the expression of "-^ [1122]. These values, bemg substituted in [4330a], give the expression of 2 c" [4330]. The formulas [4330—4360] are so frequently referred to in the work, that we have given the numerical values in centesimal, as well as in sexagesimal seconds. The values given in [4330, 4331, 4332], are altered, in [4610 — 4612], by reason of the changes in the masses of Venus and Mars. We have seen in vol. I. p. 612, note 468, that terms of the order m'e' are retained, and those of the order m'e'^, which are of the Jirst order relative to the mass m', are [4330e] [4.330/] 256 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. Coeff. eqiia. centre = 2E — t. 0",579130 — f . 0",0000207446 = 2E—t. 0% 187638 — f . 0',0000067213, 2E being this coefficient at the beginning of the year 1750, when t is ëàrth'B nothing. We have also found the sidéral longitude of the perihelion of tlie earth's orbit, namely,* Long, perih. of the earth = ^"+ i • 36",881443 + t"' • 0",0002454382 = t^' + i . 1 1',949588 + t- . ,0000795220. Lastly, the values of jf, q", at any time t, have been found respectively equal to,t p" = t. 0",2.36793 + Ï-. 0",0000665275 = t . 0',076721 + t- . O',0000215549 ; q" = — t. 1",546156 + t\ 0,0000208253 =^ — t. 0',500955 + i"-. O',0000067474. [4330] Secular equations of the earth' orbit. [4331] [4332] [4330A] d e" [4330e-] neglected, in the expression of — [1122]. If we suppose, for a rough estimate, that e' = -^'^, the neglected terms will be of the order of j^^y part of those retained; so that the neglected part in the coefficient of t [4330], may be considered as of the order j^X0%18763S =0^0004, which is much greater than the coefficient of t^ in [4330] : and at the first vjew it might be thought strange that we sliould neglect this, and yet notice the much smaller coefficient of fi, which is of the order of the square of the disturbing masses. But the reason will appear very evident from the consideration, that when t is large, the term depending on t^ becomes very great in comparison with these neglected [4330i] terms. Thus, iÇ t= 2500, the neglected term 0,0004 1 is only one second, while the term depending on t^, exceeds 42". Similar remarks may be made relative to the quantities w", jj", q" [4331,4332]. * (2615) Proceeding as in the last note, we may deduce from [3S50«], by changing [4331a] M into •zn", ^" = t^"-\- i • -j-/ — \~ i^^' "TT ' ^'^^ quantities In the second member referring to the epoch of 1750. The difierential of —— [1126], divided by dt, gives —rur [4331i] i'^ terms of e, e', &c. zs, zi' , and their first differentials. Substituting in this expression the values of the differentials [1122, 1126], it changes into a function of the finite quantities e, e', &:c. -a, -a, Sic; and by using the numerical values [4080,4081], we get the '^ values of — - , -775-) to be substituted in [4331a], to obtain [4.331]. t (2616) The expressions of —-, -—, are in [425 1 i] ; their differentials taken Vl.x.§31.] THEORY OF THE EARTH. 257 We have given, in [3100 — 3110], the expressions of the precession of the equinoxes,* and of the inclination of the equator, referred to the fixed ecliptic, and to the apparent ecliptic. In these formulas, we have supposed the values of p'\ </", to be given under the forms ;?"==-. c . sin. (§-f + (3) ; q" = ^ . c . cos. {g t + ^) [30686]. Moreover, we have seen, in [1133], that the finite expressions of //', ç", appear under these forms, and we may determine, by the method explained in [1098, &c.], the values of c, g, p. To obtain these quantities accurately, by this method, we must know the correct values of the masses of the planets; and there is considerable uncertainty relative to some of them, as we have observed in [4076, &c.]. Therefore, instead of making the tedious calculation, required by this method, it is preferable to simplify it, so as to embrace a period of ten or twelve hundred years, before and after the epoch of 1750 ; which is sufficient for all the purposes of astronomy. We may easily rectify these calculations as often as the development of the secular variations shall make known, with greater accuracy, the masses of the planets. We shall give to the values of p" and (f the following forms, which are comprised in those mentioned in [4334]. f p" = 2. c.sm.(^gt-\- (3) = c. sin./3 — c.cos.f3. s'm.gt — c.sin./3. s\n.(g't-'r}'^) ; ç"= 2. c.cos.{gt -|_/3) ^ c.cos. f3 — c.cos.(i.cos.g( — c . sin. § . COS. (g't-\- ^v) ; ff being the semi-circumference of a circle whose radius is unity. If we [4333] [4334] [4335] [4336] Assumed forms of [4337] r", 1"- , . , 1 1- • 1 1 1 , ■ ddp" ddq" relatively to t, and divided by at, give -tt ' , dp dp in terms oi ---, -; — , cic. dt at —, — , &:c. ; substitutina; the values of these last quantities [11321, we get -~- , — - , dt dt' ' ^ '■ L J' & (^^2 ' rf<a ' expressed in finite terms of p, p', &;c. q, q', &c. The values of p, p', &tc. q, q', &z;c. are given in [4251c], in terms of (jj, cp', &,c. ê, &', &,c. ; and the numerical values of these last quantities, in the year 1750, are in [4082, 4083] ; hence we obtain the numerical values of p, p', Sic. q, q', &;c. at that epoch. Substituting these in [4251 f/,e], and in the preceding values of d dp" d d q" we get tlie numerical dp" dq" ddp" ddq" values of — , -;— , — r^ dl dt' dt3 in the general values of p dt^ ' dfi -— -, at the same epoch, 1750 ; these are to be substituted q" [4250], to obtain [4332]. * (2617) The formulas, here referred to, are [3100, 3101, 3107, 3110]. t (2618) The three terms of the second member of the value of p" or q" [4337], VOL. III. 65 [4332a] [4.332i] [4.332c] [4332rf] [4333a] 258 PERTURBATIONS OF THE PLANETS ; [Méc. Ctl. develop these two functions relatively to the powers of the time t, we shall find, by comparing them with the preceding series [4332],* Valuesof c £r . COS. |3 = — 0',076721 : [4338] [4339] [4337a] [4338a] [43386 ; eg', sin. f3 = — 0',500955 ; cg\ cos. [3 = 0S0000134948 ; cg'K sin. |3 = 0',0000431098. Hence we easily obtain,! g = — 36^2808 ; g' == — 17',7502; c. sin. f3 = 582P,308; c.cos. f3 = 436%17. are deduced from those of p" or q" [4334], by changing c, g, p, respectively, into c, 0, p, in the first term ; — c . cos. p, g, 0, in the second term ; and — c.sin. p, g', J^r, in the third term. Tliese expressions of p", q", being developed according to the powers of [433/6] f^ and compared with those in [43.32], give, as in [4.339], values of c, p, g, g', which satisfy the numerical expressions of p", q" , [4332], neglecting f, and the higher powers of t : and as the values [4332] will answer for ten or twelve centuries from the epoch, it will follow, that the forms assumed in [4337] will answer for the same period, by using these values of c, p, g, g'. * (2619) We have by development, using the formulas [43, 44] Int. and neglecting terms of the order «^ sin.gt = gt; cos.gt=l — hg^i-; sm.{g't-]-iv) = cos.g'i = l — ig"^t^; cos. {g't -\-i ■ïï) = — sin. g't = — g't ; substituting these in [4337], we get, p" = i: .c . sin. {gt -i-^)=c . sin. p — c. g t .cos.fi — c.(l — i g'^t~) .s'm. p ^= — t . {c g . COS. p) + t^-{i cg'''^. sin. p) ; ç"=2. c .COS. (^^ ~\-fi) = c. cos.(3— c . {I — Ig^t-). COS. p -{- c g' t . s\n. fi =z t . {eg', sin. p) + fi. (I cg^. COS. p). Comparing the coefficients of t, in these expressions, with the corresponding ones in [4332], r4338cl "'6 &^^' without any reduction, the two first equations [4338]. In like manner, by comparing the coefficients of I i^, in [4332,43336], we get the other two equations [4338]. f (2620) Dividing the square of the first equation [4338], by the third, we get c . cos. p [4339] ; and the square of the second, divided by the fourth, gives c . sin. p [4339]. [4339a] Now, dividing the values of c^^.cos.p, cg'~. s'm.fi [4338], by those of eg. cos. g, eg', sin. p [4338], respectively, and multiplying the products by the radius in seconds, 206265% we get g, g' [4339]. VI.x.§31.] THEORY OF THE EARTH. 259 Now we have seen, in [3100], that the precession of the equinoxes +, relative precessio.. , , , relative to to the fixed ecliptic of 1 750, noticing only the secular variations, is, 'cU|,1"tf ■I = /^ + ^ + 2 . I [j— 1 j . tang, h + cot. /i 5 • y • sin. (ft + f3). [4340] First form. To obtain ^ .c .un. {ft -\- ^), we must increase the angle gt-\-^., in 2 . c . sin. {g t + f3), by the quantity 1 1 [3073', &c.] ;* making f = g-\-l [3113a] ; then we shall have, 2 . c . sin. {ft + f3) = c . sin. {lt + f^) — c . cos. (3 . sin. {gt ^l t) — c . sin. p . sin. {gtJ^U + \^) ; consequently,! [4341] [4342] * (2021) If we increase the angle gt, by the quantity lt={f — g) t [3113a], the function 2 . c . sin.(^< + P) will become 2 . c . sin. (/<-(- p), as in [4341] ; and the first equation [4337], will change into [4342] ; observing that we have ^ = [4337aJ, in the first term, or c . sin. p ;= c . sin. [0 . t -{- js), which becomes c . sin. {It -\- p), as in the first term of [4342]. [43410] t (2622) The expression 2 . c . sin. (/i + P)i in the form assumed [4342], consists of three terms. In the first of these terms, the general symbols c, f, /3, of the first [4342o] member, become c, I, 3 ; or in other words, f is changed into /, while c, (3, are unaltered ; and the corresponding term of [4340] becomes, [43426] [4342c] I \ ') Ic . 1 j . tang, h -f-cot. A> . — . sin.(/ t -\- ^) ; or simply, c.cot.h . sin. {It -\- ^); which is the first term of 4^ [4343], depending on c. The second term of [4342], — c . cos. p . sin. {gt -\- It), being compared with the general expression c . sin. {ft + p), in the first member of [4342], shows that c, /, p, must be changed into — c . cos. p, g -\-t, 0, respectively; and the corresponding term of [4340] becomes, 'i / ' , \ II 7 ) 'c .COS. (3 . , , , , — ^(^rpj— Ij-tang.A+cot.A^ .____. sm.(^i + ?0 ; [4342d] which is easily reduced to the same form as the term of [4343], depending on the angle gl-\-lt. Lastly, the «Aire/ term of [4342], — c .s\n.fi .s\n.{^t -\- It -{-\v), being r4342e] compared with the general term, in the first member of [4342J, gives for c, f, (3, the corresponding expressions, — c.sin.(3, g -\-h J*, respectively; and the resulting terra of [4340] is, — ^(^; — l)-tang.A+cot.A^ . ^±^ .ûu.{g' t-{-lt + 1^); which is easily reduced to the form of the last term of [4343]. The two first terms of [4340, 4343], represented by It -\- 1, are the same in both formulas. [4342g] 260 PERTURBATIONS OF THE PLANETS; [Méc. Cél. r;sft" ^ = It + ?, + €. cot. h . sin. Qt + p) the fixed ecliptic 01 7 r n- '\ 1750. ^ ^ ^ çjjg_ o _ S çQ^_ ^ ^_ _ ^ajj„_ /j / _ gij^_ (-rrt + lt) [4343] ^^ ^ ' •^ -" I Second — ^ . c . sin. f3 . > cot. h — ~_ . tang, h i . sin. (g't-j- ii-^ i'^)- form. l+g' [4345] îôfàiwe'"" Then by putting V /or the inclination of the equator to the fixed ecliptic of ediptkof 1750, we shall have, as in fSlOll,* 1750. ' ' L J' [4344] V = h — ^.-. COS. (ft + p). First formj J To obtain 2 . c . cos. (ft + /3), we must increase the angle gt -{- § in 2 . c . cos. (gt + P) by lt\ [3073, &c.] ; hence we shall have, 2 . c . cos. (ft -j- p) = c. cos. (I t-i- j5) — c . cos. f3 . cos. (gt -\- 1 1) — c . sin. |3 . COS. (g't J^lt-^-^-n) ; therefore, Î second J form. Y = h — C . COS. (I t + (3) -j- —— . C . COS. f3 . COS. (g t + 1 1) [4346] ^^^ + — -j.c.sin. f3.cos.(^7 4-Z< + i^). [4347] 4^' denoting the precession of the equinoxes relative to the apparent ecliptic. * (2623) This is the same as [3101], putting V for the part of ê, depending on [4344a] ^ and 2 ; or in other words, neglecting the periodical terms depending on the angles /^ + p', 2«, 2t)'. f (2624) This is done upon the principles used in [4341, &.c.]; and in the same [4345o] manner as [4342] was deduced from the first of the equations [4337], we may derive [4345] from the second of [4337]. X (2625) Proceeding as in [4342« — ■/] ; and comparing the general form of the first member of [4345], with the three terms of the second member, we find, that c, f, p, become, respectively, c, /, p, in the Jirst term ; — c . cos. p, g -\- I, 0, in the second term ; and — c . sin. p, g' -\- I, I "^j in the third term, in the terms under the sign 2 [4344], we get the three terms c the first term h, is the same in both expressions [4344, 4346]. second term; and — c.sin.p, g' -\- I, I "^j in the tAirrf term. Substituting these values in the terms under the sign 2 [4344], we get the three terms containing c, in [4346] ; VI.x.§31.] THEORY OF THE EARTH. 261 and V' the inclination of the equator to this ecUptic_; we shall have, as in [4347] recesfiion fjuiiy rola- live ic the I T ) . ^ , T \ ajiparciit eclii»tic. [4348] [4350] [3107,3110],* r^rz: 1' = lt +^ + 7^ .c.cos. |3. )cot. /i+ 7-— . tana;. /t i. sm. (irt-\-lt) + j^—, . c . sin. p . j cot. h + — ;; . tang, h I . sin. (g't-^ lt+ ^r:); V = h — j^ . c . COS. (3 . cos.(gt+lt) — j^-^, . c.sin./3. cos.(^7 + /^+i*). [««] The expression of 4-' gives,t ~ = Z + c ^ • COS. ^ . < cot. /t + -=—, — . tang, h > . cos. (£[14-1 1) dt ^ I l+g i ^^ + c §•' . sin. (3 , ) cot. h -f — — . tang, h > . cos. (g't ~\-lt -{-^■jr). If we subtract from this value of — -, when t is nothing, its value at any [4350'j other epoch, and reduce the difference of these two expressions to time ; considering the whole circumference as equal to one tropical year ; we shall get the increment of the length of the tropical year since 1750. We see, ' ^ by this formula, and by the differential of the general expression of * (2626) Retaining only the secular inequalities in 4"', ^' [3107,3110], changing also Ù' into V [3103, 4347'], we get, by a slight reduction in the term of -^J, under [43''''"] the sign 2, +' = ^^+? + 2.^cot.;i+j.tang.A^ . (^-^V c .sm.(ft + fi); [43476] V = A + 2 . (^-^ ^ . c . COS. (/i + p). [4347c] In the terms under the sign 2 [43476], we must substitute, successively, the values of the triplets of terms c, f, ^, given in [4342a, c,/], and we shall obtain [4348] ; observing I — f that the first term vanishes, because the factor — — = 0. In like manner the substitution r , ['4347f] of the same triplets of values [4346a — 6], m [4.347c], gives h [4349] ;the first term vanishing, f I on account of the factor = 0. / 1(2627) The differential of ■].' [4348], taken relatively to t, and divided by rf r, [4349a] gives [4350]. VOL. in. 66 262 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. [4350'"] [4351] [4351'] [4352] [4353] [4353'] [4353"] [4354] 4'' [3107],* that the action of the sun and moon changes considerably the law of the variation of the length of the year. In the most probable hypothesis on the masses of the planets, the whole variations, in the length of the year, and in the obliquity of the ecliptic, are reduced to nearly a quarter partf of what they would be without that action [31 15, 31 13;y]. ^ 154",63 ^ 50', 1 ; According to observation, we have in 1750, but, by what has been said, we get at this epoch, | I ~dr I -{-eg . cos. (3 . ) cot. A + l+. di tang, h V hence we obtain, l-\r c g . cos. I'S . \ cot. h -{■ 1 + ^ tang. /j \ = 154",63 = 50',1. If we neglect the square of c, in this equation, we may substitute for h, the obliquity of the ecliptic to the equator in 1750.§ This obliquity was then, by observation, 26^,0796 = 23" 28'" 17 ,9 ; hence we deduce,** I = 155",542 = 50',396 ; [4350o] [4351a] [43516] [4352a] * (2623) Tills difTerential is found in [3 11 8], and by reducing it into time, as in [3118'], we get tlie decrement of the 3'ear, using f=zg-l-l [3113a]; or the increment of the year, by changing its sign, as in [4350"]. t (2629) This subject has ah-eady been discussed in [3113a — z] ; and we have merely to remark in this place, that the values arbitrarily assumed in [4337 — 4339] do not produce such essential alterations in these variations of ■]^', V, as are mentioned in [3113iy, 4351]. This ditierence is what might be expected, taking into consideration, that the results, obtained in [4338, 4339], are restricted to values of /, which are less than 1200 [4335] ; and that for much greater values of t, the results cannot be relied upon. t (2630) At the epoch 1750, we have <;= [4329"], and then cos. (gt-{-It)=l, cos. {g t -\- 1 1 -{- ^ v) = cos. ;| ir ^ ; substituting these in [4350], it becomes as in [4352] ; putting this equal to 50',1 [4351'], derived from observation, we get [4353]. <§! (2631) The expression of V [4346] differs from h, by terms of the order c; [4353a] hence it is evident that if ws neglect terms of the order C", we may substitute indifferently, the value of V or h, for h, in [4353]. ** (2632) Substituting in [4353] the values k = 23''28'"17',9 [4353"], also the values [4354a] of eg. cos. ^, g [4338,4339], it becomes, as in the following equation, from which we easily obtain the value of / [4354], VI.x.>,^31.] THEORY OF THE EARTH. 263 then we have in 1 750,* V = A — -^ . C . COS. ^ ; [4355] which gives, h = 26°,0796 — 3460",3 = 23'^ 28" \Tfi — 1 121', 1. [4356] By means of these values we obtain the following expressions,! [wliich arc altered in 4614 — 4617], / _ 0',076T21 . cot. 23'' 28"' 17%9 — ^^7^ • tang. 23* 28" 17%9 = 154',63. [43346] * (2633) Putting ^=^0 in [4349], it becomes as in [4355]. Substituting in tiiis, V = 23'' 28'" 17%9 [4353"], also tbe values of /, g, c.cos.p [43.54,4339], it becomes, [435Ga] 23'' 28"' \V,9 = A + 1 121 ',1 ; hence we get h [4356]. t (2634) Dividing the value of c.sin.|3 [4339] by that of c.cos.js [4339], we get tang.|3=13,.34636 = tang.85''42"'54"; hence (3 = 85'' 42'" 54°' ; substituting this [4357a] in the expression of c.sin.|3 [4339], we obtain c = 5321',.308 . cosec. (3 = 5837',6. Using these values of p, c, and these of A, I, g, g" [4356, 4354, 4339], we get, [43576] c. cot.A = 13646',3; . c . COS. p . j cot. h — .j^— . tang. A ^ = — 5o52',8 ; [4357c] — -L.c. sin. p . 5cot. h — -^ . tang. hl= — 23097%7 ; '+g c t-f-g } l-\-g=z 14',115 ; Z+^^=.32',645. Substituting these in the third, fourth and fifth terms of [4343], we get the third, fifth and fourth terms of [4357], respectively. The [4357^^] term 1 1 [4343, 4354], gives the first term of [4357]. The term ^ [4343], is to be taken so as to render ■\,=^0 [4357] when « = ; whence ^ = _ 13646-,3 . sin. 85'' 42" 54' + 23097^7 = 2''38'" 9',4. [4357e] In like manner, we have, ' .c.cos.p=1.557V3; -L.c .sin.p = 8986',6 ; [4357/] substituting these and h [4356], also the preceding values [4357c], in [4346], we get [4.358]. From the same data, we have, ^ . c . cos. p . < cot. h -\- — — . tang. h>= — 4333',2 ; i-Vs " d ' l+g z^—,.c. sin.p .^cot./i + ,-r— ,. tang. A[ = — 9499',4 l+g i l+g > [4.3.57e-] [4358] [4359] 264 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. ^ = tA 55",542 + 2°,92883 + 421 1 8",3 . sin. (t . 1 55",542 + 95°,2389) — 71289",2 . COS. i.(100",757) — 16521",! .sin. (i.43",564) ^^^^'^ = t . 50^396 + 2''38"'09%4 +13646%3. sin. {t . 50^396 + So'' 42"" 54') Precession -f,;bj;- — 23097%7 . cos. {t . 32%645) — 5352',8 . sin. {t . 14',1 15) ; theeclip- ye.^"^^'" V = 26°,0796 — 3460",3 — 1 801 7",4 . cos. {t . 1 55",542 + 95°,2389) to.""'] + 4806",5 . COS. (t . 43",564) — 27736",3 . sin. (t . 100",757) == 23^ 28"'17',9 — 1 121%1 — 5837--,6 . cos. (t . 50',396 + So' 42"" 54') + 1557'-,3 . COS. (t . 14%1 15) — 8986',6 . sin. (t . 32',645) ; i' = f . 155",542 + 2°,92883 — 29288",3 . cos. t . (100",757) — 1 3374",2 . sin. (t . 43",564) =- t . 60',396 + 2''38'" 09^4 — 9489',4 . cos. (t . 32%645) — 4333',2.sin.(f . 14', 115); [Apparent! orbit. J V'= 26°,0796 — 3460",3 . ^ 1 — cos. (t . 43",564) | — 9769",2 . sin. (i . 100",757) = 23''28"'17',9 — 1121',l.jl ~cos.(i. ]4',115)i — 3165',2.sin. (<.32',645). We may determine, by means of these formulas, the precession of the equinoxes and the obliqiiity of the ecliptic, in the interval of ten or tivelve hundred years [4357A] sin. {g t -{- 1 1 -\- ^ -jr) = cos. (^-'^ -f- / ^) = cos. {( . 32%G45) ; Z< = i . 50',.396. Substituting these in [4348], it becomes as in [4359], the constant quantity 2,, being taken so as to make 4-' = 0, when t=^0 [4359] ; consequently, t*^^^'^ ^ = 9489',4 = 2-* 38"' 9%4. Lastly, by a similar calculation, we have, -f^ .c.cos.p = — 112P-,1 ; ,4— , •c.sin.3 = — 3165',2; [4357i] ' ' " cos.(g't-\-lt-]-h'^)= — sm.{g't + It)= — s\n.{t. 32^645) ; substituting these and [4356] in [4349], we get [4360]. The numerical values, given in r4357n [435T— 4360], are varied by the author in [4614 — 4617], on account of the changes made in the values of the masses of Venus and Mars. We have already given the formulas of Poisson and Bessel, in [3380^,(7]. [4360] VI.x.§3l.] THEORY OF THE EARTH. 265 before, or after the epoch of 1750; observing to make t negative, for any time previous to this epoch. We may indeed apply the formula to the observations made in the time of Hipparchus ; taking into consideration the imperfections of these observations. The preceding value of -i', gives, for the increment of the tropical year, counting from 1750, the following expression,* Increment of the year = — O''^000083568 . {1 —cos. {t . 14^1 15) \ — 0''»^00042327 . sin. {t . 32'-,645). Hence it follows, that in the time of Hipparchus, or one hundred and twenty-eight years before the Christian era, the tropical year was 12'*''-,326 [= 10,65 sexages.] longer than in 1750;t the obliquity of the ecliptic was also greater by 2832",27 = 917^66. [4361] [4362] [4363] [4363'] * (2635) Using the same data as the preceding note, we get the numerical values of the two functions [4362c, (/], expressed in sexagesimal seconds. These are turned into time by supposing the whole circumference, 360''= 1296000", to be described in one year, or 3g5da,s^242 ; hence we have, c^.cos.p. ^cot.^ + — — .tang.A?= — 0',296527=:— 0''"y,000083568 ; c g'. sin. p jcot. h ■ -r—, . tang. A ^ =— l',501877 = — 0''^^00042327. Substituting these and [4357c?], in [4350], we get the general expression of -— [4362/] ; which becomes as in [4362^], when t^O. Subtracting the first of these expressions from the second, we get the increment of the year [4350'], as in [4362], corresponding to any number t, of years after 1750. ^=1 — 0''"5',000083568 . cos. {t . 14',1 15) + 0^''y,00042327 . sin. {t . 32',645) ; ^ = / — 0'i''^00008356S. dt ' These numerical values are altered in [4618], in consequence of a change in the values of the masses of Venus and Mars. [4362o] [43626] [4362c] [4362rf] [4362e] [4362/] [43C>2g] [4.3G2/I] t (2636) In the year 128 before the Christian era, < = — 128— 17.50 = — 1878; substituting this in the two terms of the expression [4362], we find that the first terin [43(53„n becomes, —0^»y,00000069, and the second, + ©■'"".OOOl 2396 ; their sum is O^'^OGO 12327, as in [4363] nearly. The variation of the obliquity of the ecliptic, in the same time, 1^43(53^1 deduced from [4360], is nearly the same as in [436.3'], being expressed by, — 112P,1.{1 —cos.{t. 14^115)} — 3165,2.sin.(i.32^645) = — 9^,2 + 926^9 = 917%7 nearly. I'i363c] VOL. III. 67 equinox [4364] and sun's apogee coincide. 266 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. [4363"] A remarkable astronomical epoch, is (hativhen the greater axis of the earth\s hie'as'tro- orMt ivas situated in the line of the equinoxes; because the apparent and ji.ean Xt'the equinoxes then coincided. We find, by the preceding formulas, that this phenomena took place about 4004 years before the Christian era, and at this epoch most of our chronologists place the creation of the ivorld ; so that, in this ])oint of view, we may consider it as an astronomical epoch. For we have, [4364'] at that time, t = — 5754 ; and the preceding expression of 4^' gives,* [4365] ^'= — 79'' 04™ 04'; which is the longitude of the fixed equinox of 1750, referred to the equinox of that time t. The preceding expression of ra", gives, for the longitude of the perigee of the earth's orbit, or of the sun's apogee, referred to the fixed equinox of 1750,t ^"= 80M5"'ll'. This longitude, referred to the equinox of the year 4004 before the Christian [4367] era, is V II '"07; J hence it follows, that the time lohen the longitude of the suri's apogee, counted from the moveable equinox, ivas nothing, precedes, about sixty-nine years, the epoch usually assumed for the creation of the world. This difference will appear very small, if we take into consideration the imperfections of the preceding expressions of 4-'? and ro", when applied to so [4367"] distant a period, and the uncertainty which still remains relatively to the motion of the equinoxes, and to the assumed values of the masses of the planets. [4365'] [4366] [4367'] [4365a] * (2637) Putting ^=—5754, we have < ..32'-,G45 == 52''10"'39'; £ . 1 4%1 15 = 22'' 33"' SS'- ; t . 50»V396 = 80'' 32"" 59» ; substituting these in [4359], we get the value of -.j^' [4365]. t (2638) Substituting «"= 98''37"'16^- [4081], in [4331], it becomes, [4366a] T^" = 98''37'" 16" + i . 1 1',949588 + t^. 0^,000079522 ; and by putting ^ = — 5754, it is reduced to 98''37'"16'— 19''5'"58'+43"' 53"=80"50'"1I', as in [4366]. J (2639) Taking, for the fixed point, the equinox of 1750 ; the longitude of the moveable equinox, and of the solar apogee, corresponding to the year 4004 before Christ, [4367a] will be respectively 79'' 4™ 4'' , 80''15'"lp- [4365,4366]; the difference of these quantities j^rfj^jm-js represents the distance of the perigee from the equinox at that time. The [4367t] distance of these points, in the year 1750, was 98'' 37'" 16' [4081] ; so that in the period of 5754 years, they have approached towards each other, by the quantity. \l.x.^31.J THEORY OF THE EARTH. 267 Another remarkable astronomical epoch, is that when the greater axis of the Another romarka- eurth^s orbit, was perpendicular to the line of equinoxes ; for then the apparent [4367"] ana mean solstices were united. This second epoch is much nearer to our '"''<^°"'° [4368] times; it goes back nearly to the year 1250. For if we suppose t = — 500, eq: uinox and sun's the preceding formulas give 90'' 1 '",* for the longitude of the sun's apogee, [4368'] counted from the moveable equinox. Hence the time when this longitude diSr'° was 90'', corresponds very nearly to the beginning of the year 1249. The imperfections of the elements used in this calculation, leaves an uncertainty of at least one year in this result. [4309] 98''37'«16'— I'' 11"'7»= 97''26"'9^; [4367c] being at the rate of about 61* in a year ; and at this rate, the arc fll"?" will be [4367d] described in about 69 years ; so that the equinox and solar apogee must have coincided about the year 4004 -[- 69 = 4073 [4367'J before the Christian era, according to the data we have used. [4368a] * (2640) In the year 1250, we have ^ = 1250 — 1750 = — 500 ; and for this value of ^ we get, from [4359, 4366a], 4,'= — 6'' 57"'; z;i" = 96''5S'"; therefore the solar apogee, in 1250, was distant from the equinox of that time, by the quantity 96<i 58"" — Q^ 57"" = 90'^ 1 "■ ; [43686] and as the distance of these points, in 1750, was 98''37'"16'" [43676], the variation of distance, in five hundred years, is 98''37"' 16' — 90'' ]"■= 8''36"' 16^ being about 61" in a [4368c] year, as in [4367rf] ; consequently, the distance of these points must have been 90^, about one year before the year 1250, or in the year 1249. 268 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. CHAPTER XI. THEOIIY OF MARS. 32. We have, in the case of the maximum * of d V", [4370] 6 0. = — (1 —o?).ôY"'; r" [4370] supposing — = a. If we consider r'" as the onlj variable quantity in a, we shall have, ma [4371] 5 r"' = — . ( 1 — a=) . 6 V". [4371'] If we take for r", r'", the mean distances of the earth and Mars from ^S' the sun [4079], and suppose 5 V" = ± 1" = ± 0^324, we shall get, [4372] 5 r'" _ ± 0,000002076 ; may be neglected, therefore we may neglect the inequalities of the radius vector r"', whose coefficients are less than ± 0,000002. We shall also neglect the inequalities of the motion in Mars in longitude, which are less than a quarter of a centesimal second, or 0',081.t * (2641) The earth is situated, relatively to Venus, in the same manner as Mars is, r4'î70 1 relatively to the earth ; therefore we may obtain 5 V"', corresponding to Mars [4370], from the calculation made for Venus in [4297, 4298], by merely changing the accents on V, in [4298], which makes it become as in [4370], and using a [4370']. Now the ôr"'.r" variation of a [4370'], considering a, r'", as the variable quantities in ôa.= ^^tj— 5 substituting this in [4370], we get [4371]; and by putting r"=a", r"'=a" [4079], using also a [4159], .5 V" [4371'] ; it becomes as in [4372]. [4373a] * (2642) The values [4373,4374] are computed from the functions [4277a, 6], accenting the symbols so as to conform to the present example. VI. xi. §32.] THEORY OF MARS. 269 Inequalities of Mars, independent of the excentricilies. 6v"'^ (l+(^')- + 0+O + (l+f^'') 0',208754 . sin. (n't — n'" t + s— e'") - 0',024915 . s\n.2 (n't — n'" t + -'— s'") j _ 0',005000 . sin. S (n't — n'" t + t'— ^"') ( _ 0',001368 . sin.4(n'i — n!" t + t— O 6',988832 . sin. (n" t — n'" t + £"— O — 0',968689 . sin. 2(n"i — n'" < + s"— O — 0', 1830 12 . sin.3(n" t — «'"f + s"— s'") — 0',058242 . sin.4(n" t — n'" t + s"— i'"') — 0',023099 . sin. 5 (n" t — n'" t + s"- s'") — 0%010339 . sin. 6 (n" i — n'" t + s"_ /") — 0',004992 . sin. 7 (n" t — n"'t + «"— ^"') 24S440843 . sin. (n'' t — «'" < + e'"— Z") ~ — 13',598063 . sin.2(n-'' t — n'" t + e'"— s'") — r,l 80288 . sm.S(n''t — n'" t + £'"— s'") — 0%172768. sin.4(n'''ï _ n'" t + e-— e'") — 0',033166 . sin.5(n''' t — n"'t + s"— s'") — 0'-,013422 . sin. 6 (n'" i — n!" t + s''— i'") [4373] Inequali- ties inde- pendent of the excen- tricilies. + (l+(xO. P,343754 . sin. (n't- 0^,443668. sin. 2 (n^r 0%023088.sin. 3(n''« 0\001879.sin.4(n"r n"'t + 5"— e"') .n"7 + £"—£'") n"'t + ^-—s"') ■ ^"7 + 5"— £'") 4r"'= (!+,.')• 0,0000016104 + 0,0000021 947 . cos. (n' t — n"'t + e'— e'") + 0,0000001972 . cos. 2(n'i — n"'t + e'— e'") + 0,000000041 8 . cos. 2 (n't — n"'t + s'— e'") [4374] VOL. III. 68 270 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. Inequali- ties inde- pendent Oi" the ex- centrici- tiea. [4374] + (i+0 0,0000023860 — 0,0000187564, + 0,0000052387. + 0,0000011969. + 0,0000004169. + 0,0000001733. + 0,0000000796. COS. (n"t- COS. 2(?^"^ COS. 3(?^"^ COS. 4(n"r COS. 5(n"f ■ COS. G(n"t ■ Inequali- ties de- pending on the first power of the excen- tricilies. [4375] + (l+f^'0 ^—0,0000066174 + 0,0000784371 — 0,0000679436 — 0,0000069390 — 0,0000010930 — 0,0000002004 , . COS. (n'^t • .cos.2(n'7 .cos.3(nH .cos.4(n'7- . cos. 5 (n'7 • — 0,0000000520 . cos. 6 (n'^ - if't + ^'—f) ■ n"'t + i"—i") .n"'t-\-i"—s"') .n"'t + s"—s"') .n"'t + i"—s"') n"'t + s"— £'") . ,t"7 + £■"— s'") - n"7 + £"— s'") -?i"7-|-e"— /") .«'"ï + ii"' — s'") .n"7 + s»_/") + 0+^^^) — 0,0000003173 + 0,0000047062 -0,0000023275 ' — 0,0000001399 -0,0000000125 .cos. (tft- .cos.2(nU- .COS.3(M^^ . cos.4(w''^ . n"'t + 1"— .n"'t + s"— £'") £'") n Inequalities depending on the first power of the excentricities. ^ 1 ',082545 . sill. (2 /*'" ? —n't + 2 ='"— s' — ^"') \ — 0S252586 . sin. (2 n'" i — w' ï + 2 s'" — .-'_ z=') 0%698649 . sin. («,'7 + e" — ^"') — 0', 134530 . sin. (2 n'7 — 7i"'t + 2 s" — /" — t;^'") — 10', 1 14699 . sin. (2 n"'f — n"t + 2 s'" — -=" — ^"') - 5% 1 23062 . sin. (2 n"7 — ?z"/ + 2 s'" — .=" — ^/' ) + (1 + p.") . ( — 6^516275 . sin. (3 n"'t — 2 jz"i + 3 /" — 2 ." — 3.'") + 0',846004 . sin. (3 tft — 2 n"t + 3 s"' — 2 =" — ^" ) + 0',677748 . sin. (4 n"'t — 3 n"t + 4 s'" — 3 ^" — ^"') — 0',0791.55 . sin. (4 n"'t — 3 n"^ + 4 s'" — 3s" — t.") , + 0',1 19926 . sin. (5 n"'t — 4 n"t + 5 /" — 4e" — ^j'")^ VI. xi. §32.] THEORY OF MARS. 271 + (1 + f^'^) + (t+t^')- ,+ 5',490297 . sin. (irt + s'"— î^'") * — 5',367005 . sin. (n'^l + ^" — ^'0 -23\552332 . sin. (2 «'^7 — n"'t + 2 5'"— s'"— ^"') ■ 2%593100 . sin. (2 w'7 — m'"^ + 2 s"— h'"— ^'') + 2%296703 . sin. (3 n" « — 2 «'" ^ + 3 s- — 2 s'"— ^"') — 3%568875 . sin. (3 n" t — 2 n"'t + 3 e^' — 2 e'"— ^"') + 0%220149 . sin. (4> n'" t — 3 n'" t + 4> s'" — 3 s'" — vs'") ' — 0%352640 . sin. (4 n'" i — 3 n"'t + 4 e" — 3 s"' — a'") - 2S868651 . sin. (2n"'t — n"i + 2 e'"— e'^— ..'") — 0',204519 . sin. (2 n"'t — 71'" t + 2 s'"— s- — ^") + r,853159 . sin. {3n"'t — 2n'''t + 3s"'—2 e"— ^"') + 0',198136 . sin. (4> 71'" i — 3 n" t + 4> s'" — 3 s'" — z^'") I 0-, 143758 . sin. {n't + f" — ^"') — 0',696926 . sin. (Wf + £" — a^) — r,798071 . sin. (2 n-t — h'" i + 2 s' — /" — ^"') + 0',132176 . sin. (2 nU — n'" / + 2 s' — s'" — z^^) — 0',100246 . sin. (3 n't — 2 71'" t + 3 £" — 2 .'" — ^') — 0', 156784 . sin. (2 n"'t — rft + 2 £'" — £' — ^3'") \ â [4375] Inequali- lies de- pending on the fijsx. power of the exceti- tricities. C 0,0000044700 . cos. (2 n"' t—n't + 2^" — ^ — ^"') ^ W=(\ + (^')- ^_ 0,00000097 13 . cos. (2/1'"^ — ^'^ + 2.'" — e' — ^ ) ^ '—0,0000022865 . cos. (71" t + s" — ^"') + 0,0000086337 . cos. (2 n"7 — ?t"/ + 2 e'"— /' — ^"') - 0,0000031269 . cos. (2 n"'i — n"^ + 2 e"'— s" — .," ) 4- (1 + f.") . y _ 0,0000200331 . cos. (3 n"'t — 2n"t + 3 e'"— 2 s"— ra'") + 0,0000025454 . cos. (3 n"'t — 2n"t + 3 e'"— 2 h"— ^" ) + 0,0000030863 . cos. (4 7i"'t — 3 7i"t + 4 «"'— 3 s"— ^"') + 0,0000040239 . cos. (4 71'" t — 3n"t + 4 e"'— 3 s"— ^") , [4376] * (264.3) The computation of the terms [4.375, 4376], is made in the same manner as for Mercury, in [4278a] ; accenting the symbols so as to conform to the case under [4375a] consideration. [4376] 272 PERTURBATIONS OF THE PLANETS ; [Méc. Cél 0,0000035825 . cos. (n"'t + s'"—^'") — 0,0000107986 . cos. (n'7 + s'"— ^"') + 0,0000031431 . COS. (n'H + i"— ^"•) |_ 0,0000599470 . cos. (2 n'"/ — n"'t + 2 =-"— /"— ^"') _j_ n 4- ^ivN ^ 7+ 0,0000069892 . cos. (2 /t'^/ — n"'i + 2 £■'— s'"— «") ^+ 0,0000114352 . cos. (3 7rt _ 2n"7 + 36"'— 25"'— ^"') -0,0000169741 . cos. (3 n''^ — 2n"'/ + 3£"'— 2^'"- ^'^) ' '—0,0000020307. cos, (4 w'7 — 3n"'i4-4s'''— S^'"— ^'') + 0,0000087307 . cos. (2 n"'t — ?i"7 + 2 s'"— s'"— ^"') y_ 0,00^0063983 . cos. (3 n"7 — 2ra'''i + 3s"'— 2£'^— ^"') — (1 + M-') . 0,0000061906 . COS. (2 n' f — n"'t + 26" — £'" — ^"'). Inequalities depending on the squares and products of the excentricities and inclinations of the orbits.* iv'" = _ (1 + ^') . 6',899619 . sin. {3n"'t — n't + 3/" — e' + 65''26'"15') ( l',414532 . sin. (3 n"'t — n"t + 3^'" — ^' + 73"! l'"55') , Inequali- \ / ^'-„°f '"<= — (1 + f/-") . J + 4\370903 . sin. (4 n't — 2 n'7 + 4 Z"- 2 s" + 67^49"' 0') \ order. ) ( (+ 2^665900 . sin. (5 n"'t — 3 n"t + 5 f'"- 3 /' + 68''23'"00) t4377] /_ 0',462779 . sin. {n''t + n"7 + ^'^ + ^"' — 53' 07'" 48') " + (1 -f ^iv^) _ ^ _ i.<,444i22 . sin. (2 n'7 + 2.- + 60^ 07"' 02') + r,295408 . sin. {n''t —7i"'t + s'" — s"'+ 54'' 41"' 32^ ' ïl' * (2644) Using the values [4076A], we get vei-y nearly, 3 n'" — n' = — 12° = 18' n [4377a] also .3 ?i"' — »i"=238°, which is nearly equal to ii"' ; 4n"—2n" = 5l°=~; [43776] ^''"' — 3n"= — 137'^= — — - nearly. Hence it is evident, that if we proceed in the same manner as in the computation of the similar inequalities of Mercury [4282a, &.c. ], we must notice the angles depending on these coefficients, in computing the terms of [4377 — 4380]. For the second of these angles comes under the form [3732], [4377c] { ?i" -|- (2 — i) . n" = n'", supposing i = — 1 ; and the others under the form [3733], supposing successively, i = — 1, i = — 2, i = — 3. Lastly, as n" is small in VI. xi. §32.] THEORY OF MARS. 273 The last of these expressions may be connected with the following inequality, computed in [4373], and which is independent of the excentricities, (1 + (.'^■) . 24-,440843 . sin. (71" t — n"'t + e-' _ s'") ; [4378] their sum, by reduction,* gives the following term of ov'", ^ ^■" = (1 + ;V') . 25%211710 . sin. (n'^t— n'" t + £'"— s"'+ 2'^ 24"'110. [43^^] We have also, <5 r'" = — (1 + f^') • 0,0000023461 . cos. (3 ir!"t — n't + 3/"— 3' + 64''47'" 29^ îp^u.. second order. [4380] 0,0000050403 . cos. (3 n"'t — ii't + 3 s'"— 3" + 72" 47'" 00') + (1 + (.") . j +0,0000070248. cos. (4 n"7 — 2«"i+46"'— 2-="— oB^ol'-'oO') - 0,0000075032 . cos. (5 n"'t —3n"t+ 5 s"'—3^"— 68" 27'"280 ^ ( +0,0000080002 . cos. (2 n^'t + 2 s'^ + 60'' 1 T" 52^) ) ^ ^+0,0000041488.cos.(?t'7 — «"'i + s'" — 5"' + 59''8'"570^ The last of these quantities may be connected with the following inequality, which is independent of the excentricities [4374], (1 + fx") . 0,0000784371 . cos. {n"t — n'" ^ + s'" — £'" ) ; [4381] their sura gives the following term of 6 r'", &r"'= (1 + (^'') . 0,0000806432 . cos. {n"t — n"'t + £'" — £'" + 2''31'"550. [4382] The inequalities of the motion of Mars, in latitude, are hardly sensible. comparison with n'", their sum w"' + «"', is very nearly equal to n'", so that this angle comes under the form [3732] in"-\-{2 — i).n"', supposing i^l; and [4377rf] produces the term of [4377], depending on the angle ?«"'< + n"'t. If we suppose i^2, in the same expression [4377(/], it becomes 2(i'*'; now, as this is small in comparison with n'", it comes under the form [37.3.3], and produces the terms of [437T, 4380], depending on the angle 2n"t. The quantity n'" — n'" differs but little from — n"' , and comes under the first form [3732], depending on the angle n'H — n"t [4377, 4380]. * (2645) The term ( I + (x-) . 24',440843 . sin. («''' t — n'" i + s'^' — s'") [4373] may be added to the term (1 + |x'^) . l-,295403 .sin. (m'>7 — ?r< + e-— s"'+ 51'' 41"'32^); and the sum reduced to one single term [4379], by a calculation similar to that in [43o0a] [4282^ — /]. In like manner the terms of [4374, 4330], depending on the angle n^" t — r\!"t, may be reduced to one single term of the form [4382]. VOL. III. 69 274 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. [4383] [43840] [43846] Putting n" equal to the longitude of the ascending node of Jupiter's orbit upon that of Mars, we find,* ( 0',094394 . sin. («-"^ + e'" — n") ) [4384] dr=n+^").^ y ^ J f ^ ^ ^0%403269.sin.(2n''i — n"'i + 2£"'— s-'—n'")^ * (2646) The term of 5 s'", depending on the attraction of Jupiter, maybe derived from the formula [4295è], by adding two accents to the quantities s', a', n', s', a", n", s", m"; also supposing y to represent the incUnation, and II the longitude of the node of Jupiter's orbit upon that of Mars [4295c]. The terra independent of 2 produces the first term of [4.384], and the term under the sign 2, corresponding to i = 2, gives the second term ; using -B^''=-^3 -K [1006,4190]. VI.xu.§33.] THEORY OF JUPITER. 275 CHAPTER XII. THEORY OF JUPITER. 33. The reciprocal action of the planets, upon each other, and upon the sun, is most sensible in the theory of Jupiter and Saturn ; and we shall [4384'] now proceed to show that the greatest inequalities of the planetary system depend on this cause. The equation [4371], 6r"' = y.(l-a=).6V"', r corresponding to Mars, becomes for Jupiter, [4385] 6 r''' = !L . (1 — a') . 6 V". [4386] r If we take for r", r'", the mean distances of the earth and Jupiter from the sun [4079], and suppose 6 V''= ± 1"= =t 0',324, we shall obtain, 6 r" = =F 0,0000409225. [4387] Therefore we may neglect the inequalities of 6 r", which are below q= 0,000041. We shall also omit the inequalities of Jupiter's motion in [4387] longitude, or latitude, which are less than a quarter of a centesimal second, or 0',081. Inequalities of Jupiter, independent of the excentriciiies* Itiequah' ^^ ^^ , „^ ( 0'-, 120833. sin. (n"t — n'^'t + s" —s-)^ [.^^ntr iV = (1 +f^ ) . < > the ex- ^ ^— 0,000086. sin. 2 (n"Z — n'''i + ^" — ^'0) "''"'"' ceotrici- tics. * (2647) The inequalities [43S8, 43S9], are deduced from [4277a, J], increasing by four the accents on the symbols, to conform to the present case, and using the data 276 PERTURBATIONS OF THE PLANETS; [Méc. CéJ. [4388] iaequalî- ties inde- pendent of the excen- tricities. [4389] + (1 + f^O • X + (l+^'0 82',8117n. - 204',406374 . — 17%071564, — 3',926329 , — P,2 10573. — 0',428420 . — 0% 170923, — ,076086 . — 0%041273. P,051737. — 0',427296 . — 0',044085 . — O',005977 , sin. (jCt ■ sin.2(M'^ sin.3(J^^^ sin.4(n'^ sin.5(?i''^- . sin. 6 (ii" t ■ . sin. 1 (n't- . sin. 8{n''t- . sin. 9 (n'' t ■ sin. [n'"t ■ sin.2(n"7 sin.3(n"7- ■ n"t -\- s'—i") ■ n''i + e" — e'^) . n"t -\- b' — e'^) n'^t + 6" — i") n" t + e" — £'") • n^t + b" — £'") .n'''t + ^'-'—é") . n" t + £'■' — i'") n"t + e"'— £") àf — 0,0000620586 + 0,0006768760. COS. {n't— n"t + b^— b") — 0,0028966200 . cos.2(ift — n"t + £'— b") — 0,0003021367 . cos. 3 {nH — n"t + b'— e'") r^ 1 v^ /— 0,0000782514. cos.4(îi"ï — n'7 + s"— e'M - \— 0,0000258952 . cos. 5 {n^'t — n''t + «" — «'") | - 0,0000094779 . cos. 6 (?j" ^ — n''t + £^— s'") | - 0,0000037560. COS. 7 (?i^7 — n'U + £^ — s") | -0,0000014781 .cos.8(»i^^ — ?ri + E"— ^'O ' - 0,0000004799 . cos. 9 (ift — n'-t + «"— '^'O Inequalities depending on the first poiver of the excentricities. Several of these inequalities are of considerable magnitude, so that it becomes necessary to notice the variations of their coefficients ; which we [43886] [4061, &1C.]. The term depending on sin. {n'' t — n'U -{- b" — b'"), being computed, by means of the formula [4277n], is found to be nearly the same as in the first line of this page, and has the same sign ; therefore the remark made in the Philosophical Transactions for 1831, page 65, that the sign of this coefficient is negative, is incorrect. VI.xii.§33.] THEORY OF JUPITER. 277 shall do, in those terms of the expression of 6«" which exceed 100", or 32',4. The coefficients of the inequalities depending on a'', have for a factor the excentricity e" ;* therefore, by putting one of these coefficients Se'" equal to Ae'", its variation will be Ae'". -^. We shall find, in [4407], that if we include even the quantities depending on the square of the disturbing force [4i04,&c.], of which we have given the analytical expression in [3910], we shall have. [4389'] [4369"] * (2648) The terms of 5v'\ Sr'" [4392,4393], were computed from those of ÙV, &r [1021, 1020], depending on e, e' ; changing m, a, e, zs, i, n, into rlfi nictiirnti,„ ni", a", e'", -a'"', e'*', Ji'", respectively. In computing the disturbing force of Saturn, we must also change the symbols m', a', he. into m'', a", he. ; and in computing that of Uranus, we must change them Into rri", «", &c. We shall neglect the terms containing the arc of circle nt, without the signs oi sine and cosine, as is done in [1023, 1024]. In this notation, the angle ra", is evidently connected with a coefficient having the factor e'''; and the angle -a^, with the factor c" ; as in [4389', 4390']. The variations of c'", e", are given in [4407] ; and if we retain only the first power of the time t, they will be as in [4390, 4391]. For an example of the method of computing these variations, we shall take the largest term of ôv'" [4392], which arises from the substitution of the value of i = 2, in the term multiplied by e, or c" [1021] ; so that this term becomes. Substituting the values of the elements [4061,4077,4081], and that of jF® deduced from F'*' [1019], we find that the coefficient becomes, as in [4392], — 13S',373337 = A e'" [4389']. ôe This is to be multiplied by —, to obtain the expression A (5 e'". Now, i5 6'"= t . 0',329487 [4390], being divided by the radius in seconds 206265', becomes, 5 e''-=^. 0,00000 15974; dividing this by e'"' [4080J, we get, — = t. 0,0000.33226 ; multiplying this by Ae" [4390/], we finally obtain, ^JÉi^ = — <.0',004598. Connecting this with Ae'" [4390/], we get the coefficient of the term depending on the angle 2 n'i— n'^< -f 2e^— e'" — ra'' [4.392]. In the same way the variations of three of VOL. III. 70 [4390al [43906] [4390fJ [4390(f] [4390e] [4390/] [4390g] [4390/. J [4390t'] 278 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. [4390] W" ^t. 0',329487. In like manner, the coefficients of the inequalities depending on ^j", have [4390] the factor e" ; and by putting B e"" for one of the coefficients, its variation will be B e". ^^, and we shall find, as in [4407], that [4391] 6"= —t. 0%642968. This being premised, we obtain, Inequali- ties de- pending on tlie first power of the excen- tricities. 6D-=(1 + /J.'') [4392] 8',608489 . sin. — 9',692385 . sin. — (138',373337 + ï + (56S634099— i — (44',460822 + i + (84',942569-i + 7%925312.sin. — 15',629621 .sin. + r,047717 .sin. — 2',78 1664. sin. / + 0',407251 . sin. — 0,9 13302. sin. + 0', 149277. sin. — 0',325592 . sin. — 5',208122.sin. — 0%569738 . sin. + 12',876650.sin. — 0',352399 . sin. + r,287482 . sin. — 0% 172892. sin. + 0-,356627 . sin. — 0',083]89.sin. (n^'t + E^ — si'') . 0%003 1 398) . sin. (2 n^ i — n'" i + 2 £^ — e'^ — ^^ ) .0%0014776).sin.(3n''i— 2n'^i+3i''— 2>— ^") . 0^0047094) . sin. (3n'' «— 2n"^+33''— 2.'"— ^' ) — 3 n'" i -f 4 £'' — 3 5'" — zi" — 3 ?i'' ; 4- 4 £" — 3 1" — ra" — 4 n'^f + 5 £' — 4 s" — zi'" — 4 n'' « + 5 6^ — 4 s'" — zj" — ôn'''t + 6i'—ôs'"'-~-u," — 5n*'i + 6s^ — ô^'" — T.^ — 6n'"^ + 7£'- — 6 e'"— 33'" — 6 irt + 7 £" 6 s'" — -n" — rft +2 £'" — £" — ra'" — nU + 2 s'" — £" — ra" — 2 71^1 +3 i"— 2 6'— z=" — 2 71^1 + S s'"— 2 £" — 73' — 3 71"^ + 4 s'"— Si" — zs'" — 3 ?ri + 4 s'"— 3 £' — ^" 4 7ft 4- 5 t-'" 4 s' — z:'" — 4>7i''t 4- 5 £"— 4 ^^" — i^' (4 71^1 (4^^ (ôn^t- (ÔTl't (6 71" t (6 n" t (In^t ÇlrCt (2 n'H (2 7l"t (3 n'7 (3 n"« (4 n'H (4 7l"t (5 Tl'H (5 71" t the other large terms of [4392] are computed. The variations of the remaining ones are too small to be noticed. VI.xii.<^33.] THEORY OF JUPITER. 279 (1+^") .ir"=(l+,.-) 0% 123506 . sin. (n"< + s^' — x.") - 0',235240 . sin. (ti'U + i"' — ^") - 0',533079 . sin. (2 n"' t — irt + 2 1" — s" — ^") + 0', 102673 . sin. (2 ?r'/ — n'" t + 2 s"' — i" — a") — 0'-, 127963 . sin. (3 n'H —2 n"i+ 3 s" — 2i"— i^''') 0,00002061 1 1 . COS. {n'H + s'"— a") — 0,0000795246 . cos. {n" 1 + ^' — ra'") + 0,0000492096 . cos. (n" i + s" — «^ ) — 0,0002922 130. COS. (2 H'i— n''i + 2E''— e"— ^i' 1+ 0,000 1688085. COS. (2/1" Ï— n"t + 2t' — 'Z^"—^-' — 0,0004584483 . cos. (3 n^i — 2n'''i+ 3 £"— 2e"— x^" + 0,0009047822. cos. (3 n'i — 2 n''7+ 35"— 26"— ^" j-l- 0,000 1 259429 . cos. (4 n' ^ — 3 iVH + 4 £" — 3 s'"— t^" /— 0,00024244 1 3 . cos . (4 n'' i — 3 n"7 + 4 s" — 3 s'"— a" + 0,0000268383 . cos. (5 n' t — ^n"t + 5 s"— 4s"— ^" — 0,0000516048. COS. (5/1"^ — 4n"i -foe" — 4s''— ^^ + 0,0000579151 . COS. (2n''t— nU-^2s"'— I' — zs" I— 0,0001346530 . cos. (5 n'H — 2ift + 3 e'"— 2 s" — ^'^ Inequali- ties de- pending on the first power of the excen- tricitiei). [4393] Inequalities depending on the squares and products of the excentricities and inclinations* l',003681 . sin. (n"ï + n'^7 + e" +£"' + 45''29"220 — 5',578707 . sin. (2 nTt + 2s" + 15' 56" 24^ + ir,724245 . sin. (3 ri't — n"t + 3^" — e'"+ 79-^ 39" 48^ —1 8',075283 . sin. (4 n' ^ — 2 n}H + 4 e"— 2 e'"— 57" 1 2" 26') '"■•=('+-')V(,69s266896-<.0S004277).si„.(^;i-4»-|+3-^^^^^^^ + P,647140.sin. (6 n"f — 4n"< + 6 s"_4£'"— 64''25'"480 + 2'-,47 6404. sin. ( n't— n'"i-|- e"— £'"4-43^17'"0P) V — 6',287997 . sin. (2 n" i — 2 n'" i + 2 £"— 2 e" + 42 '4O'"440 Inequali- tiei of the second order. [4394] * (2649) The calculation of the six first terms in [4394] is made in exactly the same way as for Mercury, in [4282a — 6]. The coefficient of the angle 3 n''< — 5 rCt, being [4394a ] 280 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. These two last inequalities being connected with the two following, [4395] ( 82',8n711.sin. (n^ï — «'" ^ + s'— £'") ^ _ *" ^ ■ ^ _204',406374 . sin. 2 (n't — n'" t -\- b' — s'") ^ ' which are found in [4388], among the terms independent of the excentricities, produce the terms,* (_ 84',628936.sin. (?ri— n^i + s'" — s" — 1''08"'530 ) ^ ^ (+209',098224.sin. (2?^'"^— 2n^'i+2£"'— 2 e''— r'09"'58^)^ Then we have [4394^/], loequali- sic'oôS""' / 0,0000822415 . cos. (2 w^i + 2 s" + 1 1'OO"' 550 order. I ^ + 0,0000226252. cos. (3 n-t—n'^t + 3 s"— s''— 21''47"'18') ^4397^ 6r''=^(l + f^'').( —0,0001010533. cos.(4?i^^—2n'"«+4î''— 2=-''— 51''04"'040 -(0,0021 1 14502-^0,00000005323).cos.(^J;l-!J;;2o;:^ — 0,0000652204. cos. (2 ?i'7— 2rt'''<+2 e'— 2 .'"+ 54''08"'52') If we connect the last of these inequalities with the following, — (1 +f^") . 0,0028966200. cos. 2(n"i — iV't + i' — i") ; which is found in [4389], among the terms that are independent of the excentricities, we obtain the equivalent expression, [4399J 6 r"= _ (1 -|- j.^) . 0,0029251 892 . cos. (2 7i'H — 2 n''t + 2 i"—2 e^'— 1''02"'080. The preceding inequalities of iv'", are calculated by the formulas [3711, 3715, 3728, 3729] ; excepting, however, that which depends on the angle [4400] 3^'"^ — brft\ observing that bn" — 2n", is a very small coefficient, as appears from the ratio which obtains between the mean motions of Jupiter large, its variations must be noticed and computed by the metiiod pointed out in [43946] [4017 — 4021]. The other coeniclents are less than 32%4, and their variations are neglected, as in [4389', he.]. The two last terms of [4394] correspond to [3729, 3728] ; [4394c] using i=^±l, or i^±2; the values of A" being found, by means of the formulas [3753 — 3755'"], and the corresponding terms are to be connected together, like those depending on M, in [4282A — /]. In like manner, the four first terms of [4397] are [4394dJ deduced from [3711]; the last term from [3728]; noticing always the variations of the elements in the greatest coefficients, as is done with the terms of & v. [4396o] * (2650) This computation is made in the usual manner, as in [4380a]. [4398] VI.xii.§33.] THEORY OF JUPITER. 281 [4400'] [4401] and Saturn [4076/t] ; so that the angle Sn'^t — ôn^t differs but very little from n''f, as in [3712, &c.] ; in consequence of which, we have used the formulas [3714, 3715], in computing this inequality, by the method given in [4017 — 4021]. Inequalities depending on the poicers and products of three and Jive dimensions of the excentricities and inclinations of the orbits, and on the square of the disturbing force. The great inequality of Jupiter, is calculated by the formulas [3809—3868; 3910—4027]. We find, from [3836—3841], a\ ilfw ^ — 5,2439100 . m' ; a\M")= 9,6074688 . m\- a\ M<-^> = — 5,8070750 . nf ; a\M'-^^= 1,1620283. m^ a\ itfw = — 0,6385781 . m" ; a\ M' '' -= 0,3320740 . m\ mequaii^ ' tics of the Hence we find, at the epoch 1750,* orJ"- a\P= 0,0001093026; a\ P' = — 0,0010230972. [4402] We must find the values of the same quantities in 2250 and 2750. For this purpose it is necessary to determine the values of e'", e", t^'", w*', j, n, in series, ascending according to the powers of the time ; continuing the series so far as to include the second power of t. We must, in the first ^ ^ place, calculate, by the formulas [3910 — 3924], the secular variations of e'% ie", 5ra'', Szs", depending on the square of the disturbing force ; and we shall obtain, for these variations,! * (2651) The values of «'P, «^P' [4402], are deduced from [3842, 3843] ; adding four accents to the letters m, a, e, zs, m , a', e , Stc. to conform to tlie present [4402a] notation, and then using the numerical values [4061, 4077,4079, 4080, fcc.]. t (2652) The value of ^e'" [4403], is computed from the part of (S e [3910], depending on the time t, without the signs of sine and cosine ; adding four accents to the letters m, a, e, m', u', e , &:c. to conform to the case now under consideration. izi'" [4403], is VOL. III. 71 [4403a] 282 PERTURBATIONS OF THE PLANETS ; [Méc. Cél, [4403] [4404] [4405] àé" = t. 0%052278 ; <! ^" = t. 0^352941 ; ie' = — i.0%102763; 61^"= t. 3',242722. The coefficients of t, in these expressions, represent the parts of — , — ^ , — , — [4404a, 6, c], depending on the square of the disturbing force.* Ut CE t Adding them respectively to the parts of the same quantities, determined in [4246, 4247], we obtain the entire values in 1750, ^^ = 0%329487 : '^= 6',952808; dt ' ^ = _ 0',642968 ; dt ' ^-^ = 19%355448. dt ' obtained from the like parts of &-a [3911]. The expressions 6 e'-', (5 a" [440-3], are deduced from [3922, 3923], by making the same additional number of accents to the letters, and then substituting the values of these elements [4061, 4077,4079, &ic.]. * (2653) We have, as in [4330rt], e'" ==; e'" + ^ . ~ + J t^. ^ ; e" in the second member, being the value of e'', at the epoch ; and by putting for e'* — e"', its value ^e'", we get, [44046] 5e-=<.— + 1^^^^. In like manner we have, [4404c] 6e^ = t.^+it^'—+^o.; 6.^ =t . - + if^.^ + ^o. The coefficients of t, \ t^, in the second members of these expressions, correspond to the epoch. The coefficients of the first power of t, in these expressions, are composed of two parts, namely, those computed in [4246,4247], and those depending on the square of the [4404rf] disturbing masses, computed in [4403] ; the sums of the corresponding parts give the coefficients, respectively, as in [4405]. Thus, ^ = 0%052278 + Jx0%554418 = 0V329487, Sic. as in [4405]. at [4404a] VI. xii. <^33.] THEORY OF JUPITER. 283 We obtain, by the same method, their values in 1950, and find, at this epoch,* ^= 0',326172; dt IT de" It = 7',053178; = _ 0%648499 ; ~— = 19^424739. dt From these we get, as in [3850, &c. 3850c], the following expressions of e'*, îj'", e", w' ; for any time whatever ; e'* = €'■' + t. 0%329487 — f: 0^,0000082871 z," = I.- + / . 6^952808 + t-. ,0002509259 e' =€" — t. 0%642968 — f: 0%0000138275 ^" =^" +1. 19--,355448 + t\ 0',0001 732274 : the values of é", to'% e", «% m the second members of these equations, correspond to the year 1750. [4406] General values of TO", -m". [4407] [4407'] * (2654) The calculation of the annual variations of the elements [4406], for the year 1950 is made in the same manner as in [4405], using the expressions of e'^, e", «'", -a", [4406a] corresponding to 1950. These elements are obtained, very nearly, by means of the annual decrements [4405], which give, with sufficient accuracy, the required values, when t does not exceed 200. Thus the increment of e'*', corresponding to t = 200, is 200X0',329487=:65',8 nearly [4405]; |-4406J] being the same as the term depending on the first power of t, in the expression of e'» [4407]. The term depending on i^, in this last expression, is very small, being represented by — 2002 X 0^000008287 1 = — 0»-,3 nearly ; [4406e] which is about -^^sxs part of the term corresponding to the first power of t. Similar remarks maybe made relative to the values of t" , ra", ■a'' . If these calculations were to be repeated, in consequence of any changes in the assumed values of the masses of the planets, we could take into consideration the parts depending on t'^, as they are given in [4407] ; and by this [4406rf] means we might obtain, by successive operations, corrected values of the elements. This process is the same as that so frequently used by astronomers, in re-touching and correcting the elements of the orbits of the heavenly bodies. Now, from [3850c], we have. ddé" îiTT'^ in which we must substitute dé" for -—', its value 0',326172 [4406] ; also for — - , its value 0',329487 [4405] ; hence [4406f] 284 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. We may find the values of y, n, by means of the equations,* 7 . sin. n = (p" . sin. ff' — <?'"• sin. o'" ; ^^^°®^ y . cos.n = (f" . COS. r — 9'\ cos. ê'\ d y du Then we compute the values of -—, —, by taking the differentials of J , • • r dq,'" dtp" dù'^ dè^ . . . [4408] these equations, and substitutmg tor — — , - — , —— , —, then* values [4246, 4247]. We find, in this manner, in 1750, 7= IMô-'SO'; n = 125"^ 44"' 34'; ^''''^ Ç = - O',000106; dt ~=~26%09U33. dt The formulas [3935, 3936] give, for the secular variations of y and n, depending on the square of the disturbing force, ij= i.0',000184; in = — ^.0',00763. If we add the coefficients of t, in these equations, to those in the preceding d y d TÏ. values of — , — [4409], we obtain, for the complete values of these quantities in 1750, [4410] rfrffi' 0',003315 „„ ^^^„ „ , . . ,. , ^ ddei^ [4406/"] we eet ,r77Tr = TTJ^^ — = — O',0000082b / . Substituting this value of -— -„, and ■' 4di'^ 40U 2dt~ d c'^ that of -7- [4405], in (:"■ [4404^], we get the first of the equations [4407]. The values [440tig-j ^^ ^,v^ gv^ ^v^ g,,g ]om;|f| |„ ^i]g same manner, changing e" [4404«,4406e], successively, into w", e", -a", and using the values [4405,4406]. [4409a] [44096] [4409c] * (2655) Tlie equations [4408] are similar to those in [4282o], adding four accents to ç), è, cp', ê', to conform to the present case ; and changing tang.ç)"', tang. 9", into 9", ç", respectively, on account of their smallness. In this case 7 [3739] represents the langent of the inclination, or very nearly the inclination itself, of the orbit of Saturn to that of Jupiter ; and n [3746], the longitude of the ascending node of the orbit of Saturn upon that of Jupiter. Substituting in [4408] the values of 9'% é'\ ç\ é-, [4082, 4083], we get 7, n [4409]. Tlien taking the differentials of [4408], and substituting the preceding values of ç)*", è", kc. ; also those of do'", de'", df, do" [4246, 4247], we get the two last equations [4409], by making a few reductions. VI.xii.§33.] THEORY OF JUPITER. 285 '^ = 0',000078 • (It V-=-26',101764. dt We find, by the same process, in 1950, - 0',001487; dy ~dt ~~ V^=— 26',402056. dt Hence we obtain, by the method in [3850 — 3853], for any time whatever t* inciininion ^ , and y = y-^t. O',000078 — t\ 0',000003913 ; [4413] n = n— ^ . 26', 101 764 — t-. 0',000750731. [4413] longitude The values of y, n, in the second memhers of these equations, correspond to n, of the 1 750. This being premised, we find in 2250, f ascending niideofiho a\P = — 0,0000801 89 ; '^;:^ [4414] «'. F =—0,001006510; J.H,.„f and in 2750, a\ P = — 0,000260997 ; «^P'=— 0,000954603- Jupiter. [4415] * (2656) If we change the symbols y, XI [4412], for the year 1950, into y^, IT, respectively, and leave those in [4411], corresponding to the year 1750, without accents, we shall have, as in [4406e], -^^ = jJ^ . $'^ -Jf\^ ^"^ •^'" ^''001487 - 0',000078| =— 0S00000.3913 ; [4413a] also, '^''" =^a^.HlL'_— |=^^^.f_26',402056 + 26',101764} = — 0',00075073. 2^,2 — tU ■ I -jj — TT ( =î*t7- i— i=;o%4U2U5(j + '^b%1017b4| = — 0%00075073. [44136] Substituting these and the values of [4411], in the general expressions of y, n [4404«], namely, we get [4413,4413']; observing that the values, in the second members, correspond to the year 1750. t (2657) The values of a\ P, a'. F', are given in [3842, 3843], in functions of e", e", -a'", ti\ y, n, &c. ; and their values in 1750, have already been given in [4402]. [4414a] VOL. III. 72 286 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. Hence we deduce, by the method of [3850—3856],* a\ ^= — 0,000000387666 ; [4416] a\ '^ =- — 0,000000002145 ; d t a". ^= 0,000000000034734; a\^= 0,000000000141280. The part of 5v'\ given in [4023], is,t [4417] 6v" . „, , ^a'^'.dP 3a'^.ddP' ) \ ( . rfP' , 2a"'.ddP ) , , „ . ddP'i / Cm". ni*-2 ~(5,V-2,i"f\ , 2a>''.dP' Sa'\ddP a'\P (5n^—2n>^).dt i5n^ — 2n"f.dt^ ^ ^ cos.(57iH—2ii'H+5s^ -2;'") (^ . dP 2a <-.ddP ' ) , , 2 i,ddP\ . , ''( ' dt (ân-'—2n").dV2S G root * ^ ^ ' ^ iiidiuiility. This becomes, by reduction to numbers, [4418] 6î;'^=(1263%799671— t . 0',008418 — <2.0'.000019247) . sin. (5 ii''t — 2n"t -{'5s'-— 2 s") 4- ( 1 19%526S51 — t . 0%473686 + i^. 0-,0000:8562) . cos.(5 n't — 2 n"t + 5 s^ — 2 s^'). The great inequality of Jupiter includes several other terms ; thus, it contains, in [3844], the expression,! To obtain a^.P, a\ P' in 2250 [4414] ^ we must put !!= 500, in [4407,4413,44131, [44146] and substitute the corresponding values of e'", -cj'^, Sic. in [3842, 3843]. In lilce manner, by putting t = 1000, we get their vakies in 2750 [4415]. * (2658) The values of «^ P [4402,4414,4415], being substituted, respectively, dP d-P [4416a] foj. p^ p^ Pi,,m [-3856], give the values of n\--, a\— [4416]. In like manner, from a^.P' [4402, 4414,4415], we get the terms depending on the differentials of P' [4410]. t (2659) The formula [44 17], is the same as in [4023] , increasing the accents on the [4418a] elements m, a, c, &c. m', a', e', fcc. by four, to conform to the case under consideration. Substituting in [4417], the values [4402, 44 1 6] , it becomes as in [4418]. r4419al Î (2660) The expression [4419] includes the third and fourth lines of (5 «" [3844], the accents being increased as in the last note. VI. xii.§:33.] fdP 6V"'= — THEORY OF JUPITER. 287 n'V 7M< To reduce it to numbers, we must calculate the values of a"". (-^ ) ; (hW a"-. ( 4-^ )? &;c. ; and we find,* V da" J \da'^\ a''^ /dAP'^' \ '^ «" . «"^ A?.W(3V \ f'"'" iC\ V ^ /f/7»/('>~ \ f^"'" , „v9 ^«/./l/ra- = — 26,46390 . m^ ; = 65,75870 . m" ; = — 50,227 14. m''; = 1 2, 14696. m"; = _ 6,75963. m^; = 4,13173. m^ From these we deduce the values of a^-. ( -— ^- j, a'". ( ^^ )' ^^' ' ^^^"^'' are necessary in the theory of Saturn, by means of the general equation of homogeneous functions [1001a],t da'" J \ d a" [4419] [4420] d /J/^'A /d 1/t'A * (2661) The accents being increased as in [4418«], the formulas [3836 — 3841] give the a'V (2) (3) valuesof a^.W"', a'J)f"\ &c. in terms of a:= — , b,, 6 -,, &.c. and their differentials. Taking the partial differentials of these expressions relative to a'", and substituting the values [44000] (2-) (3) [420-2-4211], we get [4420]. Observing that hp h^, &c. are functions of a [964]. and if we represent any one of them by h, its partial differential, relative to n'^, will be, /■ dh \ fdh\ / do.\ /db\ 1 t (2662) Tlie general values of M^''\ M''\ iW^^', M'^\ M^'', M<5> [3836£/, 3337c, 3838A, 3840A, &ic.], are composed of functions of a'", a", of the forms, 288 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. Hence we find, in 1750,* a"2. (if. V COS. (5 7iV — 2 n'^t + 5 £"— 2 e'") — a"'~.f-—A .sin. (5ft''< — 2?i'''<+5 6^— 2;") [4422] 2 m". n'" 5 Ji"— 2 n'" ' ^ ■ „ /d P'\ [4423] [4424] [4425] = — 1 7%228862 . sin. (5 n'' i— 2n'''< + 5£'— 2 ;'0 + 5',36001 6 . COS. (5 if t — 2 n'' < + 5 s"— 2 r) ; and in 1950, it becomes, — 16',836801 . sin. (5 n" / — 2 n''^ + 5 e^— 2 s") + 6^449839 . cos. (5 n't — 2 n'"^ + 5 £"— 2 e"). Hence we obtain the following value of this function, for any time whatever t, 6 ?)'"=: — (17',228862 — /. 0',001960) . sin. (5 n't — 2 n'^t + b^-—2 i") + (5',360016 +i.0',00o449).cos.(5n^7 — 2n"'i + 5s^ — 2 s"'). all of which are homogeneous, and of the order — 1, in a'^, a" [1001', 1007'] ; i being any integral number. Hence the general value of JJi*'' is also homogeneous, and of the degree — 1, in a", a''; and the formula [lOOlo], by changing A, a, a', m, into M^", a'\ a", —1 becomes as in [4421]. * (266-3) The values of m^n" P, m" a" P', are found as in [4402n], by increasing [4422a] tbe accents of the elements in [3842, 3843] by ybwr. Taking the partial differentials of these expressions, relative to a'", we obtain the values of, [44225] ™'«^-C^)' -^"-63' expressed in functions of a'^, e'", &c- a^, e", &ic. and of the terms [4420]. Substituting these in [4419, or 4422], we get [4423], corresponding to the year 1750. Repeating this calculation, with elements computed for the epoch 1950, it becomes as in [4424] ; observing that the functions [4420], must also be computed and taken for the year 1950. Comparing the numerical coefficients of the terms [4423,4424], we find the increments, in 200 years, to be respectively represented by, — 16',8.36801 + 17^228S62=:0^392061, [4422i] and G^449839 — 5,-360016 = l-',089823. Dividing these by 200, we get the annual increments, or the coefficients of t, as in the general expression of '5 1''' [4425]. VI.xii.§33.] THEORY OF JUPITER. 289 The great inequality of Jupiter [3844] contains also the term,* ôv''' = — h He", sin. (on" t — 2 n"t-{- 5 1" — 2 s'^—zs^" + A); which, in 1750, is equal to, 0^820290 . sin. (5 1^1—2 n" t + 5 b"— 2 i'") — 1%837963 . cos.(5 n't— 2 n" t + 5s'' — 2 s'") ; and in 1950, is, 0%701624 . sin. (bn''t — 2 n" t + 5i'—2 é") — r, 840958 . cos. (5 n't — 2 71" t + 5 s^— 2 e"). Hence we find, that for any time whatever t, this term is represented by, 6 v''= (0',820290 — t . 0',000593) . sin. (5 n't — 2 n'" i + 5 s^— 2 £'') — (1 -,837963 + t . 0^000015) . cos. (5 n'^t — 2 rrt + 5 e^— 2 s''). To determine the part of the great inequality of Jupiter, depending on the products of five dimensions of the excentricities and inclinations of the orbits, we have computed, by the formulas [3860—3860'^], the values of iV*"', N^^\ &c. for the tAvo epochs 1750 and 1950, and have found. In 1750. a\iV('"= 0,00000135044 fl\iVW= 0,00000789719 a\ iV'=' = — 0,0000198552 a\iV(3)= 0,0000175127 a-\ iVH) = — 0,0000066540 a\ iV<^) = 0,0000009277 a\N^'^= 0,0000003618 a\ iV<'' ^ 0,0000003643 a\ A^<«' = — 0,0000001720 a\ iV'^' = 0,0000000730. In 1950. a\ iV<°' = 0,00000129983 fl^7V(')=:3 0,00000754771 a\ iV<"~) = — 0,0000196012 «\iV'="= 0,0000172415 a\ iV'" = — 0,0000066551 a\ N'-'^ = 0,0000009408 a\ N'-'^ = 0,0000003562 a\ iV'" = 0,0000003460 «v,jV(S) ^ — 0,0000001712 [4426] 14427] [4428] [4429] Terms of the fifth order on e, e', 7. [4430] a\ iVW 0,0000000732. * (2664) The term [4426] is the same as that depending on — |/Je [3844], accenting the symbols as in [4402a] . In this case H denotes the coefficient of, 73 VOL. III. [4431] 290 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. By means of these values* we have computed the corresponding inequality in [4430] Saturn, in [4487]. Multiplying it by the factor —~?~r-, we obtain the following inequality of Jupiter,t 6 1,'" = _ (1 2',536393 — t . 0^001 755) . sin. (5 Ji'f — 2 n'" i + 5 £" — 2 s") + (8',120963 + t . 0-,004885) . cos.(5 ift — 2n'''t + 5 £>— 2 s"). Lastly, we have computed, by the method in [4003], that part of the great inequality of Saturn, which depends on the square of the disturbing force, [4431] ^j^ij jj^ q£ ^ sensible magnitude. Then we have deduced from it the corresponding inequality of Jupiter, by multiplying it by 177^ ! which gives, for this last inequality, the following expression,^ [4426a] COS. {5n't—-3nt-\-5s'—3s-\- A), in the expression [3814], corresponding to Jupiter. Computing the value of — | He", for [44266] jj^g ygj^j.g j^j5Q^ jggQ^ ^ jjj [4427,4428], we obtain its annual increment, and the general value [4429]. * (2665) The signs of all the terms in [44-30, 4431], are different in the original work ; [4430a] ^^g j-^^yg clianged them, iii order to correct the mistake in the signs mentioned in [3860«]. t (2666) Changing, in [1208], ,g, <^', into iv", ôv^, which represent, respectively, the corresponding parts of the great inequalities of Jupiter and Saturn, we get, by using the notation of [4402«], [44306] 5«-=_^'^.5v [4430c] Substituting in this, the values w", m", a'', a^ r5 «" [4077, 4079,4487], we get [4431]. [4431o] X (2667) We have already mentioned in [4006^ — 4007rt] the difficulties which occurred in computing this part of the great inequality of Jupiter, and liave also observed, that the numbers given by the author, in [4432], are inaccurate ; the chief coefficient having a wrong sign, as Mr. Pontécoulant found by computing the most important terms, depending on the [44316] arguments contained in the table [4006m], numbered from 1 to 10, and from 1' to 10'. The parts of êv", corresponding to these terms, are given in [4431/], from the abstract, printed by Mr. Pontécoulant, in the Connaissance des Terns, for 1833 ; using, for brevity, the ^^^^^'^ symbol T5 = 5 ift — 2 n'H + 5 s' — 2 i" [3890J] . The first line of the function [443 1/] [4431rf] ig produced by the term 3 ci^ff. {ndt.dll ./d 7?) [5844] ; the other lines arise from the products of the quantities in the table [4006m], marked with the numbers on the same lines Vl.xii. ^^33.] THEORY OF JUPITER. ( ]%6U663 — t . 0%00'l68S) . sm. (5 n't — 271'" t -\- 5 e^ — 21''') — (18',461954+ i.0',001515) . cos.(5 if t — 2 n"t + 5 B''—2r). 291 [4432] respectively. The sum of all these terms is given in [4431^] ; and it differs essentially from that of La Place, in [4432] ; particularly in the term depending on cos. 1\ , which has a difl^rent sign, though it is nearly of t)ie same numerical value ; an error in the sign having been discovered in the original minutes of the numerical calculation of La Place. 5 u" = + 0',02489 . sin. T,, -f- 0',002G6 . cos. T5 1 + 0',08628 . sin. Tj — 0%01857 . cos. T5 1' — 2',00454 . sin. T^ + 0%4375- . cos. T^ 2 + 0',07587 . sin. T5 + (y.OSlQT . cos. T^ 2' + 0',39242 . sin. Tr, + 0,22555 . cos. 7^ 3 + 0%28829 . sin. T5 + 0',19273 . cos. T5 3' — 0%71831 . sin. 7^ — r,5S65S . cos. T^ 4 _ 0',14619 . sin. Tg — 0^09422 . cos. Tg 5 — 0%76290 . sin. T5 + 0',-7529 . cos. Tj 6, 6' + 2%16304 . sin. T5 +16',97139 . cos. T^ 7, i = 2, + 6',G2968 . sin. T5 — 0%80829 . cos. Tg 7, t == 1 , — 2,49438 . sin. T5 — 0,92192 . cos. Tg 8, i = 2, + 0',22613 . sin. Tg — 0^53472 . cos. Tg = 3',76028 . sin. Tg +14'',72286. cos. ïg . In computing these numbers, the mass of Saturn is supposed to be, as in [4061 J], equal to WbT^ ■> instead of t^tî^jï) used by I^a Place [4061]. To compare them with La Place's calculation [4432], given below, in [4431it], we must increase the coefficients [4431^], in the ratio of 3512 to 3359,4 ; by which means they will become as in [4431iJ ; the terms depending on t, t^, being neglected ; 5 V" = 3',93109 . sin. Tg + 15',39164 . cos. Tg ; ôv"= p-,64166 . sin. 7'g— 18',46195 . cos. Tg . The difl'erence of the two expressions [443h', ^], which we shall denote by C'", is a correction, to be applied to the formula [4433 or 4434] ; and we shall have, C =3 2',2S943 . sin. Tg + 3.3',85359 . cos. Tg. We may remark, that the number of terms of the forms 7 to 10, and 7' to 10', [4006m], is infinite ; but it is only necessary to notice a few of them, in which S r, 5 v, â r', or ô v', have sensible values. Moreover, the terms depending on ô i, were not computed by Mr. Pontécoulant,when he published the above results. The effects of the correction C" [4431/], of the terms depending on S s, and of other quantities of a similar nature, are taken into consideration in book x. chap. viii. [9037, &ic.] ; where the final results of all these calculations, relative to the inequalities of the motions of Jupiter and Saturn, are given. [4431e] Terms of the order of the square of the dia~ tuibin^ forces. [4431/] [443%] [4431A] [4431 i] [443U] CorrectioH of ihe g:reat ine- quality. [4431J] [4431m] [4431n] [4431o] [4431;?] [4431g] 292 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. Now, if we connect the several parts of the great inequality of Jupiter, we shall obtain, for its complete value,* r (126P,569155— ^0',013495— /3.0',0000I9247).sin.(5w^<— 2n'-i'+5£^— 2si')- [4433] (1+1^^).)+ (9G',466088— t0%474651+^2.0.00007S564).cos.(5«^<— 2«'7+5e>— 2£'^)J C+ function C" [4431/] -}- 2 5 v'" [4431] If we reduce these to one single term, by the method in [4024 — 4027"], we «'"'",. shall obtain, for 5 v", the following expression, inequality. ' ' o 1 ' \(1265',251781-<.0',037090+i-0',000036669).sin. ]/ [4434] (l+,^^)-r ' ' T J V-<.77»-,653 + r2.0',013581 A. (+ function C'" [4431/] +2.5 î;'" [4431] ) This inequality may require some correction, on account of the coefficient i^", depending on the value of the mass of Saturn ; and also on account of the [4434] slight imperfection in the assumed value of the divisor (on' — 2n)^; a long series of observations will remove this small source of error. We must apply this great inequality to Jupiter'' s mean motion, as we have seen in [4006"]. The square of the disturbing force produces also, in 6 v'", the inequality [3890], [4435] (5 1)'' = ^ . ^ . sm. (double argument oi the great inequality) ; which, in numbers, is, [4436] &v" = — 13",238897 .sin. (double argument of the great inequality) ; ive must also apply the inequality of a long period to the mean motion of Jupiter. The inequality [3921], [4437] 6v-= 1 . ^•^"'"^""+^^^^"^^ffJr.sin.(5>r/-10»-/ + 53'--10s--i?-2), reduced to numbers, becomes, [4438] 6 If" = _ 4',024751 . sin. (5 n'" t —10 n-t + 5 £"—10 s^' + 51" 21'" .55'). [4433a] * (2668) The expression [4433], is the sum of the terms contained in the functions [4418,4425,4429,4431,4432] multiplied by (1+ l^''')- Then, by computing this expression for the times, t = 500, and t = 1000, we may reduce the whole to one term, as in [4434], by the method explained in [4024—4027"]. VI.xii.>^33.] THEORY OF JUPITER. 293 We have also, in [3844], the inequality,* 6 v'" = ^ . Ke'\ sin. {5 n't — 4> n'" t + 5 i" — 4> s" + z^'" -J^ B) ; [4439] and by reducing it to numbers, it becomes, &v" = 10',084660.sin.(4n'^i: — 5n^-^ + 4s'^ — 5 £^ + 45''21"'440 ; [4440] if we connect this Avith the two inequalities [4392], f P,097613 . sin. (5n'i— 4?i'^'/ + 5 s"— 4 s" — z^") — 2',781664.sin. (5 n't — 4> n'" t -^ 5 1" — 4> s" — z^" ) ; we obtain the single equivalent expression, 6v" = (l + t^") .1 P,506 190 . sin. (4 n'" t — 5 n" t + 4> s"— 5 6"+ OS'' 00™ 36^). [4442] We have seen, in [3773], that the expression of d.iv'" contains a secular inequality, depending on the equation, [4441] [4439a] * (2669) The inequality [4439], is the same as the last of [3344], augmenting the accents of e, n, n', Sic. to conform to the present example. The term K, which occurs intliis expression is, by [3824 — 3826], equal to the constant term of the coefficient of the part of [4394]. depending on the angle 3 n'^t — 5 ?i'' t ; or rather on the angle ôtVt — 3n"'t. This part being nearly equal to — 1 69^265895 . sin. (5 n" t — 3 n'^'t + 5 1"— 3 s'^— 55''40"' 49^. [44394] If we compare this with [3826], putting i=5, we get, Z = — 169%265895; i? = — 55M0™ 49"' ; [4439c] and by [4081], ra"'= 10'' 21'" 4' ; hence, •TO"- -\-B=—45''l 9'" 4.5^ ; [4439(/] and [4439] becomes, ^.Ke'\sm.{5n't — 4 71'" t -{- 5 b" — 4 e'" —45'' 19™ 45') = — f . Z é\ sin. (4 n'^t — 5 n" t -\- A t"— 5 s" + 45" 19" 45"). Substituting in this, the value of K [4439c], and that of c" [4080], it becomes neai'ly as in [4440]. t (2670) These mequalities are found in the ninth and tenth lines of [4392], with a slight and unimportant variation in the first coefficient. These terms [4441] may be [4440a] connected with [4440], and reduced to one term, of the form [4442], by the method given in [4282A— Z]. VOL. III. 74 [4439e] 294 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. d.dv'" m".»'^ dt 8 Hence we deduce,* [4444] ^^"'= — 23%9441 . e''=^— 27%7951 . e'=^ + 42^,9296 . é\ e\ cos. (^-— ^"). [4444'] ^^ ''^^^Z neglect the constant part of the second member of this equation, which is confounded ivith the mean motion of Jupiter, and then we shall have,t * (2671) We have, as in [.3756a, è], 2 2 2 2 2 2 [4443a] h'" + l'" = e"' , A" -j- P = e" , li" h" + t' />' = e'" e" . cos. (to^— to'' ). Substituting these in [444-3] ; also the values of .^"", A'-^\ and their differentials, in terms of Oj, and its differentials [996 — 1001]; then the values of these quantities [4202, Sic] ; we finally get the expression [4444] . f (2672) We shall put E for the general expression of the second member of [4444], corresponding to any value whatever of t, and E for its value when ^ ^ ; then substituting the values é", e", -a'", -n' [4407], we shall obtain, [4445a] ^-#-' = ^ = E + ^-^+^°- [4444,4445a]. Multiplying this by d t, and integrating, supposing Sv'" = when t = 0, we get, [44456] 6v^''= Et + ^. — .t^ + kxi. of which the first term Et, may be neglected, being confounded with the mean motion of Jupiter ; then we have, by neglecting t^, t'^, &c. s iv 1 ''^ v2 d.Sv'" dE . ^....-. [4445c] èv'^ = i.-.f, or --^ = — .t, as m [4445]. The coefficient of t, in the second member of this last expression, represents the differential of the second member of [4444] , divided by d t, corresponding to the time of the epoch 1750. Substituting in it the values [4405], and dividing by the radius in seconds 206265^, we get, d.Sv'v [4445d] —7^ = — 0',0000013 . i, nearly. This equation being multiplied by d t, and integrated, gives [4446] ; no constant quantity being added, because it is supposed to vanish when t = 0. VI.xii.<^33.] THEORY OF JUPITER. 295 il^^ _ 03-= 9441 .t.2 e'\ — — 27^7951 . < . 2 e". ^ dt (It dt [4445] + 42^9296.ï.5(e-. ^ +c\ ^).cos.(ra-— ^"O— e'^e^'^^^ ^'.sin.(«^— ^»)^. Substituting for ^ , '-^, ^"^ '-^\ their values, given in [4405], and integrating, we obtain, ^v'- =—t~. 0',00000065. [4446] This inequality is insensible in the interval of ten or tivehe hundred years, and even as it respects the most ancient observations that have been handed [4446'] down to us ; therefore we may neglect it. It now remains to consider the radius vector of Jupiter. We have found, in [3845], that the terms depending on the powers and products of the third degree of the excentricities, add, to the expression of this radius, the quantity,* i ,-- = — H a'\ é\ COS. (5 n" ^ — 2 n''' i + 5 s' — 2 è^— ^''+ A) + Hà\ é\ COS. (4 «'''i — 5 n" Ï + 4 a-— 5 s' — ^'"—A) ^^447] ^ Correc- ,4m\n-.a-2r p .sin. (5 w'i — 2 7ii^<+5a'— 2 6-) ) 'A°dius'"" -\ . < ■ > r • vector. 5,iv_2„.v ^_[_p'.cos. (Sn^i— 2n'''<+5 6''— 2s"')!) Reducing this function to numbers, we obtain, (— 0,0003042733. cos.(5n'i—2n"'^+5s'— 2s'''— 12''08'"490) 6r'^=(l + f^').< >• [4448] ( + 0,0001001 860. cos.(4/î''i—5?i'ï+4si'— 5s' + 45n6"'470 ^ If we connect this expression with the terms computed in [4393], i,_ J 0,0000268383. cos.(5n'< — 4w"'i + 5E'—4ê''—ra''')? àr —{^ '^^'l — 0,0000516048. cos.(5r^ — 4n'7 + 5s' — 45'"-^')^' ^'^'^^^^ * (267.3) The expression [4447] is composed of the three last terms of [3845], increasing the accents as in [4383a]. The value of H is as in [4426a] ; those of P, P', as in [4402] ; the other elements are given in [4061, 4077, 4079,4080] ; hence the expression [4447a] [4447] becomes as in [4448]. Connecting this with the two terms of or", given in [4393 or 4449], and reducing by the method [4282A— Z], we obtain [4450]. 296 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. we obtain the following result, [4450] à r"= (1 + f^O • 0,0000983161 . cos. (4 n'^t — 5 n't + 4, £'^— 5 s-— 14''23'" 19'). The semi-major axis a'", which we have used in calculating the elliptical part [4450] of the radius vector, must be augmented by the quantity ^o'". m'" [4058]. Adding this to the expression of a'" [4079], we obtain, [4451] a'" = 5,20279108. Inequalities of Jupiterh motion in latitude. 34. It follows, from [3931,3931'], that the terms depending on the square of the disturbing force, add to the values of — - , — — , the following quantities,* * (2674) In deducing the differentials of 5 9, 5 è, Uc. from [.3931—3932'], in order [4452a] jQ f5|^(i jjjg increments to be applied to the values of -tt^ "tt") ^c. [4246, &tc.], we may consider 5y, II, J <p, 5 ë, to be the only variable quantities ; or, in other words, we may neglect the variations of n, è, ç, y, on account of their smallness. For the expressions of 5 7, i5 n [393.5,3936], which are independent of the periodical angles, are of the order [4452!)] m'~; consequently their differential coefficients -77") , are of the ««me order, and are therefore much greater than the terms arising from the variations of the angles n — ê, in the differentials of the expressions [3931 — 3932'] ; because these last terms depend on the [4452c] products ^7-jt^ ^'^'Jt' ^c. which are evidently of the order m'^ ; since -^, -— , [4411] are of the order m'. Hence the differentials of [3931,3931'] become, by dividing by fl t, and increasing the accents, as in [4388o] ; * —- : -^ ^ .]—r-^ .cos. (n — è") — y.—, — .sm.fn — ô'")} ; [4452e] [4452/] dt p. — ; — = : — -— ^ — . } '- . sm. (n — ô'>) + 7.— --• .cos.fn — Ô"-)}. ^ dt j«'^ y/a"' + '"'Va" i dt ^ ' ^ ' dt ^ '5 Now, from [4410], we have. [4452g] 1^ = O',000184 = ^ ; ^' = _ 0^00T631 = '^ ; substitutmg these, in [4452e,/ J, we get, — — — , — — - , which are changed mto -j-, U Z (I- t lit [4452;»] 1^", in [4452,4453]; and by using [44-52^], also the values of y, n [4409], m'\ m\ VI.xU.§34.] THEORY OF JUPITER. 297 ^'= — 0%078213 dt — 0',223251 dt de" — = 6',457092 dt ' dt Then we find, by means of the formula [42956] [4452] dt ~ ;H''Va"-fm''.v/«'' l^ ' t ^ ) ^ = - -"'-^"" .$i2'.sm.(n_r) + ,.i^.cos.(n-r)^- [4453] 6 J, en, beuig comiKited by the formulas [3931,3931']. Reduchig these functions to numbers, we obtain, ifL =_0',000073; [4454] dt ^= 0,000811. - [4*5^1 dt d(f"' d(f>y The first of these expressions must be added to the values ot — , -j^ [4246], and the second to the values of -j^, '-^ [4246] ; hence we obtain, dtpi [4456] a'", a", ê" [4061,4079,4083], they become as in [4454,4455]. Adding the expression [4454] to the first terms of -— and -y- [4246], we get their values [4456] ; also [445ai] do" rfd'" addmg [4455] to the first terms of — — and -~ [4246], we obtain the correspondmg values [4456]. * (2675) Tlie terms of us'"' [4457], are deduced from those in [4295&], by adding three accents to the symbols m", n', n", /, î", a', a", in order to conform to the case ,..._, •' ) J J 3 ; : 3 r4457nl now under consideration. 7, IT, are as in [4409]. The values of 5<'~'^= — . „ [1006], are given in [4210,4079]. VOL. III. 75 298 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. / ,564458 .sin. (n" ^ + s" — rr) \ + 0',663927 . sin. (2 fi't — n'"t + 2 s" — s- — n'") [4457] 6 5'"= (1 +^^).|-j- V,\l97S2.sm.(3n''t — 2 n"t + 3s-— 2 ^^ — n-")^^; — 0',279382 . sin. (4 n't — 3 n'H + 4 s' — 3 .'^ — n'^) — 0',269130 . sin. (2 n'-t — ift + 2 s'' — s'— n'^) n'', in this formula, being the longitude of the ascending node of Saturn's [4457'] Q^}j{t iipon that of Jupiter [42956 — c]. Lastly, we have, in [3885], the inequality,* [4458] 6 s'" = 3',941 680 . sin. (3 n'^t — 5 Ji' ^ + 3 i" — 5 s'' + 59'-' 30» 35'). Inequali- ties ill the latitude. * (2676) The quantity [4458], is deduced from [3885], reducing both terms to one, as [4458a] in|-4282A-/]. Before concluding the notes on this chapter, we may remark, that the inequalities of the motions of Jupiter and Saturn, computed in tliis book, are corrected by the author in [5974,&ic.], and afterwards more thoroughly, in book x. chap. viii. [9037,&ic.] ; where he has decreased the assumed value of the mass of Saturn [4061]. He has also computed several [4458 ] ^^^jj inequalities, which had not been previously noticed, and has given new forms to some of the arguments. Finally, the subject of these inequalities has been treated in a wholly different manner, withafrequentuseof definite integrals, by Professor Hansen, Director of the Observatory . g , at Seeberg, in a memoir, entitled, " Untersuchung ueber die gegenseitigen Storungen des Jiipiters und Saturnsf which gained, in 1830, the prize of the Royal Academy of Sciences, of Berlin, relative to the inequalities of these two planets. In this method, the true longitude is computed by means of the elements corresponding to the invariable ellipsis at the time of the [4458rf] epoch ; taking instead of t, a function of t, which corrects for the perturbations. As the inequalities of Jupiter's motion had not been completed by Professor Hansen, when he [4458c] published this memoir, we may have occasion to refer to it more particularly, after the completion of his work. Vl.xiii. §35.] THEORY OF SATURN 299 CHAPTER XIII. THEORY OF SATURN. 35. The equation [4386], r corresponding to Jupiter, becomes for Saturn, If we take for r", and r\ the mean distances of the earth and Saturn from the sun [4079], and suppose 6 V' = ± 1" = ± 0',324, we shall find, 6?- = ±0,000141326. Therefore we may neglect the inequalities of àr", below =F 0,000141. We shall also neglect the inequalities of Saturn, in longitude and latitude, which are less than a quarter of a centesimal second, or 0%081* Inequalities of Saturn, independent of the excentricilies* ,+ 3%156532.sin. (w''i — n"i + s'" — £ — 3r,493729 . sin. 2(«"i — 71" t + i'" — =' — 6',56593 \ . ûn.S{n"' t — n' t + b" — i - 1%965748 . sin. 4(/i'''i — n^t + i" — î ii)'==(l + (.''). ^ _ 0',697047 . sin. 5(n'^^ — n^i + £" — s' — 0,270789 . sin. 6 {n}" t — n't-\- s''' — s — 0-, 1 1 6291 . sin. 7 (ir t — n't + £*' — s — 0',056126 . sin. 8(n"'t — nH + i" — s" K— 0%034097 . sin. 9 (n'^i — n't + s'" — =" [4459] [4460] [4461] Terms which may be neglected. [4462] Inequali- ties inde- pendent of the ex- cent rioi- ties. [4463] * (2677) These are computed as in [4277a — 0], increasing the accents on a, n, rt, hx,. so as to conform to the present case. [4463a] 300 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. [4463] + (l+f^^') InequftTi- ties inde- pendent of the excen- tficities. ir'=(l+f^") [4464] + (l+f^'') + 9',248269 . sin. (n^H —n't + i"' — s^) —14^451913 . sin. 2(ri'H — n''t + s- — s") — l',427160 . sin. 3(»'" t — n''t + s"' — s^) — ,314960 . sin. A^Çn^'t — n" t -{- b"' — s") — 0',090690 . sin. 5 (n^'t — n" t + s^ — 0%047444 . sin. 6 (n"' t — n't + t" — 0',010686 . sin. 7(/j''7 — n't + i" — 0%003942 . sin. S (if 't— n't + i" ' + 0,0039077763 + 0,0081638400. cos. (n''i — rC / + e'^— ^\ + 0,0013838330. cos. 2 {n"t — TCt-\- e'^— e' 1 + 0,0003200673 . cos. 3 {iVt — n't + £'"— £^ '+ 0,0000992632. COS. 4 (n''ï — n"ï + £'" — £^' + 0,0000355919. COS. 5 (m'"^ — m^ï + e'^—£^ I + 0,0000135999 .COS. 6 (w'^/ — «"^ + f'^ — £' + 0,0000( f55 1 35 . COS. 7 {n'" t — ')ft+ e'" — £\ + 0,0000021631 .COS. 8(?r^ — Jt^^ + E'"— e" + 0,0000006436 . cos. 9 (n" t—7ft-^ e"_ = '—0,0000137622 + 0,0001491217. cos. (71" t — 7^ t + i" — i''' - 0,00039499 1 6 . cos. 2 (n' f — »r' i + £" — e' ' — 0,0000480303 . cos. 3 {7f 1 — 7^1+^''— e' ' - 0,00001 1 8201 . cos. 4 («^ Ï — n^ 7 + £"— £"' — 0,0000036280 . cos. 5 {if t — »" t + s^— e^ — 0,0000012501 . cos. 6 {7ft — n'H + e^ — s^ Inequalities dependiTig 07i the first poiver of the excentricities* We shall here notice the secular variations in the coefficients of those [4465] inequalities of Saturn, Avhich exceed 1 00", or 32',4 ; in the same manner as we have done for Jupiter, in [4389']. Hence we have, [4466a] * (2678) The inequalities depending on tlie first power of the excentricities, are computed in the same manner as for Jupiter [4390a, &c.]. VI.xiii.§35.] THEORY OF SATURN. 301 èv"^ (l + f^'"). + (l+f^'") — ir,509517 . sin. (trt + i' — z^) + r,258041 . sin. (rrt + s'" — tz'") — 2'-,064438 . sin. (2 n'7 — n't -\-2 i"— s- — a" + 2^,672881 . sin. (2 7rt — «"i + 2 e"— s" — ^i' — 0^292291 . sin. (3 n"t — 27ft +3 s"— 2 =' — in^ — 0',223191 . sin. (3 »"< _ 2 ?i'^ + 3 s"— 2 6"— ^' — 0%090633 . sin. (4 7rt — 3 n^i + 4 s'^— 3 1" — .3* — (1 82%068686 — Ï. 0S0101095) . shi/_^^^^J^^l, + (41 7',057741 + 1 . 0^0] 38572). sin. (_^Z^_~.!!^l;}j + (34',341627 — ^0^0019068).sin. Sn''t — 27i"t _^3sV_2çiv_-,v VOL. III. — 17'-,654164 . sin. (3 n^i — 2 ?»"< + 3 s" — 2a" — ^^'^ + 4',795080 . sin. (4 «"i — 3 ?^'^i + 4 s' — 3 s'" — k'' ■ 2%43541 . sin. (4 rt'' i — 3 n'" i + 4 s" — 3 s" — ^'^ + r,393612 . sin. (5 7ft—^n"t + 5 £"—4 s'^'—z^^ — 0-,703450 . sin. (5 ift — 4 n'H + 5 s^— 4 s'' — ra'" + 0S537161 . sin. (6 71" t — 5 n'" t J^ 6 s" — 5 s'" — z>' — 0-,25651 . sin. (6 'ïû' t — 5 n'" i + 6 a" — 5 s'" — ^'^ + 0V2] 6195 . sin. (7 n't — 6 n'^ t+l^' — Q s'^ — ^^ — 0', 1 07342 . sin. {l7i't — & ir t + 1 1'—G s'" — -sj" ,+ l',142398 . sin. (7tH + a" — tn^) — P,01 1647 . sin. {ifH + s^^ _ ^") —10^033866 . sin. (2 n" t — 7i''t + 2 s"' — s^ — z,^ ) + 2%766173 . sin. (2 >t^7 — Ji'i + 2 s^' — s'' — t^^i-^ -16^936280 . sin. (3 71'' t — 2 7^1 + 3 s"' _ 2 5'— :3^ + 25', 153348 . sin. (3 w^'i — 2 n^i+ 3 s" — 2 «'— «" + 0',559336 . sin. (4 rf't — 3 n"^ + 4 h"' — 3 s' — ^^ ' — 0',758225 . sin. (4 n^'t — 3 7ft -^4, s"' — 3 £"— ^3^' — 0-, 1 87729 . sin. (5 n'H — 4 n^' ^ + 5 s'' — 4 s' — — 0',673817. sin. (2n^i— «^■74-2?'— «''— + r,521577 .sin. (3?i^^ — 2?i'''^ + 3s^ — 2e'"— + 0^ 151 701 .sin. (4?r^ — 3 «"'^ + 4«* — 3 £"— 76 Inequali- ties de- pending on Ihe first power of iheexcen- tricities- [4466] 302 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. /— 0,0003422170 . cos. (n'i + £^ — ^n'") I — 0,0020775935 . cos. (2 ?ft~ n'H + 2 s"— e-_ w" ) . /I, i„s )+ 0,0053861750. COS. r2?rï — n'7 + 2 s^—s"—^") (5r ==( 1+ fji'^). / \ i / j+0,0011594872.cos. (3n^i — 2n'7 4-3E''— 2.'"— ^'')j [4467] j —0,0006217670 . cos. (3 n^ i — 2 n'^i + 35" — 2 £'"— ra'^) \+ 0,00021 17893 . cos. (4 7iU — 3 n'H + 45"— 3 s"— ^^^ ) , + n+ -) ^— 0,0003750767. cos. (3 r'i — 2 ?i'ï + 3£^'— 2 s^—^^)> (+ 0,0005605490. COS. (3 n'V — 2 H-' i + 3 s^'— 2e'' — ^") ^ * Inequalities depending on the squares and products of the excentricities and inclinations of the orbits* i"'n°';- / /k;,. o '.noon , n. rvno^oN • /3 n^t — n'" t + 3 s" — s'' ^nS'" / -(54^847829_^.0^00362).sln.f 84^36"'45^-^.34',55 order. 1 ^ ' ' ^_ i )+ 28',526709.sin.(«'''ï — n''^ + E'''-£^ + 84''16'"430 '-(669%682372-i.0%015469).sin.^-_j"5'g,j5!!5f_^^_4g.r5 ,,,^^, , — 2-,935793.siii.(5n'^ — 3«"Ï+5e^— 33-— 57''O9'"O80 [4468] > ' V ( + 1 •■,923552 . sin. (3 n^'t — Sn't + S s" — 3 s^— 67" 54'"43') ) ^^ ^ ^4-3^,025379. sin. (3 n"<— n"^ + 3£''— s'— 85''34'"120 ) If we connect the inequalities depending on n'^t — n^t; also those on [4468'] Sn^'t — Sn^t, with the corresponding terms which are independent of the excentricities [4463], we shall obtain for their sum, the following expression, [4469] [4470] 6r^= + (1 + ij.'^) . 28',967123 . sin. (n'U — n't + s'"— g^-f 78''03'"130 — (1 + f^'O . 1 ',9 16292 . sin. (3 n^' t — Qn-'t + S s"— 3 ï>+68''27'"07'). Then we have,t '—0,0011710805. cos. (3 n^ï — n'''^-f3i^ — £''—90''I2'"350 ^,.v^(-l^^iv-j 1-0,0005621 901. COS. ( n'H — n't+ £i''_sv_83''26'"330 (. +(0,0151990624- 1 . 0,0000003370) .cos. (^"5J.;^^:'3;3+^^/'^^^ [4468o] * (2679) Computed as in [4394«, Sic], for Jupiter. [4470(1] t (2680) Tliis computation is made as in [4394c/] . VI.xiii.§35.] THEORY OF SATURN. 303 The inequality of the radius vector, depending on the angle li'^t — rft, being connected with the similar term in [4464], which is independent of the ^ ^ excentricities, becomes, f^r" ^ (l + i^'O . 0,0081090035.cos.(/r^ — n'^i + s-— e"— 3''57™35'). [4471] Since 5 n" — 2 n" is very small, we have computed the inequality depending on 2»'''^ — ^n't, by the formulas [3714, 3715]. Moreover, as Sn^' — n" is very small, we have computed the inequality depending on the angle [-44721 on^'t — M'7, by the formulas [3711,3718]. For greater accuracy, ive must apply this last inequality to the mean motion of Saturn, on account of the length of its period. Inequalities depending on the poicers and products of three and Jive dimensions of the excentricities and inclinations of the orbits, and on the square of the disturhing force. The most considerable part of the great inequality of Saturn, is that which has (5 n' — 2 n")', for a divisor, and depends on P, and P'. It is derived [4472] from the great inequality of Jupiter, by multiplying it by — v ' wa' iv ? "^ [4473] conformity with the formulas [3844,3846].* Hence we get, for this part of the inequality of Saturn, the following expression, è v'= — i 2957^357566 — t . 0',01 9701 — A 0^00004505 1 . sin.(5 nU—2 n"t4-5 e^— 2 s") ' [44741 — |279%746590— / . P,1086.38 + î!2.0',00018387|.cos.(5w"<— 2m"Y+5ev_26'^). * (2681) If we represent, for brevity, the terms between the braces in the two first lines of [3844], by aP^, we shall find, by inspection, that the two fii-st lines of [3846], between the braces, are equal to a' Pg ; and by noticing only those terms of 5 r, ô v, which [4472a] have the small divisor (5 n' — 2 n)^, we shall get, by increasing the accents so as to confonTi to the case now under consideration, , . 6jn».n''2 15m'"'. ny^ „, r^^~oi.i ôv'^ = —- ^-^.a'v.p ; . 5v''=-- --——.a\Pl. [44/26] (5 7i>-— 2 n")2 - (5nv— 2n'^)2 ^ Hence it is evident that (> v" is easily deduced fi-om 5 v", by multiplying this last quantity by the factor [4473] ; so that we shall have, 15 m'^.n"^. a" , . ^v'-=— .^„ „„ -^v"' [4472c] as in the terms of the fifth dimension of the excentricities [3868a — cj. 304 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. inec.uaii- The great inequality of Saturn is composed of several other parts: it S. contains, in [3846] , the function,* _j_av2 _/ — N COS. (5 rCt — 27V''t 4- 3 i" — 2 £'") [4475] 6V'' = — ^^:;--nn^ • < - a" 2 . f — j . sin. (5 71' < — 2 n'" < + 5 £» — 2 s'^) [4476] [4477] [4478] [4480] [4481] 5?i' — 2 7t" Reducing this quantity to numbers, we find in 1750, 6 v''= + 52%l3n99l. COS. (5 n't — 2 n"t + ôs"— 2£'0 — 1 P,275407 . sin. {brCt—2 1^ i + .5 s^ — 2 i") ; and in 1950, 6 v^ = + 51Sl 92839 . sin. {5rC t — 2n"t + bs''—^ s'") — 14^982033 . cos. (5 n^i — 2 n"t + 5 -='— 2 s"). Hence we deduce the value of this function for any time whatever t, 6 v' = + (52^138991 — t . 0%0047308) . sin. (5 7^1 — 2 n'^t + 5 £"— 2 ;'") — (11%275407 + ^. 0',01 85331) .cos. (5 n't — 2 7i"t + ôs-'—2B"). The great inequality of Saturn contains also, in [3846], the term, [4479] 5 w^ = — 4 //' e" . sin. ( 5n''ï — 2 n'^ i + 5 £" — 2 e'"— ^» + A'). This term, in 1750, is, 6v'' = + 7%554290 . sin. (5 ift—2 n"'t + 5 e"— 2 £*") + 5',321290 . cos. (5 n't — 2 n"t+ 5 s' — 2 s") ; and, in 1950, it is, a v" = + 7%71 1294 . sin. (5 Wt — 2n"'t + 5i' — 2 s") + 4^825821 .cos.(5n''t—2n"'t + ôs'' — 2e"'). * (2682) The expression [447 o] is similar to [4419], in Jupiter's theory, and is [4475a] computed in the same manner ; namely, by finding tlie values of ( > v ')' ("TT )' ^'^^ similar to [4420] ; which maybe easily done, by means of formula [4421], and the values [4475t] [4420]. Then from [3842, 3843], we get (t-;). (j^X^'^- It is useless, however, to explain the details of this computation, as it is done in almost exactly the same way as VI. xiii. § 35.] THEORY OF SATURN. 305 Hence, for any time t, it becomes, 6 r" = + { 7'-,554290 + t . 0\000785 \ . sin. (5 n'i — 2 w" i + 5 s^- — 2 ;'^) + {5^321290 — t . 0%002477 \ . cos. (5 nH—2 n"t + 5 î'— 2 é"). The part of Saturn's great inequality, depending on the poivers and products of five dimensions of the excentricities and inclinations of the orbits, is, by [3846,4023],* [4482] for Jupiter ; we shall tlierefore only observe, that the expressions [4476, 4477, 4478, 4479, 4480, 4481, 44S-2,] correspond respectively to [4423, 4424, 4425, 4426, 4427, 4428, 4429]. * (2683) From the terms of R, of the third dimension, depending on P, P' [3810], we have deduced in the two first lines of [3844], the corresponding terms of S v; which ai-e afterwards developed in [4022,4023], according to the powers of t; and the same process may be a))plied to the two first lines of 5 v' [3846]. We may also derive these tenns of S 11', from the corresponding ones of 5 v, by multiplying by the factor — —- — , or ISm'^'.n'a.a rather by 6 m'. rfi. a as is evident by the inspection of the formulas [3844,3846]. We may [4475f] [4483a] [4483i] [4483c] proceed in exactly the same manner with the terms of R, of the fifth dimension, depending on P„ PI [3863], or with those of il', depending on P„, P,/[3865]; the only change requisite is to place the accents below the letters P, P'. Now, if we neglect the parts of [4023] , depending on t"^, ddP, ddP', and make the above-mentioned changes in the factor and in the accents of the remaining terms ; also putting P, , for P„ , and Pf, for P„' [3864è], [4483(/] we shall get, for S v'' the expression [4483], depending on quantities of the fifth order in e", e", y. In finding the values of P, , P/, we may observe that the function R [3859] is easily reduced to the form [386-3], by the method explained in [3842i,&c.] ; using the values of A'"», A"*", &ic. [4430], by means of which we obtain the expressions of a^.P , a\P;, [4434,4485], for the two epochs of 1750, 1950. The difference of these two expressions being found, and divided respectively by 200, give the values [448C] ; as is evident from the formula [3723]. Substituting [4484, 4486], in [4483], it becomes as in [4487]. The signs of all the terms [4434—4487], are different in the original work, being changed, as in [4430a], to correct the mistake mentioned in [3860a]. JMoreover, to rectify this mistake in the signs, it is necessary to add the expression 2 (S j;" [4487] to the second member of the great inequality of Saturn [4492, &c.], in the same manner as the similar value of 2ÔV''' [4431], is added to the expression of the great inequality of Jupiter [4434, &ic.]. The numerical coefficients, in [4434, 4491], are equal to those given by the author; but the corrections C', C, 2ôv'\ 2 «5 «% in the second members, are not mentioned in the original work. [4483e] [4483/] VOL. III. 77 306 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. fifth order. , -» -.^ m which m'^P^, m'^P/ [3863, 44836], express the coefficients of sm. (ôn''t — 2n''t + ôc' — 2 ;"), cos. (5 n' t — 2 trt + 5 -=^— 2 s'^), m the development of R, depending on the products of five dimensions of the excentricities and inclinations. We find, in the year 1 750, ^4484^ a\ P, = 0,0000068376 ; a\P;= 0,0000100087; and in the year 1950, ^4485j a\P, = 0,0000077132; [4486] [4487] [4489] consequently. a\P;=: 0,0000096940; a\ —' = 0,0000000043780 ; at a\ i^ =3 —0,0000000015735. dt Hence the preceding function [4483], reduced to numbers, is, 5v''=^ + j^ 29% 144591 —t. 0%004081 } . sin. (5 n't — 2 )rt + 5 s"— 2 s'") — \ 1 8',879594 + t . 0-,01 1 356 j . cos. (5 n't — 2 ir t + 5 ="— 2 s''). Lastly, we have, in [4003], the sensilile part of the great inequality of Saturn, depending on the square of the disturbing force. This, in 1750, is,* Sv' = — 3^,816537 . sin. (5k^7— 2îi'^i + 5 s'— 2 s") [4488] + 42^,92031 9 . cos. (5 n't — 2 n'^ f + 5 s" — 2 .='") + function C [4489/c] ; and, in 1950, 6 tj" = — 1 %636772 . sin. (5 n''t — 2n"t-\-5 =^' — 2 r) + 43',624686 . cos. {ôn''t—2n"t + bf—2 ^") + function C' [4489/t'] , nearly. * (2684) The expression of & v" [4003] , being developed as in [3842a,J], and then computed '■ "■' as in the last note, becomes, according to the author, in 1750 and 1950, as in [4488, 4489], Vl.xiii. §35.] THEORY OF SATURN. Therefore, in tlie time 1750 +^ this part is expressed by, 307 respectively. From these values, tlie general form [4490] is (leiluced, by the method used in [44S3e, &ic.] ; but these numerical values, of the function [4003], have the same defects as the similar expression in Jupiter's motion [4432], of which we have treated in [44896] [4005(7 — 40076, 4431 rt — A]. The corrected value of Hv^', given by Mr. Pontécoulant in the paper referred to in [443 If], is as in the following table, which is similar to that of Jupiter [4431/,&c.]. Ô v'= 2',17020 . sin. T^ + 0' ,23185 . cos. Tj 1 + 8^14230 . sin. T^ + P,8S43S . cos. Tg 1' + 4^891 14 . sin. T^ — P,067G9 . cos. Tg 2- _ 0,951 1 2 . sin. Tg — 0',54669 . cos. Tg 2 -f- 0',054SS . sin. Tg — 0',830G0 . cos. Tg 3 _ 0',2576S . sin. Tg — 0%80208 . cos. Tg 3' + P,74101 . sin. Tg + 3 ,84548 . cos. Tg 4 + 0',22091 . sin. Tg + 0',2.3748 . cos. Tg 5 + r,85702 . sin. Tg — r,18481 . cos. Tg 6, G' + 3',466n7 .sin. Tg -40%36260 . cos. Tg 7, i = 2, — 16^06895 . sin. Tg + ] %9591 4 . cos. Tg 7, i = 1, + 6%04586 . sin. Tg + 2',23454 . cos. 1\ 8, z = 2, — 0%54808 . sin. Tg + 1%29603 . cos. Tg = 10%7635G . sin. Tg -33',10557 . cos. Tg. Termg of tlie order of the square of the dis- turbing forces. [4489c] This differs very much from the expression given by La Place, in [4488] ; which is connected with the other terms of the great inequality [4491], after multiplying it by 1 -j- fx'". This multiplication, by 1 -j-f^'") is not strictly correct; because some of the terms depend on (1 +t^'') • (1 + /J-')) and others upon (1 + (-^''Tj ^'ut as jx''', ij.", are small, this difference is not of much importance in this small inequality. We shall therefore adopt this method of the author, as we have already done in the similar inequali:y of Jujiiter [4431A, &tc.] ; where the factor 1 -{-[>■'', is used for all the terms. Proceeding, therefore, as in [443lA,&;c.], we shall observe that the mass of Jupiter . [4061ffj, is used in computing [4489rf] ; and the mass ^n^?:^^n [4061], is used in computing [4488] ; and if we increase the expression [4489(/]. in the ratio of 1070,5 to 1067,09, it becomes as in [4489iJ. Subtracting the expression [4483] from [4489t], we get very nearly the correction C" [4489A;], to be applied to the formula [4491 or 4492]. We must also apply a correction, depending on à ^, similar to that of 5e [443 Ip J, in the great inequality of Jupiter ; 6 v" = 10^,79796 . sin. Tg — 3.3%21 1 37 . cos. % ; C' = 14',61450 . sin. Tg— 76',I3169 .cos. Tg. [4489rf] [4489«] [4489/] [4489g] [4489ft] Correction ofihe great ine- quality. [4489i] [4489fe] 308 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. 6v' = — {3',816537 — t . 0',01 08988 1. sin. (ôn't — 2 n'" ^ + 5 s'' — 2 e'") [44903 +{42^920319 + t . 0%00352l8].cos.(5 n't— 2n" t + 5s'' — 2^'") + function C [4489A:]. Now, if we connect together the different parts of the great inequality of Saturn, we shall obtain its complete value, which is to be applied to ike planètes mean motion ;* 1+{2931%125445— <.0^0307355— ^2.0%0000450| .sin/^'g^";"^^?) + {223»-,252793-<.lM025051+i2.0^0001838| .cos/^g^';^^^?) + function C" [4489^'] + 2 <S î;^ [4487] Great Reducing these two terms to one, by the method in [4025 — 4027'], we shall mequality. obtain, C(O939»,615848-<.0^085024+i2.0',00008421).sin. .y.-gog + fa 0-012676 ^f ' [4492] ,5t>'=-(l V) < i-t.// ,b^-\-t .u,vi.4i}/i, :)\ ^4- function C^ [4489^-]+ 2 5i;^ [4487] ^ The square of the disturbing force produces also, in [3891'], the inequality,! [4493] ôv' = ^ . "' '^"""i "* '^'^ . sin. (double of the argument of the great inequality) ; which, in numbers, is, [4494] ^«'= (30 ,688957 — /.0',001 724).sin.(double argument of the great inequality) ; and this must also be applied to the mean motion of Saturn. [4489i] [4489m] Professor Hansen, in the work mentioned in [4458c], makes this part of the great inequality of Saturn, in the year 1800, as in [4489m], using the masses m'", m' [4061]. The corresponding value of La Place's formula, is found by putting t = 50, in [4490], by which means it becomes as in [4489o]. The difference of these two expressions represents the value of C' [4489p], corresponding to the calculations of Professor Hansen, noticing all the terms of any importance ; [4489n] à V- = 15',476 . sin. Tj — 47',531 . cos. Tg ; [4489o] àv' = — 3',271 . sin. Tj + 43',096 . cos. T^ ; C>- = 1 8',747 . sin. T^ — 90',627 . cos. T^. * (2685) The function [4491] is the sum of the expressions [4474,4478,4482,4487,4490]; and this sum is easily reduced to the form [4 192],containingbut one term,by the method explained in [4025—4027']. There is a small mistake in the calculation of the term 223' ,252793 [4491], which in the preceding sum is 223',900794; the difference being 0',648 = 2". (4493a] t (2686) The term [4493] is the same as [-3891'], —H' [3891] being the great [4489;?] [4491a] VI.xiii.§35.] THEORY OF SATURN. 309 The inequality [3927],* reduced to numbers, is, 6 1"= + 8',26451 7 . sin. (4 n'" / — 9 n" / + 4 s''— 9 s^' + 51'' 49'" 37'). [4496] We have also, in [3846], the inequality,! ôv'= l;K' e' . sin. (Sn't — 2 n'^7 + 3 s" — 2 =" + ^' +5') ; [4497] inequality of Saturn, or 5' =2939%61 5848 — <. 0^085024, and :ï'=4''2r' 20% nearly [4493]: [44936] substituting this and the values of m"-', irû', a'", a" [4061,4079], and dividing by the radius in seconds 206265% for the sake of homogenity, we get ô v" [4494]. The correction in the value of H' [4483/], has a slight efiect on this result ; and the same may be observed '■ ' relative to the correction of H [4483/], in the term [4436] ; and in other terms depending on H, H- * (2687) The inequality [4495] is the same as [3927], increasing the accents as in [4388n]. Now we have nearly as in [44936], F=2939%615848, :3' = 4''21"'20' [44936] ; [4495o] and by comparing the expression [3925] with the third line of [4468], we get, by neglecting the teiTus depending on t, K = 669%682372, B' = — bQ^ 10'" 57'. [44956] Substituting these in [4495], it becomes, -f 9%2107 .sin. (4?i'>7 — 9?j^-r +4 £'' — 9 £^4-51'^49'" 37»). [4495c] In the original work the coefficient has a difterent sign, being — 25",507770 =— 8%264517, also the angle — B' — Â' , as given at first, is, — 67°,3508 = — GO-* 36"' 57'. [4495rf] These mistakes are corrected by the author in [9105], where the coefficient is made equal to +8',264517, and the angle — B — .?= 51" 49" 37' nearly. t (2688) This is the same as the last line of [3846], increasing the accents as in r4497„n [4388a]. VOL. III. 78 310 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. and by reduction to numbers, it becomes in 1750,* [4498] 6 v" = 47% 115141 . sin. (2 n'' t — Sn" t + 2 e'"— 3 s" + US'' 08'" 08^ ; and in 1950, [4499] 6 V' = 46',307169 . sin. (2 n"t — Sn''t-\-2 i""— 3 =' + U9' 41'" \&). Therefore its value for any time whatever t, is, ^^5Q(j^ 5u'=(47%115141 — <.0%0040399).sin.(27r<— 3?rt-[-26i>'— 3£^+148''08"'08'+^27%94). Connecting this expression with the following, obtained in [4466], èv" = + (34V341627 — i.0%0019).sin. (3n^t—2nrt+3s'—2^''—z^) ^''^"^^ — 1 7%6541 64 . sin. (3 7i^t — 2 rr t + 3^" — 2 s''' — «''') ; we shall obtain for their sum, the following inequality,! [4502] s 1)'=:— (24^571253— < . 0',004392).sin.(2?i''Y— 3 ?i^-<+2s'^— 3 e^+14''48'"19*— M2',38). We have found, in [3777], that Saturn's mean motion is subjected to a secular equation, corresponding to that of Jupiter in [4446], namely, [4503] êv" ^ — t"". ,00000065. The corresponding secular equation of Saturn is represented, as in [3777], by,t [4504] 6 v" = -^^ . f. 0"-,00000065 ; and is therefore, in numbers, [4505] 6v' = fA)%00000\51; which may be neglected without any sensible error. * (2689) If we retain the terras depending on t, in the values of K', B' [4495i,4468], we shall have, K' = 669',682372 — t . 0',015469 ; B'= — SG'' 10"" 57^ — i . 49'-,5 ; [.4498a] ^ = ^i2\^< 20'— t . 77%629 [4492, 3926], &ic. With these values, and those of e^, zf [4407], we may compute the function [4497], for [44986] the j-ears 1750, 1950, as in [4498, 44S9]; hence we may deduce the general expression [4500], by the same method as in [4017—4021]. [4502o] 1(2690) This reduction is made as in [42S2/t—r]. [4505o] t (2691) The integral of [3777 or 3785], being divided by m'\/a', ^ives, Secular equatiun. VI.xiii.§35.] THEORY OF SATURN. 311 It now remains to consider the radius vector of Saturn. We have seen, in [3847], that the terms, depending on the tliird power or product of the excentricities, add to the expression of the radius vector of Saturn, the quantity,* 6r' ^ — H' a\ e\ cos. (5 n't — 2 n"t + 5 £" — 2 ^'— ^' + A) + H' a\ e\ COS. (3 n't — 2 n"t + 3 s>' — 2 s- + ^^ + A) ^450^^ 10 m". n\ a'-^ C P . sin. {5)i't — 2 n'^i + 5 s^— 2 s'") ) 5 n'— 2 n" '(-\-P'. cos. (5 yt^i— 2w"7+ 5 e^— 2 s'") ^ ' Reducing this function to numbers, we obtain, , ( +0,00351994565.cos.(5n''i— 2n'''^+5 .^— 2s-+ 13^01'"490 ) 6r''^(l + (A"').< ^ >. [45071 (_0,0008553506.cos.(2/rf— 3n^i+2s"— 3s' + 35''49"'080^ ^ ^ nequQli- ties in the Connecting the last of these two inequalities with those we have found 1 in [4467], depending on the first power of the excentricities, namely, ^^^'j»» V .1, i„x ^ + 0,001 1594872. cos. (3/1^^ — 2 ?r/ + 3s'— 2 s'-'—^M) ^ ( — 0,0006217670.cos.(3n"i — 2n'''^ + 3£'— 2 s"' — ^'0^ we get,t r"= — (1 4- f^'") . 0,0013806201 . cos. (Zn'^t—S n't+2 s'"— 3 s"— 23'^ 19" 18'). [4509] the accents being increased as in [4.38Sa] • Substituting Sv'" [4503], we get Sv" [4504], which is reduced to numbers as in [4505], by using the elements m'", m"' , «'", a" [4505c] [4061,4079], This correction is only 1*,5, in 1000 years, which is hardly deserving of notice. * (2692) The function [4506] is the same as the three last terms of [3847], multiplied by a', and increasing the accents [4388a] ; the first term of [3847] being of the second order in e, e', y, is included in [4170]. H represents the part of — -;- [3848], [4506a] depending on the angle 4 n" t — 2 li" t ; P, P', are given in [4402, &ic.]. Hence the expression [4506] becomes, in numbers, as in [4507]. t (2693) The function [4508] is the same as the fourth and fifth lines of [4467]. Connecting these with the similar terms [4507], and reducing the whole to one term, by the [4509a] method in [4282A — /], it becomes as in [4509]. 312 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. The semi-major axis, which is used in calculating the elliptical part of the radius vector, must be increased as in [4058], by the quantity ^a'-m''; and hy adding it to the value of a" [4079], we obtain, [4510] «^ = 9,53881757. Inequalities of Satimi^s motion in latitude. SQ. The formula [1030] gives,* + ] %787358 . sin. {n" t + b" — n^) — 0',2501 80. sin. (2 n'-'t — n't + 2 é" — s" — n^) 0',083g46 .sin. (3 n' 't — 2n'' t + 3 s"—2s'' — n") ^^^- ] + 3% 1 43523 . sin. (2 n' t — n"l + 2 j" — s" — n') — 0%522865 . sin. (3 n't — 2 n"t + 3 .-" — 2 s" — n") — 0',083182.sin.(4n"f — 3n"'i+4,£''— 3£'^— n^) [4511] ^ Ç + 0%084871 . sin. (n'"' t + ^'■' — n") ^(l_^p.-) . 3 + 0',122203.sin.(2n"f— n^f + 2£^' — ^=^— n") / + 0',662991 . sin. (3 n^'t — 2n't+ 3 £''— 2 £^— n^') n', being the longitude of the node of Jiipiter''s orbit on that of Satnrn, [4512] and n"', the longitude of the orbit of Uranus on that of Saturn. Lastly, we have, in [3886], the inequality ,t [4513] 6s' = — 9', 163599 . sin. (2 n'^t — An't + 2s- — 4 s^+dd' 30™ 350- It follows, from [3932, 3932'], that the terms depending on the square of the disturbing force, add to the values ot — -, —, the quantities,! * (2694) The terms of 5 s''' [4511], are computed from [4295i], increasing the " accents, so that m" may be the attracted planet, and m''' or ot'' the disturbing planet. f (2695) The inequahty [4513] is the same as [3886], reduced to one term, as in [4513a] [4282/t— Z]. t (2696) The values [4514,4515], are deduced from [3932,3932'], in the same [4514a] manner as [4452, 4453], are derived from [3931,3931']. We may also derive [4514] from [4452], and [4515] from [4453], by the following method. The expressions VI.xiii.§36.] THEORY OF SATURN. 313 ■rf,v ,«ivva^ Sir,^os.(n-n-'-^.sm.(u-é^)h [4514] de" m'\ y/»'" iSj .^^ ^^ ^^^ ^_ ySn dt m'\ s/a'"-^ m\ /«" ( t '.sin.(n— (r)+ ^— .cos.(n— â')^ ^7, 6n, being determined as in [3935,3936]. Reducing the functions [4514, 4515] to numbers, we get, do?'' [4515] = _|_ 0',000154; [4516] ==._0\001873. [4517] [4518] dt de" 77 Tire expression [4516] is to be added to the valuesof -p, -^ [4247]; d è^ d è^ and the expression [4517] is to be added to the values of -jj- , — ^ [4247]. Hence we obtain, ^ =. + 0',099894 ; dt '^' = — 0% 155136; dt — =_9,007165; dt 1^ = — 19%043372. dt [3931,3931'], become the same as [3932,3932'], respectively, by changing, in the second members, è into à', and multiplying by — . This is equivalent, in the present [4514A] notation, to the change of ê", into ê", and then multiplying by the factor — . Therefore, if we perform this operation on the fonnulas [4452,4453], they become ^4514^1 respectively, as in [4514,4515]; in which we must compute (5 7, 5 IT, as in [4452A] ; and then, as in [4452/t, &c.], we obtain the other quantities [4516, 4517,4518]. We have already remarked, that the inequalities of the motion of this planet are again noticed by the author, in book x. chap. vili. [9037, &c.], and the subject is also resumed in the notes on this part of the work. VOL. III. 79 314 PERTURBATIONS OF THE PLANETS ; [Méc. Cél CHAPTER XIV. THEORY OP URANUS. 37. The equation [4460], [4519] 6r^=^.(l—a?).ô\\ corresponding to Saturn, becomes for Uranus, [4520] ôr^'^'^.n— a=) . <S V". r If we take the mean distances of the earth and Uranus from the sun, for r", and r", and sup]30se â V"'=: ± 1"= ±0,324, we shall find, [4521] Ô r" = ± 0,00057648. l^™^ Therefore we may neglect the inequalities of 5 r", below ± 0,00057 ; "sLwd. and we shall also omit the inequalities of the motion of Uranus, in [4522] longitude or latitude, below a quarter of a centesimal second, or 0%081. Inequalities of Uramis, independent of the excentricities.* +52%306055 . sin. {n"'t — n'H + e" — £^') — 0', 190366 . sin. 2(n'U — n'H + s" — ^'") 6«''=(1 + H'") . { — 0',026023 . sin. 3(n"^ — n''H + s- — s^*) ^'^^^^^ ' — 0%003593 . sin. 4 («'" t — n'H + s" — i-') — 0',000768 . sm.b{n"t — n^H + «'"— s"') * (2697) Computed as in [42T7«, &ic.], changing the accents on a, ii, n', &,c. to "•' conform to the case now under consideration. Vl.xiv. §37.] + (1+^-") THEORY OF URANUS. 315 +2P,371379 . sin. (n't — rûH -\- i' — i") — 4--,220972 . sin. 2{n''t — n'H + b" — e^') — ,8621 15 . sin. 3(n''t — n" t + s^ — s") — 0%2444U9 . sin. 4 (ît' t — n" t + e" — s") \ . [4523] ^ _ 0',08U21 1 . sin. 5 (n" t — n"" i + e" — s^') _ 0-,028931 . sin. 6 (h" t — 7t" < + s" — £"0 — 0',01 0929 . sin. l{nH — ri" i + s' — s") — 0%004148 . sin. ^{ift — n'H + ^' — e") Inequali- ties inde- pendent of thd exGÊQ~ 0,0063473160 \ fi"«i«- + 0,0048914790 . cos. {ii}'t — rf't + s'''— ^'") 5rvi=(H-^-) . / + 0,00002361 84. cos. 2 (7i"i — n^'^ + ^'^—^'0 + 0,0000030669 . cos. 3 {n'H — n'H + s-_ £") + 0,0000005044. cos. 4(w'''i — nH + s'"— ^'0 + 0,0023641285 + 0,0035433901 . cos. (n" t — n'H + B" — s"') _(.-(l + ^v) _ I ^ 0,0004061682 . cos. 2(n't — n"i + s" — s") + 0,0000889425 . cos. 3 (n" t — n^'t + e- — s^') + 0,0000255870 . cos. 4> (n" t — n"t + ^''—s'") Inequalities depending on the first power of the excentricities* — 1%233612 . sin. (n'" t + s" — ^^') + r,25954B . sin. (2 n'H — n'H + 2 8'" — /'— w'") «jy- = (1 + (^'O . j _ 3.^g3g663 _ sin. (2 r^^ — ?r< + 2 s^' — s'" — t."') — 0^221997 . sin. (2 w^'f — n'H + 2 s" — a" — to*") [4524] [4525] * (2698) These inequalties were computed in the same manner as those for Jupiter [4525a] in [4375a]. Inequali- ties de- pending on tho first tncities. 316 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. — P,402359 . sin. (n^'t + i" — w") + 0',214857 . sin. (n" t + s^ — ^^ ) — 0^219788 . sin. (2 n" t — if H + 2 s'— s" — îj^') &1 \ + 0',878763 . sin. (2 rVt — if H + 2 e>— s^'' — ^^) — (44S051575 — t . 0',000247) . sin. (_5!v,'^^^, [4525] + (1 + f^O • ( _j_ (1 49',807764 — i . 0^008306) . sin. (_^S7~J[^^ + 2%486191 . sin. (3 «"i — 2 ?i^'^ + 3 s"— 2 s" — ^") — r,642451 . sin. (3 ji^'/ — 2 ?j'7 + 3 s''— 2 s" — ^^) + 0',422729 . sin. (4 n"ï — 3 n-' ^ + 4 s" — 3 s' — «^') — 0',281 800 . sin. (4 n'H — 3 «"i + 4 e'' — 3 s' — ^^ ) + 0^ 1 26493 . sin. (5 n"'^ _ 4 »^7 + 5 s^' — 4 e^ — ::i") ( _ 0,0016092001 . COS. (2n'H — ift + 26^' — s^— ^^') ) [4526] Ar^'=('l+M''). < >. ^ ^ J + 0,0061835858. COS. (2n^'i — ?r^ + 2 s^'—E"—.^-) C Inequalities depending on the squares and products of the excentricitîes and inclinations of the orbits.* ^(132^508872-t0^0145205).sin.(i-37 -^^^^^^^ f^Tg'^^) \ [4537] 6r^"=(l+0.j _!_ i^-,7i3455.sin.(4?i^'i — 2?r^+4/'— 25'— 38''34"'54'')1- J^oAt ( + 8^380157 . sin. (n" t — ii''H + a' - /' + 88"29"' 40^ j second The first of these inequalities must be applied to the mean motion of the planet, on account of the length of its period. The last of these inequalities, being connected with the corresponding one in [4523], which is independent of the excentricities, gives the following,! [4528] & r^' = (1 + f^') • 23^156281 . sin.(n^^ — ii''' t + i^'— e^' +21^1" 05^. [4527o] * (2699) Computed as in [4377a, &c.], for Jupiter. t (2700) The term + (1 + f^"). 2^,37 1379. sin. (n'< — n^V + s'— £>') [4523], [4528a] being connected with the last term of [4527], by the method used in [4282A — /], becomes as in [4528]. Vl.xiv..^3S.] THEORY OF URANUS. 317 Tiieu we have,* ô ,•'■' = _- (1 + (x^) . 0,0007553840 . cos. (3 n'H — n't + 3 s"— 5^+75" 00" 42^. [4529] Inequalities depending on the poiver.t and products of three dimensions of the excentricities and inclinations of the orbits.f twîd'""' order. S r'- ^ — (1 + M-^) . 0',964688 . sin. (5 n" t — 2n't + ô s"— 2 s' + OS'' 23" 3P). [4.53oi Inequalities of the motion of Uranus in latitude. 38. From the formula [1030], we obtain,t 6 5^' = (1 + ,.-) . 0%638393 . sin. (ra'^ t + s"— n'^) Inequali- ties in the . latitude. ( 0',9 15741. sin. (w^^ + s^—n") ) [4531] ^ ( + 2',921052.sin.(2w^-i — n^ï+2£"— s''— m)^ n" being here the longitude of the ascending node of Jupiter's orbit upon that of Uranus, and n' the longitude of the ascending node of Saturn's orbit ^ ^ upon that of Uranus. * (-2701) Computed as in [4394«, Sic] for Jupiter. r4529al t (2702) This computation is made as in [4417, &c.] for Jupiter ; changing the accents to conform to the present notation. [4530a] t (2703) The terms [4531] are computed from the formula [4295è], altering the accents to conform to the present case. [4531a] VOL. III. 80 318 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. CHAPTER XV. ON SOME EQUATIONS OF CONDITION BETWEEN THE INEttUALITIES OF THE PLANETS, WHICH MAV BE USED IN VERIFYING THEIK NUMERICAL VALUES. 39. The inequalities of a long period, produced by the reciprocal action of two planets m, and m', are nearly in the ratio of m'\/a' to — m\/a [1208] ; so that to obtain the perturbations of this kind, corresponding, in the motion of m', to those in the motion of m, loe need only to multiply the last [4532] lyy ^^_ T\ù% result is most to be relied upon, in those cases, in which the ratio of the mean motions of the two planets is such, as to render the period of these inequalities great, in comparison with the times of their revolutions. We shall now, by means of this theorem, verify several of the preceding inequalities. The action of the earth on Venus produces, in [4291], the two following inequalities, whose period is about four years, àv'^— 1%5 49550 . sin. (3 n"t — 2 w'^ + 3 s"— 2 s'— ^') + 4',766332. sin. (3 n"t — 2 n't + 3=" — 2 / — ^"). By multiplying them by — ^^^ryr,? we have, for the corresponding inequality of the earth, hv"^ P, 1 33838 , sin. (3 n"t — 'Un' t -\-S i' — 2 .' — ^) — .3'-,487666 . sin. (3 n"^ — 2 ?î'i + 3 ;" — 2 / — ^■'). We have found, by a direct calculation, in [4307], that these inequalities are, hv"= r,186390 . sin. (3n'7 — 2 n'i+ 3s" — 2s'_^') ^3%667112.sin. (3h"^ — 2n'i + 3.-" — 2=-'— ^"); [4533] Venus and the Earth [4534] [4535] VI. XV. §39.] VERIFICATION OF SEVERAL INEQUALITIES. 319 which difTers but little from the preceding expression [4534]. The action of the earth upon Venus, produces also, in [4293], the following inequalitv, whose period is about eight years, 6v' = — \ -,505036 . sin. (5 n" i — 3 n' i + 5 s"— 3 e' + SO'' 54"' 26'). [453G] Multiplying it by, ,, „, we obtain, for the corresponding inequality of the earth, ôv"= l',101277.sin. (5 n"t — 3 n't + ôs" — 3 s' -\-20''54>'"2&) ; [4537] and, by a direct calculation, we have, in [4309], 6 v"= r, 125575 . sin. (5 n"t — 3n' ^+5 /'— 3 s' + 2 1-; 02"' 18'). [4538] Mars suffers, by the action of Venus, as we have seen in [4377], the following inequality of a long period, è v"'= — 6',899619 . sin. (3 n"'t — n't + 3 £'"—£'+ 65" 26'"15'). [4539] „T , • , • -, m"V«"' 1 • Multiplying It by y—- , we obtain, fnd' ™ \/« Mars. 5 v" = 2%078266 . sin. (3 n'" t—7i't + 3 /" — / + 65' 26"* 1 5') ; [4540] and the direct calculation [4293] gives, 6 v' ^ 2',009677 . sin. (37i"'t — n't + 3 /" — s' + 65' 53"' 09') ; [4541] which differs but little from the preceding. Mars suffers, from the action of the earth [4375], the two following TheEarti, inequalities, whose period is about sixteen years, mIL S v"'= — 10',1 14699 . sin. (2 n"'t — 7i"t+ 2 s'" — a"— t,'") + 5', 1 23062 . sin . (2 n'" t — n" t + 2 a"' — e" — ^") . ^''^''^^ m' \/a"' Multiplying them by — "^;^;^ ' ^"^'^ obtain, for the corresponding inequalities of the earth, 6 v" = 2',2293 . sin. (2 n'" t — n" t + 2s"' — i' — •=='") — 1 -', 1 292 . sin. (2 jj'" t — n!' t + 2 a'" — a" — t^") ; and the direct calculation gives, in [4307], [45431 320 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. [4544] 6 v" = 2% 1 37658 . sin. (2 n'" t — n"t + 2 s'" — s" — ^"') — P,095603 . sin. (2 n"'t — n"t + 2 b'" — s"— v!') ; which differ but little from the preceding. Mars also suffers, on the part of the earth, in [4377], the following inequality of a long period, f4545j i v"'= — 4%370903 sin. (4 n!" t—2n"t + 4> s'"— 2 «" + 67'' 49"' OOO- m"'\/a"' Multiplying it by iryui we obtain, for the corresponding inequality of the earth, f 4546] <5 v" = 0-,9634 . sin. (4 n'" t — 2 n" t + 4 -="' — 2 e" + 67'' 49" 00») ; which differs but little from the expression, given in [4309], [4547] 6 v" = 0-,993935. sin. (4 n'" t — 2n"t-{-4> s'" _ 2 s" + 67" 48™ 56'). ^"and" The two great inequalities of Jupiter and Saturn, are also to each other, ^""'"' nearly in the ratio of — 77f\/a'' to m'''\/a"', as is evident by comparing [4548] |-4434^ 4492]. Saturn Lastly, Urauus suffers, from the action of Saturn, the following inequality and Uranus, of a long period [4527], [4549] è V'' = — 1 32',508872 . sin. (3 n"' t — n't+Ss^'— i' — SS" 19" 05"). Multiplving it by — , we obtain, in the motion of Saturn, the inequality, [4550] 6 v" = 32-,368 . sin. (3 7^1 — n't + 3 e^'— e" — 88'' 19'" 05^) ; which differs but little from the inequality, given in [4468],* [4551] év" = 30%888288 . sin. (3 n'H — n" t + 3 e-— s' — 87" 25™ 07') ; 40. We shall now consider, in the development of R, the term of the form [3745'], * (2704) The term here referred to is the last one of the expression [4468] ; which [4550a] differs, however, a little ; the coefficient being 31',025379, instead of 30%888288 ; and the constant angf»- 85''34"' 12% instead of 87'' 25"' 07'. VI. XV. §40.] VERIFICATION OF SEVERAL INEQUALITIES. 321 R = «I'.ilf ". e e'. COS. [ / . (71' ï — n ï + e'— <=) + 2 n < + 2 s — ^ — ^') I ; [4552] supposing i. (11 — 11!) + 2 n to be very small in comparison ivith n or %'. We find, in [1286, &c.], that this term produces, in the excentricity e, of the orbit of the planet m, considered as a variable ellipsis, the following inequality, which we shall represent by,* ie=— ., /'''t" — .M''\e'.co^.\i.(n't — n i + a'— 0+ 2 n i + 2 s — « — ^'|; [45531 i.{n — n)-\-2n ' and in the position of the perihelion ra, an inequality [1294, &c.], which we shall represent by, a^=_ /"'•"" .M<".-. sin. \ i. (n't _ ,i i + 6'— + 2 n i + 2 e — ^ — ^' 1 . 14554] i.(n — ?i)-|-2 7t e ^ The expression of v contains the term 2 e . sin. Çnt -j- e — w) ; and the variation of the elliptical elements, produces, in this quantity, the following expression,! [4.555] 6v =^ 26e . sin. (n t + ^ — w) — 2e 6-si . cos. (nt+e — «) ; [4.5.5t)] * (2705) If we take the partial differential of R [1281], relative to e, and multiply it by -7;—, — . , it will produce the corresponding term of e, represented by (5 e [4553a] [4553i] [1286]. Now, if we perform the same operation on the assumed value of R [4552], and put fx = 1 [-3709] ; changing also i', i, into i, i — 2, respectively, we shall get (5 1 [4553]. Again, if we multiply the same partial differential of R [1281], relative to e, by — .andt, putting |j. = 1 , it becomes like the expression of cdzi [1294]; and by the same process we deduce, from R [4552], the expression, e d-a = — m'.andt . JH''\ e'. cos. \ i.{n' t — nt-\- s — s) -\-2nt -\- 2 s — ra — zs'\. [45.53c] Dividing this by e, and integrating, we get the part of ra, represented by w [4554] ; observing that we may consider the terms 31, e, e', of the second member, as constant quantities, in taking this integral ; always neglecting quantities of a higher order than those which are retained, and such as depend on different angles. t (2706) Since v [3834] contains the tenu 2 e . sin. {yit -\- s — ro), it is evident that the variation of v, corresponding to the increments Se, <)■&, in e, zs, respectively, is as in [4556]; and by using the symbol JV:=nt-{-s — zs [3702»], it becomes, 8v=:2Se. sin. W — 2 e ô a . cos. W. [4.557a] Now, if we put, for brevity, VOL. III. 81 322 PERTURBATIONS OF THE PLANETS ; [M^c. Cél. which gives in v the inequality, 2 jn ft vt s • It follows, from § 65 of the second book, that in the case of i.(n' — n) + 2n [4557] being very small, the expression of R, relative to the action of m upon m', contains also a term, of the following form and value, very nearly,* [4558] R :^m. M'". e e'. cos. {i.(n't — ?t ^ + s'— e) + 2 n ; + 2 s — w — ^^'1 ; since, by noticing only the two terms of this kind, in R, and R, we have, as in [1202], very nearly. [45576] T,^i.{n't — nt-\-^—i)+^nt-\-2-= — zs — zi'; M,=^ '"'" — .M'\e': i.(n — n)-|-2n the expressions [4553, 4554] become, [4557c] (5e = — ^j.cos.T,; e <5 w = — Ji, .sin.T, ; substituting these in [4557a] , we get, [4557rf) èv = 2 Jlii. { — cos.Tj. sin. JV -{- sin-Tj . cos. W\ = 2 JW,. sin.(Ti— JF) = 2 iVii- sin. { i.{n't — nt-\- s' — s) + « t -\- s — ra' | [455re] ^2M,.sm.\{i~l).{n't — nt + s'—E)Jf-n't+i'-u'\, as in [4557]. * (2707) Using the symbol Tj [45576], we get, from [4552], [4558a] B. = ni. JW">. e e' . cos. Ti . Its differential, relative to d [37056 — c], is, [45586] à.R = rri. M^^\ e e'. (i — 2) . n dt . sin. Ti ; substituting this in the differential of [4559], which gives m'. d'iî'=: — 7n.AR, and dividing by m', we obtain, [4558c] d'iî'= — m . M'-^K e e'.(i —2) . n dt . sin. Tj . Now, i.{n' — n)-\-2n, being very small [4557'], we have, very nearly, [4558rf] {i—2).ndt=^in'dt; hence, [4558e] à'R= — m. iVi">. e e'. in'dt. sin. Tj . Integrating this, relative to the characteristic d', which does not affect n t r3982al, we [4.5.58/] , r^rron •■ ■' ^ obtam, as m [4558], [4558g-] ^'= "* • -^^"'- ^ ^'- COS. Tj . VI.xv.§40.J VERIFICATION OF SEVERAL INEQUALITIES. 323 m.fàR+ m'.fd' i?'= ; [4559] therefore we have, in v', the inequality,* S v'= .,!"''"'!'' -ili'". e . sm.l(i—l).(n't — 7it -{- i' — s) + nt 4-s — 7,\. [4560] t.{n' — «) + 2n ( V / \ These two inequalities of v and v' [4557,4560], are in the ratio of m'.e'.\/a' [4560] to ?ft.e.y/« ; so that the second may be deduced from the first, by multiplying the coefficient of the first by "1;}^^^ [4660«-]. ^4560"] •^ ?»'.\/« e' *- '^ The quantity 5n" — 3 n' being small, in comparison with n' or n", we have, in v' [4557], by supposing i ^ 5, an inequality depending on the argument 5 n" ^ — 4 n' t + 5 s"— 4 /— ^" ; and in v" [4560] , an inequality [4560'"] depending on the argument 4n"i — 3n't + ^s" — 3s' — -a'. The first of these inequalities is, by [4291], 6 v' = 2^,196527 .sin. (5 n"t — ^n't + b f— 4 ^'— ^")- [45Ci] Multiplying its coefficient by / „ . -, we have, for the earth, the ve"j "* V*^ ^ thcEarlh. inequality, iv"=0\6580.sm.(^n"t — 3n't-\-4^i" — 3e—zs'). [4562] By a direct calculation, we have found, in [4307], this inequality to be, Ô v" = 0',722424 . sin. (4 n" t — 3n't + 4^s" — 3e' — ^'); [4563] which differs but little from the preceding. * (2708) We may obtain ô v' from R', by a similar process to that used in the two preceding notes ; or, more simply, by derivation, in tlie following manner. If we [45600] change, in [4552], i, m, a, n, e, v, &ic. into — i-\-2, m', a', n', e', v' he. respectively, without altering Jlf"', R changes into R [4558a,^], and the factor r.(»i' — ti)-\-2n, becomes, (— ' + 2).(m — 7«') + 2n'; [45606] which, by reduction, is easily reduced to its original form ; so that the angle T, [45576] remains unaltered. The factor M^ [45576], changes into ^^-d^^^n-^'"''-^-^ [«60c] W changes into W [.3726a]; and the second expression of 5 d [4557 </], becomes as in the first of the following expressions of lî v', which, by successive operations, is reduced to the form [4560e], as in [4560] ; 324 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. In like manner, 4 n"' — 2 n" is rather small, in comparison with n" or n'" [4076/i] ; and if we suppose i = 4, we obtain in v" [4557], an inequality depending on the argument 4 w"'i — 3 n"t + 4 s'" — 3 s" — ^"' ; [4564] and in v"' [4560], an inequality depending on the argument Sn"'t — Inl't + 3 £'"— 2 e"— ra". The first of these inequalities is, by [4307], ^45g5^ & v" = OS8071 1 1 . sin. (4 n"'t — 3 n" t + 4 e'" — 3 -="— ^"'). m"\/a" e" Multiplying its coefficient by ,„ , — . — [4560"], we get, for Mars, the The Ea- til 1 J to J m \/ o! t and *^ *'^" inequality, [4566] ^ «'" = 0',661446 . sin. (3 n'" ^ — 2 ?î"/ + 3 £'" — 2 e" — ^") ; and by direct calculation we have, in [4375], [4567] <s v'" = 0', 846004 . sin. (3 n'" i — 2 n" « + 3 .'" — 2 e" — i.") ; the difference is within the limits of the error which may be supposed to [45681 exist, taking into consideration, that the ratio 4 ?i"' — 2 m" to n'", instead of being very small, is nearly equal to |. 41. It also follows, from § 71, of the second book, that if i.{n' — n)+2 n [4569] ^^ ^''^^'2/ s»«a// in comparison with n', the inequality of m, in latitude, depending on (i — \).{n't — nt-{-s — s) -{- n' t -\- s' , is to the inequality of m', in [45<39'] latitude, depending on (i — \).{n't — nt^s — i)-{-nt + i, in the ratio of m! \/a' to — m\/a.* [4560(i] 'î d' = 2 Ma . sin. (Tj — W) = 2 .¥o . sin. \ i.{n't — n < + e'— s) + 2 « i + 2 £ — n't — s'- ra ? r4.5<jOel "^^ 2^2 • sin.{(i — \).{n' t — nt-\-i' — i) -\- nt-\- s — la^. Dividing the value of ^v' [4560] by that of &v [4557], we get, successively, by using ^"^^^^■f^ an = a-i, a'n'= «'-* [3709'], i5«' m.a'n' e m.a'~i e m.ai e ■ r^rz-^wn In applying this formula to numbers, we must vary the accents in the elements, so as to conform to the notation used in this book, as is done in [4560"', &;c.]. [4569a] * (2709) The inequality of s, here referred to, is given in [1342] ; that of s', depending VI. XV. §41.] VERIFICATION OF SEVERAL INEQUALITIES. 325 If we suppose i = 5, we shall have, in the motion of Venus in latitude venu» [4569o-, 4295], the inequality [4295], thlEani.. ôs' = — 0',312535 . sin. (5 n"t — 4 n't + ôi"—4>e'—è'). [4570] Multiplying the coefficient of this inequality by ■;yy^ [4569'], we get, in the motion of the earth in latitude, the inequality [4569/], & s" = 0',22869 1 . sin. (4 n" i — 3 n' Ï + 4 s" — 3 s' — ; f"*^^^ ^ and, by direct calculation, we have found, in [4312], the inequality, 6 s" = 0S234256 . sin. (4 n"t — 3 ti' Ï+4 s"— 3 s'— 6') ; [4572] which differs but little from the preceding. on the same angle, is similar, the accents being changed so as to adapt them to the value of s'. Instead of this formula, we may use [4295J], observing that the second line of this expression is used in computing the inequalities which are taken into consideration in ^ ' [4569 — 4576]. The expression of 5 s, deduced from this part of [4295i], may be simplified; because the divisor n^ — \n — i.{n — «')P' '^^'^Y ^^ reduced to the form [45G9c] (' . ()i — «') .\i .[n' — n) -\- 2 n] . Hence this part of i5 s becomes, _B(.-i) 6s=im'.n^.a^a'. -^ tttT-: --r-r.y .sin. h'.(»i' i — ?i i + s'— s) + n <+ s — nj ; [45G9d] 7 being the inclination, and II the longitude of the ascending node of m', upon the orbit of m. This expression may be simplified, from the circumstance, that, in the terms here [4569e] taken into consideration, the divisor i.{n — n') is very nearly equal to 2n [4569]. Substituting this, and ?irt^=l, in [45^9 (Z] ; making also a slight reduction in the ["^^'^^l arrangement of tlie terms depending on i, we get, ôs=+im'.^a'.{aa')i (■_l)_(j!^^"^^>^,^ -7 •sin-K''-l)-K^- ^^ i + s'-s)-\- n't + b'-UI [4569g:] Changing the elements ?«, a, n, s, U, &c. into m, a', n', s', n + 180'^. Sic. M,-pq.-i respectively, and altering the sign of i — 1, which does not affect 2i"~'' [956,956'], we get, 5 s'=— my a .{act') . rr—rr:-, -—, • 7 .sin.U?:— l).(?l'^— « ? + /— e) + ?i C + £ — H^ [4569i] Hence we evidently perceive, that S s is to as' as m'\/a' to — m\/a, as in [4569']. [4569A;] Now, the values n', n" [407 6A] make 5 n" — 3 ?i' quite small, in comparison with ?i'. This corresponds with the value assumed for i.{n' — 7))-{-2n [4569], supposing i = 5 ; [45G9i] hence we get [4.570 — 4572]. In like manner, 3 ?i'' — n" [4076A], is very small, in VOL. III. 82 326 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. [4573] The quantity 3 n" — if is small in comparison with n"' ; therefore, [4573] by making i =: 3 [456%, z], we obtain in ôs", an inequality depending on 3 n''t — 2 7f t -{- S s''' — 2 i" ; and in 6 5", an inequality depending on 2n''H — n''t + 2!'' — s\ The first of these inequalities is, by [4511], [4574] 6 s" = 0',662991 . sin. (3 n^'t — 2 n^t + 3 s"'— 2 1^ — n'''). n"' being the longitude of the ascending node of the orbit of Uranus upon [4574'] that of Saturn. Multiplying the coefficient of this inequality by -^y— , and we obtain in ôs"', the inequality, Uranus. ' ^ J ' [4575] 65"= — 2'',714213. sin. (2ra'"^ — w^^ + 2£^'— s"— n^'); and by [4531], this inequality becomes, by putting n^ = n"' + 1 80'' [4531', 4574'], [4576] is'' = — 2',921052 . sin. (2 n^'t — nU + 2 1"'— e" — n^') ; which differs but little from the preceding. 42. It follows, from § 69, of the second book, that if we suppose [4576'] i' n' — in to be very small relatively to n and n', and represent by,* [4577] R =3r m'.P. sin.(i'n't — int + i'e'—is)-}-m'.P'.cos.(i'n't—int + i's'—is), the part of the development of R, depending on the angle i'n't — i nt-\- i's' — i s ; it will produce, in 6v, the inequality, ..-„„ , comparison with ?r or n"' ; and this comes under the form [4569], by putting f = 3; hence we get [4574—4576] ; observing in [4576], that n^ = n" + 180<f. * (2710) Using the value Tg = i'n' t — i nt + i'^ — i s [4019a], and (j, = 1 [3709] , we find that the tenns of i2, Se, eSzs, which correspond to each other in [1287,1288,1297], become, [45776] -R = "*'• P- sin- ^9 + '">''• P- c°s- ^9 ; 14577c] , e = "^4^ . 5 _ (^) . Sin. T,- (f) . cos. T, \ ; m—m ( \de J \de J ) 2 m. an , , . . ^V=rr-r-r7.-< , h [4578] Vl.xv.H2J VERIFICATION OF SEVERAL INEQUALITIES. S27 — ( — V COS. {i'n' t — int-\-i' ^ — is — nt — s + ra) and in 6 v', the inequality,* , , ( — ( -—A . COS. (i' n' t — int4-i's' — is — n't — s' + ra') , 2m. an' \ \de'J ^ ^ ' \ 6v'^ T-; ' ^ , >. [4579] ^^" = ^$S. • 1 O •^°^- ^«- (S) • ^'"- ^^ 1 • f4577d] Substituting these in 8v [4556], using for brevity, W=znt-\-s — w [3702a], and reducing, by [22, 24] Int. we get, as in [4578], „ , (— ("— Vrsin.Tg.sin.fF+cos.rg-cos.fF), tn-in }_|./^V(sii,.2'9.C0S.fr— C0S.r9.sin.fF)< 2m'. an ( /dP\ ,_, „^, , /dP'\ . ,^ „n > * (2711) Proceeding in the same manner as in [4558a — c], and using Tg [4577a], we have, àTg= — indt, à'T^==i'n'dt; [4578a] hence the differential of It [45776], relative to the characteristic d, becomes, dK = — m'.in.{P.cos.T9 — P'.sin.rg^ [45786] Substituting this in m'.à'R'= — m.dR [4558i — c], we get, d' «' = m . i n . { P. cos. Tg — P'. sin. Tg } . [4578c] Integrating this, relatively to d', and observing that the divisor i'n' is, by hypothesis, very nearly equal to in [4576'], we get, for the corresponding terms of R', depending on the angle Tg, the following expression ; R' = m.{P. sin.Tg + F. cos. Tg]. [4578rf] From this value of R' we may compute 5 v', in the same manner as we have found S v [4578], from R [45776]. It will, however, be rather more simple to use the principle of [4578e] derivation, by observing, that if we take the differential coefficient of R [4577è], relative to e, multiply it by 2andt, then take its hitegral relative to t, and change Tg into Tg — W, it will become equal to S v [4577e]. In like manner, if we take the differential Mgyg^-, coefficient of JÎ' [4578rf], relative to e', multiply it by 2a!n'dt, take its integral relative 328 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. It follows also, from ^71, book ii. that the same terms of R [4577] produce, in 6 s, the inequality,* [4580] 5 s = .—. : ' — ^ .cos .(i'n't— i u t4- i' s' — i I — n t — e-|- n) ) — (--\ . sin. (i' 7i't — int-^i's' — is — nt — s + n) \ y being the tangent of the respective inclinations of the orbits of m and m', [4580'] and n the longitude of the ascending node of the orbit of m' upon that of m [42956— c]. If we increase the argument of the inequality of 6 v [4578], by [4580"] nt -{- s — ra, and multiply its coefficient by e ; also, if we increase the argument of the inequality of 6v' [4579], by n't + s' — ■us', and multiply [45811 its coefficient by — ^-— . e' ;t lastly, if we increase the argument of the ■■ ■' m \/a "^ inequality of 6 s [4580], by nt-\-i — n, and multiply its coefficient by — 2-/, the sum of these three inequalities will be. [4582] Qm'.an '+^•(f)+«■•(")+^Q^™'■<'■■»■'-'•»'+^"■-")l to t, and afterwards change Tg into Tg — TV [.3726a], it will produce the following expression of & v', which is equivalent to [4579] ; l"»^l *"'=l5^ h (© ■ »■ (''.-"'■) + Q • ''■•■ (T,-rr)f * (2712) If we put, for brevity, T.2 — i'n't — int-{'A—gè;, also 7 = tang, ip/ [4580a] ["1337'^ 3739] . the assumed value of R [1337"] becomes, R = m'.k .jK cqs.%. [45806] Substituting this in the expression — fi — ] . andi, we find that it becomes equal to the expression of s or 5 s [1342] ; provided the angle T, be decreased, after the integration, by the quantity v — Ô/, or by the angular distance of the body m from the ascending node of the orbit of Î»' upon that of ?« [1.337']. In the present notation v — 0/ is represented by the quantity nt-\-s — n, neglecting terms of the order e [429.5i — c]. The same process being performed upon the assumed value of R [4577], produces the expression of 6s [4580]. [4581a] t (2713) This factor is equal to ^^^'-.«'[4560/]. VI. XV. §42.] VERIFICATION OF SEVERAL INEQUALITIES. 329 Now, P and P' are homogeneous functions of e, e', 7, of the dimension i' — i, and i' is supposed greater than i ; therefore the preceding function is equal to,* '2m!.an.{i'—i) ^ ^_p ^^^(■^^,f_ j-,j^_)_^v^.,_ • ^^ _^p,_ sin. {i'n't—int+i's'—i 1 . [4583] Now we have, in àv, the inequality, [1304], '^'^— p""^ — ^ • S -f*- ^°^- (*'"'^ — î'ni-f iV — i £) — P'. sin. (iV/ — int+i's' — is) \ ; [4584] hence it follows, that if we represent by 6v = K. sin. (i' n't — i nt-\-i' i'—iB — nt — i-\- O), [4585] the inequality of 6 v, depending on the angle i'n't — in t + i'^' — i s — n t — £ ; and by 5 v' = K'. sin. {i' n' t — int + i s' — z ; — n' t — s'+ O), the inequality of &v', depending on the angle i'n't — int-\-i'i' — z's — 71' t — s'; [4586] lastly, if we represent by 6s = K". sin. (i! n't — int-^i' s — z s — nt — e + 0"), [4587] the inequality of à s, depending on the angle i'n't — i nt + i's' — is — nt — s, we shall have,t Ke . sin. (i' n't — int -{- i' i — is — « -f- O) + '^'. K' e'. sin. (i' n' t — int + i' s' — is— ^'+ O') —2 K" 7 . sin. (i' n't — int + i's' — is — n-{- O") ^''^^^^ = _ U^Illl.H. ^''"'~"'^ . sin. (if n't — i nt + i' s' — is + Q); * (2714) From [957'"] it appears, that any part of B, depending on angles of the form i'n't — int, must be composed of terms in e, e', 7, of the orders i' — i, ^ ^^ i' — Ï + 2, &IC. ; and by neglecting all, except the first, on account of their smallness, they must be of the order i' — i; and therefore homogeneous in these quantities. Now, if we r-rpoii put, in [1001a], a = e, «'= e', a"=y, m = i' — i, and then, successively, .^''':= P, d^')= P', we get, «•(f)+''-(S)+-(f)=P-^)- Substituting these in [4582], we obtam [4583]. t (2715) The first member of [4588] is equal to the sum of the inequalities ôv, èv', VOL, III. 83 330 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. [4588'] n IS 6v = H. sin. (i' n' t — i n t -\- i' s' — i « + Q) being the inequality of 6 v depending on the angle i' n' t — int + i' s — i^ . The quantity 5 n' — 2 n [4076/i] is very small in comparison with ^'^^^^"^ and we have, in àv [4282], the inequality, [4589] àv = l %690443 . sin. (5 n' < — 3 n i + 5 s' — 3 s + 43" 1 8 "■ 32'). ^,^^^^^, The inequality 6 s [4283'], depending on 5n't — 3nt + ôs' — 3e, Venus. insensible; and we have, in 6v' [4293] , the inequality, [4590] Sv' = — 0',333596 . sin. ( 4 n' < — 2 n/ + 4 s' — 2 ^ — 39' 30'" 30'). Lastly, we have, in ô v [4283], the inequality, [4591] 6t)= 8',483765 . sin. (.5 n't — 2nt + 5.-' — 2.- — 30" 1 3- 36'). In this case ?''= 5, i = 2 [4584,4591]; and we have, by what precedes [4585 — 4591], the following equation of condition; r,690i43. e . sin. (5 n' t — 2nt + 5 s' — 2 i —^ + 4^3' 18'" 32') [4592] — 0',333596.e'. -"'^.sin.(5n'i — 2ni + 5£' — 2s_^'_39''30'"300 = _ 8',483765 . ■^"'"'^" ^. sin. (5 n7 — 2 n^ + 5 / — 2 . — 30" 13"' 360- 71 The first member of this equation is,* [4593] 0%359753 . sin. (5 n' i — 2 w i -f- 5 e' — 2 s — 28" 27"" 33') ; the second member is, [4594] 0',3605 . sin. (ôn't — 2nt + ô s'— 2 s — 30" 13'" 36') ; and their difference is insensible. Ss, [4585,4586,45871; multiplied respectively by e, — - — . e', and — 2y; the [4588a] '»•« arguments being also increased by nt-\-s — w, n't-^e' — ts', nt-\-s — IT, respectively, according to the directions in [4530"— 4531]. Now, it is shown, in [4530"~4583J, that this sum is equal to the expression [458.3], which is the same as that of &v [4584], multiphed by — — . ( ) ; and if we suppose this expression of Sv to be reduced to the [45886] '^^ \ " / form [4588'], this product will be represented by the second member of [4588]. * (2716) This is easily olrtained, by reducing the two terms of the first member [4593a] of [4592] into one, by the method [4282/t — /], after substituting the values m, m', a, a', he. [4001,4079,4080]. VI. XV. §43.] VERIFICATION OF SEVERAL INEQUALITIES. 331 We may verify, by the preceding theorems, many of the corresponding inequalities of Jupiter and Saturn ; but as all the inequalities of these two [4594'] planets have been verified several times, with much care, by different computers, this last verification is unnecessary. 43. The inequality of m, produced by the action of m', and depending on the argument n' t + -' — ^'j is expressed as in book ii. ^ 50, 55, by,* à V = ^-^ — - . (0,1) . e'. sin. (n' t + i' — «')• [4595] The inequality of ?«', produced by the action of m, and depending on the argument nt-\-£ — ct, is, 6 v' = " . (1 ,0) . e . sin. (Jl / + e — ra). [4596] n.{n- — n-') ^ ' ^ [4596'] The coefficients of these two inequalities are, therefore, in the ratio of — (0,1) . 7i\ e' to (\,0).n'\ e; now we have, in [1093], therefore, if we put Q for the coefficient of the inequality 6 v [4595], we shall find, that the coefficient of the inequality ôv' [4596], will be represented by, f^.î.Q [4595/]. [4598] m. a ■' e * (2717) The term of 5 ti depending on n't-{-s' — zs', is deduced from that in [1021], depending on G'', by putting i := 1 ; whence we obtain, Sv='^ . G<". e'. sin. {71' t + s'— ^'}. [4595„] Now, from [1018, 1019, 1073], we have, in the case of i=l, ^"=-«H-^)-*«H7^)=™r„-(«'')= [45956] G-^-,£^..^-==-„7:(^)-(0.1)- [4595.] Substituting this value of G<'\ in 5v [4.595rt], it becomes as in [4595]. The value of Sv' [4596] may be directly computed in a similar manner ; or it may be obtained more simply by derivation from [4595] ; changing to, a, n, e, he. into m', a', n', e, &lc. ; and [4595rf] the contrary ; observing, that by these changes, (0,1) becomes (1,0), according to the 332 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. The inequalities of this kind have been verified, either by means of this equation of condition, or by that of the preceding expression of Q. Thus, the action of Jupiter produces, in the earth, the sensible inequality [4307], [4599] 6 v"= — 2',539884 . sin. (n" t + s"— ^"). This inequality, by what precedes, is represented by [4595], [4600] 6 v"= ^^,^^.,_^..,^ . (2,4) . e\ sin. (n- 1 + a-- «-) ; TheEarth ^^d wc havc (2,4) =6^947861 [4233]. If we substitute this, in [4600], juiuer. also the values of n", tf\ é" [4077, 4080] ; then multiply the result by the expression of the radius in seconds, we shall obtain, [4601] âî)" = — 2'-,5401.sin.(?i'-'/; + e"'— ra'^). The action of Uranus upon Saturn, produces, in the motion of Saturn, the inequality [4466] , [4602] 5 î;v ^ _ p^oil647 . sin. {itH + i"— :='')• Saturn and Uranus „v „vo pV Multiplying its coefficient by ^v73"' Ti [4'598], we obtain, in Uranus, the inequality, [4603] 5z;'i=0%214852. sin. (m'^ + s' — ^') ; and the direct calculation has given, in [4525], [4604] à v"' = 0',2 14857 . sin. (nU + i"— ^'). notation in [1085, &;c.]. Comparing the values of o v, & v' [4595, 4596] , we get the first [4595e] expression of [4595/] ; and by substituting the value of (1,0) [4597]; also n^=zcri, _9 n'^=a' ^ [-3709'], we get successively the last expression [4595/], which is equivalent to [4598]; . , (1,0) n'^e . m\/a n'^e m . a^ e rdw^n à v= • . .01) = ; — , • —r-, .ov = — -— — r- . -, .ov. L^^^^.'J (0,1) Ji3e' viVa' n3e' m'.a'^ e' [4600o] * (2718) The expression [4600] is similar to [4595], changing m, m', &ic. into m", m'", Stc. Vl.x%'i. 5,41.] ON THE MASSES OF THE PLANETS AND MOON. 333 CHAPTER XVI. OiN THE MASSES OF THE PLANETS AND MOON. [4604] 44. One of the most important objects in the theory of the planets is the determination of their masses ; and we have pointed out, in [4062 — 4076'], the imperfections of our present estimation of these values. The most sure method of obtaining a more accurate result, is that which depends on the development of the secular inequalities of the motions of the planets ; but until future ages shall make known these inequalities with greater precision, we may use the periodical inequalities, deduced from a great number of observations. For this purpose, Delambre has discussed the numerous observations of the sun, by Bradley and Maskelyne ; from which he has obtained the maximum of the inequalities produced by the actions of Venus, [4604"] Mars and the moon. The whole collection of these observations of Bradley and Maskelyne, makes the maximum of the action of Venus greater than that which corresponds to the mass we have assumed for Venus [4061], in the ratio of 1,0743 to 1 ; hence the mass of Venus is ■gyVe^-aa of that of [4605] the sun. The observations of Bradley and Maskelyne, when we take them Mass or Venue. separately into consideration, give nearly the same results ; therefore, it is probable, that this estimate of the mass of Venus is not liable to an error of a fifteenth part of its value. [460.5'] Hence it follows, incontestably, that the secular diminution of the obliquity of the ecliptic approaches very near to 154"=49',9. To reduce it, as some astronomers have done, to 105"= 34% we must decrease the [4606] mass of Venus one half;* and this is evidently incompatible with the [4606] [4604" * (2719) This appears, by substituting q"= — M', t = \00 [4606], in [4074c]; whence we get, very nearly, — .34'= — 50' — 31V' j consequently, i>-'= — h, nearly. VOL. III. 84 [4606o] 334 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. observations of the periodical inequalities, produced by Venus, in the motion [4606"] of the earth. The best modern observations of the obliquity of the ecliptic are too near to each other, to determine this element with accuracy. The observations of the Arabs appear to have been taken with much care. They [4607] made no alteration in the system of Ptolemy ; but directed their attention particularly to the perfection of the instruments, and to the accuracy of their observations. These observations give a secular diminution of the obliquity r.„«-„ of the ecliptic, which differs but very little from 154"^49%9. This [460/ ] .... . diminution is also confirmed by the observations of Cocheouking, made in China, by means of a high gnomon ; and it appears to me, that these observations may be relied upon for their accuracy. Delambre has also determined, by a great number of observations, the [4607"] maximum of the action of Mars upon the motion of the earth. He has Mass of found this action to be less than that which corresponds to the mass we have assumed for Mars [4061], in the ratio of 0,725 to 1 ; making the [4608] mass of Mars ^TTis-a^? of that of the sun. This value is probably not quite so accurate as that of the mass of Venus, because its effect is less ; but, as the data [4076], from which we have determined the mass of Mars, [4608] in [4075, &c.], are very hypothetical, it is important to ascertain the error which might result from this cause, in the theory of the sun's apparent motion. Now, the observations of Bradley and Maskelyne, combined together, or taken separately, concur in indicating a diminution in the mass of Mars ; therefore, we shall decrease the preceding inequalities, produced [4609] by Mars, in the earth's motion, in the ratio of 0,725 to unity. These changes, in the masses of Venus and Mars, produce sensible alterations in the secular variations of the elements of the earth's orbit. Longitude We find the longitude of the earth's perihelion to be represented by the 'p^ihe'Ln. following expression ;* [4610] Long, perihelion © = ^"+ 1 . 1 P,807719 + t\ 0',00008 16482 ; the coefficient of the equation of the centre of the earth's orbit is represented by. * (2720) The expression [4610] is computed as in [4331], changing the masses of [4610a] Venus and Mars, as in [4605—4608]. The formulas [4611,4612] are computed in hke manner as [4330, 4332], respectively. VI. xvi. •§, 44.] ON THE MASSES OF THE PLANETS AND MOON. 335 Coeff. equat. centre © = 2^— ï.0',171793 — ^-.0',0000068194. [461 1] Lastly, the values of p" and (f [4332], become, p'= t. 0'-,080543 + f. 0',000023 1 1 34 ; ^^^^2 q"= — ^0%521142 + i% 0',0000071196. Hence it follows, from [4074c, 461 3«], that the secular diminution of the obliquity of the ecliptic, in this century, is equal to 52',1142.* Using these data, we find, by the formulas of ^ 31, f ^^t.\ 5b", 5921 + 3M 1 1 9 + 42556",2 . sin. {t . 1 55",5927 + 95°,0733) — 73530",8. cos. (i.99", 1 227) — 1 7572",4. sin. {t . 43",0446) [4C14] = t . 50%412 + 2H7"- 57^+ 13788^2 . sin. {t . 50%412 + 85''33"'570 — 2382.3-,98.cos.(^.32%l]58)— 5693%5. sin. (i.l3^9465) ; tPixcilT . —^v, ,.... ^v..^ ,.. -. , ..y...^^ ,^.^. , .^ ,^.^^j """'-J [4615] + 5082",7. cos (^ . 43",0446)— 28463",6 . sin. (^ 99", 1227) Corrected = 23'' 28*" 17%9— 1191',2 — 5892',8.cos. (i.50%412 + 85'* 33" 57^ o7'.r precessioii + 1646%8 . cos. (i. 13%9465)— 9222»-,2 . sin.(/.32'-,1158); JJJS P- tic for the year (, ailer the |'= i .155",5927 + 3°,11019 — 3°,11019 .cos.(ï.99",1227) etocV 1750. — 14282",3 . sin. (t . 43",0446) [4616] = ï.ô0%4120 + 2'^47'"57^ — 2'*47'"57^cos. (L32%1158) — 4627^5 . sin. (M3^9465) ; pApparent"! L orbit. J V'= 26°,0796 — 3676",6 .^1— cos. (< . 43",0446) ^ — 10330",4. sin. {t. 99", 1227) [4617] = 23''28'"17%9— 119P,2.p— cos.(Ll3%9465)| — 334^,05 . sin. (^.32%1 158). * (2721) The chief term of the value of q" [4612] is — i.0%521142, and by putting < ^100, it becomes q"^ — 52',1142. This represents, by [4074a — c], the secular [4613a] variation of the obliquity of the ecliptic, corresponding to the second formula [4612]; in the original work it is printed 160",85=52',1154, and it is thus quoted in [3380n]. t (2722) The formulas [4614 — 4618], are computed in precisely the same manner as 336 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. The increment of the tropical year, counted from 1750, is, then, Increment of the represented by, [4618] Increment of the year = — 0'"'^000086354 . { 1— cos. (t . 13',9465) I — 0''=^000442198 . sm.(t . 32^1158). Hence it follows, that, at the time of HIpparchus, the tropical year icas [4618'] 10^9528 sexagesimal seconds longer than in 1750. The obliquity of the ecliptic was then greater by 955%2168. Lastly, the greater axis of the sun'' s [4618"] orbit coincided with the line of equinoxes, in the year 4089 before our era ; it tvas perpendicular to that line in 1248. The mass of the moon has been determined by the observations of the tides in the port of Brest ; and, although these observations are [4619] far from being so complete as we could wish, yet they give, with considerable precision, the ratio of the action of the moon, to that of the sun, upon the tides of that port. But, it has been observed, in [2435 — 2437'], that local circumstances may have a very sensible influence on this ratio, and also on the resulting value of the moon's mass. Several methods have been pointed out, in the second book, to ascertain this influence ; but they require very exact observations of the tides. The observations which have been made at Brest, leave, in their results, such a degree of uncertainty, as makes us fear that there may be an error of at least an eighth part, in the value of the moon's mass. Indeed, the observations of the equinoctial and solsticial tides, ^^'^^^^ seem to indicate, that the action of the moon upon these tides is augmented one tenth part, by the local circumstances of the port. This will decrease, [4621] '^y one tenth, the assumed value of ihe moon's mass ; and, in fact, it appears, by several astronomical phenomena, that the assumed value [4321] is rather too great. The first of these phenomena is the lunar equation, in the tables of the [4692] sun's motion. We have found, in [4324], 8',8298 for the coefficient of this inequality, supposing the sun's parallax to be S^S [4322]. It will be [4357— 4360, 4362], ahering the masses of Venus and Mars, as in [4605,4608]. We have previously spoken of this change of the masses of tliese two planets, in [3380rt,&;c.], [4614a] ^^^ j^^^g ^jg^ given the formulas of Poisson and Bessel [3380p,y], for the determination of the precession and the obliquity of the ecliptic. Vl.xvi. §44.] ON THE MASSES OF THE PLANETS AND MOON. 337 8',5767,* if the sun's parallax be 8^56, which is the value deduced [4622'] from the lunar theory, as will be seen in the following book. Delambre has determined the coefficient of this lunar equation, by the comparison of a very great number of observations of the moon, and has found it equal to 7',5. [4623] If we adopt this value, and also the second of the above estimates of the sun's parallax, which several astronomers have deduced from the last transit "lass. of Venus over the sun's disc, we find the mass of the moon to be ^-i ^ of [4624] the earth's mass [4622&]. The second astronomical phenomenon is the nutation of the earth's axis. We have found, in [3378a], the coefficient of the inequality of the nutation to be equal to 10',0556;t supposing the mass of the moon, divided by the [4625] cube of its mean distance from the earth, to be equal to triple the mass of the sun, divided by the cube of the mean distance of the earth from the sun [2706]. This makes the mass of the moon equal to ^i^^ of the earth's [4626] mass [4321]. Maskelyne has found, by the comparison of all Bradley's observations on the nutation, that the coefficient of this inequality is equal * (2723) The coefficient of this inequality, neglecting its sign, is -rv-—, multiplied by the radius in seconds 206265' [4.3141; and by substitutinsr — = f3-r-,and — =. ^.^' ■- -' ' •' ^ M 58,6 ' r" 3454" 1 m'snai- [4622a] [4321,432.3], it becomes ^^ X ^^A- ' ^ 206265'. Putting this parallax equal to 8',8, the coefficient becomes nearly equal to 8%8298 [4324] ; and by using the value of the parallax 8", 56 [5589], the coefficient becomes 8',58 nearly, as in [4622']. To reduce this to 7', 5, the value obtained by Delambre, we must decrease the moon's mass in the 7 5 1 1 ratio of the numbers 7', 5 to 8', 58, so that it will be equal to — ^x— ^— =7;=-, [46226] 8,58 58,6 6/ instead of -p^rK'f given by the author in [4624]. t (2724) The coefficient 31",036 = 10%0556 is computed, in [3376e], from the formula — — . — = 10',0556 ; in which X=3 [3376,-3079] represents the assumed [4625a] ratio of the lunar to the solar force on the tide. This value of X is used, in [4319], in computing the value of m [4321,4626]. Now, substituting X = 3, in [4625a], we obtain, ~ = \x 10',0556 = 13%4074 ; VOL. HI. 85 338 PERTURBATIONS OF THE PLANETS; [Méc. Cél. [4627] to 9',55 ; and this result makes the moon's mass equal to y'y of the earth's mass. Lastly, the third astronomical phenomenon is the moon's parallax. We shall see, in [5605], that the constant term contained in the expression of this parallax, when developed in a function of the moon's true longitude, is [4628] 3427%93; supposing the moon's mass to be ^^-g- of the earth's mass. Burg has computed this constant term, by means of a very great number of [4629] observations of the moon. He finds it equal to 3432',04 [5605] ; and, by the formulas given in the next book, this result will be found to correspond [4629'] with a mass of the moon, which is equal to ^l-g- of that of the earth.* [46256] substituting tliis value in the first member of the equation [4625rt], we get — -— . 13',4074, for the mitation, corresponding to any assumed value of X. If we put this equal to the vahie Q'jSS, obtained by Maskelyne [4627], we get, X 9,5500 , 9,5500 ^ ,.„ . , r , o j i. [4 625cl r~rT^= . o ,r.~,. ; hence x= — -— -r = 2,4i6, instead oi X=:3, used above; ^ !-{->■ 13,4074 cf,o574 and as the mass of the moon is proportional to X [3079] , it mil be reduced, from r^-^ [^•^^^1]'^° .5-^X3-5=71' as in [4627]. * (2725) The constant term of the parallax is — .(l-f-ee) [5311] ; and by substituting D / M \i [4629a] the value of — [5324], it becomes of the form A . f j ; A being a function of the known quantities a, e, he, which are independent of M, m. Now, by using the value of — =^j— [4628], we obtain the constant term [5330'], corresponding to the latitude whose sine is \/^ ; also the constant term 3427',93 [5605] of the horizontal parallax ; hence we have, [46296] ^. ^^^V=3427%93, and ./2=3447V32; so that the constant term of the horizontal parallax is, [46290] 344r,32.(^-^)*. Putting this equal to the constant term of Burg's tables 3442',44— 10S40^3432'',04 [5605], we get, [4629^] '^= gig) = 1,01341 = 1 +^ nearly, as in [4629']. Vl.xvi.§44.] ON THE MASSES OF THE PLANETS AND MOON. 339 Hence it appears, from all three of these phenomena, that we must decrease a little tlic mass of the moon, deduced from the observations of the tides at Brest ; therefore, the action of the moon on the tides in that port, is [4630] sensibly increased by local circumstances. For the numerous observations, both of the heights and intervals of the tides, do not permit us to suppose this action to be less than triple the action of the sun. The most probable value of the moon's mass, which appears to result from these various phenomena, is -g-i^y of the earth's mass.* By using this [4631] value, we find 7',572,t for the coefficient of the lunar equation of the solar [4632] tables, and 3430%88,t for the constant term of the expression of the [4033] moon's parallax. We also find 9',648 . cos. (longitude of the moon's node), [4634] for the inequality of the nutation, and — 18%03.sin. (long, moon's node), ^ [4C35] *■ (2726) Subsequent observations of tbe tides at Brest, induced tbe author to reduce this value of X [3079], from X = 3 to X= 2,35333 [11905]; making the mass of the [463ia] moon equal to jj.Vjrir of that of the earth [11906]; as we have ah-eady remarked in [33806', &ic.]. We may observe, that the value of X= 3 [4318,4319] corresponds with [46316] 71» 1 ..ml 5ri=rr-; [4321], and that X is proportional to m ; hence we get, m the case of — =— — M 58,6 L J' IF ' o ' M 68,5 [4631c] [4631], the value x=3. ^ = 2,566, as in [4637]. t (2727) This equation of the earth's motion is proportional to — [4314] ; and if m 1 [4632a] we suppose — = — - [4321], it becomes 8^,58 nearly, as in [4622'] ; but if we use Jrl. OQjO >K 1 58 6 [46326] Ti.=^TT [4631], this equation becomes 8^,58 X ;;3V = 7'%34; which differs a little Jn ob,o oo,o from [4632]. X (2728) Substituting M=68,5.m [4631c], in the constant term of the moon's parallax [4629f], it becomes 3447%32 . r^y= 3430^8, as in [4633]. Moreover, by [4633a] substituting X= 2,566 [4631c], in the coefficient of the nutation [4625 J], it becomes, '^ .13S4074=|^^.13',4074 = 9%648, as in [4634]. [46336] 1+X ' 3,566" § (2729) The coefficients of the inequalities in the nutation and precession are represented, in [3376e,/, 3378,3380], by _f^'' „„ — ^/^^ , .cot.2A ; which are to [4635a] [4638] 340 PERTURBATIONS OF THE PLANETS ; [Méc. Ce]. for the inequality of the precession of the equinoxes. The ratio of the [4636] moon's action on the tides to that of the sun is then 2,566 [4631c] ; and as the observations of the tides in the port of Brest make this ratio equal to 3 [46316], it appears evident that it is increased, by local circumstances, [4637] in the ratio of 3 to 2,566. Future observations, made with great exactness, will enable us to determine, with precision, these points, in which there remains, at present, some slight degree of uncertainty. Jupiter's mass appears to be well determined ; Saturn's has still some degree of uncertainty [4635c], and it is a desirable object to correct it. This may be done by observing the greatest elongations of the two outer [4638'] satellites, in opposite points of their orbits, in order to have regard to the ellipticity of the orbits. We may also use, for this purpose, the great inequality of Jupiter [4417], when the mean motions of Jupiter and Saturn shall be accurately determined ; for these mean motions have a very sensible influence upon the divisor (5 n" — 2 n"y, which affects this inequality. It appears probable, that the mean annual motion we have assigned to Jupiter, must be increased, one or two centesimal seconds ; and that of Saturn, decreased, by nearly the same quantity. The periodical inequalities of Jupiter and Uranus, produced by the action of Saturn, afford also a tolerably accurate method of determining the mass of Uranus. The value we have assigned to the mass of Uranus, depends on the [4641] greatest elongation of its satellites, which were observed by Herschel. These elongations should be verified with great care. With respect to Mercury's mass, we may use, in ascertaining its value, the inequalities it produces in the motion of Venus. Fortunately, the influence [4642] of Mercury on the planetary system is very small ; so that the error, depending on any inaccuracy in this estimate of its mass, must be nearly insensible. [4639] each other as 1 to — 2.cot.2A. Hence, if we suppose the inequality of the nutation to [46356] ^^ 9^,648, as in [4634], that of the precession will be — 2x9%648.cot.2 A; and by using 2A = 52°,1592 = 46''56"'35S8, it becomes — 18',03, as in [4635]. Before concluding this note we may observe, that the late estimates of these masses, [4635c] ^^ different astronomers, have already been given in [4061 (/—m]. VI. xvii. <§> 45.] ASTRONOMICAL TABLES. INVARIABLE PLANE. 341 [4643] CHAPTER XVII. ON THE FORMATION OF ASTRONOMICAL TABLES, AND ON THK INVARIABLE PLANE OP THE PLANETARY SYSTEM. 45. We shall now proceed to explain the method which must be used in constructing astronomical tables. We have given the inequalities, in longitude and in latitude, to a quarter of a centesimal second ; but the most perfect observations do not attain to that degree of accuracy ; so that we may simplify the calculations, by neglecting the inequalities which are less than a centesimal second. We must form, by means of a great number of observations, selected and combined in the most advantageous manner, the same number of equations of condition, between the corrections of the elliptical elements of each planet. These elements being already known, to a considerable degree of accuracy, their corrections must be so small that we may neglect their squares and higher powers ; and by this means the equations of condition become linear.* We must add together all the equations in which the coefficients of the same unknown quantity are considerable ; so that from these sums we can form the same number of fundamental equations as there are unknown quantities ; and then, by [4644] elimination, we may obtain each of the unknown quantities. We can also find, by the same method, the corrections which may be necessary in the assumed masses of the planets. If the numerical values of the planetary inequalities be accurately calculated, which may be ascertained by a careful verification of the preceding results ; we may, with each new observation, * (2730) We have given the form of an equation of this kind, in [849(f] ; and have shown, in [84 9a — r], how to combine any number of them together, by the method of the [4644a] least squares ; which process is now generally used, in preference to that in [4644]. VOL. III. 86 342 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. form another equation of condition. Then if we determine, every ten years, the corrections resulting from the combination of these equations with all the preceding ones, we may, from time to time, correct the elements of the orbits ; and by this means obtain more accurate tables of the motions ; [4645] supposing that the comets do not produce any alteration in the elements ; and there is every reason to believe that their action on the planetary system is insensible. 46. We have determined, in [1162'], the invariable plane, in which the sum of the products of the mass of each planet, by the area its radius vector describes about the sun, when projected upon this plane, is a maximum. If we put 7 for the inclination of this plane to the fixed ecliptic of 1750, and n for the longitude of its ascending node upon that plane, we shall have, as in [1162'], 2 . OT . \/a.{\ — ef) . sin. (p . siu. ê [4646] [4647] [4648] [4649] tang. 7 . sm. H: tang. 7 .cos. 11: 2 . m.^a.(i— ee).COS. 9 ' 2 . JW . ^«.(1 — ee) . sin. (p . COS. ê 2.OT.\/«.(1 — ee).COS.(p The integral sign of finite differences 2 includes all the similar terms relative to each planet. If we use the values of m, a, e, cp, and ê, given for each of these bodies, in [4061 — 4083], we shall find, by these formulas, 7= l''35'"3P; n=]02''57"'29\ Then, by substituting for e, tp, 6, their values, relative to the epoch 1950 [4081—4083, 4242, &c.], we shall obtain, 7= l''35'»3P; n= 102'^ 57™ 15'; which differ but very little from the preceding values [4648]. This serves as a confirmation of the variations we have previously computed in the inclinations and in the nodes of the planetary orbits. VI.xvm.§47.] ACTION OF THE FIXED STARS. 343 CHAPTER XVIII. ON THE ACTION OF THE FIXED STARS UPON THE PLANETARY SYSTEM. 47. To complete the theory of the perturbations of the planetary system, there yet remains to he noticed those, which this system suffers, from the [4G49'] action of the comets and fixed stars. Now, if we take into consideration, that we do not accurately know the elements of the orbits of most of the comets ; and, that there may be some, which are always invisible to us, by reason of their great perihelion distance, though they may act on the remote planets ; it must be evident, that it is impossible to determine their action. Fortunately, there are many reasons for believing, that the masses of the comets are very small ; consequently, their action must be nearly insensible. We shall, therefore, restrict ourselves, in this article, to the consideration of the action of the fixed stars. For this purpose, we shall resume the formulas [930, 931, 932], C / fl 7?\ ^ "\ exprès- a.cos.vfndt.r.ûn.v. j 2/di2 + r.r^j \ ) ^7^1, — a. sin.v .fndt.r. COS. V. < 2fdR-}-r.(--—\ [4650] General 6r:^ 1_£_2_^. ^x) [4651] t^ • \/l — ee 2r.d.ôr+dr.Sr 3a jr> , 2a „ ,^ /^7?\ 5 — 7 .ffndt.dR-] .fndt.r. ( — ) 6V^ '^^^ ^—7— '— ^^; (Y) [4652] V/l— ee ' ^ ^ ^ ,, . /dR\ . . . /dR\ a. cos. V.J ndt.r.Hva.v.y — 1 — a. sm.v. J ndt. r .cos.r.( — j S s = /./1 ^-^- (^ [4653] f^.y/l — ee 344 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. [4654] We shall put m' for the mass of the star ; x', y', z', its three rectangular co-ordinates, referred to the sun's centre of gravity ; r', its distance from that centre ; a:, y, z, the three co-ordinates of the planet m ; and r, its distance from the sun. We shall have, as in [3736], [4655] R = >»'-(^^'+yy'+~^ ') _ m! Developing the second member of this equation, according to the descending powers of r', we shall have,* [4656] ^=—7 + -jTT—i^'- ^^ ,-5 ' &C. [4656'] We shall take, for the fixed plane, that of the primitive orbit of the planet ; and we shall have, by neglecting the square of z,f [4657] X = r . cos. V ; y = r . sin. v ; z = r s. [46581 Putting I for the latitude of the star m', and U for its longitude, we obtain, Î [4659] a;' =r'. cos. Z. C0S.C7: ?/'=?•'. cos. /. sin. C/; 2' = r*. sin. /. * (2731) Putting, for brevity, xx' -\-yy'-\-zz'=zrr'.f; and, as in [914'], [4655a] a;2-|-/-f22 = 7-2, x'^+y'^ -}-z'^=r'% we find, that the last term of [4655] becomes, by successive reductions, as in [4655c] ; [46555] _„/.|(.,'_:,)2+(y'_y)^+(z'-_^)«f*=-_m^{r'2-2r'r/-^r«r*=-p.{l-2('^^')J"* [4655c] __-'_-7?:;^^^^U?.™'.f^i^*'^^f _ &c. r' r \ r'^ J 2 r' \ r'2 / Substituting this in [4655], we find that the first term of [4655] is destroyed by the second term of [4655c], and the whole expression of R becomes, by a slight reduction, as in [4656]. t (2732) The values of x, y [4657], correspond with those found in [926'— 927]. [4657a] The value oî z^rs [4657] is the same as that in [931"], changing as into s, to conform to the present notation. \ (2733) The radius vector of the body m' is /, and its latitude above the fixed [4659a] plane /. Hence it is evident, from the principles of the orthographic projection, that the projection of r', upon the fixed plane, is /.cos./; and the perpendicular z', let fall from m'. Vl.x™i447.] ACTION OF THE FIXED STARS. 346 Hence we deduce, by neglecting the descending powers of r', below Z"^,* [4059] /?= — ^' + '1^' • )<2— 3.cos.=Z— 3.cos.=/.cos.(2i;— 2C/)— 65.sin.2/.cos.(v-t7)|. [4(iG0] Now, ?"', /, and U, vary nearly by insensible degrees ; hence, if we put R^ [4661] for the part of R, divided by r'^, and neglect the square of the excentricity of the orbit of m: also, the term dependine; on 5, which is of the order of lb' [4661'] the disturbing forces, that m suffers by the action of the planets ; we shall have,t fAR==R- ^-^' . (2 - 3 . cos.^Z) ; [4662] r.(g) = 2i?, - [4662-] upon the fixed plane, is equal to /.sin./, as in [4659]. Now, this projected radius r'.cos.l, makes the angle U with the axis of x [4658, Sic], and 90'' — U with the axis of «/. Hence we easily obtain expressions of x', y', similar to those of x, y [4657], and which may be deduced from them, by changing r into r'.cos.l, and v into U, as in [4659]. [46596] [4659c] * (2734) Substituting the values of J", y, &c. [4657,4659], in the first member of [4660a], reducing, developing and neglecting terms of the order s^, we get, by using [24, 6, 31] Int. the following expressions, \xx'-\-yy'-\-zz'l^^^r"r'^.\cos.I.(cos.v.cos.U-\-s\n.v.sm.V)-{-s.sin.I\^ [4660a] =r^/^. ^cos.Z.cos.(w — [7)-)-s.sin. /}^ ^7-^r'^.{cos.^Lcos.^(« — [7)4- 2s. sin. Z.cos.Z.cos.(« — U)l =r2r'2.^cos.2Z.[|+icos.(2i; — 2Z7)]+«.sin.2/. cos. («—[/)}. [4660i] Now, the first and second terms of [4656], are the same as the first and second terms of [4660] respectively ; so that if we neglect terms of the order mentioned in [4659'J, we shall find, that the remaining part of [4656] becomes, — -^ -{xx'+y y'+z 2' p. [4660c] Substituting in this the expression [4660e], it produces the three last tenns of il [4660]. t (2735) If we use the symbol R,, we shall have, from [4660,4661], iî^=^-Ç.j2 — 3COS.2/— 3cos.2Z.cos.(2y — 2f7) — 65.sin.2/.cos.(j;— Z7)i ; [4662a] ^= — ~'+^' [46626] VOL. III. 87 346 PERTURBATIONS OF THE PLANETS; [Méc. Cél. [4662"] Then, if we put n = 1, which is nearly equivalent to the supposition, that the sun's mass is equal to unity [3709], we shall obtain from the formula [4651],* [4669e] [4662rf] [4662rf'] [4662e] [4662/] [4662e-] [4662^1] [4662i] [4662fe] [4662 i] [4662m] [4662n] [4662o] [4662o'] [4662p] [4662fl] The characteristic d affects the elements of the orbit of the body m, namely, r, », s, inc. ; but does not affect those of the body ?»', as r , I, U,hc.; hence the differential of [46626] becomes, àR=àR^. Integrating this, and adding, as in [1012'], the constant quantity m'g, to complete the integral, we get /dfi=/d/î,+ m'^. Now, as r', I, U, are nearly constant, we may neglect their variations, and then the quantity di?, will be the complete differential of /?, ; so that we may write R, for fdR/, hence the expression [4662f/] becomes fdR^R^-j-m'g. If we neglect terms of the order e^, in the expression of r [1256], it becomes as in [4664]; and if we substitute this in the expression of r^.dv [1256], we easily obtain the expression of ndt [4664]. By inadvertence, the author has given a wrong sign to the term depending on e, in the value of r [4664] , wliich in the original work is r = a.{l-j-e.cos.(D — ro)|. This affects the numerical coefficients of the formulas [4666,4666',&tc.], but does not alter the general results [4669',4673,&.c.]. Putting, for brevity, h equal to the coefficient of r"^, in the expression of R^ [4662a], we have. h= ^.{2— 3. cos.^l— 3. cos.-l.cos.{2v — 2U) — 6s. sm.2l. COS. {v—U)l R.= h.i whence dR\ :2A r = . r we obtain the Substituting this in the partial differential of R [4662è], relatively to following expression, \d^) \d^) ~ T ' multiplying this by r, we get [4662']. If we determine the constant quantity g, as in [1016",&.c.],by making the coefficient of t vanish from the expression of ôv, we shall find, by putting fj.=l, and neglecting e^, that the terms of 5v [4652], necessary to be noticed in finding the constant quantity, are, a.f{3fàR+2r.(~y.ndt. Substituting the values [4662e, 4662'], it becomes, a ./{I R,-{-3m'g).ndt ; and if we retain only the constant part of R,, the preceding expression will vanish, and we shall have the constant part ot Sv equal to nothing, by putting 7 Ri-\-3m'g = 0; or m'g=^ — i-H^r Now, the constant part of R^ is evidently obtained, by putting r^=a, and retaining only the two first terms of [4662a]. Hence we get, , 7 m'. cfi ,- n o 7\ ^*^=-"Ï27T-(^~^-'=°'-')' and fàR [4662c] becomes as in [4662]. In the original work the numerical coefficient is — \, instead of — ^^. * (2736) From [4662e, 4662'], we get. VI.xnii.§47.] ACTION OF THE FIXED STARS. 347 6 r = 4 a . COS. v .fn dt.rR,. sin. v — 4.a. sin.i; ./n dt.rR,. cos. î;. [4663] Substituting the following expressions [1256, 4662/, &c.], r = a.\\-e.cos.(v-^)\; n dt = dv .{I— 2e .cos. (v — ^)\; [4664] and neglecting under the sign /, the periodical terms, affected with the angle [4665] V, we shall have,* ndt.r.R,cos.v=-^^\{(l-hcos.H).e.cos.^-hcos.H.e.cos.(^-2U)]; [4666'] 2/d /Î + r . (^) = 4 i?,+ 2 m'g. [4663a] Substituting this in [4651], also ii=l, and neglecting c^, we get, — =4. COS. ■y./'ri(/«.r iî,. sin. i; — 4.sin.i). AitZ^.r-R^.cos.v a "^ ' ^ ' [4663a ] -(-2m'^.cos.t)./nf/<.r.sin.î) — 2mg.sm.v.f7idt.r.cos.v. This differs from [4663], in the terms multiplied by g. The two expressions would agree, if we were to take the arbitrary constant quantity g [4662d] equal to nothing ; but this J would be inconsistent with [4662?t, 4668]. * (2737) From [4662/], we obtain ndt.rR,^h.ndt.r^. Now we have, by neglecting e^ r^ = a^.\l — .3 e.cos.fw— «)| [4664]; multiplying this by ndi [4664], [4666a] we get, ndf.r^=:a^.dv.\l — 5e.cos.(« — ■a)\; hence, ndt .rR = h.a^.dv.\\ — 5e.cos.(«j — ro)}. [46666] Multiplying this successively, by s'm.v, and cos.d, we get, by reduction, ndt .r R^. sm.v^= h .a^.dv .\s\n.v — f e.sin.-a — f e.sin. (2« — ra)|; [4666c] ndt . r R^ . cos.t) =h.a^.dv .\ cos.t) — | e . cos. is — f e . cos. (2 v — o) | . [4666rf] The second of these expressions may be derived from the first, by augmenting each of the angles v, zs, U, by 90'; as appears, by making this change in the second members ; no [4666e] alteration being made in /, /, &c.; so that h [4662^-] may remain the same. If we suppose the plane of x y, to be the primitive orbit of m, the latitude « will be of the order of the disturbing forces of the planets, which is neglected in [4661'] ; and then A [4662A:] is composed of the two terms, ^.(2 — 3.COS.2/), _|^3.3.cos.2Z.cos.(2t;— St;-). [4666^] Tliese are to be substituted in [4666c], and those terms retained, which do not contain the 348 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. [4666"] which gives, by considering ïs, Z, r', U, as very nearly constant,^ 5r 3m'.a^.v [4667] - = .{(1 — I . COS.-/). e . sin.(z) — w) — a. cos.'Z. e . siu.(t)4-w — 2C7)|. [4666i] [4666fe] [4667a] [46676] [4667c] [4667d] [4667e] [4667/] [4667gr] [4667A] angle v, or its multiples [4665] ; consequently, the first of these terms of h must be combined with the second of [4666c] ; and the second of these terms of h, with the third of [4666c] ; hence we shall have, m'. a?, dv [4666A] ndt. rR, . sin. v = • 4,. '3 .f— (2 — 3.cos-2Z).Ae.sin.«— V-e.cos.2/.sin.(«— 2t7)|; which is easily reduced to the form [4666]. In like manner we may compute [4666'] ; or, we may obtain it much more easily, by derivation from [4666], by increasing the angles V, «, V, by 90'', as in [4666e]. These results are free from the error in the value of r [4662^] ; and if we compare them with those given by the author, in the original work, we find, that we must multiply his expressions by — 5, to obtain those in [4666,4666'] ; or, in other words, we must change e into — 5e, in his formulas. * (2738) Putting, for brevity, . 5m'. a^ ,, „ „,, I5m.a? „ B^ —^—- .COS. I . e ; 16r'3 we find, that the integrals of [4666, 4666'] become, very nearly, f7idt.7-R^.sïn.v= — ^î'. sin. 13 — 5u.sin.(^3 — 2U) ; fndt.r R^.cos.v= — Av .cos.zi-\-Bv.cos.(-Gs — 2U). Multiplying the first of these expressions by 4.cos.i', the second by — 4.sin.K, and taking the sum of the products ; putting — sin. ra.cos.v-j-cos.a.sin.i) = sin. [v — to) ; — sin.(« — 2U).cos.v — cos. (to — 2t/).sin.i) = — sin.(w + TO — '2U) ; we get, for the terms in the first line of [4663a'], the following expression, 4 . cos. v.fndt .rR^. sin. i' — 4 . sin. « .fn di .rR^. cos. v ^4:.A.v.sm.(v — to) — 4.B .v.s\n.{v-\--a — 2U). Again, if we multiply together the expressions of r and ndt [4664], neglecting e^, we obtain, ndt ,r =^ adv .\l — 3e .cos. [v — ^)}. Multiplying this, successively, by sin.r, cos. d; reducing and retaining only the terms, which are independent of the angle i', we get. ndt.r.s'm.v = — adv. ^e.sm. -a ; fii dt .r.s'm.v = — a w . | e . sin. to ; ndt .r.cos.v -adv .^e. COS. zi. fndt.r. COS. d = — av.^c. cos. i Multiplying these integrals, respectively, by 2?n'^.cos.D, — 2m'g.s'm.v ; taking the sum of the products, and reducing, by means of [4667 J]; then substituting the value of VI. xviii. § 47.] ACTION OF THE FIXED STARS. 349 Now we have,* Sr a — ôe . COS. (v — ^) — e (5 3 . sin. (v — ^). [4668] Secular variations Comparing together the two expressions [4667, 4668], we obtain,t iriïè excentrici- ly ami 1 'Î m' /7^ï' perihelion. 5 e = '-""■" . cos.^Z . e . sin. (2 ^ — 2U) ; [4669] S^= _ iî^!i'.p_3 . cOS.^Z — I . COS.^Z. COS. (2^ — 2U)]. [4669'] Thus the action of the star m' produces secular variations in the excentricity and in the longitude of the perihelion of the orbit of the planet m; but these variations are incomparably smaller than those arising from the action of the other [4669"] planets. For, if we suppose m to be the earth, r' cannot, by observation. m'g [466-2y], we finally get, for the second line of [4663a'], 2 m'g . cos . V .fn dt .r.s'm.v — 2 m'g . sin . î) .fn dt.r. cos. v = 2m'g.^.ave . j — sin.w.cos. v + cos.ra.sin.i) | [4667i] = m'g.^ave.sm.{v — «) = 9~^3~'^^ — f .cos.'^/j.e.sin. (w — «). [4667fc] Adding together the expressions [4667e,^-]; re-substituting the values of A, B [4667cr], we get the complete value of — [4663rt'], as in [4667]. In the original work, the author [466/^] .3 m'. a^v makes the factor, which is without the braces, equal to V^ — ) instead of „ ,, » and the numerical coefficient of the second term within the braces is erroneously printed — f instead of — J. These mistakes are the consequences of using erroneous values of g and ;■ [4662o', p]. [4667m] * (2739) In finding the variation of r [4664], we must neglect that of v, arising from the constant quantity g' [4662/i], and the expression becomes as in [4668] ; which is MQgg;,-] similar to [3876]. The signs of the terms in the second member of [4668], in the original work, are incorrect, by reason of the mistake mentioned in [4662^-]. [4669o] t (2740) From [21] Int. we have, sin.{f + 3— 2f7] =sin.[(y — îs) + (2w — 2t7)} = sin. (v — ra) .COS. (2zs — 2U)-\-cos. {v — a) .sin. (2zi — 2U). Substituting this in the last term of [4667], and then comparing separately, the coefficients of sin. (i- — 3j) and cos.(i' — w), in the two expressions [4667, 4668] ; we get, by a slight [46696] reduction, the values of 6 e, ôtz [4669,4669']. These expressions agree with those given VOL. III. 88 350 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. t 3 [4670] be supposed less than lOOOOOa, and then the term '^^-^, does not exceed,* [4671] m'^.0%000000001 ; t denoting the number of Julian years. This is incomparably less than the [4671'] secular variation of the excentricity of the earth's orbit, resulting from the The ao- action of the planets, which, by [4244], is equal to. Stars has [4672] — ^.0',093819, no sensi- ble effect ontheex- uiiloss we suDDOsej that m' has a value which is wholly improbable. Hence centrici- J L ties and perihelia of the we may conclude, tJiat the action of the stars has no sensible influence on the p'ian'e"ts. secular variations of the excentricities and perihelia of the planetary orbits. [4673] In like manner, it is evident, from the development of the formula [4653], that their action has not any sensible influence on the position of these orbits.f by Mr. Plana, in the Memoirs of the Astronomical Society of London, vol.ii. p. 354 ; which he deduced from the formulas [1253a]. Hence we see, that the method here f4669cl proposed by La Place, to find 5e, Sw, when it is correctly followed, leads to an accurate result ; and is not liable to the objection made by Mr. Plana, in the same page of that volume, namely; that it is nowise fit for the intended purpose, without taking into view other circumstances, which render the calculation more complicated. We may remark, that in [4669(/] the original work, the factor -y- [4669], is printed J ; and, in [4669'], the factors —, [4669e] — f.cos.% are changed into , — J. cos.®/, respectively *■ (2741) The value of ?-=100000a [4670], corresponds to an annual parallax of [4671a] about 2'; and we have nearly «=1295977'.^ [4077]; substituting these in — j^- [4670], it becomes as in [4671] ; or simply, by supposing 7ft'= the sun's mass = 1, ^0',000000001. The secular variation of c" [4330a], is nearly represented by, [46716] -^.<= — i.(0%187638).^=— 0',093819.!: [4244,4672]; which is much greater than the expression [4671]. t (2742) If we substitute rs = z [4657], in R^, R [4662i, «], and retain only *■ "•' the terms of R, containing z, we find, [46736] R= j-j^ .sm.2l.cos.{v — U), and f— j = — — ^ .sm.2/.cos.(t) — U) . VI.xvrii.§47.] ACTION OF THE FIXED STARS. 351 PFe shall noio examine into the influence of the attraction of the stars on the mean motion of the planets. For this purpose, we shall observe, that the formula [4652] gives, in d.6v, the term* dAv = ^andt .R/, from [4674] which we deduce the following expression,! d.iv='^ .ndt. {2 — 5. co^.H]. [4675] We shall put r'=r;. (1-aO; /=Z^. (1-/30; [467G] r' and / being the values of r' and /, in 1750, or when / = 0; we shall [407G'] have, in &v, the variation, f 6 V = ^-^ . ri — # . cos.%) . a . ,1 f— ^-^. sin. 2l.^.nt\ [4677] Substituting this in S s [4653], we find that the terms are multiphed by the very small factor of the order [4670,4671], which renders them insensible [4671']. * (2743) This expression arises from the last term of 5v [4652], which, by neglecting quantities of the order e^, and putting jj-^I [3709], becomes, 2afndt.r. (^\ = 2afndt.2R, [4662'] . [4674a] Its differential gives, in d.5v, the terra Aandt.R^, as in [4674]. This would be increased to landt.R^, by noticing the term depending on fàR [46-52], as we have [46746] seen in [4662o']. This increases the terms [4675, 4677] in the ratio of 7 to 4. m' r- t (2744) The two first and chief terms of R, [4662a], are -^ • (2— Z.cos.^l) . Substituting the value of r [4664], we obtain the part rrj • (2 — 3. cos.-/), which [4675o] does not contain v ; hence, the term of d.Sv [4674], becomes as in [4675]. Î (2745) The value of / [4676] gives cos.? =cos.(/,— p^/,) = cos./,-f-p^sin./,, [467Ga] by using [61] Int. Squaring this, neglecting i^, and putting 2 . sin. /,. cos. /,= sin. 2/, [31] Int., we get cos.^I = cos.^/^ -\-^t- sin. 2 Z, ; whence, 2 — 3.cos.2/ = 2.(l— f.cos.2/,)— 3(3/.sin.2/,. [46765] If we now substitute the value of r' [4676], in the first member of the following expression, and then develop it according to the powers of a, neglecting a*, we get, -^.ndt = —j^ .7idt.{l-lr3o-t). [4676c] 352 PERTURBATIONS OF THE PLANETS ; [Méc. Cél- We cannot ascertain, by observation, the value of aï, but may determine that of f3t. Now, if we suppose, relatively to the earth, f3=l"^ U%324, and [4678] ,,' ^^ 100000 a ; the quantity -^ . ^nf becomes, very nearly,* ^/ [4679'] which is insensible, from the time of the most early observations on record. The expression of d.&v, contains also, by what precedes, the terms, f [4680] dM^=—i.m\a\ndLfAy-^~.cos.{v—U)\—&tri;M\ndt.Sp^l.co^^^^^ Multiplying together the expressions [4676i, c], we get the value of d .&v [4675], nearly, [4676d] tZ.5u=z — JL ,ndt.{\ — %.cos.H\A —- .(1 — J.cos.^/J.an^t/i 7T-.sm.2Lsnic/;. We may neglect the first term of this formula, because we have taken the constant quantity [4676c] ^ so as to make the coefficient of t vaftish from the expression of 5« [4662«]. Integrating the other two terms of [4676c/], we get the value of h%) [4677]. * (2746) The assumed values of (3, r/, are taken within reasonable limits ; since the value of p corresponds to an annual variation in tlie latitude of the star, of about a third of a [4679a] sexagesimal second ; and the value of r/ to an annual parallax of nearly two sexagesimal seconds. To reduce the expression [4678] to numbers, we have, in the case of i=\, nt = circumference of the circle = C,2S.31 ; hence, generally, [46796] „ < = 6,2831 .t; also, p t = 0',324 . t. The product of these two expressions is, [4679c] fi7i(^ = 2',0357 . t^ Substituting this, and rf=l0^.a, in the first member of [4679], it becomes as in the [4679d] second member of that equation. This is wholly insensible in observations made 3000 years ago ; since, by putting t = — 3000, and 7«'=:1, it becomes less than O',00000002. t (2747) If we now notice only the terms of R, R, [4662rt, b'\, depending on s, we obtain, [4680a] Rz= — f ."^ .«.sin.2/.cos.(i' — U) ; whence, r.f—j^ — 3.^^.s.sin.2/.cos.(t) — ['). If we substitute the value of r [4664], and neglect terms of the order es, we get, [46806] R= — f.m'.a^. } ~^ — .cos.(« — Z7)| ; r. f — 1 = — 3. -^ .s.sm.2/.cos.(«— t'). Now, if we put (J.= l, and neglect e^ ; noticing only the terms of [4652], where R VI. xviii. §47.] ACTION OF THE FIXED STARS. 363 Now we have,* s = t .— . sin. I' — t--r-- cos.i) ; t^^^^] (It (It which gives, by neglecting the quantities multiplied by the sine or cosine of the angle v,f s.sin.2/ , y.^ ^ sin. 2/ (. dq . jj dp ^^^ tj\ . [4682] [4683] consequently, t ,. , s. sin. 21 . rr\ , sin.2Z ^dq . ^r '^ V tt} i d . -^3- • COS. (t' -C/) = ^ . -^^. I ^ . sin.t/-^^ . cos.f/ ^ . Hence we obtain, in d.6v, the term,^ ^.,^,=_l^î^^„^^^.sin.2^5^.sin.C7-^.cos.f7^ [4684] 4 r^ i dt dt 3 [4G80c] explicitly occurs, we get, for its differential, dJjv = 3a.7idt.fdR~\-2a.7idt.r. (-77 )• Substituting, in the first term of this expression, the value of R [46S0J], we get the first term of [4680] ; and we obtain the last terni of [4680], by the substitution of the second L •' 1 expression [4680&] in the last term of [4680c]. * (2748) This expression is similar to that in [3802, Sic.]. We may remark, that the author, in this article, has interchanged the usual signification of the symbols p, q [3802]. [4681n] We have rectified tliis, by changing jj into q, and q into p, in all the formulas [4681 — 4685] of the originLil work. sin.2i t (2749) If we multiply the expression [4681] by — ^.cos.(i' — U), and reduce the products by [19, 20] Int., we shall obtain the equation [4682], by retaining only the [4682a] terms which are independent of 1;; or in other words, by retaining only the terms |sin.f/, ^cos.f-^, of the expressions sin. v. cos. (» — U), and cos. v . cos. (u — U), respectively. X (2750) If we neglect the variations of r', /, U, in the second member of [4682], the sign d may be considered as the complete differential, and then the signs /d, mutually [46e3aJ counteract each other, and they may be prefixed to the first member of [4682], without altering its second member; hence we get [4683] from [4682]. §(2751) Multiplying [4683] by — ^ .m'.d^n dt, nnd [4682] by —Q.m'.({\ndt, we find, that the sum of the products, or the second member of [4680], is as in [4684]. l'1684«| Integrating this, we get, [4685]. VOL. in. 89 354 PERTURBATIONS OF THE PLANETS ; [Méc. Cél. consequentlj, we have, in 6 v, the secular inequality, [4685] 5x, = . _^,nt\ sm. 2Z. \ -/-.sm.U — — - cos.t/ S . 8 r* I (It dt 5 We have given the values of -— , -j-, [4332], relatively to the earth. If we substitute them in the preceding term of &v [4685], we shall find that it is insensible,* even in the most ancient observations. * (2752) From [4332] it appears, that — -, — -, are each less than F, and sin. 2/, sin.ZJ, cos.f^, do not exceed unity : tlierefore, sin.2/. < •—- .sin.t/ — —- .cos.U )■ , maybe [4685a] ■' ' i dt dt <, ■' considered as less than V ; and then, the expression [4635], neglecting its sign, becomes 21 m'.a? less than — . —7 — .nt^.V ; which is found to be insensible, in [4679']. 8 r'3 Other terms of the like nature with those which have been particularly examined, in this [46856] chapter, may be deduced fiom the formulas [4651 — 4653] ; but it is evident, from what we have seen, that they must be excessively small ; so that it is hardly worth the labor of a r../-o^ 1 more thoroudi examination. The author himself, seems to have considered the subject as [4685c] ° ... ... not deserving much attention, and has been quite negligent in the numerical details of this article ; so that it has been found necessary to correct the text in several places, as we have [4685(/] already remarked. In writing the notes on this volume, soon after its first publication by the author, I pointed out the mistakes in this chapter. It has since been done by Mr. Plana, in vol. ii. p. 351 of the Memoirs of the Astronomical Society of London, for 1826; and [4685e] subsequently by La Place, in the Connaissance des Terns, for the year 1829, page 250. The method used by Mr. Plana is more direct and simple than that of the author. It consists in [4685/] substituting the value of R [4660], in the formulas [5787—5791], and making the necessary reductions ; but, as the process is simple, it is unnecessary to enter minutely upon it. Mr. Plana remarks, in page 355 of the work above-mentioned, that the action of the fixed stars affects the mathematical accuracy of the equation [1114], [4685g-] ''^- '» • \/» + e'^. m'.\/a'-\- kc. = constant ; as we have already remarked in [11 146]. This is evident ; for, if we increase the quantity e, in the first member, by the expression &e [4669], the second member will be increased by the quantity, [4685?,,] ■2f.5e = ^^^ . cos.a/.t-.s.n.(2:3 — 2t/), nearly; which destroys the constancy of the second member. The same defect exists in the equation [1134 or 1155]. VI.xviii.§47.] ACTION OF THE FIXED ST^VRS. 355 It is easy also, to satisfy ourselves, that the preceding results hold good, relatively to those planets which are the most distant from the sun. Hence it a|)pears, that the action of the stars upon the planetary system, is so much [4680] decreased, by reason of their great distance, that it is wholly insensible. It now remains to compare with observations, the formulas of the planetary perturbations, given in this book, and particularly those of the two great inequalities of Jupiter and Saturn. This comparison requires too much detail for the limits of the present work; we shall, therefore, merely remark, that before the discovery of these great inequalities, the errors of the ^ best tables sometimes amounted to thirty-five or forty minutes ; and now they do not exceed a minute. Halley had concluded, by the comparison of modern observations, the one witli the other ; and also, by comparing the modern with the ancient observations, that Saturn's motion is retarded, and Jupiter's accelerated, from age to age. On the other hand, Lambert ascertained, from the comparison of modern observations alone, that Saturn's motion was then [4088] accelerated, and Jupiter's motion retarded. These phenomena, apparently opposed to each other, indicated, in ihc motions of the two planets, great inequalities of a long period, of which it was important to know the laws and the cause. By submitting to analysis their mutual perturbations, I discovered the two principal inequalities [4i34, 4492] ; and perceived, that the phenomena, observed by Halley and Lambert, naturally arise from them ; and, that they represent, with remarkable accuracy, both ancient and modern observations. The magnitude of these inequalities, and the great length of the period of revolution, to complete v.'hich requires more than nine hundred years, depend, as we have seen, on the nearly commensurable ratio which obtains between the mean motions of Jupiter and Saturn. This ratio gives rise to several other important inequalities, which I have determined, and these inequalities have given to the tables the precision they now have. The same analysis, being applied to all the other planets, has enabled me to discover, in their motions, some very sensible inequalities, which have been confirmed by observation. I have reason to believe, that the preceding formulas, computed with particular care, will give a still greater degree of precision to the tables of the motions of the planetary bodies. IVrii-iS cl' diflcit'iii ufilerà. [4693] SEVENTH BOOK, THEORY OF TriE MOON. The theory of the moon has difficulties peculiar to itself, arising from the magnitude of its numerous inequalities, and from the slow convergency of the series by which they are determined. If tlie body were nearer to the earth, the inequalities of its motion would be less, and their approximations more converging, But, in its present distance, these approximations depend on a very complicated analysis; and it is only by a very particular attention, and a nice discrimination, that we can determine the influence of the successive integrations, upon the various terms of the expression of the disturbing force. The selection of co-ordinates is not unimportant for the [4692] success of the approximations. The sun's disturbing force depends on the sines and cosines of the moon's elongation from the sun, and on its multiples. Their reduction to sines and cosines of angles, depending on the mean motions of the sun and moon, is troublesome, and has but little convergency, on account of the moon's great inequalities. It is, therefore, advantageous to avoid this reduction, and to determine the moon's mean longitude in a function of the true longitude, which may be usciul on several occasions. We may, then, if it be required, determine accurately, by inverting the series, the true longitude, in a function of the mean longitude. It is in this way we shall consider the lunar theory. To arrange conveniently these approximations, we shall divide the inequalities, and the terms which compose them, into several orders. We shall consider as quantities of the first order, the ratio of the sun's mean motion to that of the moon, the excentricity of the orbit of the moon or earth, and the inclination of the moon's orbit to the ecliptic. Thus, in the expression of the mean longitude, in a function of the true longitude [5574 — 5578], the principal term of the moon's equation of the centre is of the first order [5574]. The second order includes the second term of that equation : the vil. Introd.] INTRODUCTION. 357 reduction to the ecliptic ; and the three great inequalities, known under the names of variation, erection, and annual equation [5575]. Tlie ^ *' ■' inequalities of the third order are fifteen in number [5576]. The present tables contain all these inequalities, together with the most important ones of the fourth order; and it is on this account, that they correspond with the observations made on the moon, with a degree of accuracy that it will be difficult to surpass ; and to this great correctness we are indebted for the important improvements in geography and nautical astronomy. The object of this book is to shoio, in the first place, that the law of universal gravity is the only source of all the inequalities of the lunar motions; and then, to use this law as a method of discovery, to perfect the theory of these inequalities, and to deduce from them several important elements of the system of the ivorld ; such as the secular equations of the moon, the parallaxes of the moon and sun, and the oblateness of the earth. A judicious choice of the co-ordinates, and well conducted approximations, with calculations made carefully, and verified several times, ought to give the same results as those derived from observation ; if the law of gravity, inversely as the square of the distance, be the law of nature. We have, therefore, endeavored to satisfy these conditions ; which require [4(i95] the consideration of some very intricate points ; the neglect of which is the cause of the discrepances, that have been observed in the previously known theories of the moon. It is in these points, that the main difficulty of the problem consists. We may easily conceive of a great many diffijrent and new methods of expressing the problem by equations ; but it is the discussion of all those terms, which are of themselves very small, and acquire a sensible value, by the successive integrations, which constitutes the important and difficult part of the process, when we endeavor to make the theory agree with observation ; which should be the chief object of the analysis. We have determined all the inequalities of the first, second and third orders, and the most important ones of the fourth order, continuing the approximation to quantities of the fourth order inclusively ; and retaining those of the fifth order, which arise in the calculation. For the purpose of comparing this analysis with observation, w^e may observe, that the coefficients of Mason's lunar tables are the result of the comparison of the theory of gravity with eleven hundred and thirty-seven observations [\g^m;-] of Bradley, made between the years 1750 and 1760; that the eminent VOL. III. 90 [4697] [4698] 358 THEORY OF THE MOON ; [Méc. Cél. astronomer Burg has rectified these tables, by means of more than three thousand of Maskelyne's observations, from 1765 to 1793; and, that the corrections he has made are small ; with the addition of nine equations, indicated by the theory. The tables of both these astronomers are arranged in the same form as those of Mayer, of which they are successive improvements : and we ought, in justice to this celebrated astronomer, to observe, that he was not only the first, who constructed lunar tables, sufficiently correct to be used in the solution of the problem of finding the longitude at sea, but also, that Mason and Burg have deduced, from his theory, the methods of improving their tables. The arguments are made to depend on each other, in order to decrease the number of them. We have reduced them, Avitli particular care, to the form which is adopted in the present theory ; that is, to sines and cosines of angles, increasing in proportion to the moon's true longitude. By comparing these results with the coefficients of the present theory, we have the satisfaction of perceiving, that the greatest difference, which, in Mayer's theory, one of the most accurate heretofore published, amounts to nearly one hundred centesimal seconds [=32',4], is here reduced to thirty [9',8], relative to the tables of Mason, and to less than twenty-six centesimal seconds [^8',3], relative to the still more accurate tables of Burg. We could diminish this difference, by noticing quantities of the fifth order, which have some influence, as may be known by inspecting the terms of this kind already calculated. This is proved by the computation of the two inequalities [5286'", &c.], in which we have carried on the approximation to quantities of the fifth order. The present theory agrees yet better with the tables, relative to the motion in latitude. The approximations of this motion are more simple and converging than those of the motions in longitude ; and the greatest difference between the coefficients of my analysis and those of the tables, is only six centesimal seconds [= l',9], so that we may consider this part of the tables as being founded upon the theory itself. As to the third co-ordinate of the moon, or [4700] its parallax, we have preferred, without hesitation, to form the tables by the theory alone, which, on account of the smallness of the inequalities of the lunar parallax, must give them more accurately than they can be olitained by observation. The differences between the results of the present theory and those of the tables, express, therefore, the differences between this theory and that of Mayer, which has been adopted by Mason and Burg. These differences are so small that they are hardly deserving of notice ; but, as the [4699] vil. Introil.] INTRODUCTION. 359 present theory agrees better with observation than Mayer's, in the motion in longitude, there is also reason to believe, that it possesses the same advantage ' ' relative to the inequalities in the parallax. The motions of the perigee and nodes of the lunar orbit, afford also a method of verifying the law of gravity. In the first approximation to the value of the motion of the perigee, by the theory of gravity, it was found, by mathematicians, only one half of what it was known to be, by observation ; and Clairaut inferred, from this circumstance, that we must modify the law of r4702'| gravity, by adding to it a second term. But he afterwards made the important remark, that by continuing the approximations to terms of a higher order, the theory would be found to agree nearly with observation. The motion, deduced from the present analysis, differs from the actual motion only a four hundredth part [5231 J ; the difference is not a three hundred and fiftieth part ^ in the motion of the nodes [5233']. Hence it incontestabhj follows, that the laiv of universal gravitation is the sole cause of the lunar inequalities. Now, if we consider the great number and extent of these inequalities, and the proximity of the moon to the earth, we must be satisfied, that it is, of all the heavenly bodies, the best adapted to confirm this great law of nature, as well as to show the power of [47041 analysis, that wonderful instrument, without the aid of which it would be impossible for the human mind to penetrate into so complicated a theory, and that can be used, as a means of discovery, as sure as by direct observation. Among the periodical inequalities of the moon's motion in longitude, that which depends on the simple angular distance of the moon from the sun is [1705] important, on account of the great light it throws on the sun's parallax. It has been determined by the theory ; noticing quantities of the fifth order, and also the perturbation of the earth by the moon, which are indispensable [4'OC] in this laborious research. Burg found this inequality to be 122,38, by the comparison of a very great numl)er of observations. If we put this equal to the result by the theory, we obtain 8',56, for the sun's mean parallax ; being [4707] the same as several astronomers have found, from the last transit of Venus over the sun [5586]. An inequality, which is not less important, is that which depends on the longitude of the moon's node. Mayer discovered it by observation, and Mason fixed it at 7',7 ; but, as it did not appear to depend on the theory [4708] 360 THEORY OF THE MOON ; [Méc. Cél. of gravity, it was neglected by most astronomers. A more thorough examination of this theory led me to the discovery, that its cause is the oblateness of the earth. Burg found it, by a great number of Maskelyne's [4709] observations, to be 6,8 ; which corresponds to an oblateness of -a-ëi.-ôT [5593]. We may also determine this oblateness, by means of an inequality in the moon's motion in latitude ; which I discovered also by the theory ; and [4710] which depends on the sine of the moon's true longitude. It is the result of a nutation in the lunar orbit, produced by the action of the terrestrial spheroid, and corresponds to that produced by the moon in our equator ; so that the one of these nutations is the reaction of the other : and, if all the particles of the earth and moon were firmly connected together, by inflexible right lines, void of mass, the whole system Avould be in equilibrium about the centre of gravity of the earth, in virtue of the forces producing these two nutations : the force, acting on the moon, compensating for its smallness, by the length of the lever to which it is attached. We may represent this inequality in latitude, by supposing the lunar orbit, instead of moving uniformly on the ecliptic, with a constant inclination, to move, with the same conditions, upon a plane but little inclined to the ecliptic, and which always passes through the equinoxes, between the ecliptic and equator : a phenomenon which occurs in the theory of Jupiter's satellites, in a still more striking manner. Thus, this inequality decreases the [4712] inclination of the moon's orbit to the ecliptic, when the ascending node of that orbit coincides with the vernal equinox. This inclination is increased, when the ascending node coincides with the autumnal equinox, which was the case in 1755; in consequence of which, the inclination, as it was found by Mason, from 1750 to 1760, is too great. This point has been determined by Burg, by observations made during a much longer interval, noticing the preceding inequality ; and he has found the inclination [^^^"^1 to be less, by 3 ,7. At my request, this astronomer has undertaken to determine the coefficient of this inequality, by a very great number of observations ; and he has found it to be equal to — 8". The oblateness of [4714] the earth, deduced from it, is ^^t.t [5602], being very nearly the same as that which is computed from the preceding inequality of longitude, Thus, the moon, by the observation of her motions, renders sensible to modern astronomy the ellipticity of the earth, whose roundness was made VII. Introd.] INTRODUCTION. 361 kiioAvu to the early astronomers by her eclipses. The experiments on the pendulum seem to indicate a less oblateness,* as we have seen in the third book. Tills difference may depend on the terms by which the earth varies from an elliptical figure ; m hich may have some small effect in the expression of the length of the pendulum, but is wholly insensible, at the distance of the moon. The two preceding inequalities deserve every attention of observers ; because they have the advantage over geodetical measures, in giving the oblateness of the earth, in a manner which is less dependant on the irregularities of its figure. If the earth were homogeneous, these inequalities would be much greater than they are found to be by observation. They l"*'^^! concur, therefore, with the phenomena of the precession of the equinoxes, and the variation of gravity at the surface of the earth, to exclude its homogeneity. It results also, that the moori's gravity towards the earth, is composed of the attractions of all the particles of the earth; ivhich furnishes another proof of the attraction of all the particles of matter. Theory combined wàth experiments on the pendulum, the geodetical measures, and the phenomena of the tides, make the constant term of the expression of the moon's parallax less than by Mason's tables. It differs but [4716] very little from that which Burg computed from a great number of observations of the moon, of eclipses of the sun, and of occultations of stars by the moon. It is only necessary to decrease a little the mass of the moon, which was determined by the phenomena of the tides, to make this constant term coincide with the result of that skilful astronomer. This diminution is also [4717] indicated by the observations of the lunar equation of the solar tables, and by the nutation of the earth's axis. This seems to prove, that in the port of Brest, the ratio of the moon's action on the tides to that of the sun, is sensibly increased by local circumstances. Future observations of all these phenomena will remove this slight degree of uncertainty. One of the most interesting results of the theory of gravity, is the knowledge of the secular inequalities of the moon. Ancient eclipses * (2753) Later and more accurate observations give a different result, as may be seen, [4715a] by referring to [201 7ji, 2056î, &:c.]. VOL. III. 91 [4718] 362 THEORY OF THE MOON ; [Méc. Cél. [4719] indicated, in the moon's mean motion, an acceleration ; the cause of which was sought for a long time in vain. Finally, I discovered, by the theory, that it depends on the secular variations of the excentricity of the earth's orbit. The same cause decreases the mean motions of the perigee and nodes of the moon, while her mean motion is increased ; so that the secular equations of the mean motions of the moon, the perigee and the nodes, [4720] are always in the ratio of the numbers 1, 3 and 0,74 [5235]. Future ages ivill develop these great inequalities, which are periodical, like the variations of the excentricity of the eartli's orbit, upon which they depend. These will finally produce variations which amount, at the least estimate, to a fortieth part of the circumference [d''], in the moon's secular motion; [4721] and to a twelfth of the circumference [30''], in that of the perigee. Observations have already confirmed these secular inequalities in a remarkable manner. The discovery of them induced me to believe, that we must diminish, by fifteen or sixteen centesimal minutes, the present secular motion of the moon's perigee, which astronomers had determined, [4722] by comparing modern observations with ancient ones. All the observations, which have been made during the last century, have put beyond doubt, this result of analysis. We see, in this, an example of the manner in which the phenomena, as they are developed, throw light upon their true causes. When [4723] the acceleration of the moon's mean motion only was known, it could be attributed to the resistance of the ether, or to the successive transmission of gravity ; but analysis shows us, that both these causes produce no sensible alteration, either in the mean motion of the nodes, or in that of the lunar perigee : this is a sufficient reason for rejecting them, even if we were ignorant of the true cause. The agreement of the theory with observations, proves, that if the moon's mean motion is affected by any causes, besides the action of gravity, their influence is very small, and is not yet perceptible. [4724] This agreement establishes, with certainty, the constancy of the duration of a day ; which is an essential element in all astronomical theories. If this duration were now one hundredth part of a centesimal second [or 0',864] [4725] more than in the time of Hipparchus, the duration of the present century would be greater than in his time, by 365i centesimal seconds [or 315',576]. [4725] In this interval, the moon would describe an arch of 173',2, and the present mean secular motion of the moon, would appear to be augmented by the VII. Introd.] INTRODUCTION. 363 same quantity. This would add 4,4* to the secular equation, which is [472G] Ibuiid, by the theory, to be 10',1 81621 [5543], in the first century after the year 1750. This augmentation is incompatible with the best observations, which do not permit us to suppose, that the secular equation can exceed, by V,62, the result of the analysis [5543]. We may, therefore, conclude, that the duration of the day has not varied a hundredth part of a centesimal [4727] second, since the time of Hipparchus ; which confirms what has been found a priori, in book v. ^ 12 [3347,&.c.],by the discussion of all the causes which could alter it. To omit nothing which can have an influence on the moon's motion, we have considered the direct action of the planets upon this satellite, and have found, that it is of very little importance. But the sun, by transmitting to the moon the action of the planets on the elements of the earth's orbit, renders their influence on the lunar motions very remarkable, and makes it much [4728] greater than on the elements themselves ; so that the secular variation of the excentricity of the earth's orbit is much more sensible, in the moon's motion, than in the earth's. It is in this manner, that the moon's action on the earth, which produces, in the earth's motion, the inequality known by the name of the hmar equation, is, if it may be so expressed, reflected back to the [4729] moon, by means of the sun, but decreased in nearly the ratio of five to nine [5226]. This new consideration adds some terms to the action of the planets on the moon, which are of more importance than those depending on their direct action. We have investigated the principal lunar inequalities, resulting from the direct and indirect actions of the planets upon the moon ; [4/30] * (2754) If we neglect tlie term of the secular equation [5543], depending on P, and put (7=10', 181621, we may represent the moon's mean motion, in i centuries after 1750, by ni -{-ai-. If we substitute in this successively, i^ — J, i = -j-|, and take the "' difference of the two results, it will be found equal to n, whicii must, therefore, represent the motion between 1700 and 1800. In like manner, by putting successively i= — 20, i^ — 19, and taking the difference of the two results, we get n — 39 a, for the motion in the century included between the years 250 and 150 before the Christian era. The difference of these two results 39 a, represents the augmentation of the secular motion between these two epochs; and, if this quantity were increased 173%2, as in [4725'j, we must increase the ' '^■' value of a by ^VX 173',2 = 4%4, as in [4726]. [4720i] 3^ THEORY OF THE MOON ; [Méc. Cél. and, if we take into view the accuracy to which the lunar tables have been carried, it must be considered useful to introduce these inequalities. The moon's parallax, the excentricity and the inclination of the lunar orbit to the apparent ecliptic, and, in general, the coefficients of all the lunar inequalities, are likewise subjected to secular variations ; but, up to the [4731] present period, they are hardly sensible. This is the reason why we find now, the same inclination, that Ptolemy obtained from his observations ; although the obliquity of the ecliptic to the equator has sensibly decreased since the time of that astronomer; so that the secular variation of the obliquity affects only the moon's declination. However, the coefficient of the annual equation, having for a factor, the excentricity of the earth's orbit, its variation is sufficiently great to be noticed, in computing ancient eclipses. [4732] The numerous comparisons, which Burg and Bouvard have made, of Mason's tables, with the observations of the moon ; at the end of the seventeenth century, by LaHire and Flamsteed ; in the middle of the eighteenth century, by Bradley ; and the uninterrupted series of observations of Maskelyne, from the time of Bradley to the year 1800, give a result which was wholly [4733] unexpected. The observations of LaHire and Flamsteed, being compared witli those of Bradley, indicate a secular motion, exceeding by fifteen or twenty centesimal seconds, that which is inserted in the third edition of La Lande's astronomy ; which, in a hundred Julian years, exceeds a whole number of [4734] revolutions, by 307'^53"'12^ Bradley's observations, being compared with the last ones of Maskelyne, give, on the contrary, a smaller secular motion, by at least one hundred and fifty centesimal seconds. Lastly, the observations [4735] made within fifteen or twenty years, prove, that the diminution of the moon's motion is now decreasing. Hence, it becomes necessary to vary incessantly the epochs of the tables ; and it is an object of importance to correct this imperfection. This evidently indicates the existence of one or more unknown ^ ^ inequalities of a long period, which the theory alone can point out. By a careful examination, I have not been able to discover any such inequality, depending on the action of the planets. If there were one in the rotation of the earth, it could be perceived in the moon's mean motion, and might introduce the observed anomalies : but an attentive examination of all the causes which can alter the rotation of the earth, has more fully convinced [4737] ^^^ ^j^^^ j^g variations are insensible. Returning back, therefore, to the VII. Intiod.] INTRODUCTION. S6b exaniin;itioii of the sun's action on the moon ; I have discovered, that this action produces an inequality, whose argument is double the longitude of the node of tiie lunar orbit, jdus the longitude of its perigee, minus three times the longitude of the sun's perigee. This inequality, whose period is 184 [4738] vears, depends on the products of these four quantities, namely ; the square of the inclination of the moon's orbit to the ecliptic ; tlie excentricity of that orbit ; the cube of the excentricity of the sun's orbit, and the ratio of the sun's parallax to that of the moon. Hence it would seem, that it ought to [4739] be insensible ; but the small divisors it acquires by integration, may render it sensible, especially, if the most important terms, of wliich it is composed, are affected with the same sign. It is very difficult to obtain its coefficient by the theory, on account of the great number of terms, and the extreme [4740] difficulty of appreciating them ; the difficulty being much greater in this than in the other inequalities of the moon. This coefficient has, therefore, been ascertained by means of the observations made during the last century ; and I have found it to be nearly equal to 15',39. Its introduction in the tables [4741] must change the epoch and mean motion ; and I have also found, that we must decrease, by 31'',964, the mean secular motion, in the third edition of [4742] LaLande's astronomy, and have determined the following formula, which must be applied to the mean longitude given by these tables, the epoch [4743] of which, in 1750, is 188" 17'" 14',6 ; Equnt,,,,, nf 184 Correction of moon's mean long. = — 12',78 — 31 ',964 . i + 15',39 . ûn.E ; T4744] i being the number of centuries elapsed since 1750, and E the double of the longitude of the node of the lunar orbit, plus the longitude of its perigee, [47451 minus three times the longitude of the sun's perigee. This formula represents, with remarkable precision, the corrections of the epochs of those tables, which have been determined, by a very great number of observations, for the six epochs of 1691, 1756, 1766, 1779, 1789 and 1801. By a most scrupulous examination of the theory, I have not been able to discover any other lunar inequality Avith a long period ; hence, it appears to me certain, that the [4746] anomalies observed in the mean motion of the moon, depend on the preceding inequality ; and I do not hesitate, therefore, to propose it to astronomers, as the only means of correcting these anomalies.* * (2755) It has not been found necessary to introduce this equation in the new tables of Damoiseau, pviblished in 1824; since the elements lie has used, give very nearly the L4<46a] VOL. III. 92 366 THEORY OF THE MOON ; [Méc. Cél. We see, by this exposition, how many interesting and delicate elements hai^e been deduced, by analysis, from observations of the moon, and how [4747] important it is to multiply and improve them. Since, by the greatness of their number, and by their correctness, we may more and more confirm the various results of analysis. The error of the tables formed from the theory, which is given in this book, does not exceed a hundred centesimal seconds, except in very rare cases; [4748] therefore, these tables will give, with sufficient accuracy, the longitude at sea. It is very easy to reduce them to the form of Mayer's tables ; but, as in the problem of the longitude, it is proposed to find the time corresponding [4749] to an observed longitude of the moon, there is some advantage in reducing into tables, the expression of the time in a function of the apparent longitude. Considering the extreme complication of the successive approximations, and the correctness of modern observations, the greatest part of the moon's inequalities have heretofore been better determined by observations than by analysis. Thus, by deriving from the tlieory those coefficients which it gives with accuracy, and also the forms of all the [4750] arguments ; then rectifying, by the comparison of a great number of observations, the coefficients which it gives by approximations, with some degree of uncertainty : we must finally obtain very accurate tables. This is the method which has been used with success by Mayer and Mason, and lately by Burg, who, by pursuing it, and profiting by the late improvements in the lunar theory, has constructed tables, whose greatest errors fall short of forty centesimal seconds. However, it would be useful, for the perfection of astronomical theories, if all the tables ^^^^^^ could be derived solely from the principle of universal gravity; without borrowing from observation any, except the indispensable data. 1 am induced to believe, that the following analysis leaves but little wanting to procure this advantage to the lunar tables ; and that, by carrying on farther the approximations, we may soon obtain the required degree of correctness, at least, as it respects the periodical inequalities ; for, however great the accuracy of the calculations may be, the motions of the nodes and same mean longitudes, at the epochs 1756, 1770, 1801 and 1812, as Burckhardt has deduced from the observations made in that interval. VII. Intiod.] INTRODUCTION. 367 perigee will always be best determined by observation.* [4752] * ('2756) Since the publication of tliis volume, two very important works on the lunar theory have been published ; the one by Baron Damoiseau, in the first volume of the Mémoires présentés par divers sai'ans à F Académie Royale des Sciences ; the other by Messrs. Plana and Carlini. We shall have occasion to speak of these works in the notes L^'^^"] on this book, and shall now merely remark, that the object of them is to carry on the approximation to such a degree of accuracy, as to be able to deduce all the inequalities from the theorv alone. 368 THEORY OF THE MOON ; [Méc. Cél. CHAPTER I. INTEGRATION OF THE DIFFERENTIAL EOUATIONS OF THE MOON'S MOTION. 1. Resuming the differential equations [525], we shall put them under the following forms,* [4753] dt= "^^ General "•'••\/' + ^/(^?)-S' dv 1 /dq\ s fdq h^u'\du) h^w^ '\ds In these equations, t denotes the time, and we have, as in [499', 397] ; M-\- m m', (x .■?/+ y ij'-\- z z') m! (L) [4756] Q = i/(x'-xy-{-(y'-yr+(z'-zr * (2757) The equation [4753] is the same as the first of [525], and if we multiply the other two equations [525] by they willbecome as in [4754, 4755]. VII. i. §1.] GENERAL DIFFERExNTIAL EQUATIONS. 369 M is the mass of the earth ; [4757] m the mass of the moon ;* [4757'] m' the mass of the sun ; [4757"] T, w, ~, the rectangular co-ordinates of the moon, referred to the centre of [47581 gravity of the earth, and to the ecliptic of a given epoch, taken as ^ , , Symbols. a fixed plane ; x, y', -', the rectangular co-ordinates of the sun, referred to the same centre [4758'] and plane ; r the radius vector of the moon ; [4759] r* the radius vector of the sun ; [4759] s the tangent of the moon's latitude above the fixed plane ; [4759"] - the projection of the moon's radius vector r, upon the fixed plane ; [47G0] V the angle formed by this projection of r and the axis of x ; [4760'] h^ a constant quantity [518 — 519], depending chiefly on the moon's [47G0"] distance from the earth [4825, &c.]. In the value of Q [4756], the earth and moon are supposed to be spherical. To obtain the true value, corresponding to the actual forms of these bodies, we shall observe, that, by the properties of the centre of gravity, we must ^ ' J transfer to the moon's centre of gravity the following forces ; first, all the forces by which each of its particles is urged by the action of the particles of the earth, and divide the sum by the whole of the moon's mass ; second, the force by which the centre of gravity of the earth is urged, by the moon's action, taking it in a contrary direction. This being jjremised, it is evident, that (131 being a particle of the earth, and dm a particle of the moon, whose distance from the particle dM is /, we shall have the forces by which the moon's centre of gravity is urged, in its relative motion about the earth, by means of the ])artial differentials of the double integral, f (M+rn) ^ dM.dm Mm -^-^ / ' * (2758) This value of to is used in the two first sections of this book ; but its signification is changed in [4793], so that, in the rest of the book, 7nt represents the sun's mean motion. [4762] [4762'] t (2759) If we substitute, in [455], the value of dJ\l [452], also VOL. III. 93 370 THEORY OF THE MOON ; [Méc. Cél. taken relatively to the co-ordinates of the moon's centre. Therefore, we [4764] must substitute this function for , in the expression of Q [4756]. If the moon were spherical, we might suppose the whole mass to be collected in the centre of gravity [470'"] ; and then, by putting V equal to the sum of [4765] the quotients, formed by dividing each particle of the earth by its distance from the moon'' s centre, we shall have [4767«], [4766] ^ ff ^^m.V. [4763a] f = \/\{=o'-^f+{y'-yf + {^-zf] [455«], it becomes, F=:/y; and then, the corresponding force of the body M on the particle dm, in the direction — x, — ) [455']. This accelerative force, acting on the single particle dm, is to be decreased in the ratio of dm to m, to obtain the corresponding effect [47636] Qp {]jg whole body m, of which it forms a part ; by which means it becomes — f — — . Integrating this, so as to include all the particles dm, of which the body m is composed, it becomes,- pdm ^ dM 1 ^-dM.dm [47636'] J — J ^^ O'"' »^./ ^^ ' which represents the value of V, to be used in finding the accelerative force of the body m, from the attraction of the body M. If we change m, M into M, m respectively, we shall get — Cr '- — • , for the value of V, to be used in finding the accelerative force of the body M, from the attraction of the body m. Adding these two parts together, we [4763c] obtain the complete value of F= T- + ^ j .yy '——, corresponding to the whole accelerative force of m towards M, supposing M to be at rest. This is easily reduced to the form [4763] ; and its partial differentials, relative to the co-ordinates x, y, z, give the r -«/.<% ., accelerative forces parallel to those co-ordinates respectively. Now, when the bodies M, m [4763a] are spherical, these accelerative forces -— 7-, -—, -—^, are represented by the ^ [4763rf'] partial differentials of Q, taken relatively to x, y, z [499], retaining in Q [4756] only the term Q^^= " , which is independent of the disturbing mass m' . Therefore, r [4763e] to notice the non-spherical forms of the bodies M, m, we have only to substitute the expression [4763], m the place of , in the function Q [4756]. VII. i. § 1] EFFECTS OF THE OBLATENESS OF THE EARTH AND MOON. 371 * V would be equal to — if the earth were spherical ; hence, if we put ôV= V ; [4767] m,&V will be the part of the integral ff ^— , depending on the non- [4768] sphericity of the earth. In like manner, if the earth be supposed spherical, and we put V equal to the sum of the quotients, formed by dividing each particle of the moon by its distance from the centre of gravity of the earth, we shall have, rr ^J^-^^ ^ M.V; [4770] [4769] and if we put m 6 F = V , [4770'] r M. sV will be the part of the integral ff — - — , depending on the non- [4771] sphericity of the moon ; hence we shall have, very nearly,! ^TT — • / / ? = ■ h (M4-m) . { ^Ti -\ ■ > . [4772] Mm ^-^ f r ^ ' ■' \ M in S * (2760) If the mass m were collected in its centre of gravity, the integral ff — - — dM . dM [4767a] would become mf -^ ; and, by putting f —-^V [4765], it changes into m.V, as in [4766]. The expression [4770] is found in a similar manner. t (2T61) If we suppose m to be spherical, we shall have /•^dM.dm „dM . ^ ,^„„ -, JJ ^ — = "U-T' as in [4 /67a]; and if ^f also be spherical, [4772o] .dM M , ^^dM.dm m M / y = 7 ; hence, ff—j^— = Adding to this the parts m.SV, M.SV [4768,4771], depending on the non-sphericity, we obtain the complete value of ff — - — = — \-m.ôV-{-M.5V'. [47721] ■»«■ 1 ■ 1 • 1 ■ , M-\-m , . , , , M-\-m ^^dM. dm .___, , . , Alultiplymg this by — — , we obtain the value of -jr. — .JJ — [4/72]; which 372 THEORY OF THE MOON ; [Méc. Céî. Therefore, in the preceding expression of Q [4756], we must augment the M-\-m term — ■ — , by the quantity, [4773] <^M+m).\~ 4-~l= increment of Q [4756], J^n=/«- in order to notice the effect of the non-sphericity of the earth and moon. fron^t'be foEï ^' ^^ shall, in the first place, suppose both bodies to be spherical, and "frîhMd shall develop the expression of Q in a series. Now, we have,* moon. [4774] [4775] 1 1 II we develop the second member of this expression, according to the descending powers of ?', it becomes, 1 (xx'+yy'+z^^-lr^) {xx'+yy'+zz'-lr^f +..^^y^.z^^^+^,. Taking for the unit of mass the sum M-{-m of the masses of the earth and ■' moon, we shall have,t j\t-\-7tl is to be substituted for — ; — in the function Q [47636,4756]; and by this means the general value of Q [4756] will be increased by the function [4773]. * (2762) The development [4774,4775], is the same as in [4655?», c], rejecting the factor — m', which is common to all the terms. We may remark, that if we use the values [4774a] 0Ï R, M-\-m [4655,4775"], the expression of Q [4756] becomes Q = ^ — ^, which will be of use hereafter. I (2763) If we put I for the latitude of the moon, we shall have, as in [4759"], ^''''"^ [31',34"'] Int., [4776t] tang.Z=.; sin./=^^^; cos.Z=^^^^. If we proceed, as in [4659, Sic], changing r' into r, and U into v, we get, [4776c] a; = r.cos.Z.cos.t); y = r.cos. Z.sin. ij; s = 7'.sin.Z= ?-5.cos.?. [4776rfl Now, the projection of?-, upon the plane ot xy, is represented by r.cos.Z = - [4659a,4760]; VII. i. §2.] DEVELOPMENT OF Q. 373 1 = M+ m = iJ- ; r = .r ^ 2/ = u COS. D sin.f U ^ y/I+Tg M u [4775"] [477C] Lunar co- ordinates. [4777] [4778] [4779] We shall mark toith one accent, for the sun, the quantities u, s and v, u^-g^r relative to the earth.* Then we have,t 1 + f. {« »'. COS. (i)'— v)-\-uu'.ss' — hu'-. (l-)-ss)|2 (H-s'9)3.m4 Q__ » I '»'•"' / , ^ |mM^C03.(«'— «)+««'.Ss'— àM'2.(l-|-ss)|3 2.(1+s'2).m3 Value of [4780] substituting in this the value of cos. I [4776e], we get [4776] ; moreover, by substitutino- the value of r.cos. Z [4776dl] in the expressions of x, y, z [4776c], they become as in [4777—4779]. * (2764) By this means the solar co-ordinates become, r' the radius vector of the sun ; s' the tangent of the sun's latitude above the fixed plane ; — the projection of the sun's radius vector upon the fixed plane ; v' the angle formed by the projection of ?•' and the axis of x, or a;' ; r \/i+«y m' COS. v' x' «' ' sin. v' y =^ / 5 t (2765) Substituting the value of R [4656], in [4774a], we get, VOL. III. 94 [4777a] [47776] [4777c] [4777i] [4777e] Solar co- ordinates. [4777/] [4777^:] [4777/i] 374 THEORY OF THE MOON ; [Méc. Cél. [4781] "^'^^ sun's distance from the earth is nearly four hundred times as great as that of the moon ; so that îi' is very small, in comparison with u ; and we [4782] may, therefore, neglect terms of the order u'^, in the lunar theory. We may also simplify the calculations, by taking the ecliptic for the plane of projection. It is true, that this last plane is not fixed ; but, in its secular motion, it carries the moon^s orbit with it ; so that the mean inclination of the moon'' s orbit, upon the variable ecliptic, remains constant, and the phenomena, depending on their respective inclinations, are always the same. [47S3] 3. To prove this, we shall observe, that, from 5j 59, book ii., s' is equal to [47841 . i ' ' 1 a series of terms of the form A; . sin. {v' -\-it -\- s) ; we shall represent it by* ^ 1 , m' m'.r- , „ , [xx'-\-yu'4-zz' — è r2)2 (xx'-\-yif-\-zz' — i r"2)3 L J ^ )■ J-' 2/-3 ' - c'o ' - r'^ ' Now, if we substitute the values [4776 — ^4779,4777e — A], in the first members of [47S0i,c], they become, by shght reductions and using [24] Int., the same as in the second members of those expressions, [4780i] *^'+yy+~-^'= — -, • {cos.i!.cos.«'+sin.'y.sin.r'+«s'| = — 7.{cos.(r' — v)-\-ss'\; , ,o cos.{v'—v]-^ss' A.(l+s2) n ii'. cos.( n'— v)-\-%i u'.s s'— I uK( \^ss) [4/80C] xx^yyArZz'-lr-= — ■ —— = :;^^^^ . By means of these values the expression of Q [4780a] becomes as in [4780] . For the first and second terms of [4780a] correspond, respectively, to the first and second of [4780] ; [4780rf] jjjg jj-ii,.^ Qf [4780a] gives the last of [4730] ; finally, the terms of [4780«], connected with the factors | ot', ^m', by the substitution of [4780c], become respectively equal to the terms connected with the factors f , J, in [4780]. *, (2766) Using the same notation as in [4230], we shall have, for the earth's latitude s", above the fixed ecliptic, the expression, [478g„] ,"=r/.sin.."-iy'.C0S.^" [1335']. Substituting in this the values of jj", q" [4334], and observing, that [4785a'] sin.t)".cos.(,§-< + (3)— cos.j)".sin.(^< + |3) = sm.{v"—gt — p), we get the earth's latitude, [47856] s" = ^.c.sm.{v"-gt-^). Changing v" into the sun's longitude v' [4777f/], we get the sun's latitude, [4765c] s' = S.c.^m.{v' — gt — fi). This is of the same form as [4785], the constant quantities c, g, p, being changed into [4785c'] k, — i, —s, respectively. Hence, the coefficient i is of the same order as the quantities VII. i. §3.] INCLINATION OF THE LUNAR ORBIT TO THE ECLIPTIC. 375 s' = 2 . ^ . sin.. (v'-\- it -}- s) ; [4785] i being a very small coefficient [4785f?], whose product, by m'î«'^ we shall neglect. The value of s, neglecting quantities of the order s^, may be [4785'] represented by* s = s, + 2 . A; . sin. (v + it + ; [4786] s^ being the tangent of the moon's latitude, above the apparent ecliptic. This being premised, we have,t [4780'] g, g", he, which are very small [4339,3113^]. The values [4339] are nearly g= — 36% [4785d] g'= — 18*; these quantities may serve to give an idea of the magnitude of g, g', Sic., though they are not computed strictly by the method given in [1098, &ic.]. * (2767) If the moon were to move in the apparent ecliptic, her latitude above the fixed plane, or its tangent, corresponding to the longitude v, would be ^.k.s'm.[v-\-i(-\-s) [4785]. M-gf i Adding to this the quantity s, [4786'], we get, very nearly, the tangent of the moon's latitude s, above the fixed plane, as in [4786]. t (2768) The quantity Q occurs in the first member of [4787], under a linear form only ; therefore, we may take each term of Q [4780] separately, and compute its effect. In making the substitution of any term of Q, we may consider the quantity M.(l-|-««)^, and its powers, as constant. For, if we put (^=A.\u.(l-\-ss)~^'', for any terra of Q, neglecting, for a moment, the variable parts contained in Jl, and taking the differential of log. Q, we shall get, (i Q J du , s ds ~Q ~ 'Tt~ ' l+ss ' hence. T?)-o^ du J u ^ ' ds l + ss ^ [4787a] [47876] [4787c] [4787(f| Substituting these in the first member of [4787], we find, that the terms mutually destroy each other. Hence, it is evident, that we may neglect the first term of Q [4780], which corresponds to b^\, A=l; the second term, which corresponds to b = 0, and the last term, which corresponds to b= — 2, A = - (!+*'«')»' """-"-"- ' -I- ^- -' -— 2.[\-\-s's')û' Then using, for brevity, the following abridged symbol B, we get from [4780], \uit'. COS. (w — v')-\-uu'.s s'—hu'^.[\-\-ss)\ B = dq (l + s's')i 3 m'. u' (i+s'TjJ' {l+s's').u^ ' \^B^ + ^B^+hc.l; \B-\-^B^-^kc.].dB + hc. [4787e] [4787/] [4787g:] [4787/i] 376 THEORY OF THE MOON ; [Méc. Cél. [4787] [4788] „ , ,, CcOS.ft! v') ") Cs. COS. (v 1'') U^ À , 5«' ,' ■ o i i ''* • / /\ M f -4--— .cos.-(v — t)') + &c. 1 f -.sin.(i) — v ) — s Substituting, in the second member of this equation, the values of s', s, [4785,4786], we get,* Substituting the partial difterentials of Q, in the first member of [4787], it becomes, 3m'. «' ,„ , „^, C</s (dB\ fdB\ . , , /rf-B\ > [4787V] — — --. B + iB^ . \ - . (—)—"« -(-r ) — (! + ««)• (-7-) \ • ^ •■ (l+«s')} ' ' - ^ \dv \dv) \du) ^ ^ ' \ds)') The part of this expression depending on lŒ, in the last factor, is of the same form as the first member of [4787], changing Q into 5; therefore, it has the property mentioned in [4787i] [4787 i] ; that is to say, we may consider the powers of m.(1 -{-««)"" as constant. Now, the last term of J9 [4787/] corresponds to the power — 2 of that quantity ; therefore, we may neglect its partial difterentials, and, in finding AB, may use the remaining terms as in the following expression ; [4787/fc] B = 7^x77, • 1 ""' "'• COS. {v — v')-\- M-' m'. s s' I . The partial differentials of this expression give, ds /'dB\ «' C ds . , ,.7 [4787m] -"^-O^dzS'T.»- l^-cos.(i— i-O+^'-^^'l du J (\-\-s's').u' ^.'l. [4787,v] -(^+'')-(^) = (i-j-^yj-M Adding these three expressions together, we find, that the terms depending on s-s' destroy each other, and we get, ds /dli\ fdB\ ,, , . /dB\ u' ( , ds ,. ,-) Now, if we retain, explicitly, the terms of B [4787/], w hich do not contain s, s', we obtain, [4787;,] B + fS^ = '^'. ^^cos. {v — v')-~+^£. cos.2(^ _ ^') + &ic. ^ . Substituting the expressions [4787o,p] in [4787A'], and neglecting terms of the third order in s, s', it becomes as in the second member of [4787]. * (2769) If we substitute the values of s', s, [4785,4786], in the last factor of [4787], VII. i. ^ 3] EFFECT OF THE SECULAR MOTION OF THE ECLIPTIC. 377 *"' Jcos.fî' — v') — 7: \-'— .cos.^ft! — î)') + &ic. ;.<«,. COS. (î) — V) — '.sin.(i; — v')>. [4789] ifi I ^ ^ 2u ' 2u ^ ' ' ) ( ' ^ ' dv ^ ')) Hence the equation [4755] becomes,* elds , , i.ml.u'^Si-\-hc. ^^lv-^'+—^ Z^T^— ; [4790] or, dds '¥+^sO^ ' [4791'] 0=1-2 + 5 + ^-IT^ + &C. [4790'] If we neglect the excentricities and inclinations of the orbits, we shall have M = -, u'=— [4826,4833]; a' and a being the mean distances of the [4791] sun and moon from the earth. We shall see, in the following article [4826], that h-== a, very nearly ; therefore, we shall have [4791 (/], we shall find, that the terms depending on k mutually destroy each other. For these terms produce, without reduction, the following expression, neglecting quantities of the order mentioned in [4785'] ; 2.t.|sln.(t)-|-i^-|-s).cos.(j; — v') — cos.(« -|-*'^ + ^) -s'"^- (" — ^') — sin. ('u'-|-i<-|-e)|. The two first terms, between the braces, are reduced by [22] Int. to sm.{{v-{-it-{-s) — (t) — v')l = sin. [v'-\-it-\-s) ; which is destroyed by the third term. The remaining terms of [4785, 4786] are «'=0, s z^ s, ; substituting these in the last factor of [4787], we obtain the expression [4789]. [4789o] [47896] [4789e] * (2770) Multiplying together the two factors of [4789], we find, that the product of the term cos. (« — v') by x,.cos. (d — v'), produces Js, , disconnected from the periodical [4791a] angle v — 1; ; so that we may put the expression under the form -2 — '. JJl — '; as we [47916] shall soon see, that it is not necessary for the present object to mention particularly the parts included in the general term + &c. This represents the value of the function in the first member of [4787], and if we divide it by h^u-, it produces the three last terms of [4755] ; which will, therefore, be represented by J-^Ll^^!/ + ^- . Substituting this in [4755], [4791c] and dividing by 1 + 75''/(t)--^' we get [4790]. Reducing the denominator of the last term of this expression into a series ; neglecting m'^, and observing, that idi) l'^^^^^ 'S of the order m'u'^, it becomes as in [4790']. Finally, substituting in [4791rf] this the values of u, u', h^ [4791, 4791'], we get [4792]. VOL. III. 95 378 THEORY OF THE MOON ; [Méc. Cél. [4799] [4793] Change ia m. [4794] [4795] [4796] [4797] [4798] We shall put mt for the Burl's mean motion ; so that m will no longer denote the moon's mass; we shall have, by ^ 16 of the second book, »r = Then, if we sui^pose the time t to be represented by the moon's mean motion, which can always be done, we shall have -^ := 1 ; therefore, = ^. + 5 + l.m^s,+ &c. Substituting, in this equation, the value of s [4786], and observing, that we may, in this case, change it into iv, we shall have,t = — ; + (1+ f . m-) .s^+2.k.{l — {i +\yi. sin.(« + i î) + + &c.; which gives, for the part of s, relative to the secular motion of the ecliptic, Î * (2771) If we change, in the equation [605' or 3700] , a into a', and n into m, to [4794a] conform to tlie notation [4791, 4793], we get m^^= tx.a'~^ ; ij. being tlie sum of tlie masses of the sun and earth. If we neglect the mass of the earth, in comparison with that of the sun, we have |A = )ft' [4757"], and the preceding expression becomes as in [4794]. In the moon's motion about the earth, the equation [605'] becomes n-=^{M-\-m). a~^ [4757,4757']; and, as the moon's mean motion nt, is here represented by t [4794], we have M=l ; substituting tliis, and M-\-m = \ [4775"], in the preceding value of n^, we obtain 1 =«"3 ^g in [4795]. Dividing the value of nv'' [4794] by this last expression, [4794c] [4794d] we get iir substituting this in [4792], it becomes as in [4796]. [4798a] [47986] [4798c] t (2772) The terms neglected, by writing iv for it, are of the order of the excentricities and inclinations, multiplied by the very small quantity i, and connected with terms containing sin. cv, s'm.gv, and their multiples, as is evident from [4828, 4794c]. All the neglected terms are considered as being included in the general expression +&ic. Now we have, (Ills s—s,-\--2.ksm.{v-\-iv-\-s) [4786,4797]; hence — substituting these in [4796], we get [4798]. :'^-2.t.(»+l)=.sin.(«+n.+£); J (2*73) This equation is of the same form as [865], which is solved in [871] ; changing y, fl^ t, m into «, , l + |m^ v, 1 + i, respectively ; and putting for a Q, or [4799a] aK, the terms under the sign 2 [4798]. These changes being made in [871], it becomes as in [4799], by a slight reduction, and changing a; the signs in the numerator and denominator. vil. i. § 3.] DEVELOPMENT OF Q AND ITS DIFFERENTIALS. 379 S.{'ii-\-P).k.s\n.{v-{-iv + i) This last quantity is insensible ; for i v, at the most, does not exceed fifty centesimal seconds [ = 16',2] in a year;* and ^nrv expresses very nearly, as we shall hereafter see [4800f/J, the retrograde motion of the nodes, which exceeds 19' [3373] ; therefore fm^ is at least four thousand times as great as i ; so that we may neglect the term, ^.k.\\—{i+\f].ûn.{v + iv-^î), in the differential equation [4798] ; and then this equation becomes independent of every thing connected with the secular motion of the ecliptic. The mean inclination of the moon's orbit to the apparent ecliptic, is one of the arbitrary quantities of the integral of this equation ; hence we perceive, that on account of the rapidity of the motion of the moon'' s nodes, this inclination is constant; and the latitude s^ of the moon, above the apparent ecliptic, is the same as if the ecliptic loere immoveable. We may, therefore, suppose s' ^= 0, in the following investigations ; which will simplify the calculations. [4799] [4800] [4801] [4802] Inclination (if the lu- nar orbit tu the apparent ecliptic. [4803] [4804] Therefore, we have, by neglecting quantities of the order m' u'^ s\ m' ?<'^,t [4805] * (2774) This agrees nearly with the remarks made in [4785(/], relative to the value of i. Moreover, tlie retrograde motion of the nodes is expressed by (g — 1) .v [4817], and the values of m, g [5117], give g — l^^m^ nearly ; therefore, the retrograde motion of t!ie nodes is nearly equal to ^m^.v, as in [4800]. The same result may be obtained analytically; for, if we neglect terms of the order p"^, e'^, the motion of the nodes [5059] becomes \p".v. Now, by comparing the coefficients of sin.(^D — <)), in [5053, 5049], and retaining only the first term of each of them, we get, V $ m [5094] ; [4800a] [48005] [4800c] [4800rf] [4800c] whence, the motion of the nodes becomes iii".v = ^nfi.v. This exceeds 19'' in a year [3373] ; which is more than 4000 times the value of iv, assumed in [4785rf] ; hence the term of s, [4799] must be insensible, and we may, therefore, neglect the corresponding terms of [4798], which are given in [4802]. Then all the remaining terms of [4798], which are included in the expression -|-&c. [47986], maybe considered as independent of the secular terms arising from i. t (2775) Substituting s'^0 [4304] in the value of Q [47S0], it becomes, without any reduction, as in [4806a]. Developing the powers, and neglecting terms of the orders [4800/] mentioned in [4805], it becomes as in [48066]. This is reduced to the form [4806c] by 380 THEORY OF THE MOON ; [Méc. Cél. [4806] + !!LL!i_.^3.(l— 4s^).cos.(îJ— 'd') + 5.cos.(3î;— 3t0|. [4807] Hence we get, by neglecting quantities of the order m'u'^s^,* \duj ^ u \clsj (1+s')^ 2m3 ' ' ^ ^* [4808] _ ^'^'* . I (3 _ 4s"-) . COS. (v—v') + 5 . COS. (3 ^— 3i;') | ; using [6, 7] Int. ; and if we connect the terms depending on the same powers of ?t' it becomes as in [4806J ; Cl+^^-[uu'.cos.{v—v') — iu'^.{l+ss)f [4806a] q= - \-m'u'.{ " n_L ws [48065] =^ + mV. j ,3 _^,, 3,,'2 3ji'3 0+^2-[i+àcos.2(î,-^')]-^-(l+")-cos.(.-î)'), « [4806c] =-—^ + rm, + ^.Ucos. (,,-,;') +icos.3(i;-i;')]--—7,— * (2776) The partial differentials of Q [4806], taken relatively to v, s, u, become, without any reduction, as in [4809,4810,4810a], respectively. Multiplying [4810] by -, we get [48106] ; adding together the expressions [4810a, Zi], and making some slight u reductions, we get [4808] ; 1 m' m'3 [4810a] V(i«y ~ \ 3 m' «'1 ^ ^ ■* ^-.[(3— 12s2).cos.(t) — t)') + 5cos.(3^ — 3u')]' s /dQ\ ss m'.u'^s^ 3 m'. «"Isa . ,, [48105] û-UJ=-fî+^)î ^ ^^.cos.(.-.) [du) i VII. i. §4.J APPROXIMATE VALUES OF s, u, t. 381 I -^] =z -— .sin.C2t> — 2i)) \(lv J 2»- ^ •^ [4809] — '!l^^-.l3.(l—As-).sm.(v—v')-^l5.sm.(3v—3v')\ (1Q\ us m'.u'^s 3m'. u"^ s , ,, r^oim 4. 7*0 integrate the equations [4753 — 4755], we shall observe, that, by excluding the sun's disturbing force, the moon will describe an ellipsis, in which the earth occupies one of the foci. We shall then have, as in [532,533], [4810'] S = /.sin.('y — 0; [4811] u^j^,j^y\n + ssy+e.co..iv-.)\. ;^4812] 5, 11, in an iiiv iriahle eliipsia. In these equations, y is the tangent of the inclination of the lunar orbit ; d the longitude of its ascending node [533"] ; e and w are tivo arbitrary quantities, depending chief y , on the excentricity of the orbit, and on the ^ position of the perihelion [534']. y and e are very small quantities. If we neglect the fourth power of 7, we shall have,* [4813] [4815] U = /,3.(l + y3) -n + l>' + e.cos.(p— ^) — iy^cos.(2t;-20|. [4816] In this value of u the ellipse is supposed to be immoveable ; but we shall soon see, that in consequence of the sun'' s action, the nodes and perigee of this ellipsis are in motion. Then putting, (1 — c).v = the direct motion of the perigee ; {g — \).v = the retrograde motion of the nodes ; ['iS,\é] * (2777) Developing (l + s.s)i, according to the powers of s, substituting [4811], neglecting s^, and reducing, by means of [1,3] Int., we get, successively, (l+,s)i = l+J,a_,,4 = l + |4i — 2Cos.(2y— 20)^ — Ç.^f — |.cos.(2« — 20)+|cos.(4j;— 4â)f = (1 + ^7"— sV/) — (ir'— tf/)-cos.(2« — 20) — ^ij^4.cos.(4t,— 4ô). Substituting this in [4812], and neglecting y'^, it becomes as in [4816]. We have retained the terms of the order 7^. in [4812a], because they are required hereafter. '■ ^ VOL. III. 96 [4812a] 382 THEORY OF THE MOON : [Méc. Cél. we shall have, from [4811,4816],* [4818] s = '/.sin.(gv—è); 1 [4819] u^—j---^.{l + l7'+e.cos.(cv — ^) — {';~.cos.(2gv—2^)\. Assumed ^ \ I J forma of movcabiJ" If we substitute this value of «, in the expression of dt [47531, observing, ellipsis. L J o [4820] x}asX, if we neglect the solar attraction, \f] vanishes ; we shall have, ( l + l.fe'^+j.^)— 2e.(]+|e^+f7")-cos.(c«— ^) [4821] dt = h^. dv . ) +|.e'.cos.(2cv— 2?^)— e^cos.(3ci)— 3n)+i7^cos.(2^i'— 2i') \. —^.e7^.{cos.{2gv+cv — 2t) — i^)+cos.(2^î;— ct' — 2d+j:) \ / [4891a] [48216] [4821c] [482W] [4821e] [4821/] c'd — [4821gr] [4891A.] [4821i] [48914] [4821?] [4821m] * (277S) The object of this article is to obtain approximate vakies of m, w', s, v', expressed in terms of v ; for the purpose of substituting them in Q, and in its differentials ; as is observed in [4838']. Now, s, ii [4818,4819], are the approximate values of s, v, corresponding to the equations [4755,4754], noticing two of the most important perturbations, namely ; the mean motions of the perigee and nodes. Substituting these in [4753], we get tlie approximate values of dt, t [4821, 4822], which are afterwards corrected in [5081,5095]. In finding the approximate value of dt [4821], from [4753], tlie term glected, and then [4753] becomes dt = -—;^; in which we must substitute the value of u [4819]. In making these substitutions, we shall put for a moment, for brevity, f — 1 y2 A^^. cos. (2^ « — 2d) ; and, during the process of the calculation, we shall omit the symbols ê, w, ra', ivhich are connected respectively with the angles gv — è, cv — ra, ^'^ c' mv — zi'; taking care to re-substitute them at the end of the o2}eration. This abridged form of writing the angles, will be used frequentlij, in the notes which follow ; it saves considerable labor, renders the formulas more simple, and cannot be attended with any inconvenience. Hence, the preceding expres.?ion of tZ^ [4821rf] becomes as in [4821A]; developing the factors, and neglecting Z^, fe^, eS 7^ he, we get successively [4821 i,fc,Z]. Substituting the value of / [4821e], and reducing, by means of [6,7,20] Int., we get [4821w]: connecting together the terms depending on the same angles, we obtain [4821]; whose integral is as in [4822] : dt=P.{l+7'')-.dv .\l + {f+e .cos.cv)]-^ ^p(^lJ^2y^}.dv.\l—'2{f+e.cos.cv)-j-3(f+e.cos.cvf—'i(/+e.cos.cv)^ ^p(^lj^2y^}.dv.\l — 2e.cos.C!) + 3e2.cos.^c«— 4e^.cos.3ct) — 2/+6/e.cos.c?)} =P.dv.{{l-^2y^) — 2e{l + 2f).cos.cv-{-2e~.cos.^cv — Ae^.cos.hv— 2 f-\-6fe. cos. cv\ (l_|-Oya)_2fi(l-f27^) .cos.ciJ + ae^. (1 + C0S.2CI')— e^(3cos.c!;+cos.3ci))^ =li^-^'i'-^_i^^,^j^^^2_f,f^^^2gv+iey^.cos.cv—^ef.[cos.{2gv4-cv)+cos.{2gv-cv)]<^' VII. i. § 4] APPROXIMATE VALUES OF s, u, t. 383 This gives, by integration, /=constant+/i^f.(I+|e^+|r)— — .(l+|e=+|r).sin.(CT— ^) .sin.(2c!; — 2^) -— .sin.(3cj; — 3sî)+— ^.sin.(2^tj — 2^) [4822] 4c 3c ^ ^S — r7T-f-l-sin-(%»'+<^«— 2^— '')— T7^--N-sin.(2ffî}— «;— 20+ra) ; 4. (-2^+0) ^ ^ '^•(2^ — the coefficients of this equation are modified a little by the sun's action, as we shall hereafter see [5081, 5095]. In the elliptical hypothesis, the coefficient of v, in this expression, is, by Mgoo'i 3, [541' — 543], equal to a^ ; which gives,* /t3.(l + |e^+|^2)=a^; [4823] « being the semi-major axis of the ellipsis ; hence we have, [4824] h = a^,(l — ie"'-hy); [4825] consequently, u = --^.{l + e-+lf-\-e.(l + ec).cos.(cv—^) — ly-.cos.(2gv—2ê)}. [4826] _3_ Then, by putting n= a ^ [482Sa], we get,t [4827] * (2779) Substituting (x=rl [4775"], in [541'J, we get n= a ^; hence [543] gives «-f-a^e=«-y + &c. ; in which the coefficient of îj is a-". To make this conform to the ' "'' result of the elliptical theory [4822], we must put the coefficients of v equal to each other; hence we get [4823]. Dividing this equation by the coefficient of h^, and taking the [48236] cube root, we obtain h [4825], neglecting terms of the fourth order in e, j. This value of A gives A=.(l + y^) = a.(l-eS); whence, ^^-_ ==:l.(i+e^) ; substituting ^4823^] this in [4819], we get [4826]. t (2780) Multiplying [4823] by 1— 1-/, and neglecting /, we get substituting this in the third term of the second member of [4822] ; also [4823], in the 3 second term, and putting the constant quantity equal to — oF^c-, we shall obtain for these 384 THEORY OF THE MOON ; [Méc. Cél. [4828] nt + s=^ v — ~.(l—^f).sm.(cv—z,)+'^-^.sin.{2cv—2^) Ac 2 mate .^ .Sill, (o CV O 'CJ ) H . oui. \ ^ii u -i-o i value of 3C '^ 4^ ^ ^ ^ nt-\-s. Q 2 O 2 ~ 77iï^T~^ • sin- (25- î' + c tJ — 2 d — ^) — — -""-^^^ . sin. (2 £• 15 — c I' — 2 1' + ta ) :• 4.(25-4-0) vol ^ 4.(2^— c) ^ ° ' ^' £ being an arbitrarj constant quantity. In substituting nt-\-e, we may r482Ql suppose c and g to he equal to unity [5117], and neglect quantities of the order e^, or ey^, in the coefficients of the sines. Thus we shall have, by retaining the term depending on sin.(2^t' — cv — 2i-\-vi), which will be useful hereafter [4828f/] ; [4830] ^*^ + ^ = « — 2e.sin.(ctJ — a)-f-ie^sin.(2c?;— 2^) -\- \y^.sm.{2gv — 2è) — fey^ sin.(2^w — cv — 2<) + ^). [4831] If Yve mark Avith one accent for the sun, the symbols relative to the moon Approxi- •^ riucsof [4779'], and observe, that /= [4804], we shall have,* t, u'. [4832] n't + s' = v'—2e'.sm. (c'v'—^Z) + 2 e'^.siu. (2c'v'— 2^') ; u'= -,. n + e'^+e'.n + e'^).cos.(c'v'—z,')\. [4833] a' ' ' ' \ ' -> V /5 [4834] The origin of the time t being arbitrary, we may suppose ; and s nothing, 3 3 2e a [48286] three terms, the expression — a- e-\-a- v .«^.(1 — j}^).sin.(fi.'— to). Substituting this in [4822], then multiplying the first member by n, and the second by its equivalent expression a ^ [4823a], it becomes, by slight reductions, as in [4828] ; observing, that, in the second _3 [4828c] ^n*^! 'h'^*^ ''"^^ °f [4822], we may put h^a - ^ 1 [4823], since these terms are of the second or third orders in e, j. Now, putting c and g equal to unity, in the coefficients of [4828], and retaining terms of the second order in e, /, also the term depending on the r4828(/l angle 2^t) — cw, we get [4830]. The reason for retaining this term, is on account of the smallness of the divisors introduced by it, in consequence of 2°- — c being very nearly equal to unity. For the values of c, g, m [5117], give very nearly, [4828e] c=l— #w^ ^=l+f'K^ 2^— c=l+3m-. * (-2781) The values [4832,4833], relative to the sun, are deduced from those of the [4832»] moon [4830,4826], by merely accenting the symbols, as in [4779']; observing also, that s'=0 [4804], corresponds to y'=^ [4818]. vu. i. §4 ] INVESTIGATION OF v', u', IN TERMS OF v. 386 If and then putting - = m, the comparison of the values of nt and n't will [18351 give,* n * v' — 2e'. sin.(c'i''— ^') + f e'^ sin.2(c'îj'— ^') = in V — 2m e . sin. (c v — ^) + t »« e-. sin. ('2 c» — 2 ra) ^ ^ ■* ^ ^ [4836] + { m.y. sin. (2gv — 2 o) — ;^ mey~. sin.(2o-r — cv — 2 â + w). Hence we deduce, by observing, that c' varies but very little from unity,t [4836'] * (278-2) If we take, for the origin of i, the moment when the bodies are in their mean conjunction, or ni-\-s equal to n't-\-i', we sliall have s^e'=0. Substituting these in [4834a] [4830,483:2], we get the values of 7it, n't. Multiplying the former by m, and substituting mn = n' [4835], we get an expression of n'l, wiiich is to be put equal to that in [4832] ; [48345] hence we get [4836]. t (2783) We may obtain v' from [4836], by means of the theorem of La Grange [629c], which, by changing ■\'X into x, then x into v and t into t, becomes, v'—F{v') = t; ^4837„^ ,_,,,,, d.Fiif , d2.F(t)3 „ i,'=t+F(t) + è.^ + i.-Ai+&c. [48376] Comparing the equations [4836,48.37a], we find, that t represents the second member of the equation [4836], and, tliat F{v') == Se'.sin. {c' v'—-:) — ^é^.sm.{2c'v'—2i^'). ^483^, Changing v' into t, we get F{t) [4837e], its powers [4837/], and the differentials [4837^], omitting, for brevity, the symbol — tt/, which is connected with c't ; the reductions being made by means of [1,2, 17] Int. Substituting these in the second member of [48376], we L^SJ/^/] get v' [4S37A] ; F(t) = 2fc'.sin.(c't — ^') — ^e'-. sin.(2c't — 2ûj') + &ic.; [4837,] F{xf=2<:'K (l-cos.2c't) — |e'3. cos.c't+&:c. ; F{lf=6e'\ sin.c't + &c. ; [4837^] è--^=2É'-2.sin.2c't + fe'='.sin.c't + &c.; è- -5^ =— e'^. sin-c^t+Sic. ; [4937^] „' = t + (2e'-^ «'=*). sin.(c't-t.')+|e'2.sin.(2c't- 2^')- [48.37;^] Now, t represents the second member of [4836], and the substitution of this value in the first term of [4837A] produces the four first terms, or the two first lines of the second [4837i] member of [48-37]. The last term of [4S37AJ produces the last term of [4837], by putting for t the first term mu of the second member of [4836] ; it being unnecessary to take any other term of t, because m is of the same order as e, or e'. To obtain the value of the [403711 second term of v' [483TÂ], we must have the expression of sin. (c't — -a'). Now, as this VOL. III. 97 386 THEORY OF THE MOON ; [Méc. Cél. [4837] v'^m V — 2me . sin. (c v — in) + i m e". sin. Ç2.C v — 2 ra) + i m 7 -. sin. (2gv — 2è) — ^ m e f. sin. (2gv — c v — 2 è -{- -.->) Approxi- TilTesof + 2e'. (1— ie'^) .sin. (c'mv— ^') — 2mee'. sin. (c?; + c'mv — ra — xa') r , M . — 2 m e e'. sin. (cv — c'mv — ^ + ^') + f ^'^' ^i"* (^ c' w î' — 2 -/) , 1 C l + e'.(l— ie'2).cos.(c'?»i'— w') + «'^-cos.(2c'mî; — 2ra') )* «' ' ( -t~''''fi6'.cos.(c« — c'mv — zs-^-z}') — me e'. COS. {cv-{- c'mv — s — u') ) ' 5. We must substitute these values of u, u', s and v', in the expression [4838'] of Q [4806], and of its partial differentials [4808—4810], which will, by this means, be developed in sines and cosines of angles proportional to v ; but it is necessary, for this development, to establish some principles relative to term is of the order e', it will be sufficient to take the two first terms of [4836], namely ; [4837i] tz=mv — 2me.sm.{cv — a); whence, f't — ■ûi'=:(^c'mv — ra') — 2c'me.sm.{cv — «). Developing the sine of this expression, by means of [60, 18] Int., neglecting e-, we get, successively, [4837m] sin, (c' t — ra') == sin. (c'm v — to') — 2c' m e . sin. {c v — zs) . cos. {c'mv — ra') [4837n] :=sin.(<''mf — ~/) — c'ine .sm.{cv-\-c'mv — ra — a') — f'me .sln.(f v — c'mv — ■a-\-z!'). Multiplying this by its coefficient 2 e' — i e'^, or 2 e'. (1 — ^ e'^), neglecting terms of the fourth order, and putting c'= 1, we get the sixth, seventh and eighth terms of [4837]. * (2784) To obtain u, we must substitute the value of v' [4837] in [4833] ; and, as we retain terms of the third order in e, c', /, m, in [4838], it is necessary to retain those of the second order in v' [4837]. Hence, if we put for a moment, for brevity, [4838a] ~ = 2 c'. sin. {c' mv — ra') -|- J e'^. sin.(2c'?rt v — 2 to') — 2m e . sin. {cv — to) ; and observe, that c' is very nearly equal to unity, we shall have, from [4837], [48386] v'=7nv+z, and c' v'—zi'—{c' mv — to') + s. Its cosine, reduced by formulas [23,4.3,44] Int., becomes, by neglecting z^, [4838c] cos. (c' v' — w') = cos. z . cos. (c' mv — to') — sin. z . sin. {c' m v — to') [4S38(i] = (1 ~ 2 ~^) • cos. {cm v — to') — z. sin. {dmv — to') ; hence, e. (l + e'2).cos. (c'jj — to') [4838e] =e'. (1 + ê'-) .cos. (c' mv — to') — | e' z^. cos. (c'mv — to') — e' z . sin. {c'mv — to'). Now, substituting the value of z [4838a], in the first members of [4838^, A], neglecting VII. i. §5.] REMARKS ON THE DIFFERENT ORDERS OF THE TERMS. 387 the magnitudes of the quantities which enter into these functions, and on the [4839] influence of the successive integrations upon the different terms. The value of m [o\\l\ is very nearhj equal to the fraction ^\ \ loe shall [4840] consider it as a very small quantity of the first order. The excentricities of o,do,. or the orbits of the sun and moon, and the inclination of the lunar orbit to the ecliptic, are nearly of the same degree of smallness [5117, 5194]. Thus, %oe g^^ shall regard the squares and products of these quantities, as very small quantities of the second order ; their cubes and products of three dimensions, as very small qua7ititics of the third order; and so on for others. The sun's m' «'•* disturbing force is of the order* -A5-, and we have seen, in ^ 3, that this [4842] quantity is of the order m", or of the second order. The fraction -, being very nearly equal to ^i^, may be considered as of the second order. We [4843] shall carry on the approximation to quantities of the third order inclusively ; terms of the fourth order, also those depending on the angle Sdrnv — 3 ro', we get, successively, by using [31, 17,2] Int., the following expressions; omitting, for brevity, the [4838/] symbols -n, n', as in [4821/] ; y — y'z^. COS. (c' mv — •n') z= — e' '. (2 sin. d mv . cos. c m v) . s'm. d m v = — e'^. sin. 2 c' m V . sm. c' m V :=^ — i e' ^. cos. d mv: [4838g-] — e'z.sin. {d mv — z/) = — e'^. (1 — cos. 2 c' m «) — |e'^. cos. cm « -\-me e'.cos. [cv — d inv) — mee'. cos. (cv -\- dmv). Substituting [4838^, h'\ in [4833e], we get, by connecting the terms, e'. (1+ c'2) . cos. (c » — w') = — c'^+ e'. (1— i e'^) . cos. d mv -\- e'"-. cos. 2 d m v -\-mcd. cos. {cv — dmv) — mc e'.cos. (cv + dmv). Finally, by the substitution of this, in [4833], we get [4838]. [4838/i] [4838i] * (2785) The accelerative forces [4763rf'J, are represented by the partial differentials of Q, relative to the co-ordinates. Thefe partial difltrentials occur in the general equations [4753 — 4755], and are computed in [4807— 4810J. Now, if we compare the part of ^^ '* "^ [4808 or 4810], which does not contain the disturbing mass m', with the chief term of the same equation, depending on this disturbing mass, we shall find, that it is of the order ^4-' °^ '^ [4791]; which, by means of [4794, 4795], is of the order m^. l'*S'*26] 388 THEORY OF THE MOON ; [Méc. Ctl. and in the calculation of these inequalities, toe shall take notice of quantities [4844] of the fourth order;* but we must take particular care not to omit any quantities of that order in the integrals. The equation [4754] becomes, by development, of the following form,t „ (Ida , -.-TT [4845] = — ^+iV-. M + n ; [4845'] N^ differs from unity but by a quantity of the order «r [4845c], and n is a series of cosines, of the form /t.cos. (/ y + ;) [4961]. The part of «, [4o4o] relative to this cosine, is represented, as in [870', 871], by [4847] „__A_.cos. (it' + O- Now, it is evident, that if r differs from unity by a quantity of the order m, [4848] j^j^g jgj.,^ k.cos.Çiv -{- ô) acquires, by integration, a divisor of that order; which increases the term considerably ; so that it will become of the order ^ ^ r — 1, if it be of the order r, in the differential equation. We shall see [4850] hereafter, that the greatness of the inequality named the evection, arises from this cause. t * (2786) The angles connected with coeflicients, as far as the third order inclusively, [4844a] are retained ; and, in computing the coefficients of these terras, the approximation is carried on, so as to include terms of the fourth order. t (2787) The chief inequality of M [4819], is that depending on cos. (ci; — ra), which we shall represent by e.cos. (cd — ra) ; putting the other terms equal to Su, so that iAaA^„-\ . dilu „ , d-. (]u iwioai „^e_cos_(-c!) — w) + <Sit. Its difierential gives — = — c^ e. cos. (cv — îi) + -^ . Multiplying the first equation by c^ and adding the product to the second equation, we get, (Wu , „ d'^.Au , U c~ u= \- c~. au. [48455] dfi ^ dr2 ~ Putting the second member of this last equation equal to — n, we get, ''''" 12 IT. \- c" u = — IT [4845c] dl^ ' and this is of the same form as [4845] ; N^ being changed into c-, which differs fiom unity by a quantity of the order 3 m- [4828e]. X (2788) The evection depends on the angle 2v — 2mv—cv-\--m, and its cosine is multiplied by ./3/"e, in the expression of <5k [4904]. Now, in finding ^/i>, from the [4850a] equation [4999], we must divide by the factor 1 — (2 — 2/« — c)^ which is of the order m ; and by this division its value is very much increased. VII. i. § 5.] REMARKS ON THE DIFFERENT ORDERS OF THE TERMS. 389 The terms where i is very small, and which depend only on the sun's [4850'] motion, do not increase, by integration, in the value of u ;* but, it is evident, from the equation [4753], that these terms acquire, by integration, [4850"] the divisor /, in the expression of t ;t we must, therefore, pay great attention to these terms. It is on them, that the magnitude of the annual [1851] equation depends. The terms of the form k .dv.sm(iv-{-!), in the expression of (~^)-~Tj [4852] [4753, 4754] acquire, by the integration of that differential expression, a divisor of the order i, in the value of u. Hence, it would seem, that in the expression of the time t, these terms ought to acquire a divisor of the order r, which would render them very great when i is very small ; but, it is essential to [4853] dbserve, that this is not the case, and that, ifive only notice the first poioer of the disturbing force, these terms will not have the divisor r, in the expression of the time. To prove this, we shall observe, that by [1195, Sic], the expression of v, in a function of the time, cannot acquire a divisor of the order r, except by means of the function — 3af)idtfdQ;t in which the [4854] [4853'] * (2739) When i is very small, the divisor i^ — JV^ [4847] becomes nearly equal to — JY'^, which is of the order — 1 [4845'] ; consequently, the term [4847] is not increased by this division. [48506] t (-2790) If the development of the denominator of ilt [4753] contain a term of the form A,-.cos.(/ 1'-|~-')> arising from u^, it would introduce in dt a term of the form Jc [4851a] k.dv.cos.{iv-{-;) ; whose integral would introduce in t a term of the form 7 .sin.(w'-[-;), having the small divisor /, as in [4851]. I I (2791) The differential of Q [4774fr], relative to the characteristic d, gives, d-R=— ^— dQ; hence fdR=^-.—fdq. [4854a] Substituting this, and ij.=1 [4775"] in ^ [1195], we get, ^ = 3 « .fn dt.~—3a .fn dt .J\\ q. [45546] Now, the first term of this expression has only one sign of integration, and can, therefore, introduce only the first power of the divisor i [1 196', &c.] ; and, if we neglect tliis term, we shall have, I = _3 a ./n dt ./d q, as in [4854]. ^^^^*'^ VOL. III. 98 390 THEORY OF THE MOON ; [Méc. Cél. differential dQ refers only to the co-ordinates of the moon. If Q contain a [48o5] term of the form k .cos. (it -{-;), i being very small; this term cannot acquire a divisor of the order r, except dQ does not acquire a multiplicator of the order i. The part of this angle it, relative to the moon, must depend solely on the mean motions of the moon, and on those of her perigee and nodes, when we neglect the square of the disturbing force. If i be very small, this part of / does not depend on the moon's mean motion ; it must, therefore, depend only on the motions of the perigee and nodes. In this case, dQ acquires a factor of the same order as the motions of the perigee and nodes, that is, of [4850] the second order [4817,4828e] ; which causes the term in question to lose its divisor of the order r. Therefore, the angles increasing slowly have, in the expression of the true longitude in a function of the time, a divisor of the order i only ; and it is evident, that this likewise holds good, in the expression [48571 ^^ ^''^ time in a function of the true longitude. But, if Ave notice the square of the disturbing force, the part of the angle it, relative to the moon's co-ordinates, may contain the sun's mean motion ; and then, the differential [4657'] dQ acquires only a factor of the fust order, or of the order m. From these principles ice can judge of the order, to which the several terms of the differential equations are reduced, in the finite expressions of the co-ordinates. 6. Upon these considerations ive shall develop the different terms of the equation [4754]. In the elliptical hypothesis, the constant part of u is represented by,* [4858] - • { 1 + C" -f 5- 7^ + 13 5 := constant part of u ; [4858'! [3 being a function of the fourth dimension in e, y, we also have, [4859] ^^' = « . { 1 — e-— 7' + |3'}; [4859] ^' being likewise a function of the fourth dimension in e and 7. The sun's [48C0] action alters this constant part of u [4858, 4964] ; but a being arbitrary, * (2792) Neglecting terms of the fourth order, we have, in [4826], the constant part of [4858a] u equal to - -{l + e^ -\-ll^; and, from [4825], h~ = a .\\ — e- — y^. Adding to these the functions of the fourth order, depending on p, (S', they become respectively, as in [4858,4859]. Vn.i.^^6.] TERMS OF q IN THE DIFFERENTIAL EQUATION IN u. 391 we may suppose, that - .p + f'~ + T7" + (3| [4858] always represents the constant part of u. In this case, we shall no longer have h- = 0.(1— e=— >H|3') [4859] ; and we shall then put, a being an arhitrarij quantity which becomes equal to a, if we exclude the sun^s action. We shall then put, m This being premised, the term m'.u'^ Q,o 3 5 or the expression becomes, by development, as follows ;* 1 Â2' dq du s JFu '§) [4808], '3 m. Il 2li\u^ m 2a. l + e^^+i^^ + le'^ — 3e.(l+ie^+fe'^).cos.(cj; — t.) + 3e'. (l + e= + ir2+|fi'2).cos.(c'OT» — ^') — f . (3 + 2 m) . e e'. cos. (cv-{-c'mv — ■is — ■^') — f. (3 — 2m).ee'. cos.(cj; — c'mv — -^ + w') + 3e-.cos.(2cv—2^) + ^7--cos.(2gv — 2è) -f- |-e'^cos.(2c'mt?— 2'o') — I- e y-. cos. (2^« — cv — 2^ + ra) [4861] [4862] a,. [4863] [4864] m. [4865] [4865'] [4866] * (2793) If we separate the terms of the expression of - [4826], into different classes; using the abridged symbols Xi, x^, x^ [4866i], whose indices represent respectively the orders of the terms, we shall have u [4866c], from which we obtain — r4866c?l, neglecting terms of tlie fourth order in e, 7 ; Xi= e. COS. (cv — zi); x.2 = e^-\-iy^— {7^.cos.{2gv—2ê) ; X3 = e^.cos. (cv—-a)', [48666] [4866e] [4866rf] u = a-\\l-\-Xi-\-x.2-{-X3\ ; ■.a^l—3.{x^-\-X2+X3)-\-6.{Xl''-{-2x^X2)—l0x^^. 392 THEORY OF THE MOON ; [Méc. Cél. To develop the term ^"'"''^ 'SS-^°^-(2'^-2^0, of the expression of- ^,.('J)-^^.('||)[^^^^^ Now, substituting the values of x^, x^, x^ [48666], in tlie first members of [4866/— i], and reducing the products, by means of [6,20, 7] Int., we obtain the second members of these [4866e] expressions respectively ; always neglecting terms of the fourth order, and those depending on the angles 2gv -\-cv, 3cv, which are not retained in [4866] ; and using the abridged notation [4821/] ; [4866/] 1 — 3.(.Ti+Xo+.i:3) = l—3e^—f7'^—3e.{l-lre^).cos.cV'}-iy^.cos.2gv ; [iS66g] + 6<-= +3e^ + 3e^cos.2ci;; [4866fc] + 12a;ia;3= — 3e.(— 46^ — 7-).cos.ct) — f e7^cos.(2^i'-a') ; [4866i] — 10^1^= — 3c.(|ca).cos.ci-. The sum of these four expressions being multiplied by a^, gives the value of m"^ [4866(/,fc]. Moreover, from [4863], we get J/t"^ [4866/] ; the product of the two expressions [4866A:,?] gives [4866??t], neglecting terms of the fourth order ; [4866fc] u-^=^a\ { l—iy''—3e.{l—ie''—7^)-cos.cv-\-3c"-.cos.2cv-{-h^.cos.2gv—^ef-cos.{2gv—cv)l; [4866i] ih-^=har\\l + e^+7 21 . r4866ml A- ^.^' .{l+e^+j7^-3e.(l+èe')-cos.Cf+3e^cos.2«,+372.cos.2^«-|e72.cos.(2^r-ft>) } «3 [4866n] ~2â'^'"^^^' X beino- put, for brevity, to denote all the terms between the braces in [4866m], except the first, or unity. We may proceed, in the same manner, to find u'-^. For, by using the symbols y^, y^. j/3 [4866^], the expression of 7t' [4838] becomes as in [4866;-]; omitting, as above, the [4866o] [4866p] angles •a, ■a', in the rest of the calculation. From this value of u' we get u'-^ [4866*]. The terms, composing the factor of this expression, are found in [4866< — ?«] ; whose sum, multiplied by a'-^, gives m'= [4866s], as in [4866x] ; neglecting the terms depending on the angle Sc'mv — 3ra' ; y, = c'.cos.c'?ftt) ; y^=e'^.cos.2c^mv; [4866g] y^ = Jc'3. cos.c'TOti+OTee'.cos.(cD — dmv) — mee'. cos. {cv-\-c'tnv) ; [4866r] «' = «'-' .\l + yi-\- 2/2+ y 2 ] ; [4866*] m'3 = «'-3.^ i+3.(yi+y2+y3)+3.(yi=^+2y,y,) + </i=h VII. i. §6] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u. 393 we shall first give the development of 3 m' . u' ''. COS. (2 ?; — 2 v') . [4866"] This term, being developed, becomes, [4866/] 1 + 3. (yi+ y-i + i/s) = 1 + 3 m c e'. cos. (cv — c'm v)—3 m ee'. cos. (cu+cW) + (3e'— |e'3).cos.c'mu -\-3e'^.cos.2c' mv; 3i/^^= +2e'2 +|e'2.cos.2c'mr; [4866u] 6)/i(/o= +-2/.e".cos. c'tod; [4866d] J/i^= + 1 e' 3. cos. c'm 2'; [4866u>] , , Cl + 3e'2 + 3e'.(I + |e'2).cos.c'mt>+8e'3.cos.2c'mr ) M'3 = a'-3.) y- ~ V ' \ o ' ^ I ' ^^ L4866a:] ( -["3»ict -cos-^CD — c mv) — 3mee. cos. {cv-\-cmv)\ = «'-3. Sl + r|; [4866)/] l+Y being used, for brevity, to denote all tlie terms between the braces, in [4866x]. [48662] Multiplying together the expressions [4866», y], and their product by m' ; then substituting _a m [4865], we get. 2 A2. ((3 = ^ .{1 + X+ Y+XYl. [4866a] Now, XY is of the second order; and, in finding its value, retaining the same angles and terms as in [4866], we may use the following expressions, which comprise the chief terms of X, Y [4866«,y]; X^e--j-^7^ — 3e.cos.cv; F=|e'^+ 3e'.cos.c'mv. [4866p] Now, taking the terms of Y. and multiplying them separately by X, we get, |ê'2. X= — lee'^.cos.ci' ; [4866y] .3e'.cos.c'mD.X=.3e'. (c-4-ï7^) . cos. c'm r — % ee'. cos. {cv-\-c'mv) — |ee'.cos.(CT — c'mv). [48666] The sum of the expressions [4866y, (5] is equal to the value of X Y, which is to be substituted in [4866a] ; moreover, the sum of the terms hctiveen the braces in [4866m, a?], decreased hij unity, is equal to the value of l-{- X-\- Y. Hence we find, that the terms of [4866a, or 4866], between the braces, are equal to the sum of the terms between the braces [4866e] in [4866m, 1], added to the second members 0/ [4866y, 5], and decreased by unity. Connecting the similar terms, we find the result of this calculation to be the same as in [4866]. VOL. III. 99 394 THEORY OF THE MOONj [Méc. Cél, /(l_|e'2— 4mV) .cos.(2i'— 2my) + 1 e'. cos . (2 V — 2 mv — c'mv-\-^') -^e'.cos.(2 V — 2mv-j-c'mv — -'') + 2me.cos.(2î; — 2mv-\-cv — w) — 2 me. COS. (2 « — 2m v — cv-\--^) \ + y e'l COS. (2 V — 2 m v — 2 c'm d+2 i^') I — y mee'.cos.(2î; — 2?tt« — cv — c'mw+^+^')\ [4867] 3»i'.M'^cos.(2u -2t;')=^-( + V ^^«'- cos.(2w— 2mt)+c«— c'm«— ^+53') ^* \+ ^mee'. cos.(2'«; — 2m w — cw-f c'm«+ra — -n')/ - 1 mee'. cos.(2« — 2 m tJ+c i)-f c'm?) — œ — 33') ) ' + |m.(3+8m).e^cos.(2cz;— 2v+2»ii;— 2^) — Jm.(3— 8m).e-.cos.(2c«+2«— 2m?;— 2^) + ^my". cos.(2gv—2v-{-2mv—2à) — J:m7^cos.(2^?;-f2y — 2mv — 2è) \ — |-me>^cos.(2«— 2mîJ— 2^-?;+ci;4-2t'— c=)/ 3 m' 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 [4867a] [48676] [4867c] [4867rf] [4867e] [4867/] [4867g-] [të67h] [4867«] * (2794) Using, for brevity, the value of v^ [4867e], putting also «^ equal to all the remainin'^ terras of the second member of [4837], except the first mv, we shall have v', as in [4367/] ; always omitting, for brevity, the symbols a, ra', as in [4821/]. Substituting this value of v' in the first member of [4867^], and developing by means of [24,43, 44] Int., it becomes as in [4867A] ; observing, that v^ is of the first order, v.^ of the second order, and, that some terms of the third order are neglected. Substituting in [4867/t] the value 2v^= Sm^.e^. sin.^cu + 167« ee'.sin. c«. sin. c'?n« — 8e'^.sin.2c'?»i) [4867e], and reducing it, by means of [l,17]Int. ; also, 2v^+2v2 = '2v'—2mv [4867/], it becomes as in [4867 i] ; tijr= — 2me.sin.ci) + 2e'.sin.c'm!;; cos.(2i) — 2«')=cos.{(2t) — 2mî;) — (2ui+2î)o)| = cos. (2w,+ 2i'2).cos.(2« — 2m«)4-sin.(2ri4-2î)2).cos.(2a— 2mj)) = (^l—2vi^).cos.{2v — 2mv) + {2v,-i-'2v.2).sm.{2v—2mv) _f (l_4»iV-4e'2)+4mV.cos.2CT+4c'2.cos.2cW) ^^^ __2,ftt,) I +8mee'.cos.(c« — c'mv) — Smve.cos.{cv-\-c'mv) y + {2v' — 2mv\.sin.{2v — 2?fti'). VII. i. >^S 6] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN w. 395 We must multiply this function by 2/1= ; and we have this factor, by [4868] We must substitute, in the last line of this expression, the value of 2v' — 2m v, whicli is easily deduced from the second member of [4837], by neglecting the first term mv, and [48674] doubhng the remaining eight terms. We must then reduce the products of the sines and cosines of this function, by means of [17, 20] Int., as in the following table ; in which, the terms of column 1, corresponding to the different angles, are taken in the same order as in [4867;'], namely ; the first five terms in the same order as in the first and second lines of [4867i] ; and the remaining eight lines as in 2 i'' — 2mv [4837, 4867A:]. We may observe, that a term is neglected in line 9, depending on the angle 2v — 2mv-\- 2gv — cv, which is [4867i] not expressly retained in [4867] ; also a term, of the order e'^, inline 10,&ic.; (Col.l.) 1 2 3 4 5 6 7 8 9 10 11 12 13 (Col. 9.) ( 1 —4)n^e^ — 4c'2).cos.(2y— 2m«) 4-|-mV.cos.(2CT — 2v-\-2mv)-\-^.m.^e^.cos.(2cv-\-2v — 2mv) -\-2e'^.cos.{2v—2mv—2c'mv)-'r2e'^.cos.{2v—2mv-i-2c'mv) -}-4mee'.cos.{2v — 2mv — cv-{-cfmv)-{-4mee'.cos.{2v — 2mv-\-cv — c'mv) — 4mee'.cos.{2v—2niv — cv — cfmv) — 4mee'.cos.(2u — 2mi;-f-c«+c'»i«) 4-2?«e.cos.(2ii — 27nv-\-cv) — 2me.cos.(2y — 2mv — cv) -|-?'«c^.cos.(2ci; — 2v-\-2mv) — fme^.cos.(2c2;+2i' — 2mv) -{-imy^.cos.{2gv — 2v-{-2mv) — i7Hy~.cos.{2gv-^2v — 2mv) — .^mej'^.cos.(2f — 2mv — 2^y-j-cu)-j- Sic. -|-2e'.cos.(2y — 2mv — c'mv) — 2e'.cos.(2r — 2mv-j-c'mv)-}-hc. —2mee'.cos.{2v~2mv—cv—c'mv)-^2mee'.cos.{2v — 2mv-\-cv^c'mv) — 2ffiee'.cos.(2« — 2mv—cv-{-c'mv)-{-2mee'.cos.{2v—2mv-{-cv — c'mv) -{-îe"^.cos.{2v—2mv—2c'mv)—^e'^.cos.{2v—2mv-^2c'mv). [Terms of C03.(2i7-20'). J [4867m] Toobtain the expression [4867], we must multiply this value of cos. (2d — 2d') [4867m], by Sm'.u^, or 3m'. a'-^. (l + Y) [4866(/] ; by this means all the terms will have the common . 3m' ,., , . , • r . -, [4867n] lactor —, like that without the braces m [4867] ; and the terms of this expression within the braces will be obtained, by multiplying the function [4S67m] by 1 + F; or, in other words, by multiplying the functions [4867m] by Y [4866x, y], and reducing the products [4867o] as in [4867r], then adding together the two functions [4867?«, r]. In the first column of [4867?-], we have given the terms of Y [4866-r,y] ; and, in the second column, the terms of [4867m], by which they are multiplied: the third column contains their products, respectively. The numbers in column 2, refer to the numbers in the margin of the lines t^^^'^-P] of [4867m], putting one accent to denote the first term of any line, tivo accents for the 396 THEORY OF THE MOON ; [Méc. Cél. putting e' equal to nothing, in the preceding development of 2h\i [4866], [4860] aiTid \^y multiplying this last quantity by —, We shall thus have, very nearly, by neglecting quantities which remain of the order m' after the [4869'] [4867c] integration,* [4867r] second term of the same line, &i'C. Thus, 6' denotes the term 2ffîe.cos.(2i' — 2mv-\-cv)i and 6", the term — 2me.cos.{2v — 2mv—cv). This method of distinguishing the terms ivill he frequently used. (Col. 3.) Products of these terms. -j-fe'^.cos.(2?; — 2 niv) -(-|é'.cos.(2ii — 2mv — c'mu )-|-fe'.cos.(2t) — 2mv-\-c'mv) -\-'3mee' .cos.{2v-2mv-\-cv-c'mv)-\-^mee' .cos.{2v-2mv-\-cv-\-dmv) — ^mee'.cos.(2,v-2mv-cv-c'mv)—Zmee'.cos.{2v-2mv-cv-\-c'mv) 4— 3e'^.cos.(2y — 2mv — 2c'mt>)-|-3e'^.cos.(2i' — 2mv) — 3e'^.co3.(2u — 2mv-\-2dnn^ — 3e'^.cos.(2y — 2mv) 4-|e'2.cos.(2i;— 2my— 2c'my)+fe'2.cos.(2u— 2?nj)4-2c'?ni;) -\-^mee .cos.{2v-2mv-cv-{-c'mv)-\-^inec' .cos.(2,v-2mv-\-cv-c'mv) — i^mee' .COS. {2v-2mv~cv-c'mv) — ^mee .cos,.[2,v-'2'mv-\-cv-{-c mv^ . Connecting together the terms of [4867m, r], depending on the same angles, we find, that the coefficient of cos.(2y — 2?nu + 2c'»i«^) vanishes, and the rest become equal to the function between the braces in [4867], conformable to [4867o]. * (2795) The method given by the author, in [4869], is evidently correct. For, if we m'.u' 3 (Col. 1.) (Col. 3.) Terms of Y [48C6r]. Terms of [4867,n] + |e'2 r -[- 3 e'. cos. c'm y 1' 6' 6" 10' 10" |c'^.cos.2(;'7rtw 1' -|-3?î2ee'.cos. (ct) — dmv) 1' — 3mee'.cos.(cy-|-c'mt') 1' I [4869a] [48696] , we get u = -,, whence, ——7 —, ; multiplying this by 2/t9.„3 We shall not, however, be under the necessity of using this process, [4869c] put e' = 0, in 1 eives J77T — ;. 1 a3 because we have already given the value of 2^;5~^ = ^ • (1 + -^) [4866m, 71] ; and, if we multiply this by the function [4867], we shall obtain [4S70]. In the first place, the factors without the braces -^, ;^, being multiplied together, produce, 3 m'.a3 .3 _2 , — . -— - = — . «I [4865] ; VII. i.{.G] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u. 397 / il+e^+iy^—èe'^).cos.{2v—2mv) 1 _i(34-lm).t.(l-}-^e-— Je'2).cos.(2y— 2mi'— CD+n) \ 2 — i(3 — 4m).c.cos [2v — 2mv-\-cv — w) ;{ -|-Je'.cos.(2r — 2m v — c'm v-j-:^') 4 — ie'. cos.(2u — 2mv-]-c'mv — ra) / 5 — --^J-(14-2?«).ee'.cos.(2i' — 2mv — cv — c'mn-f a-f-^') -'\'~{ 1 — 2 m) .e c'. COS. (2 v — 2 m «-j-c v — c'mv — ra-f-^) 6 7 ^-.COS.(2«-2w')=±!^./ + i (3+2m).ce'.cos.(2t)— 2mj;— CD+c'mD-J-za— Î3') \ . 8 [4870] 2li\u 2a + T (3 — 2/?«).ec'.cos.(2^; — 2 m d-(-c y -f t''« ^ — « — si') l+-V.e'2.cos.(2r— 2?nt)— 2c'mz)+2îî') +i(6+15m+8ffl2).e2.cos.(2ct)— 2t)+27«i'-2i3) +K6— 15m+8waj.g2_£.os.(2cr4-2t)— 2mj;— 2-51) +Ï (3+2 ?«) .y-. cos.(2^ j;— 2 t)-f 2 m i-— 2 è) +} {3—2m).y~.cos.{2g v-\-2v—2mv—2ê) 9 10 II 12 jl3 14 The term ^^^,.cos.(.-.), of the expression -,^.(f — f (2+m).e7a.cos.(2z!— 2m«— 2^c4-ct;+2()— ^) / 15 hh lis [4808], [4871] which is the same as the common factor of [4870] . Moreover, the terms between the braces in [4870], are represented by the product of the terms between the braces in [4867], by l-{-X [4866n] ; or, in other words, this product is equal to the terms between the ^ ■' braces in [4867], added to the function [4S69c]. TJiis last function being the result of the product of these terms of [4867] by the the quantity X; and it is obtained in the following table, which is similar to [4867r]. The first column contains the terms of X; the second, [48fi9t/'] the terms of [4867], and the third, the corresponding products, reduced in the usual manner, and using the accented number 1', to denote the first term of the first line of [4867], as in [4867 5] ; VOL. III. 100 398 THEORY OF THE MOON ; [4871'] gives t!îe following ;* [Mtc. Cv\. [4869e] [4869/] [4869e-] (Cl. 1.) Terms of X [4866jn,n] — 3e.cos.cv -f-3e^. cos.âcw -{-^■y^.cos.2gv — |cy^.cos.(2^y — cv) (Col. 2.) Terms of [4867]. 1' 1' 1' 2 3 4 5 13 1' 1' 1' 4 1' (Cul. 3.) Products of these terms. [4870a] [48706] [4870c] [4870rf] [4870e] -f-e^.cos.(2 V — 2 mv) -j-.l7^-cos.(2); — 2mv) -Jc.cos.(2t>— 2 m y+CD)—fe. (1—1 e'2).cos.(2t>— 2m r—cr) -?^^ee'.cos.{2v-2mv-cv-c'mL-) — --j'-ee'.co3.(2tf — 2mv-}-cv—c'mv) 4-.?Ée'.cos.(2« — 2mt;-cz)-fc'mi')+3ee'.cos.(2î) — 2mv-\-cv-\-c'mv) -3 m e^. COS. (2 V — 2/»!;) — 37ne^.cos.{2cv-\-2v — 2 m») -\-3me^.cos.(2v — 2m,v)-\-3me^.cos.(2cv — 2v-\-2mv) -f m e y^.cos.(2 v — 2 m v — 2^- v-\-c v) ■|(?.cos.(2i' — 2 my — cv) -|-5 e-.cos. (2 c I' — 2 1'-|-'~ ni v) -\-^ e^.cos.(2f y-j-2 v — 2 m v) -\-^y-.cos.{2gv — 2v-{-2mv)-\-^y^.cos.(2gv-{-2v — 2m v) -|-|-me7^.cos.(2!; — 2mv — 2gv-\-cv) -|e7^.cos.(2D — 2mv — 2gv-\-cv). Now, adding tlie function [4839e] to the terms between tlie braces in [4867], we get very nearly, the expression between the braces [4870]. Tiiere are some shght differences, of the same order as that of tlie terms which we have usually neglected. Thus, the term — 'im^e^, in the coefficient of line 1 [4867], is neglected in [4870]. The term — 2me, in hne 5 [4867], is connected with the ftctor (l+lt^ — i^'^) in line 2 [4870], which arises from the chief terms of this coefficient in [48G9e] ; but this merely introduces terms of the sixth order. Finally, we may observe, that a similar factor might be introduced in the coefficient of line 3 [4870]. * (2796) Proceeding in the same manner as in note 2793, and retaining terms of the second order only, we get, from [4866c] u~*=câ.\l — 'i.{xi-\-x.2)-\-l0xi^\; substituting in this the value of 10ccj^:= 10 e^.cos.^<:« = 5 e^+5e^cos.as; ; also the value of x^-j-x.^ [4866i], we get, u~' = a'. \ l-\-e^ — j^ — 4e . cos. cv-\-5 e^. cos. 2 c j; -[-}'"• cos. 2gvl. Multiplying this by 9 m' ■ \l + e"~^y~) [4863], we obtain, 9) „,„ , = -—^ — .ll-\-2e^ — 4e.cos.ct)-)-5e^.cos.2ct)4->'^.cos.2fi-i'i 8^2. ,j4 8(,^ ( I I I / to s Again, from [48667,?-], we have successively, 6w^^ = 3 e'--j--3e'^.cos. 2c'm y ; = a'-'.|l+3e'2-|-4e'.cos.c'm»-[-7c'2.cos.2c'm»}. VIl.i.§6.] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN m. 399 8«, ■ a __.cos.(«— r)=<^^ji^ji_e'_cos.(i'— mz)+c'm«— ^') /• 2 [4872] , 2~lh a , . t . i\ -1 -.e. cos.(i' — m V — c m v-I-t^ ) 8 a, a [4870/] If we denote tlie factors between the braces in [4870c, e] by 1-|-X,, l + ^u respectively, their product will be l-f-^i+î^'i+^i^V» ^Y noticing only the chief terms of X,, F,, we have, Xi Yi= ( — 4e.cos.ci').(4e'.cos.c'HM') = — Sec'. cos. (c« — c'mv) — 8e e' .cos. [cv-\-c'mv). [4870g] Adding these terms of Xj Fj, to those of l+^u l + ^i [4870c, e], and decreasing the sum by unity, we get the expression of 1-j-Xj-f- Fj + Xj Fj, to be used in the product of the functions [4870c, c], which becomes, C 1 -j-2 e--)-3 e'^4-4 c'. cos. c'm« — 4e.cos.cv '\ 9m'. u"^ 9m'. a' ),-g ^ ,0 „ 1 n "> 0/ V g^-^ =g^^^ • ) +^^-cos.2cv-j-f.cos.2gv-frle~.co5.2c'mv'> . [i870h] (_ — 8 c e'. cos. (c V — c'in v) — 8 e e'. cos.(c v-{-c'm v) ) m . w Substituting the value of —7— [4865], in the first factor of this expression, it becomes, 9m'. a* a — ^O' 8^4- l'A; -o [4870,-] which is of the fourth order [4842,4843] ; therefore, in finding the value of cos.(«) — v'), we need only to retain, in general, the terms of the first order ; except in those depending on the angle v — mv ; in which greater accuracy is required [4874]. Hence we may neglect [4870/i;] «0 [4867/"], and we shall have the value of cos.(«; — v') [4870»i], by proceeding as in [4867^,A]. Substituting in this the value of I'l^^ 2e'. sin.c'wt) [4S67e], it becomes as in [4870n]. It being unnecessary to notice other terms of a higher order, or such as depend on [4870i] angles which differ from those in [4872] ; cos.(r — r')=(l — irj^).cos.(i' — ?«t))-|-i'i-sin.(v — mv) r4870 1 = (I — fc'^).cos.(t) — mv) — c'.cos.(« — mv-{-c'mv)-\-c' .cos.[v — mv — dm v). r4870n] The four terms of which this expression is composed, being multiplied by the terms between the braces in the function [4870/i], produce respectively the terms in the four lines [4870o — r]. Their sum is given in [4870^]; to which we must annex the common factor 9m'. vJ^ [4870ï] , and we shall obtain the corresponding terms of ' ,^ .cos.{v — v), as in [4872]. We shall hereafter, in [4870/ — w], see, that the neglected terms have much less effect, in the value of u, than those we have explicitly retained ; 400 THEORY OF THE MOON ; [Méc. Cél. [4872'] - being, hy the preceding article [4843], of the order m' ; the two first of [4873] these terms become of the order »t^ by the integrations. The inequality, depending on the angle v — mv, is remarkably loell adapted to the determination of the Slut's parallax, by means of the ratio -. It is, therefore, important [4874] [4870o] 1 [4870;>] 2 [48709] 3 [487 0<] (Coi.i.) {Col. a.) (l-j-2c^-f-3e'^).cos.(«) — mv)-\-2e'.co'3,.[v — mi'-f-c'mt')-l-2e'. cos.(r — mv — c'mv) — e'^. cos.(i) — mv) — e' . COS. (v — m v-\'C'm v) [4870r] 4 j J^e' .cos.{v—mv—c'mv). [4870«] (l-|-2e"-4-2e'-).co3.(i! — mv)-\-e'.cos.{v — mv-\-c'mv)-\-^e' .cos.{v — mv — c'mv). If we compare the terms [4872] with the assumed form [4846], we find the values of i, corresponding to them respectively, are i^l — in, i:=l — m-|-c'm, i=l — m — cm; and, as c' hardly differs from unity, they are very nearly represented by i^\ — m, i=l, i = \—2iii. The corresponding divisors, in the value of u [4847], are of the orders [4870m] (1 — m)~ — JV^ \—N^, (1— 2m)2 — N~ ; and, as JV- differs from unity by quantities of the order m^ [4815'], these divisors will be respectively of the orders m, m^, m. In consequence of these divisors, the part of the first term [4872] which is independent of c, e', is reduced from tlie fourth to the third order ; the second term is reduced from the fifth to the third order ; and the third term is reduced from tlie fifth to the fourth order. Several terms of the function [4870t", or 4872], are not increased so sensibly in the value of m, and they are [4870tt] therefore neglected. Thus, the term — 4e.cos.cv [4870/(1, being multiplied by the first term of [4870»], produces, in the function [1872], the following expression, [4870jr] |.^ . -.( — 4c.cos.cy).cos.(!; — mv) = — ~.^ . -.2c.{cos.(ci' — v-\-mv)-\-cos.{cv-\-v — mv)\. r4670vl corresponding, in [4846], to i = c — l-f'"> i^c-\-\ — m, and as c=l — f»«~ nearly [4828e], these terms will not render the divisor i^ — JV^ small [4847]. We may observe, that the term treated of in [4871], occurs in [4S08], under the form 3wi' w'M .(3— 4«-).cos.(v — v'), and in [4754], with a dilierent sign, and under the !brm [4870^'] 8ui 3m'.i('-l ,_ , o, , ,, y»ï'.!('' / , . n\ / /\ 1 • 1 1 1 • o .(3 — 4s-).cos.(t' — v), or, .(1 — *;.s-j.cos.(y — v ) ; whicli, by neglectmg s", becomes as in [4871]. Now, by [4618], wc have, [4870Z] _4,a__£-/.cos.2(5-f-d) = _f ,.2_£y5.cos.(25-.-2<!) ; which contains the constant quantity — f^-; so that we might multiply the function [4871] [4870z'] by 1— §r, which would change the foctor (l+2(;'-+2e'-2) [4872] into l-l-2e-+2c'2— f/^. VII. i. §6.] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u. 401 to determine this inequality with particular care ; and, for this purpose, we [4875] shall carry on the approximation so as to include terms of the order m*. We shall now develop the term \j)-j;r~Yj~^ of the equation [4754]. In the first place, this term contains the following,* — ^-^^.--.s'm.(2v—2v'). tl87C] .sin. (2d — 2v'), by increasing 2v by a right angle, f [4876'] We shall have * (2797) This is produced by the first term of [4809]. t (2798) We may change 2y mto any other angle, as 2 v in [4867^ — r, 4867,4870], without altering the angles mv, gv, cv, c'mv, as is evident by the mere inspection of the process of calculation in [4767^, &.c.]. This change being made in [4870], and then putting 2 V = 2 1"-]- 90'', its first member becomes, -^u^-'^-^-'-^''^^ ^^ In [4876']. In the second member of [4870], we must, by the same process, change any term of the form cos.(2!;-l-p) into — sin.(2«4-|3) ; and any one of the form cos.(3 — 2r) Into -|-sin.(3— 2v). Hence we get, by changing the signs of all the terms of [4870], and neglecting the symbols è, ct, -ô/, as In [4821/], ?^';sin.(2«-2.')=|^ 2h\u^ ^ ^ 2a, /(I +e2+i 5-2—1 e'2).sln. (2 v— 2 m«) — |(3+4»?).e.(l+|e2— Je'2).sin.(2î>-2mD-cî)) -5(3 — 4m).e .sin. (2 \i — 2 m vA-c v) -|-Je'. sin.(2t) — 2mv — (^mv) I — J e'. sin. (2 v — 2 m v-\-(!m. v) \ — -i-{\ -|-2 ni) .e e'. sin . (2 « — 2 mv — c v — dmv) I — %'-( 1 — 2 ni).ee'. sin .(2 v — 2 m v-\-c v — c'mv) -\- 1 (.3-l-2OT).eÊ'. sln.(2y — 2mv — cv-{-dmv) \ -\- i (3 — 2 m) .e c'. sin . (2 v — 2 m v-\-c v -f c'm v) I +-¥• e' ^. si n. (2 r — 2 7n v — 2 c'm v) -l(6+15?n+8m2).e2.sin.(2ct)— 2i}+27n«) +i{6—l5m4-8m^).e^.s:ii.{2cv+2v—2mv) -i (3+2 m).y~. s\n.{2g v—2 v-\-2 m v) +^{S—2m).y-.sm.{2gv+2v—2mv) \ — ^ (6-|-3 in).ey'^. sin. (2 v — 2 m v — 2^r-f-cy) [4875a] [4876o] [4876t] [4876c] [4876rf] 1 2 3 4 5 6 7 8 [4876c] 9 10 11 12 13 14 15 VOL. III. 101 402 THEORY OF THE MOON ; [Méc. Cél- 3m' u'^ ill the preceding development of rr^— 5.cos.(2« — 2v') [4870]. We must then mutiply this development by,* •— ce.(l+ie^ — J-7-).sin.(ct; — 3j) v i + icelsin.(2ctJ— 2^) I 2 [4878] rfM I , ^ . ,„ q . ( „ «</«; 1 I + ia-7^sin.(2^i;— 20) | 4 , — ±ef.sm.(2gv—cv—2ê+z!) I 5 * (2799) The differential of [4826], relative to v, gives, by neglecting «, 6, as in [4821/], [4878a] ~ = a-K{—ce.(l+e^).sm.cv+^gf.sm.2gvl; dv and if we neglect terms of the third order in all the coefficients, except those which are connected with the angle 2gv—cv, we obtain from u [4866c], the following value [48786] of - [4878c, tZ], by observing, that a-i^E^.cos.^cf =ie2+ie^.cos.2cy [4866è]. w We may remark, that the author has retained, in the coefficient of cos. cj;, a term of the third order e^, but has neglected others of the same order, as will be seen in [48846] ; [4878c] i^a.{ l-(x^+x,+x,)+{x,i-x,+x,y-{.T,-j-r,+x,f+hc.l u [4878d] =a.\{l—ie-—lf)—e.(l-\-e").cos.cv+he^.cos.2cv+if.cos.2gv\. Multiplying together the two expressions [4878a,rf], we find, that the factor without the braces becomes a-^.a = 1 ; so, that we have only to notice the product of the factors between the [4878e] jjj.j^(.g3 -pj-jjg jg jjQj^g jjj (i^g following table ; in which is given, in column 1, each of the four terms of the function [4S78(f] ; and the corresponding products, by the function [4878a], are given in column 2, on the same lines respectively ; (Col. 2.) — ce.(l-\-ie'^ — ^■)'^).sm.cv-\-^g'y^. sin.2^v 4-i c e^. sin.2 c v — | g c -f. sin. (2^ v — c v) — &;c. ■\-\c e^. sin.cu — \ ce^. sin.3c v -\-\ct -f. sin. (2^ t — ci))-f-&ic. Connecting together the similar terms, and putting c = l, g^=\, in those of the order e 7-, it becomes as in [4878]. (Col. 1.) [4878/] l-|c"-i7' [4878^] — c.(l4-«^)-cos.cw [4878/.] -|-Je2.cos.2c«; [4878i] 4-^7®.cos.2^w VII.i.§6.] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u. Then we shall have,* /cc.(l-f^.[2— 19m].e=— |e'2).cos.(2u— 2m»— cv+ra) / — ce.cos.(2« — 2mv-^cv — «) I + J.cce'.cos.(2t) — 2mv — cv — c'Mt)-j-ra-|-w') \ — |-.cee'.cos.(2i; — 2mv-\-cv — cfmv — ra-j-w') I — ^.cee'.cos.(2D — 2mv — cv-\-c'mv-\-zi — •ra') 3m'A3 du . ,^ ^ ,, 3in +h-cee.cos.{2v—2mv-\-cv-{-c'mv—-a—a') — nT^T" -T •sin.(2u — 2t)')=: — ./ 2''-"' ''" 4"' \— 2c.(I+m).e2.cos.(2c?;— 2«+2mu— 2«) 403 4-2c.(l— m).e^.cos.(2ci)4-2t) — 2mv — 2a) -{- 4 m c. e^. COS. (2 v — 2 m v) — tgf.co5.{2gv—2v-\-2mv—2è) -}-igy^.cos.{2gv-\-2v—2mv-\-2è) _|_ J.(2 — 5m) .e y^. cos.(2u — 2mv — 2gv-^cv-\-2ô — ro)/ 1 2 3 4 5 6 7 8 9 10 11 12 [4879] * (2S00) If any term of [4876t;], be represented by . A . sin. V, 2a, ' and any term of [4878], by .^'.sin.F', the product of these two terms, changing its sign, will represent the corresponding part of — ' ^ ■— .sin.(2t; — 2v') [4879], which, by reduction, becomes, ^^.{AA'.cos.{V-\-V')—AA'.cos.{Vy^V')l. The factor of this expression, without the braces, is the same as in [4879] ; consequently, the terms within the braces, must arise from the terms A A', cos. ( V+ V) —A A'. cos.( F«= V) . These terms are computed in the following table, neglecting quantities of the third order in e, e', y, except they depend on the angles 2 V — 2 m vzhc v-\- zs, 2 v — 2 m v — 2g u-j-c v-\-2 ê — «. The numbers in the first column refer, respectively, to the five terms or lines of [4878] ; and those in the second column, to the terms or lines of [4876e] ; in the third column are the corresponding terms of the function [4879/"] ; and the sum of all of them represents the terms between the braces in [4879] : [4879o] [48796] [4879c] [4879rf] [4879e] 404 THEORY OF THE MOON ; [Méc. Cél. [4880] The terms,* '«'4 m.u 8}Au= -..I3.sïn.(v—v')+lô.s'm.(3v—3v')] du dv [4879/] (Col. a.) Function [4870(/]. ce.cos.(2y — 2mv-\-cv)-l[-ce.(l-\-^e^ — Je'2).cos.(2« — 2mv — cv) 1 +J(3+4w).fe2.cos.(2y-2my)— i(3+4ïK).ce2.cos.(2CT-2«+2«y) 2 i(;3— 4w).cc2.cos.(2y-2mi')+^(3-4m).cc2.cos.(2cy4-2i;-2/Hw) 3 — icce' .COS. (2v-2mv-\-cv — c'mv)-^?rcee' .cos. C2v -'Hmv- cv — c'mv) 4 -f-T7Ccc'.cos.(2i'— 2);u'-|-fi'-[-f'mi') — ^'^cc' .cos. {^v —2mv —cv-^c'inv) 5 6 7 8 9 10 11 12 13 14 15 (Col. 1.) (Col. 3. A' [4878]. [4876c -ce{\+le^-m. sin.cz' 1 o 3 4 5 11 12 13 -j-Jce^-sin.2c» 1 o 3 — ^ce^.s'm.Scv . . -{-igf.sin.2gv 1 3 — ^e'y^.s)n.{2gv- —cv) 1 [4879^] [4879fc] [4879q [48794] — |(G-(-15?k).cc3.cos.(2« — 2mv — cu)-j-&c. -[-i(6 — 15m).ce^.cos.(2y — 2mv-\-cv)-\-&c. -i(3_[_2,„).ej,2.cos.(2y— 2my— 2^v+cd)+&,c. -f-ice^.cos (2eD-j-2u — 2mv) — ice-.cos.(2cu — 2v-{-2?nv) — J-(3-j-4»i).fe3.cos.(2y — 2mv-^cv)-\-&L,c. -j-J-(3 — im).ce^.cos.{2v — 2mv — c«)-)-&,c. . . neglected. -\-igy~-cos.{2gv-\-2v — 2mv)—lg7-.cos.{2gv — 2v-\-2mv) -|-i(3 — im).c}'^.cos.[2v — 2niv — 2gv-\-cv)-\-&,c. -|-|e7".cos.(2y — 27iiv — 2gv-\'Cv)-\-&bc. Connecting the terms of this expression, we obtain the factors between the braces in [4S79], neglecting terms of the tliird order, connected with the angle 2v — ■2 m v-^-cv, or with other angles differing considerably from v. To estimate roughly one of these neglected terms, we shall observe, that y^ « ^ e' [51 H, 5120] ; therefore, the greatest product of the third order, which can be made of these three quantities, and can occur in the above function, is ey-; and, if this be multiplied by the factor i^ [4879], or its equivalent expression |m^, it becomes |m^. ey®. Substituting the values [.5117, .5120], and multiplying by the radius in seconds 206265", we get ^ m^. e7^ = 0",38 ; whidi represents the order of the greatest neglected term in [4879]. This may be somewhat increased by integration in this value of u [4847], by means of the divisor i^ — JV^ ; for which reason the author has retained the last term of the function [4879], which depends on the factor ey~. We may observe, that the factor l-|-|e- — f e'~, which occurs in the second term of the first line of [4879/], might also be connected with the first term in that line. * (2801) Substituting, in Cy)'W^v [4'754], the term of [4809], depending on ?A [4880a] it becomes as in [4880] ; neglecting the very small teim depending on s-. We have, in VII. l.s^6.] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u. 405 in the expression of ( '-7^ i.,~-i-) produce no inequality of the third order [4881] '■ \dv J Irirdv in the integrals. Lastly, we shall develop T~^-f'^'-\ [-4754]. This function contains [4881'] dv u '3 m' y,u'^.dv the following term,* — j2~-J^ — i — .sin. (2îj — 2v'). The development of [1883) ft It 3m'. U'^ ._ -, ,, ^)rl^/^^ • 1 r 3m'. u'^ ■ ,n ^ ,^ ___ . COS. (2 v-1 V') [4870] , gives that of p- - . sin. (2 v—2 v'), [4883] by increasing the angle 2*; by a right angle [4883rt], and multiplying it byf [4883'] [4872], the expression of " ^ .cos.(t) — v') ; in wliich we may change v into t)-|-90'', as in [48765, c], witliout altering m v, c'mv ; and we shall obtain the expression of ~8Â5:^-""-("~^)- [4880J] This being multiplied by one third part of the expression [4878], gives the value of -S5-^-^'"-(^-''')-S [4880]. ^4g,0c] Now, the chief term of [4872] has the factor ^.nfi.-, [5094] ; and that of [4878] is ce, or c, nearly, neglecting its sign. Hence, the greatest coefficient of this product, is, i-.w2.^,.e = 0,0000004 [5117,5120]; [4680rf] which, in seconds, is less than 0',09. This is insensible, and it is not increased by integration in [4847]. The same may be inferred, relative to the term of [4880], depending [4880e] on the angle 3v — 3v'. Hence, we may conclude, that the expression [4880] maybe neglected, as in [4881]. * (2802) The first term of (^^) [4809], being substituted in [4881'], produces the expression [4882] ; and we have already seen, that the expression [4870] gives that in [4883a] [4876e]; by changing 2» into 2i)-f90'', according to the method proposed in [4876'] or [4883']. t (2803) Retaining terms of the third order in [48786], and multiplying by 2, we get, 2 — = 2a.\l—{xi+x,,+.r.j)-^xi'i-^2xi x.^—x^^. [4884a] Substituting the values [48666], we obtain, VOL. III. 102 406 THEORY OF THE MOON; [Méc. Cél. — e.(l — le^ — lf).cos.(cv — ^) [4884] ï^2a.^ +ie^cos.(2ct; — 2^) + i?2.cos.(2g-?;— 2ô) —lef.cos.(2gv—cv—2è-Jr-:s) Hence we shall have,* [48846] [4885a] [48856] [4885c] 1 — {xi-{-x.2-\-X3) = 1 — c^ — i>^ — c.[l-\-e^).cos.cv -\-ly^.cos.2gv 2xxi= — c.( — 2c^ — i7~)-cos.cv -rriey^.cos-i^gv — cv) — a:j3__ — j,_^ Sga ).cos.cv. The sum of these, gives the terras between the braces in [4884a, 4884]. * (2804) Multiplying together the second members of [4876e, 4884], we obtain the expression of 3»i'.m'3 --^ a -.sin.f2D — 2v'): and the factor witliout the braces becomes 3 m ■—, as in Ifi.u* ^ ' a. [4885] . The products of the terms between the braces, are found in the following table ; in which the first column contains the terms of [4884] ; the second column, the terms of [4876e] ; and the third column, their respective products, reduced by [ 1 8, 19] Int. ; using the abridged notation [4821/] ; (Col. 1.) [4884]. 1 — f.cos.(;« -j-e(ie^+j72)cos.c« -j-ie^-cos.2cy -}-57®.cos.2^y. — Je72.cos.(2jO-y-cD) (Col. a.) [487Ge] All the terms. 1 2 1 o 3 4 5 11 13 1 1 3 1 3 1 (Col. 3.) _. ,. . 3m'.«'3 . ,„ „ ,. Correspondinsr terms of -r= .sin.fiJw— at) ). :■ the whole function [4876e] between the braces (- 4^2— i72).sin.(2D— 2»i«) 1 2 -j->(3-|-4;rt).e.(|e24-iy2).sin.(2i'— 2»u'— ft') 3 -|_|c.(l_}-e2-|-|y2-|e'2).{— sin.(2y— 2mw+cw)-sin.(2K-2»i«-cy)] 4 -|-i(3-]-4,„).e2.|sin.(2y— 2?nu)— sin.(2cD— 22)+2)rM))} 5 _f_.(:3_4„j).f2^sin.(2y— 2mw)4-sin.(2cw+2»— 2»iu)} 6 ■\-\ce'.\ — sin.(2u — 2mw+ct) — Cmw) — sin.(2u — 2nîD — cv — drnv) \ 7 _j_ic(;'.^_|-sin.(2u— 2«u'-j-c2J+c'OTt))+sin.(2« — 1mv—cv-\-c'mv)\ 8 9 10 11 12 13 14 15 16 __i_(G+1.5/n+8M2).e3,gin.(2y— 2my— cv) — _i^(3-|-2/n).cy2.sin.(2w — 2mu — 1gv-\-cv) +(^c3+iey2).sin.(2y— 2/«K— cr ) — ica.sin.(2cw— 2tf+2»!!;)+J:e2.sin.(2cv+2i'— 2m') _|(3_4Hi).e3.sin.(2w— 2;ny— cu) +^>2sin.(2g'î,-j-2t'— 2mw)— ^y2.sin.(2^y_2t,_j-2»iy) _^i^(3_4,n).e72.sin.(2H— 2mw— 2^«+cd) — i-cy2.sin.(2i'— 2»!y— 2§-y+cv). VlI.i.^G.J DEVELOP^rENT OF THE DIFFERENTIAL EQUATION IN u. 407 -^ n (l_}-2e2— |e'2) o o \m .COS. (2d — 2m2;) 2.(l+>») 2— 2/«— c 2.(1— m) 2— 2m+c 7e' {1+fe-— i/'— |e'-|.e,cos.(2t)— 2mî?— cî;+ra) \ 2 e.cos.(2« — 2mv-\-cv — -ji) + oTS o— r.C0S.(2v — 2ot» — c'mV-^-:r!) ~-{.i — dm) ^ ^ . COS. (2 V — ^mv^c'mv — si') cos.(2v — 2 mv — cv — c'inv^^-\-ui') COS. (2 V —2 m t^+c v — c'm v — ra+ro) 2.(2— m) 7.(2+3 m).ce' 2.(2— 3ot— c) ~.(2— 3m).ee' ' 2^(2— 3 m+c) (24-m).ee' ■?;:;.« / , ['■-i+m).ee' -é.m.~./J^ .A^^^-J^^,cos.(2v—2mv—cv+c'mv+z^—z.') (2—m).ee' + "oTo r-T-.cos.(2t) — 2m«+cu+c'mv — •a — îî') (10+19/«+8m2) „ ■ 4:(2^2-+^-^-^°^-C2'^^-2t'+2m.-2.) , (10-19m+Sm.2) „ + -4:(2^^-^^-'^««-(2-^+2.-2m.-2.) — 4:^^24:^ •'''•cos-(2^«-2«+2mt;— 20 I (2— m) "^ 4.(2o-+2-2m) -^^-^"^-^^^"+^^~^^^— ^^) 17e'3 + 2. (2 —4m) • ^Qs- (2 D— 2 m ?;— 2 c'm v+2z>') I (5+m) „ / 4.(2-2m— 2g+c) • ^ '^ • cos-(2«— 2mt;— 2g-z;+c^+2a-^)/ 15 The first line of this table includes the terms of the function [4876e], and by adding them to ... ,, , 3m' «3 [4885rfl theremammg terms of [4885c], we get the terms of —-^.sm.(^v—2v') ; which ought to be 408 THEORY OF THE MOON ; [Méc. Cél The terms of this formula, depending on the angles 2cv — 2v-T-2mv — 2ra and [4886] 2gv — 2v-{-2mv — 2^, have divisors of the order m ; and they again acquire these divisors, by integration, in the expression of tlie moon's mean longitude ; wliich reduces them to the second order ; and this, it would seem, ought to make the inequalities relative to these angles become great. But we must observe, that, by [4853, &ic.], the terms having for a divisor the square of the coefficient of v, in these angles, nearly destroy each other, in the expression of the mean longitude ; so, that the inequalities in question, become of the third order, conformably to the result of observations, as will be seen hereafter [5576]. We may, therefore, for this reason, dispense [4887] with the calculation of the terms multiplied by e* [4886'] [4886"] „9 9 e 7 , because the [4885e] [4885/] [4885g] [4885A] [4885;] [4885fc] [4885i] equal to the differential of [4885] divided by — dv ; or, in other words, it ought to be equal to the terms between the braces in [4885], changing cos. into sin., and neglecting the divisors 2 — 2m, 2 — 2m — c, he, which are introduced in [4885], by the integration. The comparison of the sums of the terms of [4876e, 4885c], with those of [4885], may be made, in most cases, by inspection, or by very slight reductions ; and they will be found to agree, neglecting some terms of the third order, depending on angles which are not expressly included in [4885] ; or, on angles, whose coefficients are not much Increased by integration ; as 2v — 2mv-{-cv, 2v — 2)nv-\-c'mv, he. The reductions, relative to the terms depending on the angle 2v — 27nv — cv, are rather more complicated than the others, on account of the great number of its terms. We have, therefore, placed these terms in the following table [4885/], in the order in which they occur in the functions [4876e,4835[] ; and have found their sum in [4885?n]. Comparing this sum with the corresponding coefficient — 2.(1+ÎH,),(1+Î< -h7'- \e^).e, in the second line of [4885], we find that they nearly agree ; their difference being equal to the very small quantity 2me.^^e~, which maybe considered as of the fifth order ; and, as this is to be multiplied by the factor without the braces, which is of the order nfi, or of the second order, it becomes of the seventh order, which is usually neglected in this coefficient : — 2e.(5+,»5e2 _.|e'2)— 2me.(l+Je2 — |e'2) [4876e],line2 [4885c], line 3 4 9 11 13 -2e.{—i,e"—i,y^ — 2e.a+ite2+A7^. —2e.{+,%e^ — 2e.(-,\e^— A7=^ — 2e. )—2me.{ —I «2—172 c'-2) )— 2me.( +I^e2 ) )— 2me.( — i1;e2 ) )• [48S5m] Sum is =-2e.(l + ie^-i?^-|e'2)-2me.n+jJe^-|7--ie'-). VII.i.§6.] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u. 409 quantities of the fourth order, which result, after integration, nearly destroy each other. The intes;ral tt^-TI ]-'^ [4754], contains also the following ° h- -^ \ dv J ir [4887'] term,* -T-/-^;^-Si»-(^'— ^')- [4888] 4 II This quantity, by development, produces the following expression,! * (2805) The second term of (—] [4809], namely, — ^ . sin. (tJ — v'), being , . ,. , , 2,fo , . 2 /dq\ dv . 3m' u't.dv . , ,. multiplied by — ,, produces, m "^^i" • (^^^j ,7> t^e term, — — .-^.sm.(«— 1> ) ; [4887a] whose integral is as in [4888]. t (2806) We may change v into v-\-^(f, in [4872], in the parts which are not connected with m^v, or c'mu, upon the same principles as in [4876a, &c.]. By this means, Mggg^n the expression [4872], with the addition of tlie two terms [4870.c], becomes as in [48896]. Multiplying [4884] by ^, we get [4889c] ; always using the abridged notation [4821/], ..gg which ivill frequently be done, in the commentary oil this hooJc, without any particular notice, that the angles -a, •s/, ê, are omitted; _2 r{l-\-<2e^-\-2e"^).sm.{v—mv) 9)ii'. «'■* . 9 nt rt ) . \ ^ • / 1 \ — ^-p, — r.sm.ti; — V )^ — ; — . — . < +2e.sm.(cu — v-\-mv) — 2e.sm.[cv-\-v — mv) , , ^. „„, 87i2.«4 ^ '' 8a, a' \ ' '^ ' ^ ^ ^ C [4889i] ' (_ -j-e'.3in.(y — mv-\-c'mv)-\-3e'.sm.{v — mv — c'mv) , 3-== 3 « • Î (1-* e2_i7.2)_e.cos.c .+ &c. S . ^^ggg^^ The product of these two expressions, retaining terms of the same form and order as in [4889], becomes as in [4889A]. For the product of the two factors without the braces, is evidently equal to — •-• -, as in [4S89A]. We shall now multiply the terms between [4889rf] the braces in [48895], by those in [4889c]. The first line of [48895], being multiplied by the factor (1 — 56^ — iy^) [4889c], produces the expression, {\ + le^—lf-+2e'^).sm.{v-mv) ; ^^^^^^^ and the term — c.coscd [4869c], being multiplied by each of the terms depending on e, in the second line of [48895], produces a term of the form e-.sin.(y— mi') ; adding these two terms to those in [4889c], we get, {l + ie^-iy^+2e'^).sm.{v-mv), as in [4889A]. ^^gg^^^ VOL. III. 103 410 THEORY OF THE MOON; [Méc. Cél. 2 I I— m .cos.(w — mv) \ 1 . , .u„..,^ „ , . ' -\-p'.r.nfi.Cv — «?? [4889] — — .r — —.sin.ft; — w)= — .-.-,.( +e.cos.(« — mt-+c'mw — -') ^- 2 [4890] [4891] 3 ft' \ -4 .cos.fi' — mv — c'mv-X--J) j 3 the other terms of the integral [4887'] may, in this part, be neglected. This being premised, if we observe, that the expression of u [4826] gives,* l+e"+l/ \ 1 ^'1,, _i) +(1— c-).e.cos.cî)— ^) . 2 a ) g I the term ( -^ + m ) • y-a'/f -y^) • '"i ? of the equation [4754], will produce, by its development,! Lastly, the first term, or unity [4889c], being multiplied by the terms in the third line of [4889i], produces those depending on e', in [4889A] ; r{l+le'^—ly^-}-2e'~).sm.{v—mv)] 3m' u'i - a a y , . . , , ' [4889A] ~\}fi-^'^^'''^^"~'^'^~^'''^^"^''^"\ + c-sin.(t!— ?w«)-j-cw«— n) \ -\-Ze! .%\\\.{y — mv — c'mi;-(- si') Multiplying this by ih, integrating, and putting in the divisors c'=l, it becomes as in [4889i] [4889]. We may remark, that the term — §7^, which we have connected with the factor (l-|-o<^^2_|_2g/2j^ i„ [4870^', 4872], ought also to be connected with that in [4889/i, 4889] ; '■^^^^^^ so that, instead of l+Je^—i 5.24.2 e' 2, we may write l+J t^— J-i^2_|_2e/a_ * (2807) The second differential of u [4826], taken relatively to d, and divided by dv^, gives, [4890a] ''^'^^=^.l-r^e.(l+.s).cos.(c.-.) + ^V-cos.(2,o-.-2ô)r Adding this to the expression [4826], and neglecting terms of the fifth order (1 — <?).^ [48906] [43O8e],weget[4890]. t (2808) The terms of the integral yJr/;^-^' are contained in [4885,4889]. [4892a] These two functions must be multiplied by the expression of ——-\-u [4890]; and the Vll.i.^se.] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u. 411 ddii \ 2 ^f/Q dv dv Ifi '-^ dv ' «3 (l4-3ea+j-v3— ge'a) 2— 2m .cos.{2v—2mv) ^H(l— "0 2-2m-c V T^* 2 >'\ ^ ' ^ 2(1 — m) /r» rt . '^ -.e.cos.(2îJ — 2'mv-{-cv — ra) — :^~7^ — K • COS. (2 V — 2mv-\-c'mv — -a') 2.(2 — m) \ — „ ,!l'^ —.eé.cos.(2v — 2 mv — cv — c'mtj+ra+ra') \ 2.(2— 3 m — c) ^ ' ' / ' o '/7^~^ — r~: • ^ ^'- COS. (2 v —2 m «4-<^ « — c'm v — ro+w) 2.(2 — Jot-j-c) ^ ' , (2+ot) , + -^5 -.ee.cos.(2» — 2m« — cv-\-c'mv-{--a — ra') /*. { Ài Tïl C) . (2-m) 2 /+ ITTTi — t--^^-cos.(2î; — 2'mv-{-cv-\-c'mv — tx — n') "' \- 4:(2^2+2^-^'-^°«-(2c.-2.+2m.-2.) + X(2^2=:2;^-^-'^''^-(^^^'+^^— 2»^^— 2^") + ^-6:(ï=^3 -i:(^qk)5-^""-^«^-C2â-.-2«+2m«-20 c (4^2 — 1) /2 m) ) + Jï6:n=7,ô"^ 4.(2^^+2-2,«)] •'''•cos.(2^«+2i'-2mv-20 Ij (5-j-m) 3.(1— m) ) ^ "U.(2-2«-2^+7)+4(2:^^5-''"-^°^-(2^~2mi;-2gt;+ci;+2«-^) + — rfi T— .-.cos.(zj — mt)) 4.(1 — m) a ^ ■' + J- . — .e'. cos.(t' — mv+c'7nv — ^') \^ 3 4.(1— 2/«) -. e'.cos.(w — mi' — c'mt;+ra') [4892] 18 sum of the products will be equal to the function [4892]. In finding the products of the [4892a'] 412 THEORY OF THE MOON ; [Méc. Cél. [4893] [4894] 7. The term >.(l+,sjl' of the expression _1 f'^3\__L f'R h~ \ du J h^ ds [4808], [ 48926] [4892c] _ __ [4889], which is of the ybwr^A order ; by this means, these terms become so small, functions [4889,4890], we may neglect the second and third lines of [4890] ; for (1 — c^).e is of the third order, 7® is of the second order ; and these are to be multiplied by the factor 2 a [4892rf] [4892e] [4892/] [4892g:] that they may be neglected, and the function [4890] is reduced to its first term -.(l+e^-j-iy^). MultijDlying this by the terms in [4889], lines 1,2,3, we obtain respectively the terms in [4892], lines 16, 17, 18. In the term depending on cos.(j; — mv), in line 16. we may, for greater accuracy, decrease the factor l-|-|e"+2c'^, by |>^, as in [4889i]. We shall now compute the product of the functions [488.5, 4890]. In the first place, the product of the factors, without the braces, is 2 3 wf . ^ X - = — ; as in [4892]. a, a a, The multiplication of the factors, between the braces, is made, term by term, as in the following table ; in which, the first column contains the terms of [4890] , the second column the terms of [4885], and the third column the corresponding products of the terms between the braces, in these lines of the two functions respectively: observing, that 4^^ — 1^3, nearly: [4892^1] (Col. 1.) Terms of [4890]. 1 (1 — c').e.cos.cv (4g^-l) 2 .y^.C0S.2^D (Col. 2.) Terms of [4885]. whole of [4885] 1 2 1 1 3 (Col. 3.) Products of these terms. whole function [4885] between the braces .cos.(2î) — 2mv) 2— 2nt 2.(l-}-jrt) 2-2m-c •{e^-\-i')'^)-e.cos.(2v — 2m v — cv) (1— cS) ■-TTz --.e.coa.C^v — 2 mi' — C1O+&C. 4.(1 — m) ^ ' ' 1 2 3 4 ^y ^^ .; 2. {cos.(2^i)-2y+2)«D)+cos.(2^f+2r-2mî)) \ 5 — — .c>^.cos.(2d — 2wv — 'i. g v-\-cv)-\- hx.. 6 4.(2— 2wi+f) Connecting the terms from lines 2 to 6 of this table, with those in line 1, or the lines between the braces of [4835] ; we get the corresponding terms between the braces, of the function [4892]. VII. i. §7.] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u. 413 becomes, by neglecting quantities of the fourth order,* a. ( 4 ) h~ |3" being a function of the fourth dimension in e, y ; and 6s the part of s arising from the disturbing force. We shall see, in [5596], that ^5 is of the following form ;t [4895] [4896] [48936] * (2809) Developing the expression [4893], according to the powers of s, it becomes — /i~-.(l — J«^+-r«'* — Sic). If we substitute in this the value of 5 [4818], augmented by [4893a] the term ôs, and neglect terms of the order ôs^, which are noticed in [4958, &;c.], we shall find, that the part of the function [4893], depending on Ss, is equal to the differential of the expression [4893«], relative to S, which is — Jr^.{-3sSs~\-J^-s^Ss-&Lc.). Neglecting terms of the order s^5s, it becomes 3h~^.sSs, as in the last terra of [4895]. Now, the value of s [4818] gives, by means of [1, 3] Int., l—^s^={l—iy^)+if.cos.2gv; -'^s^ = M7^— i|7^cos.25-ô+&c.; [4893c] 1— §s^+W— &ic.= {l—t/-)+Î7^-{l—i7^)-cos.2gv-\-teTms of the 4th order. [4S93rf] And, from h^ [4863] , we get, — ^-2. =: .^i_|_e2-]-y2^_j_ terms of the 4th order}. [4893el Multiplying together the two expressions [4893£/,e], we get the part of the function [4893a], which is independent of Ss, as in [4895]. f (2810) The form here assumed for 5s is easily obtained from a comparison of the equations [4754, 475-5], by which u, s, are determined, with the preceding development of [^^'''"J the terms of M. Forthe equation [4754] contains the function — r3[~r ) — rô^-( T )' whose [48976] terms have been developed in [4866, 4870, 4872, &tc.] ; and the equation [4755], by which s IS determined, cont3.\ns the same function, multiplied by -. Now, the chief term of r4897ci the factor - is equal to a-,.sm.{gv — ê), as is evident from [4818, 4791] ; and, if we multiply the terms we have just mentioned [48G6, 4870, 4872, fee] by a7.sin.(o-j) — ê), we [4897rf] shall obtain the most important terms of [4755], depending on the function [4897c]. Thus, the first term of [4866] produces a term depending on sm.(gv—ê), which may be considered as being included in the form [4818]. The second term of [4866] produces the '■ ^' angles gvzhcv [4897], lines 3, 4. The third term of [4866] produces the angles gv±(fmv [4897], lines 8, 9. The first term of [4370] produces the angles 2v — 2mvzizgv [4897], lines 1,2. The second term of [4870] produces the angles 2v — 2mvdizgv — cv [4897], lines 6, 7. The third line of [4870] produces the fifth line of [4897] ; and so on, [4897^] VOL. III. 104 414 THEORY OF THE MOON ; [Méc. Cél 6s=^ B^^^K J'. sin. (2 V— 2m V— g v+ô) 1 -i-B^^'\y.s'm.(2v — 2mv+gv—ê) + B^^-^ . e -/.sin. {gv-\-cv — ê — ct) -j- B^'-^\ ey .sin.Çgv — cv — ^+^) -{-B^^''\ey.sin.(2v — 2mv — gv~\-cv-^Ê — to) ~i-B^'-^\er.s'm.(2v — 2mv+gv — cv — ^+ra) -\-B^^'^\ej.sin.(2v — 2mv—gv — ct)+â+ra) -{-B^'-''\e'-)'.s'm.(gv-\-c'mv — ê — z=') +B^^^Ke'r.sm.(gv — c'mv — ^+ra') 9 -\-Bf\ eV-sin.(2« — 2mv—gv-i-c'mvi-ê — ra) 10 +B[^''Ke'y.s'm.(2v — 2mv—gv—c'mv-\-É-{-^') +Bl''\e"-y.sm.(2cv—gv—2zi+è) Atisumed form of 2 3 4 5 3+^) 6 7 8 ') 11 12 16 +Bl''\e"-y.sm.(2cv—gv—2zi+è) + B\'^Ke^y.sïn.(2v—2mv—2cv+gv+2r,—D) 13 +B<'-^Ke''y.s\n.(2cv+gv—2v+2mv—2^—è) 14 +5^''''.-.7.sin.(^ti — v+mv — ê) 15 + 5''^^.-,.7.sin.('£-«+î? — mv — ^). a ^^ for other terms. Hence we see, that the forms of the angles in [4897], are given a priori [4897ft] by the theory; and they agree with the results of observation [5596]. The differential equation in s [4755], is similar to that of u [4754], and may be reduced to the form [4897m], which is similar to [4845]. For the chief term of s is given in [4818], and if we [4897i] suppose the other terms of « to be represented by 5s, we shall have «=7.sin.(^i' — â)+i3s. dds d-.Ss Its difFerential gives "7^= — g^.y.sin.(gv — ^)-\-~pr- Multiplying the fu'st of these [4897A] expressions by g--. and adding it to the second, we get -—-\-g^.s=-—-j~g^.5s; and if [4897t] we put the second member of this expression equal to — n', we shall get, [4897m] ^^+^9., + n'=0. This is of the same form as [4845] , g taking the place of JV, and differing from unity by quantities of the order m^ [4828c, 4845']. Moreover, n' may be considered as a series of terms, whose general form is k'.sm.{iv — 6), like that in [4846] ; and the part of s, relative to this sine, is represented as in [4847, Sic] by VII. i-sW] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN m. 415 The number placed beloio any one of the letters B, indicates the order of that letter. Thus, Bf is of the second order ; ^t"' is of the first order ; and t'*^^^! B'-g'^ is finite. We may observe, that this takes phice according as the number [4898'] by wliich v is multiplied, in the corresponding sine, differs from unity, by a finite number, by a quantity of the order m, or by a quantity of the order nf, [4899] respectively ; because the integration [4897o] causes the terms to acquire a [4900] divisor of the same order. This being premised, we shall have,* -JV2 . sin. {iv — 6) ; so that these terms may be much increased by this integration, when i is nearly equal to unity. From the similarity of the equations [4754, 4755] it is evident, that the terms of n' _2 [4897m], depending on the disturbing force of the sun, must have the same factor m , as the functions [4866, 4870, 4872, &c.] ; and in is of the order «i^ [5094], or of the second order. This factor is divided by i^ — JV~, in finding the value of s [4897o], or that of &s [4897] ; and, as i^ — JV^ may be considered as of the same order as i~^g'^^i^-\-^m^ [4828e] ; the order of the symbol B will ie represented by Hence, it i2— 1— 3m2 appears, that if { differs considerably from unity, tlie corresponding symbol B will be of the second order, as in [4897], lines 2, 3, 4, 5, Sic. ; using the values of c, g [4828e]. In the first term of [4897], the coefEcient of u is i=2 — 2m — g^=l — 2m nearly; hence, i^ — 1 — I'm^ is. of the order m, and the corresponding value of B [4897r] is of the order m, represented by Bf; and the same occurs in lines 8 — 11 [4897]. In line 12 we have, i=^2c—g = 1 — if'-rri^ [4828e] ; hence, the divisor of the expression [4897 r] becomes of the order m^, and the corresponding value of B is reduced to the order m", or a finite order, as it is called by the author in [4898'], and is represented by i?},'". If we compare the indices of B [4897], with their values, computed in [5122—5214], we shall find they generally agree ; but the term B'i^' [5179] is nearly oï the first, instead of the second order ; i?i'-' is of the second order, fee. * (2811) Substituting in the firstmember of [4901], the values of A-s, s [4893e,4897i], and neglecting terms of the order &^ we get [4901a]. If we also neglect terms of the fifth order, it becomes as in [4901e] ; 3s.6s 3 , . ^ — = -•7<^*-sin.(^r— â)x |l+e^+/2+terms of the fourth order} = -.Yh.sm.{gv—è). We must substitute in this last expression, the value of vs [4897], and we shall get [4901]. If any term of & be represented by C.sin.F, the two corresponding terms of [49016] [4897o] [4897p] [4897?] [4897r] [4897s] [4897<] [4897u] [4901a] [4901i] [4901e] 416 THEORY OF THE MOON ; [Méc. Cél. ^==-~l Bf -B^^ ] .y^cos.(2 v-2 m v) 1 + ^.B^p.7~.cos.(2v—2mv—2gv-\-2è) 2 — --.Bf\e y^cos.r2 &■ v—c v—2 é+^) 4 2a, ^ + -.B^,'Key^cos.(2v—2mv—2gv+cv-^2ê—^) 5 [4901] -i- ~AB^'^—B'i'>l.ef-.cos.(2v—2mv—cv+^) 6 + ^.lB[-^+B['^.e'7''.cos.(c'mv—^) 7 — — .£f'.e'7^cos.(2 ?;— 2 m v+c'm v—z,') 8 ^.S'"').e'7^.cos.(2 1>— 2 m v—c' m v+^') 9 2a, — —.B^^'\e^y^.cos.(2cv—2 ^) 10 + —SB'-}*^+B^!'^.-.7^cos.(v—m v). 11 2a, "^ "a. will be [4901(i] ^.y.C.cos.{(^D— â)«>F| —^.y.C.cos.^^t)— â+F^ ; but it is not, in general, found to be necessary to notice more than one of these terms. The [4901c] only cases in which the author has noticed both terms, are those depending on Bf\ Bf* [4897], lines 1 — 4. The neglected terms are generally smaller than those which are retained, or they are such as depend on angles that have not been usually noticed, because their coefficients do not increase by the integrations. For, the function [4901] forms part of r490in ^^^ expression of n [4902, or 4845] ; and its coefficients may be increased by the divisor t2 — JV^ [4847, 8iC.], when i differs but little from unity ; as is the case in lines 3 — 6,11 [4901]. To estimate roughly the order of the terms, which are not increased by the integrations, and are neglected as in [4901], we may observe, that they produce terms of a [4901g] similar order in u [4847], and in the lunar parallax [5309, &.c.]. Now, if we put - equal VII. i.'^^ T.] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u. 417 If we connect together the different terms which we have developed, we shall find, that the equation [4754] becomes of the following form,* cJc/u = "IP + " + " ; [4902] n being a rational and integral function of constant quantities, and of sines and cosines of angles proportional to v ; but, as loe propose to notice all the [-1903] to the constant term of the lunar parallax 3424', 16 [5331], and use the values of c, e', 7 [5194,5117], also f^u [5221], we shall get, very nearly, _3 2 a, :40' TTZ-n-' :2%3; :^.e'Y' 0%7: — - e^ y~ := 0' 1 • — - . - >2 2 a. 2 a, a' OM. The first of these expressions, being multiplied by the very small quantity .Bf,') [5177], becomes insensible; and it is retained in [4901] line 1, merely because there is no inconvenience in doing it, since it is found necessary to notice the angle 2d — 2mv, in consequence of the magnitude of the other term i?J'\ In like manner, the term -^ .e>=2.^|'=— 0',01 2o, [5178, 490 U], [490U] [4901i] is nearly insensible ; but it is retained in [4801] line 3, because the coefficient c, in the angle cv — -us, diflers but very little from unity [4828e], and it is increased by integration ; which is not the case with the coefficient depending on the other angle 2gv-\-cv — 2d — ra, with which -B^-' is connected. One of the largest of the values o{ B, is that denoted by 3 J?f> = 0,07824 [5183]; multiplying it by the coefficient — .e'j2 = 0',7, with which [4901A] it is connected in [4901] line 7, it becomes 0',05 ; this is retained in the angle c'mv — ■n' [4901] line 7, because the divisor i- — N^ [4847] is nearly equal to unity ; but it is neglected in the angle 2gv-\-c'mv — 2d — ra' ; because it is considerably decreased by the divisor i- — JV^, which is nearly equal to 3. We may also observe, that it is of more importance 10 retain the terms depending on the angle c'mv — -a, than those on 2gv-\-c'mv — 2d — -sj' ; because the terms introduced by the former, in the value of dt [4753], are increased by integration, in finding the value of t, in consequence of the smallnessof the coefficient c'm of the angle v. Similar remarks may be made relative to the other terms, which are neglected or retained. [490K] [4901»; * (2812) Connecting together the terms [4866,4870,4872,4892,4895,4901, &c.], depending on Q, and putting the sum equal to n; then adding it to the terms of [4754], [4909o] which are independent of Q, it becomes as in [4902]. VOL. III. 105 418 THEORY OF THE MOON ; [Méc. Cél. inequalities of the third order, and the quantities of the fourth order connected [4903'] with them, ive must add to the preceding terms all those which depend on the square of the disturbing force, and become of these orders by integrations. We shall now examine these new terms. r4<»03"i ^' ■^°^' *^"^ purpose xve shall suppose ou to be the part of u arising from the disturbing force ; and, that we have,* aàu = AJ-^K COS. (2» — 2 mv) 1 + J/^'. e . COS. (2 w— 2 m v—c v+^n) 2 +A^'-^\e.cos.{2v — 2mv-\-cv — ra) 3 +^/'. e'. COS. (2 V — 2 m v-{-c'mv — ^') 4 Assumed ^ ,,-, , ,^ ^ * . /x r form of + J,< ^ e. COS. (2 V — 2 m v — cmv-\-^) o ÔU. ~ \ -^A^'-^le'. cos.(c'mv — W) 6 -|-^/'^'. e e'.cos. (2 v — 2 m v—c v+c'm ?;+« — -') 7 +^/''.ee'.cos.(2 «; — 2 m v — cv — c'mtJ+ro+za') 8 +^i''*^e e'.cos. (c v-^-c'mv — « — t^') 9 +yi/^'.ee'.cos.(cD — c'mv — -zs+ijj') 10 [4904] +4'°). e^cos.(2ct;— 2^) 11 +4'^'. C-. cos.(2cv—2v-i-2mv—2^) 12 +4'->. '/. cos.(2 o-v— 2 Ù) 13 +4'='.7^ cos.(2^«— 2«+2ot«— 20) 14 +4»'. e'^cos.(2 c'mi)— 2 ^) 15 +4'^'.e7^cos.(2o-i)— CÎ)— 20+ra) 16 +4'«.e7^cos.(2j;— 2mw— 2^î)+cv+2â— T.) 17 +^["'. -.cos.(î; — mv) 18 +4^'- -.e'.cos.(« — tnv-\-c'mv — -n') 19 a +4°). -,. e'.cos. (î; — m^; — c'miJ+ts') 20 a [4904a] * (2813) The terms of a<5tt [4904] are evidently of the same form as those of tlie function Vll.i. §S.] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u. 419 The number 0, 1, or 2, placed below any one of the letters A, denotes, that it is of the order zero, or of the order m, or of the order m^, respectively. [4905] We shall here take into consideration the inequalities of the third order, and those of the fourth order, which can produce terms of the fourth order in the coefficients of the inerjualities of the third order. We shall continue the approximation to a greater degree of accuracy, relative to the inequality [4906] depending on cos.(j' — mv). This being premised, we find, that the term *"' "'^ , . ,3ot' u^ (5m [4865'] gives, by its variation, the expression ^~T~ ? f^""'^ [4907] [4905'] which we deduce the following function ;* n [4902a]. Tlie order of the coefficient A may be found by the formula ia— l-f3ma ' [49046] whicli is similar to that in [4S97/-], using for JV~ the value of c-=l — Zm", instead of g^, which is used in [4897 5-, r] ; i being the coefficient of v, in the angle corresponding to the [4904cl coefficient.^. Thus, for «/if"' [4904], we have i = 2— 2/«; hence ^<"> is of the order m^, or 2. For A'^^, we have i = 2 — 2 m — c=l — m, nearly; hence ^^'^ is of the order m, or 1; and so on, for the other coefficients of [4904]. If we compare these indices of A, with the values obtained by numerical calculation in [5122 — 5213], we shall find, that in [4904rf] general they are correctly marked. * (2314) The expression [4907], whose value is to be determined, may be put under the form 3 2 m'. m'3 ~Ya^7r'i}fi.u^^"' "' [4908a] in which the second and third factors have been already computed in [4884, 4866] ; we shall 3 first find the product of these two factors, and then multiply it by and a Su. Now, if we multiply the factors without the braces, in [4884j 4866], by , the product becomes 2 _a m ^3^ 3 m as in the second member of [4908/] . The products of the terras between the braces, in [4884, 4866], are found in the following table ; in which the first column gives the terms of [4884] ; the second column, the terms of [4866] ; and the third column, the products of these terms respectively ; using the abridged notation [482 1/J, and neglecting the same terms and angles as we have usually done ; [49086] [4908c] 420 THEORY OF THE MOON ; [Méc. Cél. [4908] [4908'] [4909] Sm'.u'^ôu .3j7r.(l+fe'2) 2 F. 2a. a,ôu [4904] ^ 1 —2A<^'\e.cos.(2v—2mv—cv+^) |2 -2A\'\e".cos.(2v—2mv—2cv-{-2^) 13 +1 ^J" .ee'.cos. (2v—2mv—cv+c'mv+zi—ô,') J 4 +|J('>.ee'.cos.(2w— 2my— cy— c'mi>-f-a-f^') v ;, +|J|'^'. - . e'.cos.(t' — mv-j-c'mv — -53') « f7 +|-J,"^\ -.e'. COS. ft» — mi; — c'mt'+ra') to ' ' a ^ ' ^° +14"'- -,.e'-.cos.(i'— m«) / 9 ?{' varies by means of the variation of v' , Avhich depends on the time ^, and on its inequalities in functions of v [4822, or 4828] ; but these inequalities are multiplied by m, in the exjjression of v' [4837], and also, by e', in the expression of v! [4838] ; we may, therefore, at first, neglect (5m', without [4908d] (Col. 1.) Terms of [4884]. 1 -e{ l-ic"2-i72)cos.«' (Col. 2.) Terms of [4866]. dioleof[4866] 1 — '3c.cos.cv -\-2e'.cos.c'mv — 3c.cos.cv -\-3c'. COS. c'mv -%-ce' .COS. {cv-\-c'mv)\ fee'.cos.(cy-c'»iy) +3e2.cos.2fi' -^^y^.cos.2gv I — 3e.cos.fi' 1 — 3e.cos.cv (Col. 3.) Products of these terms, whole of the function [4866] — ic2 a.72 -|.(_[_je3+fc;2).cos.cy -}-( — ^c~e' — ^e'}'~).cos. c'niv — ( l+f 62_j.y2_|_^e/2) .c.cos.cr -|-fc2 -\-^e^.cos.2cv — §ee'.cos.(c!) — c'7itv) — ^tc'. cos. (cv-^c'iiiv) -j-fe-e'.cos.c'my-|-&c. -\-^e~e'.cos.c'mv-\-&c. ^C'^.COS.CV-^&LC. -|«»/2.cos.(2g'i'-cr)-)-&.c . -j-5e^.cos.2c« 1 — 3e.cos.cv — ^e^.cos.cv -\-ie^.cos.2cv~\-&Lc. -\-\y^.cos.^gv. 1 — 3e.cos.c« ly-.cos.2gv —^ey~.cos.{'2gv-cv)-\-&i.c. Connecthig together the terms which are e.Kphcitly given in this tahle, with those between r4908('1 *'^*^ braces in [4866], wliich are included in the first line of this table; the sum becomes equal to the expression between the braces in [4908/"] ; and the factor of a ou [4908a] becomes as in the second member of [4908/"] : VII. i. ^8.] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN m. 421 any sensible error. We shall hereafter [4947, &c.] notice the term of this variation, which depends upon the action of the moon upon the earth. [4909'] 3 m 27, ~\- ( — 4 e — 3 e' — 6ee' ®+e y^) -cos.ct +3e'. (l+2e2+fe'^).cos.c'mj; — 3.(2-|-m) .ce', cos. (cv-\-dm v) -3.(2 — m).ee'.cos.(cD — cfinv) -\-5e^.cos.2cv -\-y^.cos.2gv +fe'^.cos.2c'7«t) K — |e7^-cos.(2^« — cv) Muhiplying this by a Su, we obtain the value of the function [4908c(, or 4907]. To reduce this to the form [4908], we may divide the terms, between the braces, by l-j-#e'^, and connect this with the factor without the braces; and, by neglecting terras of the fourth order in e, e', y, between the braces, we get, l+2e2 -j-( — 4e — Bt^-\-e 2^).cos.cv +3e'. {l-\-2e^—^e'^).cos.c'mv -3. {2-\-m).ee'. COS. (cv-{-c'mv) — 3.(2 — m).ee'.cos(c« — c'mv) \ . aàu. -f-5e^.cos.2cD -|-7^.cos.2^i; -l-Se'^.cos.2c'mD 3m'.u^.U 31h.{l +ie'^) 2a. ' „, . 3m'.(l+|e'a) . , 1 he factor —- — - is the same as 2a le 7^.CGS. (2^ t) — cv) [4908]. The term 1, between the braces in [4908/] [4908g:] [4908^], being multiplied by the external factor aSu, produces the term aSu in the first line of [490S]. Now, if we neglect this term 1, between the braces in [4908^], and [4908A] multiply the remaining terras by aSu [4S04], it will produce the terms of [4908], between the braces, which contain A explicitly. In performing this multiplication, it will only be necessary to retain the two following terms of [4908jg-] ; namely, — 4e.cos.cr-|-3e'.cos.c'mu. [4908t] For, the other terms, between the braces, are of the second order ; and these are multiplied VOL. III. 106 422 THEORY OF THE MOON ; [Méc. Cél. [4909"] [4910] 3 m' m" The term „,„' , . cos. (2 y —2^') [4870], has, for its variation, 9m'.M'3 3m'.u' /„'3 , ^ . 6u. COS. (2v — 2v') + -— ^ .iv\sm.(2v—2v'). If we substitute the preceding value of &u, we shall find, that the first of these terms produces the function,* [4908fc] [4910a] [49106] [49i0c] [4910rf] [4910e] [4910/] [4910^] [4910A] [4910{] by m, of the second order, and by a 5m, of the «etorao, order ; proaucing terms of the sixth order; some of which may be reduced to the Jifih by integration [4847]. The terms, depending on the angle » — mv, of higher orders, are retained as in [4874, &lc.]. The two terms [4908J] evidently produce those in [4908], which depend explicitly on the symbol A, neglecting the terms which have been usually rejected. * (2815) If we take the differential of [4885], relative to dv, and multiply it by ia.dv , we shall obtain the expression of 9ot'.u'3 ' ihKu->.a .sin.(2« — 2v'). The effect of this 2 operation will be to change the factor 3 m.— [4885] into — - — , as in [4910 Ar] ; moreover, it will take away the divisors 2 — 2m, 2 — 2m — c, he, which were introduced by the integration, and will change, in tlie second member, cos. into sin. When the function is reduced to this form, we may change 2v into 2 y + 90'', as in [4876a — d] ; and we shall obtain the expression of 9m'. u' 3 4hKu'>.a .cos.{2v—2v') [4910k']. If an angle, in the second member of [4885], be of the form cos.(2î)-}-(3), it becomes, in [4910f/], sin.(2i;+(3); and, in [4910c], it changes into sin.(2«4-|3-|-90''), or cos.(2j)-f-^) ; which is the same as its original form in [4885]. But, if it be of the form cos.(3 — 2v), the successive changes are sin.(p — 2v), sin.((3— 2« — 90''), and — cos.(p — 2v) ; this last being the same form as the original, but with a different sign. Hence we easily derive the expression [4910^] from [4885], by using the factor neglecting the denominators 2 — 2 m, Sic. [4910c], and changing the signs of the terms depending on angles of the form cos. ((3 — 2d) ; VIL i. «5- s.] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u. 423 ^'"'■%.5u.cos.(2v—2v') '2h^.u* 9m 4a. 4°>.(l-|e'=^) + { Ji"-44''^ +4='-Mf -e'^+J^r'-f ''} •«•(1-f «'')-cos. (c» -«) + { S^o'+^^'J+Jt^' I . e'. COS. (c' m v—^) i_[_^4') — iJ<'>].ee'.cos.(cw+c'/«i' — ^ — ^) + -^f • e e'. COS. (2 v — Imv — c t) — c'mî;+«-}-a') + ^<^' . e e'. COS. (2 « — 2 m w — c v-\-dm v-\--a — to') \+{^J""+i(2+?tt). J;"-2( 1 +«0.^"' I .e/^cos.(2o-zj-a'-2i)+TO)^' 1+4'^'. 67^. COS. (2 z? — 2mt5 — 2^w+ct)+2â — a) + { Jf^— i^'^'.e'^j. -,. cos.(«— OT z)) + {4"'— è4"'l.-.e'.cos.(i;— m2;+c'm?)— to') + {4'3)+^'J(/^)^.-,.e'.cos.(«— mtJ— c'mtJ+TO') 1 2 3 4 5 6 7 [4911] 8 9 10 11 12 4 A^ vr.a ^ ' 9m_ 4a, /(l+2e2— |c'2).cos.(2d— 2mD) V 1 -2(l+m).(l+Je2-iy2-fe'2).e.cos.(2«-2mj;-c!;) \ 2 — 2(1 — m).e.cos.(2y — 2mv-\-cv) I 3 +Je'.cos.(2i; — 2m« — c'otw) I 4 -Je'.cos.(2j; — 2m«-j-c'?n«) f 5 ^ — J(2+3m) .ee'.cos.(2v—2mv—cv — c'mv) I 6 -f(2— 3m).ee'.cos.(2u — 2mv-\-cv — c'înv) 1 7 +|(2-j-m) .ee'.cos.(2« — 2«t)— cjj-fc'mv) \- 8 [4910/t] 1+1(2 — m) .ee'.cos.(2«) — 2mv-{-cv-{-c'mv) / 9 /+ï(10+19/«+8OT2).e2.cos.(2ct;— 2j;+2mi') 10 '+i{l0~19m^8m%e^cos.(2cv-\-2v—2mv)\ll +i(24-m).j^.cos.{2gv—2v-\-2mv) Il2 +i(2—m). 72.003.(2^ D+2 v—2mv) Il3 +^.e'3.cos.(2î)— 2mi;— 2c'7«») /l4 \ — î(5-j-m).ey®.cos.(2î; — 2mt) — 2gv-\-cv) / 15 424 THEORY OF THE MOON ; [Méc. Cél. [4911'] aèu contains a term, depending on cos. (3t) — Smv), which we have [4910i] Multiplying the first member of this expression by 2.a6u, and the second by its equivalent expression [4904], we shall obtain, by making the usual reductions, the value of the first term of [4910], as in the second member of [4911]. For, the factor, without the braces, a 97/1 [4910m] — ' — , is the same in both these functions ; we shall, therefore, neglect the consideration 4 a, of it in the remainder of this note ; and, in speaking of the functions [4910A:, 4911], shall [4910n] refer exclusively to the terms between the braces ; and, shall separately investigate the results arising from each line of the function 2-a&u [4904], by the ivhole of the function [4910A:]. First. We shall take into consideration the product of the term 2 ^g'*. cos. (2 a; — 2mv), by the whole of the function [4910fc] ; and shall reduce the products by formula [20] Int., retaining the same angles as in [4911]. The first line of [4910A:] produces the term (l-[-2e^ — Je'^)..42""; the part depending on cos.(4t) — 4 ??(«;) being neglected. This 2 corresponds to the first line of [4911], neglecting the part depending on 7/t .c^.^,'*, of [4910o] the sixth order, as is done generally in the rest of this calculation ; the term, depending on fe'®, is retained, on account of its importance in the secular equations of the moon's motion [4932, 5059, 5087, &.c.]. Again, if we neglect e^ y^ in the factor [4910/1] line 2, and introduce the factor (1 — Je'~) in [4910À:] line .3, according to the directions in [4869^, fee], we shall find, that these terms, when multiplied by 2A.j°\cos.{2v — 2mv), produce respectively the terms — 2.(l+m).(l— Je'S).^/'. e.cos.ct;, — 2.(1— m).(l -Ae'^) .^^w.e.cos.ct; ; whose sum is — 4.(1— fe'2).^2°'icos.ct), as in [4911] line 2. In like manner, the terms in [4910^] lines 4, 5 being multiplied by 2jÎ^°\cos.{2v — 2mv), produce respectively the terms ^A.;^°\e .COS. cm V, — l^^'^'.c'. cos. c'mo ; whose sum is 3AP. e'. cos. c'm v, as in [491 1] line 3. the remaining terms of the function [4910fc] may be neglected, on account of their smallness, and the forms of the angles. Second. We shall now compute the terms produced by multiplying 2.^,<'\ e .cos.(2v— 2 m v—cv) [4904], by the terms of [4910/*:]. The first line of [4910A-] produces .^i<'\ e . (1— Je'^) . cos. c v, as in [4911] line 2. The second and third lines of [491 OAr] depend on c^, which is neglected. [4910/)] The fourth line of [4910^::] gives iee'.Ai^'\cos.(cv—c'tnv), as in [4911] line 4 ; the fifth line, — ^ e b. Jli-'\ COS. {cv-]-c' m v), as in [4911] line 5; and the twelfth line } {2-\-m) .ey^.cos.{2gv — cv), as in [4911] line 8. VII. i. ■§ 8] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u. 425 neglected,* on account of its smallness in [4904] ; but, as it may have an influence in the term depending on cos.(îj — mv), we shall take notice t*^^^'^ The other terms, depending on ►?/'>, are neglected, on account of their smallness, &ic. Third. The product of 2J.r-''.e.cos.{)îv—2mv-\-cv) [4904], by the first term of [4910t], produces the term A./~le. {l—ie"'}.co5.cv, as in [4911] line 2. This is the [^910?] only term depending on ^.f\ which requires attention ; the other terms being small, or of forms which are unnoticed. Fourth. The product of 2 Jl/^Ke'. cos. {2 v— 2 7JIV+ cm v) [4904], by the first term of [4910A], produces the term Ji.^^^'.e'. cos. c'mv [4911] fine 3; the other terms maybe neglected. In like manner, 2^a<".e'.cos.(2«—2/?iu—f'mî;) [4904], produces w3^«.e' cos.c'mw [4910r] [4911] line 3; and 2.^2'^'. e'. cos. c'otd [4904], gives nothing deserving of notice. Fifth. The terni 2.,'î/''' ce'.co3.(2(; — 2mv — cv-\-c'mv) [4904], being multiplied by the first term of [4910Â:], produces ^/'".ee'.cos.(ct' — c'mv) [4911] fine 4; and the same term, being multiplied by the fifth term of [4910A-], produces — \ee'^.A['^\cos.cv; which is nearly the same as in [4911] line 2. In like manner, the term [4910s] 2.4/''. e e . cos.(2î) — 2mv — cv — c'niv), being multiplied by the first and fourth terms of [4910A-], produces the terms Ap.cc'.cos.{ci'-\-c'mv), and -{-lJlp\ee'^.cos.cv; as in [4911] lines 5, 2. Sixth. The terms depending on Ai^\ .4/°' [4904], being combined with the first term of [4910A-], produce the terms [4911] lines 6, 7. Those depending on ./^a'"^ ./3/'", ^.2"^', produce small terms, which are not noticed. The term 2A\^'>K',^.cos.{2gv—2v+2mv), being combined with the term — 2.(l-j-7?i).c.cos (2« — 2mv — cv) [4910A-]line 2, produces the term depending on ^4,"^' [4911] line 8. The term depending on A.^^*^ [4904], produces nothing of importance. Seventh. The terms 2.â^^^^\ey-.cos.{2gv—cv), 2Ai^"^\ef.cos.(2v—2mv—2gv-j-cv) [4904] , being combined with cos.(2y — 2mv) [4910/i:], produce respectively the terms in [-lOlOu] [4911] fines 9, 8, depending on .^J'^', .4/"". Eighih. The term|,-2.,3/'^'.cos.(D— mu), being combined with the terms in [4910/^-] lines 1, 5, 4, produces the terms depending on .4/'"', in [4911] lines 10, 11, 12, [4910r] respectively. JVinth. The first term of [49101], being combined with the terms of 2.aûu [4904], depending on .^o*'^'' -^o™. produces the corresponding terms of [4911] fines 12, 11. [4910u>] [4910<] * (2816) This term occurs in [4808], and must, therefore, be found in the differential equation in u [4754] , and in its integral 5u, or a ou. VOL. III. 107 [4911a] 426 THEORY OF THE MOON ; [Méc. Cél. of it. For this purpose, we shall put it under the following form ; [4912] [4912'] [4913] [4914] [4914'] [4915] [4916] Term of aiu=^'^^ Substituting this in the expression it produces the term,* 2 9 w ( Aa, ^ i - .cos.(3t;— 3«'). ^^,.6u.co^.{2v—2v') [4910], To develop the variation A2.m3 • -. cos.f» — mv). a ^ iv'.ûn.{2v — 2v') [4910], we shall observe, that iv' contains, in [4837], the same inequalities as the expression of the moon's mean longitude, in terms of the true longitude ; but they are multiplied by the small quantity m. It is sufficient, in this case, to notice the terms in which the coefficient oïv differs but little from unity;t and it is evident that as the term e.cos.(ct) — ^n), of the expression of «i< [4826], gives, in v', the termj — 2me.s'm..{cv — ra) ; any term, whatever, of af>u, such as A;.cos.(ù"-|-£), r4913ol * (^Sl"^) Substituting the values of m, u' , [4791], and h^ = a, [4863], also v'=^mv [4837] nearly, in the expression [4912'], it becomes [49136] 9) 2a,.<i'3 —.aSii.cos.(2v — 2d') = — "-— .a5u.cos.{2v—2mv) [4865]. If we now substitute the term of aSu [4912], we obtain that in [4913], and also one depending on the angle 5 v — 5 m v, which may be neglected. t (2818) We shall see, in [4918], that the terms of this form, in which the coefficients [4914a] of V are nearly equal to unity, produce only small quantities of the fifth or sixth order. These terms are noticed, because they are much increased, by integration, in finding the [49146] value of u [4841] ; but this does not happen with the terms in which the coefficient of » differs considerably from unity ; and we may also observe, that, in this last case, the terms [4914c] may also be decreased by the integration in [4822]. Hence, we see the propriety of noticing only the terms mentioned by the author in [4915]. X (2819) If we inspect the calculation in [4812 — 4837], we shall find, that the term [4915a] c.cos.(c«;— ra), which occurs in u [4812,4816, 4819, 4826], is introduced into dt [4821], and by integration, produces in t [4822], or rather, in nt-\-s [4830], a term — 2c. sin. (en — w). [49156] rpjjjg jg j-,;iu]^piied by m in the second member of the equation [4836] ; and it finally produces in v' [4837], the term — 2?»e.sin.(cj; — 13), as in [4916]. This may be derived [4915c] f,.Q,j^ {],g preceding term of u, by changing cos. into sin. and multiplying the result by VII.i.§8.] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN m. 427 ill which i differs but little from unity, gives very nearly, in dif, the term 2mk.sm.(iv-{-s). Thus we find, that the preceding term [4914] gives, by its development, the function,* [4917] — 2 m. Tlie same method of derivation may be used with any other term of u, in which ihe coefficient of v diflers but little from c, or from unity [48286] ; as is the case with the [4915d] term ^•.cos.(^•»-)-s) of u [4916], which produces, in Sv', the term — 2mk.sm.(iv-{-s) 14917]. * (2820) Instead of the angle iv-\-s [4916, Sic], we shall, for brevity, use iv, omitting s, as we have to, -n', 6, in [4821/], and re-substituting it at the end of the calculation. Then, if we represent any term of ai5w [4904], in which i differs but little from unity, by a5u = Ji:cos.iv [4916], the corresponding term of 5v' will be very nearly represented by ^i)'= — '2m k.s'm.iv [4917]. Moreover, if we represent any term between the braces of the second member of [4876e], by As'm.V ; or, in other words, any term of the function A2.u3 — .sin. (2» — 2v') by .^.sin.F; and then multiply it by the preceding expression of 5v', we get, by using [17] Int., —^ .5i''.sin.(2c — 2î)')= .^AmJc.cos.(iv'-^T^) — Am'k.cos.{i v-{-V )\ . The factor, without the braces, is the same as in [4918] ; consequently, the terms, between the braces, in [4918], must arise from the other factor of [4918/] ; namely, A mk.cos.{iv>xy^ — A mk.cos.{iv-}-V) ; in which we must substitute the terms of a&u [4904], for k.cos.iv; and, the terms between the braces in [4876e], for .^.sin. K; neglecting the terms which are insensible from their smallness, or those, where the coefficients of v, in the angles, vary much from unity [4915]. We shall, in the first place, compare the terms of the function [4918^], with the terms between the braces in [4918], taking successively, for k, the coefficients of the terms [4904], which are retained by the author. First. The term .^/".e.cos.(2« — 2mv — cv) [4904], corresponds to /i;=./î/'^e, iv = 2v — 2mv — cv; combining this with the first line of [4876e], neglecting e^+i)-^, we find that this first term of [4918^] produces the first line of [4918]. If we combine the same term of [4904] with the first term in line 13 [4876e], we find, that the second term of [4918^] produces the second hne of [4918]. It is unnecessary to notice the products of the other terms of [4876e], by the term [4918A:] ; because the coefficients are small, or the angles are different from those which are usually retained. Second. The term Af)^^^\e'y^.cos.[2gv — cv), being combined with the first tenu of [4876e], produces, by means of the first term of [4918o'], the third line of [4918]. [4918a] [49186] [4918c] [4918rf] [4918e] [4918/] [4918gr] [4918^] [491 8i] [4918A:] [4918i] [4918m] [4918n] 428 THEORY OF THE MOON 5 [Méc. Cél. m.J/".e.(l— |e'-).cos.(ci'— î3) . 1 +^.m.A[^\ef.cos.(2gv — cv — 20+13) | 2 3wtt^ , . .^ ^ . 3w / +'/».^''"'.e?^ .COS. , , CI. ° 1 I 3 [4918] ■^j^.'V.sin.(2t;— 2i;')-=— ^^-^ ' " ' V +cv+2l—^ J \ -\-m.A\"\-.cos.(v — mv) a ' -j-m.Al^^K-.e'.cos.(v — mv — c'm^7+3') The other terms of this development are insensible. The terms a m' ni' i .\2.cos.{v—v')+5.co&.(3v—Qv')\, [4919] Qh^.u^ of the expression |.^gjg Third. The term AP''\ -, . cqs.{v — mv), [4904], combined with the first temi of [4876e], produces, in like manner, the fourth line of [4918]. Fourth. The term r4918ol •^o"*'- ",-fi'-cos.(i; — mv-\-c'mv — is') [4904], combined witii tlie same first term of [4876eJ, produces the fifth line of [4918]. [4918c] [4918r] It appears, from [4840, &c.], that the terras in the five lines of the function [4918], are of the orders 5, 7, 6, 6, 6, respectively. The integration [4847], introduces divisors of the order m^ [4828e], in the first and second lines of [4918], and of the order m, in the other three lines. By this means, the first line of [4918] produces, in the value of u, a term of the third order, and the other lines produce terms of the fifth order ; which are within the limits proposed in [4905', &c.]. With respect to the order of the terms which have been neglected, we may observe, that, in calculating in [4918Z] the quantity produced by one of [4918s] the ^rea<es< terms of [4904] ; namely, .^"\e.cos.(2y — 2mu — cv), when combined with the greatest term of [4876e], contained in its first line, we have noticed only the first term of the function [4918^], and neglected its second. This second term has the same coefficient of the fifth order, as in the first line of [49! S], but the quantity cos.c» is changed into cos.(4w — Amv — cv) ; making 2^4 — 4m — c^=.3, nearly [4846] ; and the divisor P — JV^ [4918u] [4847] becomes so large, that the corresponding term is much decreased, so that it may be neglected. Similar results will be obtained relative to the other neglected terms. [4918<] VII.i.^3] DEVELOPiMENT OF THE DIFFERENTIAL EQUATION IN u. 429 have, for variation,* 2 — '"^ "'/""" . -, . {3.cos.(«— miO+5.cos.(3 v—Smv)]. [4921] Substituting AfKcos.(2v — 2mv), for aôu, we obtain the term,t [4921] 2 a, - «, The variation of the term [4876], 3 m'. U'^ du . ,n r, IS jj^^.-.sm.(2v-2v), [4923] * (2821) The variation of [4919], relative to u, which is the most important part of this expression, as we shall see in [4922;], is —'-rrr^-^uA3.cos.(v—v')4-5.cos?(v—v')]. [4921al If we neglect terms of the order e, we may substitute the values of u, ii' [4791], h^ = a, 2 [4863], and 7/1 [4S65], in the factor, without the braces, and it will become, 2 3m'.u'*.Su „ m'.«3 aiiu a 37n .aôu a . ,..„„ -, o^o 5 =—i-— 7^ • — ■-,= 7, •-' as m [4921]. [49216] 2/i2.M5 ■^ a 3 a, a' '■Za^ a ■■ -■ '■ -' ]Moreover, by putting v'^mv [4837], in the terra between the braces [4921 «], h becomes [4921c] as in [4921]. t (2822) Taking, for aSu, its first term [4904]; namely, aSu^A':?\cos.{2v—2mv), we get, by noticing only the angle v — mv, which requires particular attention, as is observed [4922a] in [4874, &ic.], we obtain, n5«.3.cos.(u — mv) = ^Aj^Kcos {v—mv) ; aki.5.cos.{3v—3mv)z=^A:i°\cos.{v—mv); [49226] whose sum is AA.2^''\cos.{v — mi-). Substituting this in [4921], it becomes as in [4922]. [4922c] The remaining terms of aùu are of the second, third, &c. orders; and, when multiplied by 2 a the factor »» • ^, they become of the sixth, seventh, &ic. orders, which are usually [4922rfl neglected. If we notice the variation of v', in [4919], it will produce terms of an order equal to those in [4921], multiplied by the factor — , which factor is of the order m " [4922e 1 [4916,4917]; so that, the terms produced by oV, will be less than those retained in [4921,4922], and may, therefore, be neglected. VOL. III. 108 430 THEORY OF THE MOON ; [Méc. Cél. [4924] may be reduced to the following terms ;* 6:ii.iP (In ÔU . ^ ^ ,^ 3m'. u'^ dSii dv u + Sm'.u'Uv' du --.cos.(2î;-2î;'); A^.i dv these terms, bj development, produce the following expression ;t [4922/] [4923a] [4923i] [4923c] [4923rf] [4923e] [4923/] [4923^] [4923/i] [4923i] * (2823) The term [4923], is the same as that whose approximate value is computed in [4876,4879]. Its variation, considering u, du, v', as variable, and neglecting &u', as in [4909], becomes as in [4924]. t (2324) Multiplying the equation [4S84] by — 2 Su, we get, by using the abridged notation [4821/], 4 rÎM 4 nSu f c . , , „ J or i^ftdM.j — 4-4-4 e.cos.ci'+&ic. (. Multiplying this by the function [4879], we get the expression of tlie first term of [4924]. Now, the function [4879] is of the third order, and ahi [4S04] is of ihe second order; therefore, if we retain only the two terms — A-\-Ae.cQs.cv of tlie factor [4923«], the final product will be correct, in the sixth order. We may even neglect the term 4 c.cos.îj ; because, when it is multiplied by tlie two greatest terms of [4879] lines 1, 2, it produces terms depending on e-. cos. (2d — 'i.mv), which mutually destroy each other; also, terms of the order c-, connected with the angles 2;; — 2/««j;2c«, which do not increase by integration, and are neglected in [491 1 ,&c.]. Hence, the first term of [4924], is represented as in [4923a, i], by the following function ; 6m. «3 du iu -TTT-r ■ -r- ■ — .sm.(2i' — 2v) =■ -4.« (5m X function [4879]. It is only necessary to notice the terms A.^\ Ji[^^, ^J'^\ in the value of a i5m [4904]; because, the function [4879] is of the third order, and the other terms A'w^'e, A.^^^c, &.C. are of the third, or higher orders; so that their products are of the sixth, or higher orders, which are neglected. The reason for retaining the term .^/'^' is, because it is connected with the angle 'igv — cv, and is much increased by integration [4828r/]. Now, the part of — 4. «(5m [-1904], depending on A"", is — 4^o"'\cos.(2u — 2mv). If we multiply this by the first line of [4879], between the braces, neglecting c^, we shall get the term — 2 cc^2W'.(l— fe'2).cos.(c«)— ra) ; and the second line of [4879], retaining the factor [4879^'], produces the same term, with a different sign ; so that these terms mutually destroy each other. The other terms produced by .^o'"', are too small to be noticed, or depend on angles which may be neglected. The product of the term — 4.^i'''e.cos.(2w — 2mv — cv), in — 4.aSu [4904], by the tenth line of [4879], between the braces, produces gA^'^ ej^. cos. {2 gv — cv). Finally, the product of VII. i. §8] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u. 431 2.(1— m).2r.(l— |e'2) l+^{^-3»i—c).J'p.e'^—l{2-m-c).^f'.e"' S + J6.(1— mJ.^f+(2— M)-2f+(2— 3m)..^|'"}.e'.cos.(c'm«-^') _)_ I (2_3m_c) .^;')_à (2— 2m— c) ../^i" I .ee'.cos. (cD+c'my -a -ra') \-\-l{2—m—c).J.f-\-l{2-~2m—c).J'-p}.ee'.cos.{cv—c'mv-a+-a') '-j-(c — m)..4i^'.ec'.cos.(2w — 2m!; — cv-{-c'mv-\--a — to') 3ml -\-{c-}-m).^f\ee'.cos.{2v — 2mv — ct— c'mr+tô+ra') ^"'"\ , Ci(4-+4+m-2c)^f"— 2(1— 2m).^i'3'j ' " ^ _[_ (2— 2m— 2^-f c)-^l'«' < /+^i5).C"^2_cos.(2i>— 2m«— 3^w+c«+23— n) . ey^. cos.(2o-i; — c» — 2è-\-a)' -{-\{\-2m).^™—h{'^—m).A[^''^.-,.e'.cos.{v—mv+c'mv—-a') , _|_^ JTO1+ J {l—m).Â[^'^ .-,.e'.cos.{v—mv—c'mv-^T^') 4^/'3y2.cos.(2^r— 2r4-2mr), in —A.aSu [4904], by the first term of [4S79], between the braces, produces — 2^,'^'^''e-y^.cos.{2gv—cv). Substituting these two terms in the second member of [4923e], we get, 6m'. u'3 du du 3 h^. u* dv u .sm.{2v — 2v) = ~ .{{gA^'''^—2AP^^^).ey^.cos.{2gv—cv)]. [4923ft] The third term of [4924], A a, 3m'.M'3.'Iu' du /r, n r\ i i . — . COS. {2.V — 2v), produces only a very [4923^] /Au'î ' dv small quantity, depending on the same angle as in the preceding' expression [4923^]. Now, without taking the trouble to compute the whole development of this third term, we may form a satisfactory idea of its value, by taking the product of the two functions [4878,4918]; which gives the expression of 3m'. u'3. iv' du . ,_ _ ,, . -- .sm.(2D — 2v) ; A3. M-» dv [4923m] and, as this differs from [49237] only by the change of cos. into sin. in its last factor, it is evident, that the two functions will produce terms of the same forms and orders ; so that, what may be neglected in the one, may also be neglected in the other. Now, the greatest term of [4878], independent of its sign, is ce.sin.cw ; and, if we multiply it by the terms 432 THEORY OF THE MOON ; [Méc. Cél. [4926] The expression of (j^)-j^j [4754], contains also the following of [4918], we obtain only quantities of the sixth order, depending on angles which may be neglected. The remaining terms of [4878] are of the second or higher orders, producing terms of the seventh or higher orders ; therefore, they may all be neglected, excepting one, depending on the angle 2gv-cv, which is retained for the reasons stated in [4828(/]. A term of this form is produced in the function [4923m], by multiplying the term in line 4 [4878], which is nearly equal to ^■)'^.s'n\.2gv, by the term depending on ^'^h, in the expression of ^'"'" "\sin. (2y— 2u') .5v' [4918] line 1. [4923»ï] Hence, it is evident, by a similar process, that the terms of the function [4923/], depending [4923o] on the angle 2gv — cv, may be found, by multiplying ^-y^.s'm.Qgv, by the terms depending on A'-^''e , in the function ., - ,-, [4923^.] 3m^ _ ^^^_ (2«_2t,') . &v'. Now, the term depending on ^/"e, in the expression of aSu [4904], is a 5u=: ^/*'.e .cos.(2y — 2mv — cv) ; the corresponding term of &v' [4916,4917], is [4923;?'] &v' = —2 ^/I'.m e.sm.{2v—2mv—cv). Multiplying this by the chief term of Âa " •<^os.(2i) — 2v') [4870], which is, ^-^.cos.(2« — 2/?ii'), we get, in the function [4923j(], the term _a . A.^^'.me.sm.cv. Finally, multiplying this by the factor iy^.sm.2gv [4923o], we get, for the third tenu of [4924], the following expression ; [4923g] 3,Môv' du _ ^ ^ .lm.A,^^\ef.cos.(2gv-cv)]. We shall now develop the second term of [4924], which is the most important. It may be put under the following form ; 3m'.«'3 dhn C 3w'.«'3 . ,„ ' „ > d.[ahu) The factor between the braces, in the second member of this expression, connected with the negative sign, is evidently equal to the differential of the first member of [4885], divided by 2.aàv ; and if we perform this process on the second member of [4885], we shall find, that 2 o — [4923s] the division by 2a, makes the factor, without the braces, become — — . Moreover, by taking VlI.i.^,8.J DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u. 433 variation ; the differential of the terms between the braces, the divisors 2 — 2m, 2 — 2m — c, &ic., which were introduced by the integration, are effaced, and cos. is changed into — sin. ; so that, if we represent any term, between the braces in ['1885J, after effiicing the divisors, by Ic'.cos.v' the corresjionding term of the first factor of the second member of [4923r], will be represented by a series of terms, of the form _2 3 »i .— IJlL.k'.sm.v' [4923«,m]. 2a Now, putting aSu equal to a series of terms of the form k.cos.(iv-\-!) [4916], or, for brevity, A-.cos.?"« [49I8i], the corresponding term of d.{a6u) dv will be ■ik. Multiplying this by the first Aictor, which is given [492;3y], we get the following expression of the function [4923r], or, of the second term of [4924] ; 2 ^m'.u'^ d 6u . , 3 in — . sm. (2 v—2 v)— —. \ikk'.cos.(iv w v')—{kk'. cos.{iv-\-v') ] . _s ' 3 m The factor without the braces is the same in all three terms of the functions [4923.1-, y.x] ; and is equal to that in [4925]; we shall, therefore, neglect wholly the consideration of this factor; and, in speaking of these functions, shall limit ourselves exclusively to the terms within the braces. These terms, of the function [4923i], are represented by, ik. jt'. cos.(/y m v') — t'.cos.(it)-f-v') | ; m wliich k.cos.iv represents the terms of [4904], cui'l k'.cos.V the terms between the braces in [4885], rejecti7ig the divisors 2 — 2m, 2 — 2m — c, he. which ivere introduced by the integration. We shall now take, for ^.cos.iy, the terms of the function [4904] ; so as to combine successively each of the symbols Xj'"', .^/", &.C. with all the terms of [4885]. We shall neglect the terms which appear to be insensible, and shall compare those which are retained with the function [4925] ; taking the terms, depending on .^a"", ^^o'") -^é^'K ^c. in the order in which they occur in [4904] ; and, noticing also the terms [4923Ar, y], depending on the angle 2^w — cv. First. The first line of [4904] gives k = J.f\ i=2—2m; substituting this in [4923r], it becomes, {2—2m).A.i''K\k'.cos.{[2—2m\omv')—k'.cos.{2v—2mv-\-v')\. The first line of [4885], neglecting e^, gives A:'= 1 — ^e'^, v'=2u — 2mv; substituting these in the first term of [4924c], we get the first line of [492.5] ; the other term of [4924c] depends on the angle (Av — 4;nu), which is neglected. In like manner, the second line of [4885], gives k'^ — 2(l-)-m).(l — |e'2).e; v'= 2;; — 2mv — cv ; hence, the first terra of [4924c] becomes, —{2—2m).Ai''\2[\+m).{\—y).e.cos.cv=—A{\+m).\{\—m)Jl.p.{\—^e'^).e.cos.cv\; and, by the same process, we get, from the third line of [48S5], by using the factor 1 — Je'^ VOL. III. 109 [4926'] [4923t] [4923u] [4923w] [4923u>] [4923x] [4923j,l [4923:] [4924o] [49246] [4924c] [4924<i] [4924e] [4924/] 434 THEORY OF THE MOON [Méc. Ct [4927] — - — -.\3.sia.(v — m r) +15. sin. (3 1) — 3m.v)l.- 8a,a' civ [4924^] [4924i] [4924*] [4924?] [4924m] [4924nl [4924o] [4924j>] [4924?] [4924r] [4924«] [4924<] [4924<'] [4S79)t], the term — 4(1 — m).\{l — m).A.}°\{l — Je'^).e.cos.cy} . The sum of tliese two terms is — 8|(1 — m).Jl2^°\{l — ^e'^).e.cos.cv], as in the second hne of [49'25]. It is unnecessary, in this case, to notice the second term of [49:24c], because tlie coefficient of v is so large, that the term becomes insensible. Proceeding in the same manner with the fourth line of [4885], which gives A:'=J«', v'=2v — 2mv — c'mv ; also, with tlie fifth line of [4885], which gives k'^ — ie', v'=^2v — 2mv-\-c'mv, we find, that the terms corresponding to the first of the functions [49'24c], are, respectively, -\-{2— 2in). A2^^\^e'. COS. c'mv, — {2— 2m). A^^^lie'. cos. c'mv ; whose sum is 6.(1 — in) . ^n"'. e'.cos.c'm v, as in [4925] line 4. The remaining terms of the function [4S85], being of the seco?u/ or higher orders in e, e', 7, multiplied by Wt of the second order, and ^o"" of the second order, produce only terms of the sixth and higher orders, which may be neglected. Second. The second line of [4904] gives Ar=.^/".e, i^2 — 2m — c, hence [4923^] becomes, (2—2m—c).A^'^\e.\lc'.cos.{[2—2m—c] VMv')—'k'.cos.{2v—2mv—cv-^v') \ . Substituting, in the first term of this function, the values [4924f/], corresponding to the first line of [4885], we get the term (2 — 2n — c)Jl^''>.r.{l — |Ê'^).cos.cit, as in the second line of [4925]. The second and third lines of [4S85], produce terms having the factor A[''.m.e^, of the fifth order; but they do not increase by integration, and are therefore neglected. The fourth and fifth lines of [4835] correspond to the values [4924A], and by substituting them in the first term of [4924/], we get the two terms, ie'.{2—2ni—c).J['''.i .cos. {cv— c'mv), —ie'.{2—2m—c).â['Ke.cos.{cv+cmv), as in [4925]lines 6, 5. All the remaining terms of [4885], excepting that in line 12, ma}' be neglected as in [4924A:]. This line corresponds to ^"':= — i(2-j-m).y^, v'^=2gv-2v-J[-2mv, and produces, by means of the second term [4924/], the expression, +i{2-{-m).{2—2m—c).A^'\ey^.cos.{2gv—cv). Connecting this with tlie terms, between the braces in [4923^, q], depending on A';'\ they become \g-\-m-{-l{2-\-m).(2 — 2m — c)l.A['Ke-)'^.cos.{2gv — cv) : and, as c is nearly equal to 1, we may, by neglecting m^, put jm.(2 — 2m — c)=:.^m; consequently, the first -c)+]^ = i{4g + 4-}-m-2c) factor of the expression becomes, ^-i-'«+f(2 — 2;«- which is the same as the coefficient of A'l\ in [4925] line 9. Third. The term Jlf\e.cos.{2v—2mv-]-cu) [4904], combined with [4885] line 1, gives the term depending on .^2'-' [4925] line 2. In like manner, we may combine the terms of [4904], depending on ^.P\ A"') ^^'tli ''^e same terms of [4885], to obtain the terms depending on Ai'^\ A^''^ [4925] line 4 ; observing, that, as c' is nearly equal to 1, we have very nearly 2 — 2m-\-c'm.=:2 — m, 2 — 2m — c'm=2 — 3m. The term depending on ^j^^' produces nothing of importance. VII. i. § 8.] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u. 435 *hence, wc obtain the quantity, [4927'] Fourth. The term depending on ^f [4904] gives k^Af^ce', i=2 — 2m — c-\-cm , or nearly t=2 — m — c. Substituting tiiis in [49232] it becomes, ^ "' (-2 — 7n—c).ji'°\ec'.\f<f.cos.([2—2/)i — c-\-c'm] v r» v') — A;'.cos.(2« — 2mv—cv+c'mv-{-v')l. [4924i)] The first line of [48S5] produces, in the first term of [4924y], the quantity depending on .4''' [49-25] hne 6 ; and the fifth hne of [4885], produces the terms depending on Jl['^\ in Une 3 [4925]. In like manner, the term depending on A'-p [4904], combined with [4885] "'' lines 1,4, produce tliose in [4925] lines 5,2, depending on A'-p. Also, the terms depending on Af\ A'p [4901], being combined with the first term of [4885], produce the [4925a] corresponding terms in [4925], lines 8,7. Fifth. The terms of [4904] depending on ^2<"», ^;"\ Ai''^\ produce nothing of importance. The terra in line 14 [4904], gives k = A[^^>.y^; i = 2g—2+2m^2m [4925t] nearly; and the first term of line 2 [4885], gives ]i'= — 2e, v'^2v — 2mv — cv. Substituting these in the second term of [4923:r], it produces 4m.C7'^.^,"3'. cos.(2^t' — cv). [4925c] Connecting this witii tlie second term of [4923/c], we obtain -2(l-2m).A[^^''.ey^.cos.{2gv-cv), Mg^Sj^i as in [4925] line 9. The term depending on -^a^'^'.e'^ [4904] produces nothing of importance. Sixth. The term in [4904] line 16, gives k=A^-^^Key^, i=2g—c=l nearly; and the [4925e] first term of [4835] line 1, makes k'=l, v'—2v—2mv ; hence, the first term of [49232] [4925/] produces ^„'''''. e7^.cos.(2v — 2mv — 2gv-\-cv), as in [4925] line 11. The same values of ]c', v', being combined with the term in [4904] line 17, produce (^2—2m—2g+c).AP'^\ey\cos.{2gv—cv), as in [492.5] line 10. [4925g] Seventh. From [4904] line 18, we have k = A["\-, i=l — m. Combining these [4925A] with k', v' [4925/], we get the term {l—m).J["\-.cos.{v—mv) [4925] line 12. If we combine the same values of k, i, with the term in line 4 [4885], we get the term rjqor-i depending on A^''' [4925] line 14 ; and if we combine them with that in line 5 [4885], we obtain the term depending on A^p\ in [4925] line 13. Eighth. From [4904] line 19, we have k= A^^^\-,.e', i = \ — m-^c'm=l nearly. Combining this with k', v' [4925/J, we get the term depending on ./îo"*' [4925] line 14. [4925t] If we combine these values of k, i, with the term in [4885] line 5, we get the term depending on A'^^''> [4925] line 12. JVinth. From [49041 line 20, we have fc=./3P\-,.e', i= 1— ?«— c'ot=1— 27« nearly. "• ^ " a' ' •' [4925/] Combiningthis with the values yt', v'[4925/'],we get the terms depending on ./3,"'' [4925] line 13. Tejith. The term of a (5m [4912], gives k=\.-,i^3 — 3m. Combining this with r^^^Sm] the values [4925/], we obtain the term depending on Xn, in [4925] hne 12. Thus, we have obtained all the terms of the function [4925], as they are given by the author ; and, it is evident, from the details of the calculation in this note, that, in general, [4925n] the neglected terms are such as have been usually rejected. * (2825) Having found, in the preceding note, the variation of the first term of [4928] * 2 9/« 436 THEORY OF THE MOON ; [Méc. Ce). 2 9 »" 1 N Jim " ^ \ -;; — . ( 1 m) .^'"' .-.COS.fv — OT V). 4o, ^ "^ a ^ ^ ( 7 ) ■ Z^^iv ' '^°"'*'"^'^ '" [4876], we shall now proceed to the calculation of the next ■ temi, which is given in [4860] ; and, if we put, for brevity, t'*^^^"] A=— ^^.|3.sin.(t>— t,')-f 15.sin,(3y— 3i;')}; this part becomes ^. — . Its variation, considering u, du, v', as variable, and neglecting i5m', [49276] as in [4909, fee], is ('L^] ^ ^^^/l' , f^) s,' ^a ■ ^ ^ \ du J dv '~ \ dv' J ' ' dv ' dv ' The factor ^^. in the value of A [4927rt], is of the order iïi.- . -, [4921 è], [4927c] which is of the /oMrtA order ; therefore, (-r-j, (-p) are of the same order. Moreover, 5m [4904] is of the «cconrf order ; — [4878] is of the^îr^i order; Sv' is of the third order [4916,4917]; consequently, (-— ).(5m. — is of the ici'OiiA order; and (—].Sv'. — V ait / du \ dv/ dv of the eighth order ; so that, by rejecting these terms, the function [4927i] is reduced to A. --7- of the sixth order. Then, by neglecting terms of the seventh order, we may use in A [4927a], the values [4921a — c], and the preceding expression becomes as in [4927]. * (2826) The differential of [4904], divided by dv, gives, -^^ =— (2— 2m).^i''\sin.(2i;— 2mt)) — (2— 2m— c) ../?/". e . sin.(2u— 2mM— cy) — &c. ; which is to be substituted in [4927]. In the first place, the terms depending'on .^o"" [4928a], produce, in [4927], the following expression ; [49286] 5"^'-(2 — 2m)..^o°\{3.sin.(t) — M«)-]-15.sin.(3i; — 3mt>) |.sin.(2t) — 2mr)). As this is of the sixth order, we need only notice the resulting terms which depend on the angle (y — mo). Now, 3.sin.(t) — mr).sin.(2u — 2mu) =^.cos.(i' — mv) — kc. ; 15.sin.(3y — 3mw).sin.(2u — 2my)^-L5-.cos.(4) — mv) — Sic.; whose sum is 9.cos.(« — m«) — Sic; hence, it is evident, that the term [4928i] is equal to [4928c] ^^.(2— 2m).^2<''i.9.cos.(D— mu) ; [4927d] [4928a] VII. i. •§. 3 ] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u. 437 The function [4891], contains, in the first place, the term, - (S + ") • / ^^ • ^•"- (2 «-2 «') [4882] ; [4930] its variation is,* which is easily reduced to the form [4928] . We may proceed in the same manner with the terms of «(5it [4901], deiiendins on /7'j".c, Jl:?\c, &ic. ; but, as these terms produce only cjuantiiie? of the sixth, seventh, &ic. orders, they may be neglected. * (2827) We shall put, for brevity, V=~+u, JV='-'i^.sm.(2v—2v'); [4929a] then, we shall have the development of F', in the second member of [4890] ; and the expression [4930] will become — V.fW.dv. Now, as V, W, contain the variable quantities u, u' , v , the variation of the function — V.fW.dv, will be denoted by — ^^■■fl O ■'^" + (?)•''"' \ .dv~5V.fJ'V.dv— V.f(^-£).âu'.dv. [49296] The three different integrals, of which this expression is composed, correspond respectively to the three integrals in [4931], as we shall find by the following investigation ; in which we shall U5e the abridged notation [4821/]. If we substitute the values of (y-j, (yr)) deduced from that of JV [4929a], in the first of thc^ integrals [4929'], it becomes, -^•'^K^}'"+(^j-'"T''"=-i:^-/-,7r- J -•^'"•(2«-2''')+|5«'-cos.(2t;-2«')^ ; [4929c] in which the terms under the sign /, are the same as in the first term of [4931]. If we substitute the values of c, g [482Se], in V [4890], and neglect terms of the order m% m^'r, e^ iA we obtain, ['•^29^] V=-^.ll-^^f.C0S.'2gv]. [4929e] Substituting this in the factor, without the sign / [4929c], it becomes as in the first term of [4931]. As the terms of nôu [4904], are of the second or higher orders, it follows, from [4908°-]. that the terms depending on Su, under the sign / [4939c], are of the fourth or higher orders ; and when these are multiplied by the terms of V, which we have neglected [4929/] m [4929f/], they will produce only terms of the siith or seventh orders. Those of the sixth VOL. II[. 110 438 THEORY OF THE MOON ; [Méc. Cél. ^^'"' ' 1+f / ^-008.(2^^-2 Oj./—,- . 5 -.si.i.(2«— 2i;')-|-èV.cos.(2o— 2i0?i U f It J h^a r,.r,oii /(hUu , \ .Zm'.u'^.fh . ,r, o '\ [4931] — ( _f oM \ . f — — .sin.(2i'— 2t7'_) rr- . / ■ .dv.s\n.(2v — 2v). * order are produced by c^. ^ ^- ['IGSOc/], and do not depend on the angles v — m i\ and '\lgv — CD, whose coeflicients are required to a great degree of accuracy; hence, we see the propriety of neglecting the above-mentioned terms of V [49ii9(/]. In making this estimate, we have omitted the consideration of hv' [4929c], because it is [4929g-] of the order tn.aou [491G, 4917], and must, therefore, produce terras of still less importance than those of «<3m, which we have neglected. Again, the value of J^ [4929«] gives 5V='-—--\-&u; substituting this in —(SP'.//r.rfw [4929i], it becomes as in [4931] line 2. Lastly, taking the partial di.Terentlal of ÏV [4929fl], relative to m', and substituting it in the third integral [49296] , it becomes , Now, from [4833], we have nearly, a' a' = e'.coscv'^^ whose variation is, a'&u' = — c' e'. (]v'. sin. c' v' ; and, as 6v' is of the order m.ahi [4929j], this quantity will be of the order me'. a Su. or of [4929fc] the fourth order [4904]. If we retain only the chief term of [4929e], we get V= - and, by using the value [4921i,&.c.], we find, that --— is of the order [4929Z] ^^^ • a a' —m .a a' [4865] ; [4929t'] V. f (^~\ . Su!, de = —V.f ^-^ . Su'.dv sin. (2«— 2u'). consequently, the function [4923/] is of the sixth order ; and, by neglecting terms of the seventh order, we may subnltute the value of V [4929;.], in [4929J] ; by which means it becomes as in third line of [4931]. * (2828) In computing the value of the function [4931], we shall retain termsof the fifth [4931o] order in e, e', y, (V); also, in the coefficient of cos.cv, we shall retain the factor l—^e'^. [49316] In the terms depending on the angles '2gv—cv, v—mv, v—mv±c'mv, we shall retain terms VII. i. §8] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u. 439 The development of these terms, observing, that c is nearly equal [4932] of the sixth order ; observing, that the divisors, arising from the integration, 2§* — 2-|-2m, 2c — 2-(-2m, which occur in the terms depending on -^3'"', -^j,"'*' [4934], are of the order m ; so that, independent of these divisors, these terms must be taken to include quantities of the sixth order. We shall first compute the term 12 ni' ./iq;l".^.sin.(2.-2.') [4931]. To obtain this, we shall take the differential of the equation [4885], and then multiply it by 2 -, neglecting such terms as we have usually done, and using the abridged notation [4821/] ; hence we get, [4931c] [4931rf] [4931e] [4931/] 6 m' 2— .sm.(2« — 2v) =^ .dv. ^(1— I e'2).sin.(2t)— 2m«) — 2(l+w).(l — Je'2).e.sin.(2«— 2mt)— cr) — 2(1— m).(l— Je'2).e.sin.(2D— 2mt;+ci') -|-Je'.sin.(2« — 2m j) — c'mv) I — i d . sin. (2 v — 2 m v-\-c'm v) \ — J (2-|-37n) .ee'.sin.(2D — Imv — cv — (imv)\ -^ (2 — 3m) .ee'.sin.(2t) — 2mv-\-cv — c'mv) ~\-l (2+m) .ee'.sin.(2v — 2mv — cv-{-c'mv) \~\'i (2 — m) .ee'.sin.(2« — 2mv-\-cv-\-c'mv) I_|(104-19m).e2.sin.(2cj)— 2«+2mr) +i(10— 19m).e2.sin.(2ct!+2t)— 2?n?;) — ^ (2-fm) . 7=.sin.(2^tJ— 2 y+ 2m v) -f-î(2 — m) .^-.sin.(2^«-|-2D — 2mv) 4-^^-.e'^.sin. (2v — 2mv — 2c'mv) \ — 1 (5-{-m) . e 7^. sin. (2 v — 2 m v — 2gv-\-c v) 1 2 3 4 5 6 7 8 9 10 11 |l2 1 13 14 15 This is to be multiplied by the expression of - [4884] , to obtain the value of the function in the first member of [4931fc]. By this means, the product of the factors, without the braces, becomes. a 12 7n dv, as in [4931 A:] ; and the products of the terms, between the braces, are found as in the following table ; in which, the first column contains the terms of [4884] ; the second, those of [4931^] ; and the third, those of [4931A:], respectively ; [4931g] [493U] 440 THEORY OF THE MOON ; [Méc. Cél. [4933] to 1 — fm^ and, that g is very nearly equal to \-\-^m^ [4828e], is, [4931t] [493U] (C..1.3.) Products, or terras of [49314]. whole function [4931^] between the braces . . . .neglected — J-e.(l— |e'a). \ sin.(2y— 2H;u-(-a))+sin.(2i)— 2mi;— «>) \ -(l+«i).t2.sin.(2cD— 2î)4-2nu))-l-&c. -|-(1 — în).c2.sin (2cî)-j-2i) — 2inv) — &c. — j-ec'.Jsin.(2u — 2«iD-|-fi;— cm«)-|-sin.(2i)— 2mti — cv — c'mv)\ -\-iee'Asm.[2v—'imv-\-cv-\-c'mvY\-sm.[2v — 2mv—cv-\-c'mv)\ . . . .neglected -)-J-c2.|sin.(2cu-(-2v— 2mD)— sin.(2cu— 2«+2mu)| -|-gL>2.{sin.(2g'i)+2u— 2mw)— sin.(2g-u— 2i;-t-2mD)} — jfi 2.(i_j_,„jsin_(2j, — 2mD-|-2g-i- — cv) — &,c. — je}2.(l — Hi).sin.(2« — 2niv — 2gv-\-cv) — &c. Substituting, in the third column of this table, the value of its first hne, which is equal to the terms between the braces in [4931^] ; and then connecting together the terms of tiie same forms, it becomes equal to the terms between the braces in the second member of [4931 A:] ; and the external factor is as in [4931A] ; hence we get. by retaining terms of the usual forms and orders, (Col. 1.) (Col. a.) Terms of [4884]. Terms of [4931f J. 1 whole of [4931g-] -y^-if same — e.cos.cw (l_|e'a).sin.(2r— 2mi>) -2(l+m)f.sin(2v-2Hit)-a') -2(1-7)1 )e.sin{2iJ-2nn; + cii) 4-Je'.sin {2v—2mv—c'mv — 4e'.sin.(2D— 2;)iD-[-c'mD; -c{-^e2-i72)cos.«) whole of [4931g-] -\-^e'^.cos.2cv -t-sin.(2y-2mD) +ly,Kcos.2gv -(-sin.(2t)— 2m») -2( 14-)n)e.sin(2i)-2mv-CD) -2(l-m)e.sin(2D-2nn)-|-cv) 12)ra' u'Hv 1 . ,^ „ ,, 12m , 7r-5 •-^-•-•sin.(2» — 2v )= . dv. (1— Je'2).sin.(2«— Smt)) — (t+2ff?) (1 — fe'^).e.sin.(2« — 2mv — cv) —{i—2m).{l—he~).e.sm.{2v—2mv-{-cv) -|- Je'. sin {2v — 2mv — c'wd) — 2 e'.sin.(2» — 2mv-^c'mv) — f (|-|-3/n).ee'.sin.(2« — 2mv — cv — c'mv) — i (I — 3 m)- e e'. sin. (2 v — 2 m v-j-c v — c'm v) \ _j-i(A-[-m).eÉ'.sin.(2 Î) — 2 jn v — cv-\-c'mv) f -)-^(J — jn).ee.im.{2 v — 2mv-\-c v-\-c'm v) —l{\5-\-2^m).e^.im.{2cv—2v-\-2mv) J^l{\b—23in).e^.sm.{2cv-\-2v—2mv) — {{h-\-m).y^.sm\2gv—2v-\-2mv) -\-l{h — m) 7^.sin.(2^t)-|-2y — 2 mv) -}-y- e'^. sin. (2 v — 2 m v — 2 c'm v) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 [49311] This is to be multiplied by a&u [4904], and then integrated, to obtain the value of the term [4931e]. Now, if we suppose any term of aiu to be represented, as in [49l8i], by a<5u=A:.cos.iu ; and any term of the second member of [4931fcJ, by .dv.Icsm.i'v ; VII.i.§S.] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN m-I 441 3 M 4«,.(1 — m) {4.(1— m)^— 1 l.A°\(l—ie'^) Z_!!L) 4.(1— m) ' < V IS l2-2,n~c ' 2-2H^-[-c3 ~ >.c(l-Je'»). 6m n. + ^ {4J/'+J/'-4'^'-10^/'>e=+|(J/^-J/^'.).6^ _ — . ( 4 . 1— ,« —1 ..Jf'. ] — — - — S ^^ .if2=;,'_l\^(3)4.('ôi:3;^'_l\^(4) 4a,.(l — m) 3 w a. >.e'.cos.(c'm«; — to') "*■ ' .e e'.cos.(2 « — 2mv — cv — c'mw+is-(-TO') ee'.cos.(2t) — 2m a; — c«+c'?«u+w — -n') a,.(2— 3 m— c) o 2 cos(cu-w) 3 4 6 6 [4934] a,.(2 — m — c) Bevelop- 7 nient of the varia- tion(4931). 8 9 the product of these two terms will be represented by .dvJck'. \ sin.(i't) — i v) -\-sm.{i'v-\-iv') Its integral gives the corresponding term of [4931c] ; namely, [4931m] l'2m' u's.rfc ill . Chi C kk' ., . kk' /•/ ■ • \> "Ai^--^"lir----^'"-(2^— 2?')=— • J— ^Tr^.-cos.(ii;— It))— — .cos.(it) + ii;) J; [4931n] all of which have the common factor , and the terms between the braces ; namely, kk' k k' — T7-:-cos.(J'y — iv) — ^r--cos.{i'v-\-iv), are computed in the following table; in which, [4931o] the first column represents the terms of a 5m ; the second, the terms of [4931^]; and the third, the terms of the function [4931o] : the operation being performed for each term separately, putting c and g equal to unity, in several of the small coefficients. When i' = i, the first term of [4931m] vanishes, and the function [493 1 o] is reduced to its second term kk' .cos.Sii,'. This case occurs in the first line of [493I7j1, which is reduced to a term, [4931o'] 2i depending on the angle 4v — 4my, that may be neglected. VOL. III. Ill 442 THEORY OF THE MOON ; [Méc. Cél. [4934] Continued* 2 6 vi a^.[c — 7Ji)' {^f +J J,'"] . ec'.cos.(c?;— cVtt I'— ro+T^i') 10 Develop- ment of tlie first term of the func- Uon[4931]. [4931;?] (Col. 1.) Terms o( aôu [4904]. Ji^ ' .e.co3.(2v—^mv — cv) A ^ Ke'.co9.(2y — ^mv-\-dmv) A ^ '.e'.cos.{2i; — ?mi' — cmx) ^j .ee'.cos.CSw— 2m«— cv-t-cwii') A ^"^.ee'.cog.CSu— 2fflw— cw— c'nitJ) A ^ .ec'.co3.(cr4-c'wip) A ^ .ee'.co3.(ci* — c'twv) ^ f '^'.c".co3.(2cîî— 2«-i-2mr) ^(13) ,j2 ^.^g_^3^„_Oy_j_nn[v) a (14) g— gpg Oc'my 2 __ ^'^^e>**.cu3.(2^v— cr) ^ f '^\c>~.cos.C:!u— 9/nu— 2uru-f-cv) ^l'^\",.C03.(i'-Jni.) 1 a ^ ' ^^ , - .e'.co3.(u — ïnî)-j-c'mu) a' (19) a X ,-.co3.(3u — 3mtJ) (Col. 2.) Termg of [493JA]. A— l-.c'-Vsin.Coy— 2011)) second term third term -t-^.c'.sin.(2u — 2mw — c'mv) ^X.e .sin.(2« — 2mv-|-c'ffiv) /'l_|..c'^Vsin.(2o— 2mv) -|-X.e'.3in.(2u — 9m I' — c'mr) ^X.e'.3in.(*3ii — 2mi;-|-c';nu) 4-^.ec'.sin.(2a — 9mw — cw-l-c'm«) — |. /|.-|- m) .■),^.sin .(2«^>— 2u-f-2mv) (\ — A.c }.sin.('2« — 2jnu) sin. (2)1 — 2hi«) 8in.(2i; — '^mv) sin.(2u — 2mt5) sin.(2H — 2mzj) ^i. e . sin.(2(' — 2H(y — ce) ^_l..e'.sin.{2u — 2;rty-|-c'nn)) Rin.(2u — 2fflv) -|-l..e^.sin.(2t'— 2mu— c my) 3in.(2u — 2m(,') sin.(2y— 2Hiy) 8in.(2y — 'imv') sin.(2u — 2ïnu) sin.(2y — 2/hu) ^A.e.9in.(3u — 2)nw — cy) [terms of 41)3IA] 8in.(2y — 2my) Bin.(9u — 2my) 3in.(0]' — 2»ni) -[-- Z..e'.sin.(2y — 2m y — c'my) ^X.e'.sin.(-3(5 — 2mu-|^'ïnu) sin.(2« — 2mi') ^-l-.e.sin.(2u — 2wu-j-c'mp) sin.(2ïi — Omy) sin.(2y — 2my) (Col. 3.) Factors of ^^^ [493l7i]. . . . .neglected +|.^<°'.^,.c,„.c'„,» 4"4-'*^0 •- .C05.C'7ffU -Jj(').e.(:_|.e2|eo3.c« ' 7 fl(l) «C , , . — -2-^j '.^-3;^.COS.(ci:— c'ïniO +4-'*^, •— r- .cns.(cr'4-c'7ni') ^ ] (■-t""ï ~~-r'rl ', .COS.C/BV * 1 )« _5 ^(M e^f' ^j--*^ '. .C0S.C7WU (+|-+l.'»)..4^j'^.É>°.cos.(2û^y— cy) +^^~^.c.Ci— A.c^^.cos.cy +.4^' ' S- .C09. c'/nu — -ï ' .- .cos.f j/iy . . .neglected — A jî' .^^ — ^ .cos.c'?«i? I 1 _^(6).«c'^.cos.cw ^2" 1 ^^^f^) ,_^^ .co3.(cîî-|-e'mv) 1 r-j-/i/ +4-^' û).'- —l-.A 1 1 2-3'»-c (9) gg' •*1 "a-m-c «(10). ..co3.(2y — 2mi'— CI! — c'mu) .co9.(9« — '^mv — cy-f-c'mu) . .cos.(2cv— 2y+9mi') "2 9C-2+2/I . . . .neglected Jf-A^ '-\ ^ ' .cos. (2;fv— 2u4-27nr) -f-A.^j'3)/.j2c^s.(2^î)— cu) . . . .neglected All thpse N terrnshiive J I the fcim- f mon factor ^ -Jt (13) "0 "s-a»! .79 ..COS. (aï— 2mf— S^'-|-c!)) — .>?( ^ .€ y^.cos. (2^u — cu) J 17) 5 1 — .y? ■ '/^^ ^_^ — .cos.fv — Tnu) « 1 — m 2 1 ■^o^'-S'Trsr,,;'^"''-^''""''-'^'""') ia(18)a c'a .^ .cos.(t' — my — c'mv) a' .e'.'i .cos.(i' — mu+c'my) ~«('0!.'i,..<-m.(,._, ('• — ïttr-f-c'mu) +^5-S'TÎ;:-=°'-("-'"'')- 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 96 27 28 29 30 31 33 33 34 35 36 37 V1I.1.§8.J DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u. 443 ^"' _^jP)_ijo)l,ee'.cos.(cv+c'7nv—-^—^') 11 a^.{c-{-m) 2 _2 [4934] H !'""l"'l X -r- cos.(2o-t^— 2«+2OTt>— 20 13 cu„.inuc,i. a,.(25-— 2+2/n) ' ^ * ' ^ + ^.^5 2J','=''-J[«"+^.4"i.e^^.cos.(2^r— ci'-2é+^) 14 «, 8 2 fi - ^(15) Continua. — ,, ; . ,.ef-.cos.(2v—2mv—2gv+cv+2ê—v^) 15 .rdfvei- ' ^ '^ ' ' • Ihe func- 2 tion[4931J. — . ^/r ,.?a+3tfO-4"^— 24"'-e'-— l.[l— (1— m)n.x,].-, cos.(î;— m«) 16 + ^.M'/') — 2 4'^)}.-.e'.cos.(y— miJ+c'mî;— t.') 17 a, ' ' ' ^ a' ^ We may remark, that the sum of the terms in lines 2, 3, is reduced to 4m. (I ^e'-).^fe.COS.CV; (38) continued. the sum of tiiose, in lines 4, 5, to 4^'2?.-.cos.c'/«« ; and the sum of those, in lines 9, 10, " m e9e' . ''"' to —10.^/"'. — . cos. c'm r. Moreover, the term neglected in line 25, of the form m e2 — -(4/"\-— .cos.(2c!; — 2n), will be used hereafter in a different calculation ; also, the term [AdiXq] 2 i.^/".e2.cos.2cy, arising from the combination of [4904] linel, with the first term in [493ir] line 3 [493 iq. The function [4931^] is also multiplied by ^y^.cos.{2gv — 2é), in [4931]; but the only term of [4931p], which requires any notice, is — ^y\e.cos.ct), in line 6 ; because the r493j,i product of these two terms produces a quantity, depending on the angle 'igv — cv, of the ugsj^i following form ; ° ' SeconrJ o term of -^-.|f .cos.(2^,;-2J)./ !^ . — .sin. (2.-2.')=- — • \\.'Ai'\ef.cos.{2gv-cv)\- -"H»n ''^a -^ «'' « a, <-« ) [4931«] 444 THEORY OF THE MOON ; [Méc. Cél. [4935] We must observée, that Cf'.sin.(2j; — 2vm) is the inequality depending on [4931i] The next term of [4931] is -^^ . f ^^-^.iôv'.cos.(2v—2v'] : which is of the Sv' order — j-, or ot [4922f/, c], in comparison with the terms produced by a du in [4931^]; and, as tliis last function may be considered as of ihefoaith order, that in [4931t)] [4931m'J may be supposed of the fifth or a higher order, in all the angles which require any notice ; so that it will only be necessary to retain the terms depending on the angles, whose coefficients increase considerably by integration ; as cv, 2gv — cv, v — mv. These are produced by the terms of aoii [4904], depending on ^4/'', ^J'''; which give, by the process in [4916, 4917], the following terms of M ; namely, W^^A Sv' = — 2;n.^/"e.sin.(2t)— 2my— ci')— 2m.^i'''.-,.sin.(î>— mi>). Now, if we multiply — ^.Sv'.dv by the first member of [49107c], and prefix the sign /, it produces the term [4931 !)]. Performing the same operation on the second member of [4910A:], we find, that it becomes, 2 '^ ^' — .y"^(5y'. f/tiX terms between the braces in [4910^] I . The first term of ôv [4931:c], being combined with the first line of [4910A:], neglecting e^, [4931:1 produces the term [4932a] line 1 ; the same term, combined with 1^^. cos. (2^1- — 2i'-f-2mD) [49 10A-] line 12, gives [4932a] line 2. The second term of [4931 x], being combined with the first of [4910A;], produces [4932«] line 3 ; hence we have, Third t^'otoc - f — m.AJ".e.(\ — Pje''^).cos.cv ~\ ^ the fuiic- 2» '\-/ i A [4932a] ./ .|5u'.cos.(2y— 2i) )= . < -Ti.^-^^ .cy .cos. (zgv—cv) ''"''' "' "' I ™ /J>17)" / N \ I ; .Ji .^".-.cos.iv — mv) \ 3 I 1 — m a ^ 'I 2 These terms are the most important ones of those depending on Sv', and they are only of [4932i] ti^e fifth or sixth order; therefore, it will not be necessary to notice the terms arising from the multiplication of these by the factor ^■y^.cos.'igv [4931]. _,, . ^, /dd6u , , \ -3 m'. u'3. (/w [49326'] The next terms of [4931] are —f— -+oi«j./ — —- — .sin.(2«— 2d') ; which will evidently be obtained, by multiplying the function [43S5], by the factor ( -,—-{- <5m j. [4932f] Now, any term oï mhi [4904,4912], being represented by aSu = k. cos. {((-{-s), the [4932c'] corresponding term of this factor will be ■ {r — l).cos.(i<-|- f) ; ^ind the product of the terms of this kind, by the corresponding ones in [4885], are computed in the following table ; putting c^l, ^=^1, in some of the small terms; but, in the term depending on VII. i. §8.J DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u. 445 siu.(2t' — 2mv), in the expression of the moon's mean longitude in terms of [4930] the angle 2gv — cv [ 193-2/line 7], we must use f=l — |m-, g^zlJ^îm^ [4932,4933], which give, very nearly. (0_^,„,),-2 -(2+«)-)' l+jni\ (l—im). 4.(2o-— 2-t-2/H) 4\-2m-{-^m-) 4m. Vl+lmy 4m by which means the coefficient of the tenn, in col. 2, line 7 [4932/], becomes —(l-im).~. •I in Moreover, the fictor — (1- — l)./i: [4932c'] becomes, in this case, by neglecting dv', — {(2-2/«-c)--ir.2/"e=— Kl-2'«+l'«')'-U--4/'^c = (4m-7«/2).^/n,^4^(l_Z;„')^(,)g_ Multiplying tills by the factor _^^*^ [4932e], we get —{l—2m).A'-^\ef for the factor of cos.(2^y — cv), in line 7 [4932/]. (ci. 1.) (Col. a.) (Ci.i. 3.) Terms of adit [4904]. Terms of [4885]. Corresponding terms of the function [49326']. .,-2 •Sill [4939a'l [4932d'] [4933e] [4932e'] .Î2"".cos.(2i;— 2mr) ^i"e.cos. (2v — 2mv — cv' w3.J'e.cos.(2u— 2/nt)+ci;) ^Ip'e'.cos (2«-2m«+c'?rtî) ; A'^^e .co5[2v-'2mv-c' mv) A[^^Y.cos{2gv-'2v+2mv) •-I, '.-,.cos.(î) — mv) -^ ' — .cos.f2i'-2mj;) 2— 2m ^ ' -2(1 -f-w) -9(l-,«) 2-2m+c 7c' ,e.cos(2u — 2mv—cv e.cos(2w— 2mi'-f-ci'' >.a.-.cos.(3y — 3mv) 2(2-3m) e' ~2(2-m) (i-yg ; 2-2m - .c os(2t! — ^mv-c'm v ] Thrse terms have tlie factor J4(l-m)3-]| 2(1— jn) ,^^»>.(1— |e'2) cos.(2u-2nM)-[-c'ni!j) .cos. (2d — 2mv) --- !— 1- .cos(2g-i;-2u+2mv 4(%--2-f2m) 2— 2fft 1 2— 2ffi 1 cos.(2d — 2mv) cos.(2ti — 2mv) — 2e-cos.(2«— 2mv—cv) .cos.(2y — 2mv) 2—2/, ' 2_o cos (2y — 2mv) + |4(l-;»)2-l } .gb^^^^fo, e.eos.c. 2 + {4(l-m)2-l }.^<i=^.^|.)e.cos.c« 3 <c—^m-\-c .l4{l-my'-l}.^^^.AI'V.cos.c'mv 4 + {4( 1_m)2— I |.-i— .^^o,e',cos.c'm« 5 ( (2-2m-c)a-l ) ^. , , — (1— 2m).^i"c;.2_cos.(2^y_ct,) — 4(1— Ae'2).^.^(2^e.cos.cy nearly ..^,'3', e .COS. cmv .cos.c mv VOL. III. 112 ^ (2-m)g- l ^ i 2(l-m) ^ C (2— 3m)2— 1 ) ( 2(l-m) ^ ^-^ —2A\'^'>ey^.cos.{2gv—cv) , C 1 — 9.(1— m )9) a + { -2=2;;r~ \ ■'^-^■^^A^^^^)- 13 7 8 9 10 11 Develop - inent of the fourth term of tlio func- tion [49J1 I [4932/J 446 THEORY OF THE MOON ; [Méc. Ctl. [^•^3fi'] its true longitude [5095]. The last term of the function [4931] is, [4932gr] 9m' „H'2.(5tt' -, . ,^ r4933A1 '^° develop it, we have, by retaining only the first power of e', a'M'=l-)-e'.cos.c't;' [4833], whose variation is a'&u'= — c'e'.Sv'.s'm.c'v'^ — e'&v'. sin. c'mv, nearly; and, by substituting the value of &v' [4931 r], we find, that Su' is of ihe fourth order; consequently, [4939i'] the expression [493'2o-] is composed of terms of the sixth and higher orders ; and, as the integration, in [4932§-], does not have the effect to increase essentially these terms of the sixth order, the whole expression may be neglected. AVe have thus computed all the terms of the function [4931]. Nothing now remains, but to connect togetlier the terms which depend on the same angles, as they are found in the [4932A:] functions [493 1|j, w, 49323,/"]. The sum of these four functions ought to be equal to the development of the expression given in [4934], neglecting, for a moment, the consideration of the terms depending on C [4935, &lc.], which will be noticed in [4937«,8>:c.]. In finding the sums of these coefficients, it will be necessary to make some slight alterations, to reduce them to the forms adopted by the author in [4934]. This will be done in the remainder of this note. [4.9321] [4939m] [4933)t] First. The term in [4932/'line I], which is independent of any angle, corresponds to [4934 line I], without any reduction. o — " Second. The second term of [4934] has the factor — c.(l — ^e'^) .cos. (cv — to) common to all its terms ; and the terms by which this factor is multiplied, in the functions which we have mentioned in [4932Ar], are collected in the following table, in the order in which they occur, without any reduction, except, that the two terms [4931p lines 2,3], are reduced to one in line 38. [493 Ip] lines 38, 6, 12,18,21 [4932rt] line 1 [4932/] lines 2, 3 [4932/] lines 6, 8 +8;n..^o"'+2.^;"— 2^^->— ^i«.e'2+7./î<;'.e'2 1 +2m.^;'i 2 _H,(,-,.).-,s.^,-i£-^+,j=:^|,A». 3 The coefficient of A[^\ in this table, is ^ , ^ , (2-2m— c' 4.(1— Î" and, by neglecting the term m^, in the numerator, which produces only terms of the sixth (2-2m-c)2-l 7-8m5-H2-2m-c)a [4ft52o] 2+2m+— --^-- = ^^j-^^^j VII. i. ^^8.] DEVELOP.AIENT OF THE DIFFERENTIAL EQUATION IN u. 447 C,<«'.e'.sin.(2 v — 2 m v j-c'm v—^') and C^"''.c'.sin.(2 v — 2 m v—c'm v-{-a) [4937] order in [4934], which are usually rejected, it becomes equal to the coefticient of A'-p, in [4934 line 2]. We may also omit the term Qm.A':^ [4932?i line ]], which is of the same order; and then, the remaining terms, connected with A^^\ in line 3, are the same as in [4934 line 2J. The terms depending on .^/~' [4932ij lines 1,4], mutually destroy each other. The remaining terms, depending on ./3/''', A'-p, are as in [4934 line 3]. Third. The third term of [4934] has the factor e'. cos. {c'liiv — w') common to all the terms. The coefficients of this factor, in the functions mentioned in [4932Z:], are given in [4932«], in the order in which they occur ; observing that the two terms in [4931/3 lines 4, 5], as well as those in lines 9, 10, are reduced to one in [4931^ line 39]. Moreover, the terms of [4931yj], depending on the angle c'mv — ro', have the divisor m, wiiich is introduced by the integration ; and they have also the common factor — ; so that they are all multiplied by 6 m a,.m fi.fma— em") fim 3m3 ,^^^,^ 6m = = — 1 5094 ; or — nearly 3m3 neglecting the term , which produces only terms of the sixth order in [4934]. Hence the factor of c'.cos.(c'm2; — ra') becomes, without any other reduction, as in the following table ; Cm , „, [4931/7] hues 39, 13,14, 17,20 [4932/] lines 4, 5 [4932/] lines 9, 10 .\AA'f—\OA'^\e~-\-A.}''^- 3SÎ '4 -. j A.(\-mf—\\jU->\ \ — î_ \ By altering a little tlie arrangement of the terms in the first line of this table, it becomes as in [4931 line 4] ; the second and third lines of the table, correspond respectively to [4934] lines 5, 6. The terms relative to C, in [4934 line 7], are discussed in the next note. Fourth. The eighth and ninth lines of [4934], correspond to [493 Ip lines 22, 23], respectively. The tenth line of [4934], depends on [4931/? lines 7, 16]. The eleventh Une of [4934]. depends on [4931/; lines 8,19]. The twelfth and thirteenth lines of [4934], correspond, respectively, to [493Ip lines 24, 26]. _a Fifth. The factors of — .672.003.(2^1)— cy), in the functions mentioned in [4932it], are contained in the following table; [49.32/)] [4932^] [4932r] [4932»] [4932<] [4932u] 448 THEORY OF THE MOON ; [Méc. Ce!. [4937'] *are the inequalities depending on tlie angles 2v — 2inv-'rc'mv — 35' and [493^!''] [4932i>] [493 1 p] lines 11, 27, 30 [493 lit] [4932a] line 2 [4932/] lines 7, 1 1 +im.A['^ 3 Sum = +|m../?,''4-2^^'3'— ^["5>. This sum agrees with the coefficient in [4934 line 14], except in the term depending on «^'j'', which is Jm.^'/' instead of ^m.Jl[^K The difference is of the seventh order only, and is hut of little importance, producing only terms of the fifth order, after integration, in [4847]. This discrepancy appears to have arisen from putting ^=1, c=l, in the [493ar'l calculation [4932e, e'], instead of the values [4932,4933]. For, by using ^ = 1, the / I-l-A m) factor r4932rf',e] becomes . -, and the fiictor r4932e'l is 4ni -' — \{2—2m—cf—l\ = _{(l_2m)2— 1* =4in—4m^=4m.{l—m). The product of these two factors is nearly equal to — (1 — i»i), instead of • — (1 — 2//i) [4932w] [49;3-2y Ijne 7]. Hence, the coefficient of m is decreased to one quarter part of its former value, and the term |m..^j'' [4932y], will be decreased in the same ratio, so as to become Jjn-.^'j'' ; by which means, the sum of all these terms ^m.A\'^ [4932i)], is reduced to |m.^,'\ as in [4934 line 14]. Sixth. The term in [4934 line 1.5], corresponds to that in [4931p line 29]. The factors of — . — .cos.(y — mv), in the functions mentioned in r4932/i:], are 2«,.(l— m) a ^ ' contained in the following table. The sum of these factors corresponds to that in [4934] line 16, neglecting terms of the order m^.A["\ + 4.4f/'>-2.4<'«.e'^— 4>2 +4m...^i>^i -I { l—{l-mf\..f,^'> —X,. \ i—l{l-my^ [49.32.T] [4931p] lines 31, 35, 37 [4932«] [4932/] lines 12,13 Sum = {4+3m).J[''^—2J]',"'\e'^—i\_. \ 1— (l-m)*2| . Seventh. The terms in [4934 line 17], correspond to those in [4931/? lines 33, 36] ; and [4933i/] the terms of [4934 line 18J, correspond to [4931^ lines 32, 34]. Hence it appears, that all the terms we have computed, agree with those in [4934]. * (2829) If we compare the value of nt-^s [4828] heretofore used, with the form [4937al finally adopted in [5095], we shall find, that the terms depending on ft", O,'', hc.C^P, p.qg-,, have been neglected ; and, if we put C for the sum of these terms, we must add C to the value of nl-\-e [4828], which will introduce in the second member of [4836] the term VIL i. ^^ 3.] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u. 449 2v 2mv — c'mv-'r^', ia the same expression. We may also observe, that [4937"] the term, Cm ; and the same quantity in the second member of [4837] ; and we shall represent this increment of v', by the expression 5v'=^ Cm. Substituting this in a'Su' [4932^], we get a'ôu'^ — Cme'.sm.c'mv. Now, if we select the chief terms of [4910, 4931], depending on Sc', Su', they will become J«'.sm.(2!) — 2y)+— r- . / • — —.tM.cos.{9,v — 2«') — — ./ — — .rfv.sin.(2u-2w'). fc3. u3 [4937c] [4937rf] [4937e] We have neglected the last term of [4924], depending on &v', because it is multiplied by — , which is of the order e [48T8] ; so that this will be of the same order as the product dv of the first term of [4937e] by e, which, as we shall soon see, may be neglected [4937À:]. Now, substituting the values of &v', Su' [4937c, rf], in [4937e], it becomes, by merely altering the arrangement of each of the terms, so as to bring them under the forms we have already computed, { 'âS-^^-^^^-^'^ ] •2C^+4^:./'^".cos.(2.-2.').f C. , 9?»' «M- . ^ /\ "'C ^ . , + --— . / —.dv.sm.(2v — 2v).—r.C.sm.cmv. The value of C, to be substituted in this expression, is easily deduced from [5095,4937tl, and is represented by C=Cf.sm.(2v—2mv)-{-C[''\e.sm.{2v—2mv—cv)-\-Cf\e.sin.(2v—'2mv-\-cv) -\-C'^\e'.5m.(2v~2mv-j-c'mv)-\-C^"'\e'.s'm.{2v—2mv—c'mv) -\-Ci"Ke'.sm.c'mv-{-hc. . . .-{-C['^\~.cos.{v—mv)+hc. If we multiply together the two functions [4876e, 4937A], and the product by 2m, we shall get the first term of tiie function [4937^J. These terms of this product are of the Jlfth and higher orders ; so that it will only be necessary to retain those which depend on the angles cv, v — mv. These tenns are found by multiplying the first term of [4876e], 2 O — namely, '^.sm.{2v—2mv), by the terms of 2mC [4937/;] depending on C,"', C['^^; 4a, from which we get, in which we have neglected some terms of the sixth order, depending on C^'^', and on the angle cv. [4937/] [4937g] [4937/i] [4937»] [4937*] VOL. III. 113 450 [4938] THEORY OF THE MOON ; [Méc. Cél. 6m .{^A'^'>-\-Af--Afl.e'.cos.(c'mv—^'), [4937«] The next term of [49375-] is found by multiplying together the functions [4910Â:,4937/i], and the product by — ^m.dv ; and then integrating the result ; as in the following table ; Terms of [4937/t]. -\-Cf\sm.{2v—2mv) [4937m] + C-p .e.s\n.{2v — 2m« — cv) + Cf''e'.s'm.{2v—2mv+c'mv) -f- C™^e'.sin(2u — 2mv—c'mv) -\-a^'>\-,.sm.{v—niv) Terms of [4910Â:]. 4-cos.(2« — 2niv) — 2e .cos . {2v — 2mv — cv ) — 2e.cos.(2w — 2mv-{-cv) + Je'.cos.(2î) — 2mv — c'i7iv) — I e'.cos.(2i) — 2mv-\-cmv) -j-cos.(2y — 2mv) -|-cos.(2t; — 2mv) +cos.(2i) — 2mv) +cos.(2« — 2mv) Terms of [4934]. _3 These terms have the factor — • a, . . . .neglected 1 -\-2Cf\7nc.cos.cv 2 — 2Cj'-''.OTe.cos.ct) 3 — i Cf^.e'. COS. dmv 4 — 2 C f'.e'. COS. c'mv 5 4- O^^ .me.cos.cv 6 — CfKe'.cos.c'mv 7 -(-C^^°'.e'.cos.c'?n« g -\-C[^^\m.-,.cos.{v — mv). 9 The last term of [4937^] being very small, we may substitute in it the values [4937n] 14 = - ; u'=z -, ; ■ mv\ h^=a, [4921«— c]; by which means it becomes, [4937o] [4937p] [4937g] '""" .me', f dv.s\n.(2v — 2mv').sm.c'mvy, C ; and, by using [4S65], it may be reduced to the form, — . me'. fCclv. {cos.{2v — 2mv — c'toî)) — cos.(2t) — 2mv-\-c'mv) \ . Now, substituting the value of C [4937A], it produces terms of the sixth order, before integration ; and some of them may be reduced to the fifth, after integration, if they be connected with the angle c'mv; we shall, therefore, retain this angle only. These terms are found, by substituting, in [4937o], the part of C [4937A] represented by C^''\sin.(2î) — 2mv). Combining this with each of the terms of [4937o], it produces a term, rhdv.sm.c'mv = .cos.c'mv ; so that both terms, taken together, produce the following expression ; — . f-l dv.s\n.(2v-2v').^ . C. sm.c'm v = — A— ^Cf\e'. cos c'mv]. VII. i. §8.] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u. 451 [4938'] appears to be of the order ?«', which would produce a quantity of the order w^^ in the expression of the moon's mean longitude ; but this term is, in fact, only of the order m^. For, we shall see, by means of the values of 4°', Jf, 4^'* [5157,5160,5161], that the function 4Jf +4'— 4<'' [^«39] is of the order m^ ; which produces, in the expression of the mean [4939'] longitude, a term of the order m'^ only. We shall, however, retain it here, because we have imposed on ourselves the condition of including terms of that order, in the calculation of the terms of the third order. For this reason, it is indispensable, in the development of j^— .J —^ .sm.(2v — 2v') [4930], [4940] to carry on the approximation to terms of the order 6ii^ ; hence we obtain the terni,t Connecting together the quantities contained in [49.37A:,??î,y] , we get the terms of the function a [4937 e] depending on C. The coefficients of — .Cf\é.cos.c'mv, in [4937m lines 4,5], and in [4937(7], being connected together, become, — J— I— 1 = — ^S as in [4934 line 7] ; [4937r] and the terms in the same line, corresponding to Ci''^ C^"', agree with those in [4937?»] lines 7, 8. The term depending on Cf' [4937ot lines 2, 3] , mutually destroy eacli other. The quantities we have mentioned include all the terms retained by the author ; who has not noticed those in [4937^], and in lines 6, 9 of [4937?«], whose sum is — .m. } C[''\c.cos.cv-\-0^^'>,-.cos.{y^mv) > . [4937s] These neglected terms are of the fifth or sixth order, increasing also by the integration in [4847] ; and are of the same orders as the terms which are usually retained with these [4937t] angles ; but, as we did not wish to alter the numerical calculations of the author, we have not introduced them into [4934]. * (2830) These values are nearly represented by ^/''i= 0,0071, .;32")__o^o030, ^»iz=: 0,0285; whence, AAf-\-Jl'-^^—A'-^=—Qfi^Z, nearly. This is less than m~ [4938a] [5117], but can hardly be called of the order m?, as in [4939'] ; however, as it is multiplied by e', which is much smaller than e, 7, in, we may consider the whole term [49385] [4938] as of the order m^. t (2831) The factor eu- is of the fourth order [4904], and, as all the terms we have 452 THEORY OF THE MOON ; [Méc. Cél. [4941] 30m'. u „u'3.fe2 , . ,^ „ ,. This term produces the following ;* computed [4910, 4924, &IC.] have the factor ?n', or m, except where the sign of integration [49327-)-] has introduced the divisor m; it follows, that these terms depending on Su^, are generally of the sixth order ; but some of tliem may he reduced to the Jifth order, by the integration we have just mentioned. Tlierefore, we need only notice those terms wliere the variations are connected with the signs of integration ; so that we may neglect the second powers or products oftlie variations in the terms [4909", 4921, 4924,4927 ,4931, &c.], and, in fact, only retain the chief term of [4930 or 4931], which depends on Su^. For, we need not notice the terms depending 5u.Sv', Su.oii. Sv'^, Su'^, &;c. ; because &u is of the second order [4904], 5v' is of tlie third order [4929^], Su' is oï the fourth order [4929i — it] ; hence, the terms depending on hi.dv, Su.Su', &ic. must generally be much less than those depending on Su^ ; therefore, we shall only notice this last quantity. We have already [4941e] found, by Taylor's theorem [610. &c.], in [49296], the increment of the function — V.ffVdv , arising fi'om the increments Su, Sv', Su', in the values of m, v, u', respectively ; and, by the same theorem, the term depending on om^, will evidently be represented by —J V.f (^-^'\ . Su\dv [610, 49296] . Substituting the value of W [4929«], it becomes, [4941a] [4941i] [4941c] [4941rf] [4941/] [4942a] [49426] [4942c] -30m'. F ^u'^.ixfi , . .^ 7l2 mO and, by using the value of F = - = w, nearly [4929Zr,4937w], it becomes as in [4941] ; neglecting in V terms of the order em^, e^, y^. * (2832) As the function [4941] is of the sixth order, before integration [49416] ; we may, by neglecting terms of the seventh order, substitute in it the values [4937n] ; by this means, it becomes. 30)n'.a3 . .„ . ■;^.J dv.^aSuy. %in.{2v — 'Hmv) ■■ 30? .fdv.{aSuf.sm.{'iv—'imv) [4865]. If we retain only the term of {aSuf, of t